1
1 4
1?ü? 4?D?
1?ù Ctê
§1,?êVg
1,)e? a§?x?xμ
(1)?2 < 1x+ 2
(2) (x?1)(x+ 2)(x?3) < 0
(3) 1x?1 <a
(4) 0 lessorequalslant cosxlessorequalslant 12
(5)
braceleftbigg x2?16 < 0
x2?2xgreaterorequalslant 0
)μ
(1) x<?52?x>?32
a45
a27a24
0 x-1-2-3 a98a98
(2) 1 <x< 3?x<?2
a45
a27 a24a24
0 x-1-2 31 2a99 a99a99
(3) a> 0?§x< 1?x> 1 + 1a?
a45
a24a27
0 x1 1 + 1
a
a99 a99
a< 0?§1 + 1a <x< 1
a45
a27 a24
0 x11 + 1
a
a99a99
a = 0?§x< 1
a45
a24
0 x1a99
2
(4) 2kpi+ pi3 lessorequalslantxlessorequalslant 2kpi+ pi2?2kpi? pi2 lessorequalslantxlessorequalslant 2kpi? pi3(k∈Z)
a45a3a0a3a0a3a0a3a0 0 x
(5)?4 <xlessorequalslant 0?2 lessorequalslantx< 4
a45a11 a8 a11 a80 x-4 2 4a99 a99
2,y2eyé aμ
(1) |x?y|greaterorequalslant||x|?|y||
(2) |x1 +x2 +x3 +···+xn|lessorequalslant|x1|+|x2|+···+|xn|
(3) |x+x1 +···+xn|greaterorequalslant|x|?(|x1|+···+|xn|)
y2μ
(1)?|x||y|greaterorequalslantxy§K(x?y)2 greaterorequalslant (|x|?|y|)2§u′|x?y|greaterorequalslant||x|?|y||
(2) ^ê?8B{y2.
(i) n = 2?§d|x1 +x2|lessorequalslant|x1|+|x2|§(?¤á.
(ii) bn = k?(?¤á§=k|x1 +x2 +x3 +···+xk|lessorequalslant|x1|+|x2|+···+|xk|.
Kn = k+ 1?§|x1 +x2 +x3 +···+xk+1|lessorequalslant|x1 +x2 +x3 +···+xk|+|xk+1|lessorequalslant|x1|+|x2|+
···+|xk|+|xk+1|
nt?§ég,ên§|x1 +x2 +x3 +···+xn|lessorequalslant|x1|+|x2|+···+|xn|t¤á.
(3) |x+x1 +···+xn|greaterorequalslant|x|?|x1 +x2 +x3 +···+xn|greaterorequalslant|x|?(|x1|+···+|xn|)
3,)eyé a§?x?xμ
(1) |x|>|x+ 1|
(2) 2 < 1|x| < 4
(3) |x|>A
(4) |x?a|<η,η?~ê§η> 0
(5)
vextendsinglevextendsingle
vextendsinglevextendsinglex?2
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle> x?2
x+ 1
(6) 2 < 1|x+ 2| < 3
)μ
(1) x<?12
a45
a24
0 x-1 a98
(2)?12 <x<?14?14 <x< 12
a45
a11a8a11a8
0 x1
2-
1
2
a101a101 a101 a101
3
(3) Agreaterorequalslant 0?§x<?A?x>A
a45
a24 a27
0 xA-A a101a101
A< 0?§x∈R
(4) a?η<x<a+η
a45
a27 a24
a0 xa+ηa?η a101a101
(5) a dux?2x+ 1 < 0§K?1 <x< 2
a45
a27 a24
0 x-1 1 2a98 a98
(6)?53 <x<?3252 <x<?73
a45
a7a4 a7a4
0 x-1-2-3 a101a101 a101a101
4,|e?ê9§3:t?ê?μ
(1) y = f(x) =?x+ 1x9f(?1),f(1)úf(2)?
(2) y = f(x) = √a2?x29f(0),f(a)úf
parenleftBig
a2
parenrightBig
(3) s = s(t) = 1te?t9s(1),s(2)?
(4) y = g(α) = α2 tanα9g(0),g
parenleftBigpi
4
parenrightBig
,g
parenleftBig
pi4
parenrightBig
(5) x = x(θ) = sinθ+ cosθ9x
parenleftBig
pi2
parenrightBig
,x(?pi)
(6) y = f(x) = 1(x?1)(x+ 2)9f(0),f(?1)
4
)μ
(1)?êX = (?∞,0)uniontext(0,∞)§f(?1) = 0,f(1) = 0,f(2) =?32
(2)?êX = [?|a|,|a|]§f(0) = |a|,f(a) = 0,f
parenleftBig
a2
parenrightBig
=
√3
2 |a|
(3)?ê(?∞,0)uniontext(0,∞)§s(1) = 1e,s(2) = 12e2
(4)?ê
braceleftBig
x
vextendsinglevextendsingle
vextendsinglex∈R,xnegationslash= kpi+ pi2,k∈Z
bracerightBig
§g(0) = 0,g
parenleftBigpi
4
parenrightBig
= pi
2
16,g
parenleftBig
pi4
parenrightBig
=?pi
2
16
(5)?êX = (?∞,∞)§x
parenleftBig
pi2
parenrightBig
=?1,x(?pi) =?1
(6)?êX = (?∞,?2)uniontext(?2,1)uniontext(1,+∞)§f(0) =?12,f(?1) =?12
5,|e?ê9μ
(1) y = √2 +x?x2
(2) y = √cosx
(3) y = ln
parenleftBig
sin pix
parenrightBig
(4) y = 1sinpix
)μ
(1)?êX = [?1,2]§
bracketleftbigg
0,32
bracketrightbigg
(2)?ê
bracketleftBig
2kpi? pi2,2kpi+ pi2
bracketrightBig
(k∈Z)§[0,1]
(3)?ê
parenleftbigg 1
2k+ 1,
1
2k
parenrightbigg
(k∈Z)§(?∞,0]
(4)?ê(n?1,n)(n = 0,±1,±2,···)§(?∞,?1]uniontext[1,+∞)
6,f(x) = x+ 1,?(x) = x?2§á)?§|f(x) +?(x)| = |f(x) +|?(x)|
)μd?§f(x)?(x) greaterorequalslant 0=(x+ 1)(x?2) greaterorequalslant 0§Kxgreaterorequalslant 2?xlessorequalslant?1.
7,f(x) = (|x|+x)(1?x)§|÷ve?ax?μ
(1) f(0) = 0
(2) f(x) < 0
)μ
(1)?f(x) = 0§K|x|+x = 0?1?x = 0§=xlessorequalslant 0?x = 1
(2)?|x|+xgreaterorequalslant 0§K?f(x) < 0§1?x< 0=?§=x> 1
8,?1-5L?>3|V!?>{R0ú?C>{R|¤>′.3?mS§A,Bü:m>?V?±w
¤~t.|?>6Iú?C>{R?êa.
)μd?9?n?￡§V = I(R0 +R).
9,3
/NìS?,?M?§T
/Nì.′a§p?h§?M?pY′x￡?1-6¤,T
M?NèVúx?m?ê'XV = V(x)§??§ú.
)μd?§V = pia2x§§[0,h]§[1,pia2h]
10.,/Y± ?è′F/§X?1-7§.°2?§>??45o§CDL?Y?§| ?ABCD?
èS?Y h?ê'X.
)μd?9?§S = h(h+ 2).
11,k? ?H?3§X^Rò?±z|¨ωlY?Yl?3S?L-?§|-?./?
lsú?mt?ê'X￡?1-8¤.
)μd?9?§s = H?ωRt
parenleftbigg
t∈
bracketleftbigg
0,Hωt
bracketrightbiggparenrightbigg
12,y = f(x) =
braceleftbigg 1 +x2,x< 0
x?1,xgreaterorequalslant 0 §|f(?2),f(?1),f(0),f(1)úf
parenleftbigg1
2
parenrightbigg
.
)μd?§f(?2) = 5,f(?1) = 2,f(0) =?1,f(1) = 0,f
parenleftbigg1
2
parenrightbigg
=?12.
5
13,x(t) =
0,0 lessorequalslantt< 10
1 +t2,10 lessorequalslanttlessorequalslant 20
t?10,20 <tlessorequalslant 30
§|x(0),x(5),x(10),x(15),x(20),x(25),x(30)§?x?ùê?/.
)μd?§x(0) = 0,x(5) = 0,x(10) = 101,x(15) = 226,x(20) = 401,x(25) = 15,x(30) = 20
14,e]y′&?-tx?ê.Uìe?5?§éuIS ×2&§U&?-t§z-20?AGe]8?§?
v20±20?O?.&?-t360?±S?§á?ùêL?a§?x?§?/.
)μd?§y = f(x) =
8,0 <xlessorequalslant 20
16,20 <xlessorequalslant 40
24,40 <xlessorequalslant 60
15,óàu)ì)n?§ù?/X?1-9§ê'Xu = u(t)(0 lessorequalslanttlessorequalslant 20).
)μd?9?§u = u(t) =
braceleftbigg 1.5t,0 lessorequalslanttlessorequalslant 10
30?1.5t,10 <tlessorequalslant 20
16,e?êfú?′ §oo
(1) f(x) = xx,?(x) = 1
(2) f(x) = x,?(x) = √x2
(3) f(x) = 1,?(x) = sin2x+ cos2x
)μ
(1)?f(?∞,0)uniontext(0,+∞)§?(?∞,+∞)§ùüê .
(2)?f(x) = x,?(x) = |x|§ùüê?êL?a§Kùüê .
(3)(x) = sin2x+ cos2x = 1e¤á§ùüê? .
17,y2éuêf(x) = ax + b§egCê?x = xn(n = 1,2,···)|¤?
ê§KéA?ê
yn = f(xn)(n = 1,2,···)?|¤?
ê.
y2μxm?1,xm,xm+1′xn￥3ê(2 lessorequalslantmlessorequalslantn)
aK?§2xm = xm?1 +xm+1
qyn = f(xn) = axn + b§Kym?1 = axm?1 + b,ym = axm + b,ym+1 = axm+1 + b§u′2ym =
2axm + 2b,ym+1 +ym?1 = axm+1 +b+axm?1 +b = 2axm + 2b§l
2ym = ym?1 +ym+1
qxm?1,xm,xm+1′xn￥3ê§Kym?1,ym,ym+1′yn￥3ê§u′yn = f(xn)(n =
1,2,···)?|¤?
ê.
18,XJ-?y = f(x)t^u?pu§¤?l￡?1-10¤§y2? af(x1) +f(x2)2 >f
parenleftBigx1 +x2
2
parenrightBig
é
u¤kx1,x2(x1 negationslash= x2)¤á￡kt?A5?ê?à?ê¤.
y2μ3-?t?ü:A(x1,f(x1)),B(x2,f(x2))§?AB§ù￥:C(xC,yC)§Kf(x1) + f(x2) =
2yC,x1 +x2 = 2xC
q-?txD = x1 +x22 ¤é:p?I?yD = f
parenleftBigx1 +x2
2
parenrightBig
§KxC = xD
q-?y = f(x)t^u?pu§¤?l?x1,x2?u?l:§KyC > yD=f(x1) +f(x2)2 >
f
parenleftBigx1 +x2
2
parenrightBig
éu¤kx1,x2(x1 negationslash= x2)¤á.
a45
a54
a28
a28
a28
a28
a28
a28
a28a28
0 x1 x2 x
A
C
B
xD
y
f(x)
19,y2eê3¤mS′üNO\?êμ
(1) y = x2(0 lessorequalslantx< +∞)
(2) y = sinx
parenleftBig
pi2 lessorequalslantxlessorequalslant pi2
parenrightBig
y2μ
6
(1) 0 lessorequalslantx1 <x2
Ky2?y1 = x22?x21 = (x2 +x1)(x2?x1) > 0§u′?êy = x20 lessorequalslantxüNO\.
(2) ?pi2 lessorequalslantx1 <x2 lessorequalslant pi2
Ky2?y1 = sinx2?sinx1 = 2cos x2 +x12 sin x2?x12
q?pi2 lessorequalslant x1 < x2 lessorequalslant pi2§K?pi2 < x1 +x22 < pi2,0 < x2x
1
2 lessorequalslant pi2§u′cos x1 +x22 > 0,sin x2?x12 >
0§l
y2?y1 > 0=?êy = sinx?pi2 lessorequalslantxlessorequalslant pi2üNO\.
20,y2e?ê3¤mS′üN~?êμ
(1) y = x2(?∞<xlessorequalslant 0)
(2) y = cosx(0 lessorequalslantxlessorequalslantpi)
y2μ
(1) 0 lessorequalslantx1 <x2
Ky2?y1 = x22?x21 = (x2 +x1)(x2?x1) < 0§u′?êy = x2xlessorequalslant 0üN~.
(2) 0 lessorequalslantx1 <x2 lessorequalslantpi
Ky2?y1 = cosx2?cosx1 =?2sin x2 +x12 sin x2?x12
q0 lessorequalslant x1 < x2 lessorequalslant pi§K0 < x1 +x22 < pi,0 < x2x
1
2 lessorequalslant pi2§u′sin x1 +x22 > 0,sin x2?x12 > 0§l
y2?y1 < 0=?êy = cosx0 lessorequalslantxlessorequalslantpiüN~.
21,e?ê?ó5μ
(1) y = x+x2?x5
(2) y = a+bcosx
(3) y = x+ sinx+ex
(4) y = xsin 1x
(5) y = sgnx =
1,x> 0?
0,x = 0?
1 x< 0?
(6) y =
2
x2,
1
2 <x< +∞?
sinx2,? 12 lessorequalslantxlessorequalslant 12?
1
2x
2,?∞<x<?1
2?
)μ
(1)?y = f(x) = x+x2?x5§Kf(?x) =?x+x2 +x5§f(?x) negationslash= f(x),f(?x) negationslash=?f(x)§u′d?ê
′ó?ê.
(2)?y = f(x) = a+bcosx§Kf(?x) = a+bcos(?x) = a+bcosx = f(x)§u′d?ê′ó?ê.
(3)?y = f(x) = x+sinx+ex§Kf(?x) =?x?sinx+e?x§f(?x) negationslash= f(x),f(?x) negationslash=?f(x)§u′d
ê′ó?ê.
(4)?y = f(x) = xsin 1x§Kf(?x) =?xsin 1?x = xsin 1x = f(x)§u′d?ê′ó?ê.
(5)?y = f(x) =
1,x> 0?
0,x = 0?
1 x< 0?
§
Kf(?x) =
1,?x> 0?
0,?x = 0?
1 ?x< 0?
=
1,x> 0?
0,x = 0?
1 x< 0?
=?f(x)§u′d?ê′ê.
7
(6)?y = f(x) =
2
x2,
1
2 <x< +∞?
sinx2,? 12 lessorequalslantxlessorequalslant 12?
1
2x
2,?∞<x<?1
2?
§
Kf(?x) =
2
(?x)2,
1
2 <?x< +∞?
sin(?x)2,? 12 lessorequalslant?xlessorequalslant 12?
1
2(?x)
2,?∞<?x<?1
2?
=
1
2x
2,1
2 <x< +∞?
sinx2,? 12 lessorequalslantxlessorequalslant 12?
2
x2,?∞<x<?
1
2?
§
f(?x) negationslash= f(x),f(?x) negationslash=?f(x)§u′d?ê′ó?ê.
22,áyü?ó?ê|è′ó?ê§üê|è′ê§êó?ê|è′ê.
y2μf1(x),f2(x)3(?a,a)(a > 0)Só?ê§g1(x),g2(x)3(?a,a)(a > 0)S
ê§F1(x) = f1(x)f2(x),F2(x) = g1(x)g2(x),F3(x) = f1(x)f2(x)
Kf1(?x) = f1(x),f2(?x) = f2(x),g1(x) =?g1(x),g2(?x) =?g2(x)§u′
F1(?x) = f1(?x)f2(?x) = f1(x)f2(x) = F1(x)
F2(?x) = g1(?x)g2(?x) = (?g1(x))(?g2(x)) = g1(x)g2(x) = F2(x)
F3(?x) = f1(?x)g1(?x) = f1(x)(?g1(x)) =?f1(x)g1(x) =?F3(x)
l
F1(x)′ó?ê?F2(x)′ó?ê?F3(x)′ê.
23,f(x)3(?∞,+∞)Sê§y2F1(x) ≡f(x) +f(?x)′ó?ê§F2(x) ≡f(x)?f(?x)′?
ê.?éAue?êF1(x),F2(x)μ
(1) y = ax
(2) y = (1 +x)n
y2μ?F1(?x) = f(?x) +f(x) = F1(x)§KF1(x) = f(x) +f(?x)′ó?ê
qF2(?x) = f(?x)?f(x) =?F2(x)§KF2(x) = f(x)?f(?x)′ê.
(1) F1(x) = f(x) +f(?x) = ax +a?x,F2(x) = f(x)?f(?x) = ax?a?x
(2) F1(x) = f(x) +f(?x) = (1 +x)n + (1?x)n,F2(x) = f(x)?f(?x) = (1 +x)n?(1?x)n
24,`2e?ê=
′±ê§?|?±?μ
(1) y = sin2x
(2) y = sinx2
(3) y = sinx+ 12 sin2x
(4) y = cos pi4x
(5) y = |sinx|+|cosx|
(6) y = √tanx
(7) y = x?[x]
(8) y = sinnpix
)μ
(1)?y = sin2x = 12? 12 cos2x§KT = 2pi2 = pi
(2) by = sinx2±ê?T = ω> 0
a±ê§éx ∈ (?∞,+∞)§ksin(x + ω)2 = sinx2§AOéx = 0?AT¤á§
Ksinω2 = 0§u′ω2 = kpi,ω = √kpi(k∈Z+)
qéx = √2ω = √2kpi?¤á§sin(√2ω + ω)2 = sinω2 = 0§K(√2 + 1)2kpi = npi(n ∈ Z+)§u
′(√2 + 1)2 = kn(k,n∈Z+)
q(√2 + 1)2 = 3 + 2√2 ∈Q?§
kn ∈Q+§Kb?¤á§=?êy = sinx2?′±ê.
(3)?y1 = sinxT = 2pi?y2 = 12 sin2xT = pi§Ky = sinx+ 12 sin2xT = 2pi.
(4) T = 2pipi
4
= 8
8
(5)?f(x) = |sinx|+|cosx|,f
parenleftBig
x+ pi2
parenrightBig
=
vextendsinglevextendsingle
vextendsinglesin
parenleftBig
x+ pi2
parenrightBigvextendsinglevextendsingle
vextendsingle+
vextendsinglevextendsingle
vextendsinglecos
parenleftBig
x+ pi2
parenrightBigvextendsinglevextendsingle
vextendsingle= |cosx|+|sinx| = f(x)
a2§y = |sinx|+|cosx|T = pi2.
(6)?f(x) = tanxT = pi§Ky = √tanxT = pi.
(7)?y = x?[x] = (x)§Ky = x?[x]T = 1.
(8) T = 2pinpi = 2n
9
§2,Eü?êúê
1,e?êU?¤Eü?êy = f(?(x))§XJU
¤KdEü?êúμ
(1) y = f(u) = 2u,u =?(x) = x2
(2) y = f(u) = lnu,u =?(x) = 1?x2
(3) y = f(u) = u2 +u3,u =?(x) =
braceleftbigg 1,x?knê?
1,xnê?
(4) y = f(u) = 2§U1§u =?(x)§X§U2
(5) y = f(u) = √u,u =?(x) = cosx
)μ
(1)?y = f(u) = 2u(?∞,+∞)§u =?(x) = x2[0,+∞)
Kd?êU¤Eü?êy = 2x2§§(?∞,+∞)§[1,+∞)
(2)?y = f(u) = lnu(0,+∞)§u =?(x) = 1?x2(?∞,1]
Kd?êU¤Eü?êy = ln(1?x2)§§(?1,1)§(?∞,0]
(3)?y = f(u) = u2 +u3(?∞,+∞)§
u =?(x) =
braceleftbigg 1,x?knê?
1,xnê? {?1,1}
Kd?êU¤Eü?êy =
braceleftbigg 2,x?knê?
0,xnê? §§(?∞,+∞)§{0,2}
(4)?y = f(u) = 2U1§u =?(x)U2
U1intersectiontextU2 negationslash= φ?§d?êU¤Eü?êy = 2§§à?N?ê
§{2}?
U1intersectiontextU2 = φ?§d?ê?U¤Eü?ê
(5)?y = f(u) = √u[0,+∞)§u =?(x) = cosx[?1,1]
Kd?êU¤Eü?êy = √cosx§§
bracketleftBig
2kpi? pi2,2kpi+ pi2
bracketrightBig
(k = 0,±1,±2,···)§
[0,1]
2,f(x) = ax2 +bx+c§y2f(x+ 3)?3f(x+ 2) + 3f(x+ 1)?f(x) ≡ 0
y2μd?§
f(x+3)?3f(x+2)+3f(x+1)?f(x) = a(x+3)2 +b(x+3)+c?3[a(x+2)2 +b(x+2)+c]+3[a(x+1)2 +
b(x+1)+c]?(ax2 +bx+c) = a[(x+3)2?x2]+b(x+3?x)?3a[(x+2)2?(x+1)2]?3b[x+2?(x+1)] =
6ax+ 9a+ 3b?3a(2x+ 3)?3b≡ 0
3,(1) y = f(x) = a+bx+ cx§|f
parenleftbigg2
x
parenrightbigg
(2) y = f(x) = x2 ln(1 +x)§|f(e?x)
(3) y = f(x) = √1 +x+x2§|f(x2)9f(?x2)
(4) y = f(t) = 1√a2 +x2§|f(atanx)
)μ
(1)?y = f(x) = a+bx+ cx§Kf
parenleftbigg2
x
parenrightbigg
= a+ 2bx + c2
x
= a+ 2bx + cx2 = cx
2 + 2ax+ 4b
2x
(2)?y = f(x) = x2 ln(1 +x)§Kf(e?x) = (e?x)2 ln(1 +e?x) = ln(e
x + 1)?x
e2x
(3)?y = f(x) = √1 +x+x2§Kf(x2) = √1 +x2 +x4,f(?x2) = √1?x2 +x4
(4)?y = f(t) = 1√a2 +x2§Kf(atanx) = 1radicalbiga2 + (atanx)2 = 1√a2 sec2x = 1|asecx|
4,ef(x) = x2,?(x) = 2x§|f(?(x))9?(f(x)).
)μ?f(x) = x2,?(x) = 2x§Kf(?(x)) = (2x)2 = 22x = 4x,?(f(x)) = 2x2
5,e?(x) = x3 + 1§|?(x2),(?(x))29?(?(x)).
)μ(x) = x3 + 1§K
(x2) = (x2)3 +1 = x6 +1,(?(x))2 = (x3 +1)2 = x6 +2x3 +1,?(?(x)) = (x3 +1)3 +1 = x9 +3x6 +3x3 +2
10
6,f(x) = 11?x§|f(f(x)),f(f(f(x))),f
parenleftbigg 1
f(x)
parenrightbigg
.
)μ?f(x) = 11?x§K
f(f(x)) = 1
1? 11?x
= x?1x,f(f(f(x))) = 1
1? 1
1? 11?x
= 1
1? x?1x
= x,f
parenleftbigg 1
f(x)
parenrightbigg
= 11?(1?x) =
1
x
7,|e?êê9êμ
(1) y = x2(?∞<xlessorequalslant 0)
(2) y = √1?x2(?1 lessorequalslantxlessorequalslant 0)
(3) y = sinx
parenleftbiggpi
2 lessorequalslantxlessorequalslant
3
2pi
parenrightbigg
(4) y =
x,?∞<x< 1?
x2,1 lessorequalslantxlessorequalslant 4?
2x,4 <x< +∞?
)μ
(1)?y = x2(?∞<xlessorequalslant 0)§Kx =?√y(0 lessorequalslanty< +∞)§l
d?êê?y =?√x(0 lessorequalslanty< +∞)
(2)?y = √1?x2(?1 lessorequalslantxlessorequalslant 0)§Kx =?radicalbig1?y2(0 lessorequalslanty lessorequalslant 1)§l
d?êê?y =?√1?x2(0 lessorequalslant
xlessorequalslant 1)
(3)?y = sinx
parenleftbiggpi
2 lessorequalslantxlessorequalslant
3
2pi
parenrightbigg
§Kx = pi? arcsiny(?1 lessorequalslant y lessorequalslant 1)§l
d?êê?y = pi?
arcsinx(?1 lessorequalslantxlessorequalslant 1)
(4)?y =
x,?∞<x< 1?
x2,1 lessorequalslantxlessorequalslant 4?
2x,4 <x< +∞?
§Kx =
y,?∞<y< 1?√
y,1 lessorequalslanty lessorequalslant 16?
log2y,16 <x< +∞?
§l
d?êê
y =
x,?∞<x< 1?√
x,1 lessorequalslantxlessorequalslant 16?
log2x,16 <x< +∞?
.
11
§3,? D ?ê
1,re3[0,1)t?êò??￠?t§|§¤?±1?±??êμ
(1) y = x2
(2) y = sinx
(3) y = ex
)μ
(1) ò? ?ê?y = (x?n)2(nlessorequalslantx<n+ 1,n∈Z)
(2) ò? ?ê?y = sin(x?n)(nlessorequalslantx<n+ 1,n∈Z)
(3) ò? ?ê?y = ex?n(nlessorequalslantx<n+ 1,n∈Z)
2,re3[0,+∞)t?êò??￠?t§(a)|§?¤ê?(b)|§?¤?ó?êμ
(1) y = x2
(2) y = sinx
)μ
(1) ò? ?ê?μ
(a) f(x) =
braceleftbigg x2,xgreaterorequalslant 0
x2,x< 0
(b) f(x) = x2
(2) ò? ?ê?μ
(a) f(x) = sinx
(b) f(x) = sin|x|
3,?e?ê?/μ
(1) y = sgncosx
(2) y = [x]?2
bracketleftBigx
2
bracketrightBig
)μ
(1)
a45
a54
0 pi x-pi
y
1
-1
a113a113 a113a113
a98a98
a98 a98
a98
a98
a98
(2)
a45
a54
0 1 2 3-1-2-3 x
y
1a99 a98
a98 a98a98a98 a98
4,êy = (x)?/.
)μ
a45
a54
a0
a0a0
a0
a0a0
a0
a0a0
a0a0
a0a0
a0
a0a0
a0
a0a0a0
0 1 2 3-1-2-3 x
y
1a99 a98a98 a98a98a98a98
5,êy = [x]?x?/.
)μ
a45a54
a64
a64a64
a64
a64a64
a64
a64a64
a64a64a64
a64a64
a64
a64a64
a64
a64a64a64a64
0 1 2 3-1-2-3 x
y
-1a99 a98a98 a98a98a98a98
12
6,ê′^e{μ3z?mn lessorequalslant x < n + 1(ù￥n?ê)Sf(x)′?5?f(n) =
1,f
parenleftbigg
n+ 12
parenrightbigg
= 0§á?d?ê?/.
)μ
a45
a54
a1
a1
a1
a1a1
a1
a1
a1
a1a1
a1
a1
a1
a1a1
a1a1a1
a1
a1
a1a1
a1
a1
a1
a1a1
a1
a1
a1
a1a1a1a1
0 1 2 3-1-2-3 x
y
1
-1
a99 a98a98 a98a98a98a98
7,êy = |sinx+ 2cosx|?/.
)μ
a45
a54
0 pi
2
pi?pi
2
pi x
y
√5
-1
8,e??êf(x) = tanx§?e?ê?/μ
(1) y = f(2x)
(2) y = f(kx+b)(knegationslash= 0)
(3) y = f
parenleftBigx
2
parenrightBig
1
)μ
(1)
a45
a54
0 pi
4
pi
2
3pi
4-
pi
4-
pi
2-
3pi
4
x
y
(2) (k,b> 0)
13
a45
a54
0 pi?2b2k
pi?b
k 3pi?2b
2k
-bk
-pi+2b2k-pi+bk x
y
(3)
a45
a54
0 2pipi
-pi
-2pi x
y
9,e??êy = f(x)?/§êy1 = |f(x)|,y2 = f(?x),y3 =?f(?x)?/§?`2y1,y2,y3?/
y?/'X.
)μy = f(x)?/Xeμ
a45
a54
0 x
y
Ky1?/?μ
a45
a54
0 x
y
Ky2?/?μ
a45
a54
0 x
y
Ky3?/?μ
a45
a54
0 x
y
y1?/f(x) < 0y?/'ux?é?§f(x) > 0y?/??
y2?/?y?/'uy?é?§
y3?/?y?/'u:é?§
14
10,e?f(x),g(x)?/§áêy = 12{f(x)+g(x)+|f(x)?g(x)|}?/§?`2y?/?f(x),g(x)?
/'X.
)μy = max{f(x),g(x)}
a45
a54
0 x
y
g(x) f(x)
11,éu3[0,pi]t?êy = x§kr§ò?[0,2pi]|§'ux = pi?é?§, 2r?ò?[0,2pi]t?
êò??￠?t|?ê?±2pi?±??ê.
)μ¤|?ê?:f(x) =
x,x∈ [0,pi]
2pi?x,x∈ [pi,2pi]
x?2npi,x∈ [2npi,(2n+ 1)pi](n = ±1,±2,···)
2npi?x,x∈ [(2n?1)pi,2npi](n = 0,?1,±2,···)
= pi
vextendsinglevextendsingle
vextendsinglexpi?2
bracketleftBigx+pi
2pi
bracketrightBigvextendsinglevextendsingle
vextendsingle
a45
a54
a0
a0
a0
a0a0a64a64
a64
a64a64
a64
a64
a64
a64a64a0a0
a0
a0a0
0 pi?pi 2pi
-pi
-2pi x
y
pi
15
1ù 4Y
§1,ê4?út
1,?eêco?μ
(1) xn = 13n sinn3
(2) xn = m(m?1)···(m?n+ 1)n! xn
(3) xn = 1√n2 + 1 + 1√n2 + 2 +···+ 1√n2 +n
(4) x1 = a> 0,y1 = b> 0,xn+1 = √xnyn,yn+1 = xn +yn2
(5) x2n = 1 + 12 +···+ 1n (n = 1,2,3,···)
x2n+1 = 1n (n = 1,2,···)
):
(1) x1 = 13 sin1,x2 = 16 sin8,x3 = 19 sin27,x4 = 112 sin64
(2) x1 = mx,x2 = m(m?1)2 x2,x3 = m(m?1)(m?2)6 x3,
x4 = m(m?1)(m?2)(m?3)24 x4
(3) x1 = 1√2,x2 = 1√5 + 1√6,x3 = 1√10 + 1√11 + 1√12,
x4 = 1√17 + 1√18 + 1√19 + 1√20
(4) x1 = a,x2 = √ab,x3 =
radicalbigg√
aba+b2,
x4 = 8√ab· 4
radicalbigga+b
2 ·
√a+√b
2
y1 = b,y2 = a+b2,y3 = (
√a+√b)2
4,
y4 = (
√a+√b)2
4 +
4√abradicalbig2(a+b)
16
(5) x2 = 1,x3 = 1,x4 = 32,x5 = 12
2,Uy2±eêtμ
(1) n+ 1n2 + 1
(2) sinnn
(3) n+ (?1)
n
n2?1
(4) 1n!
(5) 1n? 12n + 13n?···+ (?1)n+1 1n2
(6) (?1)n(0.999)n
(7) 1n +e?n
16
(8) e
n
n
(9) √n+ 1?√n
(10) 1 + 2 + 3 +···+nn3
y2:
(1) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglen+ 1
n2 + 1?0
vextendsinglevextendsingle
vextendsinglevextendsingle = n+ 1
n2 + 1 <
2n
n2 =
2
n§?|
vextendsinglevextendsingle
vextendsinglevextendsinglen+ 1
n2 + 1?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§2
n < ε=?"
N =
bracketleftbigg2
ε
bracketrightbigg
+ 1,Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsinglen+ 1
n2 + 1?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±n+ 1
n2 + 1 → 0(n→∞)
(2) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglesinn
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsinglesinn
n
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsinglesinn
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle<ε§1
n <ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§
Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsinglesinn
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±sinn
n → 0(n→∞)
(3) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglen+ (?1)n
n2?1?0
vextendsinglevextendsingle
vextendsinglevextendsingle = n+ (?1)n
n2?1 <
n+ 1
n2?1 =
1
n?1§?|
vextendsinglevextendsingle
vextendsinglevextendsinglen+ (?1)n
n2?1?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§?
1n?1 < ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§Kn > N?§
vextendsinglevextendsingle
vextendsinglevextendsinglen+ (?1)n
n2?1?0
vextendsinglevextendsingle
vextendsinglevextendsingle< εo¤á§¤±n+ (?1)n
n2?1 →
0(n→∞)
(4) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle 1
n!?0
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
n! <
1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle 1
n!?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§1
n < ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§K
n>N?§
vextendsinglevextendsingle
vextendsinglevextendsingle 1
n!?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤± 1
n! → 0(n→∞)
(5) Sn = 1n? 12n + 13n?···+ (?1)n+1 1n2
é?ε> 0§duSn = 1n(1? 12 + 13?···+ (?1)n+1 1n)
δn = 1?12+13?···+(?1)n+1 1n§KSn = δnnn = 2k+1?§k0 <δn = 1?(12?13)?(14?15)?···?
( 12k? 12k+ 1) < 1?n = 2k?§k0 <δn = 1?(12?13)?(14?15)?···?( 12k?2? 12k?1)? 12k < 1"
o?§k0 <δn < 1l
|Sn?0| = Sn = δnn < 1n?||Sn?0|<ε§1n <ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+1§
Kn>N?§|Sn?0|<εo¤á§¤±1n? 12n + 13n?···+ (?1)6n+ 1 1n2 → 0(n→∞)
(6) é?ε > 0§dun > lnn§Ken > n§u′e?n < 1n§l
vextendsinglevextendsingle
vextendsinglevextendsingle1
n +e
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
n + e?n <
2
n§?
||(?1)n(0.999)n?0|<ε§(0.999)n <ε=?"N =
bracketleftbigg
2500ln 1ε
bracketrightbigg
+1§Kn>N?§|(?1)n(0.999)n?
0|<εo¤á§¤±(?1)n(0.999)n → 0(n→∞)
(7) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle1
n +e
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
n! <
1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle1
n +e
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle< ε§2
n < ε=?"N =
bracketleftbigg2
ε
bracketrightbigg
+
1§Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsingle1
n +e
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±1
n +e
n → 0(n→∞)
(8) é?ε > 0§due?n < e0 = 1§K
vextendsinglevextendsingle
vextendsinglevextendsinglee?n
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle = e?n
n <
1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsinglee?n
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§1
n < ε=?"
N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsinglee?n
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±e?n
n → 0(n→∞)
(9) é?ε> 0§du|√n+ 1?√n?0| = 1√n+ 1 +√n < 12√n§?||√n+ 1?√n?0|<ε§ 12√n <
ε=?"N =
bracketleftbigg 1
4ε2
bracketrightbigg
+1§Kn>N?§|√n+ 1?√n?0|<εo¤á§¤±√n+ 1?√n→ 0(n→
∞)
(10) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 2 + 3 +···+n
n3?0
vextendsinglevextendsingle
vextendsinglevextendsingle = n+ 1
2n2 <
2n
2n2 =
1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle1
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle< ε§1
n < ε=?"
N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§Kn > N?§
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 2 + 3 +···+n
n3?0
vextendsinglevextendsingle
vextendsinglevextendsingle < εo¤á§¤±1 + 2 + 3 +···+n
n3 →
0(n→∞)
17
3,T~`2e'ut′μ
(1) éε> 0§3N§n>N?§¤áxn <ε?
(2) éε> 0§3xn§||xn|<ε.
):
(1) ~Xμê{?1 + (?1)n+1}(?{?n})={0,?2,0,?2,···} (?{?1,?2,?3,···})÷vt?^?§?′
t?
(2) ~Xμê{1,12,1,13,···,1,1n,···}÷vt?^?§?′t"
4,Uy2μ
(1) lim
n→∞
3n2 +n
2n2?1 =
3
2
(2) lim
n→∞
(0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9) = 1
(3) lim
n→∞
√n2 +n
n = 1
(4) xn = 11·2 + 12·3 +···+ 1(n?1)·n → 1(n→∞)
(5) lim
n→∞
rn = 1§d?rn =
n?1
n n?óên+ 1
n nê
(6) lim
n→∞
rn = 1§d?rn =
3 n = 3k(k = 1,2,3,···)
3n+ 1
n n = 3k+ 1
2 + 1 +n3?√n+n n = 3k+ 2
y2μ
(1) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle3n2 +n
2n2?1?
3
2
vextendsinglevextendsingle
vextendsinglevextendsingle = 2n+ 3
4n2?2 <
4(n+ 1)
4(n+ 1)(n?1) =
1
n?1(n greaterorequalslant 2)§?|
vextendsinglevextendsingle
vextendsinglevextendsingle3n2 +n
2n2?1?
3
2
vextendsinglevextendsingle
vextendsinglevextendsingle<
ε§ 1n?1 < ε=?"N = max(
bracketleftbigg1
ε
bracketrightbigg
+ 1,2),Kn > N?§
vextendsinglevextendsingle
vextendsinglevextendsingle3n2 +n
2n2?1?
3
2
vextendsinglevextendsingle
vextendsinglevextendsingle < εo¤á§¤
±3n
2 +n
2n2?1 →
3
2(n→∞)
(2) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle = (0.1)n =
1
10n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle< ε§
1
10n < ε=?"N =
bracketleftbigg
lg 1ε
bracketrightbigg
+ 1,Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9 → 1(n→∞)
(3) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle
√n2 +n
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle =
√n2 +n?n
n =<
1√
n2 +n+n <
1
2n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle
√n2 +n
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle<ε§?
12n <ε=?"N =
bracketleftbigg 1
2ε
bracketrightbigg
+ 1,Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsingle
√n2 +n
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±
√n2 +n
n → 1(n→
∞)
(4) é?ε> 0§duxn = 1? 12 + 12? 13 +···+ 1n?1? 1n = 1? 1n§K|xn?1| = 1n§?||xn?1|<ε§
1n <ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1,Kn>N?§|xn?1|<εo¤á§¤±xn → 1(n→∞)
(5) é?ε> 0§du|rn?1| =
vextendsinglevextendsingle
vextendsinglevextendsinglen±1
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
n§?||rn?1| <ε§
1
n <ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1,K
n>N?§|rn?1|<εo¤á§¤±rn → 1(n→∞)
(6) é?ε > 0§du|r3k?3| = 0,|r3k+1?3| = 1n,|r3k+2?3| =
√n?2
3?√n+n =
n?4
n√n+n+√n+ 6 <
n
n√n =
1√
n§?||rn?3| < ε§
1
n < ε?
1√
n < ε=?"N = max
parenleftbiggbracketleftbigg1
ε
bracketrightbigg
+ 1,
bracketleftbigg 1
ε2
bracketrightbigg
+ 1
parenrightbigg
,K
n>N?§|rn?3|<εo¤á§¤±rn → 3(n→∞)
18
5,(1) Uy2§ean →a(n→∞)§Kég,êk§an+k →a(n→∞)
(2) Uy2§ean →a(n→∞)§K|an|→|a|.q′?¤áo
(3) e|an|→ 0§áˉan →a′¤áooo
y2μ
(1) duan → a(n → ∞)§é?ε > 0,?N ∈ Z+§n > N?§|an?a| < ε§Ké?k ∈ Z+,n + k >
N?§|an+k?a| < ε§u′é?ε > 0,?N ∈ Z+§n + k > N?§|an+k?a| < ε§l
an+k →
a(n→∞)
triangled(?`2μKêc?k§K?ù5"
(2) (i) duan → a§é?ε > 0,?N ∈ Z+§n > N?§|an?a| < ε.qvextendsinglevextendsingle|an|?|a|vextendsinglevextendsingle< |an?a|§u′
é?ε> 0,?N ∈Z+§n>N?§vextendsinglevextendsingle|an|?|a|vextendsinglevextendsingle<ε¤á§=|an|→|a|(n→∞
(ii)¤á"
~μ
(a)?¤áμan = (?1)n§K|an|→ 1§
an?4
(b) ¤áμan = 1n§K|an|→ 0,an → 0
(3) du|an| → 0§é?ε > 0,?N ∈ Z+§n > N?§vextendsinglevextendsingle|an|? 0vextendsinglevextendsingle < ε§q|an? 0| = vextendsinglevextendsingle|an|? 0vextendsinglevextendsingle§u
′é?ε > 0,?N ∈ Z+§n > N?§|an? 0| < ε¤á§=an → 0(n → ∞)"l
e|an| → 0§
Kan → 0¤á"
6,Uy2§exn →a§?a>b§K3N§n>N?§¤áxn >b.
y2μduxn → a§é?ε > 0,?N ∈ Z+§n > N?§|xn? 0| < ε§=a?ε < xn < a+ε.qa > b§
a?b> 0§Kε = a?b> 0§l
N ∈Z+§n>N?§kxn >a?ε = a?(a?b) = b.=3N§
n>N?§¤áxn >b.
7,e{xnyn}§U{xn},{yn}.
)μ?U"
~μxn = (?1)n,yn = (?1)n(n = 1,2,···),xnyn ≡ 1(n = 1,2,···)§K{xnyn}§{xn},{yn}t
"e{xnyn}§?U{xn},{yn}.
8,|^4?5?9O?y2μ
(1) lim
n→∞
parenleftbigg 1
n2 +
1
(n+ 1)2 +···+
1
(2n)2
parenrightbigg
= 0
(2) lim
n→∞
parenleftbigg 1
√n2 + 1 + 1√n2 + 2 +···+ 1√n2 +n
parenrightbigg
= 1
(3) |^(1 +h)n =
nsummationtext
k=0
Cknhk = 1 +nh+ n(n?1)2 h2 +···+hn
y2μ
(i) lim
n→∞
n
an = 0(a> 1)
(ii) lim
n→∞
n5
en = 0(e≈ 2.7)
y2μ
(1) é?n∈Z+§k0 lessorequalslant 1n2 + 1(n+ 1)2 +···+ 1(2n)2 lessorequalslant n+ 1n2 §? lim
n→∞
n+ 1
n2 = 0§K limn→∞
parenleftbigg 1
n2 +
1
(n+ 1)2 +···+
1
(2n)2
parenrightbigg
=
0
(2) é?n ∈ Z+§k nn+ 1 < 1√n2 + 1 + 1√n2 + 2 + ··· + 1√n2 +n < nn = 1? lim
n→∞
n
n+ 1 = 1§
K lim
n→∞
parenleftbigg 1
√n2 + 1 + 1√n2 + 2 +···+ 1√n2 +n
parenrightbigg
= 1
(3) (i) a = 1 +h(h> 0)§du0 < nan = n(1 +h)n = n
1 +nh+ n(n1)2 h2 +···+hn
< nn(n?1)
2 h
2
=
2
(n?1)h2§q
2
h2§
1
n?1 → 0(n→∞)§K
2
(n?1)h2 → 0.l
limn→∞
n
an = 0
19
(ii) e = 1 + h(h ≈ 1.7)§du0 < n
5
en =
n5
(1 +h)n =
n5
1 +nh+C2nh2 +···+hn <
n5
C6nh6 <
720n5
(n?5)6h6§q
720
h6§
n5
(n?5)6 → 0(n→∞)§K
720n5
(n?5)6h6 → 0(n→∞)§l
limn→∞
n5
en =
0
9,|e4?μ
(1) lim
n→∞
3n3 + 2n2?n+ 1
2n3?3n2 + 2
(2) lim
n→∞
6n2?n+ 1
n3 +n2 + 2
(3) lim
n→∞
parenleftbigg
1? 1n√2
parenrightbigg
cosn
(4) lim
n→∞
1 + 12 +···+ 12n
1 + 14 +···+ 14n
(5) lim
n→∞
bracketleftBigg
(sinn!)
parenleftbiggn?1
n2 + 1
parenrightbigg10
parenleftbigg 1
1·2 +
1
2·3 +···+
1
(n?1)·n
parenrightbigg2n2 + 1
n2?1
bracketrightBigg
(6) lim
n→∞
(?2)n + 3n
(?2)n+1 + 3n+1
)μ
(1) lim
n→∞
3n3 + 2n2?n+ 1
2n3?3n2 + 2 =
3
2
(2) lim
n→∞
6n2?n+ 1
n3 +n2 + 2 = 0
(3) dun√2 → 1(n→∞)§1? n√2 → 0(n→∞)§q|cosn|lessorequalslant 1§l
lim
n→∞
parenleftbigg
1? 1n√2
parenrightbigg
cosn = 0
(4) lim
n→∞
1 + 12 +···+ 12n
1 + 14 +···+ 14n
= lim
n→∞
1?(12)n+1
1? 12
1?(14)n+1
1? 14
= 24
3
= 32
(5) du{sinn!}?k.ê§
parenleftbiggn?1
n2 + 1
parenrightbigg10
→ 0,1? 1n → 1,2n
2 + 1
n2 + 1 → 2(n→∞)§
lim
n→∞
bracketleftBigg
(sinn!)
parenleftbiggn?1
n2 + 1
parenrightbigg10
parenleftbigg 1
1·2 +
1
2·3 +···+
1
(n?1)·n
parenrightbigg2n2 + 1
n2?1
bracketrightBigg
=?2
(6) lim
n→∞
(?2)n + 3n
(?2)n+1 + 3n+1 = limn→∞
(?23 )n + 1
(?2)(?23 )n + 3
= 13
10,exn →a> 0§áyμ
(1) √xn →√a
(2)
radicalbig
a0xmn +a1xm?1n +···+am?1xn +am →radicalbiga0am +a1am?1 +···+am?1a+am
(ù￥a0am +a1am?1 +···+am?1a+am > 0)
y2μ
(1) duxn → a > 0§é?ε > 0,?N ∈ Z+§n > N?§|xn? a| < √aε§?|√xn? √a| =vextendsingle
vextendsinglevextendsingle
vextendsingle
xn?a√
xn +√a
vextendsinglevextendsingle
vextendsinglevextendsingle < |xn?a|√
a < ε§=ét?ε > 0,?N ∈ Z
+§n > N?§|√xn? √a| < ε§l
√xn →√a(n→∞)
20
(2) duxn →a(n→∞)§a0xmn +a1xm?1n +···+am?1xn+am →a0am+a1am?1+···+am?1a+am > 0§
Ka(1)radicalbig
a0xmn +a1xm?1n +···+am?1xn +am →radicalbiga0am +a1am?1 +···+am?1a+am
11,éê{xn}§ex2k →a(k→∞),x2k+1 →a(k→∞)§y2μxn →a(n→∞)
y2μ?ε> 0§?x2k →a(n→∞)§?K1 ∈Z+§|k>K1?§|x2k?a|<ε¤á"
q?x2k+1 →a(n→∞)§?K2 ∈Z+§|k>K2?§|x2k+1?a|<ε¤á"
N = max{2K1,2K2 + 1}§Kn>N?§en?óê§n = 2k>N greaterorequalslant 2K1,k>K1,|xn?a| = |x2k?a|<
ε§
enê§n = 2k+ 1 >N greaterorequalslant 2K2 + 1,k>K2,|xn?a| = |x2k+1?a|<ε§
dxn →a(n→∞)
12,|^üNk.7k4?§y2 lim
n→∞
xn3§?|?§μ
(1) x1 = √2,···,xn = √2xn?1
(2) x0 = 1,x1 = 1 + x01 +x
0
,···,xn+1 = 1 + xn1 +x
n
y2μ
(1) w,x1 < x2§bxn?1 < xn§Kxn = √2xn?1 < √2xn§d8B{§{xn}′üNO\§qxn =√
2xn?1§x2n = 2xn1 lessorequalslant 2xn§u′xn lessorequalslant 2§={xn}dt."l
lim
n→∞
xn3§P lim
n→∞
xn = l§
3x2n = 2xn?1ü>-n→∞§l2 = 2l§)?l = 2§= lim
n→∞
xn = 2"
(2) w,xn greaterorequalslant 1§k^?xn = 1 + xn?11 +x
n?1
= 2? 11 +x
n?1
< 2§{xn}k."qx1 = 1 + x01 +x
0
=
1 + 11 + 1 = 32 > 1 = x0§bxn1 < xn§Kxn = 2? 11 +x
n?1
< 2? 11 +x
n
= xn+1§d
8B{§{xn}′üNO\"l
lim
n→∞
xn3§P lim
n→∞
xn = l§3xn = 2? 11 +x
n?1
ü>-
n → ∞§l = 2? 11 +l§=l2 = 1 + l§)l1 = 1 +
√5
2,l2 =
1?√5
2 ￡?üK?§¤§
= lim
n→∞
xn = 1 +
√5
2 "
13,ex1 = a> 0,y1 = b> 0(a<b),xn+1 = √xnyn,yn+1 = xn +yn2 §y2μ lim
n→∞
xn = lim
n→∞
yn.
y2μdu√xnyn lessorequalslant xn +yn2?da? ?=xn = yn§xn+1 lessorequalslant yn+1 ò¤á?=xn =
yn.q0 <a<b§x1 <y1§Kd4íúa§xn+1 <yn+1?xn > 0,yn > 0(n∈Z+).
xn+1 = √xnyn >
√x
nxn = xn,yn+1 =
xn +yn
2 <
yn +yn
2 = yn§Kxn <xn+1 <yn+1 <yn.qdx1 = a> 0,y1 = b> 0§
a < xn < xn+1 < yn+1 < yn < b§`2{xn}?{yn}?′üNk.ê§l
{xn},{yn}tk4?§
lim
n→∞
xn = α,lim
n→∞
yn = β§qdxn+1 = √xnyn§x2n+1 = xnyn§3 aü>-n → ∞§α2 = αβ q
d0 <a<xn <xn+1§?k0 <alessorequalslantα§l
α = β=k lim
n→∞
xn = lim
n→∞
yn.
14,|^üNk.7k4?y2±eê7k4?μ
(1) xn = 1 + 122 +···+ 1n2
(2) xn = 13 + 1 + 132 + 1 +···+ 13n + 1
(3) xn = n
k
an(a> 1,k?ê)
(4) xn = n√a (0 <a< 1)
y2μ
(1) duxn+1?xn = 1(n+ 1)2 > 0§xn+1 > xn§K{xn}?üNO\,q1 < xn < 1 + 11˙2 + ··· +
1
n˙(n+ 1) = 1 +
parenleftbigg
1? 12 +···+ 1n
1
1n
parenrightbigg
= 2? 1n < 2§{xn}k.§u′{xn}34?"
(2) duxn+1?xn = 13n+1 + 1 > 0§xn+1 >xn§K{xn}?üNO\,q14 <xn < 14+ 132 +···+ 13n <
1
3 +
1
32 +···+
1
3n =
1
3
1? 13
= 12§{xn}k.§u′{xn}34?"
21
(3) dua > 1,k?ê§xn = n
k
an > 0§K{xn}ke."q
xn+1
xn =
parenleftbigg
1 + 1n
parenrightbiggk
a =
1
a
parenleftbigg
1 + 1n
parenrightbiggk
→
1
a(n → ∞) < 1§?N ∈ Z
+§n > N?§kxn+1
xn < 1§KlN + 1?mkxn+1 < xn§u
′{xn}?üN~(n>N)§l
{xn}34?"
(4) dulnxn = 1n lna = yn,0 <a< 1§{yn}′üNO\§l
dxn = n√a = eyn{xn}′üNO\
"q0 <xn = n√a< n√1 = 1§{xn}k.§u′{xn}34?"
15,y2μexnt,§yneü§
xn?ynt§Kxnúyn7kó?4?"
y2μdxnt,§x1 lessorequalslant x2 lessorequalslant ··· lessorequalslant xn lessorequalslant ···§qyneü§y1 greaterorequalslant y2 greaterorequalslant ··· greaterorequalslant yn greaterorequalslant ···§qxn?yn
t§{xn?yn}k.§|xn?yn|lessorequalslantC(n = 1,2,···)￡ù￥C?,~ꤧK?C lessorequalslantxn?yn lessorequalslantC=xn lessorequalslant
yn +C lessorequalslant y1 +C§u′{xn}kt.§l
{xn}34?"qyn greaterorequalslant xn?C greaterorequalslant x1?C§u′{yn}ke.§l
{yn}34?§K lim
n→∞
xn? lim
n→∞
yn = lim
n→∞
(xn?yn) = 0§u′ lim
n→∞
xn = lim
n→∞
yn.
16,x￠ê§qyn(x) = sinsin···sinbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
n
x§y2{yn(x)}4?3§?|d4?.
y2μk0 lessorequalslant x lessorequalslant pi§K0 lessorequalslant sinx lessorequalslant x§l
kyn+1(x) = sinyn(x) lessorequalslant yn(x)§{yn(x)}′±0?e.ü
Neü?ê§7k4?§Ké?x0 ∈ [0,pi]§k0 lessorequalslant u0 = lim
n→∞
yn(x0) = sin
parenleftBig
lim
n→∞
fn?1(x0)
parenrightBig
= sinu0§
Ku0 = 0§l
é?x∈ [0,pi],lim
n→∞
yn(x) = 0.
ón?yx∈ [?pi,0]k lim
n→∞
yn(x) = 0.
2d±?5? lim
n→∞
yn(x) = 0
17,e lim
n→∞
xn = a§áyμ lim
n→∞
x1 +x2 +···+x+n
n = a
y2μd lim
n→∞
xn = a§é?ε> 0,?N1 ∈Z+§n>N1?§k|xn?a|< ε2§Kk
vextendsinglevextendsingle
vextendsinglex1 +x2 +···+xnn?a
vextendsinglevextendsingle
vextendsingle=vextendsingle
vextendsinglevextendsingle
vextendsingle
(x1?a) + (x2?a) +···+ (xn?a)
n
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant |x1?a|+|x2?a|+···+|xN1?a|+|xN1+1?a|+···+|xn?a|
n <
|x1?a|+|x2?a|+···+|xN1?a|
n +
n?N1
n ·
ε
2 <|x
1?a|+|x2?a|+···+|xN1?a|
n +
ε
2(∵
n?N1
n < 1)
M = max(|x1?a|,|x2?a|,···,|xn1?a|)§K|x1?a|+|x2?a|+···+|xN1?a|n lessorequalslant N1 ·Mn §qN1·M?
§KN1 ·Mn → 0(n→∞)§
u′ét?ε> 0,?N2 =
bracketleftbigg2N
1 ·M
ε
bracketrightbigg
∈Z+§n>N2?§k
|x1?a|+|x2?a|+···+|xN1?a|
n <
ε
2
N = max(N1,N2)§Kn>N?§k
vextendsinglevextendsingle
vextendsinglex1 +x2 +···+xnn?a
vextendsinglevextendsingle
vextendsingle< ε2 + ε2 = ε§
=k lim
n→∞
x1 +x2 +···+x+n
n = a
5μe lim
n→∞
x1 +x2 +···+xn
n = anotdblarrowright limn→∞xn3"
~μxn = (?1)n?1(n = 1,2,···)§Kw,lim
n→∞
x1 +x2 +···+xn
n = 0§ limn→∞xn?3"
18,y2μe lim
n→∞
an = a,lim
n→∞
bn = b§K lim
n→∞
a1bn +a2bn?1 +···+anb1
n = aby2μ
(1) a = 0§y lim
n→∞
a1bn +a2bn?1 +···+anb1
n = 0
d lim
n→∞
bn = b§Ka?n4(P38)§?M > 0§||bn|lessorequalslantM(n∈Z+)
d lim
n→∞
an = 0§Ké?ε> 0,?N1 ∈Z+§n>N1?§k|an|< ε2M.N = max
braceleftbiggbracketleftbigg2(|a
1|+···+|an|)M
ε
bracketrightbigg
+ 1,N1
bracerightbigg
§
u′ngreaterorequalslantN(greaterorequalslantN1)?§k
vextendsinglevextendsingle
vextendsinglevextendsinglea1bn +a2bn?1 +···+anb1
n
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsinglea1bn +a2bn?1 +···+aN1bn?N1+1 +aN1+1bn?N1 +···+anb1
n
vextendsinglevextendsingle
vextendsinglevextendsingle
lessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsinglea1bn +a2bn?1 +···+aN1bn?N1+1
n
vextendsinglevextendsingle
vextendsinglevextendsingle+
vextendsinglevextendsingle
vextendsinglevextendsingleaN1+1bn?N1 +···+anb1
n
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant (|a1|+···+|aN1|)M
n +
(n?N1)· ε2M ·M
n <
22
ε
2 +
ε
2 = ε§l
limn→∞
a1bn +a2bn?1 +···+anb1
n = 0
(2) anegationslash= 0,bnegationslash= 0?§d lim
n→∞
bn = b§ lim
n→∞
bn +bn?1 +···+b1
n = bnegationslash= 0§q limn→∞an = a§ limn→∞(an?a) = 0
d(1) lim
n→∞
(a1?a)bn +···+ (an?a)b1
n = 0§
u′ lim
n→∞
a1 ·
bn
n +···+an ·
b1
n
bn +···+b1
n
a
= lim
n→∞
(a1?a)· bnn +···+ (an?a)· bnn
bn +···+b1
n
= 0b = 0§
= lim
n→∞
a1 · bnn +···+an · b1n
bn +···+b1
n
= a§
l
lim
n→∞
a1bn +a2bn?1 +···+anb1
n = limn→∞
a1 ·
bn
n +···+an ·
b1
n
bn +···+b1
n
· bn +···+b1n
= ab
19,Uy2eêtμ
(1) √n
(2) n!
(3) lnn
(4) n
2 + 1
2n+ 1
(5) n
2 + 1
2n?1
(6) 1 + 12 + 13 +···+ 1n
y2μ
(1) é?G > 0§?||√n| > G§n > G2=?.N = [G2]§Kn > N?§|√n| > Go¤á§
{√n}′t"
(2) é?G > 0§du|n!| > n§?||n!| > G§n > G=?.N = [G]§Kn > N?§|n!| > Go¤
á§{n!}′t"
(3) é?G > 0§?||lnn| > G§n > eG=?.N = [eG]§Kn > N?§|lnn| > Go¤á§
{lnn}′t"
(4) é?G > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle > n2
3n =
n
3§?|
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle > G§n
3 > G=?.N = [3G]§Kn >
N?§
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle>Go¤á§{n2 + 1
2n+ 1}′t"
(5) é?G > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n?1
vextendsinglevextendsingle
vextendsinglevextendsingle > n2
2n =
n
2§?|
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n?1
vextendsinglevextendsingle
vextendsinglevextendsingle > G§n
2 > G=?.N = [2G]§Kn >
N?§
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n?1
vextendsinglevextendsingle
vextendsinglevextendsingle>Go¤á§{n2 + 1
2n?1}′t"
(6) é?G > 0§du lim
n→∞
parenleftbigg
1 + 1n
parenrightbiggn
= e?
parenleftbigg
1 + 1n
parenrightbiggn
üNO\§K
parenleftbigg
1 + 1n
parenrightbiggn
< e§u′ln
parenleftbigg
1 + 1n
parenrightbigg
<
1
n§l
1+
1
2+
1
3+···+
1
n > ln2+ln
3
2+···+ln
parenleftbigg
1 + 1n
parenrightbigg
= ln(n+1) > lnn§K?|
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 1
2 +
1
3 +···+
1
n
vextendsinglevextendsingle
vextendsinglevextendsingle>
G§lnn > G=?.N = [eG]§Kn > N?§
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 1
2 +
1
3 +···+
1
n
vextendsinglevextendsingle
vextendsinglevextendsingle > Go¤á§{1 + 1
2 +
1
3 +···+
1
n}′t"
20,y2μe{xn}′t§xn negationslash= 0(n = 1,2,3,···)§K
braceleftbigg 1
xn
bracerightbigg
′t"
y2μdu{xn}′t§é?ε> 0,?N ∈Z+§n>N?§k|xn|<ε
23
qxn negationslash= 0(n = 1,2,3,···)§ 1x
n
3?
vextendsinglevextendsingle
vextendsinglevextendsingle 1
xn
vextendsinglevextendsingle
vextendsinglevextendsingle> 1
ε
qε′§1ε?′§l
braceleftbigg 1
xn
bracerightbigg
′t"
21,y2μe{xn}t§{yn}?k.Ct§K{xn ±yn}t"
ddO?e4?μ
(1) lim
n→∞
parenleftbigg
sinn+ n
2
√n2 + 1
parenrightbigg
(2) lim
n→∞
(n?arctann)
(3) lim
n→∞
bracketleftbigg
n+ (?1)n
parenleftbigg 1
1·3 +
1
3·5 +···+
1
(2n?1)(2n+ 1)
parenrightbiggbracketrightbigg
qμütú4?NoáU?/"
i)y2μdu{yn}?k.Ct§73êM§||yn| lessorequalslant M§q{xn}t§é?G > M >
0,?N ∈ Z+§n > N?§k|xn| > G§Kn > N?§k|xn ±yn| greaterorequalslant |xn|?|yn| > G?M.dG5
9G>M > 0§?G?M > 0?G?M′§l
{xn ±yn}t"
ii))μ
(1) du lim
n→∞
n2√
n2 + 1 = ∞?|sinn|lessorequalslant 1§K limn→∞
parenleftbigg
sinn+ n
2
√n2 + 1
parenrightbigg
= ∞
(2) du lim
n→∞
n = ∞?|arctann|lessorequalslant pi2§K lim
n→∞
(n?arctann) = ∞
(3) xn = (?1)n
parenleftbigg 1
1·3 +
1
3·5 +···+
1
(2n?1)(2n+ 1)
parenrightbigg
§Kxn = (?1)
n
2
bracketleftbigg
1? 13 + 13? 15 +···+ 12n?1? 12n+ 1
bracketrightbigg
=
(?1)n
2
parenleftbigg
1? 12n+ 1
parenrightbigg
= (?1)
n
2 ·
2n
2n+ 1 =
(?1)n
2 + 1n
§k13 <|xn|< 12.qd lim
n→∞
n = ∞§l
lim
n→∞
bracketleftbigg
n+ (?1)n
parenleftbigg 1
1·3 +
1
3·5 +···+
1
(2n?1)(2n+ 1)
parenrightbiggbracketrightbigg
= ∞
iii))μ
(1) xn = n→ +∞,yn = 2n→ +∞;xn +yn = 3n→ +∞
(2) xn =?n→?∞,yn =?2n→?∞;xn +yn =?3n→?∞
(3) xn =?n→?∞,yn = 2n→ +∞;xn +yn = n→ +∞
(4) xn = n→ +∞,yn =?2n→?∞;xn +yn =?n→?∞
(5) xn = n+a→ +∞,yn =?n→?∞;xn +yn = a￡~t¤
(6) xn = n+ (?1)n → +∞,yn =?n→ +∞;xn +yn = (?1)n?4?
22,tútú!
!?4??/"
)μ
(1) ú!
μ?yn → 0(n → ∞)§{yn}k."qxn → ∞(n → ∞)§KdtK(?§k{xn ±yn}
t"
(2)?μxn negationslash= 0,yn negationslash= 0?§duxn →∞,yn → 0(n→∞)§Kkyn · 1x
n
→ 0§=ynx
n
→ 0,xny
n
→∞
23,T~`2tút|è?Uu)/"
)μ
(1) xn = n→ +∞,yn = 1n2 → 0(n→∞);xn ·yn = 1n → 0(n→∞)
(2) xn = n2 → +∞,yn = 1n → 0(n→∞);xn ·yn = n→ +∞(n→∞)
(3) xn = n→ +∞,yn = an → 0(n→∞);xn ·yn = a￡~t¤
(4) xn = n(?1)n →∞,yn = 1n → 0(n→∞);xn ·yn = (?1)n?4?k.
(5) xn = n2n(?1)n → ∞,yn = 1n → 0(n→ ∞);xn ·yn = n·n(?1)n = n1+(?1)n?4?§?.￡′?
t¤
24
24,exn →∞,yn →anegationslash= 0§y2xnyn →∞
y2μduxn → ∞(n → ∞)§ 1x
n
→ 0(n → ∞)?qyn → a negationslash= 0(n → ∞)§ 1y
n
→ 1a(n → ∞)§u
′ 1x
n
· 1y
n
→ 0(n→∞)§l
xnyn →∞(n→∞)
25,exn → +∞,yn →?∞§y2xnyn →?∞.
y2μ?xn → +∞§Ké?G1 > 0,?N1 ∈ Z+§n > N1?§kxn > G1?qyn →?∞§Ké?G2 >
0,?N2 ∈ Z+§n > N2?§k?yn > G2 > 0,N = max(N1,N2)§Kn > N?§k?xnyn > G1G2§
=xnyn <?G1G2.dG1,G25§G1G2′?G1G2 > 0§Kxnyn →?∞.
26,exn → +∞§y2x1 +x2 +···+xnn → +∞
y2μ?xn → +∞§Ké?G > 0,?N1 ∈ Z+§n > N1?§kxn > 3G§u′x1 +x2 +···+xnn =
x1 +···+xN1
n +x
N1+1 +···+xn
n >
x1 +···+xN1
n +
n?N1
n ·3G§
M = max(|x1|,···,|xN1|)§K
vextendsinglevextendsingle
vextendsinglex1 +·+xN1n
vextendsinglevextendsingle
vextendsingle lessorequalslant |x1|+···+|xN1|n lessorequalslant N1 ·Mn §u′ét?G > 0§
N2 =
bracketleftbigg2N
1 ·M
G
bracketrightbigg
§Kn>N2?§k
vextendsinglevextendsingle
vextendsinglex1 +···+xN1n
vextendsinglevextendsingle
vextendsingle< G2§l
x1 +···+xN1n >?G2"q limn→∞ n?N1n =
1§éuε = 12,?N3 ∈Z+§n>N3?§k
vextendsinglevextendsingle
vextendsinglevextendsinglen?N1
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle< 1
2§l
n?N1
n >
1
2§N = max{N1,N2,N3}§
Kn>N?§kx1 +x2 +···+xnn >?G2 + 3G2 = G§ddx1 +x2 +···+xnn → +∞(n→∞).
25
§2,?ê4?
1,^y2μ
(1) lim
x→?1
x?3
x2?9 =
1
2
(2) lim
x→3
x?3
x2?9 =
1
6
(3) lim
x→1
x?1√
x?1 = 2
(4) lim
x→1
(x?2)(x?1)
x?3 = 0
(5) lim
t→1
t(t?1)
t2?1 =
1
2
(6) lim
x→∞
x?1
x+ 2 = 1
(7) lim
x→3
x
x2?9 = ∞
(8) lim
x→∞
x2 +x
x+ 1 = ∞
y2μ
(1) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x+ 3?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle x+ 1
2x+ 6
vextendsinglevextendsingle
vextendsinglevextendsingle§?x→?1§|x+ 1|< 1§K?2 <x<
0§l
2 < |2x + 6| < 6§u′
vextendsinglevextendsingle
vextendsinglevextendsingle x+ 1
2x+ 6
vextendsinglevextendsingle
vextendsinglevextendsingle < |x+ 1|
2 §?|
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§|x+ 1|
2 < ε=?"
δ = min{2ε,1}> 0§K0 <|x?(?1)|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§ lim
x→?1
x?3
x2?9 =
1
2
(2) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
6
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x+ 3?
1
6
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle x?3
6x+ 18
vextendsinglevextendsingle
vextendsinglevextendsingle§?x → 3§|x? 3| < 1§K2 < x <
4§l
30 < |6x + 18| < 42§u′
vextendsinglevextendsingle
vextendsinglevextendsingle x?3
6x+ 18
vextendsinglevextendsingle
vextendsinglevextendsingle < |x?3|
30 §?|
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
6
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§|x?3|
30 < ε=
"δ = min{30ε,1}> 0§K0 <|x?3|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
6
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§lim
x→3
x?3
x2?9 =
1
6
(3) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x?1?2
vextendsinglevextendsingle
vextendsinglevextendsingle = |√x+ 1?2| = |√x?1| =
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle§?x→ 1§|x?1|< 1§
K0 < x < 2§l
1 < |√x + 1| < √2 + 1§u′
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle < |x? 1|§?|
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x?1?2
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§?
|x? 1| < ε=?"δ = min{ε,1} > 0§K0 < |x? 1| < δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x?1?2
vextendsinglevextendsingle
vextendsinglevextendsingle < εo¤á§
lim
x→1
x?1√
x?1 = 2
(4) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle(x?2)(x?1)
x?3?0
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle
parenleftbigg
1 + 1x?3
parenrightbigg
(x?1)
vextendsinglevextendsingle
vextendsinglevextendsingle§?x → 1§|x? 1| < 1§K0 <
x < 2§l
0 <
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle < 2
3§u′
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle < 2
3|x? 1|§?|
vextendsinglevextendsingle
vextendsinglevextendsingle(x?2)(x?1)
x?3?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§?
23|x?1|<ε=?"δ = min
braceleftbigg3
2ε,1
bracerightbigg
> 0§K0 <|x?1|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsingle(x?2)(x?1)
x?3?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo
¤á§lim
x→1
(x?2)(x?1)
x?3 = 0
(5) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglet(t?1)
t2?1?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle t
t+ 1?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle t?1
2t+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle§?t→ 1§|t?1|< 1§K0 <t< 2§
l
2 < |2t + 2| < 6§u′
vextendsinglevextendsingle
vextendsinglevextendsingle t?1
2t+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle < |t?1|
2 §?|
vextendsinglevextendsingle
vextendsinglevextendsinglet(t?1)
t2?1?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§|t?1|
2 < ε=?"
δ = min{2ε,1}> 0§K0 <|x?(?1)|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsinglet(t?1)
t2?1?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§lim
t→1
t(t?1)
t2?1 =
1
2
26
(6) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglex?1
x+ 2?1
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle 3
x+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle§?x→∞§|x|> 2§K|x+2|>|x|?2§u′
vextendsinglevextendsingle
vextendsinglevextendsingle 3
x+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle<
3
|x|?2§?|
vextendsinglevextendsingle
vextendsinglevextendsingle 3
x+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§ 3
|x|?2 < ε=?§=|x| >
3
ε"X =
3
ε + 2§K|x| > X?§ò
k
vextendsinglevextendsingle
vextendsinglevextendsinglex?1
x+ 2?1
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§ lim
x→∞
x?1
x+ 2 = 1
(7) é?G > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle x
x2?9
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle x
x+ 3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle§?x → 3§|x? 3| < 1§K2 < x < 4§l
27 <
vextendsinglevextendsingle
vextendsinglevextendsingle x
x+ 3
vextendsinglevextendsingle
vextendsinglevextendsingle < 4
5§u′
vextendsinglevextendsingle
vextendsinglevextendsingle x
x+ 3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle > 2
7
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle§?|
vextendsinglevextendsingle
vextendsinglevextendsingle x
x2?9
vextendsinglevextendsingle
vextendsinglevextendsingle > G§2
7
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle > G=
"δ = min
braceleftbigg 2
7G,1
bracerightbigg
> 0§K0 <|x?3|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsingle x
x2?9
vextendsinglevextendsingle
vextendsinglevextendsingle>Go¤á§lim
x→3
x
x2?9 = ∞
(8) é?G > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglex2 +x
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle = |x|§?x → ∞§X = G > 0§K|x| > X?§òk
vextendsinglevextendsingle
vextendsinglevextendsinglex2 +x
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle> Go¤
᧠lim
x→∞
x2 +x
x+ 1 = ∞
2,|4?μ
(1) lim
x→0
x2?1
2x2?x?1
(2) lim
x→1
x2?1
2x2?x?1
(3) lim
x→∞
x2?1
2x2?x?1
(4) lim
x→0
(1 +x)(1 + 2x)(1 + 3x)?1
x
(5) lim
t→1
t2(t?1)
t2?1
(6) lim
t→1
t2?√t√
t?1
(7) lim
x→3
√1 +x?2
x?3
(8) lim
x→0
(1 +x)5?(1 + 5x)
x2 +x5
(9) lim
x→0
(1 +mx)n?(1 +nx)m
x2 ￡m,n?g,ê¤
(10) lim
x→3
x2?5 + 6
x2?8x+ 15
(11) lim
x→∞
x2 + 3x
x2
(12) lim
x→∞
5x?7
2x+√x
)μ
(1) lim
x→0
x2?1
2x2?x?1 = 1
(2) lim
x→1
x2?1
2x2?x?1 = limx→1
(x?1)(x+ 1)
(2x+ 1)(x?1) = limx→1
x+ 1
2x+ 1 =
2
3
(3) lim
x→∞
x2?1
2x2?x?1 =
1
2
(4) lim
x→0
(1 +x)(1 + 2x)(1 + 3x)?1
x = limx→0(6 + 11x+ 6x
2) = 6
(5) lim
t→1
t2(t?1)
t2?1 = limt→1
t2
t+ 1 =
1
2
(6) lim
t→1
t2?√t√
t?1 = limt→1
√t(√t?1)(t+√t+ 1
√t?1 =
lim
t→1
√t(t+√t+ 1) = 3
27
(7) lim
x→3
√1 +x?2
x?3 = limx→3
1√
1 +x+ 2 =
1
4
(8) lim
x→0
(1 +x)5?(1 + 5x)
x2 +x5 = limx→0
10x2 + 10x3 + 5x4 +x5
x2 +x5 = 10
(9) lim
x→0
(1 +mx)n?(1 +nx)m
x2 =
lim
x→0
(C2nm2?C2mn2)x2 + (C3nm3?C3mn3)x3 +···+mnxn?nmxm
x2 = C
2
nm
2?C2
mn
2 = n2m?m2n
2
(10) lim
x→3
x2?5 + 6
x2?8x+ 15 = limx→3
(x?2)(x?3)
(x?3)(x?5) = limx→3
x?2
x?5 =?
1
2
(11) lim
x→∞
x2 + 3x
x2 = 1
(12) lim
x→∞
5x?7
2x+√x =
5
2
3, R(x) = P(x)Q(x)
a￥P(x)úQ(x)?xa§P(a) = Q(a) = 0§ˉlim
x→a
k=
U?o
)μduP(x)úQ(x)?xa?P(a) = Q(a) = 0§
KP(x) = (x?a)mP1(x),Q(x) = (x?a)nQ1(x)(P1(a) negationslash= 0,Q1(x) negationslash= 0)§u′lim
x→a
R(x) = lim
x→a
P(x)
Q(x) =
lim
x→a
(x?a)mP1(x)
(x?a)nQ1(x)
μ
(1) n = m?§lim
x→a
R(x) = P1(a)Q
1(a)
(2) n>m?§lim
x→a
(x?a)m?n = ∞?lim
x→a
P1(x)
Q1(x) =
P1(a)
Q1(a) negationslash= 0§limx→aR(x) = ∞
(3) n<m?§lim
x→a
(x?a)m?n = 0?lim
x→a
P1(x)
Q1(x) =
P1(a)
Q1(a)§limx→aR(x) = 0
4,|e4?μ
(1) lim
x→0
sin2x?sin3x
x
(2) lim
h→0
cos(x+h)?cosx
h
(3) lim
x→+∞
(√x2 + 1?x)
(4) lim
x→?∞
(√x2 + 1?x)
(5) lim
x→0
x2
1?cosx
(6) lim
x→+∞
radicalBig
x+radicalbigx+√x
x+ 1
(7) lim
x→0
cosx?cos3x
x2
(8) lim
x→0
sin5x?sin3x
sin2x
(9) lim
x→1
(1?x)tan pix2
(10) lim
x→a
sinx?sina
x?a
(11) lim
x→0
(√1 +x2 +x)n?(√1 +x2?x)n
x
(12) lim
x→0
x
bracketleftbigg1
x
bracketrightbigg
)μ
28
(1) lim
x→0
sin2x?sin3x
x = limx→0
sin2x
x? limx→0
sin3x
x = 2?3 =?1
(2) lim
h→0
cos(x+h)?cosx
h = limh→0
2sin 2x+h2 sin h2
h = limh→0
sin h2
h sin
2x+h
2 =?sinx
(3) lim
x→+∞
(√x2 + 1?x) = lim
x→+∞
1√
x2 + 1 +x = 0
(4) lim
x→?∞
(√x2 + 1?x) = +∞
(5) lim
x→0
x2
1?cosx = limx→0
x2
x2
2
= 2
(6) lim
x→+∞
radicalBig
x+radicalbigx+√x
x+ 1 = 0
(7) lim
x→0
cosx?cos3x
x2 = limx→0
2sinxsin2x
x2 = 4
(8) lim
x→0
sin5x?sin3x
sin2x = limx→0
2sinxcos4x
2x = 1
(9) -y = x? 1§Klim
x→1
(1?x)tan pix2 = lim
y→0
ytan
parenleftBigpi
2(1 +y)
parenrightBig
= lim
y→0
ycot pi2y = lim
y→0
ycos pi2y
sin pi2y
=
lim
y→0
y
pi
2y
= 2pi
(10) lim
x→a
sinx?sina
x?a = limx→a
2cos x+a2 sin x?a2
x?a = cosa
(11) lim
x→0
(√1 +x2 +x)n?(√1 +x2?x)n
x =
lim
x→0
2C1n(1 +x2)n?12 x+ 2C3n(1 +x2)n?32 x2 +···
x =
lim
x→0
bracketleftBig
2n(1 +x2)n?12 + 2C3n(1 +x2)n?32 x+···
bracketrightBig
= 2n
(12) du
bracketleftbigg1
x
bracketrightbigg
= 1x?
parenleftbigg1
x
parenrightbigg
0 lessorequalslant
parenleftbigg1
x
parenrightbigg
< 1§
Klim
x→0
x
bracketleftbigg1
x
bracketrightbigg
= lim
x→0
braceleftbigg
1?x
parenleftbigg1
x
parenrightbiggbracerightbigg
= 1? lim
x→0
x
parenleftbigg1
x
parenrightbigg
= 1
5,e lim
x→x0
f(x) = A,lim
x→x0
g(x) = B§3δ> 0§0 <|x?x0|<δ?kf(x) greaterorequalslantg(x)§y2AgreaterorequalslantB.
qe0 <|x?x0|<δ?f(x) >g(x)§′¤áA>B
y2μ
(1) ^?y{"bA<B§Kd lim
x→x0
f(x) = A,lim
x→x0
g(x) = B95?1§?δ0 > 0§|0 < |x?x0| <
δ0?§kg(x) > f(x)"ùμ?δ > 0§0 < |x?x0| < δ?§kf(x) greaterorequalslant g(x)g?§b?¤
á§=AgreaterorequalslantB¤á"
(2)"~μ
(i) ¤á"f(x) = 2(x
2 + 3x4)
x2,g(x) = x
2 + 3x4x2,?δ > 0§0 < |x| < δ?§kf(x) > g(x)"
qA = lim
x→x0
f(x) = 2,B = lim
x→x0
g(x) = 1§A>B¤á"
(ii)?¤á"f(x) = x
2 + 3x4
x2,g(x) = x
2 +x4x2,?δ > 0§0 < |x| < δ?§kf(x) > g(x)"qA =
lim
x→x0
f(x) = 1,B = lim
x→x0
g(x) = 1§kA = B"
6,e3:x0?Skg(x) lessorequalslant f(x) lessorequalslant h(x)§g(x)úh(x)3x04?3 uA§y2 lim
x→x0
f(x) =
A.
y2μXJéxn,xn →x0,xn negationslash= x0 §bxn ∈O(x0,δ)?{x0}§kg(xn) lessorequalslantf(xn) lessorequalslanth(xn)
±9g(xn) → A,h(xn) → A(n → ∞) §dê4?5?μf(xn) → A(n → ∞) §ùòy2
f(x) →
A(x→x0).
29
7,e lim
x→x0
f(x) = A,lim
x→x0
g(x) = B negationslash= 0§y2 lim
x→x0
f(x)
g(x) =
A
B.
y2μ
vextendsinglevextendsingle
vextendsinglevextendsinglef(x)
g(x)?
A
B
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingleBf(x)?Ag(x)
Bg(x)
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingleBf(x)?AB +AB?Ag(x)
BG(x)
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant |B||f(x)?A|+|A||g(x)?B|
|B||g(x)| §
du lim
x→x0
f(x) = A,lim
x→x0
g(x) = B§é?ε> 0,?δ1 > 0§0 <|x?x0|<δ1?§k|f(x)?A|<ε?ét
ε> 0,?δ2 > 0§0 <|x?x0|<δ2?§k|g(x)?B|<ε
qa|{$?μ lim
x→x0
Bg(x) = B2 > B
2
2 §Ka5?3§?δ3 > 0§0 <|x?x0|<δ3?§kBg(x) >
B2
2
δ = min{δ1,δ2,δ3}§0 <|x?x0|<δ?§k
vextendsinglevextendsingle
vextendsinglevextendsinglef(x)
g(x)?
A
B
vextendsinglevextendsingle
vextendsinglevextendsingle< (|A|+|B|)ε
B2
2
= 2(|A|+|B|)B2 ε
u′§é?ε> 0,?δ> 0§0 <|x?x0|<δ?§k
vextendsinglevextendsingle
vextendsinglevextendsinglef(x)
g(x)?
A
B
vextendsinglevextendsingle
vextendsinglevextendsingle< 2(|A|+|B|)
B2 ε§l
limx→x0
f(x)
g(x) =
A
B.
8,(1) f(x) =
0 x> 1
1 x = 1
x2 + 2 x< 1
|f(x)3x = 1?m4?"
(2) f(x) =
xsin 1x x> 0
1 +x2 x< 0
|f(x)3x = 0?m4?"
)μ
(1) lim
x→1?0
f(x) = lim
x→1?0
(x2 + 2) = 3,lim
x→1+0
f(x) = 0
(2) lim
x→?0
f(x) = lim
x→?0
(1 +x2) = 1,
lim
x→+0
f(x) = lim
x→+0
(xsin 1x) = 0
9,`2e?ê3¤?:?m4/μ
(1) y =
1
2x 0 <xlessorequalslant 1
x2 1 <x< 2 (3x = 1.5,2,1n:)
2x 2 <x< 3
(2) y = x·sin 1x￡3x = 0:¤
(3) y = 2
1
x + 1
21x?1
￡3x = 0:¤
(4) y = 1x?
bracketleftbigg1
x
bracketrightbigg
￡3x = 1n:¤
(5) D(x) =
braceleftbigg 1 x?knê
0 xnê (3:)
(6) y = (x?1)(?1)
[x]
x2?1 (3x =?1¤
)μ
(1) lim
x→1.5?0
y = lim
x→1.5+0
y = 2.25,
lim
x→2?0
y = lim
x→2?0
x2 = 4,lim
x→2+0
y = lim
x→2+0
(2x) = 4
lim
x→1?0
y = lim
x→1?0
1
2x =
1
2,limx→1+0y = limx→1+0x
2 = 1
(2) lim
x→+0
y = lim
x→+0
y = 0
(3) du lim
x→+0
1
x = +∞,limx→?0
1
x =?∞§
K lim
x→+0
21x = +∞,lim
x→?0
21x = 0§
u′ lim
x→+0
y = lim
x→=0
21x + 1
21x?1
= lim
x→+0
parenleftbigg
1 + 2
21x?1
parenrightbigg
= 1,lim
x→+?0
y = lim
x→?0
21x + 1
21x?1
=?1
30
(4) lim
x→1n+0
y = lim
x→1n+0
parenleftbigg1
x?
bracketleftbigg1
x
bracketrightbiggparenrightbigg
= n?(n?1) = 1
lim
x→1n?0
y = lim
x→1n?0
parenleftbigg1
x?
bracketleftbigg1
x
bracketrightbiggparenrightbigg
= n?n = 0
(5) d?ê3:?m43"
x0?Rt:§dknêú?nê3ê?tè?5§?knS{x(1)n } → x0 + 0§?nS
{x(2)n }→x0 + 0§
lim
x(1)n →x0+0
D
parenleftBig
x(1)
parenrightBig
= 1,lim
x(2)n →x0+0
D
parenleftBig
x(2)
parenrightBig
= 0§l
d?ê3:m43
ón§d?ê3:?43
l
d?ê3:?m43"
(6) y = (x?1)(?1)
[x]
x2?1 =
(?1)[x]
x+ 1? limx→?1+0[x] =?1,limx→?1?0[x] =?2
K lim
x→?1+0
y =?∞,lim
x→?1+0
y =?∞
10,e4?μ
(1) lim
x→∞
sinx
x
(2) lim
x→∞
ex sinx
(3) lim
x→∞
xarctanx
(4) lim
x→∞
xtanx(xnegationslash= npi+ pi2)
)μ
(1) du lim
x→∞
1
x = 0?sinx′k.t§ limx→∞
sinx
x = 0
(2) du lim
x→+∞
ex = +∞§exn = 2npi → +∞(n → ∞)§Kexn sinxn = e2npi sin2npi = 0 → 0(n →
∞)?exn = pi2 + 2npi → +∞(n → ∞)§Kexn sinxn = epi2 +2npi sin
parenleftBigpi
2 + 2npi
parenrightBig
= epi2 +2npi →
+∞(n→∞)§ lim
x→+∞
ex sinx?3§l
lim
x→∞
ex sinx?3.
(3) du lim
x→?∞
arctanx =?pi2,lim
x→+∞
xarctanx = pi2§
K lim
x→?∞
xarctanx = +∞,lim
x→+∞
xarctanx = +∞§l
lim
x→∞
xarctanx = +∞
(4) xn = npi → ∞(n → ∞)§k lim
n→∞
xn tanxn = lim
n→∞
npitannpi = 0?,xn = pi4 +npi → ∞(n →
∞)§k lim
n→∞
xn tanxn = lim
n→∞
parenleftBigpi
4 +npi
parenrightBig
tan
parenleftBigpi
4 +npi
parenrightBig
= lim
n→∞
parenleftBigpi
4 +npi
parenrightBig
= +∞§ lim
x→∞
xtanx(xnegationslash=
npi+ pi2)?3.
11,l^? lim
x→∞
parenleftbiggx2 + 1
x+ 1?ax?b
parenrightbigg
= 0§|~êaúb.
)μdu lim
x→∞
parenleftbiggx2 + 1
x+ 1?ax?b
parenrightbigg
= lim
x→∞
(x2 + 1)?ax(x+ 1)?b(x+ 1)
x+ 1 = limx→∞
(1?a)x2?(a+b)x?b+ 1
x+ 1 =
0§Kk
braceleftbigg 1?a = 0
a+b = 0 §l
braceleftbigg a = 1
b =?1
12,l^? lim
x→?∞
(√x2?x+ 1?a1x?b1) = 0,lim
x→?∞
(√x2?x+ 1?a2x?b2) = 0§|~êa1,b1,a2,b2.
)μdu lim
x→?∞
(√x2?x+ 1?a1x?b1) = lim
x→?∞
(1?a21)x2?(1 + 2a1b1)x+ 1?b21√
x2?x+ 1 +a1x+b1 = 0§K
braceleftbigg 1?a2
1 = 0
1 + 2a1b1 = 0 §
u′
braceleftBigg a
1 = ±1
b1 =?12,
qa^μea1 = 1§K lim
x→?∞
(√x2?x+ 1?a1x?b1) = +∞§l
braceleftBigg a
1 =?1
b1 = 12 §ón
braceleftBigg a
2 = 1
b2 =?12,
13,e lim
x→+∞
[f(x)?(kx+b)] = 0§Ky = kx+b′-?y = f(x)x→ +∞ìC?.|^ù§í?ì
C?37^?.
y2μe-?3ìC?§Kk
lim
x→+∞
[f(x)?(kx+b)] = 0,(1)
31
f(x)x = 1x[f(x)?kx?b] +k+ bx§-x→ +∞üà45?(1)a§
lim
x→+∞
f(x)
x = k (2)
.
Q|?
k§2l(1)a|
b = lim
x→+∞
[f(x)?kx] (3)
.
§e(2)!(3)üa¤á§á=?w?^?(1)¤á.
-?y = f(x)x → +∞?3ìC?y = kx+b7?^?′4? lim
x→+∞
f(x)
x = k! limx→+∞[f(x)?
kx] = bt¤á.
14,e lim
x→?∞
f(x) = A> 0§y23X > 0§|x<?X¤áμA2 <f(x) < 32A.
y2μdu lim
x→?∞
f(x) = A > 0§éε = A2 > 0,?X > 0 §x <?X?§k|f(x)?A| < A2§
=A2 <f(x) < 32A.
15,e lim
x→+∞
f(x) = A,lim
x→+∞
g(x) = B§y2 lim
x→+∞
f(x)g(x) = AB.
y2μdu lim
x→+∞
f(x) = A§é?ε > 0,?X1 > 0§x > X1?§k|f(x)?A| < εX2 > 0,M > 0§
x>X2?§k|f(x)|<A.
q lim
x→+∞
g(x) = B§ét?ε> 0,?X3 > 0§x>X3?§k|g(x)?B|<ε.
X = max{X1,X2,X3}§ét?ε> 0§x>X?§
k|f(x)g(x)?AB| = |f(x)g(x)?f(x)B+f(x)B?AB|lessorequalslant|f(x)||g(x)?B|+|B||f(x)?A|lessorequalslantMε+|B|ε =
(M +|B|)ε§= lim
x→+∞
f(x)g(x) = AB.
16,y2 lim
x→+∞
f(x) = A^?′μéêxn → +∞,f(xn) →A.
y2μ
du lim
x→+∞
f(x) = A§é?ε> 0,?X > 0§x>X?§k|f(x)?A|<ε.
qxn → +∞(n → ∞)§ét?X > 0,?N ∈ Z+§n > N?§kxn > X§l
|f(xn)?A| < ε§u
′ lim
n→∞
f(xn) = A.
^?y{"b lim
x→+∞
f(x) negationslash= A§K?ε0 > 0§é?X > 0§?kxprime§xprime >X?§k|f(xprime)?A| greaterorequalslant
ε0.
AO/§X?1,2,3,···§?xprime1,xprime2,xprime3,···§|
xprime1 > 1?§k|f(xprime1)?A|greaterorequalslantε0?xprime2 > 2?§k|f(xprime2)?A|greaterorequalslantε0?xprime3 > 3?§k|f(xprime3)?A|greaterorequalslantε0?···
l?>?±w?xprimen → +∞(n → ∞)§
lm>w? lim
n→∞
f(xprimen) negationslash= A§g?§Kb?¤á§
lim
x→+∞
f(x) = A
17,y2 lim
x→x0+0
f(x) = +∞^?′μéêxn,xn >x0,xn →x0§kf(xn) → +∞.
y2μ
du lim
x→x0+0
f(x) = +∞§é?G> 0,?δ> 0§0 <x?x0 <δ?§kf(x) >G.
qxn > x0,xn → x0(n → ∞)§ét?δ > 0,?N ∈ Z+§n > N?§k0 < xn?x0 < δ§l
f(xn) >
G§u′ lim
n→∞
f(xn) = +∞.
^?y{"b lim
x→x0+0
f(x) negationslash= +∞§K?G0 > 0§é?δ > 0§?kxprime§0 < xprime?x0 < δ?§
kf(xprime) lessorequalslantG0.
AO/§δ?1,12,13,···§?xprime1,xprime2,xprime3,···§|
0 < xprime1? x0 < 1?§kf(xprime1) lessorequalslant G0?0 < xprime2? x0 < 12?§kf(xprime2) lessorequalslant G0?0 < xprime3? x0 < 13?§
kf(xprime3) lessorequalslantG0?···
l?>?±w?xprimen > x0,xprimen → x0§
lm>w? lim
x→x0+0
f(x) negationslash= +∞§g?§Kb?¤á§
lim
x→x0+0
f(x) = +∞
18,Tüe?|f(x)
(1) f(+0) = 0,f(?0) = 1
32
(2) f(+0)?3§∞,f(?0) = 0
(3) f(+∞) = 0,f(?∞)?3
(4) f(+∞) = f(?∞) = A￡~ê¤
(5) f(x0 + 0)úf(x0?0)3
(6) f(x0 + 0) = +∞,f(x0?0) =?∞
(7) f(x0 + 0) = 1,f(x0?0) = +∞
(8) f(+∞)?3§∞,f(?∞) =?∞
)μ
(1) f(x) =
braceleftbigg 0 x> 0
1 xlessorequalslant 0
(2) f(x) =
braceleftBigg
sin 1x x> 0
0 xlessorequalslant 0
(3) f(x) = e?x
(4) f(x) = Ax+ 1x
(5) f(x) = sin 1x?x
0
(6) f(x) = 1x?x
0
(7) f(x) = 1 +e?
1
x?x0
(8) f(x) =
braceleftbigg sinx xgreaterorequalslant 0
x x< 0
33
§3,?Y?ê
1,Uy2e?ê3S?Yμ
(1) y = √x
(2) y = 1x
(3) y = |x|
(4) y = sin 1x
y2μ
(1) x0?(0,+∞)S:§|√x?√x0|< |x?x0|√x+√x
0
lessorequalslant |x?x0|√x
0
é?ε> 0§δ = √x0ε§|x?x0|<δ?§k|√x?√x0|< |x?x0|√x
0
<ε§y = √x3x0:?Y.
qdx03(0,+∞)￥5§Ky = √x3(0,+∞)S?Y.
x0 = 0?§ét?ε> 0§δ = ε2§0 <x?x0 <δ?§k|√x?√x0|<√x<ε§f(+0) = 0 =
f(0)§
l
y = √x3[0,+∞)S?Y.
(2) x0?(0,+∞)S:§|x?x0| < x02 §Kx> x02,xx0 > x
2
0
2 §u′|
1
x?
1
x0 | =
|x?x0|
xx0 <
|x?x0|
x20
2
ex0?(?∞,0)S:§|x?x0|<?x02 §Kx< x02,xx0 > x
2
0
2 §u′|
1
x?
1
x0 | =
|x?x0|
xx0 <
|x?x0|
x20
2
x0?(?∞,0)uniontext(0,+∞)S:§
é?ε > 0§δ = min
braceleftbigg|x
0|
2,
x20
2 ε
bracerightbigg
> 0§|x?x0| < δ?§k
vextendsinglevextendsingle
vextendsinglevextendsingle1
x?
1
x0
vextendsinglevextendsingle
vextendsinglevextendsingle = |x?x0|
xx0 > ε§y =
1
x3x0:?Y
qdx03(?∞,0)uniontext(0,+∞)S5§y = 1x3(?∞,0)uniontext(0,+∞)S?Y.
(3) x0?(?∞,+∞)S:§||x|?|x0||lessorequalslant|x?x0|.
é?ε> 0§δ = ε> 0§|x?x0|<δ?§k||x|?|x0||lessorequalslant|x?x0|<ε§y = |x|3x0:?Y
qdx03(?∞,+∞)S5§y = |x|3(?∞,+∞)S?Y.
(4) x0?(0,+∞)S:§|x? x0| < x02 §Kx > x02,xx0 > x
2
0
2 §u′
vextendsinglevextendsingle
vextendsinglevextendsinglesin 1
x?sin
1
x0
vextendsinglevextendsingle
vextendsinglevextendsingle =
2
vextendsinglevextendsingle
vextendsinglevextendsinglesin x+x0
2xx0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglecos x?x0
2xx0
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant |x?x0|
xx0 <
|x?x0|
x20
2
ex0?(?∞,0)S:§|x?x0| <?x02 §Kx < x02,xx0 > x
2
0
2 §u′
vextendsinglevextendsingle
vextendsinglevextendsinglesin 1
x? sin
1
x0
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
|x?x0|
xx0 <
|x?x0|
x20
2
x0?(?∞,0)uniontext(0,+∞)S:§
é?ε > 0§δ = min
braceleftbigg|x
0|
2,
x20
2 ε
bracerightbigg
> 0§|x?x0| < δ?§k
vextendsinglevextendsingle
vextendsinglevextendsinglesin 1
x?sin
1
x0
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant |x?x0|
xx0 < ε§
y = sin 1x3x0:?Y
qdx03(?∞,0)uniontext(0,+∞)S5§y = sin 1x3(?∞,0)uniontext(0,+∞)S?Y.
2,|^?Y?ê$?§|e?ê?Yμ
(1) y = tanx
34
(2) y = 1xn
(3) y = secx+ cscx
(4) y = 1√cosx
(5) y = ln(1 +x)x2?2x
(6) y = [x]tanx1 + sinx
)μ
(1)?y = tanx = sinxcosx§Kcosxnegationslash= 0?§y = tanx?Y§y = tanx?Y
parenleftBig
pi2 +kpi,pi2 +kpi
parenrightBig
(k∈
Z).
(2) en > 0§Ky = 1xn?Y(?∞,0)uniontext(0,+∞)?en lessorequalslant 0§Ky = 1xn?Y§=§?Y
(?∞,+∞).
(3)?secx?Y
parenleftbigg
k? 12
parenrightbigg
pi < x <
parenleftbigg
k+ 12
parenrightbigg
pi(k = 0,±1,±2,···)§cscx?Ykpi < x <
(k+ 1)pi(k = 0,±1,±2,···)§
y = secx+ cscx?Y
parenleftBig
kpi? pi2
parenrightBiguniontextparenleftBig
kpi,kpi+ pi2
parenrightBig
((k = 0,±1,±2,···).
(4) cosx> 0?§y = 1√cosx?Y§y = 1√cosx?Y
parenleftBig
pi2 + 2kpi,pi2 + 2kpi
parenrightBig
.
(5)?ln(1+x)x>?1Y§ 1x2?2xxnegationslash= 0,xnegationslash= 2Y§y = ln(1 +x)x2?2x ?Y(?1,0)uniontext(0,2)uniontext(2,+∞).
(6)?y = [x]tanx1 + sinx = [x]sinx(1 + sinx)cosx§Ksinx negationslash= 1,cosx negationslash= 0,x /∈ Z/{0}?§y = [x]tanx1 + sinx?Y§
y = [x]tanx1 + sinx?Yx∈
parenleftBig
kpi? pi2,kpi+ pi2
parenrightBig
x /∈Z/{0}(k∈Z).
3,e?ê?Y5§?x?ù?/.
(1) y =
x2?4
x?2,exnegationslash= 2
4,x = 2
(2) y =
vextendsinglevextendsingle
vextendsinglevextendsinglesinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle,xnegationslash= 0
1,x = 0
(3) y ==
sinx
|x|,xnegationslash= 0
1,x = 0
(4) y=[x]
)μ
(1)?lim
x→2
y = lim
x→2
x2?4
x?2 = limx→2(x+ 2) = 4§?x = 2?§y = 4§?ê3x = 2?Y
xnegationslash= 2?§y = x
2?4
x?2 = x+ 2w,?Y§
y =
x2?4
x?2,exnegationslash= 2
4,x = 2
3(?∞,+∞)S?Y.
(2) x negationslash= 0?§y =
vextendsinglevextendsingle
vextendsinglevextendsinglesinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle = sinx
x?y =?
sinx
x w,?Y"qlimx→0
vextendsinglevextendsingle
vextendsinglevextendsinglesinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle = 1 = f(0)§?ê3x = 0?
Y§
u′y =
vextendsinglevextendsingle
vextendsinglevextendsinglesinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle,xnegationslash= 0
1,x = 0
3(?∞,+∞)S?Y.
35
(3)? lim
x→+0
y = lim
x→+0
sinx
|x| = 1,limx→?0y = limx→?0
sinx
|x| =?1 §limx→0y?3"qx> 0?§y =
sinx
|x| =
sinx
x §x< 0?§y =
sinx
|x| =?
sinx
x §w,?Y§d?ê3?0 ?Y§=3(?∞,0)
uniontext(0,+∞)S
Y.
(4)? lim
x→k+0
y = lim
x→k+0
[x] = k,lim
x→k?0
y = lim
x→k?0
[x] = k? 1(k ∈ Z)§Klim
x→k
y?3§x = k(k ∈
Z)?y = [x]m?:§3m?:?m?Y
k<x<k+ 1(k∈Z)?§y = [x]w,?Y§d?ê3?k(k∈Z) ?Y.
4,ef(x)?Y§|f(x)|úf2(x)′Yoqe|f(x)|?f2(x)?Y§f(x)′Yo
)μ
(1) f(x)3ùIt?Y§x0?It:
f(x)3x0:?Y§é?ε> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|<ε
||f(x)|?|f(x0)||lessorequalslant|f(x)?f(x0)|<ε§=é?ε> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|<ε§
|f(x)|3x0:?Y
qdx03It5§|f(x)|3ItY
ó|f2(x)? f2(x0)| = |f(x)? f(x0)||f(x) + f(x0)| = |f(x)? f(x0)||f(x)? f(x0) + 2f(x0)| lessorequalslant
|f(x)?f(x0)|(|f(x)?f(x0)|+ 2f(x0)) <ε(ε+ 2f(x0))§f2(x)3x0:?Y
qdx03It5§f2(x)3ItY
(2)?L5§e|f(x)|?f2(x)?Y§f(x)Y.
(i)Y"~μf(x) =
braceleftbigg 1,xgreaterorequalslant 0
1,x< 0 §|f(x)| = 1úf
2(x) = 1t3(?∞,+∞)S?Y§f(x)3x =
0:Y?
(ii)?Y"~μf(x) = x,Kf(x)!|f(x)|!f2(x)3(?∞,+∞)St?Y"
5,(1)?êf(x)x = x0Y§
êg(x)x = x0Y§ˉd?êú3x0:′Yo
(2) x = x0êf(x)úg(x)Y§ˉd?êúf(x) +g(x)3?:x0′?7Yo
):
(1) ^?y{"bf(x) +g(x)3x0:?Y"
f(x)x = x0Y§Kd?Y?ê5?§g(x) = [f(x) +g(x)]?f(x)x0Yg?"
b?¤á§=f(x) +g(x)3x0:?Y.
(2)"
(i)?Yμ~μf(x) =
braceleftbigg 1,xgreaterorequalslant 0
1,x< 0,g(x) =
braceleftbigg?1,xgreaterorequalslant 0
1,x< 0 3x = 0Y§f(x) +
g(x) = 03x = 0?Y.
(ii)Yμ~μf(x) = g(x) = 1x3x = 0Y§f(x) +g(x) = 2x3x = 0Y.
6,(1)?êf(x)3x0?Y§
êg(x)3x0Y?
(2) x = x0êf(x)úg(x)Y§ˉd?ê|èf(x)g(x)3?:x0′?7Yo
):
(1)"
(i)?Yμ~μf(x) = 03x = 0?Y§g(x) =
braceleftbigg 1,xgreaterorequalslant 0
0,x< 0 3x = 0Y§f(x)g(x) = 03x =
0?Y.
(ii)Yμ~μf(x) = x3x = 0?Y§g(x) = 1x23x = 0Y§f(x)g(x) = 1x3x = 0Y.
(2)"
(i)?Yμ~μf(x) =
braceleftbigg 1,xgreaterorequalslant 0
1,x< 0,g(x) =
braceleftbigg?1,xgreaterorequalslant 0
1,x< 0 3x = 0Y§f(x)g(x) =
13x = 0?Y.
(ii)Yμ~μf(x) = g(x) = 1x3x = 0Y§f(x)g(x) = 1x23x = 0Y.
7,ef(x)3[a,∞)?Y§ lim
x→∞
f(x)3§y2f(x)3[a,∞)k..
y2μdu lim
x→∞
f(x)3§ lim
x→∞
f(x) = A
Kéε = 1,?X > 0§x>X?§k|f(x)?A|<ε = 1¤á§l
|f(x)| = |f(x)?A+A|lessorequalslant|f(x)?A|+
36
|A|< 1 +|A|
X1 = max{X,a+ 1}§Kf(x)3(X1,∞)Sk.§?|f(x)|<|A|+ 1,x∈ (X1,∞)
qduf(x)3[a,X1]t?Y§f(x)3[a,X1]tk.§ù.?M > 0§=?x∈ [a,X1]§k|f(x)|lessorequalslantM
G = max{|A|+ 1,M}§K?x∈ [a,∞),f(x) lessorequalslantG§
=f(x)3[a,∞)k..
8,eéε> 0§f(x)3[a+ε,b?ε]?Y§ˉμ
(1) f(x)′?(a,b)3?Yo
(2) f(x)′?3[a,b]?Yo
)μ
(1)?x0 ∈ (a,b)§ε = min
braceleftbiggx
0?a
2,
b?x0
2
bracerightbigg
§Kx0 ∈ [a+ε,b?ε]
éε> 0§f(x)3[a+ε,b?ε]?Y§f(x)3x0:?Y
dx0 ∈ (a,b)5§f(x)3(a,b)S?Y.
(2)Y"
(i)Y"~μf(x)3[0 +ε,1?ε](ε> 0)S?Y§f(x)3[0,1]tY§3x = 0:?m.
(ii)?Y"~μf(x)3[1 +ε,2?ε](ε> 0)S?Y§?f(x)3[1,2]t?Y.
9,ef(x)3x0:?Y§f(x0) > 0§y23x0δ?O(x0,δ)§x ∈ O(x0,δ)?§f(x) greaterorequalslant c> 0§c?,
~ê.
y2μduf(x)3x0:?Y§?f(x0) > 0§Kf(x0) >c> 0
éε = f(x0)?c> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|<ε = f(x0)?c§Kf(x0)?[f(x0)?
c] lessorequalslantf(x)§=f(x) greaterorequalslantc> 0.
10,y2e?Y?ê3kn:?ê0§Kd?êe?0.
y2μf(x)?￠?t?Y?ê§x0n:.
dkn:3ê?tè?5§?±?nê{xn}§|xn →x0(n→∞).
f(x)3x0?Y§Kf(x0) = lim
n→∞
f(xn) = 0§
dx0:5§f(x)3¤k?n:?ê0.
qf(x)3kn:?ê0§Kd?êe?0.
11,ef(x)3[a,b]?Y§e§Uy2 1f(x)3[a,b]?Y.
y2μduf(x)3[a,b]?Y§e§Kf(x)3(a,b)?Y§f(x) > 0§ 1f(x)3§x∈ [a,b]
x0?(a,b)S:§Ké?ε> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|<ε.
qf(x)3[a,b]?Y§Kd4?m?Y?ê5?2§?f(x)3[a,b]t?m > 0§=f(x) greaterorequalslant m,x ∈
[a,b]§u′
vextendsinglevextendsingle
vextendsinglevextendsingle 1
f(x)?
1
f(x0)
vextendsinglevextendsingle
vextendsinglevextendsingle= |f(x)?f(x0)|
f(x)f(x0) <
ε
m2§ limx→x0
1
f(x) =
1
f(x0)§l
1
f(x)3x0?Y.
dx03(a,b)S5§f(x)3(a,b)S?Y.
qf(a+ 0) = f(a) > 0§K 1f(a+ 0) = 1f(a)§f(x)3[a,b)?Y?
qf(b?0) = f(b) > 0§K 1f(b?0) = 1f(b)§f(x)3[a,b]?Y.
12,ef(x)úg(x)?3[a,b]?Y§áy2max(f(x),g(x))±9min(f(x),g(x))?3[a,b]?Y.
y2μduf(x)úg(x)?3[a,b]?Y§f(x)?g(x)úf(x) +g(x)?3[a,b]?Y.
d14K(?§k|f(x)?g(x)|3[a,b]?Y.
-?(x) = max(f(x),g(x)) = 12(f(x) +g(x) +|f(x)?g(x)|),
ψ(x) = min(f(x),g(x)) = 12(f(x) +g(x)?|f(x)?g(x)|)§
?(x),ψ(x)?3[a,b]?Y.
13,ef(x)′?Y§y2éc> 0§?êg(x) =
c,ef(x) <?c
f(x),e|f(x)|lessorequalslantc
c,ef(x) >c
′?Y.
y2μdug(x) = max(?c,min(f(x),c))
qduf(x)?Y§?éc> 0§?(x) = c?Y§ψ(x) =?c?Y§
KdtK(?§min(f(x),c)?Y§l
2dtK(?§g(x)?Y.
14,e?êY:5?￡=Y:¤μ
(1) y = x(1 +x)2
37
(2) y = 1 +x1 +x3
(3) y = x
2?1
x3?3x+ 2
(4) y = xsinx
(5) y = cos2 1x
(6) y = [x] + [?x]
(7) y = 1lnx
(8) y = x
2?x
|x|(x2?1)
(9) y =
1
q,x =
p
q(q> 0,q,p?p?ê)
0,xnê
(10) y =
braceleftbigg x,|x|lessorequalslant 1
1,|x|> 1
(11) y =
braceleftBigg
cos pix2,|x|lessorequalslant 1
|x?1|,|x|> 1
(12) y =
braceleftbigg sinpix,x?knê
0,xnê
)μ
(1)? lim
x→?1?0
x
(1 +x)2 =?∞§x =?1?1aY:￡m?:¤.
(2)? lim
x→?1
1 +x
1 +x3 =
1
3§y3x =?1:vk§x =?1￡Y:.
(3)?y = x
2?1
x3?3x+ 2 =
(x?1)(x+ 1)
(x?1)(x2 +x+ 1)?3(x?1) =
(x?1)(x+ 1)
(x?1)(x2 +x?2) =
(x?1)(x+ 1)
(x?1)2(x+ 2)§
q lim
x→1?0
y =?∞,lim
x→?2?0
y =?∞§x =?2,x = 1?1aY:.
(4)?lim
x→0
x
sinx = 1y3x = 0§x = 0￡Y:?
q lim
x→kpi
k∈Z,knegationslash=0
x
sinx = ∞§x = kpi(k∈Z,knegationslash= 0)?1aY:.
(5)?lim
x→0
cos2 1x3[0,1]m
§?
.4?§d43§u′x = 0?1aY:.
(6)?x→k+ 0?§?x→?k?0§ lim
x→k+0
y = lim
x→k+0
([x] + [?x]) = k+ (?k?1) =?1?
q?x→k?0?§?x→?k+ 0§ lim
x→k?0
y = lim
x→k?0
([x] + [?x]) = k?1 + (?k) =?1(k∈Z)
qx = k?§y = [x] + [?x] = [k] + [?k] = 0(k∈Z)§ê:t￡Y:.
(7)? lim
x→1+0
1
lnx = +∞§x =?1?1aY:?
lim
x→?0
1
lnx?3§x = 0?1aY:.
(8) y = x(x?1)|x|(x?1)(x+ 1)
lim
x→1
y = 12y3x = 1§x = 1￡Y:?
lim
x→+0
y = 1,lim
x→?0
y =?1§x = 0?1?aY:￡am?:¤?
lim
x→?1+0
y =?∞§x =?1?1am?:.
(9)?d?ê′±1?±??ê§?3?m[0,1]§ù§?m?/?daq.
3[0,1]t§?1?1knêkü?μ01,111?2knêkμ12?
1?3knêkü?μ13,231?4knêkü?μ14,34?
1?5knêko?μ15,25,35,451?6knêkü?μ16,56?···
38
o?§?1Lkknê?êllessorequalslant 2 + 1 + 2 + 3 +···+ (k?1) = k(k?1)2 + 2§=?1Lkk
nê?kk"
e?5y§3:x0 ∈ [0,1]§x→x0?§y → 0.
é?ε> 0§k =
bracketleftbigg1
ε
bracketrightbigg
§3[0,1]t§?1Lkknê?r1,r2,···,rl.
δ = min
limits1lessorequalslantilessorequalslantl|ri?x0|§K0 <|x?x0|<δ§=x /∈{r1,r2,···,rn}§?ò′xnê§k
nêpq§?q greaterorequalslantk+ 1 >k?§òk|y?0| =
1
q lessorequalslant
1
k+ 1,x?knêx =
p
q,q>k
0 <ε,xnê
.
lim
x→x0
y = 0 §u′μn:?′d?ê?Y:§kn:?′d?ê?￡Y:.
(10)? lim
x→?1+0
y =?1,lim
x→?1?0
y = 1§x =?1?1?aY:.
(11)? lim
x→?1+0
y = 0,lim
x→?1?0
y = 2§x =?1?1?aY:.
(12) (i) x0 negationslash= n,n∈Z§
kn:rn →x0?rn >x0§K lim
rn→x0+0
f(rn) = sinpix0 negationslash= 0?
?n:xn →x0?xn >x0§K lim
xn→x0+0
f(xn) = 0"
f(x0 + 0)?3§l
xnegationslash= n(n∈Z)ê1aY:.
(ii) x0 = n,n∈Z§
xnê?§|f(x)?f(n)| = 0?
x?knê?§|f(x)?f(n)| lessorequalslant pi|x?n|§é?ε > 0,?δ = εpi > 0§||x?n| < δ?§k|f(x)?
f(n)|<ε§f(x)3x = n(n∈Z)?Y.
15,x = 0?e?êf(x)§áf(0)ê?§|f(x)3x = 0?Yμ
(1) f(x) =
√1 +x?1
3√1 +x?1
(2) f(x) = tan2xx
(3) f(x) = sinx·sin 1x
(4) f(x) = (1 +x)1x
)μ
(1)?lim
x→0
f(x) = lim
x→0
√1 +x?1
3√1 +x?1 = limx→0
3radicalbig(1 +x)2 + 3√1 +x+ 1
√1 +x+ 1 = 32§
f(0) = 32.
(2)?lim
x→0
f(x) = lim
x→0
tan2x
x = 2§
f(0) = 2.
(3)?lim
x→0
f(x) = lim
x→0
sinx·sin 1x = 0§
f(0) = 0.
(4)?lim
x→0
f(x) = lim
x→0
(1 +x)1x = e§
f(0) = e.
16,ef(x)3[a,b]?Y§a<x1 <x2 <···<xn <b§K3[x1,xn]￥7kξ§|f(ξ) = f(x1) +f(x2) +···+f(xn)n,
y2μM = max
1lessorequalslantilessorequalslantn
f(xi),m = min
1lessorequalslantilessorequalslantn
f(xi)
Kf(x1) +f(x2) +···+f(xn)n lessorequalslantM?
ónf(x1) +f(x2) +···+f(xn)n greaterorequalslantm.
duf(x)3[x1,xn]? [a,b]t?Y§d0n§7?ξ ∈ [x1,xn]? [a,b]§|f(ξ) = f(x1) +f(x2) +···+f(xn)n,
17,^Yyμ
39
(1) f(x) = 3√x3[0,1]t′Y?
(2) f(x) = sinx3(?∞,+∞)t′Y?
(3) f(x) = sinx23(?∞,+∞)tY.
y2μ
(1) éx1,x2 ∈ [0,1]§| 3√x1? 3√x2| = |x1?x2|3radicalbigx2
1 + 3
√x
1x2 + 3
radicalbigx2
2
= |x1?x2|3
4(
3√x1 + 3√x2)2 + 1
4(
3√x1? 3√x2)2 lessorequalslant
|x1?x2|
1
4(
3√x1? 3√x2)2§
=14| 3√x1? 3√x2|3 lessorequalslant|x1?x2|§?=| 3√x1? 3√x2|lessorequalslant 3radicalbig4|x1?x2|
é?ε> 0,?δ = ε
3
4 > 0§|é?x1,x2 ∈ [0,1]§|x1?x2|<δ?§ok|
3√x1? 3√x2|lessorequalslant 3
radicalbig4|x
1?x2|<
ε
l
f(x) = 3√x3[0,1]t′Y.
(2) éx1,x2 ∈ (?∞,+∞)§|sinx1?sinx2| = 2
vextendsinglevextendsingle
vextendsinglecos x1 +x22 sin x1?x22
vextendsinglevextendsingle
vextendsinglelessorequalslant 2
vextendsinglevextendsingle
vextendsinglex1?x22
vextendsinglevextendsingle
vextendsingle= |x1?x2|§
é?ε > 0,?δ = ε > 0§|é?x1,x2 ∈ (?∞,+∞)§|x1?x2| < δ?§ok|sinx1? sinx2| lessorequalslant
|x1?x2|<ε
l
f(x) = sinx3(?∞,+∞)t′Y.
(3) ε0 = 1§éδ> 0§xprimen =radicalbig2npi+ pi2,xprimeprimen =radicalbig2npi? pi2§|xprimen?xprimeprimen| = |radicalbig2npi+ pi2?radicalbig2npi? pi2| =vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle piradicalbig2npi+ pi
2 +
radicalbig2npi? pi
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle→ 0(n→∞)
n§k|xprimen?xprimeprimen|<δ§
|sin(xprimen)2?sin2(xprimeprimen)2| = |1?(?1)| = 2 > 1 = ε0
l
f(x) = sinx23(?∞,+∞)tY.
40
§4,tút
1,|etx→ 0? úì?ü?μ
(1) x3 +x6
(2) 4x2 + 6x3?x5
(3) √x·sinx
(4) radicalbigx2 + 3√x
(5) √1 +x?√1?x
(6) tanx?sinx
(7) ln(1 +x)
)μ
(1) dulim
x→0
x3 +x6
x3 = limx→0(1 +x
3) = 1§§′3 t§§ì?üx3.
(2) dulim
x→0
4x2 + 6x3?x5
4x2 = limx→0(1 +
3
2x?
x3
4 ) = 1§§′2 t§§ì?ü4x
2.
(3) dulim
x→0
√x·sinx
|x| = limx→0
radicalbigg
sinx
x = 1§§′1 t§§ì?ü|x|.
(4) dulim
x→0
radicalbigx
2 + 3√x
6√x = limx→0
radicalBig
x53 + 1 = 1§§′16 t§§ì?ü 6√x.
(5) dulim
x→0
√1 +x?√1?x
x = limx→0
2x
x(√1 +x+√1?x) = 1§§′1 t§§ì?ü?
x.
(6) dulim
x→0
tanx?sinx
x3
2
= lim
x→0
2tanx?sinxx3 = lim
x→0
2(1?cosx)
cosx·x2 = limx→0
x2
x2 = 1§§′3
t§§ì?üx
3
2,
(7) dulim
x→0
ln(1 +x)
x = 1§§′1 t§§ì?üx.
2,x→∞?§|eCt úì?ü?μ
(1) x2 +x6
(2) 4x2 + 6x4?x5
(3) 3
radicalbigg
x2 sin 1x
(4)
radicalBig
1 +radicalbig1 +√x
(5) 2x
5
x3?3x+ 1
)μ
(1) du lim
x→∞
x2 +x6
x6 = 1§§′6 t§§ì?üx
6.
(2) du lim
x→∞
4x2 + 6x4?x5
x5 = 1§§′5 t§§ì?üx
5.
(3) du lim
x→∞
3
radicalbigg
x2 sin 1x
3√x = limx→∞
3√x
3√x = 1§§′
1
3 t§§ì?ü
3√x.
(4) du lim
x→∞
radicalBig
1 +radicalbig1 +√x
8√x = limx→∞
radicaltpradicalvertex
radicalvertexradicalbtparenleftbigg1
x
parenrightbigg1
4 +
radicalBiggparenleftbigg
1
x
parenrightbigg1
2 + 1 = 1§§′1
8 t§§ì?
ü 8√x.
41
(5) du lim
x→∞
2x5
x3?3x+ 1
2x2 = limx→∞
x3
x3?3x+ 1 = 1§§′2 t§§ì?ü2x
2.
3,áyμ?x→ 0?
(1) o(?xm) +o(?xn) = o(?xn)(m>n> 0)
(2) o(?xm)o(?xn) = o(?xm+n)(m,n> 0)
(3) |f(x)|lessorequalslantM§Kf(x)o(?x) = o(?x)
(4)?xm ·o(1) = o(?xm)
y2μ
(1) du?x→ 0§?xm → 0,?xn → 0§u′o(?x
m)
xm → 0,
o(?xn)
xn → 0
qm>n> 0§?x
m
xn =?x
m?n → 0 §u′o(?xm) +o(?xn)
xn =
o(?xm)
xm ·
xm
xn +
o(?xn)
xn → 0§
l
o(?xm) +o(?xn) = o(?xn)
(2) du?x→ 0§?xm → 0,?xn → 0§u′o(?x
m)
xm → 0,
o(?xn)
xn → 0
u′o(?x
m)o(?xn)
xm+n =
o(?xm)
xm ·
o(?xn)
xn → 0§
l
o(?xm)o(?xn) = o(?xm+n)
(3)?x → 0§o(?x)?x → 0§q|f(x)| lessorequalslant M§f(x)k.§u′f(x)o(?x)?x = f(x)o(?x)?x → 0§l
f(x)o(?x) = o(?x).
(4) do(1)u′t§Ko(1) → 0§u′?x
m ·o(1)
xm =
xm
xmo(1) = o(1) → 0§l
x
m ·o(1) =
o(?xm).
42
1ü? 4?Y?
1nù 'u￠ê? ?n9
4?mt?Y?ê5?y2
§1,'u￠ê? ?n
1,luy2e(.5.
y2μα,αprime?′ê8Ee(.§u′?x∈E§?kxgreaterorequalslantα§=α′Ee.?xgreaterorequalslantαprime§=αprime′Ee..
duα′Ee(.§′e.￥§l
kα greaterorequalslant αprime?ódαprime′Ee(.§kαprime greaterorequalslant α.dd
α = αprime.
2,β = supE,β /∈E§áygE￥?àê{xn}§ù4β?qeβ ∈E§K?/X?o
y2μ
(1) duβ = supE,β /∈E§Kdt(.§
(i) é?x∈E§?kx<β?
(ii) é?ε> 0§?3êx0 ∈E§|x0 >β?ε.
εn = 1n§éz?εn?kxn ∈ E§|β > xn > β?εn§=0 < β?xn < εn§u′?àê
{xn}?E.
q lim
n→∞
(β?εn) = β? lim
n→∞
εn = β?β greaterorequalslant lim
n→∞
xn greaterorequalslant lim
n→∞
(β?εn) = β§ lim
n→∞
xn = β.
(2) β ∈ E?§·K¤á"~μ?¤á"E = (1,12,13,···,1n,···),β = supE = 1,1 ∈ E.
q1n → 0(n→∞)§KE￥f4?t?0§β ∈E?§·K?¤á"
¤á"E =
braceleftbigg
sin pi8,sin 2pi8,···,sin npi8,···
bracerightbigg
,β = supE = 1,1 ∈E§xn = sin 16n+ 48 pi§K lim
n→∞
xn =
1§β ∈E?§·K¤á"
3,T~μ
(1) kt(.?e(.ê?
(2) 1kt(.?1ke(.ê?
(3) Q1kt(.q1ke(.ê?
(4) Q?1kt(.§q?1ke(.ê§ù￥t!e(.?k?.
)μ
(1) {xn} = {?n},sup{xn} =?1
(2) {xn} = {1n},sup{xn} = 1 ∈{xn},inf{xn} = 0 /∈{xn}
(3) {xn} = {1 + (?1)n},sup{xn} = 2 ∈{xn},inf{xn} = 0 ∈{xn}
(4) E =
parenleftbigg
1,12,1 + 12,13,1 + 23,···,1n,1 + n?1n
parenrightbigg
,supE = 2 /∈E,infE = 0 /∈E
4,áyê7kt(.úe(.§au+∞ê7ke(.§au?∞ê7kt(..
y2μ
(1) éue?~êê§w,t!e(.t.
éu?e?~êê§?{xn}§={xn}k4?§Kd1ù§1?n4§ê{xn}′k.ê.
l
d ù?nn§ê{xn}kt!e(.§=ê7kt!e(..
5μy2μt!e(.β,α￥?káu{xn}.
ˉ￠t§eα = β§Kα = β = xn,n = 1,2,···
eαnegationslash= β§?α /∈{xn}§KdSK2§3f
braceleftBig
x(1)nk
bracerightBig
u�3f
braceleftBig
x(2)nk
bracerightBig
uβ§
{xn}§ù{xn}g?§α,β￥?káu{xn}.
(2)?{xn}′au+∞ê§K?N ∈Z+§n>N?§ekxn >x1§u′x1,x2,···,xN￥??§=
{xn}e(."
43
(3)?{xn}′au?∞ê§K?N ∈Z+§n>N?§ekxn <x1§u′x1,x2,···,xN￥§=
{xn}t(."
5,|ê{xn}t!e(.:
(1) xn = 1? 1n
(2) xn =?n[2 + (?2)n]
(3) x2k = k,x2k+1 = 1 + 1k(k = 1,2,3,···)
)μ
(1) α = 0￡¤§β = 1￡¤
(2)? lim
k→∞
bracketleftbig?2kparenleftbig2 + (?2)2kparenrightbigbracketrightbig=?∞,lim
k→∞
bracketleftbig?(2k+ 1)parenleftbig2 + (?2)2k+1parenrightbigbracketrightbig= +∞§{x
n}?t!e(..
(3)? lim
k→∞
x2k = lim
x→∞
k = +∞§{xn}?t(.?
q?x2k greaterorequalslant 1,k = 1,2,3,··· ;x2k+1 > 1?min{x2k} = 1§inf{xn} = 1￡¤.
6,y2μüN~ke.ê7k4?.
y2μdu{yn}ke.§{yn}7ke(..
de(.kμ(i)yn greaterorequalslantα(n = 1,2,3,···)?(ii)é?ε> 0§?kyN ∈{yn}§|yN <α+ε.
du{yn}′üN~ê§n > N?§kyn < α+ε§=n > N?§k0 lessorequalslant yn?α < ε§u′yn →
α(n→∞).
l
üN~ke.ê7k4?.
7,ám@?n^?μeò4?mU?m?m§(JX?oeò^?[a1,b1]? [a2,b2]? ···K?ò^
bn?an → 0K§(JNoáT~`2.
)μ
(1) 3?m@?n￥§eò4?mU?m?m§=
(i) (an+1,bn+1)? (an,bn)?
(ii) lim
n→∞
(bn?an) = 0
K?±y2{an},{bn}Euó?4?ξ§= lim
n→∞
an = lim
n→∞
bn = ξ§d?ξ?U? ?áuù
m
m§=ξ /∈ (an,bn)(n∈Z+)§?=ξ?U(an,bn)ú:.
~μm?m
braceleftbigg
(0,1n)
bracerightbigg
§
(i)
parenleftbigg
0,1n+ 1
parenrightbigg
parenleftbigg
0,1n
parenrightbigg
(ii) lim
n→∞
parenleftbigg1
n?0
parenrightbigg
= lim
n→∞
1
n = 0?
an = 0 → 0(n→∞);bn = 1n → 0(n→∞)§Kξ = 0 /∈
parenleftbigg
0,1n
parenrightbigg
§=(¤á.
(2) eò^?[an+1,bn+1]? [an,bn]K§=?k^?bn?an → 0¤á§K?Uy{an}?{bn}.
~μ4?m
bracketleftbigg
n? 1n,n+ 1n
bracketrightbigg
′@,lim
n→∞
bracketleftbigg
n+ 1n?
parenleftbigg
n? 1n
parenrightbiggbracketrightbigg
= lim
n→∞
2
n = 0§
limn→∞
parenleftbigg
n+ 1n
parenrightbigg
lim
n→∞
parenleftbigg
n? 1n
parenrightbigg
.
?3ξ?{an},{bn}ú4?§=(¤á.
(3) eò^?bn?an → 0K§=?k^?[an+1,bn+1]? [an,bn]¤á.K?±y2{an},{bn}￡m
@?ny2?¤§?Uy lim
n→∞
bn = lim
n→∞
an¤á§l
[an,bn]ú:§$yú
?m.
~μ4?m
bracketleftbigg
1? 1n+ 1,2 + 1n+ 1
bracketrightbigg
bracketleftbigg
1? 1n,2 + 1n
bracketrightbigg
,n∈Z+§? lim
n→∞
bracketleftbigg
2 + 1n?
parenleftbigg
1? 1n
parenrightbiggbracketrightbigg
= 1.
d lim
n→∞
an = 1,lim
n→∞
bn = 2§[1,2]?
bracketleftbigg
1? 1n,2 + 1n
bracketrightbigg
,n∈Z+§=(¤á.
8,e{xn}?.§t§K73ü?fx(1)nk →∞,x(2)nk →a(a?,k?ê).
y2μky
braceleftBig
x(1)nk
bracerightBig
′t.
du{xn}?.§é￠êM > 0§?knprime ∈Z+§||xnprime|>M.
M = 1§K73n1§|
vextendsinglevextendsingle
vextendsinglex(1)n1
vextendsinglevextendsingle
vextendsingle > 1?M = 2§K73n2§|
vextendsinglevextendsingle
vextendsinglex(1)n2
vextendsinglevextendsingle
vextendsingle > 2?···?M = K§K7
44
3nK >nK?1§|
vextendsinglevextendsingle
vextendsinglex(1)nK
vextendsinglevextendsingle
vextendsingle>K§···.
K??f
braceleftBig
x(1)nk
bracerightBig
§é?M ∈Z+§K = M§Kk>K?§òk
vextendsinglevextendsingle
vextendsinglex(1)nk
vextendsinglevextendsingle
vextendsingle>M§k limk→∞x(1)nk = ∞.
d?{xn}?′t§Kd§?M0 > 0§é?N ∈ Z+§?km ∈ Z+§m > N?§
k|xm|<M0.
yN = m0parenleftbigm0 ∈Z+parenrightbig§K?km1 >m0§||xm1|lessorequalslantM0
2N = m1§K?km2 >m1§||xm2|lessorequalslantM0§···
Xd?1e§K??mtμm1 < m2 < ··· < mt < ···§||xmt| lessorequalslant M0§=f{xmt}?|xmt| lessorequalslant
M0(mt ∈Z+)§ù`2f{xmt}k.§d5?n§k.f{xmt}7kf.
Pùf?{x(2)nk}§§?′{xn}f?§ua.= lim
k→∞
x(2)nk = a￡a?,k?ê¤.
9,k.ê{xn}e§K73ü?fx(1)nk →a,x(2)nk →b(anegationslash= b).
y2μdu{xn}k.§Kd5?n§7kfx(1)nk →a.
du{xn}§3ε0 > 0§3(a?ε0,a+ε0) k{xn}§¤{xn}f§P?
braceleftBig
x(2)n
bracerightBig
.
du
braceleftBig
x(2)n
bracerightBig
k.§3fx(2)nk →b§w,anegationslash= b.
10,e3?m[a,b]￥ü?ê
braceleftBig
x(1)n
bracerightBig
9
braceleftBig
x(2)n
bracerightBig
÷vx(1)n?x(2)n → 0(n→∞)§K3düê￥Ué?k?óv
Inkf§|x(1)nk →x0,x(1)nk →x0(k→∞).
y2μ?
braceleftBig
x(1)n
bracerightBig
[a,b]§K
braceleftBig
x(1)n
bracerightBig
k.ê§Kd5?n§
braceleftBig
x(1)n
bracerightBig
7kf§P?
braceleftBig
x(1)nk
bracerightBig
§
lim
k→∞
x(1)n = x0.
3
braceleftBig
x(2)n
bracerightBig
￥
braceleftBig
x(1)nk
bracerightBig
k?óvIf
braceleftBig
x(2)nk
bracerightBig
.
x(1)n?x(2)n → 0(n→∞)§K lim
k→∞
parenleftBig
x(1)nk?x(2)nk
parenrightBig
= 0§
u′ lim
k→∞
x(2)nk = lim
k→∞
bracketleftBig
x(1)nk?
parenleftBig
x(1)nk?x(2)nk
parenrightBigbracketrightBig
= lim
k→∞
x(1)nk? lim
k→∞
parenleftBig
x(1)nk?x(2)nk
parenrightBig
= x0?0 = x0.
11,|^?üneê5μ
(1) xn = a0 +a1q+a2q2 +···+anqn(|q|< 1,|ak|lessorequalslantM)
(2) xn = 1 + sin12 + sin222 +···+ sinn2n
(3) xn = 1? 12 + 13?···+ (?1)n+1 1n
y2μ
(1) n>m§K|xn?xm| =vextendsinglevextendsingleam+1qm+1 +am+1qm+1 +···+anqnvextendsinglevextendsinglelessorequalslantMparenleftbig|q|m+1 +|q|m+2 +···+|q|nparenrightbig=
M|q|m+1 1?|q|
n?m
1?|q| <M|q|
m+1 1
1?|q| → 0(m→∞)
é?ε> 0,?N ∈Z+§n>m>N?§kM|q|m+1 11?|q| <ε§l
k|xn?xm|<ε.
d?ün§{xn}7.
(2) m > n§é?ε > 0￡ε < 12¤§du|xm?xn| =
vextendsinglevextendsingle
vextendsinglevextendsinglesin(n+ 1)
2n+1 +
sin(n+ 2)
2n+2 +···+
sinm
2m
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
1
2n+1 +
1
2n+2 +···+
1
2m =
1
2n+1
parenleftbigg
1 + 12 +···+ 12m?n?1
parenrightbigg
= 12n+1
1?
parenleftbigg1
2
parenrightbiggm?n
1? 12
< 12n§e?|xm?
xn|<ε§ 12n <ε=?.
N =
lnε
ln 12
∈Z+§m>n>N?§k|x
m?xn|<ε.
d?ün§{xn}7.
￡?μ3￡1¤￥-a0 = 1,ak = sink,q = 12§Kd￡1¤=￡2¤¤.
(3) é?ε> 0§é?k∈Z+§du|xn+k?xn| =
vextendsinglevextendsingle
vextendsinglevextendsingle(?1)n+2
n+ 1 +
(?1)n+3
n+ 2 +···+
(?1)n+k?1
n+k
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle 1
n+ 1?
1
n+ 2 +···+
(?1)k?1
n+k
vextendsinglevextendsingle
vextendsinglevextendsingle=
1
n+ 1?
parenleftbigg 1
n+ 2?
1
n+ 3 +···+
(?1)k
n+k
parenrightbigg
< 1n+ 1 < 1n§e?|xn+k?xn|<ε§1n <ε=?.
45
N =
bracketleftbigg1
ε
bracketrightbigg
§Kn+k>n>N?§k|xn+k?xn|<ε.
d?ün§{xn}7.
12,|^k?CX?ny2?dA.d?n.
y2μ{xn}?k.ê§K73a,b§|alessorequalslantxn lessorequalslantb.
^?y{"b{xn}f§Kéx0 ∈ [a,b]§?kε0 > 0§|3O(x0,ε0)￥?1
k{xn}k.
Ké?ε> 0§3O(x0,ε)￥1k{xn}.
εn = 1n§w,3O(x0,εn)￥?1k{xn}§K3{xn}￥??μxn1 ∈ O(x0,1)§q?
xn2 ∈O
parenleftbigg
x0,12
parenrightbigg
(n2 >n1)§Xd?1e§?{xn}f{xnk},|xnk?x0|< 1k§é?M ∈Z+§
K = M§Kk>K?§òk|xnk?x0|< 1k < 1K < 1M§Kxnk →x0(k→∞)ù?bg?.
dx0 ∈ [a,b]5§é[a,b]￥z?:?kù?§|d1{xn}k§¤kù
?
¤[a,b]mCX.
dk?CX?n§K3kCX[a,b]§?
[a,b]1k{xn}k§ùxn ∈ [a,b]g
§b?¤á§K{xn}7kf.
13,|^?dA.d?ny2üNk.ê7k4?.
y2μ{xn}?üNO\k.ê§x1 lessorequalslantx2 lessorequalslant···lessorequalslantxn lessorequalslant···lessorequalslantM
a?dA.d?n§3f{xnk},lim
k→∞
xnk = a.
eyμ lim
n→∞
xn = a.
kyxn lessorequalslanta,n = 1,2,···.e?,§?N ∈Z+§|xN >a.
dunk →∞(k→∞)§k§7knk >N§l
xnk greaterorequalslantxN >a§u′a = lim
k→∞
xnk greaterorequalslantxN >ag?.
2y lim
n→∞
xn = a.
é?ε> 0,?k0§|
vextendsinglevextendsingle
vextendsinglexnk0?a
vextendsinglevextendsingle
vextendsingle= a?xnk0 <ε.
N = nk0§Kn>N?§kxn greaterorequalslantxnk0 = xN§l
k|a?xn| = a?xn lessorequalslanta?xnk0 <ε§ lim
n→∞
xn = a.
=üNO\k.ê7k4?.
ón?§üN~k.ê7k4?§l
üNk.ê7k4?.
14,(1) y2üNk.?ê3?!m4
(2) y2üNk.?êY:1?aY:.
y2μ
(1) d??f(x)3(a,b)tüNO\k.§?x0 ∈ (a,b)§β(x0) = sup
a<x<x0
f(x)§
dt(.§é?ε> 0§?kxprime ∈ (a,x0)§|f(xprime) >β(x0)?ε=f(xprime) +ε>β(x0)
δ = x0?xprime > 0§?f(x)3(a,b)tüNO\§δ>x0?x> 0=xprime <x?§kf(xprime) <f(x)§u′
kf(x) +ε>β(x0)=0 lessorequalslantβ(x0)?f(x) <ε§l
|β(x0)?f(x)|<ε
`2 lim
x→x0?0
f(x) = β(x0).=f(x)3x03?4?.
ón?§f(x)3(a,b)tüN~k.?§f(x)3x03?4?§l
üNk.?ê3?4?.
ón?§üNk.?ê3m4?.
(2) x0?f(x)Y:§Kd￡1¤(?f (x0?0)úf (x0 + 0)3§d?f (x0?0) negationslash= f (x0 + 0)"
K§f (x0?0) = f (x0 + 0)§df(x)üN5§7kf (x0) = f (x0?0) = f (x0 + 0).
ù`2x0′?Y:§g?§f (x0?0) negationslash= f (x0 + 0)§l
x0′f(x)1?aY:.
15,y2 lim
x→+∞
f(x)37?^?′μéε > 0§3X > 0§xprime,xprimeprime > X?ek|f(xprime)?
f(xprimeprime)|<ε.
y2μ lim
x→+∞
f(x)3§ lim
x→+∞
f(x) = A.
é?ε> 0,?X > 0§x>X?§k|f(x)?A|< ε2
xprime,xprimeprime > X?§k|f(xprime)?A| < ε2,|f(xprimeprime)?A| < ε2§K|f(xprime)?f(xprimeprime)| = |f(xprime)?A? (f(xprimeprime)?A)| lessorequalslant
|f(xprime)?A|+|f(xprimeprime)?A|<ε§l
éε> 0§3X > 0§xprime,xprimeprime >X?ek|f(xprime)?f(xprimeprime)|<ε.
3f(x)S§àê{xn}§|xn → +∞(n→∞)
d?§é?ε> 0,?X > 0§xprime,xprimeprime >X?§ek|f(xprime)?f(xprimeprime)|<ε.
q?xn → +∞§u′ét?X > 0§N ∈ Z+§n > N?§kxn > X§l
n,m > N?§ò
kxn >X,xm >X§?
k|f(xn)?f(xm)|<ε.
46
d?ün§ lim
n→∞
f(xn)3§ lim
n→∞
f(xn) = A
dxn59?ê4ê4?'X§ lim
x→+∞
f(x) = A= lim
x→+∞
f(x)3.
16,y2 lim
x→x0
f(x)37?^?′μéε> 0§3δ > 0§0 < |xprime?x0| <δ,0 < |xprimeprime?x0| <
δ?§ek|f(xprime)?f(xprimeprime)|<ε.
y2μ lim
x→x0
f(x)3§ lim
x→x0
f(x) = A.
é?ε> 0,?δ> 0§0 <|x?x0|<δ?§k|f(x)?A|< ε2
0 < |xprime?x0| < δ,0 < |xprimeprime?x0| < δ?§k|f(xprime)?A| < ε2,|f(xprimeprime)?A| < ε2§K|f(xprime)?f(xprimeprime)| =
|f(xprime)?A? (f(xprimeprime)?A)| lessorequalslant |f(xprime)?A| + |f(xprimeprime)?A| < ε§l
éε > 0§3δ > 0§0 <
|xprime?x0|<δ,0 <|xprimeprime?x0|<δ?§ek|f(xprime)?f(xprimeprime)|<ε.
3f(x)S§àê{xn}§|xn →x0?xn negationslash= x0(n→∞)
d?§é?ε > 0,?δ > 0§xprime,xprimeprime ∈ D(f)§?0 < |xprime?x0| < δ,0 < |xprimeprime?x0| < δ?§òk|f(xprime)?
f(xprimeprime)|<ε.
q?xn →x0,xn negationslash= x0(n→ ∞)§u′ét?δ > 0§N ∈Z+§n>N?§k0 <|xn?x0|<δ§l
n,m>N?§òk0 <|xn?x0|<δ,0 <|xm?x0|<δ§?
k|f(xn)?f(xm)|<ε.
dê?ün§ lim
n→∞
f(xn)3§ lim
n→∞
f(xn) = A
d{xn}′±x0?4?ê?xn negationslash= x0 9?ê4ê4?'X§ lim
x→x0
f(x) = A= lim
x→x0
f(x)
3.
17,y2f(x)3x0:?Y7?^?′μéε> 0§3δ> 0§|xprime?x0|<δ,|xprimeprime?x0|<δ?§e
k|f(xprime)?f(xprimeprime)|<ε.
y2μf(x)3x0:?Y§Ké?ε> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|< ε2
|xprime? x0| < δ,|xprimeprime? x0| < δ?§k|f(xprime)? f(x0)| < ε2,|f(xprimeprime)? f(x0)| < ε2§K|f(xprime)? f(xprimeprime)| =
|f(xprime)?f(x0)? (f(xprimeprime)?f(x0))| lessorequalslant |f(xprime)?f(x0)| + |f(xprimeprime)?f(x0)| < ε§l
éε > 0§
3δ> 0§|xprime?x0|<δ,|xprimeprime?x0|<δ?§ek|f(xprime)?f(xprimeprime)|<ε.
xprime = x0,xprimeprime = x§Kd?§é?ε> 0,?δ> 0§|x?x0|<δ?§òk|f(x)?f(x0)|<ε.
l
f(x)3x0:?Y.
47
§2,4?mt?Y?ê5?y2
1,y2μeüNk.?êf(x)?f(a),f(b)?m§Kf(x)3[a,b]?Y.
y2μf(x)?üNO\k.?ê.
d ù§1,14K￡1¤§f(x)3[a,b]à:a(b)?m￡?¤4?3§d?f(a) = f(a+ 0)(f(b) = f(b?
0))§
e?,§7kf(a) <f(a+ 0) = inf
a<x<b
f(x)(f(b) >f(b?0) = sup
a<x<b
f(x))§u′df(x)?f(a)?f(b)?
m§éf(a) < y < f(a + 0)(f(b? 0) < y < f(b))§7kx ∈ (a,b)§|f(x) = y§d
f(a+ 0) = inf
a<x<b
f(x)(f(b?0) = sup
a<x<b
f(x))g?.
dd?f(x)3a(b)m￡?¤?Y.
ekx0 ∈ (a,b)§|f(x)3x0:Y"d§1,14(2)(?§x07?1?am?:§=f(x0 + 0)úf(x0?
0)3§f(x0 + 0) negationslash= f(x0?0).
q?f(x)?üNO?ê§f(x0?0) lessorequalslantf(x0) <f(x0 + 0)?f(x0?0) <f(x0) lessorequalslantf(x0 + 0)§ù?f(x)?
(f(x0?0),f(x0 + 0))?méuf(x0)?§ùg?§b?¤á.
u′f(x)3[a,b]?Y.
ón§f(x)?üN~k.?ê?§f(x)3[a,b]?Y.
l
f(x)3[a,b]?Y.
2,y2μ?êf(x)3(a,b)?Y§f(a+ 0),f(b?0)3§Kf(x)?f(a+ 0)úf(b?0)?m(?U?
uf(a+ 0),f(b?0)).
y2μduf(a+ 0),f(b?0)3§Kf(a) = f(a+ 0),f(b) = f(b?0).
qf(x)3(a,b)?Y§Kf(x)3[a,b]?Y§?
f(x)3[a,b]t7kMú??m.
2d0n§f(x)?±Múmm.
eM = f(a+ 0)(?f(b?0))§m = f(b?0)(?f(b?0))§ù?f(x)?(f(a+ 0),f(b?0))￥(
U? uf(a+ 0),f(b?0)).
eM >f(a+ 0)(?f(b?0))§m<f(b?0)(?f(b?0))§ù?f(x)?(f(a+ 0),f(b?0))￥(?
U uf(a+ 0),f(b?0)),f(x)?f(a+ 0)úf(b?0)?m(?U? uf(a+ 0),f(b?0)).
3,y2(a,b)t?Y?êY7?^?′μf(a+ 0),f(b?0)3.
y2μ?f(x)?(a,b)t?Y?ê
f(a+ 0),f(b?0)3§Kf(a) = f(a+ 0),f(b) = f(b?0)§u′f(x)3[a,b]?Y§Kdx÷?
n§f(x)3[a,b]tY§l
f(x)3(a,b)tY.
f(x)3(a,b)tY§Kd§é?ε > 0,?δ(ε) > 0§x1,x2 ∈ (a,b)?|x1?x2| < δ(ε)?§
k|f(x1)?f(x2)|<ε.
éa§0 <x1?a< δ(ε)2,0 <x2?a< δ(ε)2?§|x1?x2| = |(x1?a)?(x2?a)|lessorequalslant|x1?a|+|x2?a|<δ(ε)§
Kk|f(x1)?f(x2)|<ε.
d?ün§ lim
x→a+0
f(x)3§=f(a+ 0)3?k?.
ón§f(b?0)3?k?.
4,e?êf(x)3(?∞,+∞)tk?4?mt?Y§K§3(?∞,+∞)tk?m?mtY.
y2μ(a,b)?(?∞,+∞)tk?m?m§K[a,b]?(?∞,+∞)tk?4?m.
f(x)3[a,b]t?Y§Kdx÷?n§f(x)3[a,b]tY§?
f(x)3(a,b)tY.
d(a,b)5§f(x)3(?∞,+∞)tk?m?mtY.
5,?êf(x) = x23(?∞,+∞)9(?l,l)t(l> 0)′Yo
)μ
(1) f(x) = x23(?∞,+∞)tY.
x1 >x2 > 0§?x1,x2 ∈ (?∞,+∞),|f(x1)?f(x2)| = |x21?x22| = |x1+x2||x1?x2| = (x1+x2)(x1?
x2) > 2x2(x1?x2)§3ε0 > 0§é?η> 0§x2 = 2ε0η,x1 = x2 + η2§
w,kx1 >x2 > 0?|x1?x2| = η2 <η§|f(x1)?f(x2)|> 2x2(x1?x2) = 2· 2ε0η · η2 = 2ε0 >ε0§
l
f(x) = x23(?∞,+∞)tY.
(2) f(x) = x23(?l,l)(l> 0)tY.
f(x)3[?l,l](l> 0)t′?Y§Kdx÷?n§f(x)3[?l,l]tY§l
f(x) = x23(?l,l)t
Y.
6,ef(x)3(a,b)Sk§é(a,b)Sx§3x,??Ox§|f(x)3OxSk..ˉμf(x)3(a,b)S
′?k.oqeò(a,b)U?[a,b]§X?o
y2μ
48
(1) f(x)3(a,b)k..
~μ?.μf(x) = 1x3(0,1)Sk§?é?x∈ (a,b)?Y§7?ük.§=3x?Ox(O(x,δx))§
|§3Ox(O(x,δx))Sk.§§3(0,1)S?..
k.μf(x) = sinx3
parenleftBig
0,pi2
parenrightBig
k§é
parenleftBig
0,pi2
parenrightBig
Sx§3x,??Ox§|f(x)3OxSk
.?f(x)3
parenleftBig
0,pi2
parenrightBig
tk.§?0 <f(x) < 1.
(2) f(x)3[a,b]k..
f(x)3[a,b]Sk§Kμf(x)3(a?δ,a)f(a)§f(x)3(b,b+δ)f(b).
d?é[a,b]Sx§3x,??Ox§|f(x)3OxSk.§=?M > 0§||f(x)| lessorequalslant M§?
d3[a,b]tz?:?ù?￡?=m?m¤§ù
m?m N¤m?m8§§CX
[a,b].
dk?CX?n§3ù
m?m8￥7kkm?mCX
[a,b]§Pùkm?m?(x1?
δ1,x1 +δ1),(x2?δ2,x2 +δ2),···,(xk?δk,xk +δk)§?AM?OP?M1,M2,···,Mk§X8
M? = max{M1,M?2,···,Mk}.
é[a,b]t:x§d?mCXVg§3ùk?m?mO(xi,δi)(i = 1,2,···,k)￥?k1x§
P§?O(xi,δi)§?3ù?m?mt§k|f(x)|lessorequalslantMi§|f(x)|lessorequalslantMi lessorequalslantM?.
dux?[a,b]t:§K3[a,b]to¤á|f(x)|lessorequalslantM?§l
y2
f(x)3[a,b]tk..
7,y2(a,b)tY?ê7k..
y2:?f(x)?(a,b)tY?ê§KdSK3§f(x)3(a,b)t?Y?f(a+0),f(b?0)3§u′
μf(a) = f(a+ 0),f(b) = f(b?0)§Kf(x)3[a,b]t?Y§u′f(x)3[a,b]tk.§l
f(x)3(a,b)t
k..
8,Uy2§üY?êúEY.kˉμüY?êèX?o
y2μ
(1) f(x)?g(x)3mXtY.
f(x)3?mXtY§Kdé?ε> 0,?δ1 > 0§é?mXSü:xprime,xprimeprime§|xprime?xprimeprime|<
δ1§òk|f(xprime)?f(xprimeprime)|< ε2.
q?g(x)3?mXtY§Kdét?ε > 0,?δ2 > 0§é?mXSü:xprime,xprimeprime§|xprime?
xprimeprime|<δ2§òk|g(xprime)?g(xprimeprime)|< ε2.
δ = min{δ1,δ2}§K|xprime?xprimeprime|<δ?§k|f(xprime)+g(xprime)?(f(xprimeprime)+g(xprimeprime))| = |f(xprime)?f(xprimeprime)+(g(xprime)?
g(xprimeprime))|lessorequalslant|f(xprime)?f(xprimeprime)|+|g(xprime)?g(xprimeprime)|< ε2 + ε2 = ε.
l
f(x)3?mXtY.
(2) (i) e?mX?km§K(?¤á.
f(x),g(x)3?mXtY§KdtK(?§3~êL> 0,M > 0§||f(x)|<L,g(x) <
M(x∈X).
qd§?ε > 0,?δ1 > 0§é?mXSü:xprime,xprimeprime§|xprime?xprimeprime| < δ1§ò
k|f(xprime)?f(xprimeprime)|< ε2M.
ó§ét?ε > 0,?δ2 > 0§é?mXSü:xprime,xprimeprime§|xprime? xprimeprime| < δ2§òk|g(xprime)?
g(xprimeprime)|< ε2L.
δ = min{δ1,δ2}§K|xprime?xprimeprime|<δ?§òk|f(xprime)?f(xprimeprime)|< ε2M,|g(xprime)?g(xprimeprime)|< ε2Ló?¤
á.
dd?§|f(xprime)g(xprime)?f(xprimeprime)g(xprimeprime)| =
|[f(xprime)?f(xprimeprime)]g(xprime) +f(xprimeprime)[g(xprime)?g(xprimeprime)]| lessorequalslant |f(xprime)?f(xprimeprime)||g(xprime)| + |f(xprimeprime)||g(xprime)?g(xprimeprime)| <ε
2M ·M +L·
ε
2L =
ε
2 +
ε
2 = ε.
l
f(x)g(x)3?mXtY.
(ii) f(x),g(x)3(?∞,+∞)Y?§f(x)g(x)3(?∞,+∞)tY.
~μ
(a)Y.
f(x) = g(x) = x§?é?ε > 0§9x1,x2 ∈ (?∞,+∞)§δ = ε§|x1? x2| < δ?§
k|x1?x2|<ε§f(x) = g(x) = x3(?∞,+∞)tY.
f(x)g(x) = x2§d15K?f(x)g(x)3(?∞,+∞)tY.
(b)Y.
f(x) = 1§?é?ε> 0§éx1,x2 ∈ (?∞,+∞)§δ = ε§|x1?x2|<δ?§k|f(x1)?
f(x2)|<ε§f(x) = 13(?∞,+∞)tY.
49
g(x) = x§Kd?g(x) = x3(?∞,+∞)tY§?f(x)g(x) = x3(?∞,+∞)t
Y.
50
1? üCt?è
1?ü? üCt
1où ê
§1,êú
1,L-?y = x2tü:A(2,4)úB(2+?x,2+?y)??§?O|??x = 19?x = 0.1???§?|
-?3A:?.
)μkAB = (2 +?x)
2?22
x = 4 +?x
?x = 1?§kAB = 5??x = 0.1?§kAB = 4.1
-?3A:k = lim
x→0
kAB = lim
x→0
(4 +?x) = 4.
2,|y = x23A(1,1):ú3B(?2,4):§ú{§.
)μ?yprime = 2x§3:A(1,1)μk1 = 2§§?μy? 1 = 2(x? 1)=2x?y? 1 = 0?{§
y?1 =?12(x?1)=x+ 2y?3 = 0
3:B(?2,4)μk2 =?4§§?μy? 4 =?4(x + 2)=4x + y + 4 = 0?{§?y? 4 = 14(x +
2)=x?4y+ 18 = 0
3,ey = f(x) = x3§|
(1) L-?t:x0,x0 +?x???￡x0 = 2,?x?O?0.1§0.01§0.001¤?
(2) 3x = x0?-?.
)μ
(1)?k = f(x0 +?x)?f(x0)?x = (x0 +?x)
3?x3
x = 3x
2
0 + 3x0?x+ (?x)
2§
μ?x = 0.1?§k = 12.61??x = 0.01?§k = 12.0601??x = 0.001?§k = 12.006001.
(2) fprime(x) = lim
x→0
f(x0 +?x)?f(x0)
x = 3x
2§
u′fprime(x0) = 3x20
4,es = vt? 12gt2§|
(1) 3t = 1,t = 1 +?t?m2t?Y￡?t = 1,0.1,0.01¤?
(2) 3t = 1]Y.
)μ
(1)?ˉv =
v(1 +?t)? 12g(1 +?t)2?
parenleftbigg
vt? 12gt2
parenrightbigg
t = v?g?
1
2g?t
2§
μ?t = 1?§ˉv = v? 32g??t = 0.1?§ˉv = v? 2120g??t = 0.01?§ˉv = v? 201200g.
(2) 3t = 1]Yv = lim
t→0
ˉv = v?g.
5,y = x23=?:21uy = 4x?5o3=?:R?u2x?6y+ 5 = 0o
)μy = 4x? 5k = 4§Kdfprime(x) = 2x = k§x = 2§=(2,4):21uy =
4x?5?
2x?6y + 5 = 0k = 13§Kdfprime(x) = 2x =?1k =?3§x =?32§=(?32,94):R?
u2x?6y+ 5 = 0.
6,|e?ê3¤?:?y?xμ
(1) y = √x(x = 2,?x = 0.01)
(2) y = 1x(x = 4,?x = 0.04)
51
)μ
(1)?y?x =
√x+?x?√x
x =
√2.01?√2
0.01 = 100
parenleftBig√
2.01?√2
parenrightBig
= 1√2.01 +√2
(2)?y?x =
1
x+?x?
1
x
x =?
1
x(x+?x) =?
1
4(4 + 0.04) =?
25
404
7,y2μ
(1)?(f(x)±g(x)) =?f(x)±?g(x)
(2)?[f(x)·g(x)] = g(x+?x)·?f(x) +f(x)·?g(x)
y2μ
(1)?(f(x)±g(x)) = [f(x+?x)±g(x+?)]?[f(x)±g(x)] = [f(x+?x)?f(x)]±[g(x+?x)?g(x)] =
f(x)±?g(x)
(2)?[f(x)·g(x)] = f(x+?x)·g(x+?x)?f(x)·g(x) = f(x+?x)·g(x+?x)?f(x)·g(x+?x)+f(x)·g(x+
x)?f(x)·g(x) = [f(x+?x)?f(x)]·g(x+?x)+f(x)[g(x+?x)?g(x)] = g(x+?x)·?f(x)+f(x)·?g(x)
52
§2,{ü?êê
1,dê|y = cosxê.
)μyprime = lim
x→0
cos(x+?x)?cosx
x = lim?x→0
2sin 2x+?x2 sin?x2
x =? lim?x→0sin
parenleftbigg
x+?x2
parenrightbiggsin?x
2
x
2
=
sinx§=(cosx)prime =?sinx.
2,dê|y = 3√xê.
)μyprime = lim
x→0
3√x+?x? 3√x
x = lim?x→0
x13
bracketleftBiggparenleftbigg
1 +?xx
parenrightbigg1
3?1
bracketrightBigg
x = lim?x→0
x?23
bracketleftBiggparenleftbigg
1 +?xx
parenrightbigg1
3?1
bracketrightBigg
x
x
= x
23
3 =
1
3 3√x2§=(
3√x)prime = 1
3 3√x2
3,Uy2μ?ó?êù?ê′ê§?êù?ê′ó?ê.
y2μf(x)ó?ê§Kf(?x) = f(x)?g(x)ê§Kg(?x) =?g(x)
u′fprime(?x) = lim
x→0
f(?x+?x)?f(?x)
x = lim?x→0
f(xx)?f(x)
x = limx→0
[f(xx)?f(x)]
x =
fprime(x)=?ó?êù?ê′ê?
gprime(?x) = lim
x→0
g(?x+?x)?g(?x)
x = lim?x→0
g(xx) +g(x)
x = limx→0
g(xx)?g(x)]
x = g
prime(x)=?
êù?ê′ó?ê.
4,Uy2μ?±ê§ù?êE?±ù?ê.
y2μf(x)±T?ê§Kf(x+T) = f(x)§
u′fprime(x+T) = lim
x→0
f(x+T +?x)?f(x+T)
x = lim?x→0
f(x+?x)?f(x)
x = f
prime(x)=?±ê§
ù?êE?±ù?ê.
53
§3,|{K
1,|^?2êúa§|e?êêμ
(1) y = x5
(2) y = x11
(3) y = x6
(4) y = 2x
(5) y = log10x
(6) y = 10x
)μ
(1) yprime = (x5)prime = 5x4
(2) yprime = (x11)prime = 11x10
(3) yprime = (x6)prime = 6x5
(4) yprime = (2x)prime = 2x ln2
(5) yprime = (log10x)prime = 1xln10
(6) yprime = (10x)prime = 10x ln10
2,|e?êêμ
(1) f(x) = 2x2?3x+ 1§?|fprime(0),fprime(1)
(2) f(x) = x5 + 3sinx§?|fprime(0),fprime
parenleftBigpi
2
parenrightBig
(3) f(x) = ex + 2cosx?7x§?|fprime(0),fprime(pi)
(4) f(x) = 4sinx?lnx+x2
(5) f(x) = anxn +an?1xn?1 +···+a1x+a0§?|fprime(0),fprime(1)
)μ
(1) fprime(x) = 4x?3§fprime(0) =?3,fprime(1) = 1
(2) fprime(x) = 5x4 + 3cosx§?|fprime(0) = 3,fprime
parenleftBigpi
2
parenrightBig
= 5pi
4
16
(3) fprime(x) = ex?2sinx?7§?|fprime(0) =?6,fprime(pi) = epi?7
(4) fprime(x) = 4cosx? 1x + 2x
(5) f(x) = nanxn?1 + (n1)an?1xn?2 +···+a1§?|fprime(0) = a1,fprime(1) =
nsummationtext
i=1
iai
3,|e?êêμ
(1) y = x2 sinx§?|fprime(0),fprime
parenleftBigpi
2
parenrightBig
(2) y = xcosx+ 3x2§?|fprime(?pi)úfprime(pi)
(3) y = xtanx+ 7x?6
(4) y = ex sinx?7cosx+ 5x2
(5) y = 4√x+ 1x?2x3
(6) y = (3x2 + 2x?1)sinx
)μ
(1) yprime = 2xsinx+x2 cosx§fprime(0) = 0,fprime
parenleftBigpi
2
parenrightBig
= pi
(2) yprime = cosx?xsinx+ 6x§fprime(?pi) =?1?6pi,fprime(pi) =?1 + 6pi
(3) yprime = tanx+xsec2x+ 7
(4) yprime = ex sinx+ex cosx+ 7sinx+ 10x = ex(sinx+ cosx) + 7sinx+ 10x
54
(5) yprime = 2√x? 1x2?6x2
(6) yprime = (3x2 + 2x?1)cosx+ (6x+ 2)sinx
4,|e?êêμ
(1) y = 2 + sinxx
(2) y = cotx
(3) y = 3x
2 + 7x?1
√x
(4) y = (1 +x
2)sinx
2x
(5) y = xlnx1 +x
(6) y = xe
x?1
sinx
)μ
(1) yprime = x(2 + sinx)
prime?(x+ sinx)
x2 =
xcosx?sinx?2
x2
(2) yprime =
parenleftBigcosx
sinx
parenrightBigprime
= sinx(cosx)
prime?cosx(sinx)prime
sin2x =?
1
sin2x =?csc
2x
(3) yprime =
√x(3x2 + 7x?1)prime?(√x)prime(3x2 + 7x?1)
x =
√x(6x+ 7)? 3x2 + 7x?1
2√x
x =
9x2 + 7x+ 1
2x√x =
9
2x
1
2 + 72x?
1
2 + 12x?
3
2
(4) yprime = 2x[(1 +x
2)sinx]prime?2(1 +x2)sinx
4x2 =
2x[2xsinx+ (1 +x2)cosx]?2(1 +x2)sinx
4x2 =
(x2?1)sinx+x(1 +x2)cosx
2x2
(5) yprime = (1 +x)(xlnx)
prime?xlnx
(1 +x)2 =
(1 +x)(lnx+ 1)?xlnx
(1 +x)2 =
x+ lnx+ 1
(1 +x)2
(6) yprime = sinx(xe
x?1)prime?(sinx)prime(xex?1)
sin2x =
ex sinx(x+ 1)?cosx(xex?1)
sin2x
5,|e?êêμ
(1) y =
√x+ cosx
x?1?7x
2
(2) y = xsinx+ cosxxsinx?cosx
(3) y = x2ex sinx+ 3 +x
2
√x?xlnx+ 8x2
(4) y = sinx1 + tanx
(5) y = xcosx?lnxx+ 1
(6) y = 1x+ cosx
)μ
(1) yprime =
(x?1)( 12√x?sinx)?(√x+ cosx)
(x?1)2? 14x =
(x?1)(1?2√xsinx)?(2x+ 2√xcosx)
2√x(x?1)2? 14x
(2) yprime = (xsinx?cosx)(sinx+xcosx?sinx)?(xsinx+ cosx)(sinx+xcosx+ sinx(xsinx?cosx)2 =?2(sinxcosx+x)(xsinx?cosx)2 =
2x+ sin2x(xsinx?cosx)2
55
(3) yprime = 2xex sinx+x2ex sinx+x2ex cosx+
2x√x? 3 +x
2
2√x
x? lnx? 1 + 16x = xe
x(2sinx+xsinx+
xcosx) + 3x
2?1
2x√x?lnx?1 + 16x
(4) yprime = cosx(1 + tanx)?sinx·sec
2x
(1 + tanx)2
(5) yprime =
(x+ 1)(cosx?xsinx? 1x)?(xcosx?lnx)
(x+ 1)2 =
xcosx?(x2 sinx+ 1)(x+ 1) +xlnx
x(x+ 1)2
(6) yprime =? 1?sinx(x+ cosx)2 = sinx?1(x+ cosx)2
6,|-?y+ 1 = (x?2)33:A(3,0)?§9{§.
)μ?y+ 1 = (x?2)3§Ky = (x?2)3?1§u′yprime = 3(x?2)2§K¤|k = y prime|x=3 = 3§
l
¤|§?μy = 3(x?3)=3x?y?9 = 0?¤|{§?μy =?13(x?3)=x+ 3y?3 = 0.
7,|-?y = lnx3:(1,0)?§ú{§.
)μ?y = lnx§Kyprime = 1x§u′¤|k = y prime|x=1 = 1§
l
¤|§?μy = x?1=x?y?1 = 0?¤|{§?μy =?(x?1)=x+y?1 = 0.
8,y = x2?2x+ 43=?:21ux?o3=?:x??45oo
)μ?y = x2?2x+ 4§yprime = 2x?2.
q21ux?k = 0§K2x?2 = 0§u′x = 1§=¤|:?(1,3)?
q?x??45ok = 1§K2x?2 = 1§u′x = 32§=¤|:?
parenleftbigg3
2,
13
4
parenrightbigg
.
9,÷$??N§ù$§?s = 3t4? 20t3 + 36t2§|ù?Y§?ˉ?Nc$?o $
o
)μ?s = 3t4?20t3 + 36t2§v = sprime = 12t3?60t2 + 72t.
v> 0=0 <t< 2?t> 3?§?N?c$v< 0=2 <t< 3?§?N? $?.
10,du ^§?￥÷XtE§D?Y?5§$§?s = 5t?t2§áˉd￥meEo
)μ?s = 5t?t2§v = sprime = 5?2t§v = 0=t = 52?§￥meE.
11,3x = 2?§?-?y = 0.1x3§áˉ: §d-3o
)μ?y = 0.1x3§yprime = 0.3x2§u′3x = 2?§k = y |x = 2 = 1.2§l
d-?3?
:(2,0.8)?§?y?0.8 = 1.2(x?2)§=6x?5y?8 = 0?d
braceleftbigg y = 0.1x3
6x?5y?8 = 0 §x
3?12x+16 =
0§K(x?2)2(x+ 4) = 0§)x1 = x2 = 2,x3 =?4§Kd-3:(?4,?6.4).
12,-?y = xn￡n?ê¤t:(1,1)?x?u:(ξn,0)§| lim
n→∞
y(ξn).
)μ?y = xn§Kyprime = nxn?1§Kd-?3x = 1?k = y prime|x=1 = n§u′d-?3:(1,1)?
§?y?1 = n(x?1)=y = nx?n+ 1.
y = 0?§x = n?1n =ξn = n?1n §K lim
n→∞
y(ξn) = lim
n→∞
parenleftbiggn?1
n
parenrightbiggn
= lim
n→∞
parenleftbigg
1? 1n
parenrightbiggn
= 1e.
13,§?y = x2 +ax+b§áˉ:(x0,y0)?u§?±l:(x0,y0)édü^
^§o
)μ(x0,y0)?2?t:§(x,y)?L(x0,y0):.
d?§?k = yprime = 2x+a§K¤|y?y0 = (2x+a)(x?x0)=y0?y =
(2x?a)(x0?x)§
qy = x2+ax+b§Ky0?(x2+ax+b) = (2x+a)(x0?x)§x2?2x0x+y0?ax0§K? = 4x20?4(y0?b?ax0)
? > 0=y0 < x20 + ax0 + b?§ü^? = 0=y0 = x20 + ax0 + b?§^
? < 0=y0 >x20 +ax0 +b?§.
14,ˉ.êao§y = xaU?éê-?y = logaxo3o
)μdK?§xprime = (logax)prime§=1 = 1xlna§Kx = 1lna§u′y = 1lna.
qdu3?:§ùp?I7L? §K logax = 1lna§u′x = e§K?lna = 1e§=a = e1e=.
êa = e1e?§y = xaU?éê-?y = logax§3:(e,e).
56
§4,Eü?ê|{
1,|e?êêμ
(1) y = 2sin3x
(2) y = 4cos(3t?1)
(3) y = 3e2x + 5cos2x
(4) y = (x+ 1)2
(5) y = (1?x+x2)3
(6) y = 3e?2t + 1
(7) y = ln(x+ 1)
(8) y = (3x+ 1)4
(9) y = √1 +x2
(10) y =
parenleftbigg
1? 1x
parenrightbigg2
(11) y = tan x2 + sin3x
(12) y = lnsinx
(13) y = x√1 +x2
(14) y = 1√2pie?3t2
)μ
(1) yprime = 6cos3x
(2) yprime =?12sin(3t?1)
(3) yprime = 6e2x?10sin2x
(4) yprime = 2(x+ 1)
(5) yprime = 3(1?x+x2)2(2x?1)
(6) yprime =?6e?2t
(7) yprime = 1x+ 1
(8) yprime = 12(3x+ 1)3
(9) yprime = x√1 +x2
(10) yprime = 2
parenleftbigg
1? 1x
parenrightbigg
·
parenleftbigg
1x2
parenrightbigg
= 2(x?1)x3
(11) yprime = 12 sec2 x2 + 3cos3x
(12) yprime = cosxsinx = cotx
(13) yprime =
√1 +x2? x2√
1 +x2
1 +x2 =
1
(1 +x2)32
(14) yprime =?3
√2t
√pi e?3t2
2,|e?êêμ
(1) y = sin3 2x
(2) y = (at+b)e?2t(a,b?~ê)
(3) y = e2t sin3t+ t
2
2
57
(4) y = ln 1?x
2
1 +x2
(5) y = e
kt sinωt
1 +t (k,ω?~ê)
(6) y = 4(x+ cos2x)2
(7) y = e?t(cost+ sint)
(8) y = x√1 + cos2x
(9) y = (x?1)√x2 + 1
(10) y = (2 + 3t)sin2t+ 7t2?7
)μ
(1) yprime = 6sin2 2xcosx = 3sin4xsin2x
(2) yprime = ae?2t?2(at+b)e?2t =?(2at+ 2b?a)e?2t
(3) yprime = 2e2t sin3t+ 3e2t cos3t+t = e2t(2sin3t+ 3cos3t) +t
(4) yprime = 1 +x
2
1?x2 ·
2x(1 +x2)?2x(1?x2)
(1 +x2)2 =
4x
x4?1
(5) yprime = (1 +t)e
kt(?ksinωt+ωcosωt)?(e?kt sinωt
(1 +t)2 =
(kt+k+ 1)e?kt sinωt+ω(1 +t)e?kt cosωt
(1 +t)2
(6) yprime =?4[(x+ cos2x)
2]prime
(x+ cos2x)4 =?
8(1?2sin2x)
(x+ cos2x)2
(7) yprime =?e?t(cost+ sint) +e?t(?sint+ cost) =?2e?t sint
(8) yprime =
√1 + cos2x?x?2sinxcosx
2√1 + cos2x
1 + cos2x =
1 + cos2x+xsinxcosx
(1 + cos2x)32
(9) yprime = √x2 + 1 + (x?1) 2x2√x2 + 1 = 2x
2?x+ 1
√x2 + 1
(10) yprime = 3sin2t+ 2(2 + 3t)cos2t+ 14t
3,|e?êêμ
(1) y = e?kt(3cosωt+ 4sinωt)(k,ω?~ê)
(2) y = xarctanx
(3) y = (2x2 + 1)2e?x sin3x
(4) y = e
t sin3t
√1 +t2
(5) y = (3t+ 1)et(cos3t?7sin3t)
(6) y = tarcsin3t+ 7e?2t lnt+ 8t
(7) y = x√a2?x2 + x√a2?x2 (a?~ê)
)μ
(1) yprime =?ke?kt(3cosωt+4sinωt)+e?kt(?3ωsinωt+4ωcosωt) = e?kt[(4ω?3k)cosωt?(3ω+4k)sinωt]
(2) yprime = arctanx+ x1 +x2
(3) yprime = 4x(2x2 + 1)e?x sin3x? (2x2 + 1)2e?xsin3x+ 3(2x2 + 1)2e?xcos3x = e?x(2x2 + 1)[(?2x2 +
8x?1)sin3x+ 3(2x2 + 1)cos3x]
(4) yprime =
e?t(?sin3t+ 3cos3t)√1 +t2?e?t sin3t t√1 +t2
1 +t2 =
e?t[3(1 +t2)cos3t?(t2 +t+ 1)sin3t]
(1 +t2)32
58
(5) yprime = 3et(cos3t?7sin3t) + (3t+ 1)et(cos3t?7sin3t) + (3t+ 1)et(?3sin3t?21cos3t) =?et[(60t+
17)cos3t+ (30t+ 31)sin3t]
(6) yprime = arcsin3t+ 3t√1?9t2?14e?2t lnt+ 7e
2t
t + 8
(7) yprime = √a2?x2? x
2
√a2?x2 +
√a2?x2 + x2√
a2?x2
a2?x2 =
(a2?2x2)(a2?x2) +a2
(a2?x2)32
4,|e?êêμ
(1) y = sinnxcosnx
(2) y = sinhnxcoshnx
(3) y = e?x2+2x
(4) y = (sinx+ cosx)n
(5) y = arcsin(sinx·cosx)
(6) y = ln
radicalbigg(x+ 2)(x+ 3)
x+ 1
(7) y = arctan 2x1?x2
(8) y = xa2√a2 +x2
)μ
(1) yprime = nsinn?1xcosxcosnx?nsinnxsinnx =
nsinn?1xcos(n+ 1)x
(2) yprime = nsinhn?1xcoshxcoshnx+nsinhnxsinhnx =
nsinhnxcosh(n+ 1)x
(3) yprime =?2(x?1)e?x2+2x
(4) yprime = n(sinx+ cosx)n?1(cosx?sinx) = n(sinx+ cosx)n?2 cos2x
(5) yprime = cos2xradicalbig1?(sinx·cosx)2 = 2cos2x√4?sin22x
(6)?y = ln
radicalbigg(x+ 2)(x+ 3)
x+ 1 =
1
2[ln(x+2)+ln(x+3)?ln(x+1)]§y
prime = 1
2
bracketleftbigg 1
x+ 2 +
1
x+ 3?
1
x+ 1
bracketrightbigg
=
x2 + 2x?1
2(x+ 1)(x+ 2)(x+ 3).
(7) yprime = 1
1 +
parenleftbigg 2x
1?x2
parenrightbigg2 · 2(1?x
2) + 4x2
(1?x2)2 =
2
1 +x2
(8) yprime =
√a2 +x2? x2√
a2 +x2
a2(a2 +x2) =
1
(a2 +x2)32
5,|^éê2|?{§|e?êêμ
(1) y = x
radicalbigg1?x
1 +x
(2) y = x
2
1?x
radicalbigg x+ 1
1 +x+x2
(3) y = (x?α1)α1(x?α2)α2 ···(x?αn)αn
(4) y = (x+√1 +x2)n
(5) y = xmmx
)μ
59
(1)?y = x
radicalbigg1?x
1 +x§K lny = lnx+
1
2 ln(1?x)?
1
2 ln(1 +x)§ü>éx|§
1
yy
prime = 1
x +
1
2(1?x)?
1
2(1 +x)§
Kyprime = 1?x?x
2
(1 +x)√1?x2 (0 <|x|< 1)
(2)?y = x
2
1?x
radicalbigg x+ 1
1 +x+x2§K lny = 2lnx? ln(1?x) +
1
2 ln(x+ 1)?
1
2 ln(1 +x+x
2)§ü>éx|
§1yyprime = 2x + + 11?x + 12(1 +x)? 1 + 2x2(1 +x+x2)§
Kyprime = x
2
1?x
radicalbigg x+ 1
1 +x+x2
parenleftbigg2
x + 11?x+ 12(x+ 1)?
2x+ 1
2(1 +x+x2)
parenrightbigg
(3)?y = (x?α1)α1(x?α2)α2 ···(x?αn)αn =
nproducttext
i=1
(x?αi)αi9y3éê?òS§Aproducttext nlim
i=1
(x?αi)αi >
0§Klny =
nsummationtext
i=1
αi ln|x?αi|§ü>éx|ê§1yyprime =
nsummationdisplay
i=1
αi
x?αi§
Kyprime =
nsummationtext
i=1
αi
x?αi
nproductdisplay
i=1
(x?αi)αi(x∈D)ù￥D =
braceleftbigg nproducttext
i=1
(x?αi)αi > 0
bracerightbigg
(4)?y = (x + √1 +x2)n§K lny = nln(x + √1 +x2)§ü>éx|§1yyprime = n
1 + x√1 +x2
x+√1 +x2 =n
√1 +x2§Kyprime = n√1 +x2 (x+
radicalbig
1 +x2)n
(5)?y = xmmx§Klny = mln|x| +xlnm§ü>éx|§1yyprime = mx + lnm§Kyprime = xm?1mx+1 +
xmmx lnm
6,f(x)′éx?|?ê§|dydx.
(1) y = f(x2)
(2) y = f(ex)·ef(x)
(3) y = f(f(f(x)))
)μ
(1) dydx = 2xfprime(x2)
(2) dydx = exfprime(ex)·ef(x) +fprime(x)f(ex)ef(x) = ef(x)(exfprime(ex) +f(ex)fprime(x))
(3) dydx = fprime(f(f(x)))fprime(f(x))fprime(x)
7,?(x),ψ(x)?éx?|?ê§|dydx.
(1) y =radicalbig?2(x) +ψ2(x)
(2) y = arctan?(x)ψ(x)(ψ(x) negationslash= 0)
(3) y =?(x)radicalbigψ(x)(?(x) negationslash= 0,ψ(x) > 0)
(4) y = log?(x)ψ(x)(?(x) > 0,ψ(x) negationslash= 0)
)μ
(1) dydx =?(x)?
prime(x) +ψ(x)ψprime(x)
radicalbig?2(x) +ψ2(x)
(2) dydx =?
prime(x)ψ(x)?ψprime(x)?(x)
2(x) +ψ2(x)
(3) dydx =?(x)
radicalbig
ψ(x)
parenleftbigg ψprime(x)
(x)ψ(x)?
prime(x)lnψ(x)
2(x)
parenrightbigg
60
(4) dydx =
ψprime(x)
ψ(x) ln?(x)?
prime(x)
(x) lnψ(x)
(ln?(x))2 =
ψprime(x)
ψ(x)ln?(x)?
prime(x)lnψ(x)
(x)(ln?(x))2 =
log?(x)ψ(x)
bracketleftbigg ψprime(x)
ψ(x)lnψ(x)?
prime(x)
(x)ln?(x)
bracketrightbigg
8,|?4-7¤?-Y?\?w?$??Y.
)μ?s =
radicalbig
l2?r2 sin2ωt?rcosωt§v = sprime = rωsinωt? r
2ωsin2ωt
2
radicalbig
l2?r2 sin2ωt
.
9,|-?y = √1?x23x = 12?§ú{§.
)μ?yprime =? x√1?x2§K3x = 12?k =?
√3
3 §
u′¤|§?μy?
√3
2 =?
√3
3
parenleftbigg
x? 12
parenrightbigg
=x+√3y?2 = 0?
¤|{§?μy?
√3
2 =
√3parenleftbiggx? 1
2
parenrightbigg
=√3x?y = 0.
10,|-?y = e?xt?:§|LT:y =?ex21§??T:{§.
)μ?k = yprime =?e?x =?e§Kx =?1§KL(?1,e):y =?ex21§LT:{§
y?e = 1e(x+ 1)=x?ey+e2 + 1 = 0.
11,|-?y = √1?x2tY2.
)μ?k = yprime =? x√1?x2 = 0§Kx = 0§u′d-?3(0,1)?Y2§§?y = 1.
12,|-?y = 12(1 + 2x2 ±
radicalbig
1 + 4x2)tIx = U:?§.ù-?uo
)μ?yprime = 2x ± 2x√1 + 4x2§K-?3x = U?k = 2U ± 2U√1 + 4U2§u′d-?3?
:(U,12(1 + 2U2 ±
radicalbig
1 + 4U2))?§?y? 12(1 + 2U2 ±
radicalbig
1 + 4U2)) = (2U ± 2U√1 + 4U2 )(x?U)§
=2U(√1 + 4U2 ±1)x?√1 + 4U2y± 12 + 12(1?2U2)
radicalbig
1 + 4U2 = 0§d-?u?
U(
√1 + 4U2 ±1)
√1 + 4U2,12
1 + 2U2(
√1 + 4U2 ±1)2
1 + 4U2 ±
radicalBigg
1 + 4U
2(√1 + 4U2 ±1)2
1 + 4U2
.
13,y = f(x)3x0?§P?(t) = f(x0 +at)§a?~ê§|?prime(0).
)μea = 0§K?(t) = f(x0)§K?prime(0) = 0
eanegationslash= 0§K?prime(x) = lim
t→0
(x)(0)
t = limt→0
f(x0 +at)?f(x0)
t = alimt→0
f(x0 +at)?f(x0)
at = af
prime(x
0).
61
§5,9ù$?
1,|e?ê3:μ
(1) y = anxn +an?1xn?1 +···+a0§|dy(0),dy(1)
(2) y = secx+ tanx§|dy(0),dy
parenleftBigpi
4
parenrightBig
,dy(pi)
(3) y = 1a arctan xa§|dy(0),dy(a)
(4) y = 1x + 1x2§|dy(0.1),dy(0.01)
)μ
(1)?dy = [nanxn?1 + (n?1)an?1xn?2 +···+a1]dx§Kdy(0) = a1dx,dy(1) =
nsummationtext
i=1
iaidx
(2)?dy = (tanxsecx+ sec2x)dx§Kdy(0) = dx,dy
parenleftBigpi
4
parenrightBig
= (√2 + 2)dx,dy(pi) = dx
(3)?dy = dxa2 +x2dx§Kdy(0) = dxa2dx,dy(a) = dx2a2dx
(4)?y =?x+ 2x3 dx§Kdy(0.1) =?2100dx,dy(0.01) =?2010000dx
2,|e?êy = y(x)μ
(1) y = x? 12x2 + 13x3? 14x4
(2) y = x2 sinx
(3) y = x1?x2
(4) y = xlnx?x
(5) y = (1?x2)n
(6) y = √x+ lnx? 1√x
(7) y = lntanx
(8) y = sinaxcosbx
(9) y = eax cosbx
(10) y = arcsin√1?x2
)μ
(1) dy = (1?x+x2?x3)dx
(2) dy = (2xsinx+x2 cosx)dx
(3) dy = 1 +x
2
(1?x2)2dx
(4) dy = lnxdx
(5) dy =?2nx(1?x2)n?1dx
(6) dy = x+ 2
√x+ 1
x32
dx
(7) dy = 2sin2xdx
(8) dy = (acosaxcosbx?bsinaxsinbx)dx
(9) dy = eax(acosbx?bsinbx)dx
(10) dy =? x|x|√1?x2dx
3,|e?êyμ
(1) y = sin2t,t = ln(3x+ 1)
62
(2) y = ln(3t+ 1),t = sin2x
(3) y = e3u,u = 12 lnt,t = x2?2x+ 5
(4) y = arctanu,u = (lnt)2,t = 1 +x2?cotx
)μ
(1) dy = 3sin(2ln(3x+ 1))3x+ 1 dx
(2) y = 3sin2x3sin2x+ 1dx
(3) y = 3(3x
2?2)
2(x3?2x+ 5)e
3
2 ln(x
2?2x+5)dx
(4) y = 2ln(1 +x
2?cotx)(2x+ csc2x)
[1 + (ln(1 +x2?cotx))4](1 +x2?cotx)dx
4,eu,v,w?xê§|?êydyμ
(1) y = u·v·w
(2) y = u·wv2
(3) y = 1√u2 +v2
(4) y = ln√u2 +v2
(5) y = arctan uv
)μ
(1) dy = (uprime ·v·w+u·vprime ·w+u·v·wprime)dx
(2) dy = v
2(uprimew+uwprime)?2uvvprimew
v4 dx
(3) dy =? uu
prime +vvprime
(u2 +v2)32
dx(u2 +v2 > 0)
(4) dy = uu
prime +vvprime
u2 +v2 dx
(5) dy = u
primev?uvprime
u2 +v2 dx(v negationslash= 0)
63
§6,ê9?ê?§¤Lê|{
1,|eêêdydxμ
(1) x
2
a2 +
y2
b2 = 1§ù￥a,b?~ê
(2) y2 = 2px§ù￥p?~ê
(3) x2 +xy+y2 = a2§ù￥a?~ê
(4) x3 +y3?xy = 0
(5) y = x+ 12 siny
(6) x23 +y23 = a23§ù￥a?~ê
(7) y?cos(x+y) = 0
(8) y = x+ arctany
(9) y = 1?ln(x+y) +ey
(10) arctan yx = ln
radicalbig
x2 +y2
)μ
(1) 3?§üàéx|ê§?5?y′x?ê§òk2xa2 + 2yy
prime
b2 = 0§Ky
prime =?b
2x
a2y(y negationslash= 0).
(2) 3?§üàéx|ê§?5?y′x?ê§òk2yyprime = 2p§Kyprime = py(y negationslash= 0).
(3) 3?§üàéx|ê§?5?y′x?ê§òk2x+xyprime +y+ 2yyprime = 0§Kyprime =?2x+yx+ 2y.
(4) 3?§üàéx|ê§?5?y′x?ê§òk3x2 + 3y2yprime?xyprime?y = 0§Kyprime = 3x
2?y
x?3y2,
(5) 3?§üàéx|ê§?5?y′x?ê§òkyprime = 1 + y
prime
2 cosy§Ky
prime = 2
2?cosy.
(6) 3?§üàéx|ê§?5?y′x?ê§òk23x?13 + 23y?13yprime = 0§Kyprime =? 3
radicalbiggx
y.
(7) 3?§üàéx|ê§?5?y′x?ê§òkyprime+(1+yprime)sin(x+y) = 0§Kyprime =? sin(x+y)1 + sin(x+y).
(8) 3?§üàéx|ê§?5?y′x?ê§òkyprime = 1 + y
prime
1 +y2§Ky
prime = 1 +y
2
y2,
(9) 3?§üàéx|ê§?5?y′x?ê§òkyprime =?1 +y
prime
x+y +y
primeey§Kyprime = 1
(x+y)ey?x?y?1.
(10) 3?§üàéx|ê§?5?y′x?ê§òkxy
prime?y
x2 +y2 =
x+yyprime
x2 +y2§Ky
prime = x+y
x?y.
2,|eê3:êdydxμ
(1) y = cosx+ 12 siny§:
parenleftBigpi
2,0
parenrightBig
(2) yex + lny = 1§:(0,1)
)μ
(1) 3?§üàéx|ê§?5?y′x?ê§òkyprime =?sinx+ y
prime
2 cosy§Ky
prime = 2sinx
cosy?2§u′3
:
parenleftBigpi
2,0
parenrightBig
§yprime =?2.
(2) 3?§üàéx|ê§?5?y′x?ê§òkex(y +yprime) + y
prime
y = 0§Ky
prime =? y
2ex
yex + 1§u′3
:(0,1)?§yprime =?12.
64
3,|-?x32 +y32 = 163:(4,4)§ú{§.
)μ3?§üàéx|ê§?5?y′x?ê§òk32x12 + 32y12yprime = 0§Kyprime =?
radicalbiggx
y§u′y
prime|x=4
y=4
=
1§l
§?y?4 =?(x?4)§=x+y?8 = 0
{§?y?4 = x?4§=x = y.
4,|e?ê?§3¤?:êμ
(1)
braceleftbigg x = acost
y = bsint 3t =
pi
3ú
pi
4?
(2)
braceleftbigg x = t?sint
y = 1?cost §3t =
pi
2,pi?
(3)
braceleftbigg x = 1?t2
y = t?t3 §3t =
√2
2,
√3
3?
(4)
braceleftbigg x = a(t?sint)
y = a(1?cost) (a′~ê)§3t = 0,
pi
2?
)μ
(1)?xprime(t) =?asint,yprime(t) = bcost§Kdydx = y
prime(t)
xprime(t) =?
b
a cott§u′§t =
pi
3?§y
prime =?
√3b
3a?
t = pi4?§yprime =?ba
(2)?xprime(t) = 1? cost,yprime(t) = sint§Kdydx = y
prime(t)
xprime(t) =
sint
1?cost§u′§t =
pi
2?§y
prime = 1?t =
pi?§yprime = 0
(3)?xprime(t) =?2t,yprime(t) = 1? 3t2§Kdydx = y
prime(t)
xprime(t) =
3t2?1
2t §u′§t =
√2
2?§y
prime =
√2
4?t =√
3
3?§y
prime = 0
(4)?xprime(t) = a(1? cost),yprime(t) = asint§Kdydx = y
prime(t)
xprime(t) = cot
t
2§u′§t = 0?§y
primet =
pi
2?§y = 1
5,|e?ê?§êμ
(1)
braceleftbigg x = acosht
y = bsinht
(2)
braceleftbigg x = sin2t
y = cos2t
(3)
braceleftbigg x = acos3t
y = asin3t
(4)
braceleftbigg x = e2t cos2t
y = e2t sin2t
)μ
(1) dydx = y
prime(t)
xprime(t) =
asinht
bcosht =
a
b cotht
(2) dydx = y
prime(t)
xprime(t) =
2costsint
2sintcost =?1
(3) dydx = y
prime(t)
xprime(t) =
3sin2tcost
3cos2tsint =?!tant
(4) dydx = y
prime(t)
xprime(t) =
e2t(2sin2t+ 2sintcost)
e2t(2cos2t?2costsint) = tant·
sint+ cost
cost?sint
6,?
I/Nì§ 10o§to
4o￡?4-11¤μ
(1) /\Y?§|YNèVéY?pYhCz
(2) |NèVéNì ?
RCz?.
65
)μ?NèV?Nì ?
R§Y?pYh'X?V = 13piR2h§?d?§R4 = h10=h = 52R§u
′
(1) V = 13pi
parenleftbigg2
5h
parenrightbigg2
h = 475pih3§l
dVdh = 425pih2?
(2) V = 13piR2 · 52R = 56piR3§l
dVdR = 52piR2.
7,?
I/Nì.t?X§§o?2arctan 34§8?p??,N§
(1) ?Nr?3§O\?Ydrdt?14?§NèO\?YdVdt′?o
(2) ?N6§NèO\?Y?24?§O\?Y′?o
)μ?NèVNr'X?V = 49pir3§V,r?′?mt?ê§ü>ét|§dVdt = 49pi(3r2)drdt=dVdt =
4
3pir
2dr
dt§K
(1) r = 3,drdt = 14?§dVdt = 3pi?
(2) ddrdt = 34pir2 dVdt§r = 6,dVdt = 24?§drdt = 12pi.
8,Ylp?18f?!.6f?
I/|ì6\5f?
/ùS.?|ì￥Y ?12f§
|ì￥Y?eü?Y?1f?/?§|d?
ù￥Y?t,?Y.
)μlm?|Y?2t?¨ §
I/|ì￥M? Y?xf?§
/ù￥Y?,p
yf?"d?§
|ì￥|?M?Nè?13pi·62 ·18? 13pi
parenleftBigx
18 ·6
parenrightBig2
·x = 216pi? pi27x3￡á?f?¤§
/ù￥5\
M?Nè?pi·52 ·y = 25piy￡á?f?¤"aK?§25piy = 216pi? pi27x3§y = 125
parenleftbigg
216? x
3
27
parenrightbigg
§u
′dydt = dydx · dxdt =? 1675 ·3x2 · dxdt =? 1225x2 · dxdt"x = 12(f?)?§dxdt =?1(f?/?)§u′d?
ù
￥Y?t,?Y?dydx =? 1225 ·122(?1) = 1625 = 0.64(f?/?).
9,?4-12¤?>′￥§P = i2R§ù￥>6i = Ur+R.|N?C>{R?§PCz?dPdR.
)μ?P = i2R,i = Ur+R§KdPdR = 2iR didR +i2 =?2U
2R
(r+R)3 +
U2
(r+R)2 =
U2(r?R)
(r+R)3
66
§7,?êT~
1,|e?ê3¤?:x0?êfprime?(x0)úmêfprime+(x0)μ
(1) y =
braceleftbigg x2,xlessorequalslant 0,
xex,x> 0,x0 = 0
(2) y =
braceleftBigg x
1 +e1x
,xnegationslash= 0,
0,x = 0,
x0 = 0
(3) y =
braceleftBigg
x2 sin 1x,xnegationslash= 0,
0,x = 0,
x0 = 0
)μ
(1) fprime+(x0) = lim
x→+0
xex?0
x = 1?f
prime
(x0) = limx→?0
x2?0
x = 0.
(2) fprime+(x0) = lim
x→+0
x
1 +e1x
0
x = limx→+0
1
1 +e1x
= 0?
fprime?(x0) = lim
x→?0
1
1 +e1x
= 1.
(3) fprime+(x0) = lim
x→+0
x2 sin 1x?0
x = 0?f
prime
(x0) = limx→?0
x2 sin 1x?0
x = 0.
2,|e?ê3ê?3:?!mêμ
(1) y = |ln|x||
(2) y = |tanx|
(3) y = √1?cosx
)μ
(1) y = |ln|x|| =
ln(?x),xlessorequalslant?1
ln(?x),?1 <x< 0
lnx,0 <x< 1
lnx,xgreaterorequalslant 1
dd?§?ê3x = 0,x = ±1?ê?3"
fprime+(?1) = lim
x→+0
ln[?(?1 +?x)]?ln(?(?1))
x = lim?x→+0ln(1x)
1?x = 1?
fprime?(?1) = lim
x→?0
ln[?(?1 +?x)]?ln(?(?1))
x = lim?x→?0ln(1x)
1?x =?1?
ê3x = 0:§fprime+(0)úfprime?(0)
fprime+(1) = lim
x→+0
ln(1 +?x)?ln1
x = lim?x→+0ln(1 +?x)
1
x = 1?
fprime?(1) = lim
x→?0
ln(1 +?x)?ln(1)
x = lim?x→?0?ln(1 +?x)
1
x =?1.
(2) y = |tanx| =
tanx,x∈
parenleftBig
kpi? pi2,kpi
bracketrightBig
tanx,x∈
parenleftBig
kpi,kpi+ pi2
parenrightBig k∈Z
ù￥x = kpi+ pi2(k∈Z)ê§m?:§?!mê
x = kpi(k∈Z)?ê?3:.
fprime+(kpi) = lim
x→+0
tan(kpi+?x)?(?tankpi)
x = lim?x→+0
tan?x
x = 1?f
prime
(kpi) = lim?x→?0
tan(kpi+?x)?(?tankpi)
x =
lim
x→?0
tan?x?x =?1.
(3)?yprime = sinx√1?cosxxnegationslash= 2kpi(k∈Z)?ak§x = 2kpi(k∈Z)?y = √1?cosx:.
fprime+(2kpi) = lim
x→+0
radicalbig1 + cos(2kpi+?x)?√1 + 2kpi
x = lim?x→+0
√1?cos?x
x =? lim?x→+0
radicalbigg
1?cos?x
x2 =
67
√2
2?
fprime?(2kpi) = lim
x→?0
radicalbig1 + cos(2kpi+?x)?√1 + 2kpi
x = lim?x→?0
√1?cos?x
x =? lim?x→?0?
radicalbigg
1?cos?x
x2 =
√2
2,
3,e
(1) f(x)3x0:?§g(x)3x0:§y2?êF(x) = f(x) +g(x)3x0:?
(2) f(x)úg(x)3x0:§U|?ú?êF(x) = f(x) +g(x)3x0:o
y2μ
(1) bF(x) = f(x) +g(x)3x0:?§qf(x)3x0:?§Kg(x) = F(x)?f(x)3x0:?§ù
g?§b?¤á"l
êF(x) = f(x) +g(x)3x0:.
(2)?U"~μ
(i)?μf(x) = |x|+x2,g(x) = x?|x|2 3x = 0:§§?ú?êF(x) = f(x) +g(x) =
x3x = 0:??Fprime(0) = 1?
(ii)μf(x) = |x|2,g(x) = |x|2 3x = 0:§§?ú?êF(x) = f(x) +g(x) = |x|3x =
0:.
4,3tK^?e§§?èG(x) = f(x)·g(x)??1No
)μ
(1) §?èG(x) = f(x)·g(x)3x0:?U?"
~μ
(i)?μf(x) = x3x = 0:??fprime(0) = 1?g(x) = |x|3x = 0:§§?èG(x) =
f(x)·g(x) = x|x|3x = 0??Gprime(0) = lim
x→0
G(x)
x = lim?x→0
x|?x|
x = lim?x→0|?x| = 0
(ii)μf(x) = 13x = 0:??fprime(0) = 0?g(x) = |x|3x = 0:§§?èG(x) =
f(x)·g(x) = |x|3x = 0:.
(2) §?èG(x) = f(x)·g(x)3x0:?U?"
~μ
(i)?μf(x) = |x|,g(x) = |x|3x = 0:§§?èG(x) = f(x) ·g(x) = x23x = 0?
Gprime(0) = 0
(ii)μf(x) = x23,g(x) = |x13|3x = 0:§§?èG(x) = f(x)·g(x) = |x|3x = 0:
.
5,e?êf(x)3km(a,b)￥kê§?lim
x→a
f(x) = ∞§′?7klim
x→a
fprime(x) = ∞o±~ff(x) = 1x +
cos 1x`2?.
§ef(x)3km(a,b)￥kê§?lim
x→a
fprime(x) = ∞§′?7klim
x→a
f(x) = ∞o±~ff(x) = 3√x`
2?.
)μ
(1)/`§?Uyklim
x→a
fprime(x) = ∞.
~μéu
parenleftBig
0,pi2
parenrightBig
S?êf(x) = 1x + cos 1x§w,klim
x→0
f(x) = ∞.
qfprime(x) =? 1x2? 1x2
parenleftbigg
sin 1x
parenrightbigg
= 1x2
parenleftbigg
sin 1x?1
parenrightbigg
§
éuA??Gêxn = 1
2npi+ pi2
(n = 1,2,···)§kfprime(xn) = 0§ lim
n→∞
fprime(xn) = 0?
éuxprimen = 1npi(n = 1,2,···)§kfprime(xprimen) =?n2pi2§ lim
n→∞
fprime(xprimen) =?∞§fprime(x)3x = 0:4
3§§=lim
x→0
fprime(x) = ∞?¤á.
(2)?Uy7klim
x→a
f(x) = ∞.
~μf(x) = 3√x§§3(0,b)(b> 0)tkê§?fprime(x) = 13 3√x2§lim
x→0
fprime(x) = ∞§lim
x→0
f(x) = 0.
68
6,e
(1) f(x)3x = g(x0)kê§
g(x)3x0:vkê?
(2) f(x)3x = g(x0)vkê§
g(x)3x0:kê?
(3) f(x)3x = g(x0)vkê§
g(x)3x0:?vkê?
KEü?êF(x) = f(g(x))3x0:′o
)μ
(1) Eü?êF(x) = f(g(x))3x0:?U?.
~μ
(i)?μf(u) = u2,g(x) = |x|,x0 = 0§f(u) = u23u0 = 0 = g(x0)??fprime(0) = 0§g(x) =
|x|3x0 = 0?F(x) = f(g(x)) = |x|2 = x23x0 = 0??Fprime(0) = 0?
(ii)?μf(u) = u,g(x) = |x|,x0 = 0§f(u) = u3u0 = 0 = g(x0)??fprime(0) = 1§g(x) =
|x|3x0 = 0?F(x) = f(g(x)) = |x|3x0 = 0.
(2) Eü?êF(x) = f(g(x))3x0:?U?.
~μ
(i)?μf(u) = |u|,g(x) = x2,x0 = 0§f(u) = |u|3u0 = 0 = g(x0)§g(x) = x23x0 = 0?
gprime(0) = 0?F(x) = f(g(x)) = |x2| = x23x0 = 0??Fprime(0) = 0?
(ii)?μf(u) = |u|,g(x) = x,x0 = 0§f(u) = |u|3u0 = 0 = g(x0)§g(x) = x3x0 = 0?
gprime(0) = 1?F(x) = f(g(x)) = |x|3x0 = 0.
(3) Eü?êF(x) = f(g(x))3x0:?U?.
~μ
(i)?μf(u) = 2u+|u|,g(x) = 23x?|x|3,x0 = 0§f(u) = 2u+|u|3u0 = 0 = g(x0)§g(x) =
2
3x?
|x|
3 3x0 = 0?F(x) = f(g(x)) = 2
parenleftbigg2
3x?
|x|
3
parenrightbigg
+
vextendsinglevextendsingle
vextendsinglevextendsingle2
3x?
|x|
3
vextendsinglevextendsingle
vextendsinglevextendsingle =
braceleftbigg x,xgreaterorequalslant 0
x,x< 0 =
x=F(x) = x(?x∈ (?∞,+∞)§F(x)3x0 = 0??Fprime(0) = 1?
(ii)?μf(u) = |u|,g(x) = |x|,x0 = 0§f(u) = |u|3u0 = 0 = g(x0)§g(x) = |x|3x0 = 0?
?F(x) = f(g(x)) = |x|3x0 = 0.
69
§8,p ê?p
1,y = 2x3 +x2 +x+ 1§|yprime,yprimeprime,y(3)úy(4).
)μyprime = 6x2 + 2x+ 1,yprimeprime = 12x+ 2,y(3) = 12,y(4) = 0
2,y = eαt(α?~ê)§|yprimeprime,y(3),y(n).
)μyprime = αeαt,yprimeprime = α2eαt,y(3) = α3eαt,y(n) = αneαt
3,|e?êp êμ
(1) y = x√1?x2§|yprimeprime
(2) y = xlnx§|yprimeprime
(3) y = e?x2§|yprimeprime
(4) y = arcsinx√1?x2§|yprimeprime
(5) y = x2 ·e2x§|yprimeprimeprime
(6) y = a3x§|yprimeprimeprime
(7) y = x3 sinhx§|y(30)
(8) y = x3 cosx§|y(50)
)μ
(1) yprime =
√1?x2 + x2√
1?x2
1?x2 =
1
(1?x2)32
= (1?x2)?32,yprimeprime = 3x(1?x2)?52
(2) yprime = 1 + lnx,yprimeprime = 1x
(3) yprime =?2xe?x2,yprimeprime =?2e?x2(1?2x2) = 2e?x2(2x2?1)
(4) yprime =
1 + xarcsinx√1?x2
1?x2 =
1
1?x2 +
xarcsinx
(1?x2)32
,
yprimeprime = 2x(1?x2)2 +
parenleftbigg
arcsinx+ x√1?x2
parenrightbigg
(1?x2)32 + 3x(1?x2)12 ·xarcsinx
(1?x2)3 =
3x
(1?x2)2 +
(2x2 + 1)arcsinx
(1?x2)52
(5) yprimeprimeprime = (x2e2x)primeprimeprime = x2(e2x)primeprimeprime + 3(x2)prime(e2x)primeprime + 3(x2)primeprime(e2x)prime + (x2)primeprimeprimee2x = 4e2x(2x2 + 6x+ 3)
(6) yprime = 3a3x lna,yprimeprime = 9ln2a·a3x,yprimeprimeprime = 27ln3a·a3x
(7)?(x3)prime = 3x2,(x3)primeprime = 6x,(x3)primeprimeprime = 6,(x3)(4) = ··· = (x3)(30) = 0;(sinhx)(30) = sinhx,(sinhx)(29) =
coshx,(sinhx)(28) = sinhx,(sinhx)(27) = coshx§y(30) = (x3 sinhx)(30) = x3(sinhx)(30)+30(x3)prime(sinhx)(29)+
435(x3)primeprime(sinhx)(28) + 4060(x3)primeprimeprime(sinh)(27) = xsinhx(x2 + 2610) + 30coshx(3x2 + 812)
(8)?(x3)prime = 3x2,(x3)primeprime = 6x,(x3)primeprimeprime = 6,(x3)(4) = ··· = (x3)(50) = 0;(cosx)(50) =?cosx,(cosx)(49) =
sinx,(cosx)(48) = cosx,(cosx)(47) = sinx§y(50) = (x3 cosx)(50) = x3(cosx)(50)+50(x3)prime(cosx)(49)+
1225(x3)primeprime(cosx)(48) + 19600(x3)primeprimeprime(cosx)(47) = xcosx(7350?x2) + 150sinx(784?x2)
4,|^ê?8B{y2e?úaμ
(1) (ax)(n) = ax ·(lna)n(a> 0)
(2) (cosx)(n) = cos
parenleftBig
x+n· pi2
parenrightBig
(3) (lnx)(n) = (?1)
n?1 ·(n?1)!
xn
y2μ
(1) (i) n = 1?§(ax)prime = ax lna = ax(lna)1§Kn = 1?úa¤á.
(ii) bn = k?úa¤á§=(ax)(k) = ax(lna)k¤á§
Kn = k+ 1?§(ax)(k+1) =
bracketleftBig
(ax)(lna)(k)
bracketrightBigprime
= (lna)k(ax)prime = (lna)k ·ax lna = ax(lna)k+1§u
′n = k+ 1?úa?¤á.
nüt§ng,ê?§úa(ax)(n) = ax ·(lna)n(a> 0)?¤á.
70
(2) (i) n = 1?§(cosx)prime =?sinx = cos
parenleftBig
x+ pi2
parenrightBig
§Kn = 1?úa¤á.
(ii) bn = k?úa¤á§=(cosx)(k) = cos
parenleftBig
x+k· pi2
parenrightBig
¤á§
Kn = k + 1?§(cosx)(k+1) =
bracketleftBig
(cosx)(k)
bracketrightBigprime
=
bracketleftBig
cos
parenleftBig
x+k· pi2
parenrightBigbracketrightBigprime
=?sin
parenleftBig
x+k· pi2
parenrightBig
=
cos
parenleftBig
x+ (k+ 1)· pi2
parenrightBig
§u′n = k+ 1?úa?¤á.
nüt§ng,ê?§úa(cosx)(n) = cos
parenleftBig
x+n· pi2
parenrightBig
¤á.
(3) (i) n = 1?§(lnx)prime = 1x = (?1)
1?1(1?1)!
x1 §Kn = 1?úa¤á.
(ii) bn = k?úa¤á§=(lnx)(k) = (?1)
k?1 ·(k?1)!
xn ¤á§
Kn = k + 1?§(lnx)(k+1) =
bracketleftBig
(lnx)(k)
bracketrightBigprime
=
bracketleftbigg(?1)k?1 ·(k?1)!
xk
bracketrightbiggprime
=?k· (?1)
k?1 ·(k?1)!
xk+1 =
(?1)k ·k!
xk+1 =
(?1)k+1?1 ·(k+ 1?1)!
xk+1 §u′n = k+ 1?úa?¤á.
nüt§ng,ê?§úa(lnx)(n) = (?1)
n?1 ·(n?1)!
xn?¤á.
5,|n êμ
(1) y = 1x(1?x)
(2) y = 1x2?2x?8
(3) y = amxm +am?1xm?1 +···+a0
(4) y = cos2ωx
(5) y = e
x
x
(6) y = 2x ·lnx
(7) y = eaxpn(x)§ù￥pn(x)?nga.
)μ
(1)?y = 1x(1?x) = 1x + 11?x§Ky(n) =
parenleftbigg1
x?
1
1?x
parenrightbigg(n)
=
parenleftbigg1
x
parenrightbigg(n)
+
parenleftbigg 1
1?x
parenrightbigg(n)
= (x?1)(n) +[(1?
x)?1](n) = (?1)n ·n!·x?n?1 + (?1)2n ·n!·(1?x)?n?1 = n![(?1)nx?n?1 + (1?x)?n?1]
(2)?y = 1x2?2x?8 = 1(x+ 2)(x?4) = 16
parenleftbigg 1
x?4?
1
x+ 2
parenrightbigg
§Ky(n) = 16
parenleftbigg 1
x?4?
1
x+ 2
parenrightbigg(n)
=
1
6[((x?4)
1)(n)?((x+2)?1)(n)] = 1
6[(?1)
n·n!(x?4)?n?1?(?1)n·n!(n+2)?n?1] = (?1)n
6 n!
parenleftbigg 1
(x?4)n+1?
1
(x+ 2)n+1
parenrightbigg
(3) n>m?§x(n) = (x2)(n) = ··· = (xm)(n) = 0§Ky(n) = 0?
n = m?§x(n) = (x2)(n) = ··· = (xm?1)(n) = 0,(xm)(n) = (xm)(m) = m!§Ky(n) = am ·m!?
n < m?§x(n) = (x2)(n) = ··· = (xn?1)(n) = 0,(xn)(n) = n!,···,(xm)(n) = m!(m?n)!xm?n§
Ky(n) = am· m!(m?n)!xm?n+am?1· (m?1)!(m?n?1)!xm?n?1+···+ann! =
m?nsummationdisplay
i=0
am?i (m?i)!(m?n?i)!xm?n?i.
(4)?yprime =?2ωcosωxsinωx =?ωsin2ωx§Ky(n) = (yprime)(n?1) = (?ωsin2ωx)(n?1) =?ωsin
parenleftbigg
2ωx+ n?12 pi
parenrightbigg
·
(2ω)n?1 =?2n?1ωn sin
parenleftbigg
2ωx+ n?12 pi
parenrightbigg
= 2n?1ωn cos
parenleftBig
2ωx+ n2pi
parenrightBig
(5) y(n) =
parenleftbiggex
x
parenrightbigg(n)
=
parenleftbigg
ex · 1x
parenrightbigg(n)
=
nsummationtext
k=0
Cknex
parenleftbigg1
x
parenrightbigg(k)
=
ex
bracketleftBigg
1
x +
nsummationdisplay
k=1
(?1)kn(n?1)···(n?k+ 1)xk+1
bracketrightBigg
71
(6) y(n) = (2x ·lnx)(n) =
nsummationtext
k=0
Ckn(2x)(n?k)(lnx)(k) =
nsummationtext
k=1
Ckn(ln2)n?k ·2x · (?1)
k?1(k?1)!
xk + 2
x(ln2)n lnx =
2x[(ln2)n lnx+n(ln2)n?1x?1 +···+ (?1)n?2(n?2)!·nln2·x?(n?1) + (?1)n?1(n?1)!·x?n]
(7) y(n) = (eaxpn(x))(n) = aneaxpn(x)+C1nan?1eaxpprimen(x)+···+eaxp(n)n (x) = eax[anpn(x)+C1nan?1pprimen(x)+
···+p(n)n (x)]
6,ef(x) =
braceleftBigg
e? 1x2,xnegationslash= 0
0,x = 0 y2f
(n)(0) = 0.
y2μx negationslash= 0?§fprime(x) = 2x3e? 1x2,fprimeprime(x) = e? 1x2
parenleftbigg
6x4 + 4x6
parenrightbigg
§ddí?f(n)(x) = e? 1x2Pn
parenleftbigg1
x
parenrightbigg
(x negationslash=
0)§ù￥Pn(t)′'uta"
e?y2μéên§tk·Kf(n)(x) = e? 1x2Pn
parenleftbigg1
x
parenrightbigg
(xnegationslash= 0)
n = 1?§·Kw,¤á.
bn = k?§·K¤á§=kf(k)(x) = e? 1x2Pk
parenleftbigg1
x
parenrightbigg
(xnegationslash= 0),Pk(t)′'uta§
Kn = k+ 1?§f(k+1)(x) = [f(k)(x)]prime =
bracketleftbigg
e? 1x2Pk
parenleftbigg1
x
parenrightbiggbracketrightbiggprime
=
e? 1x2
bracketleftbigg 2
x3Pk
parenleftbigg1
x
parenrightbigg
1x2Pprimek
parenleftbigg1
x
parenrightbiggbracketrightbigg
= e? 1x2
bracketleftBigg
2
parenleftbigg1
x
parenrightbigg3
Pk
parenleftbigg1
x
parenrightbigg
parenleftbigg1
x
parenrightbigg2
Pprimek
parenleftbigg1
x
parenrightbiggbracketrightBigg
= e? 1x2Pk+1
parenleftbigg1
x
parenrightbigg
§
ù￥Pk+1(t)′'ut,a.
aê?8B{?§·Kég,ênt¤á.
n = 1?§fprime(0) = lim
x→0
f(0 +?x)?f(0)
x = lim?x→0
e?( 1?x)2
x = lim?x→0
1
x
e( 1?x)2
= 0
bn = k?§f(k)(0) = 0§Kf(k+1) = lim
x→0
f(k)(0 +?x)?f(k)(0)
x = lim?x→0
e?( 1?x)2Pk
parenleftbigg 1
x
parenrightbigg
x =
lim
x→0
1
xPk
parenleftbigg 1
x
parenrightbigg
e( 1?x)2
= 0
aê?8B{?§f(n)(0) = 0.
7,f(x)? ê3§|yprimeprime9yprimeprimeprimeμ
(1) y = f(x2)
(2) y = f
parenleftbigg1
x
parenrightbigg
(3) y = f(e?x)
(4) y = f(lnx)
)μ
(1) yprime = 2xfprime(x2),yprimeprime = 2fprime(x2) + 4x2fprimeprime(x2),
yprimeprimeprime = 12xfprimeprime(x2) + 8x3fprimeprimeprime(x2)
(2) yprime =? 1x2fprime
parenleftbigg1
x
parenrightbigg
,yprimeprime = 2x3fprime
parenleftbigg1
x
parenrightbigg
+ 1x4fprimeprime
parenleftbigg1
x
parenrightbigg
,
yprimeprimeprime =? 6x4fprime
parenleftbigg1
x
parenrightbigg
6x5fprimeprime
parenleftbigg1
x
parenrightbigg
1x6fprimeprimeprime
parenleftbigg1
x
parenrightbigg
(3) yprime =?e?xfprime(e?x),yprimeprime = e?xfprime(e?x)+e?2xfprimeprime(e?x),yprimeprimeprime =?e?xfprime(e?x)?3e?2xfprimeprime(e?x)?e?3xfprimeprimeprime(e?x)
(4) yprime = 1xfprime(lnx),yprimeprime = 1x2fprimeprime(lnx)? 1x2fprime(lnx) = 1x2 [fprimeprime(lnx)?fprime(lnx)],yprimeprimeprime = 1x3 [2fprime(lnx)?3fprimeprime(lnx)+
fprimeprimeprime(lnx)]
8,y = ex sinx,z = ex cosx§y2§?÷v?§yprimeprime = 2z,zprimeprime =?2y.
y2μ?y = ex sinx,z = ex cosx§Kyprime = ex(sinx + cosx),yprimeprime = 2ex cosx;zprime = ex(cosx? sinx),zprimeprime =
2ex sinx§u′yprimeprime = 2z,zprimeprime =?2y.
72
9,y = C1eλ1x +C2eλ2x§C1,C2,λ1,λ2′~ê§y2§÷v?§yprimeprime?(λ1 +λ2)yprime +λ1λ2y = 0.
y2μ?y = C1eλ1x + C2eλ2x§C1,C2,λ1,λ2′~ê§Kyprime = C1λ1eλ1x + C2λ2eλ2x,yprimeprime = C1λ21eλ1x +
C2λ22eλ2x§
u′yprimeprime?(λ1 +λ2)yprime +λ1λ2y = C1λ21eλ1x +C2λ22eλ2x?(λ1 +λ2)(C1λ1eλ1x +C2λ2eλ2x)+λ1λ2(C1eλ1x +
C2eλ2x) = 0=yprimeprime?(λ1 +λ2)yprime +λ1λ2y = 0.
10,y = C1 sinx+C2 cosx§y2y÷v?§yprimeprime +y = 0.
y2μ?y = C1 sinx + C2 cosx§Kyprime = C1 cosx?C2 sinx,yprimeprime =?C1 sinx?C2 cosx =?(C1 sinx +
C2 cosx) =?y=yprimeprime +y = 0.
11,e?ê(x) = f(x)?f(a)fprime(a)
bracketleftbigg
1 + f(x)?f(a)fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
§|?prime(a)9?primeprime(a).
)μ(x) = f(x)?f(a)fprime(a)
bracketleftbigg
1 + f(x)?f(a)fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
§
K?prime(x) = f
prime(x)
fprime(a)
bracketleftbigg
1 + f(x)?f(a)fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
+
f(x)?f(a)
fprime(a)
bracketleftbiggfprime(x)
fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
,
primeprime(x) = f
primeprime(x)
fprime(a)
bracketleftbigg
1 + f(x)?f(a)fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
+ 2f
prime(x)
fprime(a)
bracketleftbiggfprime(x)
fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
+
f(x)?f(a)
fprime(a)
bracketleftbiggfprimeprime(x)
fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
§
K?prime(a) = 1,?primeprime(a) = 2
12,x =?(y)′y = f(x)ê§áˉX?dfprime,fprimeprime,fprimeprimeprimeprimeprimeprime(y)o
)μprime(y) = 1fprime(x)§K?primeprime(y)fprime(x) =? f
primeprime(x)
[fprime(x)]2§u′?
primeprime(y) =? f
primeprime(x)
[fprime(x)]3§
K?primeprimeprime(y)fprime(x) =?f
primeprimeprime(x)[fprime(x)]3?3[fprime(x)]2[fprimeprime(x)]2
[fprime(x)]6 §l
primeprimeprime(y) = 3[f
primeprime(x)]2?fprimeprimeprime(x)fprime(x)
[fprime(x)]5,
13,á|{Z?s = ae?λt sinωt3t?Yú\?Y§?|Y?=:.
)μ?Yv = sprime = ae?λt(?λsinωt+ωcosωt)§\?Ya = vprime = sprimeprime = ae?λt[(λ2?ω2)sinωt?2λωcosωt]?
Y?=:=v = 0§K?λsinωt+ωcosωt = 0§u′tanωt = ωλ(λnegationslash= 0).
14,|e?ê?§ êd
2y
dx2μ
(1)
braceleftbigg x = 2t?t2
y = 3t?t3
(2)
braceleftbigg x = acost
y = asint
(3)
braceleftbigg x = a(t?sint)
y = a(1?cost)
(4)
braceleftbigg x = et cost
y = et sint
(5)
braceleftbigg x = acos3t
y = asin3t
(6)
braceleftbigg x = fprime(t)
y = tfprime(t)?f(t)
)μ
(1) dydx = 3?3t
2
2?2t =
3
2(1 +t)
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
= 34(1?t)
(2) dydx = acost?asint =?cott
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
=? 1asin3t
73
(3) dydx = asinta(1?cost) = cot t2
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
=? 1
4asin4 t2
(4) dydx = e
t(sint+ cost)
et(cost?sint) =
sint+ cost
cost?sint
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
= 2et(cost?sint)3
(5) dydx = 3asin
2tcost
3acos2tsint =?tant
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
= 13acos4tsint
(6) dydx = tf
primeprime(t)
fprimeprime(t) = t
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
= 1fprimeprime(t)
15,|dê¤(? êμ
(1) ex+y?xy = 0
(2) x3 +y3?3axy = 0
(3) y2 + 2lny?x4 = 0
)μ
(1) é?§ex+y?xy = 0üà'ux|§
(1 +yprime)ex+y?y?xyprime = 0 (4)
u′yprime = y?e
x+y
ex+y?x§
2é(4)üà'ux|§yprimeprimeex+y + (1 +yprime)2ex+y?2yprime?xyprimeprime = 0§Kyprimeprime = 2y
prime?(1 +yprime)2ex+y
ex+y?x §
òyprime = y?e
x+y
ex+y?x?\ta§=y
primeprime = 2(y?e
x+y)
(ex+y?x)2?
(x?y)2ex+y
(ex+y?x)3,
(2) é?§x3 +y3?3axy = 0üà'ux|§
x2 +y2yprime?axyprime?ay = 0 (1)
u′yprime = ay?x
2
y2?ax§
2é(1)üà'ux|§2x+ 2y(yprime)2 +y2yprimeprime?2ayprime?axyprimeprime = 0§Kyprimeprime = 2ay
prime?2y(yprime)2?2x
y2?ax §
òyprime = ay?x
2
y2?ax?\ta§=y
primeprime = 2a(ay?x
2)
(y2?ax)2?
2y(ay?x2)2
(y2?ax)3?
2x
y2?ax.
(3) é?§y2 + 2lny?x4 = 0üà'ux|§
yyprime + 1yyprime?2x3 = 0 (1)
u′yprime = 2x
3y
y2 + 1§
2é(1)üà'ux|§(yprime)2 +yyprimeprime + yy
primeprime?(yprime)2
y2?6x
2 = 0§Kyprimeprime = 6x2y2 + (yprime)2(1?y2)
y(y2 + 1) §
òyprime = 2x
3y
y2 + 1?\ta§=y
primeprime = 2x
2y
(y2 + 1)3 [3(y
2 + 1)2 + 2x4(1?y2)].
74
16,|p (x′gCt)μ
(1) y = √1 +x2§|d2y
(2) y = xx§|d2y
(3) y = xcos2x§|d3y
(4) y = 1√x§|d3y
(5) y = xn ·ex§|dny
(6) y = lnxx §|dny
)μ
(1) dy = x√1 +x2dx,d2y = (1 +x2)?32dx2
(2) dy = xx(lnx+ 1)dx,d2y = xx
bracketleftbigg
(lnx+ 1)2 + 1x
bracketrightbigg
dx2
(3) d3y = (xcos2x)(3)dx3 = (x(cos2x)(3) + 3(cos2x)(2))dx3 = (8xsin2x?12cos2x)dx3
(4) d3y =
parenleftbigg 1
√x
parenrightbigg(3)
dx3 =?158 x?72dx3
(5) dny = (xn ·ex)(n)dxn =
parenleftbigg
ex
nsummationtext
k=0
Ckn n!(n?k)!xn?k
parenrightbigg
dxn
(6) dny =
parenleftbigglnx
x
parenrightbigg(n)
dxn =
parenleftbigg1
x lnx
parenrightbigg(n)
dxn =
bracketleftBigg
(?1)nn!lnxxn+1 +
nsummationdisplay
k=1
Ckn(?1)n?1 (n?k)!(k?1)!xn+1
bracketrightBigg
dxn =
(?1)n n!xn+1
bracketleftBigg
lnx?
nsummationdisplay
k=1
1
k
bracketrightBigg
dxn
17,éy = ex|d2y§?e?ü/μ
(1) x′gCt
(2) x′￥mCt?.
)μ
(1) dy = exdx,d2y = exdx2
(2) dy = exdx,d2y = ex(dx2 +d2x)
18,eu,v?x?ê§v
g§|p μ
(1) y = u(x)·v(x)§|d2y
(2) y = u(x)v(x)§|d2y
(3) y = um(x)vn(x)(m,n?~ê)§|d2y
(4) y = au(x)(a> 0)§|d2y
(5) y = lnu(x)§|d3y
(6) y = sin(u(x))§|d3y
)μ
(1) dy = (uprime(x)v(x) +u(x)vprime(x))dx,
d2y = [uprimeprime(x)v(x) + 2uprime(x)vprime(x) +u(x)vprimeprime(x)]dx2
(2) dy = u
prime(x)v(x)?u(x)vprime(x)
v2(x) dx,
d2y =
bracketleftbigguprimeprime(x)
v(x)?
u(x)vprimeprime(x) + 2uprime(x)vprime(x)
v2(x) +
2u(x)(vprime(x))2
v3(x)
bracketrightbigg
dx2
75
(3) dy = [mum?1(x)vn(x)uprime(x) +num(x)vn?1(x)vprime(x)]dx,
d2y = [m(m? 1)um?2(x)vn(x)(uprime(x))2 + 2mnum?1(x)vn?1(x)uprime(x)vprime(x) + mum?1(x)vn(x)uprimeprime(x) +
n(n?1)um(x)vn?2(x)(vprime(x))2 +num(x)vn?1(x)vprimeprime(x)]dx2
(4) dy = au(x) lna·uprime(x)dx,
d2y = au(x) lna[lna(uprime(x))2 +uprimeprime(x)]dx2
(5) dy = u
prime(x)
u(x) dx,d
2y =
bracketleftbigguprimeprime(x)
u(x)?
(uprime(x))2
u2(x)
bracketrightbigg
dx2,
d3y =
bracketleftbigguprimeprimeprime(x)
u(x)?
3uprime(x)uprimeprime(x)
u2(x) +
2(uprime(x))3
u3(x)
bracketrightbigg
dx3
(6) dy = cos(u(x))uprime(x)dx,d2y = [cos(u(x))uprimeprime(x)?sin(u(x))(uprime(x))2]dx2,
d3y = [cos(u(x))uprimeprimeprime(x)?3sin(u(x))uprime(x)uprimeprime(x)?cos(u(x))(uprime(x))3]dx3.
76
1êù? ?n9ùA^
§1,￥n
1,3¤ê?n￥§ex0mà:§áT~`2(¤á.
)μ~μ?êy = x3?m[?1,1]tk§§3à:x0 = 1?§=?x∈ [?1,1]§ekf(x) lessorequalslant
f(x0) = 1§,
yprime|x=1 = 1 negationslash= 0.
2,éux0 ∈ (a,b)§efprime(x0) > 0§K3§?!m?O?(x0,δ),O+(x0,δ)|x∈O?(x0,δ)f(x0) >
f(x)§x∈O+(x0,δ)f(x0) <f(x).
y2μ?fprime(x0) = lim
x→x0
f(x)?f(x0)
x?x0 > 0§a4?5?§3x0δ(δ > 0)?O(x0,δ)? (a,b)§
|x ∈ O(x0,δ)?§kf(x)?f(x0)x?x
0
> 0§l
x ∈ O?(x0,δ)=x? x0 < 0?§kf(x0) > f(x)§
x∈O+(x0,δ)=x?x0 > 0?§kf(x0) <f(x).
3,y2μefprime+(x0) > 0,fprime?(x0) < 0§K3x0?§|3d?Sf(x) greaterorequalslantf(x0).
y2μ?fprime+(x0) = lim
x→x0+0
f(x)?f(x0)
x?x0 > 0§Kdm4?5?§73x0δ1(δ1 > 0)m?O+(x0,δ1)§
|x∈O+(x0,δ1)=0 <x?x0 <δ1?§kf(x)?f(x0)x?x
0
> 0§l
kf(x0) <f(x)?
qfprime?(x0) = lim
x→x0?0
f(x)?f(x0)
x?x0 < 0§Kd?4?5?§73x0δ2(δ2 > 0)??O?(x0,δ2)§|
x∈O?(x0,δ2)=0 <x0?x<δ2?§kf(x)?f(x0)x?x
0
< 0§l
kf(x0) <f(x)?
δ = min(δ1,δ2)§x∈O(x0,δ)?§okf(x) greaterorequalslantf(x0).
4,ef(x)3[a,b]?Y§f(a) = f(b) = 0,fprime(a)·fprime(b) > 0§Kf(x)3(a,b)S?k":.
y2μ?fprime(a)·fprime(b) > 0§fprime(a) > 0,fprime(b) > 0(fprime(a) < 0,fprime(b) < 0?1ón?y)
qfprime(a) = fprime+(a) = lim
x→a+0
f(x)?f(a)
x?a > 0§Kdm4?5?§73aδ1(δ1 > 0)m?O+(a,δ1)§|
x∈O+(a,δ1)=0 <x?a<δ1?§kf(x)?f(a)x?a > 0§l
kf(a) <f(x)?
?x1 ∈O+(a,δ1)§Kkf(x1) >f(a)
qf(a) = 0§Kf(x1) > 0
qfprime(b) = fprime?(b) = lim
x→b?0
f(x)?f(b)
x?b > 0§Kd?4?5?§73bδ2(δ2 > 0)??O?(b,δ2)§|
x∈O?(b,δ2)=0 <b?x<δ2?§kf(x)?f(b)x?x
0
> 0§l
kf(b) >f(x)?
?x2 ∈O?(b,δ1)§Kkf(x2) <f(b)
qf(b) = 0§Kf(x2) < 0
f(x)3[a,b]?Y§3[x1,x2]Y§qf(x1) > 0,f(x2) < 0§Kd":3?n?§3[x1,x2]S?
k":§
q[x1,x2]? [a,b]§l
f(x)3[a,b]S?k":.
ón§fprime(a) < 0,fprime(b) < 0?§f(x)3[a,b]S?k":.
5,df(x+?x)?f(x) = fprime(x+θ?x)?x(0 <θ< 1)§|?êθ = θ(x,?x)§
(1) f(x) = ax2 +bx+c(anegationslash= 0)
(2) f(x) = 1x
(3) f(x) = ex
)μ
(1) fprime(x) = 2ax+b,fprime(x+θ?x) = 2a(x+θ?x) +b§
K[2a(x + θ?x) + b]?x = f(x +?x)?f(x) = a(x +?x)2 + b(x +?x) + c? (ax2 + bx + c) =
2ax·?x+a(?x)2 +b?x =
bracketleftbigg
2a
parenleftbigg
x+ 12?x
parenrightbigg
+b
bracketrightbigg
x§u′θ = 12
(2) fprime(x) =? 1x2,fprime(x+θ?x) =? 1(x+θ?x)2§
Kx(x+θ?x)2 = f(x+?x)?f(x) = 1x+?x? 1x =xx(x+?x)§l
x2θ2 + 2x·?xθ = 0§
77
u′θ =?x±
√x2 +x?x
x §d?Kò?à(θ ∈ (0,1)
§?Ak
x
x >?1(x negationslash= 0)(dx
2 +
x?x> 0§K?xx >?1)
(3) fprime(x) = ex,fprime(x+θ?x) = ex+θ?x§
Kex+θ?x?x = f(x +?x)?f(x) = ex+?x?ex = ex(e?x? 1)§l
eθ?x?x = e?x? 1§u′θ =
1
x ln
e?x?1
x §?±yθ∈ (0,1)
6,f(x)3?m[a,b]S?Y§3(a,b)?§|^?ê
Φ(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
x f(x) 1
b f(b) 1
a f(a) 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
y2.?KFúa§?QêΦ(x)A.
y2μ?Φ(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
x f(x) 1
b f(b) 1
a f(a) 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= (a?b)f(x) + (f(b)?f(a))x+bf(a)?af(b)§
qf(x)3?m[a,b]S?Y§Kd?Y?êoK$?{K§Φ(x)3[a,b]?Y?
qf(x)3(a,b)?§Kd??êoK$?{K§Φ(x)3(a,b)?.
qΦ(a) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
a f(a) 1
b f(b) 1
a f(a) 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle = 0,Φ(b) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
b f(b) 1
b f(b) 1
a f(a) 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle = 0§Kda?n§3(a,b)S?k?:ξ§
|Φprime(ξ) = 0.
Φprime(x) = (a?b)fprime(x) +f(b)?f(a)§K0 = Φprime(ξ) = (a?b)fprime(ξ) +f(b)?f(a)=fprime(ξ) = f(b)?f(a)b?a,
Φ(x)Aμn/?èúaS? = 12
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
x1 y1 1
x2 y2 1
x3 y3 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle§ù￥(x1,y1),(x2,y2),(x3,y3)L?o:?I§
KΦ(x)L?±A(x,f(x)),B(a,f(a)),C(b,f(b))?o:n/?èü.
7,áée?ê?.?KFúaf(b)?f(a) = fprime(c)(b?a)§?|c.
(1) f(x) = x3,x∈ [0,1]
(2) f(x) = arctanx,x∈ [0,1]
)μ
(1)?fprime(x) = 3x2§K3c2(1?0) = 13?03=3c2 = 1§qc∈ (0,1)§c =
√3
3,
(2)?fprime(x) = 11 +x2§K 11 +c2 (1?0) = arctan1?arctan0= 11 +c2 = pi4§qc∈ (0,1)§c =
radicalbigg4
pi?1.
8,áée?êüúaf(b)?f(a)g(b)?g(a) = f
prime(c)
gprime(c)§?|c.
(1) f(x) = sinx,g(x) = cosx,x∈
bracketleftBig
0,pi2
bracketrightBig
(2) f(x) = x2,g(x) = √x,x∈ [1,4]
)μ
(1)?fprime(x) = cosx,gprime(x) =?sinx§K
f
parenleftBigpi
2
parenrightBig
f(0)
g
parenleftBigpi
2
parenrightBig
g(0)
= f
prime(c)
gprime(c)=
1?0
0?1 =
cosc
sinc§?=cotc = 1§qc ∈
parenleftBig
0,pi2
parenrightBig
§c = pi4.
(2)?fprime(x) = 2x,gprime(x) = 12√x§Kf(4)?f(1)g(4)?g(1) = f
prime(c)
gprime(c)=
16?1
2?1 =
2c
1
2√c
§?=4c32 = 15§qc∈ (1,4)§
c =
parenleftbigg15
4
parenrightbigg2
3.
78
9,áêy = |x?1|3?m[0,3]t?/§ùpovk21uu§.?KF?n￥=?^¤
áo
)μ?ê3:x = 1§=ù?/ACBò?§dò?3C(0,1):3§.?KF?n￥
1?^?=3(0,3)S?ù?^÷v.
a45
a54
a0
a0
a0
a0
a0
a0
a0
a64
a64
a64
a64
a64
a64
a64a64
0 3 x
y
10,|^.?KFúay2? aμ
(1) |sinx?siny|lessorequalslant|x?y|
(2) x∈
parenleftBig
pi2,pi2
parenrightBig
§|x|lessorequalslant|tanx|( ò?k3x = 0?¤á)
(3) n·yn?1(y?x) <xn?yn <n·xn?1(x?y)(n> 1,x>y)
(4) x1 +x < ln(1 +x) <x(x> 0)
(5) exnegationslash= 0,ex > 1 +x(?x> 0,x< 0ü1y2)
y2μ
(1)x>y,f(t) = sint3[y,x]t?Y§3(y,x)S?§.?KF?n¤á§?
ksinx?siny =
cosξ(x? y)(ξ ∈ (y,x))§K|sinx? siny| = |cosξ(x? y)| = |cosξ||(x? y)| lessorequalslant |x? y|(?(x,y ∈
(?∞,+∞))¤á.
(2)x ∈
parenleftBig
0,pi2
parenrightBig
,f(t) = tant3[0,x]t?Y§3(0,x)S?§.?KF?n¤á§?
ktanx?
tan0 = sec2ξ(x?0)
parenleftbigg
ξ ∈ (0,x),x∈
parenleftbigg
0,pi2
parenrightbiggparenrightbigg
§Kx = cos2ξ·tanx< tanx
ón?y§x∈
parenleftBig
pi2,0
parenrightBig
§?x<?tanx.
x = 0?§|tanx| = |x|.
o?§x∈
parenleftBig
pi2,pi2
parenrightBig
§|x|lessorequalslant|tanx|¤á.
x = 0?§ ò¤á?0 <|ξ|<|x|< pi2?§0 < cos2ξ< 1§?U¤á|x|<|tanx|.
(3) f(t) = tn3[y,x]t?Y§3(y,x)S?§.?KF?n¤á§?
kxn?yn = n·ξn?1(x?y)(0 <
y<ξ<x)§
qn> 1§Kyn?1 <ξn?1 <xn?1§n·yn?1(x?y) <n·ξn?1(x?y) <n·xn?1(x?y)=n·yn?1(x?y) <
xn?yn <n·xn?1(x?y)¤á.
(4) f(t) = ln(1 +t)3[0,x]t?Y§3(0,x)S?§.?KF?n¤á§?
kln(1 +x) = ln(1 +x)?
ln1 = 11 +ξ(1 + x? 1) = x1 +ξ(0 < ξ < x)§q1 < 1 + ξ < 1 + x§K 11 +x < 11 +ξ < 1§l
k x1 +x < x1 +ξ <x(x> 0)= x1 +x < ln(1 +x) <x(x> 0)¤á.
(5) f(t) = etw,÷v.?KF?n^?.
x> 0?§éf(t) = et3[0,x]A^.?KFúa§kex?e0 = eξ(x?0)=ex?1 = xeξ(0 <ξ<x)§
0 <ξ<x§Keξ > 1§l
ex?1 = xeξ >x=ex > 1 +x?
x< 0?§éf(t) = et3[x,0]A^.?KFúa§ke0?ex = eξ(0?x)=1?ex =?xeξ(x<ξ< 0)§
x<ξ< 0§K0 <eξ < 1§l
1?ex =?xeξ <?x=ex > 1 +x.
o?§exnegationslash= 0§okex > 1 +x.
11,efprime(x) ≡k§áyf(x) = kx+b.
y2μ?F(x) = f(x)?kx
duFprime(x) = fprime(x)? k ≡ 0§a.?KF?ní?1§F(x) = f(x)? kx = b(?x ∈ (?∞,+∞)§
f(x) = kx+b.
79
12,y2?§x3?3x+c = 03[0,1]S?1küó?.
y2μ-f(x) = x3?3x+c
^?y{.f(x)3[0,1]Sküó?0 <x1 <x2 < 1.
d?f(x1) = f(x2) = 0§aa?n§73ξ ∈ (x1,x2)§|fprime(ξ) = 0=3ξ2? 3 = 0§)ξ = ±1§ù
ξ ∈ (x1,x2)? (0,1)g?.
b?¤á.=?§x3?3x+c = 03[0,1]S?1küó?.
13,e3[a,b]t|fprime(x)|greaterorequalslant|?prime(x)|,fprime(x) negationslash= 0§K|?f(x)|greaterorequalslant|(x)|.?y3
bracketleftbigg1
2,x
bracketrightbigg
t?arctanxlessorequalslant?ln(1 +x2)§d
dy23
bracketleftbigg1
2,1
bracketrightbigg
t±e? a¤áμarctanx?ln(1 +x2) greaterorequalslant pi4?ln2.
y2μ?3[a,b]t|fprime(x)|greaterorequalslant|?prime(x)|,fprime(x) negationslash= 0§f(x),?(x)3[a,b]t?§l
3[a,b]t?Y.
x,x+?x∈ [a,b],?x> 0§Kf(x),?(x)3[x,x+?x]t?Y??fprime(x) negationslash= 0.
d?ü?n§73ξ ∈ (x,x+?x)§|?(x+?x)(x)f(x+?x)?f(x) =?
prime(ξ)
fprime(ξ)=
(x)
f(x) =
prime(ξ)
fprime(ξ)§u′
vextendsinglevextendsingle
vextendsinglevextendsingle(x)
f(x)
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle?prime(ξ)
fprime(ξ)
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 1=|?f(x)|greaterorequalslant|(x)|.
(arctanx)prime = 11 +x2,(ln(1 + x2))prime = 2x1 +x2§?3
bracketleftbigg1
2,x
bracketrightbigg
t§k2x > 1§K(ln(1 + x2))prime = 2x1 +x2 >
1
1 +x2 = (arctanx)
prime > 0§f(x) = ln(1+x2),?(x) = arctanx§Kdt?(?§3
bracketleftbigg1
2,x
bracketrightbigg
t§?arctanx =
|?arctanx|lessorequalslant|?ln(1 +x2)| =?ln(1 +x2).
3
bracketleftbigg1
2,1
bracketrightbigg
t?x§3[x,1]tkarctan1? arctanx =?arctanx lessorequalslant?ln(1 +x2) = ln(1 + 12)? ln(1 +
x2)=pi4?arctanxlessorequalslant ln2?ln(1 +x2)§l
arctanx?ln(1 +x2) greaterorequalslant pi4?ln2.
14,ef(x)3?mX(d)￥?kk.ê§=|fprime(x)|lessorequalslantM§Kf(x)3X￥Y.
y2μ?ef(x)3?mXt?§l
3Xt?Y§?|fprime(x)|lessorequalslantM,M > 0
x1,x2 ∈X§x1 <x2§Kf(x)3[x1,x2]t?Y?.
d.?KF￥n§?ξ ∈ (x1,x2)§|f(x2)?f(x1) = fprime(ξ)(x2?x1)§K|f(x2)?f(x1)| = |fprime(ξ)|(x2?
x1) lessorequalslantM(x2?x1)§u′é?ε> 0§δ = εM§K|x2?x1|<δ = εM?§|f(x2)?f(x1)|lessorequalslantM(x2?x1) <
ε¤á§u′f(x)3X￥Y.
80
§2,Vúa
1,|x|?§íeCqúaμ
tanx≈x;cosx·sinx≈x; n√1±x≈ 1± xn;ex ≈ 1 +x.
y2μ
(1) -f(x) = tanx§?|x|§^Cqúax0 = 0§u′f(x0) = 0,fprime(0) = sec2xvextendsinglevextendsinglex=0 = 1§l
f(x) ≈f(0) +fprime(0)(x?0)=?tanx≈x.
(2) -f(x) = cosx·sinx§?|x|§^Cqúax0 = 0§u′f(x0) = 0,fprime(0) = (?sin2x+ cos2x)vextendsinglevextendsinglex=0 =
1§l
f(x) ≈f(0) +fprime(0)(x?0)=?cosx·sinx≈x.
(3) -f(x) = n√1±x§?|x|§^Cqúax0 = 0§u′f(x0) = 1,fprime(0) = ±1n(1±x) 1n?1
vextendsinglevextendsingle
vextendsinglevextendsingle
x=0
=
±1n§l
f(x) ≈f(0) +fprime(0)(x?0)=?n√1±x≈ 1± xn.
(4) -f(x) = ex§?|x|§^Cqúax0 = 0§u′f(x0) = 1,fprime(0) = ex|x=0 = 1§l
f(x) ≈f(0) +fprime(0)(x?0)=?ex ≈ 1 +x.
2,|tan4oCq?.
)μdtK§tanx≈x§tan4o = tan pi45 ≈ pi45 ≈ 0.0698.
3,|√37Cq?.
)μ?√37 = √36 + 1 = 6
radicalbigg
1 + 136§a11K§√37 = 6
radicalbigg
1 + 136 ≈ 6
parenleftbigg
1 + 172
parenrightbigg
≈ 6.083.
4,?5-5¤à?o§?oàR§2H§H'R.
(1) y2μ?otYD≈ H
2
2R?
(2) 2H = 50§R = 100§|D.
)μ
(1)?D = R?√R2?H2§KD = R
1?
radicalBigg
1?
parenleftbiggH
R
parenrightbigg2?
.
qH'R§
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
parenleftbiggH
R
parenrightbigg2vextendsinglevextendsinglevextendsingle
vextendsinglevextendsingle§K
radicalBigg
1?
parenleftbiggH
R
parenrightbigg2
≈ 1?12
parenleftbiggH
R
parenrightbigg2
= 1?H
2
2R2§l
D≈R
bracketleftbigg
1?
parenleftbigg
1? H
2
2R2
parenrightbiggbracketrightbigg
=
H2
2R.
(2) D = R?√R2?H2 = 100?√1002?252 ≈ 3.175;D≈ H
2
2R =
252
200 = 3.125
5,?
g30.12§?ù?
0.05.|
g ?èyé?
ú?é?
.
)μ?
èS = pi4D2§K|^êO?
§Skyé?
|?S| ≈
vextendsinglevextendsingle
vextendsinglepi2D?D
vextendsinglevextendsingle
vextendsingle = pi2 × 30.12 × 0.05 ≈
2.3656(2)é?
vextendsinglevextendsingle
vextendsinglevextendsingle?S
S
vextendsinglevextendsingle
vextendsinglevextendsingle≈
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
2D?Dpi
4D
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle2?D
D
vextendsinglevextendsingle
vextendsinglevextendsingle≈ 0.33%.
6,?7á￥ND = 10.12§?
D = 0.05.O?￥NNè9ùyé?
§?é?
.
)μ?￥NèV = pi6D3§￥NNèV = pi6(10.12)3 ≈ 542.675(3)?
|^ê?
O§Vkyé?
|?V|≈
vextendsinglevextendsingle
vextendsinglepi2D2?D
vextendsinglevextendsingle
vextendsingle= pi2 ×10.122 ×0.05 ≈ 8.044(3)?
é?
vextendsinglevextendsingle
vextendsinglevextendsingle?V
V
vextendsinglevextendsingle
vextendsinglevextendsingle≈
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
2D
2?D
pi
6D
3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle3?D
D
vextendsinglevextendsingle
vextendsinglevextendsingle≈ 1.48%.
7,|e?ê3x = 0:VDmaμ
(1) f(x) = √1 +x
(2) f(x) = 11 +x
(3) f(x) = esinx(Dm?1kx3?)
81
(4) f(x) = cosx
(5) f(x) = lncosx(Dm?1kx6?)
(6) f(x) = ln(1 +x)
)μ
(1) f(x) = √1 +x,fprime(x) = 12(1+x)?12,fprimeprime(x) =?14(1+x)?32,···,f(n)(x) = (?1)n?1 (2n?3)!!2n (1+x)12?n
rx = 0?g?\t?a§kf(0) = 1,fprime(0) = 12,fprimeprime(0) =?14,···,f(n)(0) = (?1)n?1 (2n?3)!!2n
u′?êf(x) = √1 +x3x = 0VDmaμf(x) = 1 + x2? x
2
8 +···+
(?1)n?1 (2n?3)!!2n
n! x
n +
o(xn) = 1 + x2? x
2
8 +···+
(?1)n?1(2n?3)!!
n!·2n +o(x
n)
(2) f(x) = 11 +x = (1+x)?1,fprime(x) =?(1+x)?2,fprimeprime(x) = 2(1+x)?3,···,f(n) = (?1)n·n!(1+x)?(n+1)
rx = 0?g?\t?a§kf(0) = 1,fprime(0) =?1,fprimeprime(0) = 2,···,f(n)(0) = (?1)n ·n!
u′?êf(x) = 11 +x3x = 0VDmaμf(x) = 1?x+x2?···+ (?1)nxn +o(xn)
(3) 5?sinx?x d.
Kesinx = 1+sinx+ 12! sin2x+ 13! sin3x+o1(sin3x) = 1+(x? x
3
3! +o(x
3))+ 1
2(x+o(x
2))2 + 1
6(x+
o(x2))3 +o1(sin3x) = 1 +x? x
3
6 +
x2
2 +
x3
6 +o(x
3) = 1 +x+ x2
2 +o(x
3).
(4) f(x) = cosx,fprime(x) =?sinx,fprimeprime(x) =?cosx,···,f(k)(x) = cos
parenleftbigg
x+ k2pi
parenrightbigg
rx = 0?g?\t?a§kf(0) = 1,fprime(0) = 0,fprimeprime(0) =?1,···,f(2m)(0) = (?1)m,f(2m+1)(0) =
0,···(m∈Z+)
u′?êf(x) = cosx3x = 0VDmaμf(x) = 1? x
2
2 +···+ (?1)
n x2n
(2n)! +o(x
2n)
(5) f(x) = lncosx = 12 ln(1?sin2x) =?12
parenleftbigg
sin2x+ sin
4x
2 +
sin6x
3 +o(sin
6x)
parenrightbigg
=
12
bracketleftBiggparenleftbigg
x? x
3
3! +
x5
5! +o1(x
5)
parenrightbigg2
+ 12
parenleftbigg
x? x
3
3! +o2(x
3)
parenrightbigg4
+ 13parenleftbigx+o3(x2))parenrightbig6 +o(x6)
bracketrightBigg
=?x
2
2?
x4
12?
x6
45 +o(x
6)
(6) f(x) = ln(1 +x),fprime(x) = 11 +x = (1 +x)?1,fprimeprime(x) =?(1 +x)?2,···,f(n)(x) = (?1)n?1!(1 +x)?n
rx = 0?g?\t?a§kf(0) = 0,fprime(0) = 1,fprimeprime(0) =?1,···,f(n)(0) = (?1)(n?1)!
u′?êf(x) = ln(1 +x)3x = 0VDmaμf(x) = x? x
2
2 +···+ (?1)
n?1xn
n +o(x
n)
8,|?êlnx3x = 1VDma.
)μdtK(?§lnx = ln(1 +x?1) = (x?1)? (x?1)
2
2 +···+ (?1)
n?1 (x?1)n
n +o((x?1)
n).
9,|?ê√x3x = 1VDma￡Dmx3?¤.
):f(x) = √x,fprime(x) = 12(1 +x)?12,fprimeprime(x) =?14(1 +x)?32,fprimeprimeprime(x) = 38(1 +x)?52
rx = 1?g?\t?a§kf(1) = 1,fprime(1) = 12,fprimeprime(1) =?14,fprimeprimeprime(1) = 38
u′?êf(x) = √x3x = 1VDmaμf(x) = 1 + 12(x?1)? 18(x?1)2 + 116(x?1)3 +o((x?1)3).
10,òaP3(x) = 1 + 3x+ 5x2?2x3L¤x+ 1ê?a.
)μ?P3(x) = 1 + 3x+ 5x2?2x3,Pprime3(x) = 3 + 10x?6x2,Pprimeprime3 (x) = 10?12x,Pprimeprimeprime3 (x) =?12,P(4)3 = ··· =
P(n)3 = 0
rx =?1?g?\t?a§kP3(?1) = 5,Pprime3(?1) =?13,Pprimeprime3 (?1) = 22,Pprimeprimeprime3 (?1) =?12,P(4)3 = ··· =
P(n)3 = 0
u′P3(x) = 5?13(x+ 1) + 11(x+ 1)2?2(x+ 1)3.
82
11,|^VúaO? 3√7?o?ê.
)μ 3√7 = 2
parenleftbigg
1? 18
parenrightbigg1
3 ≈ 2
bracketleftBigg
1 + 13
parenleftbigg
18
parenrightbigg
+ 12! · 13
parenleftbigg1
3?1
parenrightbiggparenleftbigg
18
parenrightbigg2
+ 13! · 13
parenleftbigg1
3?1
parenrightbiggparenleftbigg1
3?2
parenrightbiggparenleftbigg
18
parenrightbigg3bracketrightBigg
≈
1.9130
< 2· 14! · 13 · 23 · 53 · 83
parenleftbigg1
8
parenrightbigg4
≈ 2.01×10?5.
12,|^Vúa|e4?μ
(1) lim
x→0
cosx?e?x
2
2 +
1
12x
4
x6
(2) lim
x→0
ex sinx?x(1 +x)
x3
(3) lim
x→∞
bracketleftbigg
x?x2 ln
parenleftbigg
1 + 1x
parenrightbiggbracketrightbigg
(4) lim
x→0
parenleftbigg1
x?
1
sinx
parenrightbigg
(5) lim
x→+∞
( 6√x6 +x5? 6√x6?x5)
(6) lim
x→0
1?cos(sinx)
2ln(1 +x2)
)μ
(1) |^Vúa§kcosx = 1? x
2
2 +
x4
4!?
x6
6! +o(x
6),e?x22 = 1? x2
2 +
x4
8?
x6
48 +ox
6§
Kcosx?e?x
2
2 +
1
12x
4 = 7
360x
6 +o(x6)§u′lim
x→0
cosx?e?x
2
2 +
1
12x
4
x6 =
7
360.
(2) |^Vúa§kex = 1+x+x
2
2 +o(x
2),sinx = x?x3
3! +o(x
3)§Kex sinx?x(1+x) = x3
3 +o(x
3)§
u′lim
x→0
ex sinx?x(1 +x)
x3 =
1
3.
(3) |^Vúa§kln
parenleftbigg
1 + 1x
parenrightbigg
= 1x? 12x2 + 13x3 +o
parenleftbigg 1
x3
parenrightbigg
Kx?x2 ln
parenleftbigg
1 + 1x
parenrightbigg
= 12? 13x +o
parenleftbigg1
x
parenrightbigg
§
u′ lim
x→∞
bracketleftbigg
x?x2 ln
parenleftbigg
1 + 1x
parenrightbiggbracketrightbigg
= 12.
(4) |^Vúa§ksinx = x? x
3
3! + o(x
3)§K1
x?
1
sinx =
sinx?x
xsinx =
x
3
3! +o(x
3)
x
parenleftbigg
x? x
3
3! +o(x
3)
parenrightbigg =
x6 +o(x)
1? x
2
6 +o(x
2)
§u′lim
x→0
parenleftbigg1
x?
1
sinx
parenrightbigg
= 0
(5)? 6√x6 +x5? 6√x6?x5 = x
parenleftbigg
1 + 1x
parenrightbigg1
6?x
parenleftbigg
1? 1x
parenrightbigg1
6
|^Vúa§k
parenleftbigg
1 + 1x
parenrightbigg1
6 = 1 + 1
6x?
5
72x2 +o
parenleftbigg 1
x2
parenrightbigg
,
parenleftbigg
1? 1x
parenrightbigg1
6 = 1? 1
6x?
5
72x2 +o
parenleftbigg 1
x2
parenrightbigg
§
K 6√x6 +x5? 6√x6?x5 = 13 +o
parenleftbigg1
x
parenrightbigg
§u′ lim
x→+∞
( 6√x6 +x5? 6√x6?x5) = 13.
(6) |^Vúa§kcos(sinx) = 1? sin
2x
2 +
sin4x
4! + o(sin
4x),ln(1 + x2) = x2? x4
2 + o(x
4)§
K1?cos(sinx)2ln(1 +x2) =
sin2x
bracketleftbigg
1? 112 sin2x+o(sin2x)
bracketrightbigg
4x2
parenleftbigg
1? 12x2 +o(x2)
parenrightbigg §l
lim
x→0
1?cos(sinx)
2ln(1 +x2) =
1
4
83
13,α,β§| lim
x→+∞
( 4√16x4?8x3 + 10x?7?αx?β) = 0.
)μ? 4√16x4?8x3 + 10x?7 = 2x· 4
radicalBigg
1 +
parenleftbigg
12x + 58x3? 716x4
parenrightbigg
= 2x?14+ 516x2? 732x3 +ε( lim
x→+∞
ε = 0)
4√16x4?8x3 + 10x?7?αx?β = (2?α)x?
parenleftbigg1
4 +β
parenrightbigg
+ 516x2? 732x3 +ε
dd?§?| lim
x→+∞
( 4√16x4?8x3 + 10x?7?αx?β) = lim
x→+∞
bracketleftbigg
(2?α)x?
parenleftbigg1
4 +β
parenrightbigg
+ 516x2? 732x3 +ε
bracketrightbigg
=
0§7Lα = 2,β =?14.
14,A§|4?lim
x→0
nradicalbigQ(x)?A
x 3§ù￥Q(x) = a0 +a1x+···+amx
m,a0 negationslash= 0,m?g,ê.
)μlim
x→0
nradicalbigQ(x)?A
x = limx→0
n√a0 +a1x+···+amxm?A
x = limx→0
n√a0
parenleftbigg
n
radicalbigg
1 + a1a
0
x+···+ ama
0
xm?A
parenrightbigg
x =
lim
x→0
n√a0
parenleftbigg
1 + 1n
parenleftbigga
1
a0x+···+
am
a0 x
m
parenrightbigg
+o(x)?A
parenrightbigg
x 3
n√a0?A = 0=A = n√a0§d?a= a1 ·
n√a0
na0,
84
§3,?ê,ü!à5?4?
1,y2e?êüN5μ
(1) y = x?sinx
(2) y =
parenleftbigg
1 + 1x
parenrightbiggx
(x> 0)
y2μ
(1)?y = f(x)3(?∞,+∞)S?Y?§fprime(x) = 1? cosx?q?1 lessorequalslant cosx lessorequalslant 1§fprime(x) greaterorequalslant 0§u
′y = x?sinx3(?∞,+∞)üNt,.
(2)?y =
parenleftbigg
1 + 1x
parenrightbiggx
§yprime =
parenleftbigg
1 + 1x
parenrightbiggxbracketleftbigg
ln(1 +x)?lnx? 11 +x
bracketrightbigg
qx> 0§
parenleftbigg
1 + 1x
parenrightbiggx
> 0§K?I?)ò￥af?ò.
-f(x) = lnx3[x,1+x](é?x> 0)tA^.?KF?n§kln(1+x)?lnx = 1ξ(1+x?x) = 1ξ(x<ξ<
1+x)§u′1x > 1ξ > 11 +x§ln(1+x)?lnx = 1ξ > 11 +x=ln(1+x)?lnx? 11 +x > 0(?x> 0)§
dd?yprime > 0§l
y =
parenleftbigg
1 + 1x
parenrightbiggx
3(0,+∞)tüNO\.
2,üN?êê′?7?üNo
)μ.
~μy = x33(?∞,+∞)tüNt,§yprime = 3x2%?üN.
3,y2e? aμ
(1) x> sinx> 2pix
parenleftBig
0 <x< pi2
parenrightBig
(2) x? x
3
6 > sinx>x(x< 0)
(3) x? x
2
2 < ln(1 +x) <x(x> 0)
(4) tanx>x+ x
3
3
parenleftBig
0 <x< pi2
parenrightBig
(5) 2√x> 3? 1x(x> 1)
(6) 12p?1 lessorequalslantxp + (1?x)p lessorequalslant 1(0 lessorequalslantxlessorequalslant 1,p> 1)
y2μ
(1) f(x) = x?sinx§d11K§f(x)3
parenleftBig
0,pi2
parenrightBig
SüNt,§qf(0) = 0§é?x∈
parenleftBig
0,pi2
parenrightBig
§kf(x) >
f(0) = 0=x?sinx> 0§l
x> sinx
parenleftBig
0 <x< pi2
parenrightBig
g(x) = sinxx,g
parenleftBigpi
2
parenrightBig
= 2pi,gprime(x) = xcosx?sinxx2
parenleftBig
0 <x< pi2
parenrightBig
5?u(x) = xcosx? sinx
parenleftBig
0 <x< pi2
parenrightBig
u(0) = 0§duuprime(x) =?xsinx < 0
parenleftBig
0 <x< pi2
parenrightBig
§
x ∈
parenleftBig
0,pi2
parenrightBig
§u(x)üNeü=u(x) < u(0) = 0
parenleftBig
0 <x< pi2
parenrightBig
§dd§gprime(x) < 0
parenleftBig
0 <x< pi2
parenrightBig
§
g(x)3
parenleftBig
0,pi2
parenrightBig
tüNeü§u′g(x) >g
parenleftBigpi
2
parenrightBig
= 2pi=sinxx > 2pi,x∈
parenleftBig
0 <x< pi2
parenrightBig
§Ksinx> 2pix§
l
x> sinx> 2pix
parenleftBig
0 <x< pi2
parenrightBig
(2) f(x) = x? sinx§d11K§f(x)3(?∞,0)SüNt,§qf(0) = 0§é?x ∈ (?∞,0)§
kf(x) <f(0) = 0=x?sinx> 0§l
x< sinx(x< 0)?
g(x) = x? x
3
6?sinx,g(0) = 0,g
prime(x) = 1? x2
2?cosx
2h(x) = 1?x
2
2?cosx(x< 0)?h(0) = 0§duh
prime(x) =?x+sinx> 0§x∈ (?∞,0)?§h(x)ü
Nt,=h(x) < h(0) = 0(x < 0)§dd§gprime(x) < 0(x < 0)§g(x)3(?∞,0)tüNeü§u
′g(x) >g(0) = 0=x? x
3
6?sinx> 0(x< 0)§Kx?
x3
6 > sinx§l
x?
x3
6 > sinx>x(x< 0)
85
(3) f(x) = ln(1+x)?x,g(x) = ln(1+x)?x+x
2
2 (x> 0)§f
prime(x) = 1
1 +x?1 =?
x
1 +x < 0(x> 0)§
Kf(x)3(0,+∞)SüNeü§qf(0) = 0§é?x> 0§kf(x) <f(0) = 0=ln(1 +x) <x(x> 0)?
gprime(x) = 11 +x?1 +x = x
2
1 +x > 0(x> 0)
g(x)3(0,+∞)tüNt,§qg(0) = 0§u′g(x) > g(0) = 0=ln(1 + x) > x? x
2
2 (x > 0)§l
x? x
2
2 < ln(1 +x) <x(x> 0)
(4) f(x) = tanx?x?x
3
3
parenleftBig
0 <x< pi2
parenrightBig
§fprime(x) = sec2x?1?x2 = tan2x?x2 = (tanx+x)(tanx?x)§
q?(tanx?x)prime = sec2x? 1 = tan2x lessorequalslant 0
parenleftBig
x∈
parenleftBig
0,pi2
parenrightBigparenrightBig
§Ktanx?x3
parenleftBig
0,pi2
parenrightBig
SüNt,§
é?x∈
parenleftBig
0,pi2
parenrightBig
§ktanx?x> 0
parenleftBig
0 <x< pi2
parenrightBig
u′fprime(x) = (tanx+x)(tanx?x) > 0
parenleftBig
0 <x< pi2
parenrightBig
§dd?§f(x)3
parenleftBig
0,pi2
parenrightBig
tüNt,§qf(0) =
0§u′f(x) >f(0) = 0=tanx?x? x
3
3 > 0
parenleftBig
0 <x< pi2
parenrightBig
§l
tanx>x+ x
3
3
parenleftBig
0 <x< pi2
parenrightBig
(5) f(x) = 2√x?3 + 1x(x> 1)§fprime(x) = 1√x? 1x2 = x
3
2?1
x2 > 0(x> 1)§u′f(x)3(1,+∞)tü
Nt,§qf(1) = 0§u′f(x) >f(1) = 0=2√x?3 + 1x > 0(x> 1)§l
12p?1 lessorequalslant xp + (1?x)p lessorequalslant
1(0 lessorequalslantxlessorequalslant 1,p> 1)
(6) f(x) = xp + (1?x)p(0 lessorequalslantxlessorequalslant 1,p> 1)§fprime(x) = pxp?1?p(1?x)p?1§
-fprime(x) = pxp?1?p(1?x)p?1 = 0§)x = 12§'f(0) = 1,f(1) = 1,f
parenleftbigg1
2
parenrightbigg
= 12p?1§dd
min
0lessorequalslantxlessorequalslant1
f(x) = 12p?1,max
0lessorequalslantxlessorequalslant1
f(x) = 1 §l
12p?1 lessorequalslantxp + (1?x)p lessorequalslant 1(0 lessorequalslantxlessorequalslant 1,p> 1)
4,(?e?êt,!eü?mμ
(1) y = x3?6x
(2) y = 2x3?3x2?12x+ 1
(3) y = x4?2x3
(4) y = x+ sinx
(5) y = 2x1 +x2
(6) y = 2x2?sinx
(7) y = xne?x(n> 0,xlessorequalslant 0)
)μ
(1)?yprime = 3x2?6 = 3(x2?2)§7:x = ±√2
x<?√2?x>√2?§yprime > 0§?êt,??√2 <x<√2?§yprime < 0§?êeü.
l
3?m(?∞,?√2)uniontext(√2,+∞)t?êt,?3?m(?√2,√2)t?êeü.
(2)?yprime = 6x2?6x?12 = 6(x2?x?2) = 6(x?2)(x+ 1)§7:x =?1,x = 2
x<?1?x> 2?§yprime > 0§?êt,??1 <x< 2?§yprime < 0§?êeü.
l
3?m(?∞,?1)uniontext(2,+∞)t?êt,?3?m(?1,2)t?êeü.
(3)?yprime = 4x3?6x2 = 2x2(2x?3)§7:x = 0,x = 32
x> 32?§yprime > 0§?êt,?x< 32?§yprime lessorequalslant 0?=3x = 0?yprime = 0§?êeü.
l
3?m
parenleftbigg3
2,+∞
parenrightbigg
t?êt,?3?m
parenleftbigg
∞,32
parenrightbigg
t?êeü.
(4)?yprime = 1 + cosxlessorequalslant 0§?ê3(?∞,+∞)t?êt,.
(5)?yprime = 2(1?x
2)
(1 +x2)2§7:x = ±1
x<?1?x> 1?§yprime < 0§?êeü??1 <x< 1?§yprime > 0§?êt,.
l
3?m(?∞,?1)uniontext(1,+∞)t?êeü?3?m(?1,1)t?êt,.
86
(6)?yprime = 4x?cosx,yprimeprime = 4 + sinx> 0§Kyprime3(?∞,+∞)tüNt,.
qyprime(0) =?1,yprime
parenleftBigpi
2
parenrightBig
= 2pi§K3
parenleftBig
0,pi2
parenrightBig
Sk:x0÷vyprime(x0) = 0=4x0 = cosx0
x>x0?§yprime > 0§?êt,?x<x0?§yprime < 0§?êeü.
l
3?m(x0,+∞)t?êt,?3?m(?∞,x0)t?êeü.
(7)?yprime = nxn?1e?x?xne?x = xn?1e?x(n?x)
n> 0,x> 0§xn?1 > 0,e?x > 0§Kxn?1e?x > 0
0 <x<n?§yprime > 0§?êt,?x>n?§yprime < 0§?êeü.
l
3?m(0,n)t?êt,?3?m(n,+∞)t?êeü.
5,|e?ê4?μ
(1) y = x?ln(1 +x)
(2) y = √xlnx
(3) y = x+ 1x
(4) y = sin3x+ cos3x
(5) y = cosx+ coshx
)μ
(1)?yprime = 1? 11 +x = x1 +x,yprimeprime = 1(1 +x)2 > 0
d?ê(?1,+∞)§K7:?x = 0§?ê?U3ù:k4?§u′x = 0′?ê4:§4
y = 0.
(2)?yprime = 1√x + 12√x lnx = 12√x(lnx+ 2),yprimeprime =? 1
2x32
+ 1
2x32
1
4x32
lnx =?lnx
232
7:?x = e?2§?ê?U3ù:k4?§qyprimeprime|x=e?2 > 0§u′x = e?2′?ê4:§4?
y =?2e.
(3)?yprime = 1? 1x2,yprimeprime =? 1x3 > 0
d?ê(?∞,0)uniontext(0,+∞)§K7:?x = ±1§?ê?U3ùü:k4?§qyprimeprime|x=1 = 1 >
0,yprimeprime|x=?1 =?1 < 0§u′x = 1′?ê4:§4y = 2?x =?1′?ê4?:§4
y = 2.
(4)?yprime = 3sinxcosx(sinx?cosx) = 32 sin2x(sinx?cosx),yprimeprime = 3cos2x(sinx?cosx)+ 32 sin2x(cosx+
sinx)
7:?x = kpi + pi4,x = kpi2 (k ∈ Z)§qyprimeprime|x=2kpi =?3 < 0,yprimeprime|x=2kpi+pi4 = 32√2 > 0,yprimeprime|x=2kpi+pi2 =
3 < 0,yprimeprime|2kpi+pi = 3 > 0,yprimeprime|x=2kpi+5pi
4
=?32√2 < 0,yprimeprime|x=2kpi+3pi
2
= 3 > 0§
u′x = 2kpi?§k4y = 1?x = 2kpi + pi2?§k4y = 1?x = 2kpi + 5pi4?§k4?
y =?
√2
2?
x = 2kpi + pi4?§k4?y =
√2
2?x = 2kpi + pi?§k4?y =?1?x = 2kpi +
3pi
2?§k4
y =?1.
(5)?yprime = 1?sinx+ sinhx§?′|7:§d?sinx+ e
x?e?x
2 = 0′?x = 0′7:
d?sinx,sinhx3
parenleftBig
pi2,pi2
parenrightBig
üN5§ù7:′.
yprimeprime =?cosx+coshx,yprimeprime(0) = 0;yprimeprimeprime = sinx+sinhx,yprimeprimeprime(0) = 0;y(4) = cosx+coshx,y(4)(0) = 2 > 0§
u′x = 0′?ê4:§4y = 2.
6,ef(x)3:x0?k?n ?Yê§fprime(x0) = fprimeprime(x0) = ··· = f(n?1)(x0) = 0,f(n)(x0) negationslash= 0§@
onê?§f(x0)?4n?óê
f(n)(x0) > 0?§f(x0)?4n?óê
f(n)(x0) <
0?§f(x0)?4.
y2μòf(x)3x = x0:^VúaDmμf(x) = f(x0) + fprime(x0)(x? x0) + f
primeprime(x
0)
2 (x? x0)
2 + ··· +
f(n)(x0)
n! (x?x0)
n +o((x?x
0)
n)
fprime(x0) = fprimeprime(x0) = ··· = f(n?1)(x0) = 0§f(x) = f(x0) + f
(n)(x
0)
n! (x?x0)
n +o((x?x
0)
n)
87
x→x0?§o((x?x0)n) → 0§xCx0?§=|x?x0|?§f(x)?f(x0)?f
(n)(x
0)
n! (x?
x0)nk?ó?ò
ef(n)(x0) > 0§
(1) nê?§ex>x0§K(x?x0)n > 0§u′f
(n)(x
0)
n! (x?x0)
n > 0§l
f(x)?f(x
0) > 0=f(x) >
f(x0)?
ex<x0§K(x?x0)n < 0§u′f
(n)(x
0)
n! (x?x0)
n < 0§l
f(x)?f(x
0) < 0=f(x) <f(x0)
df(x0)?′4?.
(2) n?óê?§xCx0§x > x0§?′x < x0§?k(x?x0)n > 0§d?f
(n)(x
0)
n! (x?
x0)n > 0(xnegationslash= x0)§l
f(x)?f(x0) > 0§=3x0,?S§ekf(x) >f(x0)§ùL2f(x0)′
4?.
ef(n)(x0) < 0§
(1) nê?§ex>x0§K(x?x0)n > 0§u′f
(n)(x
0)
n! (x?x0)
n < 0§l
f(x)?f(x
0) < 0=f(x) <
f(x0)?
ex<x0§K(x?x0)n < 0§u′f
(n)(x
0)
n! (x?x0)
n > 0§l
f(x)?f(x
0) > 0=f(x) >f(x0)
df(x0)?′4?.
(2) n?óê?§xCx0§x > x0§?′x < x0§?k(x?x0)n > 0§d?f
(n)(x
0)
n! (x?
x0)n < 0(xnegationslash= x0)§l
f(x)?f(x0) < 0§=3x0,?S§ekf(x) <f(x0)§ùL2f(x0)′
4.
7,|e?ê3mtú??μ
(1) y = |x2?3x+ 2|,[?10,10]
(2) y = e|x?3|,[?5,5]
)μ
(1) y =
(x?2)(x?1),xlessorequalslant 1
(x?2)(x?2),1 <xlessorequalslant 2
(x?2)(x?1) x> 2
§
|§yprime =
2x?3,x< 1
3,x = 1
2x+ 3,?1 <x< 2
3,x = 2
2x?3,x> 2
§K7:x = 32§ê?3:x = 1,x = 2
qy(?10) = 132,y(1) = 0,y
parenleftbigg3
2
parenrightbigg
= 54,y(2) = 0,y(10) = 72§?ê′132§??′0.
(2) y =
braceleftbigg ex?3,xgreaterorequalslant 3
e3?x,x< 3 §|§y
prime =
ex?3,x> 3
3,x = 3
e3?x,x< 3
§w,?7:
qy(?5) = e8,y(3) = 1,y(5) = e2§?êe8§?1.
8,c′tAB??l?100úp§ó?C?A40úp§ACR?uAB.8?3AB￥m?:D?ó?C^
ú′￡?5-21¤§|l?A?B$àó?C¤^$¤.ˉD:AT3o?z?úpc′$
¤?ú′$¤?'′3:5.
)μ|AD| = xúp§K|DB| = 100?xúp?zúpc′$¤?3t§Kzúpú′$¤?5t§o$¤
yt
Kyt = √x2 + 1600(5t)+(100?x)(3t)=y = 5√x2 + 1600+3(100?x)§u′yprime = 5x?3
√x2 + 1600
√x2 + 1600,yprimeprime =
8000
(x2 + 1600)32
> 0§7:?x = 30§?x = 30?4:§D:A3?A30úp?.
9,r
7?¤Y/7^.ˉY/ú°§ ?èo
)μ
7 R§Y/!°?O?x,y§Kradicalbigx2 +y2 = 2R§u′y = √4R2?x2§l
S =
88
xy = x√4R2?x2
KSprime = 4R
2?2x2
√4R2?x2,Sprimeprime = 2x
3?12R2x
(4R2?x2)32
§7:?x = √2R§d?Sprimeprime < 0§Kx = √2R?4?:§d
x = y = √2R§Y/!°t√2R?§ ?è.
10,S = (x?a1)2 + (x?a2)2 +···+ (x?an)2.ˉx§S?o
)μSprime = 2[nx? (a1 + a2 + ··· + an)],Sprimeprime = 2n > 0§7:?x = a1 +a2 +···+ann §?x?4:§=
x = a1 +a2 +···+ann?§S?.
11,
/G?§?ùNè?V§üà?ázüèda§y?ázü?db§
ˉG?úp' u??§Edo
)μd
/G?D§p?H§KV = 14piD2H§u′H = 4VpiD2
EdG = 2a
parenleftBigpi
4D
2
parenrightBig
+bpiDH = pi2aD2+b4VD §KGprime = piaD?4bVD2 §7:?D = 3
radicalbigg4bV
api,D<
3
radicalbigg4bV
api?§G
prime <
0?D> 3
radicalbigg4bV
api?§G
prime > 0§KD = 3
radicalbigg4bV
api ′4:§l
′?:.
u′DH = D4V
piD2
= piD
3
4V =
b
a=G?p'?
b
a?§Ed.
12,^R
/c?§}
%?α
÷/?¤|ì.ˉα§|ìNèo
)μdK§{eü?
%?x = 2pi?α§|ì.±?Rx = R(2pi?α)§.Rx2pi§ùp
h =
radicalBigg
R2?
parenleftbiggRx
2pi
parenrightbigg2
= R2pi
radicalbig
4pi2?x2(x > 0)§u′|ìNè?V = 13pi
parenleftbiggRx
2pi
parenrightbigg2
· R2pi
radicalbig
4pi2?x2 =
R3
24pi2x
2radicalbig4pi2?x2(x> 0)
UK§?I?x§?êf(x) = x4(4pi2?x2).
fprime(x) = 16pi2x3?6x5,fprimeprime(x) = 48pi2x2?30x4§7:?x = 2pi
radicalbigg2
3§?f
primeprime
parenleftBigg
2pi
radicalbigg2
3
parenrightBigg
< 0§x = 2pi
radicalbigg2
3?
4?:§?
}
%A?α = 2pi
parenleftBigg
1?
radicalbigg2
3
parenrightBigg
§¤?|ìNè.
13.,?a§p?hn/§á|ùSY/?è.
)μùSY/!°?O?b,c
Kd?§ba = h?ch =b = h?ch a§u′S = bc = ach?ch = ahc?ac
2
h §KS
prime = ah?2ac
h,S
primeprime =
2ah < 0§7:?c = h2§u′c = h2?4?:§d?b = a2§l
è?S = bc = ah4,
14,?l§ár§ü?§|±ùü>¤?¤Y/?è.
)μdY/?x§K°?l?x
S = x(l?x) = lx?x2§KSprime = l?2x,Sprimeprime =?2 < 0§7:?x = l2§?x = l2?4?:§?dx = l2?§
Y/?�S = l
2
4"
15,Suy
x
2
a2 +
y2
b2 = 1§
>21u?Y/.
)μd?¤|Y/?xu(x,0)§Kù?yu
parenleftbigg
0,ba
radicalbig
a2?x2
parenrightbigg
dY/?è?S§K14S = x· ba
radicalbig
a2?x2§l
S = 4ba
radicalbig
a2?x2§
Ksprime = 4ba · a
2?2x2
√a2?x2,Sprimeprime = 4ba · 2x
3?3a2x
(a2?x2)32
§7:?x =
√2
2 a§d?S
primeprime < 0§Kx =
√2
2 a?S4:§
u′x =
√2
2 a?Y/?è§è?S = 2ab.
16,|:M(p,p)y2 = 2px?á?l.
)μ:M(p,p)y2 = 2pxt:(x,y)?l?d =radicalbig(x?p)2 + (y?p)2 =
radicalBiggparenleftbigg
y2
2p?p
parenrightbigg2
+ (y?p)2 =
radicalBigg
y4
4p2 + 2p
2?2py
89
f(y) = y
4
4p2 + 2p
2?2py§Kfprime(y) = 1
p2 (y
3?2p3),fprimeprime(y) = 3y2
p2 > 0§7:?y =
3√2p§?§ò′f(y)4
?:§?d¤|?á?l?d =
radicalBig
f( 3√2p) = |p|
radicalBig
2 + 2?23?234,
17,`E±u = 20?/??Y?àê1§ì?3ù?h = 82kˉE±v = 16?/??Y?H
ê1§ˉüE?l?Co
):x? üE?l?C§üES?§KS =radicalbig(82?16x)2 + (20x)2 = √656x2?2624x+ 6724
-f(x) = 656x2?2624x+ 6724§|ù??"Kfprime(x) = 1312x?2624,fprimeprime(x) = 1312 > 0§7:?x = 2?
§?f(x)4?:§K2? üE?l?C§d?S = 10√41.
18,2/t-?§-t?Pú6.??N?/? TXê?μ"y\F§|?Nm?￡?.ˉdY2
Y§^o￡?5-22¤o
)μaK§kF cos? = μ(PG?F sin?)=F = μPGcos?+μsin?
-y = cos?+μsin?§?|F?§|y
dyprime =?sin?+μcos?,yprimeprime =?cosμsin?§7: = arctanμ§d?yprimeprime < 0§L2? = arctanμ?§y
§l
F??§=^.
19,X?5-23¤?§k`!ˉü)èü^?C?ì§ˉC?ìMA3§¤^?>o
)μMA3?`Rl?xúp?§¤^?>?l
d?§l = √1 +x2 + radicalbig2.252 + (3?x)2§Klprime = x√1 +x2 + x?3radicalbig2.252 + (3?x)2,lprimeprime = 1
(1 +x2)32
+
2.25
(2.252 + (3?x)2)32
> 0§7:?x = 1.2§?:§=x = 1.2úp?§¤^?>.
20,y
x
2
a2 +
y2
b2 = 1ü?IOuA,Bü:§
(1) |ABü:m?l?
(2) |?OAB??è.
)μ?:?(x,y)§Kk =?b
2x
a2y§u′§?Y?y =?
b2x
a2y(X?x)§
5§?:M(x.y)31§3ü?I?t O?a
2
x,
b2
y§K
(1) ¤|ABü:m?l?d =
radicalBigg
a4
x2 +
b4
y2 = a
radicalbigg
a2
x2 +
b2
a2?x2
-f(x) = a
2
x2 +
b2
a2?x2§?|d??§?I|f(x)??.
dfprime(x) =?2a
2
x3 +
2b2x
(a2?x2)2,f
primeprime(x) = 6a2
x4 +
2a2b2 + 6b2x2
(a2?x2)3 > 0§?dux ∈ [0,a],x
2 lessorequalslant a2§K7:
÷vx2 = a
3
a+b?d?f(x)??§=d??§?á?l?d = a
radicalbigf(x) = a+b.
(2) UK§kS = 12 · a
2
x ·
ab√
a2?x2 =
a3b
2x√a2?x2§êg(x) = x
2(a2?x2)
|S??§|g(x)
dgprime(x) = 2a2x?4x3,gprimeprime(x) = 2a2?12x2§7:?x = a√2?d?gprimeprime(x) < 0§=x = a√2?g(x)?
§l
S??§??è?S = ab.
21,êxα(α> 190 <α< 1),ex,lnx,xlnx3(0,+∞)Sà5.
)μf(x) = xα,fprime(x) = αxα?1,fprimeprime(x) = α(α?1)xα?2
α> 1?§fprimeprime(x) > 0§Kxα3(0,+∞)Seà?0 <α< 1?§fprimeprime(x) < 0§Kxα3(0,+∞)Stà.
f(x) = ex,fprime(x) = ex,fprimeprime(x) = ex > 0(x> 0)§Kex3(0,+∞)Seà
f(x) = lnx,fprime(x) = 1x,fprimeprime(x) =? 1x2 < 0(x> 0)§Klnx3(0,+∞)Stà
f(x) = xlnx,fprime(x) = 1 + lnx,fprimeprime(x) = 1x > 0(x> 0)§Kxlnx3(0,+∞)Seà
22,e?êà5ú$:μ
(1) y = 3x2?x3
(2) y = a
2
a2 +x2 (a> 0)
90
(3) y = x+ sinx
(4) y = √1 +x2
)μ
(1) yprime = 6x?3x2,yprimeprime = 6?6x§yprimeprime = 0x = 1§LXeμ
x (?∞,1) (1,+∞)
yprimeprime?ò + -
y eà tà
§$:
I?(1,2)
(2) yprime =? 2ax(a2 +x2)2,yprimeprime = 2a
2(3x2?a2)
(a2 +x2)3 §y
primeprime = 0x = ±
√3
3 a§LXeμ
x
parenleftbigg
∞,?
√3
3 a
parenrightbigg parenleftbigg
√3
3 a,
√3
3 a
parenrightbigg parenleftbigg√3
3 a,+∞
parenrightbigg
yprimeprime?ò + - +
y eà tà eà
§$:?I?
parenleftbigg
√3
3 a,
3
4
parenrightbigg
,
parenleftbigg√3
3 a,
3
4
parenrightbigg
(3) yprime = 1 + cosx,yprimeprime =?sinx§yprimeprime = 0x = kpi(k∈Z)§LXeμ
x ((2k?1)pi,2kpi) (2kpi,(2k+ 1)pi) ((2k+ 1)pi,2(k+ 1)pi)
yprimeprime?ò + - +
y eà tà eà
§$:?I?(kpi,kpi)
(4) yprime = x√1 +x2,yprimeprime = 1
(1 +x2)32
§Kyprimeprime > 0§?ê′eà§l
$:.
23,y2-?y = x+ 1x2 + 1k?uótn?$:.
y2μyprime = 1?2x?x
2
(x2 + 1)2,y
primeprime = 2(x?1)(x+ 2?
√3)(x+ 2 +√3)
(x2 + 1)3
-yprimeprime = 0§x1 = 1,x2 =?2 +√3,x3 =?2?√3
x <?2?√3?§yprimeprime < 0??2?√3 < x <?2 +√3?§yprimeprime > 0??2 +√3 < x <?1?§yprimeprime < 0?
x>?1?§yprimeprime > 0
u′-?3x1,x2,x3?kn?$:A(1,1),B
parenleftbigg
2 +√3,
√3 + 1
4
parenrightbigg
,C
parenleftbigg
(2 +√3),1?
√3
4
parenrightbigg
LA,B¤?y = 14x+ 34§òC:?I?\t§§1?
√3
4 =
2?√3
4 +
3
4 =
1?√3
4 =C÷v
d?§§K-?y = x+ 1x2 + 1k?uótn?$:.
24,ef(x)′eà?ê￡eà?ꤧfprime(x0)3§K
f(x) greaterorequalslantf(x0) +fprime(x0)(x?x0)f(x) >f(x
0) +f
prime(x
0)(x?x0)
bracerightbigg
(xnegationslash= x0).
y2μx?f(x)S:§xnegationslash= x0￡x>x0¤
-x1 = x+x02 §df(x)?eà?ê§Kf(x)?f(x0)x?x
0
greaterorequalslant f(x1)?f(x0)x
1?x0
x2 = x1 +x02 §df(x)?eà?
ê§Kf(x1)?f(x0)x?x
0
greaterorequalslant f(x2)?f(x0)x
2?x0
x3 = x2 +x02 §df(x)?eà?ê§Kf(x2)?f(x0)x?x
0
greaterorequalslant f(x3)?f(x0)x
3?x0
Xd?1e§?ê{xn}§|xn?x0| = |x?x0|2n → 0(n→∞)§Kxn →x0(n→∞)§?f(xn)?f(x0)x
n?x0
greaterorequalslant
f(xn+1)?f(x0)
xn+1?x0
qfprime(x0)3§K lim
n→∞
f(xn)?f(x0)
xn?x0 = limxn→x0
f(xn)?f(x0)
xn?x0 = f
prime(x
0)
qf(x)?f(x0)x?x
0
greaterorequalslant f(xn)?f(x0)x
n?x0
§Kd4?5?§f(x)?f(x0)x?x
0
greaterorequalslantfprime(x0)§l
f(x) greaterorequalslantf(x0)+fprime(x0)(x?
x0)
ón?y§ef(x)′eà?ê§Kf(x) >f(x0) +fprime(x0)(x?x0).
25,ef(x)′eà?ê§K?f(x)′tà?ê.
y2μ?f(x)′eà?ê§Kf(x)3[a,b]t?Y§é[a,b]￥ü:x1,x2§ekf
parenleftBigx1 +x2
2
parenrightBig
lessorequalslant f(x1) +f(x2)2 §
u′?f
parenleftBigx1 +x2
2
parenrightBig
greaterorequalslant?f(x1) +f(x2)2 = (?f(x1)) + (?f(x2))2 §l
f(x)′tà?ê.
26,(1) efn(x)′eà?ê§ˉF(x) = min
n
{fn(x)}′?′eà?êo
91
(2) ef(x),g(x)′eà?ê§ˉf(x) +g(x)′?′eà?êo
(3) `2ng?ê?′eà?ê.
)μ
(1).
~μ
f1(x) = 1x,f2(x) = x2(x> 0)?§f1(x),f2(x)?′eà?ê§F(x) = min
braceleftbigg1
x,x
2
bracerightbigg
3(1,1):?÷
veà?ê§=F(x)?′eà?ê.
f1(x) = x2,f2(x) = x
2
2?§f1(x),f2(x)?′eà?ê§?F(x) = min
braceleftbigg
x2,x
2
2
bracerightbigg
= x
2
2 ′eà?ê.
(2) f(x) +g(x)′eà?ê.
f(x),g(x)′eà?ê§Kf
parenleftBigx1 +x2
2
parenrightBig
lessorequalslant f(x1) +f(x2)2,
g
parenleftBigx1 +x2
2
parenrightBig
lessorequalslant g(x1) +g(x2)2 §u′(f+g)
parenleftBigx1 +x2
2
parenrightBig
= f
parenleftBigx1 +x2
2
parenrightBig
+g
parenleftBigx1 +x2
2
parenrightBig
lessorequalslant f(x1) +f(x2)2 +
g(x1) +g(x2)
2 =
1
2[(f +g)(x1) + (f +g)(x2)]=f(x) +g(x)′eà?ê.
(3) f(x) = ax3 +bx2 +cx+d(anegationslash= 0)§Kfprime(x) = 3ax2 + 2bx+c,fprimeprime(x) = 6ax+ 2b
u′§
a> 0?§x>? b3a?§fprimeprime(x) > 0§f(x)′eà?ê?x<? b3a?§fprimeprime(x) < 0§f(x)′tà?ê
a< 0?§x>? b3a?§fprimeprime(x) < 0§f(x)′tà?ê?x<? b3a?§fprimeprime(x) > 0§f(x)′eà?ê
Kf(x)?′eà?ê§3x =? b3a?k$:.
27,X?àJ?êh> 0§?U|-?y = h√pie?h2x23x = ±σ(σ> 0,σ~ê)?k$:.
)μyprime =?2h
3
√pixe?h2x2,yprimeprime = 2h
3
√pie?h2x2(2h2x2?1)
-yprimeprime = 0§Kx = ± 1√2h
x <? 1√2h?§yprimeprime > 0§-?eà?? 1√2h < x < 1√2h?§yprimeprime < 0§-?tà?x > 1√2h?§yprimeprime >
0§-?eà
K3x = ± 1√2h?kü?$:§u′± 1√2h = ±σ§qh,σ> 0§Kh = 1√2σ.
28,|y = x
2
x2 + 14?9$:§?|$:?§.
)μyprime = 2x(1 +x2)2,yprimeprime = 2?6x
2
(1 +x2)3
7:?x = 0§?eLμ
x (?∞,0) 0 (0,+∞)
yprime?ò - 0 +
y arrowsoutheast 4?0 arrownortheast
Kx = 0§yk4?y = 0.
-yprimeprime = 0§Kx = ±
√3
3 §?eLμ
x
parenleftbigg
∞,?
√3
3
parenrightbigg parenleftbigg
√3
3,
√3
3
parenrightbigg parenleftbigg√3
3,+∞
parenrightbigg
yprimeprime?ò - + -
y tà eà tà
$:?
parenleftbigg
√3
3,
1
4
parenrightbigg
,
parenleftbigg√3
3,
1
4
parenrightbigg
.
3$:
parenleftbigg
√3
3,
1
4
parenrightbigg
§?y? 14 =
2
√3
3parenleftbigg
1
3 + 1
parenrightbigg2
parenleftbigg
x+
√3
3
parenrightbigg
=3√3x+ 8y+ 1 = 0?
92
3$:
parenleftbigg√3
3,
1
4
parenrightbigg
§?y? 14 =
2√3
3parenleftbigg
1
3 + 1
parenrightbigg2
parenleftbigg
x?
√3
3
parenrightbigg
=3√3x?8y?1 = 0.
29,e?ê?/μ
(1) y = x3?6x
(2) y = 3x1 +x2
(3) y = 5e?x2
(4) y = e
x +e?x
2
(5) y = 1x2?1
(6) y = ln 1 +x1?x
(7) y = (x?1)2(x+ 2)3
(8) y = (x?1)
3
(x+ 1)3
(9) y = x
2?2x?3
x2 + 1
(10) y = x+ arctanx
)μ
(1) (i)(?∞,+∞)§′ê§-?'u:é?.
(ii) yprime = 3x2?6,yprimeprime = 6x§x = ±√2?§yprime = 0?x = 0?§yprimeprime = 0.
(iii) LXeμ
x (?∞,?√2)?√2 (?√2,0) 0 (0,√2) √2 (√2,+∞)
yprime + 0 - - - 0 +
yprimeprime - - - 0 + + +
y tàarrownortheast 44√2 tàarrowsoutheast 0 eàarrowsoutheast 44√2 eàarrownortheast
a45
a54
-√6-√2 0
√6√2
x
y
(2) (i)(?∞,+∞)§′ê§-?'u:é?.
(ii) yprime = 3(1?x
2)
(1 +x2)2,y
primeprime = 6x(x2?3)
(1 +x2)3 §x = ±1?§y
prime = 0?x = 0,x = ±√3?§yprimeprime = 0.
(iii) LXeμ
x (?∞,?√3)?√3 (?√3,?1) -1 (?1,0) 0 (0,1) 1 (1,√3) √3 (√3,+∞)
yprime - - - 0 + + + 0 - - -
yprimeprime - 0 + + + 0 - - - 0 +
y tàarrowsoutheast?34√3 eàarrowsoutheast 4? eàarrownortheast 0 tàarrownortheast 4 tàarrowsoutheast 34√3 eàarrowsoutheast
32 32
93
(iv) x→∞?§y → 0§y = 0′-??^Y2ìC?.
a45
a54
-1-
√3
0 1 √3 x
y
(3) (i)(?∞,+∞)§′ó?ê§-?'uy?é?.
(ii) yprime =?10xe?x2,yprimeprime = 10e?x2(2x2?1)§x = 0?§yprime = 0?x = ± 1√2?§yprimeprime = 0.
(iii) LXeμ
x (?∞,? 1√2)?DF1√2 (? 1√2,0) 0 (0,1sqrt2) 1√2 ( 1√2,+∞)
yprime + + + 0 - - -
yprimeprime + 0 - - - 0 +
y eàarrownortheast 5√e tàarrownortheast 4 tàarrowsoutheast 5√e eàarrowsoutheast
5
(iv) x→∞?§y → 0§y = 0′-??^Y2ìC?.
a45
a54
- 1√2 0
5
1√
2
x
y
(4) (i)(?∞,+∞)§′ó?ê§-?'uy?é?￡ù′V-{u?êcoshx = e
x +e?x
2 ¤.
(ii) yprime = sinhx,yprimeprime = coshx§x = 0?§yprime = 0?
duyprimeprime > 0(x∈ (?∞,+∞)§y3(?∞,+∞)t′eà.?yprimeprime|x=0 = 1 > 0§ymin = 1.
a45
a54
1
x
y
(5) (i)(?∞,?1)uniontext(?1,1)uniontext(1,+∞)§′ó?ê§-?'uy?é?.
(ii) yprime =? 2x(x2?1)2,yprimeprime = 2(x
2 + 1)
(x2?1)3§x = 0?§y
prime = 0?x = ±1?§yprime?3?x =
±1?§yprimeprime?3.
94
(iii) LXeμ
x (?∞,?1) -1 (?1,0) 0 (0,1) 1 (1,+∞)
yprime +?3 + 0 -?3 -
yprimeprime +?3 - - -?3 +
y eàarrownortheast tàarrownortheast 4 tàarrowsoutheast eàarrowsoutheast
-1
(iv) x→∞?§y → 0§y = 0′-??^Y2ìCx→±1?§y →∞§x = ±1′-?
?^R?ìC?.
a45
a54
-1 10
-1
x
y
(6) (i)(?1,1)§′ê§-?'u:é?.
(ii) yprime = 21?x2,yprimeprime = 4x(1?x2)2§yprime = 0?)?x = 0?§yprimeprime = 0.
(iii) LXeμ
x (?1,0) 0 (0,1)
yprime + + +
yprimeprime - 0 +
y tàarrownortheast 0 eàarrownortheast
(iv) x→ 1§y → +∞§x = 1′-??^R?ìC
x→?1+?§y →?∞§x =?1′-??^R?ìC?.
a45
a54
-1 10 x
y
(7) (i)(?∞,+∞).
(ii) yprime = (x?1)(x?2)2(5x?7),yprimeprime = 2(x?2)(10x2?28x+19)§x = 1,x = 2,x = 75 = 1.4?§yprime =
0?x = 2,x = 14±
√6
10?§y
primeprime = 0.
(iii) LXeμ
x (?∞,1) 1
parenleftBigg
1,?14?
√6
10
parenrightBigg
14?√6
10
parenleftBigg
14?
√6
,1.4
parenrightBigg
1.4
yprime + 0 - - - 0
yprimeprime - - - 0 + +
y tàarrownortheast 4 tàarrowsoutheast -0.0154 eàarrowsoutheast 4?
0 -0.0346
95
x
parenleftBigg
1.4,14 +
√6
10
parenrightBigg
14 +√6
10
parenleftBigg
14 +√6
,2
parenrightBigg
2 (2,+∞)
yprime + + + 0 +
yprimeprime + 0 - 0 +
y eàarrownortheast -0.0186 tàarrownortheast 0 eàarrownortheast
a45
a54
10 2
-8
x
y
(8) (i)(?∞,?1)uniontext(?1,+∞).
(ii) yprime = 6(x?1)
2
(x+ 1)4,y
primeprime =?12(x?1)(x?3)
(x+ 1)5 §x = 1?§y
prime = 0?x = 1,x = 3?§yprimeprime = 0?
x =?1?§yprime,yprimeprimet?3.
(iii) LXeμ
x (?∞,?1) -1 (?1,1) 1 (1,3) 3 (3,+∞)
yprime +?3 + 0 + + +
yprimeprime -?3 - 0 + 0 -
y tàarrownortheast tàarrownortheast 0 eàarrownortheast 18 tàarrownortheast
(iv) x→?1§y → +∞§x =?1′-??^R?ìC
x→∞?§y → 1§y = 1′-??^Y2ìC?.
a45
a54
-1 10 3 x
y
(9) (i)(?∞,+∞).
96
(ii) yprime = 2(x
2 + 4x?1)
(x2 + 1)2,y
primeprime =?4(x3 + 6x2?3x?2)
(x2 + 1)3 §x =?2 ±
√5?§yprime = 0?yprimeprime = 0?
x1,x2,x3§ù￥x1 ∈ (?7,?6),x2 ∈ (?1,0),x3 ∈
parenleftbigg1
2,1
parenrightbigg
.
(iii) LXeμ
x (?∞,x1) x1 (x1,?2?√5)?2?√5 (?2?√5,x2) x2
yprime + + + 0 - -
yprimeprime + 0 - - - 0
y eàarrownortheast $,tàarrownortheast 4 tàarrowsoutheast $:√
5?1
x (x2,?2 +√5)?2 +√5 (?2 +√5,x3) x3 (x3,+∞)
yprime - 0 + + +
yprimeprime + + + 0 -
y eàarrowsoutheast 4? eàarrownortheast $,tàarrownortheast
√5?1
(iv) x→∞?§y → 1§x = 1?-??^Y2ìC?.
a45
a54
-1 10 3-7 -6 x
y
1
(10) (i)(?∞,+∞)§′ê§-?'u:éx = 0?§y = 0.
(ii) yprime = 1 + 11 +x2 > 0§-?üNt,§?4?:.
yprimeprime =? 2x(1 +x2)2§x = 0?§yprimeprime = 0?x > 0?§yprimeprime < 0?x < 0?§yprimeprime > 0§K(0,0)?$
:.
(iii) k = lim
x→∞
y
x = 1,b1 = limx→?∞(y?kx) =?
pi
2,b2 = limx→+∞(y?kx) =
pi
2§-?kü^ìC
μy = x+ pi2,y = x? pi2.
a45
a54
0 x
y
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
30,á?e?ê?/μy =
9x+x4
x?x3,xnegationslash= 0
9,x = 0
)μ
(1)(?∞,?1)uniontext(?1,1)uniontext(1,+∞).
(2) yprime =
x4 + 3x2 + 18x
(1?x2)2,xnegationslash= 0
0,x = 0
,yprimeprime =
2(x
3 + 27x2 + 3x+ 9
(x2?1)2,xnegationslash= 0
18,x = 0
§x = 0,x = 3?§yprime = 0?yprimeprime = 0x1§ù￥x1 ∈ (?27,?26)?x = ±1?§yprime,yprimeprimet?3.
97
(3) LXeμ
x (?∞,x1) x1 (x1,?1) -1 (?1,0) 0
yprime - - -?3 - 0
yprimeprime + 0 - - -
y eàarrowsoutheast $,tàarrowsoutheast tàarrowsoutheast 4?
9
x (0,1) 1 (1,3) 3 (3,+∞)
yprime +?3 + 0 -
yprimeprime -?3 - - -
y tàarrownortheast tàarrownortheast 4 tàarrowsoutheast
92
(4) x→±1?§y →∞§x = ±1′-?R?ìC?.
a45
a54
-27 0 x
y
98
§4,2?-?-?
1,|-?y = 4x?x2-?±93:(2,4)-.
)μ?y = 4x?x2§yprime = 4? 2x,yprimeprime =?2§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
yprimeprime
(1 +yprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
2
[1 + 4(2?x)2]32
§u′-
ρ = 1K = 12[1 + 4(x?2)2]32§l
3:(2,4)-ρ = 12.
2,|e-?--μ
(1) ]ó?y = acosh xa(a> 0)
(2) y2 = 2px(p> 0)
(3) ^ó?x = a(t?sint),y = a(1?cost)(a> 0)
(4) %9?ρ = a(1 + cosθ)(a> 0)
(5) VY?ρ2 = 2a2 cos2θ(a> 0)
(6) éêú?ρ = aeλθ(λ> 0)
)μ
(1)?y = acosh xa§yprime = sinh xa,yprimeprime = 1a cosh xa§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
yprimeprime
(1 +yprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
1
acosh2 xa
§u′-
ρ = 1Kacosh2 xa.
(2)?y2 = 2px§K2yyprime = 2p=yprime = py§yprimeprime =? py2yprime =?p
2
y3§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
yprimeprime
(1 +yprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
p2
(y2 +p2)32
=
p2
(2px+p2)32
= 1
p
parenleftbigg
1 + 2xp
parenrightbigg3
2
§u′-ρ = 1K = p
parenleftbigg
1 + 2xp
parenrightbigg3
2
parenleftBigg
(y
2 +p2)32
p2
parenrightBigg
.
(3)?x = a(t? sint),y = a(1? cost)§xprime = a(1? cost),xprimeprime = asint;yprime = asintyprimeprime = acost§K-
K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
xprimeyprimeprime?xprimeprimeyprime
(xprime2 +yprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
4a
vextendsinglevextendsingle
vextendsinglevextendsinglesin t
2
vextendsinglevextendsingle
vextendsinglevextendsingle
§u′-ρ = 1K = 4a
vextendsinglevextendsingle
vextendsinglevextendsinglesin t
2
vextendsinglevextendsingle
vextendsinglevextendsingle(= 2radicalbig2ay).
(4)?ρ = a(1 + cosθ)§ρprime =?asinθ,ρprimeprime =?acosθ§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ρ2 + 2ρprime2?ρρprimeprime
(ρ2 +ρprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
3
2√2aρ§u′-
R = 2
√2aρ
3,
(5)?ρ2 = 2a2 cos2θ§K2ρρprime =?4a2 sin2θ§ρprime =?2a
2 sin2θ
ρ,ρ
primeprime =?4a4 +ρ4
ρ3 §K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ρ2 + 2ρprime2?ρρprimeprime
(ρ2 +ρprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
3ρ
2a2§u′-R =
2a2
3ρ,
(6)?ρ = aeλθ§ρprime = λaeλθ = λρ,ρprimeprime = aλ2eλθ = λ2ρ§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ρ2 + 2ρprime2?ρρprimeprime
(ρ2 +ρprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
|ρ|(1 +λ2)12
§
u′-R = |ρ|√1 +λ2.
3,|-?y = 2(x?1)2?-.
)μ?y = 2(x?1)2§yprime = 4(x?1),yprimeprime = 4§K-R = 1K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1 +yprime2)32
yprimeprime
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
[1 + 16(x?1)2]32
4
|R?§K7k[1 + 16(x?1)2]32?§=x = 1?§Rmin = 14.
4,÷′?y = x
2
4000￡ü¤?:à?1§3?I:O?Yv = 140?/|§?1
N
-G = 70ú6.|dé?1
.
)μd?n?￡§?!?
±$??N¤é?%F = mv
2
R §ù￥mN?t§v?§?
99
Y§R?
.
¤|é?1
A?F = Gg+ mv
2
R §ùA
%.
aK?§k|-§yprime = x2000,yprimeprime = 12000§K-R = 1K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(20002 +x2)32
20002
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle§u′3?I
:OR = 2000￡?¤§q3?I:O?Yv = 140?/|§l
F = 1372(N).
5,-t′P§± ?v¨L?x￡?5-32¤§x?ACB′?§ùo?X.|e?LC:?éx?
.
)μ±O?:§AB?x?§CO?yá?IX§K§y =?4δl2 x2 +δ
d?n?§e?LC:?éx?F = mv
2
R cosθ+mg
aK?§k|-§yprime =?8δl2 x,yprimeprime =?8δl2 §K-R = 1K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(l2 + 8δx)32
8lδ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle§u′3:CR =
l2
8δ§q3:Cθ = pi§l
F = Pg+
Pv2
R cosθ =
gl2?8δv2
l2 P.
100
§5,.
1,|^a7?{K|e4?μ
(1) lim
x→0
tanax
sinbx
(2) lim
x→0
1?cosx2
x3 sinx
(3) lim
x→∞
pi
2?arctanx
sin 1x
(4) lim
x→∞
xb
eax
(5) lim
x→1
parenleftbigg 1
lnx?
1
x?1
parenrightbigg
(6) lim
x→pi
(pi?x)tan x2
(7) lim
x→0
ln(cosax)
ln(cosbx)
(8) lim
x→0
cos(sinx)?cosx
x4
(9) lim
x→0
ax?bx
x
(10) lim
x→1
x?1
lnx
(11) lim
x→a
ax?xa
x?a (a> 0)
(12) lim
x→pi6
1?2sinx
cos3x
(13) lim
x→0
lnx
cotx
(14) lim
x→+∞
lncx
xb (b,c> 0)
(15) lim
x→0
(1 +x)1x?e
x
(16) lim
x→0
xb lncx(b,c> 0)
(17) lim
x→0
xsinx
(18) lim
x→1
x 11?x
(19) lim
x→0
parenleftbigg1
x?
1
ex?1
parenrightbigg
(20) lim
x→+0
parenleftbigg
ln 1x
parenrightbiggx
)μ
(1) lim
x→0
tanax
sinbx = limx→0
asec2ax
bcosbx =
a
b
(2) lim
x→0
1?cosx2
x3 sinx = limx→0
1?cosx2
x4 = limx→0
2xsinx2
4x3 limx→0
sinx2
2x2 =
1
2
(3) lim
x→∞
pi
2?arctanx
sin 1x
= lim
x→∞
11 +x2
1x2 cos 1x
= lim
x→∞
x2
(1 +x2)cos 1x
= 1
101
(4) b?ê§ lim
x→∞
xb
eax = limx→∞
bxb?1
aeax = ··· = limx→∞
b!
abeax = 0
bê§K[b] lessorequalslantb< [b]+1§u′|x|
[b]
eax lessorequalslant
|x|b
eax <
|x|[b]+1
eax (|x|> 1)§
!müàx→∞?§
ty2§?40§?d§￥m40.
l
§éa,b§tk lim
x→∞
xb
eax = 0
(5) lim
x→1
parenleftbigg 1
lnx?
1
x?1
parenrightbigg
= lim
x→1
x?1?lnx
(x?1)lnx = limx→1
1? 1x
lnx+ x?1x
= lim
x→1
x?1
xlnx+x?1 = limx→1
1
lnx+ 1 + 1 =
1
2
(6) lim
x→pi
(pi?x)tan x2 = lim
x→pi
pi?x
cot x2
= lim
x→pi
1
12 csc2 x2
= 2
(7) lim
x→0
ln(cosax)
ln(cosbx) = limx→0
atanax
btanbx =
a
b limx→0
tanax
tanbx =
a
b limx→0
asec2ax
bsec2bx =
a2
b2 (bnegationslash= 0)
(8) lim
x→0
cos(sinx)?cosx
x4 = limx→0
sin(sinx)cosx+ sinx
4x3
= lim
x→0
cos(sinx)cos2x+ sin(sinx)sinx+ cosx
12x2 =
lim
x→0
sin(sinx)cos3x+ 32 cos(sinx)sin2x+ sin(sinx)cosx?sinx
24x =
lim
x→0
bracketleftbiggcos(sinx)cos4x?3sin(sinx)sin2xcosx+ 3cos(sinx)cos2x
24 +
cos(sinx)cos2x?sin(sinx)sinx?cosx
24
bracketrightbigg
=
1
6
(9) lim
x→0
ax?bx
x = limx→0
ax lna?bx lnb
1 = lna?lnb = ln
a
b(anegationslash= 0,bnegationslash= 0)
(10) lim
x→1
x?1
lnx = limx→1
1
1
x
= 1
(11) lim
x→a
ax?xa
x?a = limx→a
ax lna?axa?1
1 = a
a(lna?1)
(12) lim
x→pi6
1?2sinx
cos3x = limx→pi6
2cosx
3sin3x =
√3
3
(13) lim
x→0
lnx
cotx = limx→0
1
x
csc2x =? limx→0
sin2x
x = 0
(14) -y = lnx,Kx = ey§u′ lim
x→+∞
lncx
xb = limy→+∞
yc
eby = 0(d(4))
(15) lim
x→0
(1 +x)1x?e
x = limx→0(1+x)
1
x
bracketleftbigg 1
x(1 +x)?
1
x2 ln(1 +x)
bracketrightbigg
= e lim
x→0
x?(1 +x)ln(1 +x)
x2 = e limx→0
1?1?ln(1 +x)
2x =
e2
(16) -y = lnx,Kx = ey§u′lim
x→0
xb lncx = lim
y→?∞
ebyyc = lim
y→?∞
yc
e?by = 0(d(4))
(17) lim
x→0
xsinx = elimx→0sinxlnx§
lim
x→0
sinxlnx = lim
x→0
lnx
1
x
= lim
x→0
1
x
1x2
=? lim
x→0
x = 0§u′lim
x→0
xsinx = 1
(18) lim
x→1
x 11?x = elimx→1
lnx
1?x§
lim
x→1
lnx
1?x =? limx→1
1
x =?1§u′limx→1x
1
1?x = 1e
(19) lim
x→0
parenleftbigg1
x?
1
ex?1
parenrightbigg
= lim
x→0
ex?x?1
x(ex?1) = limx→0 =
ex?1
ex?1 +xex = limx→0
ex
2ex +xex =
1
2
102
(20) lim
x→+0
parenleftbigg
ln 1x
parenrightbiggx
= e
limx→+0xln
ln 1x
-y = 1x§K lim
x→+0
xln(ln 1x) = lim
y→+∞
ln(lny)
y = limy→+∞
1
ylny = 0§l
limx→+0
parenleftbigg
ln 1x
parenrightbiggx
= 1
2,á`2e?ê?U^a7?{K|4?μ
(1) lim
x→0
x2 sin 1x
sinx
(2) lim
x→∞
x+ sinx
x?cosx
(3) lim
x→∞
2x+ sin2x
(2x+ sinx)esinx
(4) lim
x→1
(x2?1)sinx
ln
parenleftBig
1 + sin pi2x
parenrightBig
)μ
(1)?
x2 sin 1x
sinx ?f!?1ó?éx|ê§
2xsin 1x?cos 1x
cosx §
cos 1x
cosxx→ 0?43§?da
7?{K?U·^§′4?′3"ˉ￠t§klim
x→0
x2 sin 1x
sinx = limx→0
x
sinx ·xsin
1
x = 0
(2)?x+ sinxx?cosx?f!?1ó?éx|ê§1 + cosx1 + sinx§x→∞?d?ê43§?da7?{
K?U·^§′4?′3"ˉ￠t§k lim
x→∞
x+ sinx
x?cosx = limx→∞
1 + sinxx
1? cosxx
= 1
(3) éu?óSμxprimen = 2npi+ pi29xprimeprimen = 2npi(n = 1,2,···)§n→∞?§K?ó4?1e91§l
43.
^a7?{K|)§k lim
x→∞
2x+ sin2x
(2x+ sinx)esinx =
lim
x→∞
2 + 2cos2x
(2 + cosx+ 2xcosx+ sinxcosx)esinx =
lim
x→∞
4cos2x
[2 + cosx(1 + 2x+ sinx)]esinx =
lim
x→∞
4bracketleftbigg
2
cos2x +
1
cosx(1 + 2x+ sinx)
bracketrightbigg
esinx
§?esinx greaterorequalslante?1,1+2x+sinxgreaterorequalslant 2x§K
vextendsinglevextendsingle
vextendsinglevextendsingle[ 2
cos2x +
1
cosx(1 + 2x+ sinx)]e
sinx
vextendsinglevextendsingle
vextendsinglevextendsinglegreaterorequalslant
e?1(?2 + 2|x|) → +∞(x→∞)§K lim
x→∞
2x+ sin2x
(2x+ sinx)esinx = 0.
(4)?|4lim
x→1
(x2?1)sinx
ln
parenleftBig
1 + sin pi2x
parenrightBig = 0§d4ü^a7?{K|4?^?.
103
§6,?§Cq)
1,|?§x3?x?4 = 0?§|?
L0.0001.
)μf(x) = x3?x? 4§3[1,2]m§f(1) =?4 < 0,f(2) = 2 > 0=f(1)f(2) < 0?fprime(x) = 3x2? 1 >
0,fprimeprime(x) = 6x> 0
f(2)fprimeprime(2) = 24 > 0§Kl:(2,f(2))=:(2,2)m§x0 = 2?D?.
u′x1 = 2? f(2)fprime(2) ≈ 1.81818,x2 = x1? f(x1)fprime(x
1)
≈ 1.79663,x3 = x2? f(x2)fprime(x
2)
≈ 1.79632,x4 = x3? f(x3)fprime(x
3)
≈
1.79632
x3?x4c5?ê?ó§ùLCu?°(?"?
`2°(Y§^1.7963á?e§kf(1.7963) ≈
0.00019 < 0§
f(1.79632) ≈ 0.00002 > 0§e1.7963Cq?§K?
L0.0001.
2,|?§x3?x?4 = 0?§|?
L0.0001.
)μf(x) = x3? 5x2 + 6x? 1§f(0) =?1 < 0,f(1) = 1 > 0,fprime(x) = 3x2? 10x+ 6§d?fprime(0) = 6 >
0,fprime(1) =?1 < 0§3(0,1)Sfprime(x)k":5?
√7
3 §d?f
prime(x)3
parenleftbigg
0,5?
√7
3
parenrightbigg
S??fprime(x)3
parenleftbigg5?√7
3,1
parenrightbigg
S
K.
y?O?f(x)3(0,0.7)?(0.7,1)￥?
f(0.7) = 1.093 > 0§3(0,0.7)￥7k￠?ξ§3(0.7,1)￥.
y|ξ,fprimeprime(x) = 6x?10 < 0(?x∈ (0,0.7))§?f(0) =?1,fprimeprime(0) =?10§x0 = 0?D?.
u′x1 = 0? f(0)fprime(0) ≈ 0.16667,x2 = x1? f(x1)fprime(x
1)
≈ 0.19706,x3 = x2? f(x2)fprime(x
2)
≈ 0.19806,x4 = x3? f(x3)fprime(x
3)
≈
0.19806
x3?x4c5?ê?ó§ùLCu?ξ°(?"?
`2°(Y§^0.1980á?e§kf(0.1980) ≈
0.00026 < 0§
f(0.1981) ≈ 0.01397 > 0§e0.1980Cq?§K?
L0.0001.
104
a49a19a220a169 a252a67a254a200a169a198
a49a56a217 a216a189a200a169
§1,a216a189a200a169a27a86a103a57a36a142a123a75
1,a121a178a181a101integraltext f(t)dt = F(t) +Ca167a75integraltext f(ax+b)dx = 1aF(ax+b) +C.
a121a178a181a207integraltext f(t)dt = F(t) +Ca167a25[F(t) +C]prime = f(t)a167a75
bracketleftbigg1
aT(ax+b)
bracketrightbiggprime
= 1a[F(ax+b)]prime = f(ax+b)a167a117
a180integraltext f(ax+b)dx = 1aF(ax+b) +C.
2,a166a101a15a216a189a200a169a181
(1)
integraldisplay
(2?sec2x)dx
(2)
integraldisplay parenleftbigg
x4?2x3 +
√x
2
parenrightbigg
dx
(3)
integraldisplay parenleftbigg√
x+ 3√x+ 2√x + 23√x?2
parenrightbigg
dx
(4)
integraldisplay parenleftbigg
ex + 1x + 1x2 + 1x3
parenrightbigg
dx
(5)
integraldisplay parenleftbigg
2cosx+ 12 sinx
parenrightbigg
dx
(6)
integraldisplay parenleftbigg
cosx? 21 +x2 + 14√1?x2
parenrightbigg
dx
(7)
integraldisplay parenleftbigg1
2 cosx+ sinx+ 1
parenrightbigg
dx
(8)
integraldisplay parenleftbigg
2x +
parenleftbigg1
3
parenrightbiggx
e
x
5
parenrightbigg
dx
(9)
integraldisplay
(3?x2)3 dx
(10)
integraldisplay parenleftbigg
1? 1x2
parenrightbiggradicalBig
x√xdx
a41a181
(1)
integraldisplay
(2?sec2x)dx = 2x?tanx+C
(2)
integraldisplay parenleftbigg
x4?2x3 +
√x
2
parenrightbigg
dx = 15x5? 12x4 + 13x32 +C
(3)
integraldisplay parenleftbigg√
x+ 3√x+ 2√x + 23√x?2
parenrightbigg
dx = 23x32 + 34x43?2x+ 3x23 + 4x12 +C
(4)
integraldisplay parenleftbigg
ex + 1x + 1x2 + 1x3
parenrightbigg
dx = ex + ln|x|? 1x? 12x2 +C
(5)
integraldisplay parenleftbigg
2cosx+ 12 sinx
parenrightbigg
dx = 2sinx? 12 cosx+C
(6)
integraldisplay parenleftbigg
cosx? 21 +x2 + 14√1?x2
parenrightbigg
dx = sinx?2arctanx+ 14 arcsinx+C
(7)
integraldisplay parenleftbigg1
2 cosx+ sinx+ 1
parenrightbigg
dx = 12 sinx?cosx+x+C
(8)
integraldisplay parenleftbigg
2x +
parenleftbigg1
3
parenrightbiggx
e
x
5
parenrightbigg
dx = 1ln22x? 1ln3
parenleftbigg1
3
parenrightbiggx
e
x
5 +C
(9)
integraldisplay
(3?x2)3 dx =
integraldisplay
(27?27x2 + 9x4?x6)dx = 27x?9x3 + 95x5? 17x7 +C
(10)
integraldisplay parenleftbigg
1? 1x2
parenrightbiggradicalBig
x√xdx =
integraldisplay
(x34?x?54 )dx = 47x74 + 4x?14 +C
105
§2,a216a189a200a169a27a79a142
1,a166a101a15a216a189a200a169a181
(1)
integraldisplay dx
5x?7
(2)
integraldisplay
cos(ωt)dt
(3)
integraldisplay dx
radicalbigg
1?
parenleftBigx
2 + 3
parenrightBig2
(4)
integraldisplay dx
√1?2x2
(5)
integraldisplay
tan10xsec2xdx
(6)
integraldisplay
eαx ·2xdx
(7)
integraldisplay
(2x + 3x)2 dx
(8)
integraldisplay
tanxdx
(9)
integraldisplay
tan
radicalbig
1 +x2 · xdx√1 +x2
(10)
integraldisplay
(αx2 +β)μxdx(μnegationslash=?1)
(11)
integraldisplay dx
1?cosx
(12)
integraldisplay dx
A2 sin2x+B2 cos2x
(13)
integraldisplay sinx·cosx
1 + sin4x dx
(14)
integraldisplay dx
sin2
parenleftBig
x+ pi4
parenrightBig
(15)
integraldisplay
x2 8
radicalbig
1 +x3 dx
(16)
integraldisplay sin2xcosx
1 + sin3x dx
(17)
integraldisplay 1?2sinx
cos2x dx
(18)
integraldisplay dx
ex +e?x
(19)
integraldisplay sinx+ cosx
3√sinx?cosx dx
(20)
integraldisplay 1 + sin2x
sin2x dx
(21)
integraldisplay radicalBiggln(x+√1 +x2)
1 +x2 dx
(22)
integraldisplay dx
√1 +e2x
(23)
integraldisplay dx
x2?2x+ 2
(24)
integraldisplay dx
(arcsinx)2√1?x2
106
(25)
integraldisplay x2 + 7
x2?2x?3 dx
(26)
integraldisplay x2?1
x4 + 1 dx
a41a181
(1)
integraldisplay dx
5x?7 =
1
5
integraldisplay d(5x?7)
5x?7 =
1
5 ln|5x?7|+C
(2)
integraldisplay
cos(ωt)dt = 1ω
integraldisplay
cos(ωt)d(ωt) = 1ω sin(ωt) +C
(3)
integraldisplay dx
radicalbigg
1?
parenleftBigx
2 + 3
parenrightBig2 = 2
integraldisplay dparenleftBigx
2 + 3
parenrightBig
radicalbigg
1?
parenleftBigx
2 + 3
parenrightBig2 = 2arcsin
parenleftBigx
2 + 3
parenrightBig
+C
(4)
integraldisplay dx
√1?2x2 =
√2
2
integraldisplay d(√x)
radicalBig
1?(√2x)2
=
√2
2 arcsin(
√2x) +C
(5)
integraldisplay
tan10xsec2xdx =
integraldisplay
tan10xd(tanx) =? 111 tan11x+C
(6)
integraldisplay
eαx ·2xdx =
integraldisplay
(2eα)xdx = (2e
α)x
ln(2eα) +C
(7)
integraldisplay
(2x + 3x)2 dx =
integraldisplay
(4x + 2·6x + 9x)dx = 4
x
ln4 +
2
ln66
x + 9x
ln9 +C
(8)
integraldisplay
tanxdx =
integraldisplay sinx
cosx dx =?
integraldisplay d(cosx)
cosx =?ln|cosx|+C = ln|secx|+C
(9)
integraldisplay
tan
radicalbig
1 +x2 · xdx√1 +x2 =
integraldisplay
tan
radicalbig
1 +x2 d(
radicalbig
1 +x2) = ln|sec
radicalbig
1 +x2|+C
(10)
integraldisplay
(αx2 +β)μxdx = 12α
integraldisplay
(αx2 +β)μd(αx2 +β) = (αx
2 +β)μ+1
2α(μ+ 1) +C
(11)
integraldisplay dx
1?cosx =
integraldisplay
csc2 x2 d
parenleftBigx
2
parenrightBig
=?cot x2 +C
(12)
integraldisplay dx
A2 sin2x+B2 cos2x =
1
AB
integraldisplay 1
1 +
parenleftbiggA
B
parenrightbigg2
tan2x
dAtanxB = 1AB arctan
parenleftbiggA
B tanx
parenrightbigg
+C
(13)
integraldisplay sinx·cosx
1 + sin4x dx =
1
2
integraldisplay 1
1 + (sin2x)2 d(sin
2x) = 1
2 arctan(sin
2x) +C
(14)
integraldisplay dx
sin2
parenleftBig
x+ pi4
parenrightBig =
integraldisplay
csc2
parenleftBig
x+ pi4
parenrightBig
d
parenleftBig
x+ pi4
parenrightBig
=?cot
parenleftBig
x+ pi4
parenrightBig
+C
(15)
integraldisplay
x2 8
radicalbig
1 +x3 dx = 13
integraldisplay
8radicalbig1 +x3 d(1 +x3) = 8
27(1 +x
3)98 +C
(16)
integraldisplay sin2xcosx
1 + sin3x dx =
1
3
integraldisplay d(1 + sin3x)
1 + sin3x =
1
3 ln(1 + sin
3x) +C
(17)
integraldisplay 1?2sinx
cos2x dx =
integraldisplay
sec2xdx+ 2
integraldisplay dcosx
cos2x = tanx?2secx+C
(18)
integraldisplay dx
ex +e?x =
integraldisplay dex
e2x + 1 = arctan(e
x) +C
(19)
integraldisplay sinx+ cosx
3√sinx?cosx dx =
integraldisplay d(sinx?cosx)
3√sinx?cosx =
3
2(sinx?cosx)
2
3 +C
(20)
integraldisplay 1 + sin2x
sin2x dx =
integraldisplay
csc2xdx+
integraldisplay d(sin2x)
sin2x =?cotx+ ln(sin
2x) +C =?cotx+ 2ln|sinx|+C
(21)
integraldisplay radicalBiggln(x+√1 +x2)
1 +x2 dx =integraldisplay radicalBig
ln(x+
radicalbig
1 +x2)d(ln(x+
radicalbig
1 +x2) = 23[ln(x+
radicalbig
1 +x2]32 +C
107
(22)
integraldisplay dx
√1 +e2x =?
integraldisplay de?x
√1 +e?2x =?ln(e?x +
radicalbig
1 +e?2x) +C
(23)
integraldisplay dx
x2?2x+ 2 =
integraldisplay d(x?1)
(x?1)2 + 1 = arctan(x?1) +C
(24)
integraldisplay dx
(arcsinx)2√1?x2 =
integraldisplay d(arcsinx)
(arcsinx)2 =?
1
arcsinx +C
(25)
integraldisplay x2 + 7
x2?2x?3 dx =
integraldisplay parenleftbigg
1 + 2x+ 10(x+ 1)(x?3)
parenrightbigg
dx =
integraldisplay parenleftbigg
1? 2x+ 1 + 4x?3
parenrightbigg
dx = x?2ln|x+ 1|+ 4ln|x?3|+C = x+ 2ln (x?3)
2
|x+ 1| +C
(26)
integraldisplay x2?1
x4 + 1 dx =
integraldisplay 1?x?2
x2 +x?2 dx =
integraldisplay d
parenleftbigg
x+ 1x
parenrightbigg
parenleftbigg
x+ 1x
parenrightbigg2
2
=
√2
4 ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x+ 1x?√2
x+ 1x +√2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+C =
√2
4 ln
vextendsinglevextendsingle
vextendsinglevextendsinglex2?
√2x+ 1
x2 +√2x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle+C
2,a166a101a15a216a189a200a169a181
(1)
integraldisplay sin√x
√x dx
(2)
integraldisplay (2√u+ 1)2
u2 du
(3)
integraldisplay
e
√x+1
dx
(4)
integraldisplay x2
√4?x2 dx
(5)
integraldisplay radicalbig
x2 +a2 dx
(6)
integraldisplay radicalbig
x2?a2 dx
(7)
integraldisplay dx
radicalbig(x?a)(b?x)
(8)
integraldisplay dx
x2radicalbigαx2 +β
(9)
integraldisplay xdx
√5 +x?x2
(10)
integraldisplay radicalbig
2 +x?x2 dx
a41a181
(1)
integraldisplay sin√x
√x dx =?2
integraldisplay
sin√xd(√x) =?2cos√x+C
(2)
integraldisplay (2√u+ 1)2
u2 du =
integraldisplay parenleftbigg4
u +
4
u32
+ 1u2
parenrightbigg
du = 4ln|u|?8u?12? 1u +C
(3) a45√1 +x = ta167a75x = t2?1,dx = 2tdta167a117a180
integraldisplay
e
√x+1
dx = 2
integraldisplay
tetdt = 2(t?1)et+C = 2(√1 +x?
1)e
√1+x
+C
(4)
integraldisplay x2
√4?x2 dx =?
integraldisplay 4?x2?4
√4?x2 dx =?
integraldisplay radicalbig
4?x2 dx+ 4
integraldisplay dx
√4?x2 =?x2
radicalbig
4?x2?2arcsin x2 +
4arcsin x2 +C = 2arcsin x2? x2
radicalbig
4?x2 +C
108
(5) I =
integraldisplay radicalbig
x2 +a2 dx = x
radicalbig
x2 +a2?
integraldisplay x2
√x2 +a2 dx = x
radicalbig
x2 +a2?
integraldisplay radicalbig
x2 +a2 dx+
integraldisplay a2
√x2 +a2 dx =
x
radicalbig
x2 +a2?I +a2 ln|x+
radicalbig
x2 +a2|+C1a167
a117a1802I = x√x2 +a2+a2 ln|x+√x2 +a2|+C1a167a108a13I = x2
radicalbig
x2 +a2+a
2
2 ln(x+
radicalbig
x2 +a2)+C(C =
C1
2 )
(6) I =
integraldisplay radicalbig
x2?a2 dx = x
radicalbig
x2?a2?
integraldisplay x2
√x2?a2 dx = x
radicalbig
x2?a2?
integraldisplay radicalbig
x2?a2 dx?
integraldisplay a2
√x2?a2 dx =
x
radicalbig
x2?a2?I?a2 ln|x+
radicalbig
x2?a2|+C1a167
a117a1802I = x√x2?a2?a2 ln|x+√x2?a2|+C1a167a108a13I = x2
radicalbig
x2?a2?a
2
2 ln(x+
radicalbig
x2?a2)+C(C =
C1
2 )
(7)
integraldisplay dx
radicalbig(x?a)(b?x) =
integraldisplay dx
radicalbig?[x2?(a+b)x]?ab =
integraldisplay d
parenleftbigg
x? a+b2
parenrightbigg
radicalBigg
parenleftbigg
x? a+b2
parenrightbigg2
+
parenleftbigga?b
2
parenrightbigg2 = arcsin
x? a+b2
a?b
2
+C = arcsin 2x?a?ba?b +Ca163a217a165a<ba164
(8)
integraldisplay dx
x2radicalbigαx2 +β =
integraldisplay dx
x3
radicalbigg
α+ βx2
=?12
integraldisplay d 1
x2radicalbigg
α+ βx2
=?1β
radicalbigg
α+ βx2 +C
(9)
integraldisplay xdx
√5 +x?x2 =
integraldisplay xdx
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 =
integraldisplay x? 1
2radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 +
1
2
integraldisplay dx
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 =?
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2
+
1
2 arcsin
x? 12
√21
2
+C =?
radicalbig
5 +x?x2 + 12 arcsin 2x?1√21 +C
(10)
integraldisplay radicalbig
2 +x?x2 dx =
integraldisplay radicalBigg9
4?
parenleftbigg
x? 12
parenrightbigg2
dx =
x? 12
2
radicalBigg
9
4?
parenleftbigg
x? 12
parenrightbigg2
+98 arcsin
x? 12
3
2
+C = 2x?14
radicalbig
2 +x?x2+
9
8 arcsin
2x?1
3 +C
3,a166a101a15a216a189a200a169a181
(1)
integraldisplay
x2 cosxdx
(2)
integraldisplay
x3 lnxdx
(3)
integraldisplay
lnxdx
(4)
integraldisplay
xnlnxdx(na143a20a18a234)
(5)
integraldisplay arcsinx
√1?x dx
(6)
integraldisplay
cscxdx
(7)
integraldisplay
cos(lnx)dx
(8)
integraldisplay xdx
sin2x
(9)
integraldisplay
xcos2xdx
109
(10)
integraldisplay
xsin2xdx
(11)
integraldisplay
arccosxdx
(12)
integraldisplay
(arcsinx)2 dx
(13)
integraldisplay
eaxcosbxdx
(14)
integraldisplay
ln(x+
radicalbig
1 +x2)dx
a41a181
(1)
integraldisplay
x2 cosxdx = x2 sinx? 2
integraldisplay
xsinxdx = x2 sinx + 2xcosx? 2
integraldisplay
cosxdx = x2 sinx + 2xcosx?
2sinx+C
(2)
integraldisplay
x3 lnxdx = 14x4 lnx? 14
integraldisplay
x3 dx = 14x4 lnx? x
4
16 +C
(3)
integraldisplay
lnxdx = xlnx?
integraldisplay
dx = xlnx?x+C
(4)
integraldisplay
xnlnxdx = x
n+1
x+ 1 lnx?
1
n+ 1
integraldisplay
xndx = x
n+1
n+ 1 lnx?
xn+1
(n+ 1)2 +C
(5)
integraldisplay arcsinx
√1?x dx =?2arcsinx·√1?x+ 2
integraldisplay 1
√1 +x dx =?2√1?xarcsinx+ 4√1 +x+C
(6)
integraldisplay
cscxdx =
integraldisplay dx
sinx =
integraldisplay 12cos( x
2)
tan x2
dx =
integraldisplay dparenleftBigtan x
2
parenrightBig
tan x2
= ln
vextendsinglevextendsingle
vextendsingletan x2
vextendsinglevextendsingle
vextendsingle+C
(7) a207I =
integraldisplay
cos(lnx)dx = xcoslnx +
integraldisplay
sin(lnx)dx = xcos(lnx) + xsin(lnx)?
integraldisplay
cos(lnx)dx =
xcos(lnx) +xsin(lnx)?I +C1a167a25I = x2[cos(lnx) + sin(lnx)] +C
parenleftbigg
C = C12
parenrightbigg
(8)
integraldisplay xdx
sin2x =
integraldisplay
xcsc2xdx =?xcotx+
integraldisplay
cotxdx =?xcotx+ ln|sinx|+C
(9)
integraldisplay
xcos2xdx = 12
integraldisplay
x(1 + cos2x)dx = x
2
4 +
1
2
integraldisplay
xcos2xdx = x
2
4 +
x
4 sin2x?
1
4
integraldisplay
sin2xdx =
x2
4 +
x
4 sin2x+
1
8 cos2x+C
(10)
integraldisplay
xsin2xdx =
integraldisplay
x(1? cos2x)dx = x
2
2?
integraldisplay
xcos2xdx = x
2
2?
parenleftbiggx2
4 +
x
4 sin2x+
1
8 cos2x
parenrightbigg
+C =
x2
4?
x
4 sin2x?
1
8 cos2x
(11)
integraldisplay
arccosxdx = xarccosx+
integraldisplay x
√1?x2 dx = xarccosx?
radicalbig
1?x2 +C
(12)
integraldisplay
(arcsinx)2 dx = x(arcsinx)2?
integraldisplay 2xarcsinx
√1?x2 dx = x(arcsinx)2 + 2arcsinx·
radicalbig
1?x2? 2
integraldisplay
dx =
x(arcsinx)2 + 2
radicalbig
1?x2 arcsinx?2x+C
(13) I =
integraldisplay
eaxcosbxdx = 1aeaxcosbx+ba
integraldisplay
eaxsinbxdx = 1aeaxcosbx+ ba2eaxsinbx?b
2
a2
integraldisplay
eaxcosbxdx =
1
ae
axcosbx+ b
a2e
axsinbx? b2
a2I +C1a167a75I =
acosbx+bsinbx
a2 +b2 e
ax +C
parenleftbigg
C = a
2
a2 +b2C1
parenrightbigg
(14)
integraldisplay
ln(x+
radicalbig
1 +x2)dx = xln(x+
radicalbig
1 +x2)?
integraldisplay x
√1 +x2x = xln(x+
radicalbig
1 +x2)?
radicalbig
1 +x2 +C
4,a166a101a15a216a189a200a169a181
(1)
integraldisplay x3 + 1
x3?5x2 + 6x dx
110
(2)
integraldisplay dx
(x+ 1)(x+ 2)2
(3)
integraldisplay dx
(x+ 1)(x+ 2)2(x+ 3)3
(4)
integraldisplay x2 + 5x+ 4
x4 + 5x2 + 4 dx
(5)
integraldisplay dx
(x2?4x+ 4)(x2?4x+ 5)
(6)
integraldisplay dx
x4 +x2 + 1
(7)
integraldisplay dx
(x+ 1)(x2 + 1)
(8)
integraldisplay dx
x3 + 1
(9)
integraldisplay x2 dx
1?x4
(10)
integraldisplay x6 +x4?4x2?2
x3(x2 + 1)2 dx
a41a181
(1) a207 x
3 + 1
x3?5x2 + 6x = 1 +
1
6x?
9
2(x?2) +
28
3(x?3)a167a25
integraldisplay x3 + 1
x3?5x2 + 6x dx = x+
1
6 ln|x|?
9
2 ln|x?
2|+ 283 ln|x?3|+C
(2) a207 1(x+ 1)(x+ 2)2 = 1x+ 1? 1x+ 2? 1(x+ 2)2a167a25
integraldisplay dx
(x+ 1)(x+ 2)2 = ln|x+1|?ln|x+2|+
1
x+ 2 +
C = ln
vextendsinglevextendsingle
vextendsinglevextendsinglex+ 1
x+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle+ 1
x+ 2 +C
(3) a207 1(x+ 1)(x+ 2)2(x+ 3)3 = 18(x+ 1) + 2x+ 2? 1(x+ 2)2? 178(x+ 3)? 54(x+ 3)2? 12(x+ 3)3a167
a25
integraldisplay dx
(x+ 1)(x+ 2)2(x+ 3)3 =
1
8 ln|x+1|+2ln|x+2|+
1
x+ 2?
17
8 ln|x+3|+
5
4(x+ 3) +
1
4(x+ 3)2 +
C = 18 ln|x+ 1|+ 2ln|x+ 2|? 178 ln|x+ 3|+ 9x
2 + 50x+ 68
4(x+ 2)(x+ 3)2 +C
(4) a207 x
2 + 5x+ 4
x4 + 5x2 + 4 =
5
3x+ 1
x2 + 1 +
53x
x2 + 4a167a25
integraldisplay x2 + 5x+ 4
x4 + 5x2 + 4 dx =
5
6 ln(x
2+1)+arctanx?5
6 ln(x
2+4)+C =
5
6 ln
parenleftbiggx2 + 1
x2 + 4
parenrightbigg
+ arctanx+C
(5) a207 1(x2?4x+ 4)(x2?4x+ 5) = 1x2?4x+ 4? 1x2?4x+ 5 = 1(x?2)2? 1(x?2)2 + 1a167a25
integraldisplay dx
(x2?4x+ 4)(x2?4x+ 5) =
1x?2?arctan(x?2) +C
(6) a207 1x4 +x2 + 1 = x+ 12(x2 +x+ 1)? x?12(x2?x+ 1)a167a25
integraldisplay dx
x4 +x2 + 1 =
1
4 ln(x
2+x+1)+
√3
6 arctan
parenleftbigg√3
3 (2x+ 1)
parenrightbigg
1
4 ln(x
2?x+ 1) +
√3
6 arctan
parenleftbigg√3
3 (2x?1)
parenrightbigg
(7) a207 1(x+ 1)(x2 + 1) = 12(x+ 1)? x?12(x2 + 1)a167a25
integraldisplay dx
(x+ 1)(x2 + 1) =
1
2 ln|x + 1|?
1
4 ln(x
2 + 1) +
1
2 arctanx+C =
1
4 ln
(x+ 1)2
x2 + 1 +
1
2 arctanx+C
(8) a207 1x3 + 1 = 13(x+ 1) +?x+ 23(x2?x+ 1)a167
a25
integraldisplay dx
x3 + 1 =
1
3 ln|x + 1|?
1
6 ln|x
2? x + 1| +
√3
3 arctan
√3(2x?1)
3 + C =
1
6 ln
(x+ 1)2
x2?x+ 1 +√
3
3 arctan
√3(2x?1)
3 +C
111
(9) a207 x
2
1?x4 =
1
2(1?x2)?
1
2(1 +x2) =
1
4(1?x) +
1
4(1 +x)?
1
2(x2 + 1)a167a25
integraldisplay x2 dx
1?x4 =?
1
4 ln|1?x|+
1
4 ln|1 +x|?
1
2 arctanx+C =
1
4 ln
vextendsinglevextendsingle
vextendsinglevextendsingle1 +x
1?x
vextendsinglevextendsingle
vextendsinglevextendsingle+ 1
2 arctanx+C
(10) a207x
6 +x4?4x2?2
x3(x2 + 1)2 =
x4(x2 + 1)
x3(x2 + 1)2?
4x2 + 2
x3(x2 + 1)2 =
x
x2 + 1? 2
(x2 + 1)2?x4
x3(x2 + 1)2 =
x
x2 + 1?
2
x3 +
2x
(x2 + 1)2a167a25
integraldisplay x6 +x4?4x2?2
x3(x2 + 1)2 dx =
1
x2 +
1
2 ln(x
2 + 1)? 1
x2 + 1 +C
5,a166a101a15a216a189a200a169a181
(1)
integraldisplay dx
4 + 5cosx
(2)
integraldisplay dx
sinx+ tanx
(3)
integraldisplay xdx
√5 +x?x2
(4)
integraldisplay 1
x 4√1 +x4 dx
(5)
integraldisplay xdx
√2 + 4x
(6)
integraldisplay cosx
1 + sinx dx
(7)
integraldisplay dx
x(1 + 2√x+ 3√x)
(8)
integraldisplay √x+ 1?√x?1
√x+ 1 +√x?1 dx
(9)
integraldisplay dx
3radicalbig(x+ 1)2(x?1)4
(10)
integraldisplay dx
√x(1 + 4√x)3
(11)
integraldisplay dx
√ax2 +bx+c(a> 0)
(12)
integraldisplay xdx
4radicalbigx3(a?x)
(13)
integraldisplay
x
radicalbig
x4 + 2x2?1dx
(14)
integraldisplay radicalbig
2 +x?x2 dx
(15)
integraldisplay x2 dx
√1 +x?x2
(16)
integraldisplay x2 + 1
x√x4 + 1 dx
(17)
integraldisplay
sin6xdx
(18)
integraldisplay
sin2xcos4xdx
(19)
integraldisplay
sin4xcos4xdx
(20)
integraldisplay cos4x
sin3x dx
(21)
integraldisplay dx
sin3xcos5x
(22)
integraldisplay
tanx·tan(x+a)dx
112
(23)
integraldisplay
sin5xcosxdx
(24)
integraldisplay sin2x
1 + sin2x dx
(25)
integraldisplay dx
sin(x+a)sin(x+b)
(26)
integraldisplay
xexcosxdx
(27)
integraldisplay dx
(2 + cosx)sinx
(28)
integraldisplay
ln(x+
radicalbig
1 +x2)2 dx
(29)
integraldisplay sinxcosx
sinx+ cosx dx
(30)
integraldisplay lnx
(1 +x2)32
dx
(31)
integraldisplay
xexsinxdx
(32)
integraldisplay x3 arccosx
√1?x2 dx
(33)
integraldisplay
(x+|x|)2 dx
(34)
integraldisplay
x2excosxdx
(35)
integraldisplay xex
(1 +x)2 dx
(36)
integraldisplay √
xln2xdx
(37)
integraldisplay dx
radicalbig(x?a)(b?x)
(38)
integraldisplay
xln 1 +x1?x dx
(39)
integraldisplay
xarctanx·ln(1 +x2)dx
(40)
integraldisplay
sinh2xcosh2xdx
a41a181
(1) a45tan x2 = ta167a75cosx = 1?t
2
1 +t2,dx =
2dt
1 +t2a167
a117a180
integraldisplay dx
4 + 5cosx =
integraldisplay 2
(3?t)(3 +t) dt =
1
3 ln
vextendsinglevextendsingle
vextendsinglevextendsingle3 +t
3?t
vextendsinglevextendsingle
vextendsinglevextendsingle+C = 1
3 ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
3 + tan x2
3?tan x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle+C
(2) a45tan x2 = ta167a75sinx = 2t1 +t2,tanx = 2t1?t2,dx = 2dt1 +t2a167
a117a180
integraldisplay dx
sinx+ tanx =
integraldisplay 1?t2
2t dt =
1
2 ln|t|?
t2
4 +C =
1
2 ln
vextendsinglevextendsingle
vextendsingletan x2
vextendsinglevextendsingle
vextendsingle? 14
parenleftBig
tan x2
parenrightBig2
+C
(3)
integraldisplay xdx
√5 +x?x2 =
integraldisplay xdx
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 =
integraldisplay x? 1
2radicalBigg
21
4 dx?
parenleftbigg
x? 12
parenrightbigg2 +
1
2
integraldisplay dx
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 =?
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2
+
1
2 arcsin
x? 12
√21
2
+C =?
radicalbig
5 +x?x2 + 12 arcsin 2x?1√21 +C
113
(4) a45t = 4√1 +x4a167a75x = 4√t4?1,dx = t3(t4?1)?34 dt
a117a180
integraldisplay 1
x 4√1 +x4 dx =
integraldisplay t2
t4?1 =
1
4
integraldisplay parenleftbigg 1
t?1?
1
t+ 1
parenrightbigg
dt+12
integraldisplay 1
1 +t2 dt =
1
4 ln
vextendsinglevextendsingle
vextendsinglevextendsinglet?1
t+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle+1
2 arctant+
C = 14 ln
vextendsinglevextendsingle
vextendsinglevextendsingle 4
√1 +x4?1
4√1 +x4 + 1
vextendsinglevextendsingle
vextendsinglevextendsingle+ 1
2 arctan(
4radicalbig1 +x4) +C
(5)
integraldisplay xdx
√2 + 4x = 12
integraldisplay
xd√4x+ 2 = 12x√2 + 4x? 12
integraldisplay
(2 + 4x)12 dx = x2√2 + 4x? 112(2 + 4x)32 +C
(6)
integraldisplay cosx
1 + sinx dx =
integraldisplay dsinx
1 + sinx = ln(1 + sinx) +C
(7) a45 6√x = ta167a75x = t6,dx = 6t5 dt
a117a180
integraldisplay dx
x(1 + 2√x+ 3√x) =
6
integraldisplay dt
t(1 + 2t3 +t2) = 6
integraldisplay bracketleftbigg1
t?
1
4(t+ 1)?
6t?1
4(2t2?t+ 1)
bracketrightbigg
dt
a113
integraldisplay 6t?1
4(2t2?t+ 1) dt =
3
8
integraldisplay d(2t2?t+ 1)
2t2?t+ 1 +
1
8
integraldisplay dt
2t2?t+ 1 =
3
8 ln|2t
2?t+1|+ 1
4√7 arctan
4t?1√
7 +
C1a167
a108a13
integraldisplay dx
x(1 + 2√x+ 3√x) = 6ln|t|?
3
2 ln|t+1|?
9
4 ln|2t
2?t+1|? 3
2√7 arctan
4t?1√
7 +C = 6ln|
6√x|?
3
2 ln|
6√x+1|? 9
4 ln|2
3√x? 6√x+1|? 3
2√7 arctan
4 6√x?1√
7 +C =
3
4 ln
x 3√x
(1 + 6√x)2(2 3√x? 6√x+ 1)3?
3
2√7 arctan
4 6√x?1√
7 +C
(8)
integraldisplay √x+ 1?√x?1
√x+ 1 +√x?1 dx =
integraldisplay (√x+ 1?√x?1)2
(√x+ 1 +√x?1)(√x+ 1?√x?1) dx =
integraldisplay
(x?
radicalbig
x2?1)dx = x
2
2?
x
2
radicalbig
x2?1 + 12 ln|x+
radicalbig
x2?1|+C
(9) a45 3
radicalbiggx+ 1
x?1 = ta167a75x =
t3 + 1
t3?1,dx =?
6t2
(t3?1)2 dt
a117a180
integraldisplay dx
3radicalbig(x+ 1)2(x?1)4 =?
3
2
integraldisplay
dt =?32t+C =?32 3
radicalbiggx+ 1
x?1 +C
(10) a45 4√x = ta167a75x = t4,dx = 4t3 dt
a117a180
integraldisplay dx
√x(1 + 4√x)3 = 4
integraldisplay t
(1 +t)3 dt = 4
integraldisplay bracketleftbigg 1
(1 +t)2?
1
(1 +t)3
bracketrightbigg
dt =? 41 +t + 2(1 +t)2 + C =
41 + 4√x + 2(1 + 4√x)2 +C
(11)
integraldisplay dx
√ax2 +bx+c = 1√a
integraldisplay d
parenleftbigg√
a
parenleftbigg
x+ b2a
parenrightbiggparenrightbigg
bracketleftbigg√
a
parenleftbigg
x+ b2a
parenrightbiggbracketrightbigg2
+ 4ac?b
2
4a
=
1√
a ln
vextendsinglevextendsingle
vextendsinglevextendsingle√a
parenleftbigg
x+ b2a
parenrightbigg
+
radicalbig
ax2 +bx+c
vextendsinglevextendsingle
vextendsinglevextendsingle+C
(12) a45 4
radicalbigga?x
x = t
a117a180
integraldisplay xdx
4radicalbigx3(a?x) =?
integraldisplay 4at2
(1 +t4)2 dt =
4a
integraldisplay bracketleftbigg t
(t2 +√2t+ 1)(t2?√2t+ 1)
bracketrightbigg2
dt =?a2
integraldisplay dt
(t2?√2t+ 1)2?
a
2
integraldisplay dt
(t2 +√2t+ 1)2 +a
integraldisplay dt
t4 + 1
a113
integraldisplay dt
(t2?√2t+ 1)2 =
integraldisplay d
parenleftbigg
t?
√2
2
parenrightbigg
bracketleftBiggparenleftbigg
t?
√2
2
parenrightbigg2
+ 12
bracketrightBigg2 = 2t?
√2
2(t2?√2t+ 1) +
√2arctan(√2t?1) +C
1
integraldisplay dt
(t2 +√2t+ 1)2 =
2t+√2
2(t2 +√2t+ 1) +
√2arctan(√2t+ 1) +C
2
integraldisplay 1
t4 + 1 dt =
1
2
integraldisplay t2 + 1
t4 + 1 dt?
1
2
integraldisplay t2?1
t4 + 1 dt
114
a13
integraldisplay t2 + 1
t4 + 1 dt =
integraldisplay 1 + 1
t2
t2 + 1t2
dt =
integraldisplay d
parenleftbigg
t? 1t
parenrightbigg
parenleftbigg
t? 1t
parenrightbigg2
+ 2
= 1√2 arctan t
2?1
√2t +C3,
integraldisplay t2?1
t4 + 1 dt =
1
2√2 ln
t2?√2t+ 1
t2 +√2t+ 1 +C4
a108a13
integraldisplay xdx
4radicalbigx3(a?x) =?
at3
1 +t4?
a√2
2 arctan
√2t
1?t2 +
a
4√2 ln
vextendsinglevextendsingle
vextendsinglevextendsinglet2 +
√2t+ 1
t2?√2t+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle+ a
2√2 arctan
t2?1√
2t +Ca167
a217a165t = 4
radicalbigga?x
x,
(13)
integraldisplay
x
radicalbig
x4 + 2x2?1dx = 12
integraldisplay radicalbig
(x2 + 1)2?2dx2 = x
2 + 1
4
radicalbig
x4 + 2x2?1?12 ln(x2+1+
radicalbig
x4 + 2x2?1)+
C
(14)
integraldisplay radicalbig
2 +x?x2 dx =
integraldisplay radicalBigg9
4?
parenleftbigg
x? 12
parenrightbigg2
dx = 2x?14
radicalbig
2 +x?x2 + 98 arcsin 2x?13 +C
(15)
integraldisplay x2 dx
√1 +x?x2 =?
integraldisplay radicalbig
1 +x?x2 dx+
integraldisplay x+ 1
√1 +x?x2 dx =?2x?14
radicalbig
1 +x?x2?58 arcsin 2x?1√5?
radicalbig
1 +x?x2 + 32 arcsin 2x?1√5 +C =?2x+ 34
radicalbig
1 +x?x2 + 78 arcsin 2x?1√5 +C
(16)
integraldisplay x2 + 1
x√x4 + 1 dx =
integraldisplay 1 + 1
x2radicalbigg
x2 + 1x2
dx =
integraldisplay d
parenleftbigg
x? 1x
parenrightbigg
radicalBiggparenleftbigg
x? 1x
parenrightbigg2
+ 2
= ln
parenleftBigg
x? 1x +
radicalbigg
x2 + 1x2
parenrightBigg
+C = ln
vextendsinglevextendsingle
vextendsinglevextendsinglex2?1 +
√x4 + 1
x
vextendsinglevextendsingle
vextendsinglevextendsingle+
C
(17)
integraldisplay
sin6xdx =
integraldisplay parenleftbigg1?cos2x
2
parenrightbigg3
dx = 18
integraldisplay
(1? 3cos2x+ 3cos2 2x? cos3 2x)dx = 18x? 316 sin2x+
3
16
integraldisplay
(1+cos4x)dx? 116
integraldisplay
cos2 2xdsin2x = 18x? 316 sin2x+ 316x+ 364 sin4x? 116 sin2x+ 148 sin3 2x+
C = 516x? 14 sin2x+ 364 sin4x+ 148 sin3 2x+C
(18)
integraldisplay
sin2xcos4xdx =?15
integraldisplay
sinxdcos5x =?15 sinxcos5x+
1
5
integraldisplay
cos6x =?15 sinxcos5x+ 15
bracketleftbigg 5
16x?
1
4 sin2x+
3
64 sin4x+
1
48 sin
3 2x
bracketrightbigg
+C = 116x? 120 sin2x+
3
320 sin4x+
1
240 sin
3 2x? 1
5 sinxcos
5x+C
(19)
integraldisplay
sin4xcos4xdx =
integraldisplay parenleftbiggsin2x
2
parenrightbigg4
dx = 116
integraldisplay parenleftbigg1?cos4x
2
parenrightbigg2
dx = 164
integraldisplay
(1?2cos4x+ cos2 4x)dx =
3
128x?
sin4x
128 +
1
1024 sin8x+C
(20)
integraldisplay cos4x
sin3x dx =?
1
2
integraldisplay
cos3xd 1sin2x =?12 · cos
3x
sin2x?
3
2
integraldisplay cos2x
sinx dx =?
cos3x
2sin2x?
3
2
integraldisplay dx
sinx +
3
2
integraldisplay
sinxdx =? cos
3x
2sin2x?
3
2
integraldisplay sec2 x
2
tan x2
dx2? 32 cosx =? cos
3x
2sin2x?
3
2 ln
vextendsinglevextendsingle
vextendsingletan x2
vextendsinglevextendsingle
vextendsingle? 32 cosx+C
(21)
integraldisplay dx
sin3xcos5x =
integraldisplay sin2x+ cos2x
sin3xcos5x dx =
integraldisplay dx
sinxcos5x +
integraldisplay dx
sin3xcos3x =
integraldisplay sin2x+ cos2x
sinxcos5x dx +integraldisplay
sin2x+ cos2x
sin3xcos3x dx =
integraldisplay sinx
cos5x dx+2
integraldisplay dx
sinxcos3x+
integraldisplay dx
sin3xcosx =
1
4 cos
4x+2
integraldisplay sin2x+ cos2x
sinxcos3x dx+integraldisplay
sin2x+ cos2x
sin3xcosx dx =
1
4 sec
4x+ 2
integraldisplay sinx
cos3x dx+ 3
integraldisplay dx
sinxcosx +
integraldisplay cosx
sin3x dx =
1
4 sec
4x+ sec2x+
3
integraldisplay dtanx
tanx?
1
2 csc
2x = 1
4 sec
4x+sec2x+3ln|tanx|?1
2 csc
2x+C1 = 1
4 tan
4x+ 3
2 tan
2x?1
2 cot
2x+
3ln|tanx|+C
(22)
integraldisplay
tanx·tan(x+a)dx =
integraldisplay
tanx· tanx+ tana1?tanxtana dx =
integraldisplay tan2x+ tanxtana+ 1?1
1?tanxtana dx =
integraldisplay 1 + tan2x
1?tanxtana dx?integraldisplay
dx =
integraldisplay dtanx
1?tanxtana?x =?cotaln|1?tanxtana|?x+C1 = cotaln
vextendsinglevextendsingle
vextendsinglevextendsingle cosx
cos(x+a)
vextendsinglevextendsingle
vextendsinglevextendsingle?x+C
115
(23)
integraldisplay
sin5xcosxdx = 12
integraldisplay
(sin6x+ sin4x)dx =? 112 cos6x? 18 cos4x+C
(24)
integraldisplay sin2x
1 + sin2x dx =
integraldisplay 1
csc2x+ 1 dx =
integraldisplay parenleftbigg
1? csc
2x
1 + csc2x
parenrightbigg
dx = x+
integraldisplay dcotx
2 + cot2x = x+
√2
2 arctan
parenleftbigg√2
2 cotx
parenrightbigg
+
C
(25) a23sin(a?b) negationslash= 0a167
a75
integraldisplay dx
sin(x+a)sin(x+b) =
1
sin(a?b)
integraldisplay sin[(x+a)?(x+b)]
sin(x+a)sin(x+b) dx =
1
sin(a?b)
integraldisplay bracketleftbiggcos(x+b)
sin(x+b)?
cos(x+a)
sin(x+a)
bracketrightbigg
dx =
1
sin(a?b) ln
vextendsinglevextendsingle
vextendsinglevextendsinglesin(x+b)
sin(x+a)
vextendsinglevextendsingle
vextendsinglevextendsingle+C
(26) I =
integraldisplay
xexcosxdx = xexcosx?
integraldisplay
ex(cosx?xsinx)dx = xexcosx?
integraldisplay
excosxdx+
integraldisplay
xexsinxdx =
xexcosx?sinx+ cosx2 ex+xexsinx?
integraldisplay
ex(sinx+xcosx)dx = xexcosx?sinx+ cosx2 ex+xexsinx?
sinx?cosx
2 e
x?
integraldisplay
xexcosxdx+C1 = ex(xcosx+xsinx?sinx)?I +C1a167
a75I =
integraldisplay
xexcosxdx = e
x
2 (xcosx+xsinx?sinx) +C
(27) a45tanx2 = ta167a75sinx = 2t1 +t2,cosx = 1?t
2
1 +t2,dx =
2dt
1 +t2
a117a180
integraldisplay dx
(2 + cosx)sinx =
integraldisplay 1 +t2
t(3 +t2) dt =
integraldisplay parenleftbigg 1
3t +
2t
3(3 +t2)
parenrightbigg
dt = 13 ln|t(t2+3)|+C = 13 ln
vextendsinglevextendsingle
vextendsingletan x2
parenleftBig
tan2 x2 + 3
parenrightBigvextendsinglevextendsingle
vextendsingle+
C =
1
6 ln
(1?cosx)(2 + cosx)2
(1 + cosx)3 +C
(28)
integraldisplay
ln(x+
radicalbig
1 +x2)2 dx = xln(x+
radicalbig
1 +x2)2?
integraldisplay
x· 1(x+√1 +x2)2 · 2(x+
radicalbig
1 +x2) ·
parenleftbigg
1 + x√1 +x2
parenrightbigg
dx = xln(x+
radicalbig
1 +x2)2?
integraldisplay d(1 +x2)
√1 +x2 =
xln(x+
radicalbig
1 +x2)2?2
radicalbig
1 +x2 +C
(29)
integraldisplay sinxcosx
sinx+ cosx dx =
integraldisplay sin2parenleftBigx+ pi
4
parenrightBig
12
√2sinparenleftBigx+ pi
4
parenrightBig dx =
√2
2
integraldisplay
sin
parenleftBig
x+ pi4
parenrightBig
dx? 12√2
integraldisplay dx
sin
parenleftBig
x+ pi4
parenrightBig =?
√2
2 cos
parenleftBig
x+ pi4
parenrightBig
1
2√2
integraldisplay dparenleftBigtanparenleftBigx
2 +
pi
8
parenrightBigparenrightBig
tan
parenleftBigx
2 +
pi
8
parenrightBig = 12(sinx?cosx)?
√2
4 ln
vextendsinglevextendsingle
vextendsingletan
parenleftBigx
2 +
pi
8
parenrightBigvextendsinglevextendsingle
vextendsingle+C
(30)
integraldisplay lnx
(1 +x2)32
dx =
integraldisplay
lnxd
parenleftbigg x
√1 +x2
parenrightbigg
= xlnx√1 +x2?
integraldisplay dx
√1 +x2 = xlnx√1 +x2?ln(x+
radicalbig
1 +x2) +C
(31)
integraldisplay
xexsinxdx = xexsinx?
integraldisplay
ex(sinx + xcosx)dx = xexsinx?
integraldisplay
exsinxdx?
integraldisplay
xexcosxdx =
xexsinx? sinx?cosx2 ex? e
x
2 (xcosx+xsinx?sinx) +C =
ex
2 (xsinx?xcosx+ cosx) +C
(32)
integraldisplay x3 arccosx
√1?x2 dx =?x2 arccosx
radicalbig
1?x2+2
integraldisplay
xarccosx
radicalbig
1?x2 dx?
integraldisplay
x2 dx =?x2
radicalbig
1?x2 arccosx?
x3
3?
2
3(1?x
2)32 arccosx? 2
3
integraldisplay
(1?x2)dx =?x2
radicalbig
1?x2 arccosx? x
3
3?
2
3(1?x
2)32 arccosx?
2
3
parenleftbigg
x? x
3
3
parenrightbigg
+C =?6x+x
3
9?
2 +x2
3
radicalbig
1?x2 arccosx+C
(33)
integraldisplay
(x+|x|)2 dx =
integraldisplay
(2x2+2x|x|)dx = 23x3+2
integraldisplay
x·sgnx·xdx = 23x3+23x3sgnx+C = 23x3+23x2|x|+C
(34)
integraldisplay
x2excosxdx = x2excosx?
integraldisplay
ex(2xcosx?x2 sinx)dx = x2excosx?ex(xcosx+xsinx?sinx)+
x2exsinx?integraldisplay
ex(2xsinx+x2 cosx)dx = x2ex(cosx+ sinx)?ex(xcosx+xsinx?sinx)?ex(xsinx?xcosx+
cosx)?
integraldisplay
x2excosxdx+C1 = ex(x2 cosx+x2 sinx?2xsinx+ sinx?cosx)?I +C1a167
a75I =
integraldisplay
x2excosxdx = e
x
2 (x
2 cosx+x2 sinx?2xsinx+ sinx?cosx) +C
116
(35)
integraldisplay xex
(1 +x)2 dx =?
integraldisplay
xexd 11 +x =? xe
x
1 +x +
integraldisplay
exdx =? xe
x
1 +x +e
x +C = ex
x+ 1 +C
(36)
integraldisplay √
xln2xdx = 23 ln2x·x32? 43
integraldisplay
x12 lnxdx = 23 ln2x·x32? 89x32 lnx+ 89
integraldisplay √
xdx = 23 ln2x·x32?
8
9x
3
2 lnx+ 1627x
3
2 +C = 227x
3
2 (9ln2x?12lnx+ 8) +C
(37) a45t =
radicalbiggb?x
x?aa167a75x =
b+at2
1 +t2,x?a =
b?a
1 +t2,dx =?
2(b?a)t
(1 +t2)2 dta167
a117a180
integraldisplay dx
radicalbig(x?a)(b?x) =?2
integraldisplay dt
1 +t2 =?2arctant+C =?2arctan
radicalbiggb?x
x?a +C
(38)
integraldisplay
xln 1 +x1?x dx = x
2
2 ln
1 +x
1?x?
integraldisplay x2
1?x2 dx =
x2
2 ln
1 +x
1?x?
integraldisplay dx
1?x2 +
integraldisplay
dx = 12x2 ln 1 +x1?x?
1
2 ln
1 +x
1?x +x+C
(39)
integraldisplay
xarctanx·ln(1 +x2)dx = x
2
2 arctanxln(1 +x
2)?
1
2
integraldisplay
x2
bracketleftbiggln(1 +x2)
1 +x2 +
2xarctanx
1 +x2
bracketrightbigg
dx = x
2
2 arctanxln(1+x
2)?1
2
integraldisplay
ln(1+x2)dx+12
integraldisplay ln(1 +x2)
1 +x2 dx?integraldisplay
xarctanxdx+
integraldisplay xarctanx
1 +x2 dx =
x2
2 arctanxln(1+x
2)?x
2 ln(1+x
2)+
integraldisplay x2
1 +x2 dx+
1
2 arctanxln(1+
x2)?
integraldisplay xarctanx
1 +x2 dx +
integraldisplay xarctanx
1 +x2 dx?
x2
2 arctanx +
1
2
integraldisplay x2
1 +x2 dx =
x2
2 arctanxln(1 + x
2)?
x
2 ln(1+x
2)+ 3
2x?
3
2 arctanx+
1
2 arctanxln(1+x
2)? x2
2 arctanx+C =
1
2 arctanx[x
2 ln(1+x2)+
ln(1 +x2)?x2?3]? x2 ln(1 +x2) + 32x+C
(40)
integraldisplay
sinh2xcosh2xdx = 14
integraldisplay
sinh2 2xdx = 18
integraldisplay
(cosh4x?1)dx = 132 sinh4x? x8 +C
117
a49a212a217 a189a200a169
§1 a189a200a169a27a86a103
a124a94a189a200a169a27a189a194a79a142a200a169a181
(1)
integraldisplay l
0
f(x)dxa167a217a165f(x) = ax+b,a,ba180a126a234
(2)
integraldisplay 2
1
x2 dx
(3)
integraldisplay 1
0
axdx
a41a181
(1) a207f(x)a51[0,l]a254a235a89a167a25a189a200a169a55a127a51a167a226a189a200a169a189a194a167a242a171a109[0,l]na31a169a167a75a122a152a102a171a109a27a127a143?xi = lna167
a18ξia143a122a135a102a171a109[xi?1,xi]a27a109a224a58a167a61ξi = inl(i = 1,2,···,n).
a138a200a169a218
nsummationdisplay
i=1
f(ξi)?xi =
nsummationdisplay
i=1
(aξi +b)?xi =
nsummationdisplay
i=1
parenleftbiggia
nl+b
parenrightbigg l
n =
nsummationdisplay
i=1
(nb+ia) ln2 = bl+ n+ 12n al2a167
a117a180
integraldisplay l
0
f(x)dx = lim
||x||= 1n→0
nsummationdisplay
i=1
(aξi +b)?xi = lim
n→∞
parenleftbigg
bl+ n+ 12n al2
parenrightbigg
= bl+ a2l2
(2) a207x2a51[?1,2]a254a235a89a167a25a189a200a169a55a127a51a167a226a189a200a169a189a194a167a242a171a109[?1,2]na31a169a167a75a122a152a102a171a109a27a127a143?xi =
3
na167a18ξia143a122a135a102a171a109[xi?1,xi]a27a109a224a58a167a61ξi =?1 +
3i
n =
3i?n
n (i = 1,2,···,n).
a138a200a169a218
nsummationdisplay
i=1
f(ξi)?xi =
nsummationdisplay
i=1
ξ2i?xi =
nsummationdisplay
i=1
parenleftbigg3i?n
n
parenrightbigg2 3
n =
nsummationdisplay
i=1
3
n3 (9i
2?6ni+n2) = 9
2
parenleftbigg
1 + 1n
parenrightbiggparenleftbigg
2 + 1n
parenrightbigg
9
parenleftbigg
1 + 1n
parenrightbigg
+ 3a167
a117a180
integraldisplay 2
1
x2 dx = lim
||x||= 1n→0
nsummationdisplay
i=1
ξ2i?xi = lim
n→∞
bracketleftbigg9
2
parenleftbigg
1 + 1n
parenrightbiggparenleftbigg
2 + 1n
parenrightbigg
9
parenleftbigg
1 + 1n
parenrightbigg
+ 3
bracketrightbigg
= 3
(3) a207axa51[0,1]a254a235a89a167a25a189a200a169a55a127a51a167a226a189a200a169a189a194a167a242a171a109[0,1]na31a169a167a75a122a152a102a171a109a27a127a143?xi = 1na167
a18ξia143a122a135a102a171a109[xi?1,xi]a27a109a224a58a167a61ξi = in(i = 1,2,···,n).
a138a200a169a218
nsummationdisplay
i=1
f(ξi)?xi =
nsummationdisplay
i=1
aξi?xi =
nsummationdisplay
i=1
1
na
i
n =
a1n(1?a)
n(1?a1n)
,anegationslash= 1
1,a = 1
a117a180
integraldisplay 1
0
axdx = lim
||x||= 1n→0
nsummationdisplay
i=1
aξi?xi =
lim
n→∞
a1n(1?a)
n(1?a1n)
= a?1lna,anegationslash= 1
1,a = 1
118
§2 a189a200a169a127a51a27a94a135
1,a7a228a101a15a188a234a27a140a200a53a181
(1) f(x)a51[?2,2]a254a107a46a167a167a27a216a235a89a58a1801n(n = 1,2,3,···)
(2) f(x) = sgn
parenleftBig
sin pix
parenrightBig
a167a51[0,1].
a41
(1) f(x)a51[?2,2]a254a180a140a200a27.
a207f(x)a51[?2,2]a254a107a46a167a25?M > 0a167a166|f(x)|lessorequalslantM,?x∈ [?2,2]a167a108a13a217a8a204ω(f) lessorequalslant 2M.
ε> 0a167a18a103a44a234Na247a118N =
bracketleftbigg2M
ε
bracketrightbigg
+1a167a117a180a51
bracketleftbigg 1
N,2
bracketrightbigg
a254f(x)a144a107a107a129a245a135a216a235a89a58a167a207a13f(x)a51
bracketleftbigg 1
N,2
bracketrightbigg
a254
a140a200.
a51
bracketleftbigg
0,1N
bracketrightbigg
a254a167a242a217a169a127a143a220a169a171a109?xia167a49ia135a2a171a109?xia254a27a8a204a23a143ωi(f) lessorequalslantω(f) lessorequalslant 2Ma167a75
nsummationdisplay
i=1
ωi(f)?xi lessorequalslant
2M
nsummationdisplay
i=1
xi lessorequalslant 2MN < 2M · ε2M = εa167a25f(x)a51
bracketleftbigg
0,1N
bracketrightbigg
a254a143a180a140a200a27.
a113a100a117f(x)a51[?2,0]a254a235a89a167a8a44a140a200.
a226a200a169a39a117a171a109a140a92a53a167a26f(x)a51[?2,2]a254a140a200.
(2) a214a191a189a194f(0) = 0.
a100a117f(x)a51[0,1]a254a107a46a167a113sgn
parenleftBig
sin pix
parenrightBig
a144a51x = 0,1n(n = 1,2,···)a109a228a167a25a100a29a75a1631a164a167a26f(x)a51[0,1]a254
a140a200.
2,a101a188a234f(x)a51[a,b]a254a140a200a167a217a200a169a180Ia167a56a51[a,b]a83a107a129a135a58a254a85a67f(x)a27a138a166a167a164a143a44a152a135a188a234f?(x)a167a121
a178f?(x)a143a51[a,b]a254a140a200a167a191a133a217a200a169a69a143I.
a121a178a181a45F(x) = f(x)?f?(x)a167a75F(x)a51[a,b]a254a216a85a67a10f(x)a27a188a234a138a27a107a129a135a58a9a254a1430a167a61a216a249a107a129a135
a58a9a167a188a234a235a89a167a108a13F(x)a51[a,b]a254a140a200a167a133a200a169a1430.
a113f?(x) = f(x)?F(x)a167a226a140a200a188a234a27a11a69a140a200a167
a107
integraldisplay b
a
f?(x)dx =
integraldisplay b
a
f(x)dx?
integraldisplay b
a
F(x)dx =
integraldisplay b
a
f(x)dx = I.
3,a63a216f,f2,|f|a110a246a109a140a200a53a27a39a88.
a41a181
(1) a101f(x)a51[a,b]a254a140a200a167a75f2(x)a51[a,b]a254a143a140a200.
a207f(x)a51[a,b]a254a140a200a167a25f(x)a51[a,b]a254a55a107a46,a23f(x) lessorequalslantM,Ma143a126a234(x∈ [a,b])
a51a171a109[xi?1,xi]a254a63a18a252a58xprime,xprimeprimea167a127a196f2(xprimeprime)?f2(xprime) = [f(xprimeprime)?f(xprime)][f(xprimeprime) +f(xprime)]
a18ωia76a171f(x)a51[xi?1,xi]a254a27a204a221a167a75|f2(xprimeprime)?f2(xprime)|lessorequalslant 2ωiM
a101f2(x)a51[xi?1,xi]a254a27a204a221a143?ia167a210a107?i lessorequalslant 2ωiMa167a108a13a107
summationdisplay
i
i?xi lessorequalslant 2M
summationdisplay
i
ωi?xi
a100a117f(x)a140a200a167a107
summationdisplay
i
ωi?xi → 0(λ(?) → 0)a167a75
summationdisplay
i
i?xi → 0(λ(?) → 0)a167a249a210a96a178a10f2(x)a51[a,b]a254
a27a140a200a53.
(2) a101f(x)a51[a,b]a254a140a200a167a75|f(x)|a51[a,b]a254a143a140a200.
a169a79a114a188a234f(x)a134|f(x)|a51a171a109[xi?1,xi]a254a27a204a221a80a143ωi,ω?i.
a207a233a225a117[xi?1,xi]a27a63a191a252a58xprime,xprimeprimea167a107||f(xprime)|?|f(xprimeprime)||lessorequalslant|f(xprime)?f(xprimeprime)|a167a25a107ω?i lessorequalslantωia167a117a180
summationdisplay
i
ω?i?xi lessorequalslant
summationdisplay
i
ωi?xi
a100a117
summationdisplay
i
ωi?xi → 0(λ(?) → 0)a167a210a140a177a237a26λ(?) → 0a158a167a143a107
summationdisplay
i
ω?i?xi → 0a167a249a210a96a178a10|f(x)a51[a,b]a254
a27a140a200a53.
(3) a101|f(x)|a51[a,b]a254a140a200a167a216a85a146a189f(x)a51[a,b]a254a143a140a200.
a126a181f(x) =
braceleftbigg 1,xa143a107a110a234
1,xa143a195a110a234
|f(x)| = 1a51a63a219a52a171a109a254a140a200a167a2f(x)a37a51a63a219a52a171a109a254a209a216a140a200.
119
(4) a101f2(x)a51[a,b]a254a140a200a167a216a85a146a189f(x)a51[a,b]a254a143a140a200.
a126a181f(x) =
braceleftbigg 1,xa143a107a110a234
1,xa143a195a110a234
f2(x) = 1a51a63a219a52a171a109a254a140a200a167a2f(x)a37a51a63a219a52a171a109a254a209a216a140a200.
(5) a101|f(x)|a51[a,b]a254a140a200a167a75a100(1)a167a26f2(x) = |f(x)|2a51[a,b]a254a152a189a140a200.
(6) a101f2(x)a51[a,b]a254a140a200a167a75a100|f(x)| =radicalbigf2(x)a167a9a0a188a234a235a89a167a112a0a188a234a140a200a167a26a100a188a234a55a140a200.
4,a101a188a234f(x)a51[a,b]a254a140a200a167a121a178a127a51a242a130a188a234a15?n(x)(n = 1,2,3,···)a166a26
integraldisplay b
a
f(x)dx = lim
n→∞
integraldisplay b
a
n(x)dx
a121a178a181a242[a,b]na31a169a167a23a169a58a143a = x(n)0 <x(n)1 <···<x(n)n?1 <x(n)n = ba61x(n)i = a+ in(b?a),i = 0,1,···,n
a51[x(n)i?1,x(n)i ]a254a45?n(x)a143a76a58(x(n)i?1,f(x(n)i?1))a57(x(n)i,f(x(n)i ))a27a134a130a167a61a8x ∈ (x(n)i?1,x(n)i )a158a167a45?n(x) =
f(x(n)i?1) + x?x
(n)
i?1
x(n)i?x(n)i?1
(f(x(n)i )?f(x(n)i?1))
a75?n(x)a180[a,b]a254a27a152a135a242a130a188a234a15a167a8a44a180a235a89a188a234a15a167a207a100
integraldisplay b
a
n(x)dxa107a189a194.
a101a45m(n)i,M(n)i a57ω(n)i a169a79a76a171a188a234f(x)a51[x(n)i?1,x(n)i ]a254a27a101a40a46a33a254a40a46a57a8a204a167a75a8x∈ [x(n)i?1,x(n)i ]a158a167m(n)i lessorequalslant
n(x) lessorequalslantM(n)i,m(n)i lessorequalslantf(x) lessorequalslantM(n)i a167a108a13|?n(x)?f(x)|lessorequalslantω(n)i
a117a180a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay b
a
f(x)dx?
integraldisplay b
a
n(x)dx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay b
a
|f(x)n(x)|dx =
nsummationdisplay
i=1
integraldisplay x(n)
i
x(n)i?1
|f(x)n(x)|dxlessorequalslant
nsummationdisplay
i=1
ω(n)i?x(n)i
a100a117f(x)a51[a,b]a254a140a200a167a25a8max|?x(n)i | = b?an → 0a158a167a55a107
nsummationdisplay
i=1
ω(n)i?x(n)i → 0a167a207a13
integraldisplay b
a
f(x)dx =
lim
n→∞
integraldisplay b
a
n(x)dx.
5,a101a188a234f(x)a51[A,B]a140a200a167a121a178lim
h→0
integraldisplay b
a
|f(x+h)?f(x)|dx = 0a167a217a165A<a<b<B(a249a152a53a159a161a143a200a169a27
a235a89a53)
a121a178a181a207f(x)a51[A,B]a140a200a167a100a254a75a40a216a167a233?ε> 0a167a127a51[A,B]a254a27a235a89a188a234?(x)a167a166
integraldisplay B
A
|f(x)(x)|dx<
ε
4
a207?(x)a51[A,B]a235a89a167a108a13a152a151a235a89a167a75a233a254a227ε > 0a167a127a51δ > 0a167a233[A,B]a165a63a191a252a58xprime,xprimeprimea167a144a135|xprime?
xprimeprime|<δa167a210a107|?(xprime)(xprimeprime)|< ε2(b?a)
a117a180a8|h|<δa158a167a107
integraldisplay b
a
|f(x+h)?f(x)|dx =
integraldisplay b
a
|f(x+h)(x+h)+?(x+h)(x)+?(x)?f(x)|dxlessorequalslant
integraldisplay b
a
|f(x+h)(x+h)|dx+
integraldisplay b
a
|?(x+h)(x)|dx+
integraldisplay b
a
|f(x)(x)|dxlessorequalslant
integraldisplay B
A
|f(x+h)(x+h)|dx+
integraldisplay b
a
|?(x+h)(x)|dx+
integraldisplay B
A
|f(x)(x)|dx< ε4 +
integraldisplay b
a
ε
2(b?a) dx+
ε
4 = ε
a108a13lim
h→0
integraldisplay b
a
|f(x+h)?f(x)|dx = 0
120
§3 a189a200a169a27a53a159
1,a101f(x),g(x)a51[a,b]a140a200a167a121a178f(x)+g(x)a143a51[a,b]a140a200a167a191a133
integraldisplay b
a
(f(x)+g(x))dx =
integraldisplay b
a
f(x)dx+
integraldisplay b
a
g(x)dx.
a121a178a181a207f(x),g(x)a51[a,b]a140a200a167a61
integraldisplay b
a
f(x)dx,
integraldisplay b
a
g(x)dxa127a51a167a25a233a63a191a169a123?,a = x0 < x1 < ··· <
xn = ba177a57[xi?1,xi]a165a63a191ξia167a107 lim
λ(?)→0
nsummationdisplay
i=1
f(ξi)?xi =
integraldisplay b
a
f(x)dx
a100a169a123?a57ξia27a63a191a53a167a26g(x)a51a100a63a191a169a123a101a167a233a254a227[xi?1,xi]a165a27ξia167a143a107 lim
λ(?)→0
nsummationdisplay
i=1
g(ξi)?xi =
integraldisplay b
a
g(x)dxa167
a117a180
integraldisplay b
a
f(x)dx+
integraldisplay b
a
g(x)dx = lim
λ(?)→0
nsummationdisplay
i=1
f(ξi)?xi+ lim
λ(?)→0
nsummationdisplay
i=1
g(ξi)?xi = lim
λ(?)→0
nsummationdisplay
i=1
(f(ξi)+g(ξi))?xi =
integraldisplay b
a
(f(x) +g(x))dx
a108a13f(x) +g(x)a143a51[a,b]a140a200a167a191a133
integraldisplay b
a
(f(x) +g(x))dx =
integraldisplay b
a
f(x)dx+
integraldisplay b
a
g(x)dx.
2,a23f(x) =
braceleftbigg 1,a8xa143a107a110a234
1,a8xa143a195a110a234
a121a178a181|f(x)|a51a63a219a171a109[a,b]a254a140a200a167a2f(x)a51[a,b]a216a140a200.
a121a178a181a207f(x) =
braceleftbigg 1,a8xa143a107a110a234
1,a8xa143a195a110a234 a167a25|f(x)| = 1a167
a75|f(x)|a51[a,b]a254a235a89a167a108a13|f(x)|a51[a,b]a254a140a200
a233a117a188a234f(x)a167a51[a,b]a27a63a152a220a169a171a109[xi?1,xi](i = 1,2,···,n)a254ωi = 2a167a25
nsummationdisplay
i=1
ωi?xi = 2
nsummationdisplay
i=1
xi =
2(b?a)notarrowright0(n→∞)a167a117a180a188a234f(x)a51[a,b]a254a216a140a200.
3,a23f(x)a51[a,b]a235a89a167f(x) greaterorequalslant 0a167f(x)a216a240a143a34a167a121a178
integraldisplay b
a
f(x)dx> 0.
a121a178a181a207f(x) greaterorequalslant 0a133a216a240a143a34a167a75a55a127a51x0 ∈ [a,b]a167a166a26f(x0) > 0
a100a235a89a188a234a27a219a220a2a210a53a167a127a510 <δlessorequalslant min
parenleftbiggx
0?a
2,
b?x0
2
parenrightbigg
a167a166a8x∈ [x0?δ,x0+δ]a158a167f(x) > f(x0)2 >
0a167a117a180a107
integraldisplay b
a
f(x)dxgreaterorequalslant
integraldisplay x0+δ
x0?δ
f(x)dxgreaterorequalslant
integraldisplay x0+δ
x0?δ
f(x0)
2 dx = f(x0)δ> 0.
4,a39a22a101a15a136a75a165a200a169a27a140a2a181
(1)
integraldisplay 1
0
xdx,
integraldisplay 1
0
x2 dx
(2)
integraldisplay pi
2
0
xdx,
integraldisplay pi
2
0
sinxdx
(3)
integraldisplay?1
2
parenleftbigg1
3
parenrightbiggx
dx,
integraldisplay 1
0
3xdx
a41a181
(1) a207x∈ (0,1)a158a167x>x2a167a75
integraldisplay 1
0
xdx>
integraldisplay 1
0
x2 dx
(2) a207x∈
parenleftBig
0,pi2
parenrightBig
a158a167x> sinxa167a75
integraldisplay pi
2
0
xdx>
integraldisplay pi
2
0
sinxdx
(3) a207
integraldisplay?1
2
parenleftbigg1
3
parenrightbiggx
dx =
integraldisplay 1
0
parenleftbigg1
3
parenrightbiggx?2
dx =
integraldisplay 1
0
32?xdxa133a8x∈ (0,1)a158a1672?x>xa167a2532?x > 3xa167
a108a13
integraldisplay?1
2
parenleftbigg1
3
parenrightbiggx
dx>
integraldisplay 1
0
3xdx
5,a23f(x)a51[a,b]a235a89a167
integraldisplay b
a
f2(x)dx = 0a167a121a178f(x)a51[a,b]a254a240a143a34.
a121a178a181a94a135a121a123.a98a23f(x)a51[a,b]a254a216a240a143a34a167a75f2(x) greaterorequalslant 0a133a216a240a143a34.
121
a113f(x)a51[a,b]a235a89a167a25f2(x)a51[a,b]a235a89a167
a75a226a493a75a140a127
integraldisplay b
a
f2(x)dx> 0a167a249a134a174a127
integraldisplay b
a
f2(x)dx = 0a103a241.
a117a180a98a23a134a216a167a108a13f(x)a51[a,b]a254a240a143a34.
6,a222a126a96a178a181f2(x)a51[a,b]a140a200a167a2f(x)a51[a,b]a216a140a200.
a41a181a126a181f(x) =
braceleftbigg 1,a8xa143a107a110a234
1,a8xa143a195a110a234 a167a25f
2(x) = 1a167
a75f2(x)a51[a,b]a254a235a89a167a108a13f2(x)a51[a,b]a254a140a200
a113a100a49a19a75a140a127a167a188a234f(x)a51[a,b]a254a216a140a200.
7,a23f(x),g(x)a51[a,b]a235a89a167a121a178 lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(θi)?xi =
integraldisplay b
a
f(x)g(x)dxa167a217a165xi?1 lessorequalslant ξi lessorequalslant xi,xi?1 lessorequalslant
θi lessorequalslantxi(i = 1,2,···,n),?xi = xi?xi?1(x0 = a,xn = b).
a121a178a181a207f(x),g(x)a51[a,b]a235a89a167a75f(x)·g(x)a51[a,b]a235a89a167a25f(x)·g(x)a51[a,b]a140a200a167a61 lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(ξi)?xi =
integraldisplay b
a
f(x)g(x)dx
a13 lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(θi)?xi = lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(ξi)?xi +
lim
max(?xi)→0
bracketleftBigg nsummationdisplay
i=1
f(ξi)g(θi)?xi?
nsummationdisplay
i=1
f(ξi)g(ξi)?xi
bracketrightBigg
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
i=1
f(ξi)g(θi)?xi?
nsummationdisplay
i=1
f(ξi)g(ξi)?xi
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
i=1
f(ξi)(g(θi)?g(ξi))?xi
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
i=1
|f(ξi)||g(θi)?g(ξi)|?xi lessorequalslant
nsummationdisplay
i=1
M(f)ωi(g)?xi = M(f)
nsummationdisplay
i=1
ωi(g)?xia167a217a165M(f)a76a171|f|a51[a,b]a254a27a254a46a167ωi(g)a76a171ga51[xi?1,xi]a254a27a8
a204.
a100fa27a235a89a53a218ga27a140a200a53a167a8max(?xi) → 0a158a167a254a161a216a31a170a109a224M(f)
nsummationdisplay
i=1
ωi(g)?xi → 0a167a108a13 lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(θi)?xi =
integraldisplay b
a
f(x)g(x)dx.
8,a23y =?(x)(x greaterorequalslant 0)a180a238a130a252a78a79a92a27a235a89a188a234a167?(0) = 0,x = ψ(x)a180a167a27a135a188a234a167a121a178
integraldisplay a
0
(x)dx +
integraldisplay b
a
ψ(y)dygreaterorequalslantab(agreaterorequalslant 0,bgreaterorequalslant 0).
a121a178a181a100y =?(x)a143a180a238a130a252a78a79a92a27a235a89a188a234a167?(0) = 0a127a217a135a188a234x = ψ(y)a180a238a130a252a78a79a92a27a235a89a188
a234a167a133ψ(0) = 0a167a207a13
integraldisplay a
0
(x)dx,
integraldisplay b
0
ψ(y)dya107a189a194
a45g(x) = bx?
integraldisplay x
0
(t)dta167a65a79a47a167a107
g(a) = ab?
integraldisplay a
0
(t)dt (1)
a13a133gprime(x) = b(x)
a100?(x)a180a238a130a252a78a79a92a27a235a89a188a234a167a207a100a80 < x < ψ(b)a158a167a107gprime(x) > 0a182a8x > ψ(b)a158a167a107gprime(x) < 0a182
a8x = ψ(b)a158a167a107gprime(x) = 0a167a207a100a8x = ψ(b)a158a167g(x)a18a129a140a138a167a61a107
g(a) lessorequalslant maxg(x) = g(ψ(b)) (2)
a169a220a200a169a167a26
integraldisplay ψ(b)
0
x?prime(x)dx = bψ(b)?
integraldisplay ψ(b)
0
(x)dx = g(ψ(b))a167a94a67a254a147a134y =?(x)a167a75x = ψ(y)a167a117a180
g(ψ(b)) =
integraldisplay ψ(b)
0
x?prime(x)dx =
integraldisplay b
0
ψ(y)dy (3)
a242()a33()a147a92()a210a26a20
integraldisplay a
0
(x)dx+
integraldisplay b
a
ψ(y)dygreaterorequalslantab(agreaterorequalslant 0,bgreaterorequalslant 0).
122
§4 a189a200a169a27a79a142
1,a79a142a101a15a189a200a169a181
(1)
integraldisplay 2
1
(x+ 1)(x2?3)
3x2 dx
(2)
integraldisplay pi
2
1
(asinx+bcosx)dx
(3)
integraldisplay 1
0
parenleftbiggx?1
x+ 1
parenrightbigg4
dx
(4)
integraldisplay 1
0
x2 + 1
x4 + 1 dx
(5)
integraldisplay 1√
5
0
x3(1?5x2)10 dx
(6)
integraldisplay 1
0
x2(2?3x2)2 dx
(7)
integraldisplay 1
5
15
x√2?5xdx
(8)
integraldisplay pi
2
0
sinmxcosnxdx
(9)
integraldisplay 1
3
13
18x?4√
9x2 + 6x+ 5 dx
(10)
integraldisplay √ln2
0
x3e?x2 dx
(11)
integraldisplay 1
0
xarctanxdx
(12)
integraldisplay 2pi
0
xcos2xdx
(13)
integraldisplay pi
pi
x2 cosxdx
(14)
integraldisplay 2pi
ω
0
sinωtsin(ωt+?)dt
(15)
integraldisplay 3
0
xdx
1 +√1 +x
(16)
integraldisplay 4
0
x(x+√x)dx
(17)
integraldisplay pi
pi
sinmxsinnxdx
(18)
integraldisplay pi
pi
sinmxcosnxdx
(19)
integraldisplay 0
1
(x+ 1)
radicalbig
1?x?x2 dx
(20)
integraldisplay 0.75
0
dx
(x+ 1)√x2 + 1
a41a181
(1)
integraldisplay 2
1
(x+ 1)(x2?3)
3x2 dx =
integraldisplay 2
1
x3 +x2?3x+ 3
3x2 dx =
1
3
integraldisplay 2
1
parenleftbigg
x+ 1? 3x? 3x2
parenrightbigg
dx = 13
parenleftbiggx2
2 +x?3lnx+
3
x
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
2
1
= 13?ln2
123
(2)
integraldisplay pi
2
1
(asinx+bcosx)dx = (?acosx+bsinx)|pi20 = a+b
(3) a45y = x+ 1a167a75
integraldisplay 1
0
parenleftbiggx?1
x+ 1
parenrightbigg4
dx =
integraldisplay 2
1
parenleftbiggy?2
y
parenrightbigg4
dy =
integraldisplay 2
1
parenleftbigg
1? 8y + 24y2? 32y3 + 16y4
parenrightbigg
dy =
parenleftbigg
y?8lny? 24y + 16y2? 163y3
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
2
1
= 173?8ln2
(4)
integraldisplay 1
0
x2 + 1
x4 + 1 dx =
1
2
integraldisplay 1
0
bracketleftbigg 1
x2 +√2x+ 1 +
1
x2?√2x+ 1
bracketrightbigg
dx =
1
2 [
√2arctan(√2x+ 1) +√2arctan(√2x?1)]vextendsinglevextendsingle
vextendsingle
1
0
=
√2
2 (arctan(
√2 + 1) + arctan(√2?1)) = √2
4 pi
(5) a45x = 1√5 sinu,dx = 1√5 cosudua167
a75
integraldisplay 1√
5
0
x3(1?5x2)10 dx = 125
integraldisplay pi
2
0
sin3ucos21udu =? 125
integraldisplay pi
2
0
(1?cos2u)cos21udcosu =? 125
parenleftbiggcos22u
22?
cos24u
24
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
pi
2
0
=
1
6600
(6)
integraldisplay 1
0
x2(2?3x2)2 dx =
integraldisplay 1
0
(4x2?12x4 + 9x6)dx = 23105
(7) a45u = √2?5x
a75
integraldisplay 1
5
15
x√2?5xdx = 225
integraldisplay √3
1
(2u2?u3)du = 2375(3√3?7)
(8) a8mnegationslash= ±na158a167
integraldisplay pi
2
0
sinmxcosnxdx = 12
integraldisplay pi
2
0
[sin(m+n)x+sin(m?n)x]dx = mm2?n2?
cos m+n2 pi
2(m+n)?
cos m?n2 pi
2(m?n) a182
a8m = ±na133m negationslash= 0a158a167
integraldisplay pi
2
0
sinmxcosnxdx = ±12
integraldisplay pi
2
0
sin2nxdx =? 14n cos2nx
vextendsinglevextendsingle
vextendsingle
pi
2
0
= ± 14n(1?
cosnpi) = ± 14n[1?(?1)n](n∈Za133nnegationslash= 0)a182
a8m = 0a158a167
integraldisplay pi
2
0
sinmxcosnxd = 0
(9)
integraldisplay 1
3
13
18x?4√
9x2 + 6x+ 5 dx =
integraldisplay 1
3
13
d(9x2 + 6x+ 5)√
9x2 + 6x+ 5?
10
3
integraldisplay 1
3
13
dxradicalBig
1 +parenleftbig3x+12 parenrightbig2
=
2radicalbig9x2 + 6x+ 5? 103 ln
3x+ 12 +
radicalBigg
1 +
parenleftbigg3x+ 1
2
parenrightbigg2?
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1
3
13
= 4(√2?1)? 103 ln(√2 + 1)
(10) a45u = x2integraldisplay √
ln2
0
x3e?x2 dx = 12
integraldisplay ln2
0
ue?udu = 12
parenleftbigg
ue?u
vextendsinglevextendsingle
vextendsingle
ln2
0
+
integraldisplay ln2
0
e?udu
parenrightbigg
= 14(1?ln2)
(11)
integraldisplay 1
0
xarctanxdx = x
2
2 arctanx
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
12
integraldisplay 1
0
x2
1 +x2 dx =
pi
8?
1
2 +
1
2 arctanx
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
= pi?24
(12)
integraldisplay 2pi
0
xcos2xdx = 12
integraldisplay 2pi
0
x(1 + cos2x)dx = x
2
4
vextendsinglevextendsingle
vextendsinglevextendsingle
2pi
0
+ 14xsin2x
vextendsinglevextendsingle
vextendsinglevextendsingle
2pi
0
14
integraldisplay 2pi
0
sin2xdx = pi2
(13)
integraldisplay pi
pi
x2 cosxdx = 2
integraldisplay pi
0
x2 cosxdx = 2x2 sinx
vextendsinglevextendsingle
vextendsingle
pi
0
4
integraldisplay pi
0
xsinxdx = 4xcosx
vextendsinglevextendsingle
vextendsingle
pi
0
4
integraldisplay pi
0
cosxdx =?4pi
(14)
integraldisplay 2pi
ω
0
sinωtsin(ωt+?)dt = 12
integraldisplay 2pi
ω
0
[coscos(2ωt+?)]dt = piω cos 14ω sin(2ωt+?)
vextendsinglevextendsingle
vextendsinglevextendsingle
2pi
ω
0
= piω cos?
(15)
integraldisplay 3
0
xdx
1 +√1 +x =
integraldisplay 3
0
x(1?√1 +x)
x dx =
integraldisplay 3
0
(√1 +x?1)dx = 23(1 +x)32
vextendsinglevextendsingle
vextendsinglevextendsingle
3
0
3 = 53
(16)
integraldisplay 4
0
x(x+√x)dx =
integraldisplay 4
0
(x2 +x32 )dx = 51215
124
(17) a8mnegationslash= ±n(m,n∈Z)a158a167
integraldisplay pi
pi
sinmxsinnxdx = 12
integraldisplay pi
pi
[cos(m?n)x?cos(m+n)x]dx = sin(m?n)x2(m?n)
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
pi
sin(m+n)x
2(m+n)
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
pi
= 0a182
a8m = ±n(m,n∈Z)a133mnegationslash= 0a158a167
integraldisplay pi
pi
sinmxsinnxdx = ±
integraldisplay pi
pi
sin2mxdx = ±
integraldisplay pi
0
(1?cos2mx)dx =
±pia182
a8m = 0a158a167
integraldisplay pi
pi
sinmxsinnxdx = 0
(18)
integraldisplay pi
pi
sinmxcosnxdx = 0
(19)
integraldisplay 0
1
(x+1)
radicalbig
1?x?x2 dx =?12
integraldisplay 0
1
(1?x?x2)12 d(1?x?x2)+12
integraldisplay 0
1
radicalbig
1?x?x2 dx = 12
integraldisplay 0
1
radicalbig
1?x?x2 dx =
1
4 +
5
8 arcsin
√5
5
(20) a45x = tant
a75
integraldisplay 0.75
0
dx
(x+ 1)√x2 + 1 =
integraldisplay arctan0.75
0
dt
sint+ cost =
integraldisplay arctan0.75
0
d
parenleftBig
t+ pi4
parenrightBig
√2sinparenleftBigt+ pi
4
parenrightBig =
√2
2 lntan
parenleftbiggt
2 +
pi
8
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
arctan0.75
0
= 1√2 ln
tan
parenleftbiggarctan0.75
2 +
pi
8
parenrightbigg
tan pi8
= 1√2 ln 9 + 4
√2
7
2,a79a142a101a15a200a169a181
(1)
integraldisplay pi
2
0
sin7xdx
(2)
integraldisplay pi
2
0
cos4xdx
(3)
integraldisplay pi
0
sin5xdx
(4)
integraldisplay 2pi
0
cos6xdx
(5)
integraldisplay a
0
(a2?x2)ndx
(6)
integraldisplay 1
0
(1?x2)6 dx
a41a181
(1)
integraldisplay pi
2
0
sin7xdx = 6!!7!!
(2)
integraldisplay pi
2
0
cos4xdx = 3!!4!! · pi2 = 3pi16
(3)
integraldisplay pi
0
sin5xdx =
integraldisplay pi
2
0
sin5xdx+
integraldisplay pi
pi
2
sin5xdx
a51a0a152a200a169a165a167a45x = pi? ya167a75sinx = siny,dx =?dya167a117a180
integraldisplay pi
pi
2
sin5xdx =?
integraldisplay 0
pi
2
sin5ydy =
integraldisplay pi
2
0
sin5ydy = I5
a108a13
integraldisplay pi
0
sin5xdx = 2I5 = 2· 4!!5!! = 1615
(4)
integraldisplay 2pi
0
cos6xdx =
integraldisplay pi
pi
cos6xdx = 2
integraldisplay pi
0
cos6xdx = 4
integraldisplay pi
2
0
cos6xdx = 4· 5!!6!! · pi2 = 1524pi
(5) a45x = acost,dx =?asintdta167
a75
integraldisplay a
0
(a2?x2)ndx =?a2n+1
integraldisplay 0
pi
2
sin2n+1tdt = a2n+1
integraldisplay pi
2
0
sin2n+1tdt = (2n)!!(2n+ 1)!!a2n+1
125
(6) a51a254a75a165a167a45a = 1,n = 6a167a75
integraldisplay 1
0
(1?x2)6 dx = 12!!13!!
3,a23f(x)a180a177a207a188a234a167a177a207a180Ta167a121a178
integraldisplay a+nT
a
f(x)dx = n
integraldisplay T
0
f(x)dxa167a100a63na180a20a18a234.
a121a178a181
integraldisplay a+nT
a
f(x)dx =
integraldisplay 0
a
f(x)dx+
integraldisplay T
0
f(x)dx+···+
integraldisplay nT
(n?1)T
f(x)dx+
integraldisplay a+nT
nT
f(x)dx
a233a254a227a31a170a27a129a0a152a135a200a169a167a23x?nT = ta167a75
integraldisplay a+nT
nT
f(x)dx =
integraldisplay a
0
f(t+nT)dt =
integraldisplay a
0
f(t)dt
a2331 <i<na167a127a196a200a169
integraldisplay iT
(i?1)T
f(x)dxa167a23x?(i?1)T = ta167a75
integraldisplay iT
(i?1)T
f(x)dx =
integraldisplay T
0
f(t+ (n?1)T)dt =
integraldisplay T
0
f(t)dt
a108a13
integraldisplay a+nT
a
f(x)dx = n
integraldisplay T
0
f(x)dx
4,a121a178a181(m,na143a20a18a234)
(1)
integraldisplay pi
pi
sin2mxdx = pi,
integraldisplay pi
pi
cos2mxdx = pi
(2)
integraldisplay pi
pi
cosmxcosnxdx = 0(mnegationslash= n)
a121a178a181
(1)
integraldisplay pi
pi
sin2mxdx = 1m
integraldisplay pi
pi
sin2mxdmx = 1m
integraldisplay pi
0
(1?cos2mx)dmx = pi? 12m sin2mx
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
0
= pi
a211a110a140a26a167
integraldisplay pi
pi
cos2mxdx = pi
(2)
integraldisplay pi
pi
cosmxcosnxdx = 12
integraldisplay pi
pi
[cos(m+n)x+cos(m?n)x]dx = sin(m+n)x(m+n)
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
0
+ sin(m?n)x(m?n)
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
0
= 0
5,a121a178a101a188a234f(x)a51a52a171a109[0,1]a235a89a167a75a181
(1)
integraldisplay pi
2
0
f(sinx)dx =
integraldisplay pi
2
0
f(cosx)dx
(2)
integraldisplay pi
0
xf(sinx)dx = pi2
integraldisplay pi
0
f(sinx)dx
a100a100a79a142
integraldisplay pi
0
xsinx
1 + cos2x dx
a121a178a181
(1) a45pi2?t = xa167a75dx =?dt,f(sinx) = f(cosx)a167
a117a180
integraldisplay pi
2
0
f(sinx)dx =?
integraldisplay 0
pi
2
f(cost)dt =
integraldisplay pi
2
0
f(cost)dt
a61
integraldisplay pi
2
0
f(sinx)dx =
integraldisplay pi
2
0
f(cosx)dx
(2) a23t = pi?xa167a75dx =?dta133xf(sinx) = (pi?t)f(sint)
a117a180
integraldisplay pi
0
xf(sinx)dx =?
integraldisplay 0
pi
(pi?t)f(sint)dt = pi
integraldisplay pi
0
f(sint)dt?
integraldisplay pi
0
tf(sint)dta167a752
integraldisplay pi
0
xf(sinx)dx =
pi
integraldisplay pi
0
f(sinx)dxa167a108a13
integraldisplay pi
0
xf(sinx)dx = pi2
integraldisplay pi
0
f(sinx)dx
integraldisplay pi
0
xsinx
1 + cos2x dx =
pi
2
integraldisplay pi
0
sinx
1 + cos2x dx =?
pi
2
integraldisplay pi
0
dcosx
1 + cos2x =?
pi
2 arctan(cosx)
vextendsinglevextendsingle
vextendsingle
pi
0
= pi
2
4
6,a121a178a219a188a234a27a152a131a6a188a234a27a143a243a188a234a167a243a188a234a27a6a188a234a165a107a152a143a219a188a234.
a121a178a181a23f(x)a51[?l,l]a254a107a189a194a167a133F(x)a180f(x) a27a152a135a6a188a234
a8f(x)a143a219a188a234a61a8f(?x) =?f(x)a158a167a100a117f(x) = ddxF(x),
f(?x) =? ddxF(?x)a167a25a107 ddx[F(x)?F(?x)] = 0a167a108a13a140a26F(x)?F(?x) = C1a133C1 = 0a167a117a180F(x) =
F(?x)a167a75f(x)a27a152a135a6a188a234F(x)a143a243a188a234a167a108a13f(x)a27a63a152a135a6a188a234F(x) +C(Ca143a63a191a126a234)a143a143a243a188a234
a8f(x)a143a243a188a234a61a8f(?x) = f(x)a158a167a97a113a140a26F(x) + F(?x) = C2a133C2 = 2F(0)a167a117a180f(x)a107a152a135a6a188
a234F(x)?F(0)a180a219a188a234.
126
7,a101f(x)a39a117x = Ta233a161a167a133a<T <ba167a75
integraldisplay b
a
f(x)dx = 2
integraldisplay b
T
f(x)dx+
integraldisplay 2T?b
a
f(x)dx
a121a178a181
integraldisplay b
a
f(x)dx =
integraldisplay 2T?b
a
f(x)dx+
integraldisplay T
2T?b
f(x)dx+
integraldisplay b
T
f(x)dx
a233a254a227a31a170a109a224a27a49a19a135a200a169a167a23x = 2T?ta167a75
integraldisplay T
2T?b
f(x)dx =?
integraldisplay T
b
f(2T?t)dt
a113f(x)a39a117x = Ta233a161a167a75f(2T?t) = f(t)
a117a180
integraldisplay T
2T?b
f(x)dx =?
integraldisplay T
b
f(2T?t)dt =?
integraldisplay T
b
f(t)dt =
integraldisplay b
T
f(t)dta167a108a13
integraldisplay b
a
f(x)dx = 2
integraldisplay b
T
f(x)dx +
integraldisplay 2T?b
a
f(x)dx
8,a121a178a181
integraldisplay a
0
x3f(x2)dx = 12
integraldisplay a2
0
xf(x)dx(a> 0)
a121a178a181a45t = x2a167a752xdx = dta167a117a180
integraldisplay a
0
x3f(x2)dx = 12
integraldisplay a2
0
tf(t)dt = 12
integraldisplay a2
0
xf(x)dx
9,a124a94a169a220a200a169a121a178a181
integraldisplay x
0
f(u)(x?u)du =
integraldisplay x
0
braceleftbiggintegraldisplay u
0
f(x)dx
bracerightbigg
du
a121a178a181
integraldisplay x
0
braceleftbiggintegraldisplay u
0
f(x)dx
bracerightbigg
du = u
integraldisplay u
0
f(x)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
x
0
integraldisplay x
0
uf(u)du = x
integraldisplay x
0
f(t)dt?
integraldisplay x
0
uf(u)du =
integraldisplay x
0
xf(u)du?
integraldisplay x
0
uf(u)du =
integraldisplay x
0
f(u)(x?u)du
10,a152a127a221a143la27a238a249a167a164a201a49a214a85a53a198p(x) = a+bx+cx2a169a217a167a193a100a101a227a94a135a251a189a88a234a,b,ca182a111a49a214a180P =integraldisplay
l
0
p(x)dxa167a52a140a49a214a160a11723la63a167a133a51a52a140a58a27a134a109a252a62a136a171a201a111a49a214a27a152a140.
a41a181a100a174a127a167a26
(1) P =
integraldisplay l
0
p(x)dx = al+ b2l2 + c3l3
(2) a45P(x) =
integraldisplay x
0
p(t)dta167a75Pprime
parenleftbigg2
3l
parenrightbigg
= p
parenleftbigg2
3l
parenrightbigg
= a+ 23bl+ 49cl2 = 0
(3)
integraldisplay 2
3l
0
p(x)dx = 23al+ 29bl+ 881cl3 = P2
a233a225a144a167a124
al+ b2l2 + c3l3 = P
2
3al+
2
9bl+
8
81cl
3 = P
2
a+ 23bl+ 49cl2 = 0
a166a41a167a26
a = 4lP
b =? 694l2P
c = 1358l3 P
11,a101f(x)a235a89a167a166a181
(1) ddx
parenleftbiggintegraldisplay b
x
f(t)dt
parenrightbigg
(2) ddx
parenleftBiggintegraldisplay
x2
0
f(t)dt
parenrightBigg
a41a181
(1) ddx
parenleftbiggintegraldisplay b
x
f(t)dt
parenrightbigg
= ddx
parenleftbigg
integraldisplay x
b
f(t)dt
parenrightbigg
=? ddx
parenleftbiggintegraldisplay x
b
f(t)dt
parenrightbigg
=?f(x)
(2) a207 ddx2
parenleftBiggintegraldisplay
x2
0
f(t)dt
parenrightBigg
= f(x2)a167a25 ddx
parenleftBiggintegraldisplay
x2
0
f(t)dt
parenrightBigg
= ddx2
parenleftBiggintegraldisplay
x2
0
f(t)dt
parenrightBigg
· dx
2
dx = 2xf(x
2)
12,a166a52a129a181
127
(1) lim
n→∞
parenleftbigg 1
n2 +
2
n2 +···+
n?1
n2
parenrightbigg
(2) lim
n→∞
1p + 2p +···+np
np+1 (p> 0)
(3) lim
n→∞
1
n
parenleftBiggradicalbigg
1 + 1n +
radicalbigg
1 + 2n +···+
radicalbigg
1 + n?1n
parenrightBigg
(4) lim
n→∞
n√n!
n
a41a181
(1) a188a234f(x) = xa51[0,1]a235a89a167a207a13a140a200.a242[0,1]na31a169a167a169a58a143in,?xi = 1n(i = 0,1,···,n?1)a167a51a122a135a2
a171a109[xi,xi+1] =
bracketleftbiggi
n,
i+ 1
n
bracketrightbigg
a167a18ξi = ina167a75f(ξi) = ina167
a117a180 lim
n→∞
parenleftbigg 1
n2 +
2
n2 +···+
n?1
n2
parenrightbigg
= lim
n→∞
1
n
parenleftbigg1
n +
2
n +···+
n?1
n
parenrightbigg
= lim
n→∞
n?1summationdisplay
i=0
i
n·
1
n =
integraldisplay 1
0
xdx =
1
2
(2) a188a234f(x) = xpa51[0,1]a235a89a167a207a13a140a200.a242[0,1]na31a169a167a169a58a143in,?xi = 1n(i = 1,2,···,n)a167a51a122a135a2a171
a109[xi?1,xi] =
bracketleftbiggi?1
n,
i
n
bracketrightbigg
a167a18ξi = ina167a75f(ξi) =
parenleftbiggi
n
parenrightbiggp
a167
a117a180 lim
n→∞
1p + 2p +···+np
np+1 = limn→∞
nsummationdisplay
i=1
parenleftbiggi
n
parenrightbiggp
· 1n =
integraldisplay 1
0
xpdx = 1p+ 1
(3) a188a234f(x) = √1 +xa51[0,1]a235a89a167a207a13a140a200.a242[0,1]na31a169a167a169a58a143in,?xi = 1n(i = 0,1,···,n? 1)a167a51
a122a135a2a171a109[xi,xi+1] =
bracketleftbiggi
n,
i+ 1
n
bracketrightbigg
a167a18ξi = ina167a75f(ξi) =
radicalbigg
1 + ina167
a117a180 lim
n→∞
1
n
parenleftBiggradicalbigg
1 + 1n +
radicalbigg
1 + 2n +···+
radicalbigg
1 + n?1n
parenrightBigg
= lim
n→∞
n?1summationdisplay
i=0
radicalbigg
1 + in·1n? lim
n→∞
1
n =
integraldisplay 1
0
√x+ 1dx?
0 = 23(2√2?1)
(4) a207
n√n!
n =
n
radicalbigg
n!
nn =
parenleftbigg1
n
parenrightbigg1
n
parenleftbigg2
n
parenrightbigg1
n ···parenleftBign
n
parenrightBig1
na167a25ln
n√n!
n =
1
n
parenleftbigg
ln 1n + ln 2n +···+ ln nn
parenrightbigg
a113a188a234f(x) = lnxa51(0,1]a235a89a167a127a196 lim
ξ→+0
integraldisplay 1
ξ
lnxdx.a242(0,1]na31a169a167a169a58a143in,?xi = 1n(i = 1,2,···,n)a167
a51a122a135a2a171a109[xi?1,xi] =
bracketleftbiggi?1
n,
i
n
bracketrightbigg
a167a18ξi = ina167a75f(ξi) =
parenleftbiggi
n
parenrightbiggp
a167
a117a180 lim
n→∞
ln
n√n!
n = limn→∞
1
n
parenleftbigg
ln 1n + ln 2n +···+ ln nn
parenrightbigg
= lim
n→∞
nsummationdisplay
i=1
ln in · 1n = lim
ξ→+0
integraldisplay 1
ξ
lnxdx =
lim
ξ→+0
(xlnx?x)|1ξ =?1
a108a13 lim
n→∞
n√n!
n = e
1 = 1
e
13,a138a226a1267a107 lim
n→∞
I2n
I2n+1 = 1a167a100a100a237a121
pi
2 =
2
1 ·
2
3 ·
4
3 ·
4
5 ···
2n
2n?1 ·
2n
2n+ 1 ···
a121a178a181a207I2n =
integraldisplay pi
2
0
sin2nxdx = (2n?1)(2n?3)···3·12n(2n?2)···4·2 ·pi2,I2n+1 =
integraldisplay pi
2
0
sin2n+1xdx = 2n(2n?2)···4·2(2n+ 1)(2n?1)···5·3a167
a75I2n+1I
2n
= 21 · 23 · 43 · 45 ··· 2n2n?1 · 2n2n+ 1 · 2pi
a80 lessorequalslant x lessorequalslant pi2a158a1670 lessorequalslant sinx lessorequalslant 1,sin2n+1x lessorequalslant sin2nx lessorequalslant sin2n?1xa167a75
integraldisplay pi
2
0
sin2n+1xdx lessorequalslant
integraldisplay pi
2
0
sin2nxdx lessorequalslant
integraldisplay pi
2
0
sin2n?1xdxa61I2n+1 lessorequalslantI2n lessorequalslantI2n+1a167a117a1801 lessorequalslant I2nI
2n+1
lessorequalslant I2n?1I
2n+1
a113a100a52a237a250a170In = n?1n In?2,I2n+1 = 2n2n+ 1I2n?1a61I2n+1I
2n?1
= 2n2n+ 1a167a25 lim
n→∞
I2n+1
I2n?1 = limn→∞
2n
2n+ 1 =
1a167a117a180 lim
n→∞
I2n
I2n+1 = 1a167a108a13
pi
2 =
2
1 ·
2
3 ·
4
3 ·
4
5 ···
2n
2n?1 ·
2n
2n+ 1 ···
128
14,a23f(x)a134g(x)a209a51[a,b]a140a200a167a121a178
bracketleftbiggintegraldisplay b
a
f(x)g(x)dx
bracketrightbigg2
lessorequalslant
integraldisplay b
a
f2(x)dx·
integraldisplay b
a
g2(x)dx
a113a31a170a51a219a158a164a225a186
a121a178a181a233a63a219a162a234ha167a207[hf(x)?g(x)]2 = h2f2(x)?2hf(x)g(x) +g2(x) greaterorequalslant 0
a100a200a169a27a53a159a167a26
integraldisplay b
a
(h2f2(x)?2hf(x)g(x)+g2(x))dxgreaterorequalslant 0a61h2
integraldisplay b
a
f2(x)dx?2h
integraldisplay b
a
f(x)g(x)dx+
integraldisplay b
a
g2(x)dxgreaterorequalslant
0
a100a19a103a110a145a170a154a75a27a94a135a167a26
parenleftbigg
2
integraldisplay b
a
f(x)g(x)dx
parenrightbigg2
4
integraldisplay b
a
f2(x)dx·
integraldisplay b
a
g2(x)dxlessorequalslant 0a61
bracketleftbiggintegraldisplay b
a
f(x)g(x)dx
bracketrightbigg2
lessorequalslant
integraldisplay b
a
f2(x)dx·
integraldisplay b
a
g2(x)dx
a135a166a31a210a164a225a167a144a135
parenleftbigg
2
integraldisplay b
a
f(x)g(x)dx
parenrightbigg2
4
integraldisplay b
a
f2(x)dx·
integraldisplay b
a
g2(x)dx = 0a61h2
integraldisplay b
a
f2(x)dx?2h
integraldisplay b
a
f(x)g(x)dx+
integraldisplay b
a
g2(x)dx = 0a107a173a138.
a216a148a23h0a143a144a167a27a173a138a167a75h20
integraldisplay b
a
f2(x)dx?2h0
integraldisplay b
a
f(x)g(x)dx+
integraldisplay b
a
g2(x)dx = 0a61
integraldisplay b
a
[h0f(x)?g(x)]2 dx =
0
a13a8g(x) = h0f(x)a158a167
integraldisplay b
a
[h0f(x)? g(x)]2 dx = 0(a217a165h0a143a144a167h2
integraldisplay b
a
f2(x)dx? 2h
integraldisplay b
a
f(x)g(x)dx +
integraldisplay b
a
g2(x)dx = 0a27a173a138)
129
a49a108a217 a189a200a169a27a65a94a218a67a113a79a142
§1 a178a161a227a47a27a161a200
1,a166a100a101a15a136a173a130a164a140a164a27a227a47a161a200a181
(1) y2 = 4(x+ 1),y2 = 4(1?x)
(2) y = |lnx|,y = 0,(0.1 lessorequalslantxlessorequalslant 10)
(3) y = x,y = x+ sin2x,(0 lessorequalslantxlessorequalslantpi)
(4) y2 = 1?x,2y = x+ 2
(5) a93a130r = acosθ+b(bgreaterorequalslanta)a167a8b = aa158a61a143a37a57a130
(6) r = 3cosθ,r = 1 + cosθ
(7) a94a211a130x = a(t?sint),y = a(1?cost)(0 lessorequalslanttlessorequalslant 2pi)a177a57xa182
(8) a40a47a130x23 +y23 = a23
a41a181
(1) a252a94a173a130x = y
2?4
4 a134x =
y2 + 4
4 =?
y2?4
4 a27a2a58a27a112a139a73a169a79a143?2a572a167
a117a180A =
integraldisplay 2
2
bracketleftbigg
y
2?4
4?
y2?4
4
bracketrightbigg
dy =?
integraldisplay 2
0
(y2?4)dy = 163
(2) a252a94a173a130y = |lnx|a134y = 0a27a2a58a27a238a139a73a1431a167
a117a180A =
integraldisplay 10
0.1
[ln|x|? 0]dx =
integraldisplay 1
0.1
(?lnx)dx +
integraldisplay 10
1
lnxdx =?(xlnx?x)
vextendsinglevextendsingle
vextendsingle
1
0.1
+ (xlnx?x)
vextendsinglevextendsingle
vextendsingle
10
1
=
9.9ln10?8.1 ≈ 14.69559
(3) A =
integraldisplay pi
0
(x+ sin2x?x)dx =
integraldisplay pi
0
sin2xdx = 12
integraldisplay pi
0
(1?cos2x)dx = pi2? sin2x4
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
0
= pi2
(4) a252a94a173a130a27a2a58a169a79a143(0,1),(?8,?3)a167
a117a180A =
integraldisplay 1
3
[1?y2?(2y?2)]dy =
integraldisplay 1
3
(3?y2?2y)dy = 323
(5) a164a166a161a200a143a181A = 12
integraldisplay 2pi
0
r2 dθ = 12
integraldisplay 2pi
0
(acosθ+b)2 dθ = pi2a2 +pib2
(6) a164a166a161a200a143a181A = pi
parenleftbigg3
2
parenrightbigg2
A1 = 94pi?A1
a217a165A1 = 12
integraldisplay pi
3
pi3
[9cos2θ?(1 + cosθ)2]dθ =
integraldisplay pi
3
0
[8cos2θ?1?2cosθ)dθ = pia167
a108a13A = 54pi
(7) a164a166a161a200a143a181A =
integraldisplay 2pi
0
ydx =
integraldisplay 2pi
0
a(1?cost)·a(1?cost)dt = a2
integraldisplay 2pi
0
(1?cost)2 dt = 3pia2
(8) a23x = acos3t,y = asin3ta167a217a1650 lessorequalslanttlessorequalslant pi3a167a167a233a65a117a111a169a131a152a27a161a200a167a164a166a161a200a143a217a111a21
a61A = 4
integraldisplay a
0
ydx = 4
integraldisplay 0
pi
2
(?3a2 sin4tcos2t)dt = 12a2
integraldisplay pi
2
0
(sin4x?sin6x)dx = 3pi8 a2
2,a134a130y = xa114a253a11x2 + 3y2 = 6ya27a161a200a169a164a252a220a169A(a2a27a152a172)a218B(a140a27a152a172)a167a166ABa131a138.
a41a181a100a174a127a167a26a253a11a144a167a143x
2
3 + (y?1)
2 = 1a167a75a253a11a161a200a143S = piab = √3pi
a113y = xa134a253a11x2 + 3y2 = 6ya27a2a58a27a112a139a73a1430,32
a117a180A =
integraldisplay 3
2
0
(
radicalbig
6y?3y2? y)dy = √3
integraldisplay 3
2
0
radicalbig
1?(y?1)2 dy? y
2
2
vextendsinglevextendsingle
vextendsinglevextendsingle
3
2
0
=
√3
3 pi?
3
4a167a75B = S? A =
2
3
√3pi+ 3
4a167
a108a13AB = 4
√3pi?9
8√3pi+ 9.
130
3,a166a173a130y = √1?x2 + arccosxa134xa182a57x =?1a164a140a27a161a200.
a41a181a207y1 = √1?x2a27a189a194a141a143[?1,1]a167a138a141a143[0,1]a182
y2 = arccosxa27a189a194a141a143[?1,1]a167a138a141a143[0,pi]
a75a161a200A =
integraldisplay 1
1
y1 dx+
integraldisplay 1
1
y2 dx =
integraldisplay 1
1
radicalbig
1?x2 dx+
integraldisplay 1
1
arccosxdx = 32pi
131
§2 a173a130a27a108a127
a166a101a15a173a130a27a108a127a181
1,y = x32 (0 lessorequalslantxlessorequalslant 4)
2,x = 14y2? 12 lny(1 lessorequalslantylessorequalslante)
3,a40a47a130x23 +y23 = a23 (a> 0)
4,a94a211a130x = a(t?sint),y = a(1?cost)(0 lessorequalslanttlessorequalslant 2pi)
5,a11a27a236a109a130x = a(cost+tsint),y = a(sint?tcost)(0 lessorequalslanttlessorequalslant 2pi)
6,a37a57a130r = a(1 + cosθ)(0 lessorequalslantθlessorequalslant 2pi)
a41a181
1,a164a166a108a127a143s =
integraldisplay 4
0
radicalBig
1 + [(x32 )prime]2 dx =
integraldisplay 4
0
radicalbigg
1 + 94xdx = 827(10√10?1)
2,a164a166a108a127a143s =
integraldisplay e
1
radicalbig
1 + (xprime)2 dy =
integraldisplay e
1
radicalBigg
1 +
parenleftbiggy
2?
1
2y
parenrightbigg2
dy =
integraldisplay e
1
radicalBiggparenleftbigg
y
2 +
1
2y
parenrightbigg2
dy =
integraldisplay e
1
parenleftbiggy
2 +
1
2y
parenrightbigg
dy =
e2 + 1
4
3,a100a174a127a140a23x = acos3t,y = asin3t(0 lessorequalslanttlessorequalslant 2pi)
a75a164a166a108a127a143s = 4
integraldisplay pi
2
0
radicalBig
[(acos3t)prime]2 + [(asin3t)prime]2 dt =
4a
integraldisplay pi
2
0
radicalBig
(?3cos2tsint)2 + (3sin2tcost)2 dt = 12a
integraldisplay pi
2
0
sintcostdt = 6a
4,s =
integraldisplay 2pi
0
radicalbig
[(a(t?sint))prime]2 + [(a(1?cost))prime]2 dt =
|a|
integraldisplay 2pi
0
radicalBig
(1?cost)2 + sin2tdt = 2|a|
integraldisplay 2pi
0
radicalbigg
1?cost
2 dt = 2|a|
integraldisplay 2pi
0
sin t2 dt = 8|a|
5,s =
integraldisplay 2pi
0
radicalbig
[(a(cost+tsint))prime]2 + [(a(sint?tcost))prime]2 dt =
|a|
integraldisplay 2pi
0
radicalbig
(tcost)2 + (tsint)2 dt = |a|
integraldisplay 2pi
0
tdt = 2pi2|a|
6,s =
integraldisplay 2pi
0
radicalbig
r2 + (rprime)2 dθ =
integraldisplay 2pi
0
radicalBig
a2(1 + cosθ)2 +a2 sin2θdθ = 4|a|
integraldisplay pi
0
radicalbigg
1 + cosθ
2 dθ = 4|a|
integraldisplay pi
0
cos θ2 dθ =
8|a|
132
§3 a78a200
1,a166a209a100a101a15a136a173a161a164a140a164a27a65a219a78a78a200a181
(1) a166a31a73a78a78a200a167a217a254a101a46a27a143a253a11a167a253a11a27a182a127a169a79a31a117A,Ba218a,ba167a13a112a143ha182
(2) a166a253a165a78a78a200a181x
2
a2 +
y2
b2 +
z2
c2 = 1a182
(3) a166a100a101a15a252a173a161a181x2 +y2 +z2 = a2,x2 +y2 = axa164a140a164a27a78a200a182
(4) a166a94a207a76a46a161a134a187a27a178a161a108a134a11a206a254a131a101a27a252a47a78a78a200a167a23a11a206a27a46a140a187a143aa167a46a161a144a167a143x2 +y2 lessorequalslant
a2a167a31a161a207a76xa182a254a27a134a187a133a134a46a161a164αa14.
a41a181
(1) a138a152a178a49a117a254a33a101a46a133a229a108a101a46a143xa27a31a161a167a100a31a161a143a253a11a167a217a140a182a169a79a143a181aprime = A+
parenleftBig
1? xh
parenrightBig
(a?A),
bprime = B +
parenleftBig
1? xh
parenrightBig
(b?B)
a117a180a100a31a161a161a200a143a181A(x) = piaprimebprime =
pi
bracketleftbigg
AB + (a?A)(b?B)
parenleftBig
1? xh
parenrightBig2
+ (A(b?B) +B(a?A))
parenleftBig
1? xh
parenrightBigbracketrightbigg
a108a13a164a166a78a200a143V =
integraldisplay h
0
A(x)dx = pi6[(2a+A)b+ (a+ 2A)B]
(2) a94a82a134a117Oxa182a27a178a161a31a253a165a26a31a232a143a152a253a11a167a167a51yOza178a161a254a27a221a75a143 y
2
b2
parenleftbigg
1? x
2
a2
parenrightbigg+ z
2
c2
parenleftbigg
1? x
2
a2
parenrightbigg =
1
a100a100a140a132a217a140a182a169a79a143b
radicalbigg
1? x
2
a2a57c
radicalbigg
1? x
2
a2a167a108a13a26a100a253a11a161a200a143A(x) = pibc
parenleftbigg
1? x
2
a2
parenrightbigg
a117a180a167a164a166a27a253a165a78a78a200a143a181
V =
integraldisplay a
a
A(x)dx = 2
integraldisplay a
0
pibc
parenleftbigg
1? x
2
a2
parenrightbigg
dx = 43abc
(3) z =radicalbiga2?x2?y2(a254a140a161)a167a217a67a122a137a140a143?√ax?x2 lessorequalslantylessorequalslant√ax?x2
a217a31a161a200a143A(x) = 2
integraldisplay √ax?x2
0
radicalbig
a2?x2?y2 dy = a32x12?a12x32 + (a2?x2)arcsin
radicalbigg x
a+x
a117a180a167a164a166a78a200a143a181
V = 2
integraldisplay a
0
A(x)dx =
2
integraldisplay a
0
bracketleftbigg
a32x12?a12x32 + (a2?x2)arcsin
radicalbigg x
a+x
bracketrightbigg
dx = 23a3
parenleftbigg
pi? 43
parenrightbigg
(4) y = √a2?x2,z = √a2?x2 tanαa167
a75A(x) = 12
radicalbig
a2?x2 ·
radicalbig
a2?x2 tanα = 12(a2?x2)tanα
a108a13a164a166a78a200a143a181
V =
integraldisplay a
a
A(x)dx =
integraldisplay a
0
(a2?x2)tanαdx = 23a3 tanα
2,a166a94a61a78a27a78a200a181
(1) x
2
a2 +
y2
b2 = 1a55Xa182a182
(2) y = sinx,y = 0(0 lessorequalslantxlessorequalslantpi)
(i) a55xa182
(ii) a55ya182
(3) x = asin3t,y = bcos3t(0 lessorequalslanttlessorequalslant 2pi)
(i) a55xa182
(ii) a55ya182
(4) a121a178a100alessorequalslantxlessorequalslantb,0 lessorequalslantylessorequalslanty(x)(a217a165y(x)a180a235a89a188a234)a164a140a164a27a161a200a55ya182a94a61a164a164a27a94a61a78a27a78a200a143a181
V =
integraldisplay b
a
2pixy(x)dx
133
(5) x = a(t?sint),y = a(1?cost)(0 lessorequalslanttlessorequalslant 2pi,y = 0)
(i) a55xa182
(ii) a55ya182
(iii) a55a134a130y = 2a
a41a181
(1) V = pi
integraldisplay a
a
y2 dx = pi
integraldisplay a
a
bracketleftbigg
b2
parenleftbigg
1? x
2
a2
parenrightbiggbracketrightbigg
dx = 43piab2
(2) (i) V = pi
integraldisplay pi
0
sin2xdx = pi
2
2
(ii) V = 2pi
integraldisplay pi
0
xsinxdx = 2pi2
(3) (i) V = 2pi
integraldisplay pi
2
0
y2 dx = 2pi
integraldisplay pi
2
0
b2 cos6t·3asin2tcostdt = 6a
integraldisplay pi
2
0
ab2 sin2tcos7tdt = 6piab2
integraldisplay pi
2
0
(cos7t?
cos9t)dt = 32105piab2
(ii) a124a94a233a161a53a167a144a73a242a254a170a137a89a165a,ba233a78a167a61a26
V = 32105pia2b
(4) a121a178a181a138[a,b]a27a63a191a169a123a181a = x0 <x1 <···<xn = b
a51[xi?1,xi]a165a63a18a152a58ξia167a233a65a27a188a234a138a143y(ξi)a182Ai ≈y(ξi)?xi,?Vi ≈ 2piξiy(ξi)?xia167a75V = lim
λ→0
nsummationdisplay
i=1
2piξiy(ξi)?xia167
a108a13V =
integraldisplay b
a
2pixy(x)dx
(5) (i) V = pi
integraldisplay 2pi
0
y2 dx = pi
integraldisplay 2pi
0
a3(1?cost)3 dt = 5pi2a3
(ii) V = 2pi
integraldisplay 2pi
0
a3(1?cost)2(t?sint)dt = 6pi3a3
(iii) a138a178a163y = y+ 2a,x = xa167a75a173a130a144a167a143x = a(t?sint),y =?a(1 + cost)a57y =?2a
a117a180V = pi
integraldisplay 2pi
0
[4a2?a2(1+cost)2]a(1?cost)dt = pia3
integraldisplay 2pi
0
(3?2cost?cos2t)(1?cost)dt = 7pi2a3
3,a121a178a114a161a2000 lessorequalslant α lessorequalslant θ lessorequalslant β lessorequalslant pi,0 lessorequalslant r lessorequalslant r(θ)(r(θ)a51[α,β]a254a235a89)a55a52a182a94a61a164a164a27a78a200a31a117a181V =
2pi
3
integraldisplay β
α
r3(θ)sinθdθa167a191a166a209r = a(1 + cosθ)a55a52a182a94a61a164a164a27a78a200.
(1) a121a178a181a94a135a3a123.
a207a177ra143a140a187a167a134a52a130a164θa14a27a247a47a55a52a182a94a61a152a177a164a26a27a78a200a143a181
V = pi3(rsinθ)2rcosθ+pi
integraldisplay r
rcosθ
(r2?x2)dx = 23pir3(1?cosθ)
a138[α,β]a27a63a191a169a123a181α = θ0 <θ1 <···<θn = β,?θi = θi?θi?1,λ = max
i
{?θi}
a51a122a135[θi?1,θi]a209a127a51θprimeia167a166cosθi?1?cosi =?sinθprimei(θi?1?θi) = sinθprimei?θi
a177r(θprimei)a138a2a247a47Aia27a140a187a167a75a247a47a55a52a182a94a61a152a177a0a164a26a27a78a200a143a181
Vi = 23pir3(θprimei)(1?cosθi)? 23pir3(θprimei)(1?cosθi?1) =
2
3pir
3(θprime
i)(cosθi?1?cosθi) =
2
3pir
3(θprime
i)sinθ
prime
i?θia167a75a18a135a173a62a247a47a55a52a182a94a61a26
nsummationdisplay
i=1
2
3pir
3(θprime
i)sinθ
prime
i?θi
a108a13V = lim
λ→0
nsummationdisplay
i=1
2
3pir
3(θprime
i)sinθ
prime
i?θi =
2pi
3
integraldisplay β
α
r3(θ)sinθdθ.
(2) a41a181V = 23pi
integraldisplay pi
0
a3(1 + cosθ)3 sinθdθ = 83pia3
4,a114a14a212a130y = x(x?a)a51a238a139a730a134c(c>a> 0)a131a109a27a108a227a55xa182a94a61a167a175ca143a219a138a158a167a84a94a61a78a27a78a200Va31a117
a177a117OPa55xa182a94a61a164a164a27a73a78a78a200a186(a2278-14)
a41a181a207a14a212a130y = x(x?a),xP = ca167a25P(c,c(c?a))
a75a177a117OPa55xa182a94a61a164a164a27a73a78a78a200a143a181V1 = 13pic[c(c?a)]2 = pi3c3(c?a)2
134
a164a166a27a94a61a78a78a200a143a181V2 = pi
integraldisplay c
0
[x(x?a)]2 dx = pi
parenleftbiggc5
5?
a
2c
4 + a2
3 c
3
parenrightbigg
a113V1 = V2a167a25pi3c3(c?a)2 = pi
parenleftbiggc5
5?
a
2c
4 + a2
3 c
3
parenrightbigg
a167
a108a13c = 54a
5,a114a173a130y =
√x
1 +x2a55xa182a94a61a26a152a94a61a78a167a167a51a58x = 0a134x = ξa131a109a27a78a200a80a138V(ξ)a167a166aa31a117a219a138a158a167a85
a166V(a) = 12 lim
ξ→∞
V(ξ).
a41a181a207V(ξ) = pi
integraldisplay ξ
0
parenleftbigg √x
1 +x2
parenrightbigg2
dx = ξ
2
2(1 +ξ2)pia167a75V(a) =
a2
2(1 +a2)pi
a113V(a) = 12 lim
ξ→∞
V(ξ) = 12 lim
ξ→∞
ξ2
2(1 +ξ2)pi =
pi
4a167a117a180a
2 = 1
a113a> 0a167a25a = 1.
6,a253a11b2x2 +a2y2 = a2b2a55xa182a94a61a26a152a94a61a253a165a78a167a114a167a247xa182a144a149a139a152a66a37a27a11a154a167a166a144a101a27a130a47a78a78a200a31
a117a253a165a78a78a200a27a152a140a167a251a189a125a152a27a140a187ρ(a2278-15).
a41a181a23a75a165a144a101a27a130a47a78a78a200a143Va167a253a165a78a78a200a143V1
a207b2x2 +a2y2 = a2b2a167a75y =
√a2b2?b2x2
a
a75V1 = pi
integraldisplay a
a
y2 dx = pi
integraldisplay a
a
a2b2?b2x2
a2 dx =
4
3piab
2
V = V1?2piρ2
radicalbiga2b2?a2ρ2
b?2pi
integraldisplay a
√
a2b2?a2ρ2
b
parenleftbigg√a2b2?b2x2
a
parenrightbigg2
dx =
4
3piab
radicalbig
b2?ρ2? 4pia3b ρ2
radicalbig
b2?ρ2 = 43piab(b2?ρ2)32
a100a75a191a167a26V = 12V1a6143piab(b2?ρ2)32 = 23piab2a167a41a100a144a167a167a26ρ = b
radicalBig
1?2?23
135
§4 a94a61a173a161a27a161a200
1,a166a101a15a94a61a173a161a27a161a200a181
(1) x2 = 2py+a(0 lessorequalslantxlessorequalslanta,a> 1)a55xa182a57ya182
(2) y = sinx(0 lessorequalslantxlessorequalslantpi)a55xa182
(3) a253a11x
2
a2 +
y2
b2 = 1a55ya182
(4) a94a211a130x = a(t?sint),y = a(1?cost)(0 lessorequalslanttlessorequalslant 2pi)a55xa182
(5) a86a221a130r2 = 2a2 cos2θ
(i) a55a52a130
(ii) a55a182θ = pi2
(iii) a55a182θ = pi4
a41a181
(1) y = x
2?a
2p,?
a
2p lessorequalslantylessorequalslant
a2?a
2p (p> 0)
(i) Fx = 2pi
integraldisplay a
0
x2?a
2p
radicaltpradicalvertex
radicalvertexradicalbt
1 +
bracketleftBiggparenleftbigg
x2?a
2p
parenrightbiggprimebracketrightBigg2
dx = pip2
integraldisplay a
0
(x2?a2)
radicalbig
x2 +p2 dx =
bracketleftBigg
a(2a2?4a+p2)
8p2
radicalbig
p2 +a2? p
2 + 4a
8 ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a+radicalbiga2 +p2
p
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
bracketrightBigg
pi
(ii) radicalbig1 + (xprime)2 =
radicalbigp2 + 2py+a
√2py+a
a75Fy = 2pi
integraldisplay a2?a
2p
a2p
radicalbig2py+aradicalbigp2 + 2py+a√
2py+a dy =
pi
p
integraldisplay a2?a
2p
a2p
radicalbig
2py2 +p2 +ad(2py+p2 +a) = 2pi3p(p2 +a2)32? 23p2pi
(2) F = 2pi
integraldisplay pi
0
sinx·
radicalbig
1 + [(sinx)prime]2 dx = 2pi
integraldisplay pi
0
sinx
radicalbig
1 + cos2xdx = 2√2pi+ 2piln(√2 + 1)
(3) F = 2pi
integraldisplay b
b
x
radicalbig
1 + (xprime)2 dy = 2pi
integraldisplay b
b
a
b
radicalbig
b2?y2·
radicaltpradicalvertex
radicalvertexradicalbt
1 +
parenleftBigg
a(?y)
bradicalbigb2?y2
parenrightBigg2
dy = 2piab
integraldisplay b
b
radicalbigg
b2 + a
2?b2
b2 y
2 dy =
4api
b
integraldisplay b
0
radicalbigg
b2 + c
2
b2y
2 dy = 2api
parenleftbigg
a+ b
2
c ln
a+c
b
parenrightbigg
.
(4) a207dS = radicalbig(xprime)2 + (yprime)2 dt = 2asin t2 dta167a75F = 2pi
integraldisplay 2pi
0
yds = 2pi
integraldisplay 2pi
0
a(1? cost) · 2asin t2 dt =
16a2pi
integraldisplay 2pi
0
sin3 t2 dt2 = 643 pia2.
(5) (i) y = √2a√cos2θsinθ,dS =
√2a
√cos2θ dθ
parenleftBig
pi4 lessorequalslantθlessorequalslant pi4
parenrightBig
a100a233a161a53a167a26F = 2×2pi
integraldisplay pi
4
0
2a2 sinθdθ = 4pia2(2?√2)
(ii) x = √2a√cos2θcosθ,dS =
√2a
√cos2θ dθ
parenleftBig
pi4 lessorequalslantθlessorequalslant pi4
parenrightBig
a75F = 2pi
integraldisplay pi
4
pi4
2a2 cosθdθ = 4√2pia2
(iii) x = √2a√cos2θcosθ,y = √2a√cos2θsinθ,
dS =
√2a
√cos2θ dθ
parenleftBig
pi4 lessorequalslantθlessorequalslant pi4
parenrightBig
a53a191a20a51?pi4 lessorequalslantθlessorequalslant pi4a83a167a240a107x?ygreaterorequalslant 0a167
a117a180a167a164a166a27a76a161a200a143F = 2×2pi
integraldisplay pi
4
pi4
x?y√
2 ·
√2a
√cos2θ dθ = 4√2pia2
integraldisplay pi
4
pi4
(cosθ?sinθ)dθ = 8pia2
136
2,a121a178a100x =?(t),y = ψ(t),z = χ(t)(t0 lessorequalslanttlessorequalslantT)a134Oxya178a161a109a164a129a27a206a161a161a200a31a117S =
integraldisplay T
t0
χ(t)
radicalbig
prime2(t) +ψprime2(t)dt
a121a178a181a23a173a130CDa51Oxya178a161a27a221a75a143ABa167a75ABa27a144a167a143
braceleftbigg x =?(t)
y = ψ(t) (t0 lessorequalslanttlessorequalslantT)
a51ABa254a18a169a58a181A = M0,M1,···,Mi?1,Mi,···,Mn = B
a51CDa233a65a27a169a58a181C = N0,N1,···,Ni?1,Ni,···,Nn = D
a233a65a27a235a234a181t0,t1,···,ti?1,ti,···,tn
a23Mia27a139a73a143χi =?(ti),yi = ψ(ti)a167a75MiNi = χ(ti)
a134a14a70a47MiMi?1Ni?1Nia27a161a200a181
Si = MiNi +Mi?1Ni?12 ·Mi?1Mi = χ(ti) +χ(ti?1)2
radicalbig
[?(ti)(ti?1)]2 + [ψ(ti)?ψ(ti?1)]2 =
χ(ti)
radicalbig
[?(ti)(ti?1)]2 + [ψ(ti)?ψ(ti?1)]2?
χ(ti)?χ(ti?1)
2
radicalbig
[?(ti)(ti?1)]2 + [ψ(ti)?ψ(ti?1)]2
a100a135a169a165a138a189a110a181?(ti)(ti?1) =?prime(ξi)?ti,ψ(ti)?ψ(ti?1) = ψprime(ηi)?ti,χ(ti)?χ(ti?1) = χprime(ζi)?ti
a147a92a167a26Si = χ(ti)radicalbig?prime2(ξi) +ψprime2i (ηi)?ti? 12χprime(ζi)
radicalBig
prime2(ξi) +ψprime2i (ηi)(?ti)2
a108a13a206a161a161a200a143a181
S = lim
λ→0
nsummationdisplay
i=1
χ(ti)
radicalBig
prime2(ξi) +ψprime2i (ηi)?ti? lim
λ→0
1
2
nsummationdisplay
i=1
χprime(ζi)
radicalBig
prime2(ξi) +ψprime2i (ηi)(?ti)2
=
integraldisplay T
t0
χ(t)
radicalbig
prime2(t) +ψprime2(t)dt?0 =
integraldisplay T
t0
χ(t)
radicalbig
prime2(t) +ψprime2(t)dt
137
§5 a159a37
1,a166a101a15a173a130a227a27a159a37a139a73a181
(1) a140a187a143aa167a108a127a14312aα(αlessorequalslantpi)a27a254a33a11a108a182
(2) a177A(0,0),B(0,1),C(2,1),D(2,0)a143a186a58a27a221a47a177a46a167a173a130a254a63a152a58a27a151a221a31a117a84a58a20a6a58a229a108a27a19
a21a182
(3) a233a234a218a130r = aekθ(a> 0,k> 0)a254a100a58(0,a)a20a58(θ,r)a27a254a33a108a227.
a41a181
(1) a177a6a58a143a11a37a167a108a140a178a27a229a169a62a164a51a134a130a143xa182a239a225a134a14a139a73a88a167a75a11a108a144a167a143x = acosα,y =
asinαa167a117a180xprime =?asinα,yprime = acosαa167
a108a13x =
integraldisplay α
2
0
acosα
radicalbig
xprime2 +yprime2 dα
s =
a2
integraldisplay α
2
0
cosαdα
s =
a2 sin α2
1
2aα
=
2asin α2
α
y =
integraldisplay α
2
0
asinα
radicalbig
xprime2 +yprime2 dα
s =
a2
integraldisplay α
2
0
sinαdα
s =
2a
α
parenleftBig
1?cos α2
parenrightBig
(2) a107a166a209a151a221a188a234.
ABa27a144a167a143
braceleftbigg x = 0
y = y y ∈ [0,1]a167a217a254a63a152a58a27a151a221a143
ρ = 2y
BCa27a144a167a143
braceleftbigg x = x
y = 1 x∈ [0,2]a167a217a254a63a152a58a27a151a221a143
ρ = 2√1 +x2
CDa27a144a167a143
braceleftbigg x = 2
y = y y ∈ [0,1]a167a217a254a63a152a58a27a151a221a143
ρ = 2radicalbig4 +y2
DAa27a144a167a143
braceleftbigg x = x
y = 0 x∈ [0,2]a167a217a254a63a152a58a27a151a221a143
ρ = 2x
a117a180mAB =
integraldisplay 1
0
2y
radicalbig
02 + 12 dy = 1,
xAB = 0,yAB =
integraldisplay 1
0
y·2y
radicalbig
02 + 12 dy
mAB =
2
3a182
mBC =
integraldisplay 2
0
2
radicalbig
1 +x2
radicalbig
12 + 02 dx = 2√5 + ln(2 +√5),
xBC =
integraldisplay 2
0
x·2
radicalbig
1 +x2
radicalbig
12 + 02 dx
mBC =
2(5√5?1)
6√5 + 3ln(2 +√5),
yBC = 1a182
mCD =
integraldisplay 1
0
2
radicalbig
4 +y2
radicalbig
02 + 12 dy = √5 + 4ln 1 +
√5
2,
xCD = 2,yCD =
integraldisplay 1
0
y·2
radicalbig
4 +y2
radicalbig
02 + 12 dy
mCD =
2(5√5?8)
3
parenleftbigg√
5 + 4ln 1 +
√5
2
parenrightbigga182
mDA =
integraldisplay 2
0
2x
radicalbig
12 + 02 dx = 4,
xDA =
integraldisplay 2
0
x·2x
radicalbig
12 + 02 dx
mDA =
4
3,yDA = 0
a25a100a221a47a177a46a27a159a37a139a73a143a181
x = mABxAB +mBCxBC +mCDxCD +mDAxDAm
AB +mBC +mCD +mDA
=
138
16√5 + 14 + 24ln 1 +
√5
2
9√5 + 15 + 3ln(2 +√5) + 12ln 1 +
√5
2
y = mAByAB +mBCyBC +mCDyCD +mDAyDAm
AB +mBC +mCD +mDA
=
16√5?14 + 3ln(2 +√5)
9√5 + 15 + 3ln(2 +√5) + 12ln 1 +
√5
2
(3) a173a37a27a134a14a139a73a143a181
x =
integraldisplay θ
0
x
radicalbig
xprime2 +yprime2 dx
s =
integraldisplay θ
0
rcosθ
radicalbig
[(rcosθ)prime]2 + [(rsinθ)prime]2 dθ
integraldisplay θ
0
radicalbig
[(rcosθ)prime]2 + [(rsinθ)prime]2 dθ
=
integraldisplay θ
0
rcosθ
radicalbig
a2(1 +k2)ekθ dθ
integraldisplay θ
0
radicalbig
a2(1 +k2)ekθ dθ
=
a
integraldisplay θ
0
e2kθ cosθdθ
integraldisplay θ
0
ekθ dθ
= ake
2kθ(sinθ+ 2kcosθ)?2ak2
(4k2 + 1)(ekθ?1)
a211a123a140a26y = ake
2kθ(2ksinθ?cosθ) +ak
(4k2 + 1)(ekθ?1)
a117a180a167a173a37a27a52a139a73a143a181
r =
radicalbig
x2 +y2 = ak(4k2 + 1)(ekθ?1)
radicalbig
(e4kθ + 1?2e2kθ cosθ)(4k2 + 1)
tanθ0 = yx = e
2kθ(sinθ+ 2kcosθ)?2k
e2kθ(2ksinθ?cosθ) + 1
a133a159a37a139a73a143(r,θ0)
2,a94a152a138a151a221a254a33a27a55a225a106a11a164a140a187a143aa27a140a11a108a167a51a252a224a94a211a24a27a55a225a106a26a254a252a94a131a130a163a2278-19a164a167a175a131a130
a127ba143a245a8a158a167a144a85a166a55a225a106MABNa27a159a37a20a208a51a11a37Oa186
a41a181
a45
a54
a39
a38
x
y
0
A
B
a
M
N
a23a55a225a106a27a151a221a143μa167a140a11a108a27a159a254a143a181m =
integraldisplay 3pi
2
pi
2
μds = piaμ
a140a11a108a181x = acosθ,y = asinθ
parenleftbiggpi
2 lessorequalslantθlessorequalslant
3
2pi
parenrightbigg
,ds =radicalbigxprime2 +yprime2 dθ = adθ
a75a140a11a108a27a159a37a139a73a143
x =
integraldisplay 3
2pi
pi
2
xμds
m =
μa2
integraldisplay 3
2pi
pi
2
cosθdθ
piaμ =?
2a
pi ;
y =
integraldisplay 3
2pi
pi
2
yμds
m =
μa2
integraldisplay 3
2pi
pi
2
sinθdθ
piaμ = 0
a113a252a94a131a130a27a159a37a139a73a143a181x = b2,y = 0a167a159a254a143a1812bμ
a117a180a100a174a127a167a26a159a58a88a159a37a139a73a143a181x =
2api ·piaμ+ b2 ·2bμ
piaμ+ 2bμ = 0a167a108a13b =
√2a
3,a182a12710a146a167a151a221a169a217a143ρ = ρ(x) = (6 + 0.3x)a90a142/a146a167a217a165xa143a229a182a27a152a135a224a58a27a229a108a167a166a182a27a159a254.
a41a181m =
integraldisplay 10
0
ρ(x)dx =
integraldisplay 10
0
(6 + 0.3x)dx = 75(a90a142)
4,a174a127a152a14a212a130a227y = x2(?1 lessorequalslant x lessorequalslant 1)a167a173a130a227a254a63a152a58a63a27a151a221a134a84a58a20ya182a27a229a108a164a20a39a167x = 1a63a151a221
a1435a167a166a100a173a130a227a27a159a254.
139
a41a181a100a174a127a167a26ρ(x) = k|x|
a207x = 1a158a167ρ(1) = 5a167a75k = 5a167a117a180ρ(x) = 5|x|
a113ds =radicalbig1 + (yprime)2 dx = √1 + 4x2 dxa167
a75m =
integraldisplay 1
1
ρ(x)ds = 2
integraldisplay 1
0
5x
radicalbig
1 + 4x2 dx = 256 √5? 56
140
§6 a178a254a138a33a245
1,a174a127a18a54a62a180a165a62a123Ra252a224a27a62a216a129a140a138a143Uma167a11a170a199a143ωa167a79a142a158a209a51Ra254a27a178a254a245a199(a169a140a197a18a54a218a28a197
a18a54a252a171a156a185a63a216).
a41a181a140a197a18a54a158a167a158a209a51Ra254a27a178a254a245a199a143a181P1 = 2T
integraldisplay T
2
0
P(t)dt = 2ω2pi
integraldisplay pi
ω
0
U2m
R cos
2ωtdt = U2m
2R
a28a197a18a54a158a167a158a209a51Ra254a27a178a254a245a199a143a181P2 = 1T
integraldisplay T
0
P(t)dt = ω2pi
integraldisplay 2pi
ω
0
U2m
R cos
2ωtdt = U2m
2R
2,a79a142a2a54a62a216u = Umcosωta51
bracketleftBig
0,piω
bracketrightBig
a218
bracketleftBig
pi2ω,pi2ω
bracketrightBig
a83a27a178a254a138.
a41a181a51
bracketleftBig
0,piω
bracketrightBig
a83a27a178a254a138a143a181u = ωpi
integraldisplay pi
ω
0
Umcosωtdt = 0a182
a51
bracketleftBig
pi2ω,pi2ω
bracketrightBig
a83a27a178a254a138a143a181u = ωpi
integraldisplay pi
2ω
pi2ω
Umcosωtdt = 2piUm.
3,a166a101a15a188a234a51a137a189a171a109a83a27a178a254a138a181
(1) y = sinx,[0,pi]
(2) y = xex,[0,1]
a41a181
(1) y = 1pi
integraldisplay pi
0
sinxdx = 2pi
(2) y =
integraldisplay 1
0
xexdx = 1
4,a114a6a148a46a127a164a73a27a229a134a6a148a27a27a127a164a20a39.a174a127a152a250a54a27a229a85a166a6a148a27a1271a102a146a167a175a114a6a148a46a12710a102a146a135a138a245
a8a245a186
a41a181a100a7a142a189a110a127a167a6a53a161a69a229Fa134a27a127a254xa164a20a39a61F = kx.
a100a94a135a167a127k = 1a167a207a13F = xa167a117a180a164a166a27a245a143W =
integraldisplay 10
0
F dx =
integraldisplay 10
0
xdx=50(a90a142·a102a146a164=5J
5,a63a239a140a120a120a234a158a135a107a101a140a51.a23a152a11a206a47a140a51a27a134a187a14320a146a167a89a2927a146a167a140a51a112a209a89a1613a146a167a135a114a89a196a166a167a79
a142a142a209a173a229a164a138a27a245.
a41a181a207?W = pir2 ·?x·x·103g = 105gpix?x
a75W = 105g
integraldisplay 30
3
pixdx = 4.37×108pi(J)
6,a44a89a165a27a185a128a180a152a70a47a167a254a466a146a167a101a462a146a167a11210a146a167a166a89a47a247a158a185a128a164a201a27a229.a23a89a27a39a173a1431a235/a1462.
a41a181a207?F = 2xyg?x = 2gx
parenleftBig
3? x5
parenrightBig
x
a75F = 2g
integraldisplay 10
0
x
parenleftBig
3? x5
parenrightBig
dx = 1.63×106(N)
7,a212a78a85a53a198x = ct3(c> 0)a138a134a130a36a196a167xa76a171a51a158a109ta83a212a78a163a196a27a229a108a167a23a48a159a27a123a229a134a132a221a178a144a164a20a39a167
a166a212a78a108x = 0a20x = aa158a123a229a164a138a27a245.
a41a181a207x = ct3(c> 0)a167a25v = xprime = 3ct2
a113a48a159a123a229a134a132a221a27a178a144a164a20a39a167a75a23f = kv2(ka143a126a234)a167a117a180f = 9kc2t4
a113a8xa108x = 0a20x = aa158a167ta108t = 0a20t =
parenleftBiga
c
parenrightBig1
3a167
a75W =
integraldisplay (a
c)
1
3
0
9kc2t4 ·3ct2 dt = 27kc3
integraldisplay (a
c)
1
3
0
t6 dt = 277 kc23a73
8,a140a187a143ra27a165a156a92a89a165a167a167a134a89a161a131a26a167a165a27a39a173a1431a167a121a242a165a108a89a165a18a209a167a135a138a245a8a245a186
a41a181a207?W = 1·pi(radicalbigr2?(x?r)2)2?x(2r?x) = pi(4r2x?4rx2 +x3)?x
a75W = pi
integraldisplay 2r
0
(4r2x?4rx2 +x3)dx = 43pir4
141
§7 a189a200a169a27a67a113a79a142
1,a94a14a212a130a47a250a170a166
integraldisplay 1
0
dx
1 +x2a27a67a113a138(a18n = 3).
a41a181a18n = 3a167a79a142a204a160a2a234a167a140a26a181
x0 = 0,y0 = 1.0000;x1 = 16,4y1 = 3.8919;x2 = 13,2y2 = 1.8000;x3 = 12,4y3 = 3.2000;x4 = 23,2y4 =
1.3846;x5 = 56,4y5 = 2.3607;x6 = 1,y6 = 0.5000
a124a94a14a212a130a47a250a170a167a107integraldisplay
1
0
dx
1 +x2 ≈
1
18[y0 + 2(y2 +y4) + 4(y1 +y3 +y5) +y6] = 0.7854
2,a166a44a202a46a27a161a200.a202a46a88a2278-24a164a171a167xa182a180a167a27a233a161a182a167OAa1272a146a16710a31a169a167a255a26a234a226a88a101a163a252a160a181a102
a146a164a181
x 0 20 40 60 80 100 120 140 160 180 200
y 0 8.5 11.0 11.5 10.5 10.0 8.0 6.5 4.5 2.5 0
a41a181a124a94a14a212a130a47a250a170a167a107
A≈ 2006×5[0 + 0 + 4(8.5 + 11.5 + 10.0 + 6.5 + 2.5) + 2(11.0 + 10.5 + 8.0 + 4.5)] = 203 ×224 ≈ 1493.3(cm2)
3,a51a176a14320a146a27a224a161a254a167a255a254a224a54a238a31a161a27a161a200.a88a74a108a224a27a152a87a149a233a87a122a1332a146a167a255a26a224a89a29a221a88a101a76a164a15a181
x 0 2 4 6 8 10 12 14 16 18 20
y(a89a29) 0.4 0.8 1.4 2.0 2.4 2.1 1.9 1.6 1.3 0.8 0.4
(a89a29a252a160a181a146)a166a100a224a54a238a31a161a27a161a200a163a2278-25a164
a41a181a124a94a14a212a130a47a250a170a167a107
A≈ 206×5[0.4 + 0.4 + 4(0.8 + 2.0 + 2.1 + 1.6 + 0.8) + 2(1.4 + 2.4 + 1.9 + 1.3)] = 23 ×44 ≈ 29.3(m2)
142
a49a110a159 a63a234a216
a49a152a220a169 a234a145a63a234a218a50a194a200a169
a49a202a217 a234a145a63a234
§1,a253a23a127a163a181a254a52a129a218a101a52a129
1,a121a178a181
(1) lim
n→∞
(xn +yn) lessorequalslant lim
n→∞
xn + lim
n→∞
yn
(2) lim
n→∞
(xn +yn) greaterorequalslant lim
n→∞
xn + lim
n→∞
yn
a121a178a181
(1) a23 lim
n→∞
xn,lim
n→∞
yna254a143a107a129a234a167a75{xn},{yn}a254a107a46.
a45αk = sup
n>k
{xn},βk = sup
n>k
{yn}a34a117a180a167a8n>ka158a167a107xn +yn lessorequalslantαk +βk
a108a13sup
n>k
{xn +yn}lessorequalslantαk +βk
a25 lim
n→∞
(xn +yn) = lim
k→∞
sup
n>k
{xn +yn}lessorequalslant lim
k→∞
αk + lim
k→∞
βk = lim
n→∞
xn + lim
n→∞
yn
a53a181a101 lim
n→∞
xn,lim
n→∞
yna131a152a143+∞.a126a88a181 lim
n→∞
xn = +∞a167a143a166a39a88a170a109a62a92a123a36a142a107a191a194a167a75 lim
n→∞
yna216
a26a143?∞.a249a158{xn}a195a254a46a167{xn+yn}a189a195a254a46a167a254a227a39a88a170a119a44a164a225a182a101 lim
n→∞
xn =?∞a167a143a166a39a88a170
a109a62a92a123a36a142a107a191a194a167a75 lim
n→∞
yna216a26a143+∞.a117a180{yn}a254a107a46a167a108a13 lim
n→∞
(xn +yn) =?∞a167a254a227a39a88a170
a119a44a164a225.
(2) a207xn greaterorequalslant inf{xn},yn greaterorequalslant inf{yn}a167a25xn +yn greaterorequalslant inf{xn}+ inf{yn}
a226a101a40a46a143a101a46a165a129a140a27a167a75inf{xn +yn}greaterorequalslant inf{xn}+ inf{yn}a167
a108a13inf
n>k
{xn +yn}greaterorequalslant inf
n>k
{xn}+ inf
n>k
{yn}
a75 lim
k→∞
inf
n>k
{xn +yn}greaterorequalslant lim
k→∞
parenleftbigg
inf
n>k
{xn}+ inf
n>k
{yn}
parenrightbigg
= lim
k→∞
inf
n>k
{xn}+ lim
k→∞
inf
n>k
{yn}
a61 lim
n→∞
(xn +yn) greaterorequalslant lim
n→∞
xn + lim
n→∞
yn.
2,a23xn greaterorequalslant 0,yn greaterorequalslant 0a167a121a178a181
(1) lim
n→∞
xnyn lessorequalslant lim
n→∞
xn · lim
n→∞
yn
(2) lim
n→∞
xnyn greaterorequalslant lim
n→∞
xn · lim
n→∞
yn
a121a178a181
(1) a2070 lessorequalslantxn lessorequalslant sup{xn},0 lessorequalslantyn lessorequalslant sup{yn}a167a750 lessorequalslantxnyn lessorequalslant sup{xn}·sup{yn}
a226a254a40a46a180a254a46a165a129a2a27a167a75a1070 lessorequalslant sup{xn ·yn}lessorequalslant sup{xn}·sup{yn}
a108a130 lessorequalslant sup
n>k
{xn ·yn}lessorequalslant sup
n>k
{xn}· sup
n>k
{yn}
a75 lim
k→∞
sup
n>k
{xn ·yn}lessorequalslant lim
k→∞
parenleftbigg
sup
n>k
{xn}· sup
n>k
{yn}
parenrightbigg
= lim
k→∞
sup
n>k
{xn}· lim
k→∞
sup
n>k
{yn}
a61 lim
n→∞
xnyn lessorequalslant lim
n→∞
xn · lim
n→∞
yn.
(2) a207xn greaterorequalslant inf{xn}greaterorequalslant 0,yn greaterorequalslant inf{yn}greaterorequalslant 0a167a75xnyn greaterorequalslant inf{xn}·inf{yn}greaterorequalslant 0
a226a101a40a46a180a101a46a165a129a140a27a167a75a107inf{xn ·yn}greaterorequalslant inf{xn}·inf{yn}greaterorequalslant 0
a108a13inf
n>k
{xn ·yn}greaterorequalslant inf
n>k
{xn}· inf
n>k
{yn}greaterorequalslant 0
a75 lim
k→∞
inf
n>k
{xn ·yn}greaterorequalslant lim
k→∞
parenleftbigg
inf
n>k
{xn}· inf
n>k
{yn}
parenrightbigg
= lim
k→∞
inf
n>k
{xn}· lim
k→∞
inf
n>k
{yn}
a61 lim
n→∞
xnyn greaterorequalslant lim
n→∞
xn · lim
n→∞
yn
3,a101 lim
n→∞
xna127a51a167a75a233a63a219a234a15{yn}a164a225a181
(1) lim
n→∞
(xn +yn) = lim
n→∞
xn + lim
n→∞
yn
143
(2) lim
n→∞
(xn ·yn) = lim
n→∞
xn · lim
n→∞
yna167a101 lim
n→∞
xn > 0
a121a178a181a23 lim
n→∞
xn = α
a101 lim
n→∞
yn = +∞(a189?∞)a167a75(1)a119a44a164a225,a207α> 0a167a75(2)a119a44a164a225.
a25a216a148a23 lim
n→∞
yn = βa143a107a129a234
a207 lim
n→∞
yn = βa167a25a127a51{yn}a27a102a15{ynk}a167a166 lim
k→∞
ynk = βa133βa143a164a107a194a241a102a15a27a52a129a165a27a129a140a246.
a113 lim
n→∞
xn = αa167a75 lim
k→∞
xnk = αa167a25 lim
k→∞
(xnk +ynk) = α+β,lim
k→∞
(xnk ·ynk) = αβ
a101a121α+βa143{xn +yn}a131a152a131a194a241a102a15a27a52a129a165a27a129a140a246(a94a135a121a123)
a98a23{xn +yn}a27a152a135a194a241a102a15{xnkprime +ynkprime}a167a166 lim
kprime→∞
parenleftbigx
nkprime +ynkprime
parenrightbig= γ >α+β
a75 lim
kprime→∞
ynkprime = lim
kprime→∞
parenleftbigx
nkprime +ynkprime
parenrightbig? lim
kprime→∞
xnkprime = γ?α>β
a249a134βa143{yn}a27a164a107a194a241a102a15a27a52a129a165a27a129a140a138a103a241.
a117a180α+βa210a180{xn +yn}a164a107a194a241a102a15a52a129a27a129a140a138.
a211a110a140a121a167a8α> 0a158a167α+βa143{xn +yn}a27a152a131a194a241a102a15a27a52a129a165a27a129a140a138.
a108a13 lim
n→∞
(xn +yn) = α+β = lim
n→∞
xn + lim
n→∞
yn
lim
n→∞
(xn ·yn) = αβ = lim
n→∞
xn · lim
n→∞
yna167a101 lim
n→∞
xn > 0
4,a166a101a15a234a15a27a254a52a129a134a101a52a129a181
(1) an = 12?n + (?1)n (n = 1,2,···)
(2) an = (?1)n
parenleftBigg
1 + 1n
parenrightBigg
(n = 1,2,···)
(3) an = (?1)
n
n (n = 1,2···)
(4) an = sin npi5 (n = 1,2,···)
a41a181
(1) a167a144a107a252a135a228a52a129a27a102a234a15a181a2k → 1,a2k+1 →?1 (k →∞) (k = 1,2,3···)
a117a180 lim
n→∞
an = 1,lim
n→∞
an =?1.
(2) a167a144a107a252a135a228a52a129a27a102a234a15a181a2k → 1,a2k+1 →?1 (k →∞) (k = 1,2,3···)
a117a180 lim
n→∞
an = 1,lim
n→∞
an =?1.
(3) a207 lim
n→∞
an = 0a167a25 lim
n→∞
an = 0,lim
n→∞
an = 0.
(4)?sin 25pilessorequalslant sin npi5 lessorequalslant sin 25pi
a8n = 10k+ 2 (k = 1,2,···)a158a167a10k+2 → sin 25pi (k →∞)
a8n = 10k?2 (k = 1,2,···)a158a167a10k?2 →?sin 25pi (k →∞)
a117a180 lim
n→∞
an = sin 25pi,lim
n→∞
an =?sin 25pi.
5,a101 lim
n→∞
nradicalbig|an| = αa167a75 lim
n→∞
nradicalbig|ak
0+n| = α
a100a63k0a180a63a191a27a189a27a18a234.
a121a178
(1) a207|ak0+n|1n = |ak0+n| 1k0+n
parenleftBig
|ak0+n| 1k0+n
parenrightBigk0n
a113 lim
n→∞
k0+nradicalbig|ak
0+n| = αa167a133a8α> 0a158a167 limn→∞
k0
n ln|ak0+n|
1
k0+n = 0a167a25 lim
n→∞
parenleftBig
|ak0+n| 1k0+n
parenrightBigk0n
= 1
a100a492a75(1)a167a26 lim
n→∞
nradicalbig|ak
0+n|lessorequalslantα
(2) a207 lim
n→∞
nradicalbig|an| = αa167a25a127a51a102a15{an
k}a167a166a26 limk→∞|ank|
1
nk = αa167
a133a8α> 0a158a167a107 lim
k→∞
|ank| 1nk?k0 = lim
k→∞
|ank| 1nk · lim
k→∞
parenleftBig
|ank| 1nk
parenrightBig k0n
k?k0 = α
144
a108a13 lim
n→∞
nradicalbig|ak
0+n|greaterorequalslantα
a110a220(1)(2)a167a26a8α> 0a158a167a40a216a164a225.
(3) a101α = 0a167a75a119a44a107 lim
n→∞
nradicalbig|an| = 0a167a108a13 lim
n→∞
nradicalbig|ak
0+n| = limn→∞
parenleftBig
|ak0+n| 1k0+n
parenrightBigk0+nn
= 0
a117a180a26a100a40a216a20a40.
6,a101 lim
n→∞
an = a<ba167a121a178a181a55a127a51Na167a8n>Na158a167a107an <b,a113a101 lim
n→∞
an = a<b.a156a185a88a219a186
a121a178a181
(1) a18ε = b?a2 a167a100§1a189a1101a167a26{an}a165a150a245a144a107a107a129a145a225a117(a+ε,+∞) =
parenleftBigg
a+b
2,+∞
parenrightBigg
a45a249a107a129a145a27a118a73a129a140a246a143Na167a75a8n>Na158a167a107an <a+ε = a+b2 < b+b2 = b
(2) a101 lim
n→∞
an = a<ba167a40a216a153a55a164a225.
a126a181an = 1 + (?1)n,n = 1,2,···a167a249a135a234a15a1430,2,0,2,···a167a119a44 lim
n→∞
an = 2,lim
n→∞
an = 0a167
a13 lim
n→∞
an = 0 < 1a167a2a107a195a161a245a145a2n = 2 > 1 (n = 1,2,···)
145
§2,a63a234a27a194a241a53a57a217a196a29a53a159
1,a63a216a101a15a63a234a27a241a209a53a181
(1) 11·6 + 16·11 +···+ 1(5n?4)(5n+ 1) +···
(2) 1 + 23 + 35 +···+ n2n?1 +···
(3)
parenleftBigg
1
2 +
1
3
parenrightBigg
+
parenleftBigg
1
22 +
1
32
parenrightBigg
+···+
parenleftBigg
1
2n +
1
3n
parenrightBigg
+···
(4) 11·4 + 14·7 +···+ 1(3n?2)(3n+ 1) +···
(5) cos pi3 + cos pi4 + cos pi5 +···
a41a181
(1) a207Sn = 11·6+ 16·11+···+ 1(5n?4)(5n+ 1) = 15
bracketleftBigg
1? 16 + 16? 111 +···+ 15n?4? 15n+ 1
bracketrightBigg
= 15
parenleftBigg
1? 15n+ 1
parenrightBigg
a75 lim
n→∞
Sn = lim
n→∞
1
5
parenleftBigg
1? 15n+ 1
parenrightBigg
= 15
a117a180a226a189a194a127a167a63a234 11·6 + 16·11 +···+ 1(5n?4)(5n+ 1) +···a194a241.
(2) a207 lim
n→∞
n
2n?1 =
1
2 negationslash= 0a167a25a63a234a117a209.
(3) a100a117
∞summationdisplay
n=1
1
2na134
∞summationdisplay
n=1
1
3na254a143a194a241a27a65a219a63a234a167
a25a100a234a15a63a234a53a1592a167a127
∞summationdisplay
n=1
parenleftBigg
1
2n +
1
3n
parenrightBigg
=
∞summationdisplay
n=1
1
2n +
∞summationdisplay
n=1
1
2n = 1 +
1
2 =
3
2
(4) a207Sn = 11·4+ 14·7+···+ 1(3n?2)(3n+ 1) = 13
bracketleftBigg
1? 14 + 14? 17 +···+ 13n?2? 13n+ 1
bracketrightBigg
= 13
parenleftBigg
1? 13n+ 1
parenrightBigg
a75 lim
n→∞
Sn = lim
n→∞
1
3
parenleftBigg
1? 13n+ 1
parenrightBigg
= 13
a117a180a226a189a194a127a167a63a234 11·4 + 14·7 +···+ 1(3n?2)(3n+ 1) +···a194a241.
(5) a207 lim
n→∞
cos pin+ 2 = 1 negationslash= 0a167a25a63a234a117a209.
2,a124a94a133a220a194a241a6a110a7a79a101a15a63a234a180a194a241a132a180a117a209.
(1) a0 +a1q+a2q2 +···+anqn +···,|q|< 1,|an|lessorequalslantA,(n = 0,1,2,···)
(2) 1 + 12? 13 + 14 + 15? 16 +···
a121a178a181
(1) a207a233a63a219a103a44a234pa167
|Sn+p?Sn| =vextendsinglevextendsingleanqn +an+1qn+1 +···+an+p?1qn+p?1vextendsinglevextendsinglelessorequalslantvextendsinglevextendsingleanvextendsinglevextendsingle|qn|+vextendsinglevextendsinglean+1vextendsinglevextendsinglevextendsinglevextendsingleqn+1vextendsinglevextendsingle+···+vextendsinglevextendsinglean+p?1vextendsinglevextendsinglevextendsinglevextendsingleqn+p?1vextendsinglevextendsinglelessorequalslant
A|q|n1?|q|
p
1?|q|
a113|q|< 1a167a750 < 1?|q|p < 1a167a117a180|Sn+p?Sn|<A· |q|
n
1?|q|
a108a13a233?ε> 0a167a18N =
bracketleftBigg
ln (1?|q|)εA /ln|q|
bracketrightBigg
a167a8n>Na158a167a233a63a219p = 1,2,3,···a167
a111a164a225|Sn+p?Sn|<ε
a85a194a241a6a110a167a63a234a0 +a1q+a2q2 +···+anqn +···a194a241.
146
(2) a100a63a234a143
∞summationdisplay
n=0
parenleftBigg
1
3n+ 1 +
1
3n+ 2?
1
3n+ 3
parenrightBigg
a180 <ε0 < 16a167a216a216na245a140a167a101a45p = na167a75a107
|Sn+p?Sn| = |S2n?Sn| = 13n+ 1+ 13n+ 2? 13n+ 3+···+ 16n?2+ 16n?1? 16n> 13n+ 3+ 13n+ 3?
1
3n+ 3 +···+
1
6n+
1
6n?
1
6n =
1
3
parenleftBigg
1
n+ 1 +
1
n+ 2 +···+
1
2n
parenrightBigg
> 13
1
2n+
1
2n+···+
1
2nbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
na145
=
1
6 >ε0
a207a100a63a234
∞summationdisplay
n=0
parenleftBigg
1
3n+ 1 +
1
3n+ 2?
1
3n+ 3
parenrightBigg
a117a209.
3,a23a107a20a145a63a234
∞summationdisplay
n=1
an(a61a122a152a145an > 0)a167a193a121a178a101a233a217a145a92a41a210a0a164a124a164a27a63a234a194a241a167a75
∞summationdisplay
n=1
ana189a194a241.
a121a178a181a23
∞summationdisplay
n=1
ana220a169a218a234a15a143{Sn}a167a92a41a210a0a164a124a164a27a63a234a143
∞summationdisplay
n=1
An
a217a165An = ain?1+1 +ain?1+2 +···+aina167a119a44
∞summationdisplay
n=1
Ana69a143a20a145a63a234.
a23a217a220a169a218a234a15a143{Snprime}a167a217a165Snprime = (a1 +a2 +···+ai1)+(ai1+1 +···+ai2)+···+(ain?1+1 +···+ain)
a119a44Snprime greaterorequalslantSn
a113
∞summationdisplay
n=1
Ana194a241a167a100a196a29a189a110a167a26{Snprime}a107a254a46a167a61a127a51M > 0a167a166Snprime lessorequalslant Ma167a108a13Sn lessorequalslant Snprime lessorequalslant Ma167a96
a178{Sn}a107a254a46
a75a100a196a29a189a110a167a26
∞summationdisplay
n=1
ana194a241.
4,a40a189a166a101a15a63a234a194a241a27xa27a137a140.
(1)
∞summationdisplay
n=0
1
(1 +x)n
(2)
∞summationdisplay
n=1
(lnx)n
a41a181
(1) a100a63a234a180a250a39a143 11 +xa27a31a39a63a234a167a25a8
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
1 +x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle< 1a158a63a234a194a241
a108a13a194a241a141a143x<?2a189x> 0.
(2) a100a63a234a180a250a39a143lnxa27a31a39a63a234a167a25a8|lnx|< 1a158a63a234a194a241
a108a13a194a241a141a1431e<x<e.
147
§3,a20a145a63a234
1,a7a228a101a15a63a234a27a194a241a218a117a209.
(1)
∞summationdisplay
n=1
1√
n2 +n
(2)
∞summationdisplay
n=1
1
(2n?1)·22n?1
(3)
∞summationdisplay
n=1
n?√n
2n?1
(4)
∞summationdisplay
n=1
sin pi2n
(5)
∞summationdisplay
n=1
1
1 +an,(a> 1)
(6)
∞summationdisplay
n=1
1
n· n√n
(7)
∞summationdisplay
n=1
parenleftBigg
1
2n+ 1
parenrightBiggn
(8)
∞summationdisplay
n=1
1
[ln(n+ 1)]n
(9)
∞summationdisplay
n=1
2 + (?1)n
2n
(10)
∞summationdisplay
n=1
2nsin pi3n
(11)
∞summationdisplay
n=1
nn
n!
(12)
∞summationdisplay
n=1
xn
(1 +x)(1 +x2)···(1 +xn),(xgreaterorequalslant 0)
(13)
∞summationdisplay
n=1
parenleftBigg
b
an
parenrightBiggn
a167a217a165an →a,an,b,aa27a20a234a167anegationslash= 0
a41a181
(1) a207 lim
n→∞
1√
n2 +n
1
n
= lim
n→∞
n√
n2 +n = 1a167a13a63a234
∞summationdisplay
n=1
1
na180a117a209a27
a75a100a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
1√
n2 +na189a117a209.
(2) a207 lim
n→∞
un+1
un = limn→∞
1
(2n+ 1)22n+1
1
(2n?1)22n?1
= lim
n→∞
2n?1
4(2n+ 1) =
1
4 < 1
a75a100a136a75a19a16a7a79a123a167a26a63a234
∞summationdisplay
n=1
1
(2n?1)·22n?1a194a241.
(3) a207 lim
n→∞
n?√n
2n?1 =
1
2notarrowright0a167a25a63a234
∞summationdisplay
n=1
n?√n
2n?1a117a209.
(4) a207sin pi2n lessorequalslant pi2na167a13
∞summationdisplay
n=1
pi
2na194a241a167a25a63a234
∞summationdisplay
n=1
sin pi2na194a241.
148
(5) a207 11 +an lessorequalslant
parenleftBigg
1
a
parenrightBiggn
a167a13
∞summationdisplay
n=1
parenleftBigg
1
a
parenrightBiggn
a194a241a167a25a63a234
∞summationdisplay
n=1
1
1 +ana194a241.
(6) a207 lim
x→+0
xx = lim
x→+0
elnxx = lim
x→+0
exlnx = e limx→+0xlnx = 1a167a25 lim
n→∞
1
n√n = limn→∞
parenleftBigg
1
n
parenrightBigg1
n
= 1
a113 lim
n→∞
1
n· n√n
1
n
= lim
n→∞
1
n√n = 1a167a13a63a234
∞summationdisplay
n=1
1
na180a117a209a27
a75a100a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
1
n· n√na117a209.
(7) a207 lim
n→∞
n
radicaltpradicalvertex
radicalvertexradicalbtparenleftBigg 1
2n+ 1
parenrightBiggn
= lim
n→∞
1
2n+ 1 = 0 < 1a167a25a63a234
∞summationdisplay
n=1
parenleftBigg
1
2n+ 1
parenrightBiggn
a194a241.
(8) a207 lim
n→∞
n
radicalBigg
1
[ln(n+ 1)]n = limn→∞
1
ln(n+ 1) = 0 < 1a167a25a63a234
∞summationdisplay
n=1
1
[ln(n+ 1)]na194a241.
(9) a207 2 + (?1)
n
2n lessorequalslant
3
2n a133a63a234
∞summationdisplay
n=1
3
2n a194a241
a75a226a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
2 + (?1)n
2n a194a241.
(10) a207 0 < 2nsin pi3n lessorequalslantpi
parenleftBigg
2
3
parenrightBiggn
a133a63a234
∞summationdisplay
n=1
parenleftBigg
2
3
parenrightBiggn
a194a241
a75a226a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
2nsin pi3n a194a241.
(11) a207 lim
n→∞
un+1
un = limn→∞
(n+ 1)n+1
(n+ 1)!
nn
n!
= lim
n→∞
parenleftBigg
1 + 1n
parenrightBiggn
= e> 1a167a25a63a234
∞summationdisplay
n=1
nn
n!a117a209.
(12) a207 lim
n→∞
un+1
un = limn→∞
xn+1/[(1 +x)(1 +x2)···(1 +xn)(1 +xn+1)]
xn/[(1 +x)(1 +x2)···(1 +xn)] = limn→∞
x
1 +xn+1 =
0 < 1,x> 1a189x = 0
1
2 < 1,x = 1x< 1,0 <x< 1
a75a226a136a75a19a16a7a79a123a167a26a63a234
∞summationdisplay
n=1
xn
(1 +x)(1 +x2)···(1 +xn)a194a241.
(13) a207 lim
n→∞
n
radicaltpradicalvertex
radicalvertexradicalbtparenleftBigg b
an
parenrightBiggn
= lim
n→∞
b
an =
b
a
a75a8ba< 1a61b<aa158a167a63a234
∞summationdisplay
n=1
parenleftBigg
b
an
parenrightBiggn
a194a241a182
a8ba> 1a61b>aa158a167a63a234
∞summationdisplay
n=1
parenleftBigg
b
an
parenrightBiggn
a117a209a182
a8ba = 1a61b = aa158a167a73a63a152a218a7a228a34a126a88a181a63a234
∞summationdisplay
n=1
parenleftBigg
1
n√n
parenrightBiggn
=
∞summationdisplay
n=1
1
na117a209a182a13a63a234
∞summationdisplay
n=1
parenleftBigg
1
n√n2
parenrightBiggn
=
∞summationdisplay
n=1
1
n2a194a241.
2,a101a20a145a63a234
∞summationdisplay
n=1
una194a241a167a121a178
∞summationdisplay
n=1
u2na143a194a241a167a217a95a88a219a186
a121a178a181a207
∞summationdisplay
n=1
una194a241a167a75 lim
n→∞
un = 0
a18ε0 = 1a167a75a127a51a20a18a234Na167a8n>Na158a167a107|un|<ε0 = 1a610 lessorequalslantun < 1a167a117a1800 lessorequalslantu2n <un(n>N)a167
149
a108a13a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
u2na194a241.
a217a95a216a253.a126a181
∞summationdisplay
n=1
1
n2a194a241a167a2
∞summationdisplay
n=1
1
na117a209a182
∞summationdisplay
n=1
1
n4a194a241a167
∞summationdisplay
n=1
1
n2a194a241.
3,a23
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna143a252a20a145a63a234a167 lim
n→∞
un
vn = 0a167a121a178a181a8
∞summationdisplay
n=1
vna194a241a158a167
∞summationdisplay
n=1
una143a194a241.a113a101
∞summationdisplay
n=1
vna117a209a158a167
∞summationdisplay
n=1
una88
a219a186a101 lim
n→∞
un
vn = ∞a167a64a111
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna27a241a209a53a131a109a107a159a111a39a88a186
a121a178a181
(1) a207 lim
n→∞
un
vn = 0a167
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna143a252a20a145a63a234
a18ε0 = 1a167a75a127a51a20a18a234Na167a8n>Na158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
un
vn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε0 = 1a610 lessorequalslant
un
vn < 1a167a117a180un <vn(n>N)
a113
∞summationdisplay
n=1
vna194a241a167a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
una194a241
a101
∞summationdisplay
n=1
vna117a209a167a75
∞summationdisplay
n=1
una140a85a194a241a167a143a140a85a117a209
a126a181
∞summationdisplay
n=1
1
na117a209a167 limn→∞
1
n2
1
n
= 0a167a2
∞summationdisplay
n=1
1
n2a194a241a182
∞summationdisplay
n=1
1√
na117a209a167 limn→∞
1
n
1√
n
= 0a167a133
∞summationdisplay
n=1
1
na117a209.
(2) a207 lim
n→∞
un
vn = ∞a167
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna143a252a20a145a63a234
a18G0 = 1a167a75a127a51a20a18a234Na167a8n>Na158a167a107unv
n
>G0 = 1a167a117a180un >vn(n>N)
a101
∞summationdisplay
n=1
una194a241a167a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
vna194a241a182a101
∞summationdisplay
n=1
vna117a209a167a75
∞summationdisplay
n=1
una117a209a182a101
∞summationdisplay
n=1
una117a209a167a75
∞summationdisplay
n=1
vna241
a209a53a216a189.
4,a101a252a20a145a63a234
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna117a209a167
∞summationdisplay
n=1
max(un,vn)a167
∞summationdisplay
n=1
min(un,vn)a252a63a234a88a219a186
a41a181a207a252a20a145a63a234
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna117a209a167un lessorequalslant max(un,vn)
a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
max(un,vn)a117a209.
a233a117
∞summationdisplay
n=1
min(un,vn)a241a209a53a216a189.
a126a181
∞summationdisplay
n=1
1
na167
∞summationdisplay
n=1
1
2na209a117a209a167
∞summationdisplay
n=1
min
parenleftBigg
1
n,
1
2n
parenrightBigg
=
∞summationdisplay
n=1
1
2na143a117a209a182
∞summationdisplay
n=1
1 + (?1)n
2 a167
∞summationdisplay
n=1
1?(?1)n
2 a209a117a209a167a2
∞summationdisplay
n=1
min
parenleftBigg
1 + (?1)n
2,
1?(?1)n
2
parenrightBigg
= 0 + 0 +···+ 0 +···a37a194a241.
5,a124a94a63a234a194a241a27a55a135a94a135a121a178a181
(1) lim
n→∞
nn
(n!)2 = 0
(2) lim
n→∞
(2n)!
an! = 0(a> 1)
a121a178a181
150
(1)
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
nn
(n!)2
a207 lim
n→∞
un+1
un = limn→∞
(n+ 1)n+1
[(n+ 1)!]2
nn
(n!)2
= lim
n→∞
1
n+ 1
parenleftBigg
1 + 1n
parenrightBiggn
= 0 < 1
a75a226a136a75a19a16a7a79a123a27a52a129a47a170a167a26
∞summationdisplay
n=1
nn
(n!)2a194a241a167a108a13a100a63a234a194a241a27a55a135a94a135a167a26 limn→∞
nn
(n!)2 = 0
(2)
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
(2n)!
an!
a2070 < un+1u
n
=
[2(n+ 1)]!
a(n+1)!
(2n)!
an!
= (2n+ 2)(2n+ 1)an2(n?1)! < 4(n+ 1)
2
an+1 (a> 1)
a13 lim
n→∞
nk
an = 0(a> 1,k ∈N)a167a117a180 limn→∞
4(n+ 1)2
an+1 = 0a167a108a13 limn→∞
un+1
un = 0
a75a226a136a75a19a16a7a79a123a27a52a129a47a170a167a26
∞summationdisplay
n=1
(2n)!
an! a194a241a167a108a13a100a63a234a194a241a27a55a135a94a135a167a26 limn→∞
(2n)!
an! = 0
6,a63a216a101a15a63a234a27a194a241a53a181
(1)
∞summationdisplay
n=1
1
n·(lnn)p
(2)
∞summationdisplay
n=1
1
n·lnn·lnlnn
(3)
∞summationdisplay
n=1
1
n·(lnn)1+σ lnlnn
(4)
∞summationdisplay
n=1
1
n·(lnn)p(lnlnn)q
a41a181
(1) a100a117a216a216pa143a219a234a167a8xa191a169a140a158a167a188a234f(x) = 1x(lnx)p a209a180a154a75a52a126a27a167a133
lim
n→∞
integraldisplay n
2
dx
x(lnx)p =
1
p?1(ln2)
1?p,p> 1
∞,plessorequalslant 1
a25a8p> 1a158a167a63a234a194a241a182a8plessorequalslant 1a158a167a63a234a117a209.
(2) a23f(x) = 1xlnxlnlnxa167f(x)a8xgreaterorequalslant 3a180a20a138a52a126a188a234.
lim
n→∞
integraldisplay n
3
dx
xlnxlnlnx = limn→∞(lnlnlnn?lnlnln2) = ∞a167a75a63a234
∞summationdisplay
n=1
1
n·lnn·lnlnna117a209.
(3) a207 lim
n→∞
integraldisplay n
2
dx
x(lnx)1+σ = limn→∞
1
σ
parenleftBigg
1
(ln2)σ?
1
(lnn)σ
parenrightBigg
= 1σ(ln2)σ (σ> 0)
a25a63a234
∞summationdisplay
n=2
1
n(lnn)1+σa194a241.
a113 1n·(lnn)1+σ lnlnnlessorequalslant 1n(lnn)1+σ a167a75a100a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
1
n·(lnn)1+σ lnlnna194a241.
(4) a45f(x) = 1x(lnx)p(lnlnx)q a167a8nlessorequalslant 3a158a180a20a138a52a126a188a234.
a113a207a143
integraldisplay +∞
3
dx
x(lnx)p(lnlnx)q =
integraldisplay +∞
lnln3
dt
e(p?1)ttq
a233a63a219qa167a8p? 1 > 0a158a167a200a169a194a241a167a8p? 1 < 0a158a167a200a169a117a209a182a8p = 1a158a167a101q > 1a167a200a169a194a241a167
a101qlessorequalslant 1a167a200a169a117a209.
151
a100a133a220a200a169a7a79a123a127a167a6a63a234a241a209a53a134a200a169a241a209a53a94a135a152a151
a75a6a63a234a8p> 1a158a194a241a182a8p< 1a158a117a209a182a8p = 1a158a167q> 1a158a63a234a194a241a182qlessorequalslant 1a158a63a234a117a209.
7,a101
∞summationdisplay
n=1
una180a194a241a27a20a145a63a234a167a191a133a234a15{un}a252a78a101a252a167a121a178 lim
n→∞
nun = 0.
a121a178a181a207
∞summationdisplay
n=1
una194a241a167a23S =
∞summationdisplay
n=1
una167Sn =
nsummationdisplay
k=1
uk
a75 lim
n→∞
Sn = S = lim
n→∞
S2na167a117a180 lim
n→∞
(S2n?Sn) = 0
a113{un}a252a78a101a252a167a75S2n?Sn = un+1 +un+2 +···+u2n greaterorequalslantu2n +u2n +···+u2n = nu2n
a113un greaterorequalslant 0a167a750 lessorequalslantnu2n lessorequalslantS2n?Sna167a117a180a26 lim
n→∞
nu2n = 0a167a108a13 lim
n→∞
(2n)u2n = 0
a113a207u2n+1 lessorequalslantu2n,un greaterorequalslant 0a167a750 lessorequalslant (2n+ 1)u2n+1 lessorequalslant (2n+ 1)u2n = 2n+ 12n (2nu2n) → 0(n→∞)
a117a180 lim
n→∞
(2n+ 1)u2n+1 = 0a167a108a13 lim
n→∞
nun = 0
8,a121a178a136a75a19a16a7a79a123a57a217a52a129a47a170.
a121a178a181
(1) a136a75a19a16a7a79a123a181
a207n>Na158a167a107un+1u
n
lessorequalslantq< 1a167a75uN+2u
N+1
lessorequalslantq,uN+2 lessorequalslantquN+1; uN+3u
N+2
lessorequalslantq,uN+3 lessorequalslantquN+2;··· ; uN+k+1u
N+k
lessorequalslant
q,uN+k+1 lessorequalslantquN+k lessorequalslant···lessorequalslantqKuN+1
a207q < 1a167a75
∞summationdisplay
k=1
qka194a241a167a117a180a100a194a241a63a234a27a53a1591a127a167
∞summationdisplay
k=1
qkuN+1a143a194a241a167a108a13a100a39a22a7a79a123a167
a26
∞summationdisplay
k=2
uN+ka143a194a241
a50a100a194a241a63a234a27a53a1595a127a167a86a92a107a129a145u1,u2,···,uN+1a0a26a20a27a35a63a234
∞summationdisplay
n=1
una143a194a241.
a101n>Na158a167un+1u
n
greaterorequalslant 1a167a75uN+1u
N
greaterorequalslant 1,uN+1 greaterorequalslantuNa167a249a96a178{uN}a180a252a78a79a92a27
a113un greaterorequalslant 0a167a75unnotarrowright0(n→∞)a167a117a180
∞summationdisplay
n=1
una117a209.
(2) a136a75a19a16a7a79a123a27a52a129a47a170a181
(i) a101 lim
n→∞
un
un?1 = ˉr< 1
a100a162a234a27a200a151a53a127a55a127a51ε0 > 0a167a166a26r< ˉr+ε0 < 1
a100a254a52a129a27a189a1101a27a121a178a165a167a127
braceleftBigg
un+1
un
bracerightBigg
a144a107a107a129a145a140a117ˉr +ε0a167a117a180a189a127a51a152a135a20a18a234N(a144a135a18
a107a129a145a165a101a73a129a140a27a137Na61a140)a167a166a26a8n> Na158a167a107un+1u
n
< ˉr +ε0 < 1a167a25a100a136a75a19a16a7a79a123a127
a63a234a194a241.
(ii) a101 lim
n→∞
un
un?1 = r> 1
a100a162a234a27a200a151a53a127a55a127a51ε0 > 0a167a166a26ˉr>r?ε0 > 1
a100a254a52a129a27a189a1102a27a121a178a165a167a127
braceleftBigg
un+1
un
bracerightBigg
a144a107a107a129a145a2a117r +ε0a167a117a180a189a127a51a152a135a20a18a234N(a144a135a18
a107a129a145a165a101a73a129a140a27a137Na61a140)a167a166a26a8n> Na158a167a107un+1u
n
> r +ε0 > 1a167a25a100a136a75a19a16a7a79a123a127
a63a234a117a209.
(iii) a222a126a96a178a181
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
1
na167 limn→∞
un+1
un = limn→∞
un+1
un = 1a167a2
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
1
na117a209a182
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
1
n2 a167 limn→∞
un+1
un = limn→∞
un+1
un = 1a167a2
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
1
n2 a194a241.
152
§4,a63a191a145a63a234
1,a63a216a101a15a63a234a27a194a241a53a163a157a41a94a135a194a241a189a253a233a194a241a164a181
(1) 12? 310 + 122? 3103 + 123? 3105 +···
(2) 1? 12 + 13!? 14 + 15!?···
(3)
∞summationdisplay
n=2
(?1)n?1 lnnn
(4)
∞summationdisplay
n=1
(?1)n?1n
3
2n
(5)
∞summationdisplay
n=1
(?1)n+1 n(n+ 1)2
(6)
∞summationdisplay
n=1
(?1)nsin xn (xnegationslash= 0)
(7) 1√2?1? 1√2 + 1 + 1√3?1? 1√3 + 1 +···+ 1√n?1? 1√n+ 1 +···
a41a181
(1) a207
∞summationdisplay
n=1
1
2na194a241a167
∞summationdisplay
n=1
1
102n?1a194a241a167a75
∞summationdisplay
n=1
3
102n?1a194a241
a117a180
∞summationdisplay
n=1
1
2n +
∞summationdisplay
n=1
1
102n?1a194a241a167a61
1
2 +
3
10 +
1
22 +
3
103 +
1
23 +
3
105 +···a194a241a167a108a13a6a63a234a253a233a194a241.
(2) a207
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
parenleftBigg
12n
parenrightBigg
a117a209
a113a233a63a234
∞summationdisplay
n=1
1
(2n?1)!
a207 lim
n→∞
1
(2n+1)!
1
(2n?1)!
= lim
n→∞
1
2n(2n+ 1) = 0 < 1a167a75a100a136a75a19a16a7a79a123a27a52a129a47a170a167a26a63a234
∞summationdisplay
n=1
1
(2n1)!a194a241
a117a180a6a63a234a117a209.
(3) a207
∞summationdisplay
n=2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle(?1)
n?1 lnn
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=2
lnn
n
a113 lim
n→∞
lnn
n
1
n
= lim
n→∞
lnn = +∞a133
∞summationdisplay
n=1
1
na117a209a167a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=2
lnn
n a117a209
a113a23f(x) = lnxx (xgreaterorequalslant 3)a167a75fprime(x) =
parenleftBigg
lnx
x
parenrightBiggprime
= 1?lnxx2 < 0 (xgreaterorequalslant 3)a167a117a180f(x) = lnxx a252a78a101a252a167a108
a13
braceleftBigg
lnn
n
bracerightBigg
a51ngreaterorequalslant 3a158a252a78a101a252
a113 lim
x→+∞
lnx
x = limx→+∞
1
x = 0a167a75 limn→∞
lnn
n = 0
a117a180a226a52a217a90a91a189a110a167a26
∞summationdisplay
n=2
(?1)n?1 lnnn a94a135a194a241.
(4)
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle(?1)
n?1n
3
2n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
n3
2n
a207 lim
n→∞
(n+1)3
2n+1
n3
2n
= lim
n→∞
1
2
parenleftBigg
n+ 1
n
parenrightBigg3
= 12 < 1a167a75a226a136a75a19a16a7a79a123a167a26
∞summationdisplay
n=1
n3
2na194a241
a108a13
∞summationdisplay
n=1
(?1)n?1n
3
2na253a233a194a241.
153
(5)
∞summationdisplay
n=1
(?1)n+1 n(n+ 1)2 =
∞summationdisplay
n=1
n
(n+ 1)2
a207 lim
n→∞
n
(n+1)2
1
n
= lim
n→∞
n2
(n+ 1)2 = 1a133
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
n
(n+ 1)2a117a209
a23f(x) = x(x+ 1)2 (x greaterorequalslant 2)a167a75fprime(x) = 1?x(x+ 1)3 < 0(x greaterorequalslant 2)a167a117a180a8x greaterorequalslant 2a158a167f(x)a252a78a101a252a167a108
a13
braceleftBigg
n
(n+ 1)2
bracerightBigg
a8ngreaterorequalslant 2a158a252a78a101a252
a113 lim
n→∞
n
(n+ 1)2 = 0a167a75a226a52a217a90a91a189a110a167a26
∞summationdisplay
n=1
(?1)n+1 n(n+ 1)2a194a241
a108a13
∞summationdisplay
n=1
(?1)n+1 n(n+ 1)2a94a135a194a241.
(6)
∞summationdisplay
n=1
(?1)nsin xn =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin xn
1
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle→|x|negationslash= 0(n→∞)a133
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea117a209
a113a233?x ∈ R,x negationslash= 0a167a207xn → 0(n → ∞)a167a75a127a51N ∈ Z+a167a8n > Na158a167a1070 <
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<
pi
2a167a117a180a8n >
Na158a167sin xna134xa107a131a211a27a206a210a133
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea145na79a140a13a126a2a200a167a75a100a52a217a90a91a7a79a123a167a26
∞summationdisplay
n=1
(?1)nsin xna194
a241
a108a13
∞summationdisplay
n=1
(?1)nsin xna94a135a194a241.
(7) a23a220a169a218a234a15a143{Sn}a167a75S2n =
n+1summationdisplay
k=2
parenleftBigg
1√
k?1?
1√
k+ 1
parenrightBigg
=
n+1summationdisplay
k=2
2
k?1 = 2
nsummationdisplay
k=1
1
k
a117a180 lim
n→∞
S2n = +∞a167a75a100a63a234a92a41a210a0a117a209a167a108a13a6a63a234a117a209.
2,a121a178a181a101a63a234a27a145a92a41a210a0a164a138a164a27a63a234a194a241a167a191a133a51a211a152a135a41a210a83a145a27a206a210a131a211a167a64a34a22a75a41a210a0a167a100a63
a234a189a194a241a182a191a100a100a127a9a63a234
∞summationdisplay
n=1
(?1)[
√n]
n a27a194a241a53.
a121a178:
(1) a174a127a35a63a234
∞summationdisplay
n=1
uprimen = (u1 +···+un1) + (un1+1 +···+un2) +···+ (unk?1+1 +···+unk) +···a194a241a133
a51a211a152a41a210a83a27a206a210a131a211
a23
nsummationdisplay
k=1
uk = Sn,
nsummationdisplay
k=1
uprimek = Sprimena167a75S1 prime = Sn1,S2 prime = Sn2,···,Skprime = Snk,···
a8a6a63a234a27a101a73na108nk?1a20nka158a167
∞summationdisplay
n=1
una27a220a169a218a252a78a67a122a167a61
a8unk?1+1,···,unka254a143a20a158a167a107Sk?1 prime = Snk?1 <Sn <Snk = Skprime
a8unk?1+1,···,unka254a143a75a158a167a107Sk?1 prime = Snk?1 >Sn >Snk = Skprime
a174a127
∞summationdisplay
n=1
uprimena194a241a167a61 lim
k→∞
Skprime = lim
k→∞
Sk?1 prime = Sprimea167a75 lim
n→∞
Sn = Sprimea167a117a180
∞summationdisplay
n=1
una194a241.
(2) a127a196
∞summationdisplay
n=1
(?1)[
√n]
n
a8n = k2,k2+1,···,k2+2k(k = 1,2,···)a158a167a195ana211a210.a80Ak = 1k2 + 1k2 + 1+···+ 1k2 + 2ka167
∞summationdisplay
k=1
(?1)kAka180
a2a134a63a234
a207
integraldisplay k2+2k
k2?1
dx
x lessorequalslant
1
k2 +
1
k2 + 1 + ··· +
1
k2 + 2k lessorequalslant
integraldisplay (k+1)2
k2
dx
x a61ln
k2 + 2k
k2?1 lessorequalslant Ak lessorequalslant ln
(k+ 1)2
k2 a167a108a13
a8k →∞a158a167Ak → 0
154
a113Ak?Ak+1 greaterorequalslant ln k
2 + 2k
k2?1? ln
(k+ 2)2
(k+ 1)2 = ln
k2 +k
k2 +k?2 > 0a167a75a100a52a217a90a91a7a79a123a127
∞summationdisplay
k=1
(?1)kAka194
a241a167a108a13a6a63a234a194a241.
3,a63a216a101a15a63a234a180a196a253a233a194a241a189a94a135a194a241a181
(1)
∞summationdisplay
n=1
(?1)n
n+x
(2)
∞summationdisplay
n=1
sin(2nx)
n!
(3)
∞summationdisplay
n=1
sinnx
n,(0 <x<pi)
(4)
∞summationdisplay
n=1
cosnx
np,(0 <x<pi)
a41a181
(1)
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(?1)n
n+x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
1
|n+x|
a207 lim
n→∞
1
|n+x|
1
n
= lim
n→∞
n
|n+x| = 1a167a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
1
|n+x|a117a209
a8xgreaterorequalslant 0a158a167 1n+xa252a78a126a8a167a133 lim
n→∞
1
n+x = 0a167a75a100a52a217a90a91a189a110a167a26
∞summationdisplay
n=1
(?1)n
n+xa194a241
a8x < 0a133a216a143a75a18a234a158a167a207xa143a189a234a167a75a8na191a169a140a158a167a61a127a51N ∈ Z+a167a8n > Na158a167a107n + x >
0a167a117a180
∞summationdisplay
n=N+1
(?1)n
n+xa180a2a134a63a234a167a133a100
1
n+xa252a78a126a8a57 limn→∞
1
n+x = 0a167a75
∞summationdisplay
n=N+1
(?1)n
n+xa194a241a167a108
a13
∞summationdisplay
n=1
(?1)n
n+xa194a241
a75a8xa216a143a75a18a234a158a167a100a63a234a143a94a135a194a241.
(2) a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(2nx)
n!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n!a167a133 limn→∞
1
(n+1)!
1
n!
= lim
n→∞
1
n+ 1 = 0 < 1a167a75a100a136a75a19a16a7a79a123a167a26
∞summationdisplay
n=1
1
n!a194a241
a50a226a39a22a7a79a123a167a26
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(2nx)
n!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea194a241a167a108a13
∞summationdisplay
n=1
sin(2nx)
n! a253a233a194a241.
(3) a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
sinkx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cos x2?cos 2n+12 x
2sin x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1vextendsingle
vextendsinglesin x2vextendsinglevextendsinglea133a234a15
braceleftBigg
1
n
bracerightBigg
a252a78a170a1170
a75a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
n=1
sinnx
n a194a241.
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinnx
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
sin2nx
n =
1
2n?
cos2nx
2n a133
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
cos2kx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinx?sin(2n+ 1)x
2sinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
|sinx|a57a234a15
braceleftBigg
1
2n
bracerightBigg
a252
a78a170a1170
a75a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
n=1
cos2nx
n a194a241.
a113
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
1
2na117a209a167a117a180
∞summationdisplay
n=1
parenleftBigg
1
2n?
cos2nx
2n
parenrightBigg
a117a209a167a108a13
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinnx
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea117a209
a75a63a234
∞summationdisplay
n=1
sinnx
n (0 <x<pi)a94a135a194a241.
(4) (i) a8p> 1a158a167a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosnx
np
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
npa133
∞summationdisplay
n=1
1
npa8p> 1a158a194a241a167a75a63a234
∞summationdisplay
n=1
cosnx
np (0 <x<pi)a253a233a194a241.
(ii) a80 <plessorequalslant 1a158a167a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
coskx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin x2?sin 2n+12 x
2sin x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1vextendsingle
vextendsinglesin x2vextendsinglevextendsinglea133a234a15
braceleftBigg
1
np
bracerightBigg
a252a78a170a1170
a75a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
n=1
cosnx
np a194a241.
155
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosnx
np
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
cos2nx
np =
1
2np +
cos2nx
2np a133a100a102a226a121a178a140a26
∞summationdisplay
n=1
cos2nx
(2n)p a194a241.
a75
∞summationdisplay
n=1
cos2nx
(2n)p ·2
p?1a194a241a167a61
∞summationdisplay
n=1
cos2nx
2np a194a241
a113a80 <plessorequalslant 1a158a167
∞summationdisplay
n=1
1
npa117a209a167a75
∞summationdisplay
n=1
1
2npa117a209a167a117a180
∞summationdisplay
n=1
parenleftBigg
1
2np +
cos2nx
2np
parenrightBigg
a117a209
a108a13
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosnx
np
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea117a209
a75a63a234
∞summationdisplay
n=1
cosnx
np (0 <x<pi)a80 <plessorequalslant 1a158a94a135a194a241.
(iii) a8plessorequalslant 0a158a167a207 cosnxnp notarrowright0a167a75a63a234
∞summationdisplay
n=1
cosnx
np (0 <x<pi)a8plessorequalslant 0a158a117a209.
4,a101a63a234
∞summationdisplay
n=1
ana194a241a167a191a133 lim
n→∞
an
bn = 1a167a85a196a228a189
∞summationdisplay
n=1
bna143a194a241a186
a121a178a181
(1) a101a63a234
∞summationdisplay
n=1
ana167
∞summationdisplay
n=1
bna209a180a20a145a63a234
a100a63a234
∞summationdisplay
n=1
ana194a241a167 lim
n→∞
an
bn = 1a167a75a226a20a145a63a234a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
bna194a241
(2) a101a63a234
∞summationdisplay
n=1
ana167
∞summationdisplay
n=1
bna216a152a189a209a180a20a145a63a234
a100a63a234
∞summationdisplay
n=1
ana194a241a167a216a140a228a189
∞summationdisplay
n=1
bna194a241
a126a181a63a234
∞summationdisplay
n=1
(?1)n√
n a143a52a217a90a91a46a63a234a167a75a217a194a241a133 limn→∞
(?1)n√
n
(?1)n√
n +
1
n
= 1
a100a117
∞summationdisplay
n=1
(?1)n√
n a194a241a167
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
parenleftBigg
(?1)n√
n +
1
n
parenrightBigg
a117a209.
5,a121a178a181a101
∞summationdisplay
n=1
an
nx0a194a241a167a64a34a8x>x0a158
∞summationdisplay
n=1
an
nxa143a194a241.
a121a178a181a207x>x0a167a75
1
(n+1)x?x0
1
nx?x0
=
parenleftBigg
n
n+ 1
parenrightBiggx?x0
=
parenleftBigg
1? 1n+ 1
parenrightBiggx?x0
< 1a167a75 1(n+ 1)x?x
0
< 1nx?x
0
a133 1nx?x
0
lessorequalslant 1
a117a180a234a15
braceleftBigg
1
nx?x0
bracerightBigg
a252a78a107a46a167a133 1nx?x
0
lessorequalslant 1
a113a63a234
∞summationdisplay
n=1
an
nx0 a194a241a167a75a100a67a19a16a7a79a123a167a26
∞summationdisplay
n=1
an
nxa194a241.
6,a23{nan}a194a241a167
∞summationdisplay
n=1
n(an?an?1)a194a241a167a75
∞summationdisplay
n=1
ana143a194a241.
a121a178a181a207{nan}a194a241a167a23a217a52a129a143a
a113
∞summationdisplay
n=1
n(an?an?1)a194a241a167a75a217a220a169a218a234a15
braceleftBigg nsummationdisplay
k=1
k(ak?ak?1)
bracerightBigg
a107a52a129a167a23a217a52a129a143S
a113
nsummationdisplay
k=1
k(ak?ak?1) = (a1?a0) + 2(a2?a1) +···+n(an?an?1) = nan?
n?1summationdisplay
k=0
ak
a61
n?1summationdisplay
k=0
ak = nan?
nsummationdisplay
k=1
k(ak?ak?1)a167a75 lim
n→∞
n?1summationdisplay
k=0
ak = lim
n→∞
nan? lim
n→∞
nsummationdisplay
k=1
k(ak?ak?1) = a?S
a117a180
∞summationdisplay
n=0
ana194a241a167a108a13
∞summationdisplay
n=1
an =
∞summationdisplay
n=0
an?a0a194a241.
156
7,a101
∞summationdisplay
v=1
(av?av?1)a253a233a194a241a167
∞summationdisplay
v=1
bva194a241a167a64a34
∞summationdisplay
v=1
avbva194a241.
a121a178a181a45Bn+mn =
n+msummationdisplay
v=n+1
bv
a100Abel a67a134a167a26
n+psummationdisplay
v=n+1
avbv = an+pBn+pn +
p?1summationdisplay
i=1
Bn+in (an+i?an+i+1)
a25
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
n+psummationdisplay
v=n+1
avbv
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant|an+p|
vextendsinglevextendsingleBn+p
n
vextendsinglevextendsingle+p?1summationdisplay
i=1
vextendsinglevextendsingle
vextendsingleBn+in
vextendsinglevextendsingle
vextendsingle|an+i?an+i+1|
a45Hpn = maxbraceleftbigvextendsinglevextendsingleBn+1n vextendsinglevextendsingle,vextendsinglevextendsingleBn+2n vextendsinglevextendsingle,···,vextendsinglevextendsingleBn+pn vextendsinglevextendsinglebracerightbiga167a75a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
n+psummationdisplay
v=n+1
avbv
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslantH
p
n
bracketleftBigg
|an+p|+
p?1summationdisplay
i=1
|an+i?an+i+1|
bracketrightBigg
a207
∞summationdisplay
v=1
|av?av?1|a194a241a167a25
∞summationdisplay
v=1
(av?av?1)a194a241a133
∞summationdisplay
v=1
(av?av?1) =?a0 +ana167a25 lim
n→∞
ana127a51
a207a13a127a51M > 0a167a166a233a152a131na167a107
p?1summationdisplay
i=1
|an+i?an+i+1|+|an+p|<M (4)
a113
∞summationdisplay
v=1
bva194a241a167a108a13a233?ε> 0,?N ∈Z+a167a8n>Na158a167a233a152a131p∈Z+a167a107
Hpn < εM (5)
a100(),()a127a167a8n>Na158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
n+psummationdisplay
v=n+1
avbv
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<εa167a249a76a178a63a234
∞summationdisplay
v=1
avbva194a241
8,a124a94a133a220a194a241a6a110a121a178a2a134a63a234a27a52a217a90a91a189a110.
a121a178a181a233a63a219a103a44a234pa167a107
|Sn+p?Sn| =vextendsinglevextendsingle(?1)n+2un+1 + (?1)n+3un+2 +···+ (?1)n+p+1un+pvextendsinglevextendsingle=vextendsinglevextendsingle(?1)n+2(un+1?un+2 +···+ (?1)p?1un+p)vextendsinglevextendsingle=vextendsingle
vextendsingleun+1?un+2 +···+ (?1)p?1un+pvextendsinglevextendsingle
a8pa143a243a234a158a167(un+1?un+2) +···+ (un+p?1?un+p) greaterorequalslant 0
a8pa143a219a234a158a167(un+1?un+2) +···+ (un+p?2?un+p?1) +un+p greaterorequalslant 0
a111a131vextendsinglevextendsingleun+1?un+2 +···+ (?1)p?1un+pvextendsinglevextendsingle= un+1?un+2 +···+ (?1)p?1un+p
a113a8pa143a243a234a158a167un+1?(un+2?un+3)?···?(un+p?2?un+p?1)?un+p lessorequalslantun+1
a8pa143a219a234a158a167un+1?(un+2?un+3)?···?(un+p?1?un+p) lessorequalslantun+1
a111a131un+1?un+2 +···+ (?1)p?1un+p lessorequalslantun+1
a233a63a191ε> 0a167a207 lim
n→∞
un = 0a167a75 lim
n→∞
un+1 = 0
a117a180a55a127a51N ∈Z+a167a8n>Na158a167a107|un+1?0|<εa167a75un+1 <ε
a100a100a8n>Na158a167a233a63a219a103a44a234pa209a107|Sn+p?Sn| = un+1?un+2 +···+ (?1)pun+p lessorequalslantun+1 <ε
a108a13a100a133a220a194a241a6a110a167a26
∞summationdisplay
n=1
(?1)n+1una194a241.
157
§5,a253a233a194a241a63a234a218a94a135a194a241a63a234a27a53a159
1,a23|x|< 1,|y|< 1a167a121a178
∞summationdisplay
v=1
(xv?1 +xv?2y+···+yv?1) = 1(1?x)(1?y)
a121a178a181a207|x|< 1,|y|< 1a167a75
∞summationdisplay
v=1
xv?1 = 1 +x+x2 +···+xv +··· = 11?xa253a233a194a241 (6)
∞summationdisplay
v=1
yv?1 = 1 +y+y2 +···+yv +··· = 11?y a253a233a194a241 (7)
()·()a167a26
∞summationdisplay
v=1
xv?1
∞summationdisplay
v=1
yv?1 = 1(1?x)(1?y)
a113
∞summationdisplay
v=1
xv?1
∞summationdisplay
v=1
yv?1 = (1+x+x2+···+xv+···)(1+y+y2+···+yv+···) =
∞summationdisplay
v=1
(xv?1+xv?2y+···+yv?1)a167
a75
∞summationdisplay
v=1
(xv?1 +xv?2y+···+yv?1) = 1(1?x)(1?y).
2,a121a178a181
∞summationdisplay
n=0
xn
n!
∞summationdisplay
n=0
yn
n! =
∞summationdisplay
n=0
(x+y)n
n!
a121a178a181a207 lim
n→∞
|x|n+1
(n+1)!
|x|n
n!
= lim
n→∞
|x|
n+ 1 = 0 < 1a167a75a226a136a75a19a16a7a79a123a27a52a129a47a170a167a26a63a234
∞summationdisplay
n=0
|x|n
n! a194a241
a117a180a63a234
∞summationdisplay
n=0
|x|n
n! a253a233a194a241
a211a110a167a63a234
∞summationdisplay
n=0
|y|n
n! a253a233a194a241
a140a21a164
∞summationdisplay
n=0
xn
n!
∞summationdisplay
n=0
yn
n! =
∞summationdisplay
n=0
Cn
a217a165Cn =
nsummationdisplay
i=0
xi
i! ·
yn?i
(n?i)! =
yn
n! +
x
1! ·
yn?1
(n?1)! +···+
xn
n! =
1
n!(C
0
ny
n+C1
nxy
n?1 +···+Cn
nx
n) = (x+y)
n
n!
a75
∞summationdisplay
n=0
xn
n!
∞summationdisplay
n=0
yn
n! =
∞summationdisplay
n=0
(x+y)n
n!
3,a121a178a181a140a177a138a209a94a135a194a241a63a234a27a141a83a63a234a167a166a217a117a209a20+∞.
a121a178a181a23
∞summationdisplay
n=1
una94a135a194a241
a100a189a1101a167a26
∞summationdisplay
n=1
vna218
∞summationdisplay
n=1
wna209a117a209a167a133
∞summationdisplay
n=1
vna117a209a20+∞a167
∞summationdisplay
n=1
(?wn)a117a209a20?∞
a192a18a117a209a20+∞a27a234a15{βn}a167a61 lim
n→∞
βn = +∞
a114
∞summationdisplay
n=1
vna85a94a83a152a145a152a145a92a229a53
a18m1a167a166v1 +v2 +···+vm1 >β1 +w1
a44a0a18m2a167a166v1 +v2 +···+vm1 +vm1+1 +···+vm2 >β2 +w1 +w2
a152a132a47a167a140a18a191a169a140a27mk >mk?1a167a166a26v1+v2+···+vm1+···+vm2+···+vmk >βk+w1+w2+···+wk (k =
3,4,···)
a249a24a2a134a47a152a92a152a124a20a145a218a152a135a75a145a181
(v1 +···+vm1?w1) + (vm1+1 +···+vm2?w2) +···+ (vmk?1+1 +···+vmk?wk) +··· (?)
a100a63a234a119a44a143a6a63a234a27a141a83a63a234
a207(?)a92a41a210a0a27a63a234
∞summationdisplay
k=1
(vmk?1+1 +···+vmk?wk)a27ka103a220a169a218
(v1 +···+vm1?w1) + (vm1+1 +···+vm2?w2) +···+ (vmk?1+1 +···+vmk?wk) >βk
158
a13 lim
k→∞
βk = +∞
a75
∞summationdisplay
k=1
(vmk?1+1 +···+vmk?wk)a117a209a20+∞
a100a117a209a63a234a140a63a191a22a41a210a167a75a140a177a138a209a94a135a194a241a63a234a27a141a83a63a234a167a166a217a117a209a20+∞.
159
§6,a195a161a166a200
1,a63a216a195a161a166a200a27a194a241a53a181
(1)
∞productdisplay
n=3
n2?4
n2?1
(2)
∞productdisplay
n=1
a(?1)
n
n (a> 0)
(3)
∞productdisplay
n=0
radicalBigg
n+ 1
n+ 2
a41a181
(1) a207n
2?4
n2?1 = 1?
3
n2?1,ngreaterorequalslant 3a167a133?
3
n2?1 < 0
a113 lim
n→∞
3
n2?1
1
n2
= 3a133
∞summationdisplay
n=3
1
n2a194a241a167a75
∞summationdisplay
n=3
3
n2?1a194a241a167a117a180
∞summationdisplay
n=3
parenleftBigg
3n2?1
parenrightBigg
a194a241
a108a13a226a189a1102a167a26
∞productdisplay
n=3
n2?4
n2?1a194a241.
(2)
∞summationdisplay
n=1
lna(?1)
n
n =
∞summationdisplay
n=1
(?1)n
n lna = lna
∞summationdisplay
n=1
(?1)n
n
a207a63a234
∞summationdisplay
n=1
(?1)n
n a143a52a217a90a91a46a63a234a167a75a217a194a241a167a117a180a63a234
∞summationdisplay
n=1
lna(?1)
n
n a194a241a167a108a13a195a161a166a200
∞productdisplay
n=1
a(?1)
n
n a194
a241.
(3) a100a117a220a169a166a200Pn =
radicalBigg
1
2 ·
2
3 ···
n
n+ 1 ·
n+ 1
n+ 2 =
radicalBigg
1
n+ 2 → 0(n→∞)
a25a195a161a166a200
∞productdisplay
n=0
radicalBigg
n+ 1
n+ 2a117a209a1170.
2,a121a178a181a101
∞summationdisplay
n=1
x2na194a241a167a75
∞productdisplay
n=1
cosxna194a241.
a121a178a181a207
∞productdisplay
n=1
cosxn =
∞productdisplay
n=1
parenleftBigg
1?2sin2 xn2
parenrightBigg
a1330 lessorequalslant 2sin2 xn2 lessorequalslant 2·
parenleftBigg
sinxn
2
parenrightBigg2
= x
2
n
2
a113
∞summationdisplay
n=1
x2na194a241a167a75
∞summationdisplay
n=1
2sin2 xn2 a194a241
a117a180a226a189a1102a167a26
∞productdisplay
n=1
cosxna194a241.
3,a121a178a181a101
∞summationdisplay
n=1
αna253a233a194a241a167a75
∞productdisplay
n=1
tan
parenleftBigg
pi
4 +αn
parenrightBigg
a194a241
parenleftBigg
a217a165|αn|< pi4
parenrightBigg
.
a121a178a181
∞productdisplay
n=1
tan
parenleftBigg
pi
4 +αn
parenrightBigg
=
∞productdisplay
n=1
1 + tanαn
1?tanαn =
∞productdisplay
n=1
parenleftBigg
1 + 2tanαn1?tanα
n
parenrightBigg
a207
∞summationdisplay
n=1
αna253a233a194a241a167a75 lim
n→∞
αn = 0a167a117a180 lim
n→∞
vextendsinglevextendsingle
vextendsingle 2tanαn1?tanαn
vextendsinglevextendsingle
vextendsingle
|αn| = limn→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
2
1?tanαn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
tanαn
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 2
a100
∞summationdisplay
n=1
αna253a233a194a241a167a26
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle 2tanαn
1?tanαn
vextendsinglevextendsingle
vextendsinglevextendsinglea194a241a167a117a180
∞summationdisplay
n=1
2tanαn
1?tanαna253a233a194a241
a108a13
∞productdisplay
n=1
tan
parenleftBigg
pi
4 +αn
parenrightBigg
a253a233a194a241.
160
a49a155a217 a50a194a200a169
§1,a195a161a129a27a50a194a200a169
1,a166a101a15a50a194a200a169a27a138a181
(1)
integraldisplay +∞
2
1
x2?1 dx
(2)
integraldisplay +∞
0
1
(x2 +p)(x2 +q)dx,(p,q> 0)
(3)
integraldisplay +∞
0
e?ax2xdx(a> 0)
(4)
integraldisplay +∞
0
e?axsinbxdx,(a> 0)
a41a181
(1)
integraldisplay +∞
2
1
x2?1 dx =
1
2ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x?1
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+∞
2
= 12ln3 = ln√3
(2)
integraldisplay +∞
0
1
(x2 +p)(x2 +q)dx =
1
q?p
parenleftBigg
1√
parctan
x√
p?
1√
qarctan
x√
q
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+∞
0
=
pi
2√
pq(√p+√q) =
pi
2(q√p+p√q)
(3)
integraldisplay +∞
0
e?ax2xdx =?e
ax2
2a
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+∞
0
= 12a
(4)
integraldisplay +∞
0
e?axsinbxdx =?asinbx?bcosbxa2 +b2 e?ax
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+∞
0
= ba2 +b2
2,a63a216a101a15a200a169a27a194a241a53a181
(1)
integraldisplay +∞
0
dx
3√x4 + 1
(2)
integraldisplay +∞
1
xarctanx
1 +x2 dx
(3)
integraldisplay +∞
1
sin 1x2 dx
(4)
integraldisplay +∞
0
dx
1 +x|sinx|
(5)
integraldisplay +∞
0
x
1 +x2 sin2x dx
(6)
integraldisplay +∞
0
xm
1 +xn dx,(n> 0,m> 0)
a41a181
(1)
integraldisplay +∞
0
dx
3√x4 + 1 =
integraldisplay 1
0
dx
3√x4 + 1 +
integraldisplay +∞
1
dx
3√x4 + 1
a207
integraldisplay 1
0
dx
3√x4 + 1a143a20a126a200a169a167a75a217a55a194a241
a233
integraldisplay +∞
1
dx
3√x4 + 1a167a207
integraldisplay +∞
1
dx
3√x4a194a241a167a75
integraldisplay +∞
1
dx
3√x4 + 1a194a241a167a108a13
integraldisplay +∞
0
dx
3√x4 + 1a194a241.
(2) a207 lim
x→+∞
x
1+x3 arctanx
1
x2
= lim
x→+∞
x3
1 +x3 arctanx =
pi
2a167a133
integraldisplay +∞
1
1
x2a194a241
a75a100a39a22a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
xarctanx
1 +x2 dxa194a241.
161
(3) a207 lim
x→+∞
sin 1x2
1
x2
= 1a133
integraldisplay +∞
1
dx
x2 a194a241a167a108a13
integraldisplay +∞
1
sin 1x2 dxa194a241.
(4) 11 +x|sinx|greaterorequalslant 11 +x
a207 lim
x→+∞
1
1+x
1
x
= 1a133
integraldisplay +∞
1
dx
x a117a209a167a108a13a100a39a22a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
dx
1 +xa117a209
a113
integraldisplay 1
0
dx
1 +xa143a20a126a200a169a75a194a241a167a117a180
integraldisplay +∞
0
dx
1 +xa117a209
a108a13a100a39a22a7a79a123a167a26
integraldisplay +∞
0
dx
1 +x|sinx|a117a209.
(5) a207x∈ [0,+∞)a158a167a107 x1 +x2 sin2xgreaterorequalslant x1 +x2a133
integraldisplay +∞
0
x
1 +x2 dx =
1
2ln(1 +x
2)
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
0
= +∞
a75a100a39a22a7a79a123a167a26
integraldisplay +∞
0
x
1 +x2 sin2x dxa117a209.
(6)
integraldisplay +∞
0
xm
1 +xn dx =
integraldisplay 1
0
xm
1 +xn dx+
integraldisplay +∞
1
xm
1 +xn dxa133
integraldisplay 1
0
xm
1 +xn dxa143a126a194a200a169
(i) a8n?m> 1a158a167a107 1xn?m · 11 + 1
xn
< 1xn?ma133a200a169
integraldisplay +∞
1
xm
1 +xn dxa194a241a167a25a6a200a169a194a241a182
(ii) a8n?mlessorequalslant 1a133xgreaterorequalslant 1a158a167a107 x
m
1 +xn greaterorequalslant
1
2xn?ma167a133
integraldisplay +∞
1
1
xn?m dxa117a209a167a25a6a200a169a117a209
a75a8n?m> 1a158a167
integraldisplay +∞
0
xm
1 +xn dxa194a241a182a8n?mlessorequalslant 1a158a167
integraldisplay +∞
0
xm
1 +xn dxa117a209.
3,a121a178a253a233a194a241a27a50a194a200a169a55a194a241a167a2a135a131a216a44.
a121a178a181a174a127
integraldisplay +∞
a
|f(x)|dxa194a241a167a100a133a220a7a79a6a110a167a26a233?ε> 0,?A> 0a167a8Aprimeprime >Aprime >Aa158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
|f(x)|dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<εa167a75
integraldisplay Aprimeprime
Aprime
|f(x)|dx<εa167a117a180
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
f(x)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
integraldisplay Aprimeprime
Aprime
|f(x)|dx<εa167
a108a13
integraldisplay +∞
a
f(x)dxa194a241.
a194a241a27a50a194a200a169a153a55a253a233a194a241.
a126a181
integraldisplay +∞
1
sinx
x dxa194a241a182a13
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dxa117a209(a132a214a25455a144).
4,a121a178a233a117a195a161a129a200a169a167a169a220a200a169a250a170a164a225a163a8a250a170a165a136a220a169a107a191a194a158a164
integraldisplay +∞
a
f(x)gprime(x)dx = f(x)g(x)
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
a
integraldisplay +∞
a
g(x)f prime(x)dx
a121a178a181a233a117a63a191A>aa167a164a225
integraldisplay A
a
f(x)gprime(x)dx = f(x)g(x)
vextendsinglevextendsingle
vextendsinglevextendsingle
A
a
integraldisplay A
a
g(x)f prime(x)dx
a252a62a18a52a129a167a26 lim
A→+∞
integraldisplay A
a
f(x)gprime(x)dx = lim
A→+∞
parenleftBigg
f(x)g(x)
vextendsinglevextendsingle
vextendsinglevextendsingle
A
a
parenrightBigg
lim
A→+∞
parenleftbiggintegraldisplay A
a
g(x)f prime(x)dx
parenrightbigg
a75
integraldisplay +∞
a
f(x)gprime(x)dx = f(x)g(x)
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
a
integraldisplay +∞
a
g(x)f prime(x)dx
5,a121a178a181
(1) a23f(x)a143[0,+∞)a254a27a152a151a235a89a188a234a167a191a133a200a169
integraldisplay +∞
0
f(x)dxa194a241a167a75 lim
x→+∞
f(x) = 0a182a88a74a61a61a200
a169
integraldisplay +∞
0
f(x)dxa194a241a167a177a57f(x)a51[0,+∞)a235a89a167f(x) greaterorequalslant 0a167a180a196a69a206a164a225 lim
x→+∞
f(x) = 0a186
a121a178a181a94a135a121a123.a23 lim
x→+∞
f(x) negationslash= 0a167a75?ε> 0a167a233a63a191a140a27A> 0a167a209a127a51xA >Aa167a166a26|f(xA)|greaterorequalslant 2ε.
a18a83a15An → +∞(n→∞)a167a107a83a15xn → +∞a133xn >An(n = 1,2,···)a167a166|f(xn)|greaterorequalslant 2ε
a44a152a144a161a167a100f(x)a27a152a151a194a241a53a167a233a254a227ε> 0,?δ> 0a167a166a26a8|xprime?xprimeprime|<δa158a167a107|f(xprime)?f(xprimeprime)|<ε
a207a100a167a233a152a131na167a8x∈
parenleftBigg
xn? δ2,xn + δ2
parenrightBigg
a158a167a107|f(x)?f(xn)|<εa167a61f(xn)?ε<f(x) <f(xn) +ε
162
a8f(xn) > 0a158a167|f(xn)| = f(xn) greaterorequalslant 2εa167a100a134a224a216a31a170a167a26f(x) > 2ε?ε = ε
a8f(xn) < 0a158a167|f(xn)| =?f(xn) greaterorequalslant 2εa167a100a109a224a216a31a170a167a26f(x) <?2ε+ε =?ε
a108a13a167
integraldisplay xn+δ
2
xn?δ2
f(x)dx>εδa163a8f(xn) > 0a158a164a189
integraldisplay xn+δ
2
xn?δ2
f(x)dx<?εδa163a8f(xn) < 0a158a164
a100a134
integraldisplay +∞
0
f(x)dxa194a241a103a241a167a75a98a23a216a164a225a167a117a180 lim
x→+∞
f(x) = 0.
(2) a200a169
integraldisplay +∞
0
f(x)dxa194a241a167a177a57f(x)a51[0,+∞)a235a89a167f(x) greaterorequalslant 0a167a191a216a85a2a121 lim
x→+∞
f(x) = 0.
a126a181
integraldisplay +∞
0
x
1 +x6 sin2x dx.
a167a180a253a233a194a241a27.
a207
integraldisplay +∞
0
x
1 +x6 sin2x dx =
∞summationdisplay
n=0
integraldisplay (n+1)pi
npi
x
1 +x6 sin2x dx =
integraldisplay pi
0
x
1 +x6 sin2x dx+
∞summationdisplay
n=1
parenleftbigI1
n +I
2
n
parenrightbig
a217a165I1n =
integraldisplay (n+1
2)pi
npi
x
1 +x6 sin2x dx =
integraldisplay pi
2
0
npi+z
1 + (npi+z)6 sin2z dz,
I2n =
integraldisplay (n+1)pi
(n+12)pi
x
1 +x6 sin2x dx =
integraldisplay pi
2
0
npi+pi?z
1 + (npi+pi?z)6 sin2z dz
a53a191a20a80 <z< pi2a158a1672pi< sinzz lessorequalslant 1a167a117a180(npi+z)6 sin2z greaterorequalslant (npi)6
parenleftBigg
2z
pi
parenrightBigg2
= (2pi2n3z)2,
(npi+pi?z)6 sin2z greaterorequalslant (2pi2n3z)2
a25a107I1n lessorequalslant
integraldisplay pi
2
0
(n+ 1)pi
1 + (2pi2n3z)2 dz =
n+ 1
2n3pi
integraldisplay (npi)3
0
dy
1 +y2 lessorequalslant
n+ 1
4n3
a211a110a167a107I2n lessorequalslant n+ 14n3
a207
integraldisplay pi
0
x
1 +x6 sin2x dxa143a20a126a200a169a167a75a55a194a241
a113n+ 12n3 < 1n2a133
∞summationdisplay
n=1
1
n2a194a241a167a75
∞summationdisplay
n=1
n+ 1
2n3 a194a241
a117a180
integraldisplay +∞
0
x
1 +x6 sin2x dxlessorequalslant
integraldisplay pi
0
x
1 +x6 sin2x dx+
∞summationdisplay
n=1
n+ 1
2n3 a253a233a194a241
a119a44f(x) = x1 +x6 sin2xa51[0,+∞)a254a154a75a235a89
a2a101a18xn = 2npi(n = 0,1,2,···)a167a107f(xn) = f(2npi) = 2npi → +∞(n→∞)
a75 lim
x→∞
f(x) negationslash= 0.
6,a121a178a181a101f(x),g(x)a51a63a219a171a109[a,A]a140a200a167a113a23f2(x),g2(x)a51[a,+∞)a200a169a194a241a167a64a34[f(x) +g(x)]2a218
|f(x)·g(x)|a51[a,+∞)a254a27a140a200.
a121a178a181a207f(x),g(x)a51a63a219a171a109[a,A]a140a200a167a75
integraldisplay A
a
|f(x)·g(x)|dxa127a51a167
integraldisplay A
a
[f(x) +g(x)]2 dxa127a51
a113
integraldisplay +∞
a
f2(x)dxa218
integraldisplay +∞
a
g2(x)dxa209a194a241a167a75
integraldisplay +∞
a
[f2(x) +g2(x)]dxa194a241a167
a117a180
integraldisplay +∞
a
2[f2(x) +g2(x)]dxa218
integraldisplay +∞
a
1
2[f
2(x) +g2(x)]dxa209a194a241
a113[|f(x)|?|g(x)|]2 = f2(x) +g2(x)?2|f(x)·g(x)|greaterorequalslant 0a61|f(x)·g(x)|lessorequalslant 12[f2(x) +g2(x)]
a75a100a39a22a7a79a123a167a26|f(x)·g(x)|a51[a,+∞)a254a140a200
a113[f(x) +g(x)]2 = f2(x) +g2(x) + 2f(x)·g(x) lessorequalslantf2(x) +g2(x) + 2|f(x)·g(x)|lessorequalslant 2[f2(x) +g2(x)]
a75a100a39a22a7a79a123a167a26[f(x) +g(x)]2a51[a,+∞)a254a140a200.
7,a233a195a161a129a50a194a200a169a167a63a216a178a144a140a200a218a253a233a140a200a27a39a88.a127a9a126a102a181
integraldisplay +∞
1
dx
x3/2a218
integraldisplay +∞
1
f(x)dxa167a217a165
f(x) = n2
parenleftBigg
a8nlessorequalslantx<n+ 1n4
parenrightBigg
a167f(x) = 0
parenleftBigg
a8n+ 1n4 lessorequalslantx<n+ 1
parenrightBigg
.
a121a178a181a178a144a140a200notdblarrowrighta253a233a140a200
a126a181
integraldisplay +∞
1
1
x3/2 dxa194a241a167a2
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
x3/4
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay +∞
1
dx
x3/4a117a209a182
163
integraldisplay +∞
1
1
x3 dxa194a241a167a133
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
x3/2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay +∞
1
1
x3/2 dxa194a241
a253a233a140a200notdblarrowrighta178a144a140a200
a126a181
integraldisplay +∞
1
f(x)dxa167a217a165f(x) = n2
parenleftBigg
a8nlessorequalslantx<n+ 1n4
parenrightBigg
a167f(x) = 0
parenleftBigg
a8n+ 1n4 lessorequalslantx<n+ 1
parenrightBigg
integraldisplay +∞
1
|f(x)|dx =
integraldisplay +∞
1
f(x)dx = 114 ·12 + 124 ·22 +···+ 1n4 ·n2 +··· = 1 + 122 +···+ 1n2 +··· =
∞summationdisplay
n=1
1
n2a194a241
integraldisplay +∞
1
f(x)dxa167a217a165f(x) = n4
parenleftBigg
a8nlessorequalslantx<n+ 1n4
parenrightBigg
a167f(x) = 0
parenleftBigg
a8n+ 1n4 lessorequalslantx<n+ 1
parenrightBigg
integraldisplay +∞
1
f2(x)dx = 114 ·14 + 124 ·24 +···+ 1n4 ·n4 +··· = 1 + 1 +···+ 1 +··· =
∞summationdisplay
n=1
1a117a209a182
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
x3/2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay +∞
1
1
x3/2 dxa194a241a167a133
integraldisplay +∞
1
1
x3 dxa194a241
8,a63a216a101a15a200a169a27a253a233a194a241a53a57a94a135a194a241a53a181
(1)
integraldisplay +∞
0
√xcosx
x+ 100 dx
(2)
integraldisplay +∞
1
cosx
xλ dx,
integraldisplay +∞
1
sinx
xλ dx
(3)
integraldisplay +∞
a
Pm(x)
Qn(x) sinxdx,Pm(x),Qn(x)a136a143m,na103a245a145a170a133a8xgreaterorequalslantaa158a167Qn(x) negationslash= 0
(4)
integraldisplay +∞
2
lnlnx
lnx sinxdx
a41a181
(1) a233A> 0a167a100a117
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
0
cos dx
vextendsinglevextendsingle
vextendsinglevextendsingle= |sinA?sin0|lessorequalslant 1
a113 lim
x→+∞
√x
x+ 100 = 0,
parenleftBigg √
x
x+ 100
parenrightBiggprime
= 100?x2√x(x+ 100)2 a167a8x> 100a158a167
parenleftBigg √
x
x+ 100
parenrightBiggprime
< 0a167a75
√x
x+ 100a252
a78a126a8
a117a180a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
0
√xcosx
x+ 100 dxa194a241.
a2a167a216a143a253a233a194a241.
a100a117
√x|cosx|
x+ 100 greaterorequalslant
√xcos2x
x+ 100 =
1
2
parenleftBigg √
x
x+ 100 +
√xcos2x
x+ 100
parenrightBigg
a207 lim
x→+∞
parenleftBigg
x12 ·
√x
x+ 100
parenrightBigg
= 1a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
√x
x+ 100 dxa117a209
a113
integraldisplay 1
0
√x
x+ 100 dxa143a20a126a200a169a167a75
integraldisplay +∞
0
√x
x+ 100 dx =
integraldisplay 1
0
√x
x+ 100 dx+
integraldisplay +∞
1
√x
x+ 100 dxa117a209
a157a99a140a227a27a121a178a167a140a127
integraldisplay +∞
0
√xcos2x
x+ 100 dxa194a241a167a108a13a200a169
integraldisplay +∞
0
√xcos2x
x+ 100 dxa117a209
a75
integraldisplay +∞
0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
√xcosx
x+ 100
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay +∞
0
√x|cosx|
x+ 100 dxa117a209a167a108a13a200a169
integraldisplay +∞
0
√xcosx
x+ 100 dxa94a135a194a241.
(2) (i) a8λ> 1a158a167a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosx
xλ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
|cosx|
xλ lessorequalslant
1
xλa133a8λ> 1a158a167
integraldisplay +∞
1
dx
xλa194a241a167a108a13
integraldisplay +∞
1
cosx
xλ dxa253a233a194a241
a211a110
integraldisplay +∞
1
sinx
xλ dxa253a233a194a241
(ii) a80 <λlessorequalslant 1a158
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
cosxdx
vextendsinglevextendsingle
vextendsinglevextendsingle= |sinA?sin1|lessorequalslant 2a133 1
xλa80 <λlessorequalslant 1a158a252a78a126a8a167a8x→ +∞a158a170a1170a167
164
a75a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
1
cosx
xλ dxa194a241
a2|cosx|xλ greaterorequalslant cos
2x
xλ =
1
2
parenleftBigg
1
xλ +
cos2x
xλ
parenrightBigg
a167a100a99a161a121a178a167a140a127
integraldisplay +∞
1
cos2x
xλ dxa194a241
a113
integraldisplay +∞
1
dx
xλ (0 <λlessorequalslant 1)a117a209a167a75
integraldisplay +∞
1
|cosx|
xλ dxa117a209a167a108a13
integraldisplay +∞
1
cosx
xλ dxa94a135a194a241
a211a110a167
integraldisplay +∞
1
sinx
xλ dxa94a135a194a241
(iii) a8λlessorequalslant 0a158
a207n→ +∞,2npi → +∞a167a117a180a233a63a191A> 0a167a150a8a140a177a233a20(2n+ 1)pi> 2npi>A
a18ε0 = 2a167a8(2n+1)pi> 2npi>Aa158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay (2n+1)pi
2npi
sinx
xλ dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
integraldisplay (2n+1)pi
2npi
sinx
xλ dxgreaterorequalslant
integraldisplay (2n+1)pi
2npi
sinxdx =
2 = ε0
a75a8λlessorequalslant 0a158a167
integraldisplay +∞
1
sinx
xλ dxa117a209
a211a110a167
integraldisplay +∞
1
cosx
xλ dxa117a209.
a110a220a127a167λ> 1a158a167
integraldisplay +∞
1
cosx
xλ dxa167
integraldisplay +∞
1
sinx
xλ dxa253a233a194a241a182
0 <λlessorequalslant 1a158a167
integraldisplay +∞
1
cosx
xλ dxa167
integraldisplay +∞
1
sinx
xλ dxa94a135a194a241a182
λ< 0a158a167
integraldisplay +∞
1
cosx
xλ dxa167
integraldisplay +∞
1
sinx
xλ dxa117a209.
(3) (i) a23m<n.a100a158a167a253a169a170Pm(x)Q
n(x)
a8xa118a10a140a158a167a145x→ +∞a13a252a78a101a252a170a1170
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
a
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 2(a233?A>a)a167a75a226a41a225a142a52a7a79a123a167a6a200a169a194a241
(ii) a23Qn(x) ≡ 1.a100a158a245a145a170a143Pm(x) = amxm +···+a0a167a216a148a23am > 0
a100a117 lim
x→+∞
Pm(x)
xm = am > 0a167a25a127a51bpi+pi> 0a167a166a8x>bpi+pia158a167Pm(x) =
am
2 x
m
a117a180a107
integraldisplay +∞
a
Pm(x)sinxdx =
integraldisplay bpi+pi
a
Pm(x)sinxdx+
∞summationdisplay
n=b+1
Ina167a217a165In =
integraldisplay (n+1)pi
npi
Pm(x)sinxdx
a100a158a107|In| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay (n+1)pi
npi
Pm(x)sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay pi
0
Pm(npi+z)(?1)nsinzdz
vextendsinglevextendsingle
vextendsinglevextendsinglegreaterorequalslant am
2 (npi)
m
integraldisplay pi
0
sinzdz =
am(npi)ma167a75In →∞(n→∞)
a113
integraldisplay bpi+pi
a
Pm(x)sinxdxa143a20a126a200a169a167a75a55a194a241a167a117a180
integraldisplay +∞
a
Pm(x)sinxdxa117a209
(iii) a8mgreaterorequalslantna158a167Pm(x)Q
n(x)
= R(x) +S(x)a167a217a165R(x)a143a253a169a170a167S(x)a143a18a170
a100(ii)a127a167
integraldisplay +∞
a
S(x)sinxdxa117a209a182a100(i)a127a167
integraldisplay +∞
a
R(x)sinxdxa194a241a167a25
integraldisplay +∞
a
Pm(x)
Qn(x) sinxdxa117
a209
(iv) a23Qn(x) = bnxn +···+b1x+b0
a100a117 lim
x→+∞
vextendsinglevextendsingle
vextendsinglePn(x)Qm(x) sinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingleam
bn x
m?nsinx
vextendsinglevextendsingle
vextendsingle
= 1a167a75a1008(2)a127a167a8λ = n?m> 1a158a167a200a169a253a233a194a241
a110a220a127a181mgreaterorequalslantna158a167a200a169a117a209a182m = n?1a158a167a200a169a94a135a194a241a182m<n?1a158a167a200a169a253a233a194a241.
(4) a233A> 2a167
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
2
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 2a167 lim
x→+∞
lnlnx
lnx = limx→+∞
1
lnx ·
1
x
1
x
= lim
x→+∞
1
lnx = 0a167parenleftBigg
lnlnx
lnx
parenrightBiggprime
= 1?lnlnxx(lnx)2 a167a8x>eea158a167
parenleftBigg
lnlnx
lnx
parenrightBiggprime
< 0a167a100a158a100a188a234a252a78a126a170a1170
a75a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
ee
lnlnx
lnx sinxdxa194a241
a113
integraldisplay ee
2
lnlnx
lnx sinxdxa143a20a126a200a169a167a75a55a194a241a167a117a180
integraldisplay +∞
2
lnlnx
lnx sinxdxa194a241
165
a113
integraldisplay +∞
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lnlnx
lnx sinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay n0pi
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lnlnx
lnx sinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx+
∞summationdisplay
n=n0
Ina167a217a165n0 > e
e
pia143a20a18a234
In =
integraldisplay (n+1)pi
npi
lnlnx
lnx |sinx|dx =
integraldisplay pi
0
lnln(npi+z)
ln(npi+z) sinzdz greaterorequalslant
lnln(n+ 1)pi
ln(n+ 1)pi
integraldisplay pi
0
sinzdz = 2 lnln(n+ 1)piln(n+ 1)pi
a207
integraldisplay +∞
ee+pi
lnlnx
lnx dx>
integraldisplay +∞
ee+pi
lnlnx
x dx = lnx(lnlnx?1)
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
ee+pi
= +∞a167a75a100a133a220a7a79a123a167a26
∞summationdisplay
n=n0
lnln(n+ 1)pi
ln(n+ 1)pi a117
a209a167a117a180
+∞summationdisplay
n=n0
Ina117a209a167a108a13a6a200a169a94a135a194a241.
166
§2,a195a46a188a234a27a50a194a200a169
1,a101a15a200a169a180a196a194a241a186a88a74a194a241a167a166a217a138.
(1)
integraldisplay 1
2
0
cotxdx
(2)
integraldisplay 1
0
lnxdx
a41a181
(1) a207 lim
x→+0
cotx = ∞a167a75x = 0a143cotxa27a219a58
a113
integraldisplay 1
2
0+η
cotxdx = ln|sinx|
vextendsinglevextendsingle
vextendsinglevextendsingle
1
2
η
= ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle?ln|sinη|→ +∞(η → +0)a167a75a200a169
integraldisplay 1
2
0
cotxdxa117a209.
(2) a207 lim
x→+0
lnx = ∞a167a75x = 0a143lnxa27a219a58
a113
integraldisplay 1
0+η
lnxdx = x(lnx?1)
vextendsinglevextendsingle
vextendsinglevextendsingle
1
η
=?ηlnη?1 +η →?1(η → +0)a167a75a200a169
integraldisplay 1
0
lnxdxa194a241a117?1.
2,a63a216a101a15a200a169a27a194a241a53a181
(1)
integraldisplay 1
0
sinx
x32
dx
(2)
integraldisplay 1
0
dx
3radicalbigx2(1?x)
(3)
integraldisplay 1
0
lnx
1?x2 dx
(4)
integraldisplay pi
2
0
dx
sin2x·cos2x
(5)
integraldisplay 1
0
|lnx|pdx
(6)
integraldisplay pi
2
0
1?cosx
xm dx
(7)
integraldisplay 1
0
xa?1(1?x)b?1 dx
(8)
integraldisplay 1
0
xa?1(1?x)b?1 lnxdx
a41a181
(1) x = 0a143 sinx
x32
a27a219a58
a207 lim
x→+0
x12 · sinx
x32
= lim
x→+0
sinx
x = 1a167a75a226a133a220a7a79a123a167a26
integraldisplay 1
0
sinx
x32
dxa253a233a194a241.
(2) x = 0a167x = 1a254a143a26a200a188a234a27a219a58a167
integraldisplay 1
0
dx
3radicalbigx2(1?x) =
integraldisplay 1
2
0
dx
3radicalbigx2(1?x) +
integraldisplay 1
1
2
dx
3radicalbigx2(1?x)
a207 lim
x→+0
x23 · 13radicalbigx2(1?x) = lim
x→+0
1
3√1?x = 1a167a75a226a133a220a7a79a123a167a26
integraldisplay 1
2
0
dx
3radicalbigx2(1?x)a253a233a194a241a182
a113 lim
x→1?0
(1?x)13 dx3radicalbigx2(1?x) = lim
x→1?0
1
3√x2 = 1a167a75a226a133a220a7a79a123a167a26
integraldisplay 1
1
2
dx
3radicalbigx2(1?x)a253a233a194a241
a108a13
integraldisplay 1
0
dx
3radicalbigx2(1?x)a253a233a194a241.
167
(3) a207lim
x→1
lnx
1?x2 = limx→1
1
x
2x =? limx→1
1
2x2 =?
1
2 a167a75x = 1a216a180a219a58a167a117a180a100a200a169a144a107a152a135a219a580
a113 lim
x→+0
x12 · lnx1?x2 = lim
x→+0
x12 lnx = lim
x→+0
lnx
x?12
= lim
x→∞
1
x
12x?32
=?2 lim
x→+0
x12 = 0
a75a100a133a220a7a79a123a167a26
integraldisplay 1
0
lnx
1?x2 dxa194a241.
(4) x = 0a167x = pi2a254a143a26a200a188a234a27a219a58a167a75
integraldisplay pi
2
0
dx
sin2x·cos2x =
integraldisplay pi
4
0
dx
sin2x·cos2x +
integraldisplay pi
2
pi
4
dx
sin2x·cos2x
a207 lim
x→+0
x2· 1sin2x·cos2x = 1a167a133 1sin2x·cos2xgreaterorequalslant 0a167a75a226a133a220a7a79a123a167a26
integraldisplay pi
4
0
dx
sin2x·cos2xa117a209a150+∞
a113 1sin2x·cos2xgreaterorequalslant 0a167a75
integraldisplay pi
2
0
dx
sin2x·cos2xa117a209.
(5)
integraldisplay 1
0
|lnx|pdx =
integraldisplay 1
2
0
|lnx|pdx+
integraldisplay 1
1
2
|lnx|pdx
a233
integraldisplay 1
2
0
|lnx|pdxa167a8p> 0a158a1670a143a219a58
a207 lim
x→+0
x12|lnx|p = lim
x→+0
|lnx|p
x?12
= lim
x→+0
(?lnx)p
x?12
= (?1)p lim
x→+0
plnp?1x· 1x
12x?32
= (?1)p+12p lim
x→+0
lnp?1x
x?12
= 0(a132a214a254a254231a1441.(16))
a75a100a133a220a7a79a123a167a26
integraldisplay 1
2
0
|lnx|pdxa8p> 0a158a194a241
a8plessorequalslant 0a158a167
integraldisplay 1
2
0
|lnx|pdxa143a20a126a200a169a167a75
integraldisplay 1
2
0
|lnx|pdxa167a117a180pa143a63a219a138a158a167
integraldisplay 1
2
0
|lnx|pdxa254a194a241.
a233
integraldisplay 1
1
2
|lnx|pdxa167p< 0a158a1671a143a219a58
a207 lim
x→1?0
(1?x)?p|lnx|p = lim
x→1?0
|lnx|p
(1?x)p = limx→1?0
ln 1x
1?x
p
=
parenleftBigg
lim
x→1?0
1
x
parenrightBiggp
= 1
a75a226a133a220a7a79a123a167a26a8?p< 1a610 >p>?1a158a167
integraldisplay 1
1
2
|lnx|pdxa194a241a182
a8?pgreaterorequalslant 1a61plessorequalslant?1a158a167
integraldisplay 1
1
2
|lnx|pdxa117a209
a8pgreaterorequalslant 0a158a167
integraldisplay 1
1
2
|lnx|pdxa143a20a126a200a169a167a25a194a241
a117a180a8p>?1a158a167
integraldisplay 1
1
2
|lnx|pdxa194a241a182a8plessorequalslant?1a158a167
integraldisplay 1
1
2
|lnx|pdxa117a209
a110a220a127a167a8p>?1a158a167
integraldisplay 1
0
|lnx|pdxa194a241a182a8plessorequalslant?1a158a167
integraldisplay 1
0
|lnx|pdxa117a209.
(6) a207 lim
x→+0
1?cosx
xm =
0,mlessorequalslant 0
lim
x→+0
sinx
mxm?1 =
0,0 <m< 1
lim
x→+0
cosx
m(m?1)xm?2 =
0,1 <m< 2
1
2,m = 2∞,m> 2
a75 lim
x→+0
1?cosx
xm =
0,m< 2
1
2,m = 2∞,m> 2
a108a13a8mlessorequalslant 2a158a167
integraldisplay pi
2
0
1?cosx
xm dxa143a20a126a200a169a167a25a194a241
a8m> 2a158a167x = 0a143
integraldisplay pi
2
0
1?cosx
xm dxa27a219a58
a113 lim
x→+0
xm?2 1?cosxxm = lim
x→+0
1?cosx
x2 =
1
2
168
a75a80 <m?2 < 1a612 <m< 3a158a167a200a169a194a241a182a8m?2 greaterorequalslant 1a61mgreaterorequalslant 3a158a167a200a169a117a209
a108a13a8m< 3a158a167
integraldisplay pi
2
0
1?cosx
xm dxa194a241a182a8mgreaterorequalslant 3a158a167
integraldisplay pi
2
0
1?cosx
xm dxa117a209.
(7) a8agreaterorequalslant 1a133bgreaterorequalslant 1a158a167
integraldisplay 1
0
xa?1(1?x)b?1 dxa143a20a126a200a169a167a25a194a241
integraldisplay 1
0
xa?1(1?x)b?1 dx =
integraldisplay 1
2
0
xa?1(1?x)b?1 dx+
integraldisplay 1
1
2
xa?1(1?x)b?1 dx
a233a200a169
integraldisplay 1
2
0
xa?1(1?x)b?1 dx =
integraldisplay 1
2
0
(1?x)b?1
x1?a dx
a207 lim
x→+0
(1?x)b?1
x1?a =
0,a> 1
1,a = 1
∞,a< 1
a133 lim
x→+0
x1?a(1?x)
b?1
x1?a = limx→+0(1?x)
b?1 = 1
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a81?a< 1a61a> 0a158a200a169a194a241a182a81?agreaterorequalslant 1a61alessorequalslant 0a158a167a200a169a117a209a182
a233a200a169
integraldisplay 1
1
2
xa?1(1?x)b?1 dx =
integraldisplay 1
1
2
xa?1
(1?x)1?b dx
a207 lim
x→1?0
xa?1
(1?x)1?b =
0,b> 1
1,b = 1
∞,b< 1
a133 lim
x→1?0
(1?x)1?b x
a?1
(1?x)1?b = limx→1?0x
a?1 = 1
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a81?b< 1a61b> 0a158a200a169a194a241a182a81?bgreaterorequalslant 1a61blessorequalslant 0a158a167a200a169a117a209a182
a110a254a164a227a167a8a> 0a133b> 0a158a167
integraldisplay 1
0
xa?1(1?x)b?1 dxa194a241a167a217a123a156a47a200a169a254a117a209.
(8)
integraldisplay 1
0
xa?1(1?x)b?1 lnxdx =
integraldisplay 1
2
0
xa?1(1?x)b?1 lnxdx+
integraldisplay 1
1
2
xa?1(1?x)b?1 lnxdx
a233a200a169
integraldisplay 1
2
0
xa?1(1?x)b?1 lnxdx =
integraldisplay 1
2
0
(1?x)b?1 lnx
x1?a dx
a207 lim
x→+0
(1?x)b?1 lnx
x1?a =
braceleftbigg 0,a> 1
∞,alessorequalslant 1
a133a233?c> 0a167 lim
x→+0
x1?a+c(1?x)
b?1
x1?a |lnx| = limx→+0
lnx
x?c = limx→+0
1x
cx?c?1 = limx→+0
xc
c = 0
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a81?a+c< 1a61a>c> 0a158a194a241
a113 lim
x→+0
x1?a(1?x)
b?1
x1?a |lnx| =? limx→+0(1?x)
b?1 lnx = ∞
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a81?agreaterorequalslant 1a61alessorequalslant 0a158a117a209
a233a200a169
integraldisplay 1
1
2
xa?1(1?x)b?1 lnxdx =
integraldisplay 1
1
2
xa?1 lnx
(1?x)1?b dx
a207 lim
x→1?0
xa?1 lnx
(1?x)1?b =
0,b> 0
1,b = 0
∞,b< 0
a133 lim
x→1?0
(1?x)?b x
a?1
(1?x)1?b|lnx| = limx→1?0
lnx
1?x = limx→1?0
1
x = 1
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a8?b< 1a61b>?1a158a194a241a182a8?bgreaterorequalslant 1a61blessorequalslant?1a158a117a209
a110a254a164a227a167a26a8a> 0a133b>?1a158a167a200a169
integraldisplay 1
0
xa?1(1?x)b?1 lnxdxa194a241a167a217a123a156a47a200a169a254a117a209.
3,a121a178a195a46a188a234a50a194a200a169a27a133a220a7a79a123a57a217a52a129a47a170.
a121a178a181
(1) a133a220a7a79a123a181
(i) a100|f(x)|lessorequalslant C(x?a)p(C > 0),p< 1a167a127 lim
ε→+0
integraldisplay b
a+ε
|f(x)|dxlessorequalslant lim
ε→+0
integraldisplay b
a+ε
C
(x?a)p dx =
lim
ε→+0
C
1?p(1?a)
1?p
vextendsinglevextendsingle
vextendsinglevextendsingle
b
a+ε
= lim
ε→+0
bracketleftBigg
C
1?p(b?a)
1?p? C
1?p ε
1?p
bracketrightBigg
= C1?p(b?a)1?p
a61 lim
ε→+0
integraldisplay b
a+ε
|f(x)|dxa127a51a167a25
integraldisplay b
a
f(x)dxa253a233a194a241
(ii) a207a107
integraldisplay b
a+ε
|f(x)|dx greaterorequalslant
integraldisplay b
a+ε
C
(x?a)p dx =
bracketleftBigg
C
1?p(b?a)
1?p? C
1?p ε
1?p
bracketrightBigg
→ ∞(a8p > 1,C >
169
0a133ε→ 0a158)
a113a8p = 1a158a167
integraldisplay b
a
C
x?a dxa117a209a167a108a13
integraldisplay b
a
|f(x)|dxa117a209.
(2) a133a220a7a79a123a27a52a129a47a170a181
(i) a23lim
x→a
(x?a)p|f(x)| = k(0 <k<∞)
a75a233?k>ε> 0a167a127a51δ> 0a167a166a8a<x<a+δa158a167a1070 <k?ε< (x?a)p|f(x)|<k+ε
a61a107 k?ε(x?a)p <|f(x)|< k+ε(x?a)p
a117a180
integraldisplay b
a
dx
(x?a)pa134
integraldisplay b
a
|f(x)dxa211a158a194a241a189a117a209a163a56a40a143a133a220a7a79a123a164
a108a13a8p< 1a158a167
integraldisplay b
a
f(x)dxa253a233a194a241a182pgreaterorequalslant 1a158a167f(x)a107a189a210a167a75
integraldisplay b
a
f(x)dxa117a209
(ii) k = 0a158a167a18ε0 = 1a167a75?δ> 0a167a166a8a<x<a+δa158a167
|(x?a)pf(x)| = (x?a)p|f(x)|< 1a61|f(x)|< 1(x?a)pa167
a75a100a133a220a7a79a123a167a26p< 1a158a167
integraldisplay b
a
f(x)dxa253a233a194a241
(iii) k = ∞a158a167a18G = 1a167a75?δ> 0a167a166a8a<x<a+δa158a167a107|(x?a)pf(x)| = (x?a)p|f(x)|> 1
a61|f(x)|> 1(x?a)p
a75a100a133a220a7a79a123a167a26a8pgreaterorequalslant 1a158a167
integraldisplay b
a
|f(x)|dxa117a209a182a113f(x)a107a189a210a167a108a13
integraldisplay b
a
f(x)dxa117a209.
a110a254a167a26a1010 lessorequalslant k < +∞,p < 1a167a64a34
integraldisplay b
a
f(x)dxa253a233a194a241a182a1010 < k lessorequalslant +∞,p greaterorequalslant 1a167a64
a34
integraldisplay b
a
f(x)dxa117a209.
4,a63a216a101a15a200a169a27a194a241a53a181
(1)
integraldisplay +∞
0
dx
3radicalbig(x?1)2x(x?2)
(2)
integraldisplay +∞
0
ln(1 +x)
xα dx
(3)
integraldisplay +∞
0
dx
xp +xq
(4)
integraldisplay +∞
0
arctanx
xα dx
(5)
integraldisplay +∞
1
dx
xplnqx
(6)
integraldisplay +∞
∞
dx
|x?a1|p1|x?a2|p2 ···|x?an|pn
a41a181
(1) x = 0,1,2a254a143a26a200a188a234a27a219a58integraldisplay
+∞
0
dx
3radicalbig(x?1)2x(x?2) =
parenleftBiggintegraldisplay 1
2
0
+
integraldisplay 1
1
2
+
integraldisplay 3
2
1
+
integraldisplay 2
3
2
+
integraldisplay 3
2
+
integraldisplay +∞
3
parenrightBigg
dx
3radicalbig(x?1)2x(x?2)
a233a200a169
integraldisplay 1
2
0
dx
3radicalbig(x?1)2x(x?2)
a207 lim
x→+0
x23
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3radicalbig(x?1)2x(x?2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limx→+0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
(x?1)2(x?2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3
= 0
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a200a169
integraldisplay 1
2
0
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a233a200a169
integraldisplay 1
1
2
dx
3radicalbig(x?1)2x(x?2)
170
a207 lim
x→1?0
(1?x)56
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3radicalbig(x?1)2x(x?2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limx→1?0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1?x)16
[x(x?2)]13
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 0
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a200a169
integraldisplay 1
1
2
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a211a100a121a123a167a26a200a169
integraldisplay 3
2
1
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a233a200a169
integraldisplay 2
3
2
dx
3radicalbig(x?1)2x(x?2)
a207 lim
x→2?0
(2?x)23
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3radicalbig(x?1)2x(x?2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limx→2?0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
2?x
x(x?1)2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3
= 0
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a200a169
integraldisplay 2
3
2
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a211a100a121a123a167a26
integraldisplay 3
2
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a207xgreaterorequalslant 3a158a167 13radicalbig(x?1)2x(x?2) lessorequalslant 1
(x?2)43
a133
integraldisplay +∞
3
dx
(x?2)43
a253a233a194a241
a75a100a39a22a7a79a123a167a26
integraldisplay +∞
3
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a110a254a140a127a167
integraldisplay +∞
0
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
(2)
integraldisplay +∞
0
ln(1 +x)
xα dx =
integraldisplay 1
0
ln(1 +x)
xα dx+
integraldisplay +∞
1
ln(1 +x)
xα dx
a233
integraldisplay 1
0
ln(1 +x)
xα dx
a207 lim
x→+0
ln(1 +x)
xα =
0,αlessorequalslant 0
lim
x→+0
1
1+x
αxα?1 =
0,0 <α< 1
1,α = 1
∞,α> 1
a167a75a8α> 1a158a1670a143a219a58
a113 lim
x→+0
xα?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ln(1 +x)
xα
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limx→+0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ln(1 +x)
x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 1
a75a8α? 1 < 1a611 < α < 2a158a167
integraldisplay 1
0
ln(1 +x)
xα dxa253a233a194a241a182a8α greaterorequalslant 2a158a167
integraldisplay 1
0
ln(1 +x)
xα dxa117a209a182
a8αlessorequalslant 1a158a167
integraldisplay 1
0
ln(1 +x)
xα dxa143a20a126a200a169a167a25a55a194a241
a108a13a8α< 2a158a167
integraldisplay 1
0
ln(1 +x)
xα dxa253a233a194a241a182a8αgreaterorequalslant 2a158a167
integraldisplay 1
0
ln(1 +x)xα dxa117a209
a233
integraldisplay +∞
1
ln(1 +x)
xα dx
a18λ> 1a167a8α?λ> 0a158a167a207 lim
x→+∞
xλ ln(1 +x)xα = lim
x→+∞
1
1+x
(α?λ)xα?λ?1 = 0
a75a8α?λ> 0a61α>λ> 1a158a167a200a169
integraldisplay +∞
1
ln(1 +x)
xα dxa253a233a194a241a182
a113 lim
x→+∞
xα ln(1 +x)xα = +∞a167a75a8αlessorequalslant 1a158a167a200a169
integraldisplay +∞
1
ln(1 +x)
xα dxa117a209
a108a13a81 <α< 2a158a167
integraldisplay +∞
0
ln(1 +x)
xα dxa253a233a194a241a182a217a167a156a47a167
integraldisplay +∞
0
ln(1 +x)
xα dxa209a117a209.
(3)
integraldisplay +∞
0
dx
xp +xq =
integraldisplay 1
0
dx
xp +xq +
integraldisplay +∞
1
dx
xp +xq
a233a200a169
integraldisplay 1
0
dx
xp +xqa167a23min(p,q) = p
a101plessorequalslant 0a167a75
integraldisplay 1
0
dx
xp +xqa143a20a126a200a169a167a25
integraldisplay 1
0
dx
xp +xqa194a241
171
a101p> 0a167a100a117 lim
x→+0
xp 1xp +xq = lim
x→+0
1
1 +xq?p =
1
2,p = q1,pnegationslash= q
a25a200a169
integraldisplay 1
0
dx
xp +xqa61a8p< 1a61min(p,q) < 1a158a194a241
a233a200a169
integraldisplay +∞
1
dx
xp +xqa167a23max(p,q) = q
a100a117 lim
x→+∞
xq 1xp +xq = lim
x→+∞
1
x?(q?p) + 1 =
1
2,p = q1,pnegationslash= q
a25a200a169
integraldisplay +∞
1
dx
xp +xqa61a8q> 1a61max(p,q) > 1a158a194a241
a75a200a169
integraldisplay +∞
0
dx
xp +xqa8min(p,q) < 1a133max(p,q) > 1a158a194a241.
(4) a80 <αlessorequalslant 1a158a167 lim
x→+0
arctanx
xα = limx→+0
1
1+x2
αxα?1 = limx→+0
x1?α
α(1 +x2) = 0
a8αlessorequalslant 0a167 lim
x→+0
arctanx
xα = limx→+0x
αarctanx = 0
a75αlessorequalslant 1a158a1670a216a143a219a58
a113 lim
x→+∞
xα arctanxxα = lim
x→+∞
arctanx = pi2
a75αlessorequalslant 1a158a167a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
0
arctanx
xα dxa117a209
a8α> 1a158a167a207 lim
x→+0
arctanx
xα = limx→+0
1
1+x2
αxα?1 = limx→+0
1
αxα?1(1 +x2) = +∞a167a750a143a219a58
a75
integraldisplay +∞
0
arctanx
xα dx =
integraldisplay 1
0
arctanx
xα dx+
integraldisplay +∞
1
arctanx
xα dx
a233
integraldisplay 1
0
arctanx
xα dxa167α> 1a1670a143a219a58
a207 lim
x→+0
xα?1 arctanxxα = lim
x→+0
arctanx
x = 1
a75a100a133a220a7a79a123a27a52a129a47a170a167a26α?1 < 1a61α< 2a158a200a169a194a241a182a8αgreaterorequalslant 2a158a167a200a169a117a209
a233
integraldisplay +∞
1
arctanx
xα dx
a207
pi
4
xα lessorequalslant
arctanx
xα lessorequalslant
pi
2
xαa133
integraldisplay +∞
1
pi
2
xα dxa8α> 1a158a200a169a194a241a182
integraldisplay +∞
1
pi
4
xα dxa8αlessorequalslant 1a158a200a169a117a209
a75a100a39a22a7a79a123a167a26a8α> 1a158a167
integraldisplay +∞
1
arctanx
xα dxa194a241a182a8αlessorequalslant 1a158a167
integraldisplay +∞
1
arctanx
xα dxa117a209
a111a131a167a81 <α< 2a158a167
integraldisplay +∞
0
arctanx
xα dxa194a241a182a217a123a156a47a100a200a169a254a117a209.
(5)
integraldisplay +∞
1
dx
xplnqx =
integraldisplay 2
1
dx
xplnqx +
integraldisplay +∞
2
dx
xplnqx
a127a196
integraldisplay 2
1
dx
xplnqxa167a233a63a191a27p
a100a117 lim
x→1+0
bracketleftBigg
(x?1)q 1xplnqx
bracketrightBigg
= lim
x→1+0
1
xp ·
(x?1)q
lnqx = limx→1+0
(x?1)q
lnqx =
parenleftBigg
lim
x→1+0
x?1
lnx
parenrightBiggq
=
parenleftBigg
lim
x→1+0
1
1
x
parenrightBiggq
= 1
a75a200a169
integraldisplay 2
1
dx
xplnqxa61a8q< 1a133pa143a63a191a138a158a194a241
a127a196
integraldisplay +∞
2
dx
xplnqxa167a101p> 1a167a18α> 0a191a169a2a167a166p?α> 1a167a75a233a63a191qa167
a100a117 lim
x→+∞
parenleftBigg
xp?α 1xplnqx
parenrightBigg
= lim
x→+∞
1
xαlnqx = 0
a117a180a200a169
integraldisplay +∞
2
dx
xplnqxa8p> 1a133qa143a63a191a138a158a194a241a182
172
a101plessorequalslant 1,q< 1a167a100a117
integraldisplay +∞
2
dx
xplnqxgreaterorequalslant
integraldisplay +∞
2
dx
xlnqx =
1
1?q(lnx)
1?q
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
2
= +∞
a75a100a158a200a169
integraldisplay +∞
2
dx
xplnqxa117a209
a108a13a200a169
integraldisplay +∞
1
dx
xplnqxa8p> 1a133q< 1a158a194a241.
(6) a196a107a167a26a200a188a234a39a1171xa180
nsummationdisplay
i=1
pia63a195a161a2(a8x→±∞a158)
a217a103(a216a148a23a143inegationslash= ja158a167ai negationslash= aj)
a207 lim
x→ai
bracketleftBigg
|x?ai|pi 1|x?a
1|p1|x?a2|p2 ···|x?an|pn
bracketrightBigg
= ci,0 <ci < +∞(i = 1,2,···,n)
a25a200a169
integraldisplay +∞
∞
dx
|x?a1|p1|x?a2|p2 ···|x?an|pna61a8
nsummationdisplay
i=1
pi > 1a133pi < 1(i = 1,2,···,n)a158a194a241.
5,a23f(x)a8x→ +0a158a252a78a170a149a117+∞a167a193a121a178a181a101
integraldisplay 1
0
f(x)dxa194a241a167a55a76lim
x→0
xf(x)dx = 0.
a121a178a181 a100a75a23a1270a180f(x)a27a219a58a167a61
integraldisplay 1
0
f(x)dxa180a195a46a188a234a27a50a194a200a169a167a133a8xa191a169a130a670a158a167f(x) greaterorequalslant 0a167
a51[0,x]a254a252a78a126
a113
integraldisplay 1
0
f(x)dxa194a241a167a75a100a133a220a194a241a6a110a167a233?ε> 0,?δ> 0a167a80 < x2 <x<δa158a167
a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay x
x
2
f(x)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
integraldisplay x
x
2
f(x)dx< ε2
a100a49a152a200a169a165a138a189a110a167a26
integraldisplay x
x
2
f(x)dx = f(ξ)
parenleftBigg
x? x2
parenrightBigg
= x2f(ξ) > x2f(x)
parenleftBigg
x
2 <ξ<x
parenrightBigg
a117a180 x2f(x) <
integraldisplay x
x
2
f(x)dx< ε2 a610 lessorequalslantxf(x) <εa167a108a13lim
x→0
xf(x)dx = 0.
6,a63a216a101a15a200a169a27a253a233a194a241a218a94a135a194a241a53a181
(1)
integraldisplay +∞
0
xpsinx
1 +xq dx(qgreaterorequalslant 0)
(2)
integraldisplay +∞
0
esinxsin2x
xλ dx(λ> 0)
(3)
integraldisplay +∞
0
sinparenleftbigx+ 1xparenrightbig
xn dx
a41a181
(1)
integraldisplay +∞
0
xpsinx
1 +xq dx =
integraldisplay 1
0
xpsinx
1 +xq dx+
integraldisplay +∞
1
xpsinx
1 +xq dx
a233
integraldisplay 1
0
xpsinx
1 +xq dx
a207 lim
x→+0
x?(p+1)x
psinx
1 +xq = limx→+0
sinx
x
1 +xq = 1
a75a8?(p+ 1) < 1a61p>?2a158a167
integraldisplay 1
0
xpsinx
1 +xq dxa253a233a194a241a182a8plessorequalslant?2a158a167a200a169
integraldisplay 1
0
xpsinx
1 +xq dxa117a209
a233
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
xpsinx
1 +xq
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
xq sinx
1 +xq
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
xp
1 +xqa13 limx→+∞x
q?p x
p
1 +xq = limx→+∞
xq
1 +xq = 1
a75a8q?p> 1a61q>p+ 1a158
integraldisplay +∞
1
xp
1 +xq dxa194a241a167a117a180a100a39a22a7a79a123a167a26
integraldisplay +∞
1
xpsinx
1 +xq dxa253a233a194a241
a113 2x
q
1 +xq greaterorequalslant 1,
2xp
1 +xq greaterorequalslant
1
xq?p,
xp|sinx|
1 +xq greaterorequalslant
|sinx|
2xq?pa133a10056a1448.(2)a127a167
integraldisplay +∞
1
sinx
xq?p dxa8qlessorequalslantp+1a158a154a253a233
a194a241
173
a111a131a167a8p>?2,q>p+ 1a158a167
integraldisplay +∞
0
xpsinx
1 +xq dxa253a233a194a241
a127a196
integraldisplay +∞
1
xpsinx
1 +xq dx
a8q>pa158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 2a167 x
p
1 +xqa252a78a126a170a1170(x→ +∞)
a75a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
1
xpsinx
1 +xq dxa194a241
a8qlessorequalslantpa158a167a8q = pa158a167 x
p
1 +xq → 1(x→ +∞)a182a8q<pa158a167
xp
1 +xq → +∞(x→ +∞)
a75a233a191a169a140a27xa167a240a107 x
p
1 +xq greaterorequalslant
1
3
a117a180a233?A> 1a167a55?N ∈Z+a167a166a262Npi+ pi4 >Aa133a8xgreaterorequalslant 2Npi+ pi4a158a167a240a107 x
p
1 +xq greaterorequalslant
1
3
a108a13a233Aprime = 2Npi+ pi4,Aprimeprime = 2Npi+ pi2a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
xp
1 +xq sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
1
3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
√2
6
a75a100a133a220a6a110a167a26
integraldisplay +∞
1
xpsinx
1 +xq dxa117a209
a110a254a164a227a167a8q>p+1 >?1a158a167
integraldisplay +∞
0
xpsinx
1 +xq dxa253a233a194a241a182a8p+1 greaterorequalslantq>p>?2a158a167
integraldisplay +∞
0
xpsinx
1 +xq dxa94
a135a194a241.
(2) a207 lim
x→+0
esinxsin2x
xλ = limx→+0
sin2x
xλ =
0,λlessorequalslant 0
lim
x→+0
2cos2x
λxλ?1 =
0,0 <λ< 1
2,λ = 1
∞,λ> 1
=
0,λ< 1
2,λ = 1
∞,λ> 1
a75a8λ> 1a158a1670a143a219a58integraldisplay
+∞
0
esinxsin2x
xλ dx =
integraldisplay 1
0
esinxsin2x
xλ dx+
integraldisplay +∞
1
esinxsin2x
xλ dx
a233
integraldisplay 1
0
esinxsin2x
xλ dxa167a26a200a188a234a143a20a1670a143a219a58
a207 lim
x→+0
xλ?1e
sinxsin2x
xλ = limx→+0
esinxsin2x
x = 2
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a8λ?1 < 1a61λ< 2a158a167
integraldisplay 1
0
esinxsin2x
xλ dxa253a233a194a241a182
a8λgreaterorequalslant 2a158a167
integraldisplay 1
0
esinxsin2x
xλ dxa117a209
a233
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
esinxsin2x
xλ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx
a8λ> 1a158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
esinxsin2x
xλ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
e
xλa167a75
integraldisplay +∞
1
esinxsin2x
xλ dxa253a233a194a241
a117a180
integraldisplay +∞
0
esinxsin2x
xλ dxa81 <λ< 2a158a253a233a194a241a182
a8λlessorequalslant 1a158a167e
sinx|sin2x|
xλ >
e?1 sin2 2x
xλ = e
1
parenleftBigg
1?cos4x
2xλ
parenrightBigg
= 12e
parenleftBigg
1
xλ?
cos4x
xλ
parenrightBigg
a207
integraldisplay +∞
1
dx
xλa117a209,
integraldisplay +∞
1
cos4x
xλ dxa194a241a167a75
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
esinxsin2x
xλ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dxa117a209
a233
integraldisplay +∞
1
esinxsin2x
xλ dx
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
esinxsin2xdx
vextendsinglevextendsingle
vextendsinglevextendsingle= 2
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
esinxsinxdsinx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 4e
a113 1xλa252a78a126a170a1170a167a75a226a41a225a142a52a7a79a123a167a26a233λ> 0a107
integraldisplay +∞
1
esinxsin2x
xλ dxa194a241
a117a180
integraldisplay +∞
0
esinxsin2x
xλ dxa80 <λlessorequalslant 1a158a94a135a194a241
174
a108a13a81 <λ< 2a158
integraldisplay +∞
0
esinxsin2x
xλ dxa253a233a194a241a182a80 <λlessorequalslant 1a158
integraldisplay +∞
0
esinxsin2x
xλ dxa94a135a194a241.
(3)
integraldisplay +∞
0
sinparenleftbigx+ 1xparenrightbig
xn dx =
integraldisplay 1
0
sinparenleftbigx+ 1xparenrightbig
xn dx+
integraldisplay +∞
0
sinparenleftbigx+ 1xparenrightbig
xn dx = I1 +I2
a233I1a167a45x = 1t,dx =? dtt2 a167a75I1 =
integraldisplay 1
0
sinparenleftbigx+ 1xparenrightbig
xn dx =
integraldisplay +∞
1
sinparenleftbigx+ 1xparenrightbig
x2?n dx
a239a196I2
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinparenleftbigx+ 1xparenrightbig
xn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
xna167a75a8n> 1a158a167I2a253a233a194a241
a207
integraldisplay +∞
1
sinparenleftbigx+ 1xparenrightbig
xn dx =
integraldisplay +∞
1
sinparenleftbigx+ 1xparenrightbigparenleftbig1? 1x2parenrightbig
xnparenleftbig1? 1x2parenrightbig dx
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
sin
parenleftbigg
x+ 1x
parenrightbiggparenleftbigg
1? 1x2
parenrightbigg
dx
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglecos
parenleftBigg
x+ 1x
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
A
1
lessorequalslant 2
a133
bracketleftBigg
xn
parenleftBigg
1? 1x2
parenrightBiggbracketrightBiggprime
= nxn?1?(n?2)xn?3 = xn?3[nx2?(n?2)]
a75a8n∈ (0,1]a158a167xn
parenleftBigg
1? 1x2
parenrightBigg
a252a78a79a61 1xnparenleftbig1? 1
x2
parenrightbiga252a78a126a170a1170(x→ +∞)
a117a180a100a41a225a142a52a7a79a123a167a26a80 <nlessorequalslant 1a158a167I2a194a241
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinparenleftbigx+ 1xparenrightbig
xn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
sin2parenleftbigx+ 1xparenrightbig
xn =
1
2xn?
cos2parenleftbigx+ 1xparenrightbig
2xn
a80 <nlessorequalslant 1a158a167
integraldisplay +∞
1
dx
2xna117a209a167
integraldisplay +∞
1
cos2parenleftbigx+ 1xparenrightbig
2xn dxa194a241
a75a80 <nlessorequalslant 1a158a167
integraldisplay +∞
1
sin2parenleftbigx+ 1xparenrightbig
xn dxa117a209
a117a180a100a39a22a7a79a123a167a127
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinparenleftbigx+ 1xparenrightbig
xn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dxa80 <nlessorequalslant 1a158a117a209
a108a13a80 <nlessorequalslant 1a158a167
integraldisplay +∞
1
sinparenleftbigx+ 1xparenrightbig
xn dxa94a135a194a241
a8nlessorequalslant 0a158a167a233?A> 1a167a55?k ∈Z+a167a166a262kpi+ pi6 >Aa133a8xgreaterorequalslant 2kpi+ pi6a158a167
a240a107x?nsin
parenleftBigg
x+ 1x
parenrightBigg
greaterorequalslant sin
parenleftBigg
2kpi+ pi6
parenrightBigg
= 12
a117a180a167a233Aprime = 2kpi+ pi6,Aprimeprime = 2kpi+ pi3a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
sinparenleftbigx+ 1xparenrightbig
xn dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
integraldisplay Aprimeprime
Aprime
sin
parenleftBigg
2kpi+ pi6
parenrightBigg
dx = pi12
a75a100a133a220a194a241a6a110a167a26a8nlessorequalslant 0a158a167I2a117a209
a233I1a167a100I2a27a40a216a167a26a82?n > 1a61n < 1a158a253a233a194a241a182a81 greaterorequalslant 2?n > 0a611 lessorequalslant n < 2a158a94a135a194a241a182
a82?nlessorequalslant 0a61nlessorequalslant 2a158a117a209
a111a131a167
integraldisplay +∞
0
sinparenleftbigx+ 1xparenrightbig
xn dxa80 <n< 2a158a94a135a194a241.
7,a23f(x)a252a78a101a252a167 lim
x→+∞
f(x) = 0a167a88a74a19a234f prime(x)a51[0,+∞)a254a235a89a167a64a34a200a169
integraldisplay +∞
0
f prime(x)sin2xdxa194a241.
a121a178a181a207(sin2x)prime = sin2xa167a19a234f prime(x)a51[0,+∞)a254a235a89a167 lim
x→+∞
f(x)sin2x = 0
a75a100a169a220a200a169a250a170a167a26
integraldisplay +∞
0
f prime(x)sin2xdx = f(x)sin2x
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
0
integraldisplay +∞
0
f(x)sin2xdx =?
integraldisplay +∞
0
f(x)sin2xdx
a233a117
integraldisplay +∞
0
f(x)sin2xdxa167a100a174a127f(x)a252a78a101a252a167 lim
x→+∞
f(x) = 0a57
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
0
sin2xdx
vextendsinglevextendsingle
vextendsinglevextendsingle= 1
2|cos2A?1|lessorequalslant 1
a75a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
0
f(x)sin2xdxa194a241a167a108a13a200a169
integraldisplay +∞
0
f prime(x)sin2xdxa194a241.
8,a51a195a46a188a234a27a50a194a200a169(a200a169a129a143a107a129)a165a167a121a178a178a144a140a200a152a189a253a233a140a200a167a2a135a131a216a44.
a121a178a181a100a174a127f2(x)a140a200a167a75f
2(x)
2 a143a140a200
a207(|f(x)|?1)2 = f2(x)?2|f(x)|+ 1 greaterorequalslant 0a167a75|f(x)|lessorequalslant f
2(x) + 1
2
175
a117a180a100a39a22a7a79a123a167a26|f(x)|a140a200a61a178a144a140a200a189a253a233a140a200.
a135a131a216a44.
a126a181a10057a144a1261a167a26
integraldisplay 2
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
(x?1)12
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dxa194a241a61
integraldisplay 2
1
dx
(x?1)12
a253a233a194a241
a2
integraldisplay 2
1
dx
x?1a117a209a167a61
1
x?1 a51[1,2]a254a216a140a200.
9,a79a142a101a15a200a169a27a133a220a204a138a181
(1)
integraldisplay 3
0
dx
1?x
(2)
integraldisplay +∞
∞
sinxdx
a41a181
(1) P.V.
integraldisplay 3
0
dx
1?x = limη→0
bracketleftBiggintegraldisplay
1?η
0
dx
1?x +
integraldisplay 3
1+η
dx
1?x
bracketrightBigg
= lim
η→0
bracketleftBigg
ln(1?x)
vextendsinglevextendsingle
vextendsinglevextendsingle
1?η
0
ln(x?1)
vextendsinglevextendsingle
vextendsinglevextendsingle
3
1+η
bracketrightBigg
=?ln2
(2) P.V.
integraldisplay +∞
∞
sinxdx = lim
A→+∞
parenleftbiggintegraldisplay A
A
sinxdx
parenrightbigg
= lim
A→+∞
parenleftBigg
cosx
vextendsinglevextendsingle
vextendsinglevextendsingle
A
A
parenrightBigg
= lim
A→+∞
(cos(?A)?cosA) = 0
10,a121a178a50a194a200a169a57a133a220a204a138a131a109a27a39a88a181
(1) a101
integraldisplay +∞
∞
f(x)dxa194a241a167a217a138a143Aa167a75a133a220a204a138P.V.
integraldisplay +∞
∞
f(x)dxa127a51a167a133a31a117Aa167a2a135a131a216a44a182
(2) a101f(x) greaterorequalslant 0a167P.V.
integraldisplay +∞
∞
f(x)dxa127a51a167a217a138a143Aa167a75
integraldisplay +∞
∞
f(x)dxa194a241a167a133a194a241a117A.
a121a178a181
(1) a100
integraldisplay +∞
∞
f(x)dxa194a241a167a127
integraldisplay +∞
∞
f(x)dx =
integraldisplay 0
∞
f(x)dx+
integraldisplay +∞
0
f(x)dxa194a241a167
a75a107 lim
B→?∞
integraldisplay 0
B
f(x)dx+ lim
A→+∞
integraldisplay A
0
f(x)dxa127a51a167a65a79a18B =?Aa167a107 lim
A→+∞
integraldisplay A
A
f(x)dxa127a51a167a133a31
a117A
a249a76a178P.V.
integraldisplay +∞
∞
f(x)dxa127a51a167a133a31a117A
a2a135a131a216a44.
a126a88a181P.V.
integraldisplay +∞
∞
sinxdx = lim
A→+∞
integraldisplay A
A
sinxdx = 0a167a2
integraldisplay +∞
∞
f(x)dx =
integraldisplay +∞
∞
sinxdxa216a194a241.
(2) a94a135a121a123.
a101a216a44a167a75a100a117f(x) greaterorequalslant 0a167a26
integraldisplay a
∞
f(x)dxa218
integraldisplay +∞
a
f(x)dxa165a150a8a107a152a143+∞
a117a180
integraldisplay a
A
f(x)dxa218
integraldisplay A
a
f(x)dxa165a8A → +∞a158a150a8a107a152a170a117+∞a167a13a44a152a135a140a117a31a1170a167a108a13a167a130a27a218
a170a117+∞a167a249a134a174a127P.V.
integraldisplay +∞
∞
f(x)dxa127a51a103a241a167a75
integraldisplay +∞
∞
f(x)dxa194a241.
a113a100P.V.
integraldisplay +∞
∞
f(x)dx = Aa167a75a226a52a129a141a152a53a167a26
integraldisplay +∞
∞
f(x)dx = A.
176
a49a19a220a169 a188a234a145a63a234
a49a155a152a217 a188a234a145a63a234a33a152a63a234
§1,a188a234a145a63a234a27a152a151a194a241
1,a63a216a101a15a188a234a83a15a51a164a171a171a141a83a27a152a151a194a241a53a181
(1) fn(x) =
radicalbigg
x2 + 1n2,?∞<x<∞
(2) fn(x) = x2?x2n,0 lessorequalslantxlessorequalslant 1
(3) fn(x) = sin xn
(i)?l<x<l
(ii)?∞<x<∞
(4) fn(x) = xn(1?x)a167 0 lessorequalslantxlessorequalslant 1
(5) fn(x) = nx1 +nxa167 0 lessorequalslantxlessorequalslant 1
(6) fn(x) = xn ln xna167 0 <x< 1
a41a181
(1) a8?∞<x< +∞a158a167f(x) = lim
n→∞
fn(x) = lim
n→∞
radicalbigg
x2 + 1n2 = |x|
a75||fn?f|| = sup
x∈(?∞,+∞)
|fn(x)?f(x)| = sup
x∈(?∞,+∞)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
radicalbigg
x2 + 1n2?|x|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
n→ 0(n→∞)
a117a180a100a189a1942a167a26fn(x)a51(?∞,+∞)a83a152a151a194a241a117|x|.
(2) a8x = 1a158a167fn(1) = 0,f(x) = 0a182a80 lessorequalslantx< 1a158a167f(x) = lim
n→∞
fn(x) = 0a167a75f(x) = 0(0 lessorequalslantxlessorequalslant 1)
||fn?f|| = sup
x∈[0,1]
|fn(x)?f(x)| = sup
x∈[0,1]
|xn?x2n| = max
x∈[0,1]
|xn?x2n|
a45(xn?x2n)prime = nxn?1(1?2xn) = 0a167a75a26x = 0,x = n
radicalbigg
1
2
a113fn(0) = 0,fn
parenleftBigg
n
radicalbigg
1
2
parenrightBigg
= 14,fn(1) = 0a167a75||fn?f|| = 14 negationslash= 0a167a117a180a100a189a1942a167a26a100a188a234a83a15a51a164a171a171
a141a83a216a152a151a194a241.
(3) (i) a8?l<x<la158a167f(x) = lim
n→∞
fn(x) = 0
||fn?f|| = sup
x∈(?l,l)
|fn(x)?f(x)| = sup
x∈(?l,l)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
l
n→ 0(n→∞)
a117a180a226a189a1942a167a26fn(x)a51(?l,l)a254a152a151a194a241a1170.
(ii) a8?∞<x< +∞a158a167f(x) = lim
n→∞
fn(x) = 0
a18ε0a1660 <ε0 < 1a167a216a216na245a140a167a144a135a18x = n2 pia167a210a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglef
parenleftBigg
n
2 pi
parenrightBigg
f
parenleftBigg
n
2 pi
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 1 >ε0
a75fn(x)a51(?∞,+∞)a254a216a152a151a194a241.
(4) a80 lessorequalslantx< 1a158a167f(x) = lim
n→∞
fn(x) = 0a182a8x = 1a158a167fn(1) = 0,f(1) = 0a167a75f(x) = lim
n→∞
fn(x) = 0
||fn?f|| = sup
x∈[0,1]
|fn(x)?f(x)| = sup
x∈[0,1]
|xn(1?x)| = max
x∈[0,1]
(xn?xn+1)
a45(xn?xn+1)prime = xn?1[n?(n+ 1)x] = 0a34a75a26x = 0,x = nn+ 1
a113fn(0) = fn(1) = 0,fn
parenleftBigg
n
n+ 1
parenrightBigg
=
parenleftBigg
n
n+ 1
parenrightBiggnparenleftBigg
1? nn+ 1
parenrightBigg
> 0
a75 max
x∈[0,1]
(xn?xn+1) =
parenleftBigg
n
n+ 1
parenrightBiggnparenleftBigg
1? nn+ 1
parenrightBigg
→ 0(n→∞)a61||fn?f||→ 0(n→∞)
a117a180a100a189a1942a167a26a100a188a234a83a15a51a164a171a171a141a83a152a151a194a241a1170.
177
(5) f(x) = lim
n→∞
fn(x) =
braceleftbigg 1,0 <xlessorequalslant 1
0,x = 0
a117a180f(x)a51[0,1]a254a216a235a89a167a13fn(x)a51[0,1]a254a235a89a167a75fn(x) = nx1 +nxa51[0,1]a254a216a152a151a194a241.
(6) a207 lim
t→+0
tlnt = 0a167a75f(x) = lim
n→∞
fn(x) = 0
|fn(x)?f(x)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
nln
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a233?ε> 0a167a207 lim
t→+0
tlnt = 0a167a75a127a51δ(ε) > 0a167a80 <t<δa158a167a107|tlnt?0|<ε
a18N =
bracketleftBigg
1
δ
bracketrightBigg
a167a8n>Na158a1671n<δ
a108a13a233a152a1310 <x< 1a167a1070 < xn<δa167a25|fn(x)?f(x)|lessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
n ln
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε
a108a13a100a189a1941a167a26a100a188a234a51(0,1)a83a152a151a194a241a1170.
2,a63a216a101a15a63a234a27a152a151a194a241a53a181
(1)
∞summationdisplay
n=0
(1?x)xna167 0 lessorequalslantxlessorequalslant 1
(2)
∞summationdisplay
n=1
(?1)n?1x2
(1 +x2)n a167?∞<x< +∞
(3)
∞summationdisplay
n=1
sinnx
3√n4 +x4 a167?∞<x< +∞
(4)
∞summationdisplay
n=1
x
1 +n4x2 a167?∞<x< +∞
(5)
∞summationdisplay
n=1
sinnxsinx√
n+x a167 0 lessorequalslantxlessorequalslant 2pi
(6)
∞summationdisplay
n=1
(?1)n(1?e?nx)
n2 +x2 a167 0 lessorequalslantx< +∞
(7)
∞summationdisplay
n=1
2nsin 13nxa167 0 <x< +∞
a41a181
(1) a207a220a169a218Sn(x) =
nsummationdisplay
k=0
(1?x)xk = 1?xn+1a167a75S(x) = lim
n→∞
Sn(x) =
braceleftbigg 1,0 lessorequalslantx< 1
0,x = 1
a117a180S(x)a51[0,1]a254a216a235a89a167a13Sn(x)a51[0,1]a254a235a89a167a75
∞summationdisplay
n=0
(1?x)xna51[0,1]a254a216a152a151a194a241.
(2) a207a100a63a234a143a2a134a63a234a167a133 x
2
(1 +x2)n+1 lessorequalslant
x2
(1 +x2)n a167a75a123a170a27a253a233a138a216a172a135a76a167a27a196a145a27a253a233a138a167
a61|rn(x)|lessorequalslant x
2
(1 +x2)n =
x2
1 +nx2 +···+x2n <
1
n (?x∈ (?∞,+∞))
a108a13a233?ε> 0,?N =
bracketleftBigg
1
ε
bracketrightBigg
a167a8n>Na158a167a107|rn(x)|<εa167a75a100a63a234a51(?∞,+∞)a254a152a151a194a241.
(3) a8?∞<x< +∞a158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinnx
3√n4 +x4
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n43
a240a164a225a167a133a63a234
∞summationdisplay
n=1
1
n43
a194a241
a75a100a159a188a7a79a123a167a26a63a234
∞summationdisplay
n=1
sinnx
3√n4 +x4 a51(?∞,+∞)a254a152a151a194a241.
(4) a2070 lessorequalslant (1?n2|x|)2 = 1?2n2|x|+n4x2a167a752n2|x|lessorequalslant 1 +n4x2a61 2n
2|x|
1 +n4x2 lessorequalslant 1 (x∈ (?∞,+∞))
a108a13
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
1 +n4x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
2n2|x|
2n2(1 +n4x2) lessorequalslant
1
2n2
178
a113a63a234
∞summationdisplay
n=1
1
2n2 a194a241a167a75a226a159a188a7a79a123a167a26a63a234
∞summationdisplay
n=1
x
1 +n4x2 a51(?∞,+∞)a254a152a151a194a241.
(5) a8x = 0,2pia158a167
nsummationdisplay
k=1
sinkxsinx = 0
a8xnegationslash= 0,2pia158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
sinkxsinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= |sinx|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
sinkx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant|sinx|
1vextendsingle
vextendsinglesin x2vextendsinglevextendsingle= 2
vextendsinglevextendsinglecos x
2
vextendsinglevextendsinglelessorequalslant 2
a75a80 lessorequalslantxlessorequalslant 2pia158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
sinkxsinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant 2
a113 1√n+x a233x ∈ [0,2pi]a39a117na252a78a52a126a133a100 1√n+x lessorequalslant 1√n a26a8n → ∞a158a167 1√n+x a39a117xa51[0,2pi]a254
a152a151a47a170a1170(a100a189a1942)
a75a226a41a225a142a52a7a79a123a167a26a63a234
∞summationdisplay
n=1
sinnxsinx√
n+x a51[0,2pi]a254a152a151a194a241.
(6) a100a117a233?x∈ [0,+∞)a167a1070 lessorequalslant 1?e?nx < 1a167a75
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(?1)n(1?e?nx)
n2 +x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n2
a113a63a234
∞summationdisplay
n=1
1
n1 a194a241a167a75a226a159a188a7a79a123a167a26a63a234
∞summationdisplay
n=1
(?1)n(1?e?nx)
n2 +x2 a51[0,+∞)a254a152a151a194a241.
(7) a80un(x) = 2nsin 13nx
a80 <x< +∞a158a167a100a117|un(x)|lessorequalslant 1x
parenleftBigg
2
3
parenrightBiggn
a133
∞summationdisplay
n=1
1
x
parenleftBigg
2
3
parenrightBiggn
a194a241a167a25a6a63a234a253a233a194a241a167a108a13a194a241a167a2a167
a51(0,+∞)a83a191a216a152a151a194a241.
a88a101a216a44a167a23a167a152a151a194a241a167a75a233a63a137ε > 0a167a18ε = 1a167a55a127a51N = N(ε) ∈ Z+(a167a134xa195a39)a167a166
a8n>Na158a167a233a117(0,+∞)a83a27a152a131xa167a254a107|un+1(x) +un+2(x) +···+un+p(x)| <εa167a217a165pa143a63a191a20
a18a234
a56a18p = 1,n = Na167a75a233a152a131x∈ (0,+∞)a167a65a107|uN+1(x)|<ε = 1
a113a18x0 = 23N+1pi ∈ (0,+∞)a167a143a65a107|uN+1(x0)|< 1
a2a175a162a254a37a107uN+1(x0) = 2N+1 sin 13N+1x
0
= 2N+1 > 1a249a134|uN+1(x0)|< 1a103a241
a75a98a23a216a164a225a167a61a63a234
∞summationdisplay
n=1
2nsin 13nxa51(0,+∞)a254a194a241a2a154a152a151a194a241.
3,a121a178a152a151a194a241a189a1941a218a189a1942a27a31a100a53.
a121a178a181a189a1941=?a189a1942
a174a127a233a63a137a27ε> 0a167a127a51a144a157a54a117εa27a20a18a234N(ε)a167a166n>N(ε)a158a167a107|Sn(x)?S(x)|< ε2 a233a152a131x∈Xa209a164
a225
a117a180||Sn?S|| = sup
x∈X
|Sn(x)?S(x)|lessorequalslant ε2 <εa167a108a13 lim
n→∞
||Sn?S|| = 0.
a189a1942=?a189a1941
a174a127 lim
n→∞
||Sn?S|| = 0a167a61a233?ε> 0,?N(ε) ∈Z+a167a166a8n>Na158a167a233a152a131x∈Xa167a209a107vextendsingle
vextendsingle||Sn?S||?0vextendsinglevextendsingle= sup
x∈X
|Sn(x)?S(x)|<ε
a13|Sn(x)?S(x)|lessorequalslantvextendsinglevextendsingle||Sn?S||?0vextendsinglevextendsingle<εa233a152a131x∈Xa209a164a225.
(a17a28a97a113a47a140a121a178a188a234a145a63a234
∞summationdisplay
n=1
una189a1941a189a1942).
4,a193a121a63a234
∞summationdisplay
n=1
ln(1 +nx)
nxn a51a63a219a171a109[1 +α,∞),α> 0a143a152a151a194a241.
a121a178a181a207a8h> 0a158a167ln(1 +h) <ha167a75
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ln(1 +nx)
nxn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
ln(1 +nx)
nxn <
nx
nxn =
1
xn?1 lessorequalslant
1
(1 +α)n?1 (1 +α lessorequalslant
x< +∞)
a113
∞summationdisplay
n=1
1
(1 +α)n?1 a194a241a167a75a226Ma7a79a123a167a26a6a63a234a51[1 +α,+∞)(α> 0)a254a152a151a194a241.
179
5,a101
∞summationdisplay
n=1
un(x)a27a152a132a145|un(x)| lessorequalslant cn(x)a167a191a133
∞summationdisplay
n=1
cn(x)a51Xa254a152a151a194a241a167a75
∞summationdisplay
n=1
un(x)a51Xa254a189a152a151a194a241a133a253a233
a194a241.
a121a178a181a207
∞summationdisplay
n=1
cn(x)a51Xa254a152a151a194a241
a75a100a152a151a194a241a27a133a220a191a135a94a135a167a26a233?ε > 0,?N = N(ε) ∈ Z+a167a166a8n > Na158a167a233a152a131x ∈ Xa218a63a191a27a20a18
a234pa167a107|cn+1(x) +cn+2(x) +···+cn+p(x)|<ε
a113
∞summationdisplay
n=1
un(x)a27a152a132a145|un(x)|lessorequalslantcn(x)
a75a233a254a227ε> 0a167a20a18a234N = N(ε)a167a166a8n>Na158a167a233a152a131x∈Xa218a254a227a20a18a234pa167a107
|un+1(x) +un+2(x) + ··· +un+p(x)| lessorequalslant vextendsinglevextendsingle|un+1(x)| + |un+2(x)| + ··· + |un+p(x)|vextendsinglevextendsingle lessorequalslant |cn+1(x) +cn+2(x) +
···+cn+p(x)|<ε
a100a152a151a194a241a27a133a220a191a135a94a135a167a26
∞summationdisplay
n=1
un(x)a51Xa254a152a151a194a241a133a253a233a194a241.
6,a23f0(x)a51[0,a]a254a235a89a167a113fn(x) =
integraldisplay x
0
fn?1(t)dta167a121a178{fn(x)}a51[0,a]a254a152a151a194a241a117a34.
a121a178a181a207f0(x)a51[0,a]a254a235a89a167a75a217a107a46a167a61a127a51M > 0a167a107|f0(x)|lessorequalslantM
a113fn(x) =
integraldisplay x
0
fn?1(t)dta167a75
|f1(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay x
0
f0(t)dt
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay x
0
|f0(t)|dtlessorequalslant
integraldisplay x
0
M dt = MxlessorequalslantMa
|f2(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay x
0
f1(t)dt
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay x
0
|f1(t)|dtlessorequalslant
integraldisplay x
0
Mtdt = Mx
2
2 lessorequalslant
Ma2
2.......................................................
|fn(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay x
0
fn?1(t)dt
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay x
0
|fn?1(t)|dtlessorequalslant
integraldisplay x
0
M t
n?1
(n?1)! dt = M
xn
n! lessorequalslantM
an
n!
a207 lim
n→∞
an
n! = 0a61a233?ε> 0,?N ∈Z
+a167a8n>Na158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
n!?0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
an
n! <ε
a117a180|fn(x)?0|<Mεa233a152a131x∈ [0,a]a254a164a225
a108a13a100a189a1941a167a26{fn(x)}a51[0,a]a254a152a151a194a241a117a34.
7,a121a178a63a234
∞summationdisplay
n=1
(?1)n?1 1n+x2 a39a117xa51(?∞,+∞)a254a143a152a151a194a241a167a2a233a63a219xa191a154a253a233a194a241a167a13a63a234
∞summationdisplay
n=1
x2
(1 +x2)n
a143a51x∈ (?∞,+∞)a254a253a233a194a241a167a2a191a216a152a151a194a241.
a121a178a181a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
(?1)k?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant 1a61
nsummationdisplay
k=1
(?1)k?1a51(?∞,+∞)a254a152a151a107a46
a113 1n+ 1 +x2 < 1n+x2 a167a75a188a234a15
braceleftBigg
1
n+x2
bracerightBigg
a233a117x∈ (?∞,+∞)a252a78a126
a113a233?ε> 0a167a18N =
bracketleftBigg
1
ε
bracketrightBigg
a167a75a8n>Na158a167a233a152a131x∈ (?∞,+∞)a167a209a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
n+x2?0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
n+x2 lessorequalslant
1
n<ε
a75
braceleftBigg
1
n+x2
bracerightBigg
a39a117x∈ (?∞,+∞)a152a151a194a241a1170a167a117a180a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
n=1
(?1)n?1 1n+x2 a51(?∞,+∞)a83
a152a151a194a241.
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle(?1)
n?1 1
n+x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
1
n+x2
a207 lim
n→∞
1
n+x2
1
n
= 1a133
∞summationdisplay
n=1
1
n a117a209a167a75a100a39a22a7a79a123a27a52a129a47a170a167a26
∞summationdisplay
n=1
1
n+x2 a117a209a167a117a180a233a63a219xa63a234a154a253a233
a194a241.
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x2
(1 +x2)n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
x2
(1 +x2)n
a233a27a189a27x∈ (?∞,+∞)a167a207 lim
n→∞
n
radicalBigg
x2
(1 +x2)n =
1
1 +x2,xnegationslash= 0
0,x = 0
a100a133a220a7a79a123a167a26
∞summationdisplay
n=1
x2
(1 +x2)n a51(?∞,+∞)a194a241a167a117a180a253a233a194a241.
180
a8xnegationslash= 0a158a167Sn(x) =
nsummationdisplay
k=1
x2
(1 +x2)k = 1?
1
(1 +x2)n a167S(x) = limn→∞Sn(x) = 1
a8x = 0a158a167Sn(0) = 0,S(0) = 0a167a75S(x) =
braceleftbigg 1,xnegationslash= 0
0,x = 0
a207Sn(x)a51(?∞,+∞)a254a235a89a167a13S(x)a51(?∞,+∞)a254a216a235a89a167a75
∞summationdisplay
n=1
x2
(1 +x2)n a51(?∞,+∞)a83a216a152a151a194a241.
8,a121a178a181
(1) a88a74
summationdisplay
|fn(x)|a51[a,b]a254a152a151a194a241a167a64a34
∞summationdisplay
1
fn(x)a51[a,b]a254a143a152a151a194a241a182
(2) a88a74
summationdisplay
fn(x)a51[a,b]a254a152a151a194a241a167a2
summationdisplay
|fn(x)|a153a55a152a151a194a241a167a177
∞summationdisplay
1
(?1)n(xn?xn+1),0 lessorequalslantxlessorequalslant 1a143a126
a53a96a178.
a121a178a181
(1) a100a133a220a79a75a57a75a23a167a26
a233?ε> 0,?N = N(ε) ∈Z+a167a166a8n>Na158a167a233a152a131x∈ [a,b]a218a63a191p∈Z=a167a107
|fn+1(x)|+|fn+2(x)|+···+|fn+p(x)|<ε
a108a13|fn+1(x) +fn+2(x) +···+fn+p(x)|lessorequalslant|fn+1(x)|+|fn+2(x)|+···+|fn+p(x)|<ε
a75a226a152a151a194a241a27a133a220a79a75a167a26
∞summationdisplay
1
fn(x)a51[a,b]a254a152a151a194a241.
(2) a126a181
∞summationdisplay
1
(?1)n(xn?xn+1)a51[0,1]a254a152a151a194a241
a207xn?xn+1 = 0(a8x = 0,1a158)a182a80 < x < 1a158a167xn?xn+1 = xn(1?x)a167a75xn?xn+1a51[0,1]a254a39
a117na252a78a126a8
a1001.(4)a167a26xn?xn+1 = xn(1?x)a51[0,1]a254a152a151a194a241a1170a167a75a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
1
(?1)n(xn?xn+1) a51[0,1]a254a152a151a194a241
a2
∞summationdisplay
1
vextendsinglevextendsingle(?1)n(xn?xn+1)vextendsinglevextendsingle= ∞summationdisplay
1
(xn?xn+1)a51[0,1]a254a154a152a151a194a241(a1002.(1)a26).
9,a23a122a152a145?n(x)a209a180[a,b]a254a27a252a78a188a234a167a88a74
summationdisplay
n(x)a51[a,b]a27a224a58a143a253a233a194a241a167a64a34a249a63a234a51[a,b]a254a152
a151a194a241.
a121a178a181a207?n(x)a51[a,b]a254a252a78a167a25a107|?n(x)|lessorequalslant|?n(a)|+|?n(b)| (?x∈ [a,b])
a100a117
summationdisplay
|?n(a)|a218
summationdisplay
|?n(b)|a194a241a167a75
summationdisplay
(|?n(a)|+|?n(b)|)a194a241
a75a226Ma7a79a123a167a26a63a234
summationdisplay
n(x)a51[a,b]a254a152a151a194a241.
10,a101a15a188a234a15a180a196a152a151a194a241a186
(1) fn(x) = (sinx)n,0 lessorequalslantxlessorequalslantpi
(2) fn(x) = (sinx) 1n
(i) 0 lessorequalslantxlessorequalslantpi
(ii) δlessorequalslantxlessorequalslantpi?δ
(3) fn(x) = x
n
1 +xn
(i) 0 lessorequalslantxlessorequalslant 1?ε
(ii) 1?ε<x< 1 +ε (ε> 0)
(iii) 1 +εlessorequalslantx<∞
a41a181
(1) f(x) = lim
n→∞
fn(x) =
0,0 lessorequalslantxlessorequalslantpia133xnegationslash= pi2
1,x = pi2
a207f(x)a51[0,pi]a254a216a235a89a167a2fn(x)a51[0,pi]a254a235a89a167a75fn(x) = (sinx)na51[0,pi]a254a216a152a151a194a241.
181
(2) (i) f(x) = lim
n→∞
fn(x) =
braceleftbigg 0,x = 0,pi
1,0 <x<pi
a207f(x)a51[0,pi]a254a216a235a89a167a2fn(x)a51[0,pi]a254a235a89a167a75fn(x) = (sinx) 1na51[0,pi]a254a216a152a151a194a241.
(ii) a207f(x) = lim
n→∞
fn(x) = 1,|fn(x)?f(x)| = 1?(sinx) 1n lessorequalslant 1?(sinδ) 1n
a61||fn?f|| = sup
x∈[δ,pi?δ]
|fn(x)?f(x)| = 1?(sinδ) 1n → 0(n→∞)
a75a100a189a1942a167a26fn(x) = (sinx) 1na51[δ,pi?δ]a254a152a151a194a241a1171.
(3) (i) a80 lessorequalslantxlessorequalslant 1?εa158a167f(x) = lim
n→∞
fn(x) = 0a167a75|fn(x)?f(x)| = x
n
1 +xn lessorequalslantx
n lessorequalslant (1?ε)n
a117a180||fn?f|| = sup
x∈[0,1?ε]
|fn(x)?f(x)| = (1?ε)n → 0 (n→∞)
a75a100a189a1942a167a26fn(x) = x
n
1 +xn a51[0,1?ε]a254a152a151a194a241a1170.
(ii) f(x) = lim
n→∞
fn(x) =
0,1?ε<x< 1
1
2,x = 11,1 <x< 1 +ε
a207f(x)a51(1?ε,1+ε)a254a216a235a89a167a13fn(x)a51(1?ε,1+ε)a254a235a89a167a75fn(x) = x
n
1 +xna51(1?ε,1+ε)a254
a216a152a151a194a241.
(iii) a81 +εlessorequalslantx< +∞a158a167f(x) = lim
n→∞
fn(x) = 1a167a75|fn(x)?f(x)| = 11 +xn lessorequalslant 11 + (1 +ε)n
a117a180||fn?f|| = sup
x∈[1+ε,+∞)
|fn(x)?f(x)| = 11 + (1 +ε)n → 0 (n→∞)
a108a13a100a189a1942a167a26fn(x) = x
n
1 +xn a51[1 +ε,+∞)a254a152a151a194a241a1171.
11,a121a178
∞summationdisplay
1
ne?nxa51(0,+∞)a83a235a89.
a121a178a181a63a18x0 ∈ (0,+∞)a167a75a127a51α,β > 0a167a166α<x0 <βa167a51[α,β]a2540 <ne?nx lessorequalslantne?nα
a207α > 0a167a75eα > 1a167a117a180 lim
n→∞
(n+ 1)e?(n+1)α
ne?nα =
1
eα < 1a167a75a100a136a75a19a16a7a79a123a27a52a129a47a170a167a26a63
a234
∞summationdisplay
1
ne?nαa194a241a167a108a13a226Ma7a79a123a167a26
∞summationdisplay
1
ne?nxa51[α,β]a254a152a151a194a241.
a113ne?nxa51[α,β]a254a235a89a167a108a13
∞summationdisplay
1
ne?nxa51[α,β]a254a235a89
a207x0 ∈ [α,β]a167a75
∞summationdisplay
1
ne?nxa51x0a58a235a89
a100a117x0a180(0,+∞)a27a63a191a58a167a25
∞summationdisplay
1
ne?nxa51(0,+∞)a83a235a89.
12,a121a178a188a234f(x) =
∞summationdisplay
1
sinnx
n3 a51(?∞,+∞)a83a235a89a167a191a107a235a89a19a188a234.
a121a178a181a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinnx
n3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n3 a133
∞summationdisplay
1
1
n3 a194a241a167a75a226Ma7a79a123a167a26f(x) =
∞summationdisplay
1
sinnx
n3 a51(?∞,+∞)a152a151a194a241
a113 sinnxn3 a51(?∞,+∞)a83a235a89a167a75f(x) =
∞summationdisplay
1
sinnx
n3 a51(?∞,+∞)a83a235a89
d
dx
parenleftBigg
sinnx
n3
parenrightBigg
= cosnxn2
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosnx
n2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n2 a133
∞summationdisplay
1
1
n2 a194a241a167a75a226Ma7a79a123a167a26
∞summationdisplay
1
cosnx
n2 a51(?∞,+∞)a152a151a194a241
a117a180f prime(x) = ddx
parenleftBigg ∞summationdisplay
1
sinnx
n3
parenrightBigg
=
∞summationdisplay
1
cosnx
n2
a113 cosnxn2 a51(?∞,+∞)a83a235a89a167a75
∞summationdisplay
1
cosnx
n2 a51(?∞,+∞)a83a235a89
182
a61f prime(x)a51(?∞,+∞)a83a235a89a133f prime(x) =
∞summationdisplay
1
cosnx
n2,
13,a121a178a188a234ζ(x) =
∞summationdisplay
1
1
nx a51(1,+∞)a235a89a167a191a107a235a89a136a30a19a188a234.
a121a178a181a136a145a166a19a234a164a26a63a234a143?
∞summationdisplay
n=1
lnn
nx,a101a121a167a511 <alessorequalslantx< +∞a254a152a151a235a89(aa143a140a1171a27a63a219a234)
a8alessorequalslantx< +∞a158a167a1070 < lnnnx lessorequalslant lnnna
a100a117 lim
n→∞
lnn
na
1
n(a+1)/2
= 0a133
∞summationdisplay
n=1
1
n(a+1)/2 a194a241
a75a63a234
∞summationdisplay
n=1
lnn
na a194a241a167a117a180a100Ma7a79a123a167a26a63a234
∞summationdisplay
n=1
lnn
nx a51alessorequalslantx< +∞a254a152a151a194a241
a53a191a20a122a145 lnnnx a209a180xa27a235a89a188a234a167a75a63a234
∞summationdisplay
n=1
1
nx a51a lessorequalslant x < +∞a254a140a197a145a166a19a234a167a26ζ
prime(x) =?
∞summationdisplay
n=1
lnn
nx
a133ζprime(x)a51alessorequalslantx< +∞a254a235a89
a100a> 1a27a63a191a53a167a26ζprime(x) =?
∞summationdisplay
n=1
lnn
nx a233a152a1311 <x< +∞a164a225a133ζ
prime(x)a511 <x< +∞a254a235a89a167a8a44ζ(x)a141
a511 <x< +∞a254a235a89
a124a94a234a198a56a66a123a167a191a53a191a20a233a63a219a20a18a234ka167a63a234
∞summationdisplay
n=1
(lnn)k
na (a > 1)a209a194a241a167a149a236a254a227a167a140a121a181a233a63a219a20a18
a234ka167ζ(k)(x)a511 <x< +∞a254a209a127a51a133a235a89a167a133a140a100a6a63a234a197a145a166a19a234ka103a167a26
ζ(k)(x) = (?1)k
∞summationdisplay
n=1
(lnn)k
nx (1 <x< +∞).
14,a193a121a63a234
∞summationdisplay
1
sin(2npix)
2n a51a18a135a162a234a182a254a152a151a194a241a167a2a51a63a219a171a109a83a216a85a197a145a166a135a251.
a121a178a181a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(2npix)
2n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
2na233?x∈ (?∞< +∞)a27a164a225a133a63a234
∞summationdisplay
1
1
2na194a241a167a75a226Ma7a79a123a167a26
∞summationdisplay
1
sin(2npix)
2n
a51a18a135a162a234a182a254a152a151a194a241parenleftBigg
sin(2npix)
2n
parenrightBiggprime
= picos(2npix)
a101a121
∞summationdisplay
1
picos(2npix)a51a63a219a171a109a83a209a107a216a235a89a58
a63a18x∈ (?∞,+∞)a167a111a127a51k ∈Za167a166x = k+ya167a217a1650 lessorequalslanty< 1
a242a217a147a92a167a26
∞summationdisplay
1
cos(2npix) =
∞summationdisplay
1
cos(2npiy)a167a65a79a27a167a18y = 2?mha167a217a165m∈Z+,h = 0,1,2,···,2m?1
a8n>ma158a167cos(2npiy) = 1a167a100a158a63a234a152a132a145a216a170a1170a167a75
∞summationdisplay
1
cos(2npix) =
∞summationdisplay
1
cos(2npiy)a117a209a167a117a180
∞summationdisplay
1
picos(2npix)a117
a209
a113a51a63a219a171a109a83a209a127a51x = k+ 2?mh(h = 0,1,2,···,2m?1)a249a24a27a58a167ka143xa27a129a2a18a234a220a169
a75a63a234
∞summationdisplay
1
sin(2npix)
2n a51a63a219a171a109a83a216a85a197a145a166a135a251.
15,a107a121
1?r2
1?2rcosx+r2 = 1 + 2
∞summationdisplay
n=1
rncosnx
a8|r|< 1a158a164a225a167a108a13a121a178a181 integraldisplay
pi
pi
1?r2
1?2rcosx+r2 dx = 2pi (|r|< 1)
.
a121a178a181|rncosnx|lessorequalslant|r|na233?x∈ (?∞,+∞)a209a164a225
a207|r|< 1a167a75
∞summationdisplay
n=1
|r|na194a241a167a117a180a100Ma7a79a123a167a26
∞summationdisplay
n=1
rncosnxa51(?∞,+∞)a83a152a151a194a241
183
a108a13a23f(x) = 1 + 2
∞summationdisplay
n=1
rncosnx
a2071?2rcosx+r2 negationslash= 0a167a254a170a252a224a211a166a1771?2rcosx+r2a167a75a26
(1?2rcosx+r2)f(x) = (1?2rcosx+r2)
parenleftBigg
1 + 2
∞summationdisplay
n=1
rncosnx
parenrightBigg
=
bracketleftBigg
1?2rcosx+r2 + 2
∞summationdisplay
n=1
rncosnx?2
∞summationdisplay
n=1
rn+1(2cosnxcosx) + 2
∞summationdisplay
n=1
rn+2 cosnx
bracketrightBigg
=
bracketleftBigg
1?2rcosx+r2 + 2
∞summationdisplay
n=1
rncosnx?2
∞summationdisplay
n=1
rn+1 cos(n+ 1)x?2
∞summationdisplay
n=1
rn+1 cos(n?1)x+ 2
∞summationdisplay
n=1
rn+2 cosnx
bracketrightBigg
=
bracketleftBigg
1?r2 + 2
∞summationdisplay
n=1
rncosnx?2
parenleftBigg ∞summationdisplay
n=1
rn+1 cos(n+ 1)x+rcosx
parenrightBigg
2
parenleftBigg ∞summationdisplay
n=1
rn+1 cos(n?1)x?r2
parenrightBigg
+ 2
∞summationdisplay
n=1
rn+2 cosnx
bracketrightBigg
= 1?r2
a117a180f(x) = 1?r
2
1?2rcosx+r2 a61
1?r2
1?2rcosx+r2 = 1 + 2
∞summationdisplay
n=1
rncosnx
a100a117a254a170a109a224a63a234a51[?pi,pi]a254a152a151a194a241a167a133rncosnxa51[?pi,pi]a254a235a89a167a75a254a170a63a234a140a177a197a145a200a169a167a26integraldisplay
pi
pi
1?r2
1?2rcosx+r2 dx =
integraldisplay pi
pi
parenleftBigg
1 + 2
∞summationdisplay
n=1
rncosnx
parenrightBigg
dx = 2pi+ 2
∞summationdisplay
n=1
integraldisplay pi
pi
rncosnxdx = 2pi.
16,a94a107a129a67a88a189a110a121a178a41a90a189a110.
a121a178a181a207{Sn(x)}a51[a,b]a254a194a241a117S(x)a167a25a233?ε> 0,?x ∈ [a,b],?N(ε,x) ∈ Z+a167a166a26a8ngreaterorequalslantN(ε,x)a158a167a209
a65a107|Sn(x)?S(x)|<εa167a65a79a107|SN(ε,x)?S(x)|<ε
a100SN(ε,x)(x)?S(x)a51xa58a235a89a167a26a127a51xa58a27a109a25a141Oxa167a166a26|SN(ε,x)(y)?S(y)|,?y ∈Ox
a117a180braceleftbigOxvextendsinglevextendsinglex∈ [a,b]bracerightbiga8a164[a,b]a27a109a67a88(a233a224a58a,ba140a138a235a89a242a255)
a226a107a129a67a88a189a110a167a108a165a192a209a107a129a135a109a25a141Ox1,···,Oxma211a24a67a88[a,b]a133a247a118|SN(ε,xi)(y)?S(y)| < ε,?y ∈
Oxi,i = 1,2,···,m
a18N = max
ilessorequalslantilessorequalslantm
N(ε,xi)a167a75a8n > Na158a167a233?x ∈ [a,b]a167a100{Sn(x)}a252a78a53a218
muniontext
i=1
Oxi? [a,b]a167a55a127a51a44
a135Oxia167a166x∈Oxia167a133a107|Sn(x)?S(x)|lessorequalslant|SN(x)?S(x)|lessorequalslant|SN(ε,xi)(x)?S(x)|<ε
a61{Sn(x)}a51[a,b]a254a152a151a194a241a117S(x).
17,a101Sn(x)a51ca58a134a235a89(n = 1,2,3,···)a167a2{Sn(c)}a117a209a167a75a51a63a219a109a171a109(c?δ,c)a83(δ > 0)a167{Sn(x)}a55a216a152
a151a194a241.
a121a178a181a94a135a121a123.
a98a23a127a51δ0 > 0a167a166a26{Sn(x)}a51(c?δ0,c)a83a152a151a194a241
a100a152a151a194a241a27a133a220a6a110a167a26a233?ε> 0,?N(ε) ∈Z+a167a166a26a8n>N(ε)a158a167a233?x∈ (c?δ0,c)a218?p∈Z+a167a209a65
a107|Sn+p(x)?Sn(x)|< ε2 (?)a164a225
a207a122a152a135Sn(x)a51ca58a134a235a89a167a75Sn+p(x)?Sn(x)a143a51ca58a134a235a89
a117a180 lim
x→c?0
[Sn+p(x)?Sn(x)] = Sn+p(c)?Sn(c)
a51(?)a170a252a224a45x→c?0a167a26|Sn+p(c)?Sn(c)|lessorequalslant ε2 <ε
a100a234a15a27a133a220a194a241a6a110a167a26{Sn(c)}a194a241a167a134a174a127{Sn(c)}a117a209a103a241
a25a98a23a216a20a40a167a75a51a63a219a109a171a109(c?δ,c)a83(δ> 0)a167{Sn(x)}a55a216a152a151a194a241.
184
§2,a152a63a234
1,a166a101a15a136a152a63a234a27a194a241a171a109a181
(1)
∞summationdisplay
n=1
(2x)n
n!
(2)
∞summationdisplay
n=1
ln(n+ 1)
n+ 1 x
n+1
(3)
∞summationdisplay
n=1
bracketleftBiggparenleftBigg
n+ 1
n
parenrightBiggn
x
bracketrightBiggn
(4)
∞summationdisplay
n=1
xn2
2n
(5)
∞summationdisplay
n=1
[3 + (?1)n]n
n x
n
(6)
∞summationdisplay
n=1
3n + (?2)n
n (x+ 1)
n
a41a181
(1) an = 2
n
n!
a207R = lim
n→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= +∞a167a75a217a194a241a141a143(?∞,+∞).
(2)
∞summationdisplay
n=1
ln(n+ 1)
n+ 1 x
n+1 =
∞summationdisplay
n=2
lnn
n x
n a167an = lnn
n
a100a117 lim
y→+∞
(y+ 1)lny
yln(y+ 1) = limy→+∞
y+ 1
y limy→+∞
lny
ln(y+ 1) = 1a167a75R = limn→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle = 1a167a117a180a217a194a241a171a109
a143(?1,1)
a8x =?1a158a167a6a63a234a143
∞summationdisplay
n=2
(?1)n lnnn xn
a207
parenleftBigg
lnx
x
parenrightBiggprime
= 1?lnxx2 a133a8xgreaterorequalslant 3a158a167
parenleftBigg
lnx
x
parenrightBiggprime
< 0a167a75
braceleftBigg
lnn
n
bracerightBigg
a252a78a126a8
a113 lim
n→∞
lnn
n = 0a167a75a63a234
∞summationdisplay
n=2
(?1)n lnnn xn a143a52a217a90a91a63a234a167a117a180a63a234
∞summationdisplay
n=2
(?1)n lnnn xn a194a241
a8x = 1a158a167a6a63a234a143
∞summationdisplay
n=2
lnn
n x
n
a207 lim
n→∞
lnn
n
n = +∞a167a75a226a20a145a63a234a27a39a22a7a79a123a57a63a234
∞summationdisplay
n=1
1
na117a209a167a26a63a234
∞summationdisplay
n=2
lnn
n x
n a117a209
a75a100a63a234a27a194a241a141a143[?1,1).
(3) a207
∞summationdisplay
n=1
bracketleftBiggparenleftBigg
n+ 1
n
parenrightBiggn
x
bracketrightBiggn
=
∞summationdisplay
n=1
parenleftBigg
1 + 1n
parenrightBiggn2
xna167a75an =
parenleftBigg
1 + 1n
parenrightBiggn2
a113 lim
n→∞
nradicalbig|an| = ea167a75a217a194a241a140a187a143R = 1
ea167a194a241a171a109a143
parenleftBigg
1e,1e
parenrightBigg
.
a8x = ±1ea158a167a6a63a234a143
∞summationdisplay
n=1
(±1)n
parenleftBigg
1 + 1n
parenrightBiggn2 parenleftBigg
1
e
parenrightBiggn
a167a75un = (±1)n
parenleftBigg
1 + 1n
parenrightBiggn2 parenleftBigg
1
e
parenrightBiggn
a100a226a55a136a123a75a167a26 lim
n→∞
|un| = e?12 negationslash= 0
a75a63a234
∞summationdisplay
n=1
(±1)n
parenleftBigg
1 + 1n
parenrightBiggn2 parenleftBigg
1
e
parenrightBiggn
a117a209a167a117a180a6a63a234a27a194a241a141a143
parenleftBigg
1e,1e
parenrightBigg
.
185
(4) an = 12n
a100 lim
n→∞
n2radicalbig|an|xn2 = |x| lim
n→∞
n
radicalbigg
1
2 = |x|< 1a167a26a217a194a241a140a187a143R = 1a167a194a241a171a109a143(?1,1)
a8|x| = 1a61x = ±1a158a167a6a63a234a67a143
∞summationdisplay
n=1
(±1)n2
2n
a100a117a63a234
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(±1)n2
2n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
1
2n a194a241a167a75a63a234
∞summationdisplay
n=1
(±1)n2
2n a253a233a194a241a75a194a241
a108a13a152a63a234
∞summationdisplay
n=1
xn2
2n a27a194a241a141a143[?1,1].
(5) an = [3 + (?1)
n]n
n
a207 lim
n→∞
n
radicalbigg
[3 + (?1)n]n
n = 4a167a75a63a234a194a241a140a187a143R =
1
4 a167a194a241a171a109a143
parenleftBigg
14,14
parenrightBigg
a8x = 14 a158a167a6a63a234a67a143
∞summationdisplay
n=1
[3 + (?1)n]n
n·4n =
∞summationdisplay
k=1
1
2k +
∞summationdisplay
k=1
1
(2k+ 1)22k+1
a233a63a234
∞summationdisplay
k=1
1
(2k+ 1)22k+1
a207 lim
k→∞
1
(2k+3)22k+3
1
(2k+ 1)22k+1
= 14 < 1 a167a75a226a136a75a19a16a7a79a123a167a26a63a234
∞summationdisplay
k=1
1
(2k+ 1)22k+1 a194a241
a113a63a234
∞summationdisplay
k=1
1
2k a117a209a167a75a63a234
∞summationdisplay
n=1
[3 + (?1)n]n
n·4n a117a209
a211a123a140a26a167a8x =?14 a158a167a63a234
∞summationdisplay
n=1
(?1)n[3 + (?1)
n]n
n·4n a117a209
a75a63a234
∞summationdisplay
n=1
[3 + (?1)n]n
n x
na27a194a241a141a143
parenleftBigg
14,14
parenrightBigg
.
(6) an = 3
n + (?2)n
n
a207 lim
n→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
3 a167a75a63a234a27a194a241a140a187a143R =
1
3 a167a194a241a171a109a143
parenleftBigg
43,?23
parenrightBigg
a8x =?43 a158a167a6a63a234a67a143
∞summationdisplay
n=1
(?1)n3
n + (?2)n
n
parenleftBigg
1
3
parenrightBiggn
=
∞summationdisplay
n=1
(?1)n
n +
∞summationdisplay
n=1
parenleftbig2
3
parenrightbign
n
a233a63a234
∞summationdisplay
n=1
parenleftbig2
3
parenrightbign
n
a207 lim
n→∞
parenleftbig2
3
parenrightbign+1/(n+ 1)
parenleftbig2
3
parenrightbign/n = 23 < 1 a167a75a226a136a75a19a16a7a79a123a167a26
∞summationdisplay
n=1
parenleftbig2
3
parenrightbign
n a194a241
a113a63a234
∞summationdisplay
n=1
(?1)n
n a194a241a167a75a8x =?
4
3 a158a167a6a63a234a194a241a182
a211a123a140a26a167a8x =?23 a158a167a6a63a234a117a209
a75a63a234
∞summationdisplay
n=1
3n + (?2)n
n (x+ 1)
na27a194a241a141a143
bracketleftBigg
43,?23
parenrightBigg
.
2,a166a63a234a27a194a241a140a187a181
(1)
∞summationdisplay
n=1
parenleftBigg
1 + 12 +···+ 1n
parenrightBigg
xn
(2)
summationdisplay(2n)!
(n!)2 x
n
a41a181
186
(1) an = 1 + 12 +···+ 1n
a2071 = n
radicalbigg
n· 1nlessorequalslant n
radicalbigg
1 + 12 +···+ 1nlessorequalslant n√n·1 → 1(n→∞)
a75 lim
n→∞
nradicalbig|an| = 1a167a117a180a217a194a241a140a187a143R = 1.
(2) an = (2n)!(n!)2
a207 lim
n→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limn→∞
(n+ 1)2
2(n+ 1)(2n+ 1) =
1
4 a167a117a180a217a194a241a140a187a143R =
1
4,
3.
∞summationdisplay
n=1
3n + (?2)n
n (x+ 1)
n a23a152a63a234summationdisplayanxna27a194a241a140a187a143Ra167summationdisplaybnxna27a194a241a140a187a143Qa167a63a216a101a15a63a234a27a194a241
a140a187a181
(1)
summationdisplay
anx2n
(2)
summationdisplay
(an +bn)xn
(3)
summationdisplay
anbnxn
a41a181
(1) lim
n→∞
2nradicalbig|an| = lim
n→∞
parenleftBig
nradicalbig|an|
parenrightBig1
2 =
radicalbigg
1
R =
1√
Ra167a75a217a194a241a140a187a143R1 =
√R.
(2) a23An = an +bn
a75a107 nradicalbig|An| = nradicalbig|an +bn|lessorequalslant nradicalbig|an|+|bn|lessorequalslant nradicalbig2max(|an|,|bn|) = n√2· nradicalbigmax(|an,bn|) = n√2max( nradicalbig|an|,nradicalbig|bn|)
a207 lim
n→∞
n√2 = 1
a75 1R
2
= lim
n→∞
nradicalbig|An|lessorequalslant lim
n→∞
{ n√2max( nradicalbig|an|,nradicalbig|bn|)} = lim
n→∞
{max( nradicalbig|an|,nradicalbig|bn|)} = max
parenleftBig
lim
n→∞
nradicalbig|an|,lim
n→∞
nradicalbig|an|
parenrightBig
=
max
parenleftBigg
1
R,
1
Q
parenrightBigg
a108a13a167a26R2 greaterorequalslant 1
max
parenleftBigg
1
R,
1
Q
parenrightBigg= min(R,Q).
(3) a23Bn = anbn
a75a107 nradicalbig|Bn| = nradicalbig|anbn| = nradicalbig|an|· nradicalbig|bn|
a117a180 1R
3
= lim
n→∞
nradicalbig|Bn| = lim
n→∞
{ nradicalbig|an|· nradicalbig|bn|lessorequalslant lim
n→∞
nradicalbig|an|· lim
n→∞
nradicalbig|bn|} = 1
R ·
1
Q =
1
RQ
a108a13R3 greaterorequalslantRQ.
4,a23a233a191a169a140a27na167|an|lessorequalslant|bn|a167a64a34a63a234
summationdisplay
anxna27a194a241a140a187a216a2a117
summationdisplay
bnxna27a194a241a140a187.
a121a178a181a207a233a191a169a140a27na167|an|lessorequalslant|bn|a167a75 nradicalbig|an|lessorequalslant nradicalbig|bn|a167a117a180 lim
n→∞
nradicalbig|an|lessorequalslant lim
n→∞
nradicalbig|bn|
a23a63a234
summationdisplay
anxna27a194a241a140a187a143Ra167a63a234
summationdisplay
bnxna27a194a241a140a187a143Q
a75a80 < lim
n→∞
nradicalbig|an|lessorequalslant lim
n→∞
nradicalbig|bn|<∞a158a167a100R = 1
lim
n→∞
nradicalbig|an|,Q =
1
lim
n→∞
nradicalbig|bn| a167a26RgreaterorequalslantQa182
a80 = lim
n→∞
nradicalbig|an|lessorequalslant lim
n→∞
nradicalbig|bn|a167a75R = ∞,Qlessorequalslant∞a167a117a180RgreaterorequalslantQa182
a8 lim
n→∞
nradicalbig|an|lessorequalslant lim
n→∞
nradicalbig|bn| = ∞a158a167a75Rgreaterorequalslant 0,Q = 0a167a117a180RgreaterorequalslantQ
a110a254a127a167a63a234
summationdisplay
anxna27a194a241a140a187a216a2a117
summationdisplay
bnxna27a194a241a140a187.
5,a121a178a152a63a234a27a53a1591a218a53a1592.
a121a178a181a53a1591.
a23xa143(x0?R,x0 +R)a83a63a152a58a167a111a140a177a192a180 <r<Ra167a166a26|x?x0|lessorequalslantr
a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=0
an(x?x0)na51[x0?r,x0 +r]a254a152a151a194a241
a113an(x?x0)n(n = 0,1,2,···)a51[x0?r,x0+r]a235a89a167a75a100a188a234a145a63a234a27a218a27a235a89a53a127S(x)a51[x0?r,x0+r]a235
a89a167a8a44a51xa249a152a58a235a89
187
a13xa143(x0?R,x0 +R)a254a63a152a58a167a75S(x)a51(x0?R,x0 +R)a235a89
a113a101
∞summationdisplay
n=0
an(x?x0)na51x0 + Ra194a241a167a75a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=0
an(x?x0)na51[a,x0 + R](a18a ∈ (x0?
R,x0 +R))a254a152a151a194a241
a100a117an(x?x0)n(n = 0,1,2,···)a51[a,x0 +R]a235a89a167a75a100a188a234a145a63a234a27a218a27a235a89a53a189a110a167a26
S(x)a51[a,x0 +R]a235a89a167a8a44a143a51x0 +Ra235a89a167a117a180S(x)a51(x0?R,x0 +R]a254a235a89
a211a110a101
∞summationdisplay
n=0
an(x?x0)na51x0?Ra194a241a167a75S(x)a51[x0?R,x0 +R)a254a235a89.
a53a1592.
(1) a23xa143(x0?R,x0+R)a83a63a152a58a167a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=0
an(x?x0)na51[x0,x]a254a152a151a194a241(a101x<x0a167
a75a18[x,x0 ]a61a140)
a113an(x?x0)n(n = 0,1,2,···)a51[x0,x]a235a89
a75a100a188a234a145a63a234a197a145a166a200a169a189a110a167a26integraldisplay
x
x0
S(x)dx =
integraldisplay x
x0
parenleftBigg ∞summationdisplay
n=0
an(x?x0)n
parenrightBigg
dx =
∞summationdisplay
n=0
integraldisplay x
x0
[an(x?x0)n]dx =
∞summationdisplay
n=0
an
n+ 1 (x?x0)
n+1
(2) a100a495a144a83a753(2)a127a167a101{xn}a194a241a167a133 lim
n→∞
xn = 0a167a75a233a63a219{yn}a167a107 lim
n→∞
(xn·yn) = lim
n→∞
xn· lim
n→∞
yn
a75 lim
n→∞
nradicalbig|nan| = lim
n→∞
nradicalbig|an|
a249a96a178a181
∞summationdisplay
n=1
nan(x?x0)n?1a134
∞summationdisplay
n=1
an(x?x0)na107a131a211a27a194a241a140a187R
a23xa180(x0?R,x0 +R)a83a63a152a58a167a111a140a192a18a152a580 <r<Ra167a166a26|x?x0|lessorequalslantr
a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=0
an(x?x0)na51[x0?r,x0 +r]a254a152a151a194a241a167a207a13a194a241
a113
∞summationdisplay
n=1
nan(x?x0)n?1a27a194a241a140a187a143Ra167a75a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=1
nan(x?x0)n?1a51[x0?r,x0+r]a254
a152a151a194a241
a113nan(x?x0)n?1(n = 1,2,···)a51[x0?r,x0 +r]a235a89a167a75a100a188a234a145a63a234a197a145a135a169a189a110a167a26
a51[x0?r,x0 +r]a8a44a143a210a51xa58a167a107 ddx S(x) = ddx
parenleftBigg ∞summationdisplay
n=0
an(x?x0)n
parenrightBigg
=
∞summationdisplay
n=1
nan(x?x0)n?1
a50a100xa51(x0?R,x0 +R)a27a63a191a53a167a26a51(x0?R,x0 +R)a254a170a143a164a225
(3) a23
∞summationdisplay
n=0
an
n+ 1 (x?x0)
n+1a194a241a140a187a143Rprime
a100(1)a167a26a8
∞summationdisplay
n=0
an(x?x0)na51(x0?R,x0 +R)a194a241(a194a241a20S(x))a158a167a107
∞summationdisplay
n=0
an
n+ 1 (x?x0)
n+1a51(x0?R,x0 +R)a254a194a241
parenleftbigg
a194a241a20
integraldisplay x
x0
S(x)dx
parenrightbigg
a167a64a34RlessorequalslantRprime
a44a152a144a161a167a100(2)a167a8
∞summationdisplay
n=0
an
n+ 1 (x?x0)
n+1a51(x0?Rprime,x0 +Rprime)a254a194a241
parenleftbigg
a194a241a20
integraldisplay x
x0
S(x)dx
parenrightbigg
a158a167a107
∞summationdisplay
n=0
an(x?x0)na51(x0?Rprime,x0 +Rprime)a194a241(a194a241a20S(x))a167a64a34Rprime lessorequalslantR
a117a180R = Rprime
6,a23
∞summationdisplay
0
ana194a241a117Aa167
∞summationdisplay
0
bna194a241a117Ba167a88a74a167a130a27a133a220a166a200
∞summationdisplay
0
cn =
∞summationdisplay
0
(a0bn +a1bn?1 +···+anb0)
a194a241a167a75a152a189a194a241a117AB.
a121a178a181a138A(x) =
∞summationdisplay
0
anxn,B(x) =
∞summationdisplay
0
bnxn,C(x) =
∞summationdisplay
0
cnxn
a8x = 1a158a167A = A(1) =
∞summationdisplay
0
an,B = B(1) =
∞summationdisplay
0
bn,C = C(1) =
∞summationdisplay
0
cn
188
a61a152a63a234
∞summationdisplay
0
anxn,
∞summationdisplay
0
bnxn,
∞summationdisplay
0
cnxna51x = 1a194a241
a100Abela49a152a189a110a167a26a254a227a27a152a63a234a51|x|< 1a83a253a233a194a241
a100a133a220a189a110a167a26a63a234
∞summationdisplay
0
cnxna194a241a117
parenleftBigg ∞summationdisplay
0
anxn
parenrightBiggparenleftBigg ∞summationdisplay
0
bnxn
parenrightBigg
a61C(x) = A(x)B(x)
a207
∞summationdisplay
0
anxn,
∞summationdisplay
0
bnxn,
∞summationdisplay
0
cnxna51x = 1a194a241
a100a152a63a234a97a113a53a1591a167a75A(x),B(x),C(x)a51x = 1a134a235a89
C(1) = lim
x→1?0
C(x) = lim
x→1?0
A(x)B(x) = A(1)B(1)
a75C = ABa167a117a180
∞summationdisplay
0
cn = AB.
7,a23f(x) =
∞summationdisplay
0
anxna8|x|<ra158a194a241a167a64a34a8
∞summationdisplay
0
an
n+ 1 r
n+1a194a241a158a164a225
integraldisplay r
0
f(x)dx =
∞summationdisplay
0
an
n+ 1 r
n+1
a216a216
∞summationdisplay
0
anxna8x = ra158a180a196a194a241.
a121a178a181a207f(x) =
∞summationdisplay
0
anxna8|x|<ra158a194a241a167a75a217a194a241a140a187a143Ra167a133rlessorequalslantRa167a108a13f(x)a51(?r,r)a83a194a241.
a75a226a53a1592a167a8x∈ (?r,r)a158a167a107
integraldisplay θ
0
f(x)dx =
integraldisplay θ
0
bracketleftBigg ∞summationdisplay
0
anxn
bracketrightBigg
dx =
∞summationdisplay
0
an
n+ 1 θ
n+1,θ ∈ (0,r)
a61
integraldisplay θ
0
f(x)dx =
∞summationdisplay
0
an
n+ 1 θ
n+1 θ ∈ (0,r)
a207
∞summationdisplay
0
an
n+ 1 r
n+1a194a241a167a75
∞summationdisplay
0
an
n+ 1 θ
n+1a51θ = ra194a241a167a117a180a217a218S(θ)a51ra58a134a235a89
S(r) =
∞summationdisplay
0
an
n+ 1 r
n+1 = lim
θ→r?0
S(θ) = lim
θ→r?0
integraldisplay θ
0
f(x)dx =
integraldisplay r
0
f(x)dx
a108a13a216a216
∞summationdisplay
0
anxna8x = ra158a180a196a194a241a167a254a107
integraldisplay r
0
f(x)dx =
∞summationdisplay
0
an
n+ 1 r
n+1
8,a124a94a254a75a121a178
integraldisplay 1
0
ln(1?t)
t dt =?
∞summationdisplay
n=1
1
n2,
a121a178a181a207ln(1 +x) =
∞summationdisplay
n=1
(?1)n+1x
n
n (?1 <x< 1)a167a75
ln(1?x)
x =?
∞summationdisplay
n=1
xn?1
n (?1 <x< 1a133xnegationslash= 0)
a61f(x) =
ln(1?x)
x,x∈ (?1,0)
uniontext(0,1)
1,x = 0
a167f(x) =?
∞summationdisplay
n=0
xn
n+ 1 (?1 <x< 1)
a207
∞summationdisplay
n=0
1n+1
n+ 1 =?
∞summationdisplay
n=0
1
(n+ 1)2 a194a241a167a75a100a254a75a40a216a167a26
integraldisplay 1
0
ln(1?x)
x dx =?
∞summationdisplay
n=1
1
(n+ 1)2 =?
∞summationdisplay
n=1
1
n2
9,a166f(x) =
∞summationdisplay
n=1
sin(2n ·x)
n! a27a240a142a78a21a63a234a167a96a178a167a27a240a142a78a21a63a234a191a216a76a171a249a135a188a234.
a121a178a181a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(2n ·x)
n!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n! (x∈ (?∞,+∞))a167a133
∞summationdisplay
n=0
1
n!a194a241a167a75a100Ma7a79a123a167a26
∞summationdisplay
n=1
sin(2n ·x)
n! a51(?∞,+∞)a83
a152a151a194a241a167a108a13a194a241
f(0) = 0,
∞summationdisplay
n=1
bracketleftBigg
sin(2n ·x)
n!
bracketrightBiggprime
=
∞summationdisplay
n=1
2ncos(2n ·x)
n!
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
2ncos(2n ·x)
n!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
2n
n! (x∈ (?∞,+∞))a167a133
∞summationdisplay
n=0
2n
n! a194a241a167a75a100Ma7a79a123a167a26
∞summationdisplay
n=1
2ncos(2n ·x)
n! a51(?∞,+∞)a83
a152a151a194a241
189
a1132
ncos(2n ·x)
n! (n = 0,1,2,···) a51(?∞,+∞)a83a235a89a167a75a100a197a145a166a19a189a110a167a26a51(?∞,+∞)a254
f prime(x) = ddx
bracketleftBigg ∞summationdisplay
n=1
sin(2n ·x)
n!
bracketrightBigg
=
∞summationdisplay
n=1
2ncos(2n ·x)
n!
a117a180f prime(0) =
∞summationdisplay
n=1
2n
n! = e
2?1
a88a100a101a22a167a94a234a198a56a66a123a167a26
f(m)(x) =
∞summationdisplay
n=1
2mnsinparenleftbigmpi2 + 2npiparenrightbig
n!,
f(m)(0) =
0,m = 2k
∞summationdisplay
n=1
(?1)k2
(2k+1)n
n! = (?1)
k(e22k+1?1),m = 2k+ 1
a75f(x)a27a240a142a78a21a63a234a143
∞summationdisplay
k=0
(?1)k(e22k+1?1) x
2k+1
(2k+ 1)! a217a194a241a140a187a143R = limk→∞
(e22k+1?1)/(2k+ 1)!
(e22k+3?1)/(2k+ 3)!
a2070 lessorequalslant (e
22k+1?1)/(2k+ 1)!
(e22k+3?1)/(2k+ 3)! = (2k+ 2)(2k+ 3)
e22k+1?1
e22k+3?1 lessorequalslant (2k+ 2)(2k+ 3)
e22k+1
e22k+3 =
(2k+ 2)(2k+ 3)
e6·22k a167
a133 lim
x→∞
(2x+ 2)(2x+ 3)
e6·22x = 0a167a75 limk→∞
(2k+ 2)(2k+ 3)
e6·22k = 0
a117a180R = 0a167a61a217a240a142a78a21a63a234a61a51x = 0a194a241
a2a100a99a161a140a127a217a51(?∞,+∞)a83a254a194a241a167a75a167a27a240a142a78a21a63a234a191a216a76a171a100a188a234.
10,a121a178a181
(1)
∞summationdisplay
n=0
x4n
(4n)! a247a118y
(IV) = ya182
(2)
∞summationdisplay
n=0
xn
(n!)2 a247a118xy
primeprime +yprime?y = 0.
a121a178a181
(1) an = n
radicalBigg
1
(4n)! a167R = limn→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= +∞
a75a127a233a63a152xa167a152a63a234a209a194a241a167a61a217a194a241a141a143(?∞,+∞)
a51a194a241a141a83a197a145a135a169a131a167a26
yprime =
∞summationdisplay
n=1
x4n?1
(4n?1)!,y
primeprime =
∞summationdisplay
n=1
x4n?2
(4n?2)!,y
primeprimeprime =
∞summationdisplay
n=1
x4n?3
(4n?3)!,y
(4) =
∞summationdisplay
n=1
x4n?4
(4n?4)! =
∞summationdisplay
n=0
x4n
(4n)! = y
a61y(IV) = y.
(2) an = 1(n!)2 a167a75R = lim
x→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= +∞
a75a127a233a63a152xa167a152a63a234a209a194a241a167a61a217a194a241a141a143(?∞,+∞)
a51a194a241a141a83a197a145a135a169a131a167a26yprime =
∞summationdisplay
n=1
xn?1
(n?1)!n! =
∞summationdisplay
n=0
xn
n!(n+ 1)!,y
primeprime =
∞summationdisplay
n=1
xn?1
(n?1)!(n+ 1)!
a117a180xyprimeprime +yprime =
∞summationdisplay
n=1
xn
(n?1)!(n+ 1)! +
∞summationdisplay
n=0
xn
n!(n+ 1)! = 1 +
∞summationdisplay
n=1
bracketleftBigg
1
(n?1)!(n+ 1)! +
1
n!(n+ 1)!
bracketrightBigg
xn =
1 +
∞summationdisplay
n=1
n+ 1
n!(n+ 1)! x
n = 1 +
∞summationdisplay
n=1
xn
(n!)2 =
∞summationdisplay
n=0
xn
(n!)2 = y
a61xyprimeprime +yprime?y = 0.
11,a208a109a181
(1) f(x) = 1a?x (anegationslash= 0)a164a143xa27a152a63a234a167a191a40a189a194a241a137a140a182
(2) f(x) = lnxa143(x?2)a27a152a63a234.
a41a181
190
(1) a207f(x) = 1a?x = 1a
parenleftBigg
1
1? xa
parenrightBigg
a167a133
∞summationdisplay
n=0
parenleftBigg
x
a
parenrightBiggn
= 11? x
a
a167a100a158
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
a
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle< 1
a75f(x) = 1a
bracketleftBigg ∞summationdisplay
n=0
parenleftBigg
x
a
parenrightBiggnbracketrightBigg
=
∞summationdisplay
n=0
xn
an+1 a167|x|<|a|
(2) f(x) = lnx = ln[2 + (x?2)] = ln2 + ln
parenleftBigg
1 + x?22
parenrightBigg
a207ln
parenleftBigg
1 + x?22
parenrightBigg
=
∞summationdisplay
n=1
(?1)n+1
n
parenleftBigg
x?2
2
parenrightBiggn
=
∞summationdisplay
n=1
(?1)n+1
2n ·n (x?2)
n a1670 <xlessorequalslant 4
a75f(x) = ln2 +
∞summationdisplay
n=1
(?1)n+1
2n ·n (x?2)
n a167a194a241a141a143(0,4].
12,a124a94a174a127a208a109a170a208a109a101a15a188a234a143a152a63a234a167a191a40a189a194a241a137a140a181
(1) e
x?e?x
2
(2) sin2x = 1?cos2x2
a41a181
(1) a207ex =
∞summationdisplay
n=0
xn
n! (?∞<x< +∞),e
x =
∞summationdisplay
n=0
(?1)nxn
n! (?∞<x< +∞)
a75f(x) = e
x?e?x
2 =
1
2
bracketleftBigg ∞summationdisplay
n=0
xn
n!?
∞summationdisplay
n=0
(?1)nxn
n!
bracketrightBigg
a8n = 2ka158a167f(x) = 0a182a8n = 2k+ 1a158a167f(x) =
∞summationdisplay
k=0
x2k+1
(2k+ 1)!
a110a254a140a127a167f(x) =
∞summationdisplay
n=0
x2n+1
(2n+ 1)! a167a194a241a141a143(?∞,+∞).
(2) a207cos2x =
∞summationdisplay
n=0
(?1)n(2x)2n
(2n)! (?∞<x< +∞)
a75sin2x = 1?cos2x2 =
∞summationdisplay
n=1
(?1)n+122n?1
(2n)! x
2na167a194a241a141a143(?∞,+∞).
13,a208a109 ddx
parenleftBigg
ex?1
x
parenrightBigg
a143xa27a152a63a234a167a191a237a2091 =
∞summationdisplay
n=1
n
(n+ 1)!,
a41a181a207ex =
∞summationdisplay
n=0
xn
n! (?∞<x< +∞)a167a75
ex?1
x =
∞summationdisplay
n=0
xn
(n+ 1)! (xnegationslash= 0)
a45f(x) =
ex?1
x,xnegationslash= 01,x = 0 a167a75
∞summationdisplay
n=0
xn
(n+ 1)! a143f(x)a27a152a63a234a167a217a194a241a137a140a143(?∞,+∞)
a100a152a63a234a27a197a145a166a19a189a110a167a26
∞summationdisplay
n=0
xn
(n+ 1)! a51(?∞,+∞)a83a197a145a166a19
d
dx f(x) =
∞summationdisplay
n=1
n
(n+ 1)! x
n?1,xnegationslash= 0
1
2,x = 0
a117a180 ddx
parenleftBigg
ex?1
x
parenrightBigg
=
∞summationdisplay
n=0
n
(n+ 1)! x
n?1
a207 ddx
parenleftBigg
ex?1
x
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x=1
= (x?1)e
x + 1
x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x=1
= 1a167a75
∞summationdisplay
n=0
n
(n+ 1)! x
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x=1
=
∞summationdisplay
n=0
n
(n+ 1)! = 1
14,a166a101a15a188a234a27a152a63a234a208a109a170a167a191a237a209a194a241a140a187a181
(1)
integraldisplay x
0
sint
t dt
191
(2)
integraldisplay x
0
cost2 dt
a41a181
(1) a207sint =
∞summationdisplay
n=0
(?1)nt2n+1
(2n+ 1)! a167a75
sint
t =
∞summationdisplay
n=0
(?1)nt2n
(2n+ 1)! (tnegationslash= 0)
a45f(t) =
sint
t,tnegationslash= 01,t = 0 a167a75
∞summationdisplay
n=0
(?1)nt2n
(2n+ 1)! a143f(t)a27a152a63a234a167a194a241a141a143(?∞,+∞)
a100a152a63a234a197a145a200a169a189a110a167a26
∞summationdisplay
n=0
(?1)nt2n
(2n+ 1)! a51(?∞,+∞)a83a197a145a200a169
integraldisplay x
0
f(t)dt =
∞summationdisplay
n=0
(?1)nt2n+1
(2n+ 1)(2n+ 1)!,xnegationslash= 0
0,x = 0
a75
integraldisplay x
0
sint
t dt =
∞summationdisplay
n=0
(?1)nt2n+1
(2n+ 1)(2n+ 1)! a167a217a194a241a140a187a143R = +∞.
(2) a207cost2 =
∞summationdisplay
n=0
(?1)n(t2)2n
(2n)! =
∞summationdisplay
n=0
(?1)nt4n
(2n)! a167a217a194a241a141a143(?∞,+∞)a167a194a241a140a187a143R = ∞
a100a152a63a234a27a197a145a200a169a189a110a167a26
∞summationdisplay
n=0
(?1)nt4n
(2n)! a51(?∞,+∞)a83a197a145a200a169
integraldisplay x
0
cost2 dt =
∞summationdisplay
n=0
(?1)nx4n+1
(4n+ 1)(2n)! a167a217a194a241a140a187a143R = ∞.
15,a166a101a15a63a234a27a218a181
(1)
∞summationdisplay
n=1
xn
n!
(2)
∞summationdisplay
n=1
(?1)n+1 x
n+1
n(n+ 1)
(3)
∞summationdisplay
n=1
n2xn?1
(4)
∞summationdisplay
n=0
(2n+ 1)x2n
n!
a41a181
(1) a207ex =
∞summationdisplay
n=0
xn
n! (?∞<x< +∞)a167a75
∞summationdisplay
n=1
xn
n! =
∞summationdisplay
n=0
xn
n!?1 = e
x?1(?∞<x<∞)
(2)
∞summationdisplay
n=1
(?1)n+1 x
n+1
n(n+ 1) =
∞summationdisplay
n=1
(?1)n+1
n x
n+1 +
∞summationdisplay
n=1
(?1)n+1
n+ 1 x
n+1
a207ln(1 +x) =
∞summationdisplay
n=1
(?1)n+1
n x
n (?1 <xlessorequalslant 1)
a75
∞summationdisplay
n=1
(?1)n+1 x
n+1
n(n+ 1) = x
∞summationdisplay
n=1
(?1)n+1
n x
n +
∞summationdisplay
n=2
(?1)n+1
n x
n = xln(1 +x) +
∞summationdisplay
n=1
(?1)n+1
n x
n?x =
(1 +x)ln(1 +x)?x(?1 <xlessorequalslant 1)
x =?1a158a167f(x) =
∞summationdisplay
n=1
1
n(n+ 1) = limN→∞
Nsummationdisplay
n=1
parenleftBigg
1
n?
1
n+ 1
parenrightBigg
= 1
(3)
∞summationdisplay
n=1
n2xn?1 =
∞summationdisplay
n=0
(n+ 1)2xn,an = (n+ 1)2
a75 lim
n→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 1a167a117a180a217a194a241a140a187a143R = 1
a8|x| = 1a158a167a100a117(n+ 1)2 → +∞a167a75a63a234a117a209a167a117a180a63a234a27a194a241a141a143(?1,1)
192
a8x∈ (?1,1)a158a167a45f(x) =
∞summationdisplay
n=1
n2xn?1,|x|< 1
a100a53a1592a167a26
∞summationdisplay
n=1
n2xn?1a51(?1,1)a140a197a145a200a169a167
integraldisplay x
0
f(x)dx =
∞summationdisplay
n=1
nxna167a133a217a194a241a140a187a216a67a167a69a1431.
a113a100a53a1592a167a26
∞summationdisplay
n=1
nxna51(?1,1)a254a140a197a145a200a169
integraldisplay x
0
parenleftbiggintegraldisplay x
0
f(x)dx
parenrightbigg
dx =
∞summationdisplay
n=1
integraldisplay x
0
nxndx =
∞summationdisplay
n=1
n
n+ 1 x
n+1 = x2
∞summationdisplay
n=0
xn +
∞summationdisplay
n=1
parenleftBigg
x
n
n
parenrightBigg
+x =
x2
1?x + ln(1?x) +x,|x|< 1
a75
integraldisplay x
0
f(x)dx =
parenleftBigg
x2
1?x + ln(1?x) +x
parenrightBiggprime
= x(1?x)2
a117a180f(x) =
parenleftBigg
x
(1?x)2
parenrightBiggprime
= 1 +x(1?x)3,|x|< 1
(4)
∞summationdisplay
n=0
(2n+ 1)x2n
n! = 1 +
∞summationdisplay
n=1
(2n+ 1)x2n
n! = 1 +
∞summationdisplay
n=1
2x2
(n?1)! x
2(n?1) +
∞summationdisplay
n=1
x2n
n!
a207ex2 = 1 +
∞summationdisplay
n=1
x2n
n! =
∞summationdisplay
n=1
x2(n?1)
(n?1)!
a75
∞summationdisplay
n=0
(2n+ 1)x2n
n! = (2x
2 + 1)ex2,(?∞<x< +∞)
193
§3,a37a67a189a110
1,a51a52a171a109[?1,1]a254a94a203a12a100a34a245a145a170B4(x)a37a67a188a234f(x) = x+|x|2 a167a138a209a188a234y = x+|x|2 a218y = B4(x)a27a227
a47.
a41a181a45x =?1 + 2ya167a75a80 lessorequalslantylessorequalslant 1a158a167?1 lessorequalslantxlessorequalslant 1a167a100a158y = x+ 12,1?y = 1?x2,f(x) = f(?1 + 2y)
a75f(x)a51[?1.1]a254a94a203a12a100a34a245a145a170a143Bn(x) =
nsummationdisplay
k=0
f
parenleftBigg
1 + 2· kn
parenrightBigg
Ckn(x+ 1)
k(1?x)n?k
2n
a75B4(x) =
4summationdisplay
k=0
f
parenleftBigg
1 + k2
parenrightBigg
Ck4 (x+ 1)
k(1?x)4?k
24
a113f(x)a8?1 lessorequalslantxlessorequalslant 0a158a167f(x) = 0a167
a75B4(x) = f
parenleftBigg
1
2
parenrightBigg
C34 (x+ 1)
3(1?x)
24 +f(1)C
4
4
(x+ 1)k
24 =
1
8(1?x)(x+ 1)
3 + 1
16(1 +x)
4.
a45
a54
a0
a0a0
-1 10
1
x
y
2,a23f(x)a180[a,b]a254a27a235a89a188a234a167a121a178a127a51a107a110a88a234a27a245a145a170P(x)a167a166a26 max
x∈[a,b]
|f(x)?P(x)|<ε.a217a165εa180a253a107a137
a189a27a63a191a20a234.
a121a178a181a207f(x)a180[a,b]a254a27a235a89a188a234
a75a100a37a67a189a110a167a26a233a63a191a137a189a27ε> 0a167a189a127a51a245a145a170Q(x)a167a166a26||f(x)?Q(x)|| = max
x∈[a,b]
|f(x)?Q(x)|< ε2
a217a165Q(x) = a0 +a1x+···+anxn(a0,a1,···,ana254a143a162a234)
a23C = max(|a|,|b|)a167a100a162a234a27a200a151a53a167a26a55a127a51a107a110a234bia167a166a26|bi?ai|< ε4(n+ 1)2Ci(i = 0,1,···,n)
a191a23P(x) = b0 +b1x+···+bnxn
a75|P(x)?Q(x)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
i=0
(bi?ai)xi
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
i=1
|bi?ai||x|i <
nsummationdisplay
i=0
ε
4(n+ 1)2Ci C
i < ε
2
a117a180||P(x)?Q(x)|| = max
x∈[a,b]
|P(x)?Q(x)|< ε2
a108a13||f(x)?P(x)||lessorequalslant||f(x)?Q(x)||+||Q(x)?P(x)||<ε
a61a127a51a107a110a88a234a27a245a145a170P(x)a167a166a26 max
x∈[a,b]
|f(x)?P(x)|<ε
194
a49a155a19a217 a76a112a68a63a234a218a76a112a68a67a134
§1,a76a112a68a63a234
1,a121a178a181
(1) 1,cosx,cos2x,···,cosnx,···
(2) sinx,sin2x,sin3x,···,sinnx,···
a180[0,pi]a254a27a20a2a88a182a21,cosx,sinx,cos2x,sin2x,···,cosnx,sinnx,···a216a180[0,pi]a254a27a20a2a88.
a121a178a181
(1) a207
integraldisplay pi
0
1·coskxdx = 0 (k = 1,2,···),
integraldisplay pi
0
coskx·coslxdx =
0,k negationslash= l,k,l = 1,2,···
pi
2,k = l = 1,2,···integraldisplay pi
0
12 dx = pi
a751,cosx,cos2x,···,cosnx,···a180[0,pi]a254a27a20a2a88
(2) a207
integraldisplay pi
0
sinkxsinlxdx =
0,k negationslash= l,k,l = 1,2,···
pi
2,k = l = 1,2,···
a75sinx,sin2x,sin3x,···,sinnx,···a180[0,pi]a254a27a20a2a88
a113
integraldisplay pi
0
1·sinxdx = 2 negationslash= 0a167a751,cosx,sinx,cos2x,sin2x,···,cosnx,sinnx,···a216a180[0,pi]a254a27a20a2a88.
2,a121a178a181sinx,sin3x,···,sin(2n+ 1)x,···a180
bracketleftBigg
0,pi2
bracketrightBigg
a254a27a20a2a88a167a21a209a167a27a73a79a20a2a88
parenleftBigg
a61a216a61a20a2a167a13a133a122a135a188a234a27a178a144a51
bracketleftBigg
0,pi2
bracketrightBigg
a254a27a200a169a1431
parenrightBigg
a167a191a19a209sin pix2l,sin 3pix2l,···,sin (2n+ 1)pix2l,···a180[0,l]a254
a27a20a2a88.
a121a178a181a207
integraldisplay pi
2
0
[sin(2k+ 1)xsin(2l+ 1)x]dx =
0,k negationslash= l,k,l = 1,2,···
pi
4,k = l = 1,2,···
a75sinx,sin3x,···,sin(2n+ 1)x,···a180
bracketleftBigg
0,pi2
bracketrightBigg
a254a27a20a2a88
a113a100
integraldisplay pi
2
0
bracketleftBigg
sin(2k+ 1)x
a
bracketrightBigg2
dx = pi4a2 = 1(k = 1,2,···)a167a26a =
√pi
2
a75a51
bracketleftBigg
0,pi2
bracketrightBigg
a254a167a27a73a79a20a2a88a1432sinx√pi,2sin3x√pi,···,2sin(2n+ 1)x√pi,···
a113
integraldisplay l
0
sin (2k+ 1)pix2l sin sin(2m+ 1)pix2l dx =
0,k negationslash= m,k,m = 1,2,···
l
2 negationslash= 0,k = m = 1,2,···
a75sin pix2l,sin 3pix2l,···,sin (2n+ 1)pix2l,···a180[0,l]a254a27a20a2a88.
3,a23f(t)a180a177a207a143Ta27a144a197a167a167a51
bracketleftBigg
T2,T2
bracketrightBigg
a254a27a188a234a76a171a170a143
f(t) =
E,a80 lessorequalslantt< T2 a158
0,a8? T2 lessorequalslantt< 0a158
a242a249a135a144a197a208a109a164a76a112a68a63a234.
a41a181a207ω = T2,f(t) =
E,a80 lessorequalslantt< T2 a158
0,a8? T2 lessorequalslantt< 0a158
195
a75ak = 2T
integraldisplay T
2
T2
f(t)coskωtdt = 2T
integraldisplay T
2
0
Ecos 2kpiT tdt = 0
a0 = 2T
integraldisplay T
2
T2
dt = 2T
integraldisplay T
2
0
Edt = E
bk = 2T
integraldisplay T
2
T2
f(t)sinkωtdt = T2
integraldisplay T
2
0
Esin 2kpiT tdt =
0,ka143a243
2E
kpi,ka143a219
a75f(x) ～ E2 + 2Epi
∞summationdisplay
k=1
1
2k?1 sin
(4k?2)pix
T =
E,0 <x< T2
0,?T2 <x< 0
E
2,x = 0,±
T
2
4,a23f(t)a180a177a207a143Ta27a140a197a18a54a197a167a167a51
bracketleftBigg
T2,T2
parenrightBigg
a254a27a188a234a76a171a170a143
f(t) =
Umsinωt,a80 lessorequalslantt< T2a158
0,a8? T2 lessorequalslantt< 0a158
a114a249a140a197a18a54a197a208a109a164a76a112a68a63a234.
a41a181a207ωprime = 2piT,f(t) =
Umsinωt,a80 lessorequalslantt< T2a158
0,a8? T2 lessorequalslantt< 0a158
a75a0 = 2T
integraldisplay T
2
T2
f(t)dt = 2UmωT
parenleftBigg
1?cos Tω2
parenrightBigg
ak = 2T
integraldisplay T
2
T2
f(t)coskωprimetdt = UmωT + 2kpi
parenleftBigg
1?cos Tω+ 2kpi2
parenrightBigg
+ UmωT?2kpi
parenleftBigg
1?cos Tω?2kpi2
parenrightBigg
bk = 2T
integraldisplay T
2
T2
f(t)sinkωprimetdt = UmωT?2kpi sin Tω?2kpi2? UmωT + 2kpi sin Tω+ 2kpi2
a75f(t) ～ 2UmωT
parenleftBigg
1?cos Tω2
parenrightBigg
+
∞summationdisplay
k=1
parenleftBigg
akcos 2kpiT t+bksin 2kpiT t
parenrightBigg
=
Umsinωt,0 lessorequalslantt< T2
0,?T2 <t< 0
Um
2 sin
Tω
2,t = ±
T
2
5,a23f(t)a1772pia143a177a207a167a51[?pi,pi)a83
f(t) =
braceleftbigg t,a8?pilessorequalslantt< 0a158
0,a80 lessorequalslantt<pia158
a114f(t)a208a109a164a76a112a68a63a234.
a41a181a207a0 = 1pi
integraldisplay pi
pi
f(t)dt =?pi2
ak = 1pi
integraldisplay pi
pi
f(t)cosktdt = 1k2pi[1?(?1)k]
bk = 1pi
integraldisplay pi
pi
f(t)sinktdt = (?1)
k+1
k
a75f(t) ～?pi4 +
∞summationdisplay
k=1
(?1)k+1
k sinkt+
1
pi
∞summationdisplay
k=1
1?(?1)k
k2 coskt =
pi4 +
∞summationdisplay
k=1
(?1)k+1
k sinkt+
2
pi
∞summationdisplay
k=0
1
(2k+ 1)2 cos(2k+ 1)t =
t,?pi<t< 0
0,0 lessorequalslantt<pi
pi2,t = ±pi
6,a23f(t)a180a177a207a1432pia33a112a143ha27a231a184a47a197a167a167a51[0,2pi)a254a27a188a234a76a171a170a143f(t) = h2pi ta167a242a249a135a231a184a47a197a208a109a164a76
a112a68a63a234.
196
a41a181a207a0 = 1pi
integraldisplay 2pi
0
f(t)dt = h
ak = 1pi
integraldisplay 2pi
0
f(t)cosktdt = 0
bk = 1pi
integraldisplay 2pi
0
f(t)sinktdt =? hkpi
a75f(t) ～ h2 +
∞summationdisplay
k=1
(akcoskt+bksinkt) = h2? hpi
∞summationdisplay
k=1
sinkt
k =
h
2pi t,0 <t< 2pih
2,t = 0,2pi
7,a242a176a221a143τa33a112a143ha33a177a207a143Ta27a221a47a197a208a109a164a123a117a63a234.
a41a181a51a152a135a177a207
bracketleftBigg
T2,T2
bracketrightBigg
a83a221a47a197a188a234a76a136a170a143f(t) =
0,?T2 lessorequalslantt<?τ2
h,?τ2 lessorequalslanttlessorequalslant τ2
0,τ2 <tlessorequalslant T2
a75a0 = 2T
integraldisplay T
2
T2
f(t)dt = 2hT τ
ak = 2T
integraldisplay T
2
T2
f(t)cos 2kpiT tdt = 2hkpi sin kτT pi
bk = 2T
integraldisplay T
2
T2
f(t)sin 2kpiT tdt = 0
a117a180f(t) ～ hT τ +
∞summationdisplay
k=1
2h
kpi sin
kpi
T τ cos
2kpi
T t
8,a21a209a88a22712-5a164a171a27a177a207a143Ta27a110a14a197a51
bracketleftBigg
0,T2
parenrightBigg
a83a27a188a234a76a171a170a167a191a242a167a208a109a164a20a117a63a234.
a45
a54
a64
a64
a64
a64
a64a64a0a0
a0
a0
a0a0a64a64
a64
a64
a64a64
0 T4 T2
T4
T2
y
x
a41a181a88a227a164a171a27a177a207a143Ta27a110a14a197a51
bracketleftBigg
0,T2
parenrightBigg
a27a188a234a76a136a170a143f(t) =
4E
T t,0 lessorequalslantt<
T
4
4E
T
parenleftBigg
T
2?t
parenrightBigg
,T4 lessorequalslantt< T2
a107a114f(t)a242a255a164
bracketleftBigg
T2,T2
bracketrightBigg
a254a27a188a234a167a50a226a75a191a167a132a55a76a114a167a242a255a164a219a188a234a167a117a180a0 = ak = 0
bk = 4T
integraldisplay T
2
0
f(t)sin 2kpiT tdt = 8Ek2pi2 sin k2 pi =
0,ka143a243
(?1)k?12 ·8E
k2pi2,ka143a219
a75f(t) ～ 8Epi2
∞summationdisplay
n=1
(?1)n+1
(2n?1)2 sin
2(2n?1)pi
T t
9,a51a171a109(0,2pi)a165a208a109f(x) = pi?x2 a164a76a112a68a63a234.
a41a181a207a0 = 1pi
integraldisplay 2pi
0
pi?x
2 dx = 0
ak = 1pi
integraldisplay 2pi
0
pi?x
2 coskxdx = 0
bk = 1pi
integraldisplay 2pi
0
pi?x
2 sinkxdx =
1
k
197
a75f(x) ～
∞summationdisplay
k=1
1
k sinkx =
pi?x
2 (0 <x< 2pi)
10,a51a171a109(?pi,pi)a165a208a109f(x) = pi2?x2a164a76a112a68a63a234.
a41a181a207a51(?pi,pi)a254a167f(x) = pi2?x2a143a243a188a234a167a75bk = 0
a113a0 = 2pi
integraldisplay pi
0
(pi2?x2)dx = 43 pi2
ak = 2pi
integraldisplay pi
0
(pi2?x2)coskxdx = (?1)k+1 4k2
a75f(x) ～ 23 pi2 + 4
∞summationdisplay
k=1
(?1)k+1
k2 coskx = pi
2?x2 (?pi<x<pi)
11,a242f(x) = sgn(cosx)a208a109a164a76a112a68a63a234.
a41a181a207f(x+ 2pi) = sgn[cos(x+ 2pi)] = sgn(cosx) = f(x)a167a75f(x)a180a1772pia143a177a207a27a177a207a188a234
a100f(?x) = f(x)a167a75f(x)a143a243a188a234a167a117a180bk = 0
a113a0 = 2pi
integraldisplay pi
0
sgn(cosx)dx = 2pi
bracketleftBiggintegraldisplay pi
2
0
dx+
integraldisplay pi
pi
2
(?1)dx
bracketrightBigg
= 0
ak =
integraldisplay pi
0
sgn(cosx)coskxdx = 4kpi sin kpi2 =
0,k = 2n
(?1)n 4(2n+ 1)pi,k = 2n+ 1 (n = 0,1,2,···)
a75f(x)a51(?∞,+∞)a254a140a208a143f(x) ～ 4pi
∞summationdisplay
n=0
(?1)n
(2n+ 1) cos(2n+ 1)x = sgn(cosx)
12,a65a8a88a219a114a171a109
parenleftBigg
0,pi2
parenrightBigg
a83a27a140a200a188a234f(x)a242a255a0a167a166a167a208a109a164a27a76a112a68a63a234a27a47a71a88a101a181
f(x) ～
∞summationdisplay
n=1
ancos(2n?1)x (?pi<x<pi)
a41a181a207a208a109a170a165a195a20a117a145a167a75f(x)a242a255a0a65a143a243a188a234
a23f(x)a242a255a20
parenleftBigg
pi
2,pi
parenrightBigg
a83a27a220a169a143?(x)
a207a208a109a170a165a243a234a145a27a88a234a2n = 0a61a2n = 2pi
bracketleftBiggintegraldisplay pi
2
0
f(x)cos2nxdx+
integraldisplay pi
pi
2
(x)cos2nxdx
bracketrightBigg
= 0
a75
integraldisplay pi
2
0
f(x)cos2nxdx+
integraldisplay pi
pi
2
(x)cos2nxdx = 0
a51a134a224a99a152a200a169a165a138a67a254a147a134a167a45x = pi?t
a75?
integraldisplay pi
2
pi
f(pi?t)cos2n(pi?t)dt+
integraldisplay pi
pi
2
(x)cos2nxdx = 0a61
integraldisplay pi
pi
2
[f(pi?x) +?(x)]cos2nxdx = 0
a135a166a254a170a164a225a167a75a55a76a8x∈
parenleftBigg
pi
2,pi
parenrightBigg
a158a34a107f(pi?x) +?(x) = 0a61?(x) =?f(pi?x)
a117a180a210a166a209a10a242a255a0a27a188a234a51
parenleftBigg
pi
2,pi
parenrightBigg
a83a27a76a136a170a143?f(pi?x)
a113a242a255a0a27a188a234a143a243a188a234a167a75a167a51
parenleftBigg
pi2,0
parenrightBigg
a27a76a136a170a143f(?x)a167a51
parenleftBigg
pi,?pi2
parenrightBigg
a27a76a136a170a143?f(pi+x)
a216a148a23a242a255a0a27a188a234a143ψ(x)a167a75ψ(x) =
f(pi+x),?pi<x<?pi2
f(?x),?pi2 <x< 0
f(x),0 <x< pi2
f(pi?x),pi2 <x<pi
13,a211a254a152a75a167a2a208a109a27a76a112a68a63a234a47a71a143a181
f(x) ～
∞summationdisplay
n=1
bnsin(2n?1)x (?pi<x<pi)
198
a41a181a207a208a109a170a165a195a123a117a145a167a75f(x)a242a255a0a65a143a219a188a234
a23f(x)a242a255a20
parenleftBigg
pi
2,pi
parenrightBigg
a83a27a220a169a143?(x)
a207a208a109a170a165a243a234a145a27a88a234b2n = 0a61b2n = 2pi
bracketleftBiggintegraldisplay pi
2
0
f(x)sin2nxdx+
integraldisplay pi
pi
2
(x)sin2nxdx
bracketrightBigg
= 0
a75
integraldisplay pi
2
0
f(x)sin2nxdx+
integraldisplay pi
pi
2
(x)sin2nxdx = 0
a51a134a224a99a152a200a169a165a138a67a254a147a134a167a45x = pi?t
a75?
integraldisplay pi
2
pi
f(pi?t)sin2n(pi?t)dt+
integraldisplay pi
pi
2
(x)sin2nxdx = 0a61
integraldisplay pi
pi
2
[?f(pi?x) +?(x)]sin2nxdx = 0
a135a166a254a170a164a225a167a75a55a76a8x∈
parenleftBigg
pi
2,pi
parenrightBigg
a158a34a107?f(pi?x) +?(x) = 0a61?(x) = f(pi?x)
a117a180a210a166a209a10a242a255a0a27a188a234a51
parenleftBigg
pi
2,pi
parenrightBigg
a83a27a76a136a170a143f(pi?x)
a113a242a255a0a27a188a234a143a219a188a234a167a75a167a51
parenleftBigg
pi2,0
parenrightBigg
a27a76a136a170a143?f(?x)a167a51
parenleftBigg
pi,?pi2
parenrightBigg
a27a76a136a170a143?f(pi+x)
a216a148a23a242a255a0a27a188a234a143ψ(x)a167a75ψ(x) =
f(pi+x),?pi<x<?pi2
f(?x),?pi2 <x< 0
f(x),0 <x< pi2
f(pi?x),pi2 <x<pi
14,a23f(x)a140a200a33a253a233a140a200a167a121a178a181
(1) a88a74a188a234f(x)a51[?pi,pi]a254a247a118f(x+pi) = f(x)a167a64a34a2m?1 = b2m?1 = 0
(2) a88a74a188a234f(x)a51[?pi,pi]a254a247a118f(x+pi) =?f(x)a167a64a34a2m = b2m = 0
a121a178a181
(1) a207f(x)a140a200a33a253a233a140a200a133a188a234f(x)a51[?pi,pi]a254a247a118f(x+pi) = f(x)
a75f(x)a51[?pi,pi]a254a140a200a33a253a233a140a200a133a177pia143a177a207
a117a180ak = 1pi
integraldisplay pi
pi
f(x)coskxdx = 1pi
bracketleftbiggintegraldisplay 0
pi
f(x)coskxdx+
integraldisplay pi
0
f(x)coskxdx
bracketrightbigg
a233a109a224a49a19a170a138a67a254a147a134a181t = x?pia167a75a217a67a1431pi
integraldisplay pi
0
f(x)coskxdx = 1pi
integraldisplay 0
pi
f(t)cosk(t+pi)dt
a117a180ak = 1pi
integraldisplay 0
pi
[1 + (?1)k]f(x)coskxdx
a108a13a167a26a2m?1 = 0(m = 1,2,···)
a211a110a167a26b2m?1 = 0(m = 1,2,···)
(2) a207f(x)a140a200a33a253a233a140a200a133a188a234f(x)a51[?pi,pi]a254a247a118f(x+pi) =?f(x)a167a75f(x+ 2pi) = f(x)
a117a180f(x)a51[?pi,pi]a254a140a200a33a253a233a140a200a133a1772pia143a177a207
a117a180ak = 1pi
integraldisplay pi
pi
f(x)coskxdx = 1pi
bracketleftbiggintegraldisplay 0
pi
f(x)coskxdx+
integraldisplay pi
0
f(x)coskxdx
bracketrightbigg
a233a109a224a49a19a170a138a67a254a147a134a181t = x?pia167a75a217a67a1431pi
integraldisplay pi
0
f(x)coskxdx =?1pi
integraldisplay 0
pi
f(t)cosk(t+pi)dt
a117a180ak = 1pi
integraldisplay 0
pi
[1 + (?1)k+1]f(x)coskxdx
a108a13a167a26a2m = 0(m = 1,2,···)
a211a110a167a26b2m = 0(m = 1,2,···)
15,a177a207a1432pia27a140a200a218a253a233a140a200a188a234f(x)a27a76a112a68a88a234a143an,bna167a79a142a181
(1) a188a234f(x+k) (ka143a126a234)a27a76a112a68a88a234an,bna182
(2) F(x) = 1pi
integraldisplay pi
pi
f(t)f(x?t)dta27a76a112a68a88a234An,Bna167a23a107a39a27a200a169a94a83a140a2a134.
199
a41a181
(1) a100a174a127a167a26a0 = 1pi
integraldisplay pi
pi
f(x)dx,an = 1pi
integraldisplay pi
pi
f(x)cosnxdx,bn = 1pi
integraldisplay pi
pi
f(x)sinnxdx
a75a138a147a134x+k = ya133f(x)a180a1772pia143a177a207a27a188a234a167a107
a0 = 1pi
integraldisplay pi
pi
f(x+k)dx = 1pi
integraldisplay pi+k
pi+k
f(y)dy = a0
an = 1pi
integraldisplay pi
pi
f(x+k)cosnxdx = 1pi
integraldisplay pi+k
pi+k
f(y)cosn(y?k)dy = ancosnk+bnsinnk
a61an = ancosnk+bnsinnk (n = 0,1,2,···)
a211a110a167a140a166a26bn = bncosnk?ansinnk
(2) a207f(x)a180a177a207a1432pia27a140a200a218a253a233a140a200a188a234
a75F(x+ 2pi) = 1pi
integraldisplay pi
pi
f(t)f(x+ 2pi?t)dt = F(x)a167a117a180F(x)a69a180a1772pia143a177a207a27a188a234
a113a0 = 1pi
integraldisplay pi
pi
f(x)dx,an = 1pi
integraldisplay pi
pi
f(x)cosnxdx,bn = 1pi
integraldisplay pi
pi
f(x)sinnxdx
a75A0 = 1pi
integraldisplay pi
pi
F(x)dx = 1pi2
integraldisplay pi
pi
f(t)dt
integraldisplay pi
pi
f(x?t)dx
a233
integraldisplay pi
pi
f(x?t)dxa138a147a134x?t = ya133f(x)a180a1772pia143a177a207a27a188a234a167a107
A0 = 1pi2
integraldisplay pi
pi
f(t)dt
integraldisplay pi?t
pi?t
f(y)dy = 1pi2
bracketleftbiggintegraldisplay pi
pi
f(t)dt
bracketrightbigg2
= a20
a211a110a167a140a166a26An = a2n?b2n
Bn = 2anbn
16,a88a74?(?x) = ψ(x)a167a175?(x)a134ψ(x)a27a76a112a68a88a234a131a109a107a159a111a39a88a186
a41a181a188a234?(x)a134ψ(x)a27a76a112a68a88a234a169a143an = 1pi
integraldisplay pi
pi
(x)cosnxdx,bn = 1pi
integraldisplay pi
pi
(x)sinnxdx
αn = 1pi
integraldisplay pi
pi
ψ(x)cosnxdx,βn = 1pi
integraldisplay pi
pi
ψ(x)sinnxdx
a233an = 1pi
integraldisplay pi
pi
(x)cosnxdxa109a224a138a67a254a147a134y =?xa167a191a242?(?x) = ψ(x)a147a92a167a26
an = 1pi
integraldisplay?pi
pi
(?y)cosn(?y)d(?y) = 1pi
integraldisplay pi
pi
ψ(x)cosnxdx = αn (n = 0,1,2,···)
a211a110a167a26bn =?βn (n = 1,2,···)
17,a88a74?(?x) =?ψ(x)a167a175?(x)a134ψ(x)a27a76a112a68a88a234a131a109a107a159a111a39a88a186
a41a181a188a234?(x)a134ψ(x)a27a76a112a68a88a234a169a143an = 1pi
integraldisplay pi
pi
(x)cosnxdx,bn = 1pi
integraldisplay pi
pi
(x)sinnxdx
αn = 1pi
integraldisplay pi
pi
ψ(x)cosnxdx,βn = 1pi
integraldisplay pi
pi
ψ(x)sinnxdx
a233an = 1pi
integraldisplay pi
pi
(x)cosnxdxa109a224a138a67a254a147a134y =?xa167a191a242?(?x) =?ψ(x)a147a92a167a26
an = 1pi
integraldisplay?pi
pi
(?y)cosn(?y)d(?y) =?1pi
integraldisplay pi
pi
ψ(x)cosnxdx =?αn (n = 0,1,2,···)
a211a110a167a26bn = βn (n = 1,2,···)
18,a23f(t)a51(?pi,pi)a254a169a227a235a89a167a8t = 0a235a89a133a107a252a253a19a234a167a121a178a8p→∞a158integraldisplay
pi
pi
f(t) cos
t
2?cospt
2sin t2 dt→
1
2
integraldisplay pi
0
[f(t)?f(?t)]cot t2 dt
a121a178a181
integraldisplay pi
pi
f(t) cos
t
2?cospt
2sin t2 dt =
integraldisplay 0
pi
f(t) cos
t
2?cospt
2sin t2 dt+
integraldisplay pi
0
f(t) cos
t
2?cospt
2sin t2 dt
a51a109a224a99a152a200a169a165a45t =?xa167a75
integraldisplay 0
pi
f(t) cos
t
2?cospt
2sin t2 dt =?
integraldisplay pi
0
f(?t) cos
t
2?cospt
2sin t2 dt
a147a163a6a170a167a26
integraldisplay pi
pi
f(t) cos
t
2?cospt
2sin t2 dt =?
integraldisplay pi
0
f(?t) cos
t
2?cospt
2sin t2 dt +
integraldisplay pi
0
f(t) cos
t
2?cospt
2sin t2 dt =
1
2
integraldisplay pi
0
[f(t)?f(?t)]cot t2 dt? 12
integraldisplay pi
0
f(t)?f(?t)
2sin t2 cosptdt
200
a101a121 lim
p→∞
integraldisplay pi
0
f(t)?f(?t)
2sin t2 cosptdt = 0
a207
integraldisplay pi
0
f(t)?f(?t)
2sin t2 cosptdt =
bracketleftbiggintegraldisplay δ
0
+
integraldisplay pi
δ
bracketrightbiggf(t)?f(?t)
2sin t2 cosptdt(a217a1650 <δ<pi)
a233a117
integraldisplay pi
δ
f(t)?f(?t)
2sin t2 cosptdt
a207f(t)a51(?pi,pi)a254a169a227a235a89a167 12sin t
2
a51(δ,pi)a254a235a89a167a75f(t)?f(?t)2sin t
2
a51(δ,pi)a254a169a227a235a89a207a13a140a200
a75a100a105a249a218a110a167a26 lim
p→∞
integraldisplay pi
δ
f(t)?f(?t)
2sin t2 cosptdt = 0
a233a117
integraldisplay δ
0
f(t)?f(?t)
2sin t2 cosptdt
lim
p→∞
integraldisplay δ
0
f(t)?f(?t)
2sin t2 cosptdt = limp→∞
integraldisplay δ
0
[f(t)?f(?t)]
parenleftBigg
1
2sin t2?
1
t
parenrightBigg
cosptdt+ lim
p→∞
integraldisplay δ
0
f(t)?f(?t)
t cosptdt
a207lim
t→0
parenleftBigg
1
2sin t2?
1
t
parenrightBigg
= 0a167a214a191a189a194a167t = 0a158a167a188a234 12sin t
2
1t a27a138a1430a167a75 12sin t
2
1t a180[0,δ]a254a27a235a89
a188a234
a113f(t)a143(?pi,pi)a254a27a169a227a235a89a188a234a167a75[f(t)?f(?t)]
parenleftBigg
1
sin t2?
1
t
parenrightBigg
a51[0,δ]a254a169a227a235a89a167a207a13a140a200a167a75a100a105a249
a218a110a167a26 lim
p→∞
integraldisplay δ
0
f(t)?f(?t)
2sin t2 cosptdt = 0
a207f prime(+0),f prime(?0)a127a51a167a75 lim
t→+0
f(t)?f(?t)
t = f
prime(+0) +f prime(?0)a127a51
a214a191a189a194a167t = 0a158a167a188a234f(t)?f(?t)t a138a143f prime(+0) +f prime(?0)a167a75f(t)?f(?t)t a180[0,δ]a254a27a169a227a188a234a167a207a13
a140a200a167a117a180a100a105a249a218a110a167a26 lim
p→∞
integraldisplay δ
0
f(t)?f(?t)
t cosptdt = 0
a110a254a140a26a167a8p→∞a158a167
integraldisplay pi
0
f(t)?f(?t)
2sin t2 cosptdt→
1
2
integraldisplay pi
0
[f(t)?f(?t)]cot t2 dt
19,a23Tn(x) = 12 +
nsummationdisplay
v=1
cosvx,T0(x) = 12,σn(x) = T0(x) +···+Tn(x)n+ 1
a121a178
(1) σn(x) = 12(n+ 1)
parenleftBigg
sin n+12 x
sin x2
parenrightBigg2
(2)
integraldisplay pi
pi
σn(x)dx = pi
a121a178a181
(1) a2072sin x2
parenleftBigg
1
2 +
nsummationdisplay
v=1
cosvx
parenrightBigg
= sin 2n+ 12 xa167a75Tn(x) = sin
2n+1
2 x
2sin x2
a117a180σn(x) = T0(x) +···+Tn(x)n+ 1 =
1
2 +
nsummationdisplay
k=1
Tk(x)
n+ 1 =
1
n+ 1
parenleftBigg
1
2 +
nsummationdisplay
k=1
sin 2n+12 x
2sin x2
parenrightBigg
= 12(n+ 1)sin2 x
2
parenleftBigg
sin2 x2 +
nsummationdisplay
k=1
sin x2sin 2k+ 12 x
parenrightBigg
= 12(n+ 1)sin2 x
2
bracketleftBigg
1
2?
1
2 cos(n+ 1)x
bracketrightBigg
=
1
2(n+ 1)
parenleftBigg
sin 2n+12 x
2sin x2
parenrightBigg2
(2)
integraldisplay pi
pi
σn(x)dx =
integraldisplay pi
pi
1
2 +
nsummationdisplay
k=1
Tk(x)
n+ 1 dx =
1
n+ 1
integraldisplay pi
pi
bracketleftBigg
1
2 +
nsummationdisplay
k=1
parenleftBigg
1
2 +
nsummationdisplay
v=1
cosvx
parenrightBiggbracketrightBigg
dx =
1
n+ 1
bracketleftBigg
pi+
nsummationdisplay
k=1
parenleftBigg
pi+
nsummationdisplay
v=1
integraldisplay pi
pi
cosvxdx
parenrightBiggbracketrightBigg
= pi.
201
20,a23?(x)a51[a,b]a254a143a252a78a79a92a188a234a167a121a178
(1) a88a74a = 0,b< 0a167a1071pi
integraldisplay b
a
(z) sinpzz dz →?12?(?0) (p→∞)
(2) a88a74a< 0,b> 0a167a1071pi
integraldisplay b
a
(z) sinpzz dz →?(+0) +?(?0)2 (p→∞)
a121a178a181
(1) a207?(x)a51[a,b]a254a143a252a78a79a92a188a234a167a75?(?t)a51[?b,?a]a254a143a252a78a126a8a188a234
a8a = 0,b< 0a158a167?(?t)a51[0,?b](?b> 0)a254a143a252a78a79a92a188a234
a233
integraldisplay b
a
(z) sinpzz dza138a67a254a147a134z =?ta167a75
integraldisplay b
a
(z) sinpzz =?
integraldisplay?b
0
(?t) sinptt dt
a75a100a41a225a142a52a218a110a167a26 lim
p→∞
integraldisplay?b
0
(?t) sinptt dt = pi2?(?0)a61 lim
p→∞
integraldisplay b
a
(z) sinpzz dz =?pi2?(?0)
a117a1801pi
integraldisplay b
a
(z) sinpzz dz →?12?(?0) (p→∞)
(2) a207a< 0,b> 0a167?(x)a51[a,b]a254a143a252a78a79a92a188a234a167
integraldisplay b
a
(z) sinpzz dz =
integraldisplay 0
a
(z) sinpzz dz+
integraldisplay b
0
(z) sinpzz dz
a226(1)a167a26 lim
p→∞
integraldisplay a
0
(z) sinpzz dz =?pi2?(?0)a167a75 lim
p→∞
integraldisplay 0
a
(z) sinpzz dz = pi2?(?0)
a113a100a41a225a142a52a218a110a167a26 lim
p→∞
integraldisplay b
0
(z) sinpzz dt = pi2?(+0)
a75 lim
p→∞
integraldisplay b
a
(z) sinpzz dz = pi2[?(+0) +?(?0)]
a117a1801pi
integraldisplay b
a
(z) sinpzz dz →?(?0) +?(+0)2 (p→∞)
202
§2,a76a112a68a67a134
1,a23f(x)a51(?∞,+∞)a83a253a233a140a200a167a121a178hatwidef(ω)a51(?∞,+∞)a83a235a89.
a121a178a181a233?ω ∈ (?∞,+∞)a167a111a107Aprime,Aprimeprimea167a166a26ω ∈ [Aprime,Aprimeprime ]
a100a117
vextendsinglevextendsingle
vextendsinglehatwidef(ω)
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
∞
f(x)e?iωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay +∞
∞
|f(x)|dx
a0a246a194a241a133a216a185a235a254ωa167a249a76a178a200a169hatwidef(ω) =
integraldisplay +∞
∞
f(x)e?iωxdxa51[Aprime,Aprimeprime ]a254a152a151a194a241
a226a152a151a194a241a200a169a27a235a89a53a167a26hatwidef(ω)a51[Aprime,Aprimeprime ]a254a235a89a167a108a13a51a58ωa63a235a89
a100ωa27a63a191a53a167a26hatwidef(ω)a51(?∞,+∞)a83a235a89.
2,a23f(x)a51(?∞,+∞)a83a253a233a140a200a167a121a178 lim
ω→∞
hatwidef(ω) = 0.
a121a178a181a100f(x)a51(?∞,+∞)a83a253a233a140a200a167a26a233a117a63a137a27ε > 0a167a127a51A > 0a167a166a107
integraldisplay +∞
A
|f(x)|dx < ε3
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
A
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay +∞
A
|f(x)|dx< ε3
a23f(x)a51[0,A]a83a195a215a58a167a75a51[0,A]a165a2a92a169a580 = t0 <t1 <···<tm = Aa167a191a23f(x)a51[tk?1,tk]a254a27a101a40
a46a143mka167a117a180integraldisplay
A
0
f(x)sinωxdx =
msummationdisplay
k=1
integraldisplay tk
tk?1
f(x)sinωxdx =
msummationdisplay
k=1
integraldisplay tk
tk?1
[f(x)?mk]sinωxdx+
msummationdisplay
k=1
mk
integraldisplay tk
tk?1
sinωxdx
a108a13
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
0
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
msummationdisplay
k=1
ωk?tk +
msummationdisplay
k=1
|mk||cosntk?1?cosntk|n lessorequalslant
msummationdisplay
k=1
ωk?tk + 2ω
msummationdisplay
k=1
|mk|
a217a165ωka143f(x)a51a171a109[tk?1,tk]a254a27a8a204a167?tk = tk?tk?1
a100a117f(x)a51[0,A]a254a140a200a167a25a140a18a44a152a169a123a167a166a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
msummationdisplay
k=1
ωk?tk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<
ε
3
a233a117a249a24a27a189a27a169a123a167
msummationdisplay
k=1
|mk|a143a152a189a138a167a207a13a127a51δ> 0a167a166a8ω>δa158a167a240a1072ω
msummationdisplay
k=1
|mk|< ε3
a117a180a233a254a227a164a192a18a27δa167a8ω>δa158vextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay +∞
0
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
0
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsingle+
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
A
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslantεa61 lim
ω→∞
integraldisplay +∞
0
f(x)sinωxdx = 0
a217a103a167a23f(x)a51a171a109[0,A]a165a107a215a58a167a143a123a66a229a132a167a216a148a23a144a107a152a135a215a58a133a1430
a117a180a233a63a137a27ε> 0a167a127a51η> 0a167a166a107
integraldisplay η
0
|f(x)|dx< ε3
a113f(x)a51[η,A]a254a195a97a58a167a25a65a94a254a227a40a74a140a26a127a51δa167a166a8ω>δa158a167a240a107
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
η
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsingle< ε
3
a117a180a8ω>δa158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
0
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay η
0
|f(x)|dx+
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
η
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsingle+
integraldisplay +∞
A
|f(x)|dx<ε
a61 lim
ω→∞
integraldisplay +∞
0
f(x)sinωxdx = 0
a211a123a167a26a8f(x)a51(?∞,+∞)a83a253a233a140a200a158a167a254a107 lim
ω→∞
integraldisplay +∞
∞
f(x)sinωxdx = 0
a211a123a140a121a26a8f(x)a51(?∞,+∞)a83a253a233a140a200a158a167 lim
ω→∞
integraldisplay +∞
∞
f(x)cosωxdx = 0
a117a180 lim
ω→∞
hatwidef(x) = 0.
3,a166a101a15a188a234a27a76a112a68a67a134a181
(1) f(x) =
Esinω0x,|x|< piω
0
0,|x|greaterorequalslant piω
0
(2) f(x) =
0,?∞<xlessorequalslant?pi2
2h
τ x+h,?
τ
2 <x< 0
2hτ x+h,0 lessorequalslantx< τ2
0,τ2 lessorequalslantx< +∞
203
a41a181
(1) hatwidef(ω) =
integraldisplay ∞
∞
f(x)e?iωxdx =
integraldisplay pi
ω0
piω0
Esinω0xe?iωxdx = E
integraldisplay pi
ω0
piω0
sinω0x(cosωx?isinωx)dx =
2Ei
integraldisplay pi
ω0
0
sinω0xsinωxdx = iE
integraldisplay pi
ω0
0
[cos(ω0+ω)x?cos(ω?ω0)x]dx = iE
sin(ω0 +ω)xω
0 +ω
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
pi
ω0
0
sin(ω?ω0)xω?ω
0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
pi
ω0
0
=
2Eω0i
ω2?ω20 sin
ω
ω0 pi(ω negationslash= ±ω0)
a207hatwidef(ω)a143(?∞,+∞)a83a27a235a89a188a234a167a75hatwidef(?ω0) = lim
ω→?ω0
hatwidef(ω) = ±iEpi
ω0,
(2) hatwidef(ω) =
integraldisplay ∞
∞
f(x)e?iωxdx =
integraldisplay 0
τ2
parenleftBigg
2h
τ x+h
parenrightBigg
e?iωxdx+
integraldisplay τ
2
0
parenleftBigg
2hτ x+h
parenrightBigg
e?iωxdx =
2h
τ
bracketleftBiggintegraldisplay
0
τ2
xe?iωxdx?
integraldisplay τ
2
0
xe?iωxdx
bracketrightBigg
+ 2hω sin ωτ2 = 4hτω2? 4hτω2 cos ωτ2 (ω negationslash= 0)
a207hatwidef(ω)a143(?∞,+∞)a83a27a235a89a188a234a167a75hatwidef(0) = lim
ω→0
hatwidef(ω) = hτ
2,
204
a49a111a159 a245a67a254a135a200a169a198
a49a152a220a169 a245a3a188a234a27a52a129a216
a49a155a110a217 a245a3a188a234a27a52a129a134a235a89
§1,a178a161a58a56
1,a121a178(xn,yn) → (x0,y0)a27a191a135a94a135a180a181xn →x0,yn →y0(n→∞)
a121a178a181?
a207 lim
n→∞
Mn = M0a167a75a233?ε> 0,?N ∈Z+a167a8n>Na158a167a107r(Mn,M0) <ε
a61radicalbig(xn?x0)2 + (yn?y0)2 <ε
a117a180a152a189a107|xn?x0|lessorequalslantr(Mn,M0) <ε,|yn?y0|lessorequalslantr(Mn,M0) <εa61xn →x0,yn →y0(n→∞)
a207(|xn?x0|+|yn?y0|)2 greaterorequalslant|xn?x0|2 +|yn?y0|2a610 lessorequalslantradicalbig|xn?x0|2 +|yn?y0|2 lessorequalslant|xn?x0|+|yn?y0|
a113xn →x0,yn →y0(n→∞)a167a75radicalbig|xn?x0|2 +|yn?y0|2 → 0(n→∞)a61(xn,yn) → (x0,y0)(n→∞)
2,a121a178a181a101a178a161a254a27a58a15{Mn}a194a241a167a75a167a144a107a152a135a52a129.
a121a178a181a23 lim
n→∞
Mn = M0a167a98a23a113a107 lim
n→∞
Mn = M0 prime
a100a189a194a167a233?ε> 0,?N ∈Z+a167a8n>Na158a167a107r(Mn,M0) < ε2,r(Mn,M0 prime) < ε2
a100a110a14a216a31a170a167a107r(M0,M0 prime) lessorequalslantr(Mn,M0) +r(Mn,M0 prime) <ε
a113M0,M0 primea143a27a189a27a252a58a167a100εa27a63a191a53a167a26r(M0,M0 prime) = 0a61M0 = M0 prime.
3,a121a178a181a101Mn →M0(n→∞)a167a64a111a167a27a63a219a152a135a102a15Mnk →M0.
a121a178a181a207Mn →M0(n→∞)a167a75a233?ε> 0,?N ∈Z+a167a8n>Na158a167a107r(Mn,M0) <ε
a56a18K = Na167a75a233a152a131k>Ka167a107nk >nK = nN greaterorequalslantNa167a103a44a107r(Mnk,M0) <εa61Mnk →M0(k →∞).
4,a166a101a15a58a56Ea27a83a58a167a9a58a167a62a46a58a181
(1) Ea100a247a118y<x2a27a58a164a124a164a182
(2) Ea100a247a1181 lessorequalslantx2 + y
2
4 < 4a27a58a164a124a164a182
(3) Ea100a247a1180 <x2 +y2 < 1a27a58a164a124a164a182
(4) Ea100a164a107a249a24a27a58(x,y)a164a124a164a167a217a165xa218ya209a180a107a110a234.
a41a181
(1) a133a247a118y<x2a27a58(x,y)a180Ea27a83a58a182a133a247a118y>x2a27a58(x,y)a180Ea27a9a58a182a133a247a118y = x2a27a58(x,y)a180Ea27
a62a46a58.
(2) a133a247a1181 <x2 + y
2
4 < 4a27a58(x,y)a180Ea27a83a58a182a133a247a118x
2 + y
2
4 < 1a189x
2 + y
2
4 > 4a27a58(x,y)a180Ea27a9a58a182
a133a247a118x2 + y
2
4 = 1a189x
2 + y
2
4 = 4a27a58(x,y)a180Ea27a62a46a58.
(3) a133a247a1180 < x2 + y2 < 1a27a58(x,y)a180Ea27a83a58a182a133a247a118x2 + y2 > 1a27a58(x,y)a180Ea27a9a58a182a6a58θa57a247
a118x2 +y2 = 1a27a58(x,y)a180Ea27a62a46a58.
(4) a100a107a110a234a57a195a110a234a27a200a151a53a167a26a178a161a254a164a107a58(x,y)a209a180Ea27a62a46a58.
5,a121a178a181a101M0a180a178a161a58a56Ea27a224a58a167a75a51Ea165a127a51a58a15Mn →M0(n→∞).
a121a178a181a174a127M0a180a178a161a58a56Ea27a224a58a167a18δn = 1na167a51O(M0,δ1)a165a189a127a51Ea27a58M1 negationslash= M0a182a51O(M0,δ2)a165a189a127
a51Ea27a58M2,M2 negationslash= Mi(inegationslash= 0,1)
a88a100a63a49a101a22a167a26a20a58a15{Mn}(Mn negationslash= Mi)(i = 0,1,···,n?1)a133r(M0,Mn) < 1n
a117a180a8n→∞a158a167r(M0,Mn) → 0a61Mn →M0(n→∞).
6,a121a178a178a161a58a15a27a194a241a6a110.
a121a178a181?
a23Mn →M0(n→∞)a167a75a233?ε> 0,?N ∈Z+a167a8n,m>Na158a167a107r(Mn,M0) < ε2,r(Mm,M0) < ε2
205
a100a229a108a27a110a14a216a31a170a167a26r(Mm,Mn) lessorequalslantr(Mn,M0) +r(Mm,M0) <ε
a23a58a15{Mn}a247a118a233?ε> 0,?N ∈Z+a167a8n,m>Na158a167a107r(Mn,Mm) <ε
a242{Mn}a169a79a221a75a20a252a138a139a73a182a254a167a26a234a15{xn},{yn}
a207|xm?xn|<r(Mm,Mn) <ε,|ym?yn|<r(Mm,Mn) <ε
a100R1a254a27a133a220a194a241a6a110a167a26{xn},{yn}a209a194a241
a23xn →x0,yn →y0(n→∞)a167a75 lim
n→∞
Mn = M0(M0(x0,y0))a61{Mn}a194a241.
7,a94a178a161a254a27a107a129a67a88a189a110a121a178a159a16a100a65a46a100a189a110.
a121a178a181
(1) a101{Mn(xn,yn)}a180a107a46a107a129a58a56a167a189a110a164a225a182
(2) a101{Mn(xn,yn)}a180a107a46a195a161a58a56a167a2265a167a144a73a121E = {Mn(xn,yn)vextendsinglevextendsinglen = 1,2,···}a165a150a8a107a152a135a224a58.
a135a121.a23Ea118a107a224a58.
a100a117a lessorequalslant xn lessorequalslant b,c lessorequalslant yn lessorequalslant d(n = 1,2,···)a167a13a221a47a141R = {(x,y)vextendsinglevextendsinglea lessorequalslant x lessorequalslant b,c lessorequalslant d}a180a107a46a52a171a141
a133E?R
a233?M(x,y) ∈Ra167a209a216a180Ea27a224a58a167a207a13a127a51δMa167a166a26O(M,δM)a150a245a107Ea165a107a129a135a58a167
{O(M,δM)vextendsinglevextendsingleM ∈R}a67a88R
a226a107a129a67a88a189a110a167a127a51a107a129a135a109a56O(M1,δM1),···,O(Mk,δMk)a211a24a67a88Ra167a217a165a122a135O(Mi,δMi)(i =
1,2,···,k)a165a150a245a107a107a129a135Ea165a27a58
a117a180
kuniontext
i=1
O(Mi,δMi)a150a245a185Ea165a107a129a135a58
a2a100a117
kuniontext
i=1
O(Mi,δMi)?R?Ea167a117a180a103a241.
206
§2,a245a3a188a234a27a52a129a218a235a89a53
1,a40a189a191a177a209a101a15a188a234a131a189a194a141a181
(1) u = √x?√1?y
(2) u = √x?y+ 1
(3) u = ln(?x?y)
(4) uradicalbigsin(x2 +y2)
(5) uradicalbigR2?x2?y2?z2 +radicalbigx2 +y2 +z2?r2
a41a181
(1) a189a194a141a143xgreaterorequalslant 0a133ylessorequalslant 1
(2) a189a194a141a143a247a118a216a31a170ylessorequalslantx+ 1a27a58a56
(3) a189a194a141a143a140a178a161x+y< 0
(4) a189a194a141a143a247a118a216a31a1702kpilessorequalslantx2 +y2 lessorequalslant (2k+ 1)pi(k = 0,1,2,···)a27a58a56
(5) a189a194a141a143a247a118a216a31a170r2 lessorequalslantx2 +y2 +z2 lessorequalslantR2a27a58a56
2,a166a101a15a52a129a181
(1) limx→0
y→0
x2 +y2
|x|+|y|
(2) limx→0
y→0
x2 +y2radicalbig
x2 +y2 + 1?1
(3) limx→0
y→0
1 +x2 +y2
x2 +y2
(4) limx→0
y→0
sin(x3 +y3)
x2 +y2
(5) limx→+∞
y→+∞
(x2 +y2)e?(x+y)
(6) limx→1
y→0
ln(x+ey)radicalbig
x2 +y2
a41a181
(1) a2070 lessorequalslant x
2 +y2
|x|+|y|lessorequalslant
(|x|+|y|)2
|x|+|y| = |x|+|y|a133limx→0y→0(|x|+|y|) = 0a167a75limx→0y→0
x2 +y2
|x|+|y| = 0
(2) a207 lim
t→+0
t√
t+ 1?1 = limt→+0(
√t+ 1 + 1) = 2a167a75lim
x→0
y→0
x2 +y2radicalbig
x2 +y2 + 1?1 = 2
(3) a207 lim
t→+0
1 +t
t = +∞a167a75limx→0y→0
1 +x2 +y2
x2 +y2 = +∞
(4) a2070 lessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(x3 +y3)
x2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
|x3 +y3|
x2 +y2 lessorequalslant
|x|3 +|y|3
x2 +y2 =
|x|3
x2 +y2 +
|y|3
x2 +y2 lessorequalslant|x|+|y|a133limx→0y→0(|x|+|y|) = 0
a75limx→0
y→0
sin(x3 +y3)
x2 +y2 = 0
(5) a207 lim
t→+∞
t
et = 0,limt→+∞
t2
et = 0
a75 limx→+∞
y→+∞
(x2 +y2)e?(x+y) = limx→+∞
y→+∞
bracketleftBigg
(x+y)2
e?(x+y)?2
x
ex ·
y
ey
bracketrightBigg
= 0
(6) limx→1
y→0
ln(x+ey)radicalbig
x2 +y2 = ln2
207
3,a193a121a101limy→a
x→b
f(x,y) = Aa127a51a167a13a8xa18a63a219a134aa25a67a131a138a158a167a52a129lim
y→b
f(x,y) =?(x)a127a51a167a75a19a103a52a129a127a51a167
a133a31a117Aa181
lim
x→a
lim
y→b
f(x,y) = limy→a
x→b
f(x,y) = A
.
a121a178a181a207a19a173a52a129a127a51a167a75a233?ε > 0,?δ > 0a167a8|x?a| < δ,|y?b| < δa133(x?a)2 + (y?b)2 negationslash= 0a158a167a240
a107|f(x,y)?A|<ε
a121a510 <|x?a|<δa165a27a189xa167a13a51a254a170a165a45y →ba167a61a26|?(x)?A|lessorequalslantεa167a249a210a121a178a10lim
x→a
(x) = A
a117a180lim
x→a
lim
y→b
f(x,y) = lim
x→a
(x) = A = limy→a
x→b
f(x,y)
4,(1) a193a222a209a252a135a19a103a52a129a216a131a31a27a126a102a182
(2) a193a222a209a144a107a152a135a19a103a52a129a127a51a27a126a102a182
(3) a193a222a209a19a173a52a129a127a51a167a2a19a103a52a129a216a28a127a51a27a126a102.
a41a181
(1) a126a181f(x,y) =
x?y
x+y,x+y negationslash= 0
0,x+y = 0
a51a58(0,0)a27a19a103a52a129
lim
x→0
lim
y→0
f(x,y) = lim
x→0
x
x = 1,limy→0 limx→0f(x,y) = limy→0
y
y =?1
a75lim
x→0
lim
y→0
f(x,y) negationslash= lim
y→0
lim
x→0
f(x,y).
(2) a126a181f(x,y) = xsin
1
x +y
x+y a51a58(0,0)a27a19a103a52a129
lim
y→0
lim
x→0
f(x,y) = lim
y→0
y
y = 1
a2lim
x→0
lim
y→0
f(x,y) = lim
x→0
xsin 1x
x = limx→0sin
1
x a216a127a51.
(3) a126a181f(x,y) =
xsin 1y,y negationslash= 0
0,y = 0
a51a58(0,0)a27a19a103a52a129a218a19a173a52a129
a2070 lessorequalslant|f(x,y)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglexsin
1
y
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant|x|a167a75limx→0y→0 f(x,y) = 0a61a217a19a173a52a129a127a51
lim
y→0
lim
x→0
f(x,y) = 0a167a13a8y → 0a158a167xsin 1y a52a129a216a127a51a167a61lim
x→0
lim
y→0
f(x,y)a216a127a51.
5,a63a216a101a15a188a234a51a58(0,0)a27a19a103a52a129a218a19a173a52a129a181
(1) f(x,y) = x
2y2
x2y2 + (x?y)2
(2) f(x,y) = (x+y)·sin 1x ·sin 1y
a41a181
(1) lim
x→0
lim
y→0
f(x,y) = 0,lim
y→0
lim
x→0
f(x,y) = 0
a101a85y = kx→ 0a27a144a149a18a52a129a167a75a107 lim
y=kx
x→0
f(x,y) = lim
x→0
x2k2
x2k2 + (1?k)2
a65a79a27a167a169a79a18k negationslash= 1a57k = 1a167a66a26a20a216a211a27a52a1290a571a167a207a100limx→0
y→0
f(x,y)a216a127a51.
(2) a2070 lessorequalslant|f(x,y)|lessorequalslant|x+y|lessorequalslant|x|+|y|a167a75limx→0
y→0
f(x,y) = 0a61a217a19a173a52a129a127a51
a113lim
y→0
(x + y) · sin 1x · sin 1y a216a127a51
parenleftBigg
a8xnegationslash= 1kpi
parenrightBigg
(k = ±1,±2,···)a167lim
x→0
(x + y) · sin 1x · sin 1y a216a127
a51
parenleftBigg
a8y negationslash= 1kpi
parenrightBigg
(k = ±1,±2,···)
a61lim
x→0
lim
y→0
f(x,y)a57lim
y→0
lim
x→0
f(x,y)a209a216a127a51.
208
6,a63a216a101a15a188a234a27a235a89a137a140a181
(1) u = 1radicalbigx2 +y2
(2) u = ln(1?x2?y2)
(3) u = 1sinxsiny
(4) u = ln 1(x?1)a + (y?b)2 + (z?c)2
a41a181
(1) a188a234u = 1radicalbigx2 +y2 a51a58(0,0)a195a189a194a167a25a6a58(0,0)a143a100a188a234a27a216a235a89a58a167a216a100a58a9a254a235a89a182
(2) a252a160a11a83a27a58a167a61a247a118x2 +y2 < 1a27a136a58a143a188a234u = ln(1?x2?y2)a27a235a89a58a182
(3) a235a89a137a140a143xnegationslash= mpi,y negationslash= npi(m,n = 0,±1,±2,···).
(4) a216a58(a.b.c)a9a254a235a89.
7,a121a178a188a234
f(x,y) =
2xy
x2 +y2,x
2 +y2 negationslash= 0
0,x2 +y2 = 0
a169a79a233a117a122a152a67a254xa218ya180a235a89a27a167a2a154a39a117a19a67a254a27a235a89a188a234.
a121a178a181a107a27a189y = anegationslash= 0a167a75a26xa27a188a234g(x) = f(x,a) = 2axx2 +a2 (?∞<x< +∞)
a167a180a63a63a107a189a194a27a107a110a188a234
a113a8y = 0a158a167f(x,0) ≡ 0a167a167a119a44a180a235a89a27
a117a180a8a67a234ya27a189a158a167a188a234f(x,y)a233a117a67a234xa180a235a89a27
a211a110a140a121a167a8a67a234xa27a189a158a167a188a234f(x,y)a233a117a67a234ya180a235a89a27
a138a143a19a3a188a234a167f(x,y)a143a51a216a58(0,0)a9a27a136a58a254a235a89a167a2a51a58(0,0)a216a235a89
a8a196a58P(x,y)a247a19a130y = mxa170a117a6a58a158a167a107 limy=mx
x→0
f(x,y) = lim
x→0
2mx2
(1 +m2)x2 =
2m
1 +m2
a18a216a211a27ma167a75a52a129a138a216a211a167a96a178a217a19a173a52a129a216a127a51a167a117a180limx→0
y→0
f(x,y) negationslash= f(0,0)
a75a217a39a117a19a67a254a27a188a234a51(0,0)a58a216a235a89a167a108a13a217a154a39a117a19a67a254a27a235a89a188a234.
8,a121a178a188a234
f(x,y) =
x2y
x4 +y2,x
2 +y2 negationslash= 0
0,x2 +y2 = 0
a51(0,0)a58a247a122a152a94a19a130x = tcosθ,y = tsinθ(0 lessorequalslantt+∞)a235a89a167a2a167a51(0,0)a58a216a235a89.
a121a178a181a8sinθ = 0a158a167cosθ = 1a189?1a167a117a180a8tnegationslash= 0a158a167f(tcosθ,tsinθ) = 0a167a13f(0,0) = 0
a75a107lim
t→0
f(tcosθ,tsinθ) = f(0,0)
a8sinθ negationslash= 0a167a107lim
t→0
f(tcosθ,tsinθ) = 0a167a25a107lim
t→0
f(tcosθ,tsinθ) = f(0,0)
a217a103a167a23a196a58P(x,y)a247a14a212a130y = x2a170a117a6a58a167a26 lim
y=x2
x→0
f(x,y) = 12 negationslash= f(0,0)a167a75a188a234f(x,y)a51a58(0,0)a216a235
a89.
9,a101f(x,y)a51a44a152a171a141Ga83a233a67a254xa143a235a89a167a233a67a254ya247a118a111a202a70a91a94a135a167a61a233a63a219
(x,yprime) ∈G,(x,yprimeprime) ∈G
a107|f(x,yprime)?f(x,yprimeprime)|lessorequalslantL|yprime?yprimeprime|
a217a165La143a126a234a167a75a100a188a234a51Ga83a235a89.
a121a178a181a207f(x,y)a51a171a141Ga83a233a67a254xa143a235a89a167a75a233Ga83a63a152a58(x0,y0)a167a233?ε< 0,?δ1 > 0a167a8|x?x0|<δ1a158a167
a107|f(x,y0)?f(x0,y0)|< ε2
a113a207f(x,y)a51Ga83a233ya247a118a111a202a70a91a94a135a167a75a233a63a219(x,y) ∈G,(x,y0) ∈Ga167a107|f(x,y)?f(x,y0)|lessorequalslantL|y?y0|
209
a45L|y?y0|< ε2 a167a75|y?y0|< ε2L
a18δ = min
parenleftBigg
δ1,ε2L
parenrightBigg
a167a8|x?x0|<δ,|y?y0|<δa158a167a189a107
|f(x,y)?f(x0,y0)|lessorequalslant|f(x,y)?f(x,y0)|+|f(x,y0)?f(x0,y0)|<ε
a61a100a188a234a51Ga83a235a89.
210
a49a155a111a217 a160a19a234a218a28a135a169
§1,a160a19a234a218a28a135a169a27a86a103
1,a166a101a15a188a234a27a160a19a234a181
(1) z = x2 ln(x2 +y2)
(2) u = exy
(3) z = xy+ xy
(4) u = arctan yx
(5) u = x2 +y2 +z2 + 2xy+ 2yz + 2zx
(6) u = eθ cos(θ+?)
a41a181
(1) zx = 2x
bracketleftBigg
ln(x2 +y2) + x
2
x2 +y2
bracketrightBigg
,zy = 2x
2y
x2 +y2.
(2) ux = yexy,uy = xexy.
(3) zx = y+ 1y,zy = x(y
2?1)
y2,
(4) ux =? yx2 +y2,uy = xx2 +y2,
(5) ux = 2(x+y+z),uy = 2(x+y+z),uz = 2(x+y+z).
(6) u? = eθ[cos(θ+?)?sin(θ+?)],uθ =?eθ[sin(θ+?) + cos(θ+?)].
2,a23f(x,y) = x2y2?2ya167a166fx(x,y),fy(x,y),fx(2,3),fy(0,0),fy(x,y)
vextendsinglevextendsingle
vextendsinglex=yy=x,
a41a181fx(x,y) = 2xy2,fy(x,y) = 2x2y?2,fx(2,3) = 36,fy(0,0) =?2,fy(x,y)
vextendsinglevextendsingle
vextendsinglex=yy=x = 2xy2?2
3,a23z = ln(√x+√y)a167a121a178x?z?x +y?z?y = 12,
a121a178a181a207z = ln(√x+√y)a167a75?z?x = 12√x(√x+√y),?z?y = 12√y(√x+√y)
a117a180x?z?x +y?z?y = 12,
4,a166a101a15a188a234a51a137a189a58(x0,y0)a27a28a135a169a181
(1) u = x4 +y4?4x2y2,(0,0),(1,1)
(2) u = xradicalbigx2 +y2,(1,0),(0,1)
(3) u = xsin(x+y),(0,0),
parenleftBigg
pi
4,
pi
4
parenrightBigg
(4) u = ln(x+y2),(0,1),(1,1)
a41a181
(1) a207du = 4x(x2?2y2)dx+ 4y(y2?2x2)dya167a75
a51(0,0)a58du = 0a182a51(1,1)a58du =?4dx?4dy.
(2) a207du = y
2
(x2 +y2)32
dx? xy
(x2 +y2)32
dya167a75
a51(1,0)a58du = 0a182a51(0,1)a58du = dx.
(3) a207du = [sin(x+y) +xcos(x+y)]dx+xcos(x+y)dya167a75
a51(0,0)a58du = 0a182a51
parenleftBigg
pi
4,
pi
4
parenrightBigg
a58du = dx.
211
(4) a207du = dxx+y2 + 2yx+y2 dya167a75
a51(0,1)a58du = dx+ 2dya182a51(1,1)a58du = dx2 + dy.
5,a166a101a15a188a234a27a28a135a169a181
(1) u = sin(x2 +y2)
(2) u = xm ·yn
(3) u = exy
(4) u = xy
(5) u =radicalbigx2 +y2 +z2
(6) u = ln(x2 +y2 +z2)
a41a181
(1) du = 2cos(x2 +y2)(xdx+ydy)
(2) du = xm?1yn?1(mydx+nxdy)
(3) du = exy(ydx+xdy)
(4) du = xy?1(ydx+xlnxdy)
(5) du = xdx+ydy+zdzradicalbigx2 +y2 +z2
(6) du = 2(xdx+ydy+zdz)x2 +y2 +z2
6,a121a178a181f(x,y) =radicalbig|xy|a51(0,0)a235a89a167fx(0,0),fy(0,0)a127a51a167a2a51(0,0)a58a216a140a135.
a121a178a181a100limx→0
y→0
radicalbig|xy| = 0a167a26lim
x→0
y→0
f(x,y) = 0 = f(0,0)a167a75f(x,y)a51(0,0)a58a235a89
a207fx(0,0) = lim
x→0
f(?x,0)?f(0,0)
x = 0,fy(0,0) = lim?y→0
f(0,?y)?f(0,0)
y = 0
a75fx(0,0),fy(0,0)a127a51
a2f(x,y) =radicalbig|xy|a51(0,0)a58a216a140a135.a101a140a135a167a75a107?f = fx(0,0)?x+fy(0,0)?y+o(ρ)a61?f = o(ρ)
a127a196a58P(x,y)a247y = xa170a1170a158a167a107?fρ =
radicalbig|?x?y|
radicalbig?x2 +?y2 = 1√2 negationslash→ 0(ρ → 0)a103a241a167a117a180a98a23a216a164a225a167
a75f(x,y)a51(0,0)a58a216a140a135.
7,a121a178a181f(x,y) =
xyradicalbig
x2 +y2,x
2 +y2 negationslash= 0
0,x2 +y2 = 0
a51(0,0)a58a27a25a141a165a235a89a167fx(x,y),fy(x,y)a107a46a167a2a51(0,0)a58
a216a140a135.
a121a178a181a100a117 xyradicalbigx2 +y2 a180a19a3a208a31a188a234a167a51a217a189a194a141a83a55a235a89a167a75f(x,y)a51x2 +y2 negationslash= 0a235a89
a1130 <
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
xyradicalbig
x2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
|xy|radicalbig
x2 +y2 lessorequalslant
radicalbigx2 +y2
2,f(0,0) = 0a167a75limx→0y→0 f(x,y) = f(0,0)a167a117a180f(x,y)a51(0,0)a58
a235a89a167a108a13f(x,y)a51(0,0)a58a27a63a219a25a141a83a235a89
a207fx(0,0) = lim
x→0
f(?x,0)?f(0,0)
x = 0,fy(0,0) = lim?y→0
f(0,?y)?f(0,0)
y = 0
a8x2 +y2 negationslash= 0a158a167fx(x,y) = y
3
(x2 +y2)32
,|fx(x,y)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
y3
(x2 +y2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant 1a167a75fx(x,y)a107a46
a211a110a140a26fy(x,y)a107a46
a2f(x,y)a51(0,0)a58a216a140a135.a101a140a135a167a75a107?f = fx(0,0)?x+fy(0,0)?y+o(ρ)a61?f = o(ρ)
a127a196a58P(x,y)a247y = xa170a1170a158a167a107?fρ =
x?y√
x2+?y2radicalbig
x2 +?y2 =
1
2 negationslash→ 0(ρ → 0)a103a241a167a117a180a98a23a216a164a225a167
a75f(x,y)a51(0,0)a58a216a140a135.
212
8,a23f(x,y) =
(x2 +y2)sin 1x2 +y2,x2 +y2 negationslash= 0
0,x2 +y2 = 0
a121a178fx(x,y),fy(x,y)a127a51a2a216a235a89a167a51(0,0)a58a27a63a219a25a141a165a195a46a167a2a51(0,0)a58a140a135.
a121a178a181a8x2 +y2 negationslash= 0a158a167fx(x,y) = 2xsin 1x2 +y2? 2xx2 +y2 cos 1x2 +y2
fx(0,0) = lim
x→0
f(?x,0)?f(0,0)
x = 0a167a75fx(0,0)a127a51
a127a9a51a58
parenleftBigg
1√
2npi,0
parenrightBigg
a27a160a19a234
fx
parenleftBigg
1√
2npi,0
parenrightBigg
=?2√2npi →?∞(n→∞)
a249a96a178fx(x,y)a51(0,0)a58a27a63a219a25a141a83a195a46a167a75a217a51(0,0)a58a216a235a89a167a117a180fx(x,y)a216a235a89
a211a110a140a26fy(x,y)a127a51a2a216a235a89a133fy(0,0) = 0a167a51(0,0)a58a27a63a219a25a141a165a195a46
a113?f?fx(0,0)?x?fy(0,0)?yρ =radicalbig?x2 +?y2 sin 1?x2 +?y2 → 0(ρ→ 0)
a75f(x,y)a51(0,0)a58a140a135.
9,a166a101a15a188a234a27a112a30a160a19a234a181
(1) u = xsin(x+y) +ycos(x+y)a167a164a107a19a30a160a19a234
(2) u = 12 ln(x2 +y2)a167 a164a107a19a30a160a19a234
(3) u = xln(xy)a167?
3u
x2?y
(4) u = ln(ax+by+cz)a167?
4u
x4,
4u
x2?y2
(5) u = (x?x0)p ·(y?y0)qa167?
p+qu
xp?yq
(6) u = x·y·zex+y+za167?
p+q+ru
xp ·?yq ·?zr
a41a181
(1) a207ux = (1?y)sin(x+y) +xcos(x+y),uy =?ysin(x+y) + (x+ 1)cos(x+y)
a75ux2 = (2?y)cos(x+y)?xsin(x+y),uxy = uyx = (1?y)cos(x+y)?(x+ 1)sin(x+y),
uy2 =?ycos(x+y)?(x+ 2)sin(x+y)
(2) a207ux = xx2 +y2,uy = yx2 +y2
a75ux2 = y
2?x2
(x2 +y2)2,uxy = uyx =?
2xy
(x2 +y2)2,uy2 =
x2?y2
(x2 +y2)2
(3) a207?u?y = xy a167a75?
2u
x?y =
1
y a167a117a180
3u
x2?y = 0
(4)?u?x = aax+by+cz,?u?y = bax+by+cz
2u
x2 =?
a2
(ax+by+cz)2,
2u
y2 =?
b2
(ax+by+cz)2
3u
x3 =
2a3
(ax+by+cz)3,
3u
x?y2 =
2ab2
(ax+by+cz)3
4u
x4 =?
6a4
(ax+by+cz)4,
4u
x2?y2 =?
6a2b2
(ax+by+cz)4
(5) a207?
qu
yq = q!(x?x0)
pa167a75?
p+qu
xp?yq = p!q!(p,qa254a143a103a44a234)
(6)?
p+q+ru
xp ·?yq ·?zr =
p
xp(xe
x)?
q
yq(ye
y)?
r
zr(ze
z) = ex+y+z(x+p)(y+q)(z +r)
10,a23
213
(1) u = x2?2xy?3y2
(2) u = xy2
(3) u = arccos
radicalBigg
x
y
a8a121a164a225a31a170?
2u
x?y =
2u
y?x
a121a178a181
(1) a207ux = 2x?2y,uy =?2x?6ya167a75?
2u
x?y =?2,
2u
y?x =?2a167a117a180
2u
x?y =
2u
y?x
(2) a207ux = y2xy2?1,uy = 2yxy2 lnxa167a75?
2u
x?y = 2yx
y2?1(1 +y2 lnx),?
2u
y?x = 2yx
y2?1(1 +y2 lnx)
a117a180?
2u
x?y =
2u
y?x
(3) a80 <xlessorequalslantya158a167u = arccos
radicalBigg
x
y = arccos
√x
√y
a75ux =? 12radicalbigx(y?x),uy =
√x
2y√y?x
a117a180?
2u
x?y =
1
4√x(y?x)32
,?
2u
y?x =
1
4√x(y?x)32
a108a13?
2u
x?y =
2u
y?x
a211a110a140a121a167a8ylessorequalslantx< 0a158a167?
2u
x?y =
2u
y?xa143a164a225
a110a254a167a26?
2u
x?y =
2u
y?x.
214
§2,a166a69a220a188a234a160a19a234a27a243a170a123a75
1,a166a101a15a188a234a27a160a19a234a181
(1) u = f(x,y)a167a217a165x = rcosθ,y = rsinθa167a166?u?r,?
2u
r2 a182
(2) u = f(x,y)a167a217a165x = aξ,y = bηa167a166?u?ξ,?
2u
ξ2,
2u
ξ?η,
u
η,
2u
η2
(3) u = f(x2 +y2 +z2)a167a166?u?x,?
2u
x2,
2
x?y,
u
y,
u
z
(4) u = f
parenleftBigg
x,xy
parenrightBigg
a167a166?u?x,?
2u
x2,
u
y,
a41a181
(1)?u?r = fx cosθ+fy sinθ
2u
r2 = fx2 cos
2θ+fxy sin2θ+f
y2 sin
2θ
(2)?u?ξ = afx,?
2u
ξ2 = a
2f
x2,
2u
ξ?η = abfxy,
u
η = bfy,
2u
η2 = b
2f
y2
(3)?u?x = 2xfprime(x2+y2+z2),?
2u
x2 = 2f
prime(x2+y2+z2)+4x2fprimeprime(x2+y2+z2),?
2
x?y = 4xyf
primeprime(x2+y2+z2),
u
y = 2yf
prime(x2 +y2 +z2),?u
z = 2zf
prime(x2 +y2 +z2)
(4)?u?x = f1 + 1y f2,?
2u
x2 = f11 +
2
y f12 +
1
y2 f22,
u
y =?
x
y2 f2
2,a23Φ = Φ(x,y,z),x = u+v,y = u?v,z = uva167a166Φu,Φv.
a41a181Φu = Φx + Φy +vΦz,Φv = Φx?Φy +uΦz
3,a166a101a15a188a234a27a28a135a169(a23a217a140a135)a181
(1) u = f(x+y)
(2) u = f(x+y,x?y)
(3) u = f(ax2 +by2 +cz2)
a41a181
(1) du = fprime(x+y)(dx+ dy)
(2) du = (f1 +f2)dx+ (f1?f2)dy
(3) du = 2fprime(ax2 +by2 +cz2)(axdx+bydy+czdz)
4,a8a121a101a15a136a170a181
(1) a23z =?(x2 +y2)a167a75y?z?x?x?z?y = 0a182
(2) a23u = y?(x2?y2)a167a75y?u?x +x?u?y = xuy a182
(3) a23u = x?(x+y) +yψ(x+y)a167a75?
2u
x2?2
2u
x?y +
2u
y2 = 0.
a121a178a181
(1) a207?z?x = 2x?prime(x2 +y2),?z?y = 2y?prime(x2 +y2)
a75y?z?x?x?z?y = 0.
(2) a207?u?x = 2xy?prime(x2?y2),?u?y =?(x2?y2)?2y2?prime(x2?y2)
a75y?u?x +x?u?y = x?(x2?y2) = xuy,
215
(3) a207?u?x =?(x+y) +x?prime(x+y) +yψprime(x+y),?u?y = x?prime(x+y) +ψ(x+y) +yψprime(x+y)
a75?
2u
x2 = 2?
prime(x+y)+x?primeprime(x+y)+yψprimeprime(x+y),?
2u
x?y =?
prime(x+y)+ψprime(x+y)+x?primeprime(x+y)+yψprimeprime(x+y)
2u
y2 = 2ψ
prime(x+y) +x?primeprime(x+y) +yψprimeprime(x+y)
a117a180?
2u
x2?2
2u
x?y +
2u
y2 = 0.
5,a166?u =?
2u
x2 +
2u
y2 +
2u
z2,u = f(x+y+z,x
2 +y2 +z2).
a41a181a207?u?x = f1 + 2xf2a167a75?
2u
x2 = f11 + 4xf12 + 4x
2f22 + 2f2
a226a233a161a53a167a26?
2u
y2 = f11 + 4yf12 + 4y
2f22 + 2f2,?
2u
z2 = f11 + 4zf12 + 4z
2f22 + 2f2
a117a180?u =?
2u
x2 +
2u
y2 +
2u
z2 = 3f11 + 4(x+y+z)f12 + 4(x
2 +y2 +z2)f22 + 6f2.
6,a101u = f(r),r =radicalbigx2 +y2a167a217a165f(r)a19a103a140a135a167a193a121a178
2u
x2 +
2u
y2 =
d2u
dr2 +
1
r
du
dr
a121a178a181a207?u?x = xradicalbigx2 +y2 fprime(r)a167a75?
2u
x2 =
y2
(x2 +y2)32
fprime(r) + x
2
x2 +y2 f
primeprime(r)
a226a233a161a53a167a26?
2u
y2 =
x2
(x2 +y2)32
fprime(r) + y
2
x2 +y2 f
primeprime(r)
a117a180?
2u
x2 +
2u
y2 =
d2u
dr2 +
1
r
du
dr
7,a101u,va143x,ya27a188a234a167x = rcosθ,y = rsinθa167a193a100
u
x =
v
y,
u
y =?
v
x
a121a178a31a170?u?r = 1r?v?θ,?v?r =?1r?u?θ,
a121a178a181a207u,va143x,ya27a188a234a167x = rcosθ,y = rsinθ
a75?u?r = cosθ?u?x + sinθ?u?y,?v?r = cosθ?v?x + sinθ?v?y,
u
θ =?rsinθ
u
x +rcosθ
u
y,
v
θ =?rsinθ
v
x +rcosθ
v
y
a113?u?x =?v?y,?u?y =v?xa167a75?u?r = 1r?v?θ,?v?r =?1r?u?θ,
8,a23f(tx,ty) = tnf(x,y)a167a75a107
x?f?x +y?f?y = nf
a228a107a249a24a53a159a27a188a234a167a161a143na103a224a103a188a234.a124a94a249a40a74a167a233z =radicalbigx2 +y2a167a166x?z?x +y?z?y,
a121a178a181a207f(tx,ty) = tnf(x,y)a167a75a252a224a233ta166a160a19a167a26f1(tx,ty)x+f2(tx,ty)y = ntn?1f(x,y)
a45t = 1a167a75f1(x,y)x+f2(x,y)y = nf(x,y)a61x?f?x +y?f?y = nf
a207z(x,y) =radicalbigx2 +y2a167a75z(tx,ty) = tradicalbigx2 +y2 (tgreaterorequalslant 0)
a117a180x?z?x +y?z?y = z =radicalbigx2 +y2.
9,a23?a134ψa180a63a191a27a19a30a140a19a188a234a167a121a178a181
z = x?
parenleftBigg
y
x
parenrightBigg
+ψ
parenleftBigg
y
x
parenrightBigg
216
a247a118x2?
2z
x2 + 2xy
2z
x?y +y
2?
2z
y2 = 0
a121a178a181a207?z?x =?
parenleftBigg
y
x
parenrightBigg
yx?prime
parenleftBigg
y
x
parenrightBigg
yx2 ψprime
parenleftBigg
y
x
parenrightBigg
,?z?y =?prime
parenleftBigg
y
x
parenrightBigg
+ 1x ψprime
parenleftBigg
y
x
parenrightBigg
a75?
2z
x2 =
y2
x3?
primeprime + 2y
x3 ψ
prime + y
2
x4 ψ
primeprime,?
2z
x?y =?
y
x2?
primeprime? 1
x2 ψ
prime? y
x3 ψ
primeprime,?
2z
y2 =
1
x?
primeprime + 1
x2 ψ
primeprime
a117a180x2?
2z
x2 + 2xy
2z
x?y +y
2?
2z
y2 = 0
10,a23u =?(x+at) +ψ(x?at)a167a217a165?,ψa180a63a191a27a19a103a140a135a188a234a167a166a121
2u
t2 = a
2?
2u
x2,
a121a178a181a207u =?(x+at) +ψ(x?at)a167?,ψa180a63a191a27a19a103a140a135a188a234
a75?u?t = a(?prime?ψprime),?u?x =?prime +ψprimea167a117a180?
2u
t2 = a
2(?primeprime +ψprimeprime),?
2u
x2 =?
primeprime +ψprimeprime
a108a13?
2u
t2 = a
2?
2u
x2.
217
§3,a100a144a167(a124)a164a40a189a27a188a234a27a166a19a123
1,a166a100a101a15a144a167a164a40a189a27a188a234z = f(x,y)a27a152a30a218a19a30a27a160a19a234a181
(1) x+y+z = ez
(2) xyz = x+y+z
a41a181
(1) a252a62a39a117xa166a19a167a261 +zx = zxeza167a75zx = 1ez?1 a167a117a180zx2 = e
z
(1?ez)3
a211a123a140a26a167zy = 1ez?1,zy2 = e
z
(1?ez)3,zxy = zyx =
ez
(1?ez)3
(2) a252a62a39a117xa166a19a167a26yz +xyzx = 1 +zx (?)a167a75zx = yz?11?xy
a242(?)a170a252a62a39a117xa166a19a167a262yzx +xyzx2 = zx2a167a75zx2 = 2yzx1?xy = 2y(yz?1)(xy?1)2
a211a123a140a26a167zy = xz?11?xy,zy2 = 2x(xz?1)(xy?1)2,zxy = zyx = 2z(xy?1)2
2,a166a100a101a15a144a167a164a40a189a27a188a234a27a28a135a169a189a160a19a234a181
(1) f(x+y,y+z,z +x) = 0a167a166?z?x,?z?y a182
(2) z = f(xz,z?y)a167a166dza182
(3) F(x?y,y?z,z?x) = 0a167a166?z?x,?z?y a182
(4) F(x,x+y,x+y+z) = 0a167a166?z?x,?z?y,?
2z
x2,
a41a181
(1) a252a62a39a117xa166a19a167a133z = z(x,y)a167a26f1 +f2zx +f3(zx + 1) = 0a167a75zx =?f1 +f3f
2 +f3
a211a123a140a26a167zy =?f1 +f2f
2 +f3
(2) a252a224a135a169a167a26dz = (xdz +zdx)f1 + (dz? dy)f2a167a75dz = zf1 dx?f2 dy1?xf
1?f2
(3) a252a62a39a117xa166a19a167a133z = z(x,y)a167a26F1?F2zx +F3(zx?1) = 0a167a75zx = F1?F3F
2?F3
a211a123a140a26a167zy = F2?F1F
2?F3
(4) a252a62a39a117xa166a19a167a133z = z(x,y)a167a26F1 +F2 +F3(1 +zx) = 0 (?)a167a75zx =?F1 +F2 +F3F
3
a51(?)a170a252a62a50a39a117xa166a19a167a26
F11 +F12 +F13(1 +zx) +F21 +F22 +F23(1 +zx) +zx2F3 + (1 +zx)[F13 +F23 +F33(1 +zx)] = 0
a75zx2 =? 1F3
3
[F23 (F11 + 2F12 +F22)?2F3(F1 +F2)(F13 +F23) +F33(F1 +F2)2]
a211a123a140a26a167zy =?F2 +F3F
3
3,a23a100a144a167z = x+y·?(z)a40a189a188a234z = z(x,y)a167a231?y?prime(z) negationslash= 0a167a121a178
z
y =?(z)·
z
x
a121a178a181a144a167a252a224a135a169a167a133z = z(x,y)a167a26dz = dx+?(z)dy+y?prime(z)dz
a1131?y?prime(z) negationslash= 0a167a75dz = dx+?(z)dy1?y?prime(z)
a117a180?z?y =?(z)1?y?prime(z),?z?x = 11?y?prime(z)a167a108a13?z?y =?(z)·?z?x
218
4,a121a178a100a144a167ax+by +cz = Φ(x2 +y2 +z2)a164a189a194a27a188a234z = z(x,y)a247a118a144a167(cy?bz)?z?x + (az?cx)?z?y =
bx?aya167a217a165Φ(u)a180ua27a140a135a188a234a167a,b,ca143a126a234.
a121a178a181a144a167a252a224a135a169a167a133z = z(x,y)a167Φ(u)a180ua27a140a135a188a234
a75a26adx+bdy+cdz = 2(xdx+ydy+zdz)Φprime
a117a180?z?x = 2xΦ
prime?a
c?2zΦprime,
z
y =
2yΦprime?b
c?2zΦprime
a108a13(cy?bz)?z?x + (az?cx)?z?y = bx?ay
5,a23?a143a63a191a27a140a135a188a234a167a121a178a100a144a167?(cx?az,cy?bz) = 0a164a189a194a27a188a234z = z(x,y)a247a118a?z?x +b?z?y = c.
a121a178a181a233a144a167a252a224a169a79a39a117x,ya166a19a167a133z = z(x,y)a167a26
c?1?a?1zx?b?2zx = 0,?a?1zy +c?2?b?2zy = 0
a117a180?z?x = c?1a?
1 +b?2
,?z?y = c?2a?
1 +b?2
a108a13a?z?x +b?z?y = c.
6,a121a178a100a144a167F(x+zy?1,y+zx?1) = 0a164a40a189a27a188a234z = z(x,y)a247a118x?z?x +y?z?y = z?xy.
a121a178a181a233a144a167a252a224a169a79a39a117x,ya166a19a167a133z = z(x,y)a167a26
F1
parenleftBigg
1 + zxy
parenrightBigg
+F2
parenleftBigg
zx
x?
z
x2
parenrightBigg
= 0,F1
parenleftBigg
zy
y?
z
y2
parenrightBigg
+F2
parenleftBigg
1 + zyx
parenrightBigg
= 0
a117a180?z?x = yzF2?x
2yF1
x(xF1 +yF2),
z
y =
xzF1?xy2F2
y(xF1 +yF2)
a108a13x?z?x +y?z?y = z?xy.
7,a166a101a15a144a167a124a164a40a189a27a188a234a27a19a234a189a160a19a234a189a28a135a169a181
(1)
braceleftbigg x+y+z = 0,
x·y·z = 1,a166
dy
dx,
dz
dx,
d2y
dx2 a182
(2)
x+y = u+v,
x
y =
sinu
sinv,
a166du,dva182
(3)
braceleftbigg xu+yv = 0,
yu+xv = 1,a166
u
x,
u
y,
v
x,
v
y,
2u
x?y a182
(4)
x = cosθcos?,
y = cosθsin?,
z = sinθ,
a166?z?x,?z?y a182
(5)
braceleftbigg u = f(u,x,v+y),
v = g(u?x,u2 ·y),a166
u
x,
v
x,
a41a181
(1) a233xa166a19a167a26
1 + dydx + dzdx = 0
yz +xz dydx+xy dzdx = 0
(?)
a233a225a166a41a167a26 dydx = y(z?x)x(y?z),dzdx = z(x?y)x(y?z)
(?)a170a50a233xa166a19a167a26
d2y
dx2 +
d2z
dx2 = 0
z dydx +y dzdx +z dydx +x dydx · dzdx +xz d
2y
dx2 +y
dz
dx +x
dy
dx ·
dz
dx+xy
d2z
dx2 = 0
a233a225a167a26 d
2y
dx2 =
2z dydx + 2y dzdx + 2x dydx · dzdx
x(y?z)
a242 dydx,dzdxa147a92a167a26 d
2y
dx2 =
yz[(x?y)2 + (x?z)2 + (y?z)2]
x2(z?y)3
219
(2) a242a6a170a85a21a143
braceleftbigg u+v = x+y
ysinu = xsinv a135a169a167a26
braceleftbigg du+ dv = dx+ dy
sinudy+ycosudu = sinvdx+xcosvdv
a75du = 1xcosv+ycosu[(sinv+xcosv)dx?(sinu?xcosv)dy]
dv = 1xcosv+ycosu[?(sinv?ycosu)dx+ (sinu+ycosu)dy]
(3) a135a169a167a26
braceleftbigg xdu+ydv =?udx?vdy
ydu+xdv =?vdx?udy
a117a180du = 1x2?y2[(yv?xu)dx+ (yu?xv)dy],dv = 1x2?y2[(yu?xv)dx+ (yv?xu)dy]
a75?u?x = yv?xux2?y2,?u?y = yu?xvx2?y2,?v?x = yu?xvx2?y2,?u?x = yv?xux2?y2
a117a180?
2u
x?y =
(yux?v?xvx)(x2?y2)?2x(yu?xv)
(x2?y2)2
a242?u?x,?v?xa147a92a167a26?
2u
x?y =
2(x2v+y2v?2xyu)
(x2?y2)2
(4) a100x,ya233xa166a160a19a234a167a26
1 =?sinθ·cosθ?x?cosθ·sinx
0 =?sinθ·sinθ?x + cosθ·cosx
a75?θ?x =? cos?sinθ,x =? sin?cosθ a167a117a180?z?x =?z?θ ·?θ?x =?cotθcos? =?xz
a211a110a140a26a167?z?y =?yz
(5) a233xa166a160a19a167a26
u
x = f1
u
x+f2 +f3
v
x
v
x = g1
parenleftBigg
u
x?1
parenrightBigg
+ 2vyg2?v?x
a75?u?x = f2(1?2vyg2)?g1f3(f
1?1)(2vyg2?1)?g1f3
,?v?x = g1(f1 +f2?1)(f
1?1)(2vyg2?1)?g1f3
.
8,a144a167x = u+v,y = u2 +v2,z = u3 +v3a189a194za143x,ya27a188a234a167a166?z?x,?z?y,
a41a181a207x2?y = 2uva167a75z = (u+v)(u2?uv+v2) = x2 (3y?x2)
a117a180?z?x = 32 (y?x2),?z?y = 32 x.
9,a23x = rcosθ,y = rsinθa167a67a134a144a167
dx
dt = y+kx(x
2 +y2)
dy
dt =?x+ky(x
2 +y2)
a143a52a139a73a144a167.
a41a181a100a144a167a127a167x,ya180ta27a188a234a167a108a52a139a73a67a134a127r,θa143a180ta27a188a234a167x = rcosθ,y = rsinθ
a252a224a233ta166a19a167a26
dx
dt = cosθ
dr
dt?rsinθ
dθ
dtdy
dt = sinθ
dr
dt +rcosθ
dθ
dt
a242x,y,dxdt,dydt a147a92a6a144a167a124a167a26
cosθ drdt?rsinθ dθdt = rsinθ+krcosθ·r2
sinθ drdt +rcosθ dθdt =?rcosθ+krsinθ·r2
a117a180 drdt = kr3,dθdt =?1.
10,a23x = eu cosθ,y = eu sinθ a167a67a134a144a167?
2z
x2 +
2z
y2 = 0.
a41a181a207x = eu cosθ,y = eu sinθa167a75u = ln(x2 +y2),θ = arctan yx
220
a75?u?x = xx2 +y2,?u?y = yx2 +y2 ;?θ?x =? yx2 +y2,?θ?y = xx2 +y2 a167a117a180?u?x =?θ?y,?u?y =θ?x
a113?z?x =?z?u ·?u?x +?z?θ ·?θ?x,?z?y =?z?u ·?u?y +?z?θ ·?θ?y
a75?
2z
x2 =
2z
u2
parenleftBigg
u
x
parenrightBigg2
+ 2?
2z
θ?u ·
u
x ·
θ
x +
2z
θ2
parenleftBigg
θ
x
parenrightBigg2
+?z?u ·?
2u
x2 +
z
θ ·
2θ
x2
2z
y2 =
2z
u2
parenleftBigg
u
y
parenrightBigg2
+ 2?
2z
θ?u ·
u
y ·
θ
y +
2z
θ2
parenleftBigg
θ
y
parenrightBigg2
+?z?u ·?
2u
y2 +
z
θ ·
2θ
y2
a113?
2u
x2 =
x
parenleftBigg
θ
y
parenrightBigg
=y
parenleftBigg
θ
x
parenrightBigg
=y
parenleftBigg
u?y
parenrightBigg
=
2u
y2
a211a123a140a26a167?
2θ
x2 =?
2θ
y2
a75?
2θ
x2 +
2θ
y2 =
2u
x2 +
2u
y2 = 0
a113
parenleftBigg
u
x
parenrightBigg2
+
parenleftBigg
u
y
parenrightBigg2
=
parenleftBigg
θ
x
parenrightBigg2
+
parenleftBigg
θ
y
parenrightBigg2
,?u?x ·?θ?x =u?y ·?θ?y
a75?
2z
x2 +
2z
y2 = e
2u
parenleftBigg
2z
u2 +
2z
θ2
parenrightBigg
= 0
a61?
2z
u2 +
2z
θ2 = 0.
11,a23x = rcosθ,y = rsinθa167a75f(x,y) = Φ(r,θ)a167a94Φa39a117r,θa27a160a19a234a53a76a171?
2f
x2 +
2f
y2,
a41a181a242f(x,y) = Φ(r,θ)a39a117r,θa166a160a19a167a26
f
x ·
x
r +
f
y ·
y
r =
Φ
r a61
f
x cosθ+
f
y sinθ =
Φ
r
f
x ·
x
θ +
f
y ·
y
θ =
Φ
θ a61?
f
x rsinθ+
f
y rcosθ =
Φ
θ
a75?
2Φ
r2 =
2f
x2 cos
2θ+?
2f
x?y sin2θ+
2f
y2 sin
2θ
2Φ
θ2 = r
2
parenleftBigg
2f
x2 sin
2θ
2f
x?y sin2θ+
2f
y2 cos
2θ
parenrightBigg
f?x rcosθf?y rsinθ
a117a180?
2Φ
r2 +
1
r2
2Φ
θ2 =
2f
x2 +
2f
y2?
1
r
Φ
r
a108a13?
2f
x2 +
2f
y2 =
2Φ
r2 +
1
r2
2Φ
θ2 +
1
r
Φ
r
12,a23x = eξ,y = eηa167a67a134a144a167ax2?
2z
x2 + 2bxy
2z
x?y +cy
2?
2z
y2 = 0(a,b,ca143a126a234).
a41a181a207x = eξ,y = eηa167a75ξ = lnx,η = lnya167a117a180 dξdx = 1x,dηdy = 1y
a75?z?x = 1x?z?ξ,?z?y = 1y?z?η
a117a180?
2z
x2 =
1
x2
parenleftBigg
2z
ξ2?
z
ξ
parenrightBigg
,?
2z
y2 =
1
y2
parenleftBigg
2z
η2?
z
η
parenrightBigg
,?
2z
x?y =
1
xy
2z
ξ?η
a147a92a6a144a167a167a122a123a18a110a167a26a
parenleftBigg
2z
ξ2?
z
ξ
parenrightBigg
+ 2b?
2z
ξ?η +c
parenleftBigg
2z
η2?
z
η
parenrightBigg
= 0.
13,a23ξ = x,η = x2 +y2a167a67a134a144a167y?z?x?x?z?y = 0,
a41a181a100a144a167a127za180x,ya27a188a234a167a13ξ,ηa113a180x,ya27a188a234a167a108a13za140a119a164a180a207a76a165a109a67a254ξ,ηa39a117x,ya27a69a220a188a234
a117a180?z?x =?z?ξ + 2x?z?η,?z?y = 2y?z?η
a207a13y?z?x?x?z?y = y?z?ξ
221
a207y negationslash≡ 0a167a75a100y?z?x?x?z?y = 0a167a26?z?ξ = 0.
14,a23ξ = x,η = y?x,ζ = z?xa167a67a134a144a167?u?x +?u?y +?u?z = 0.
a41a181a100a144a167a127ua180x,y,za27a188a234a167a13ξ,η,ζa113a180x,y,za27a188a234a167a108a13ua140a119a164a180a207a76a165a109a67a254ξ,η,ζa39a117x,y.za27a69
a220a188a234
a117a180?u?x =?u?ξu?ηu?ζ,?u?y =?u?η,?u?z =?u?ζ
a75a100?u?x +?u?y +?u?z = 0a167a26?z?ξ = 0
15,a23a130a53a67a134ξ = x + λ1y,η = x + λ2ya167a121a51a135a114a144a167A?
2u
x2 + 2B
2u
x?y + C
2u
y2 = 0(A,B,Ca143a126a234a167
a133AC?B2 < 0)a67a134a143?
2u
ξ?η = 0a167a121a178λ1,λ2a143a144a167Cλ
2 + 2Bλ+A = 0a27a252a135a131a201a162a138.
a121a178a181a100a144a167a127ua180x,ya27a188a234a167a207a13a140a177a114ua192a143a177ξ,ηa143a165a109a67a254a27a39a117x,ya27a69a220a188a234a167a117a180
u
x =
u
ξ +
u
η,
u
y = λ1
u
ξ +λ2
u
η
2u
x2 =
2u
ξ2 + 2
2u
ξ?η +
2u
η2,
2u
y2 = λ
2
1
2u
ξ2 + 2λ1λ2
2u
ξ?η +λ
2
2
2u
η2,
2u
x?y = λ1
2u
ξ2 + (λ1 +λ2)
2u
ξ?η +λ2
2u
η2
a207a13A?
2u
x2 + 2B
2u
x?y +C
2u
y2 =
2u
ξ2 (A+ 2Bλ1 +Cλ
2
1) + 2
2u
ξ?η [A+B(λ1 +λ2) +Cλ1λ2] +
2u
η2(A+
2Bλ2 +Cλ22) = 0
a135a166A?
2u
x2 + 2B
2u
x?y +C
2u
y2 = 0a67a134a143
2u
ξ?η = 0a167a55a76
A+ 2Bλ1 +Cλ21 = 0
A+ 2Bλ2 +Cλ22 = 0
A+B(λ1 +λ2) +Cλ1λ2 negationslash= 0
a100a99a252a135a144a167a167a26λ1,λ2a180a144a167Cλ2 + 2Bλ+A = 0a27a138
a13a100a49a110a135a144a167a167a26λ1 negationslash= λ2a167a75λ1,λ2a180Cλ2 + 2Bλ+A = 0a27a252a135a131a201a162a138
a113a207λ1 +λ2 =?2BC,λ1λ1 = AC a167a75A+B(λ1 +λ2) +Cλ1λ2 = 2C (AC?B2) negationslash= 0
a117a180a144a167A?
2u
x2 + 2B
2u
x?y + C
2u
y2 = 0a51a130a53a67a134ξ = x + λ1y,η = x + λ2ya101a40a162a67a134a143
2u
ξ?η = 0a167
a133λ1,λ2a143a144a167Cλ2 + 2Bλ+A = 0a27a252a135a131a201a162a138.
16,a121a178a46a202a46a100a144a167?w ≡?
2w
x2 +
2w
y2 = 0a51a67a122x =?(u,v),y = ψ(u,v)
parenleftBigg
a167a130a247a118u =?ψ?v,v =ψ?u
parenrightBigg
a101
a47a71a2a177a216a67.
a121a178a181a108a144a167a127ωa180x,ya27a188a234a167x,ya180u,va27a188a234a167a75wa180a177x,ya143a165a109a67a254a27u,va27a188a234
a117a180?w?u =?w?x ·u +?w?y ·?ψ?u,?w?v =?w?x ·v +?w?y ·?ψ?v
2w
u2 =
2w
x2
parenleftBigg
u
parenrightBigg2
+ 2?
2w
x?y ·
u ·
ψ
u +
2w
y2
parenleftBigg
ψ
u
parenrightBigg2
+?w?x ·?
2?
u2 +
w
y ·
2ψ
u2
2w
v2 =
2w
x2
parenleftBigg
v
parenrightBigg2
+ 2?
2w
x?y ·
v ·
ψ
v +
2w
y2
parenleftBigg
ψ
v
parenrightBigg2
+?w?x ·?
2?
v2 +
w
y ·
2ψ
v2
a53a191a154a242a122a94a135u =?ψ?v,v =ψ?u a167a75?
2?
u2 =
2ψ
u?v,
2?
v2 =?
2ψ
v?u,
2ψ
u2 =?
2?
u?v,
2ψ
v2 =
2?
v?u
a242?
2w
u2,
2w
v2 a131a92a167a191a242a254a227a136a170a147a92a167a26
2w
u2 +
2w
v2 =
parenleftBigg
2w
x2 +
2w
y2
parenrightBiggbracketleftBiggparenleftBigg
u
parenrightBigg2
+
parenleftBigg
v
parenrightBigg2bracketrightBigg
a207?
2w
x2 +
2w
y2 = 0a167a75
2w
u2 +
2w
v2 = 0
a249a76a178a46a202a46a100a144a167?w ≡?
2w
x2 +
2w
y2 = 0a51a67a122x =?(u,v),y = ψ(u,v)a101a47a71a2a177a216a67.
17,a23ξ = x?at,η = x+ata167a67a134a144a167?
2u
t2 = a
2?
2u
x2,
222
a41a181a100a144a167a127ua180t,xa27a188a234a167ξ,ηa143a180t,xa27a188a234a167a25a140a242ua192a143a177ξ,ηa143a165a109a67a254a27a39a117t,xa27a188a234
a75?u?t =?a?u?ξ +a?u?η,?u?x =?u?ξ +?u?η
2u
t2 = a
2?
2u
ξ2?2a
2?
2u
ξ?η +a
2?
2u
η2,
2u
x2 =
2u
ξ2 +
2u
ξ?η +
2u
η2
a117a180a100?
2u
t2 = a
2?
2u
x2 a167a264a
2?
2u
ξ?η = 0
a113anegationslash≡ 0a167a75?
2u
ξ?η = 0.
18,a138a103a67a234a218a207a67a234a27a67a134a167a18u,va143a35a27a103a67a234a167w = w(u,v)a143a35a27a207a67a234a181
(1) a23u = x2 +y2,v = 1x + 1y,w = lnz?(x+y)a167a67a134a144a167
y?z?x?x?z?y = (y?x)·z
(2) a23u = x+y,v = yx,w = zxa167a67a134a144a167
2z
x2?2
2z
x?y +
2z
y2 = 0
(3) a23x = u,y = u1 +uv,z = u1 +u·w a167a67a134a144a167
x2?z?x +y2?z?y = z2
(4) a23u = xy,v = x,w = xz?ya167a67a134a144a167
y?
2z
y2 + 2
z
y =
2
x
a41a181
(1) a100a174a127a167a26du = 2xdx+ 2ydy,dv =? 1x2 dx? 1y2 dy,dw = 1z dz? dx? dy
a44a152a144a161a167dw =?w?u du+?w?v dv
a751z dz? dx? dy =?w?u (2xdx+ 2ydy) +?w?v
parenleftBigg
1x2 dx? 1y2 dy
parenrightBigg
a18a110a167a26dz =
parenleftBigg
2xz?w?u? zx2 ·?w?v +z
parenrightBigg
dx+
parenleftBigg
2yz?w?u? zy2 ·?w?v +z
parenrightBigg
dy
a242a254a170a164a40a189a27?z?x,?z?y a147a92a6a144a167a167a26z
parenleftBigg
x
y2?
y
x2
parenrightBigg
w
v = 0
a113z
parenleftBigg
x
y2?
y
x2
parenrightBigg
negationslash≡ 0a167a75?w?v = 0.
(2) a100a174a127a167a26du = dx+ dy,dv =? yx2 dx+ 1x dy,dw =? zx2 dx+ 1x dz
a44a152a144a161a167dw =?w?u du+?w?v dv
a75? zx2 dx+ 1x dz =?w?u (dx+ dy) +?w?v
parenleftBigg
yx2 dx+ 1x dy
parenrightBigg
a18a110a167a26dz =
parenleftBigg
x?w?u? yx ·?w?v + zx
parenrightBigg
dx+
parenleftBigg
x?w?u +?w?v
parenrightBigg
dy
a75?z?x = x?w?u? yx ·?w?v + zx,?z?y = x?w?u +?w?v
223
a45R =?z?xz?y = w?(1 +v)?w?v
a75?
2z
x2? 2
2z
x?y +
2z
y2 =
x
parenleftBigg
z
x?
z
y
parenrightBigg
y
parenleftBigg
z
x?
z
y
parenrightBigg
=?R?xR?y =?R?u
parenleftBigg
u
x?
u
y
parenrightBigg
+
R
v
parenleftBigg
v
x?
v
y
parenrightBigg
=v
bracketleftBigg
w?(1 +v)?w?v
bracketrightBiggparenleftBigg
yx2? 1x
parenrightBigg
= (1 +v)
2
x
2w
v2 = 0
a207(1 +v)
2
x negationslash≡ 0a167a75a6a144a167a67a143
2w
v2 = 0.
(3) a207x = u,y = u1 +uv,z = u1 +u·w a167a75u = x,v = 1y? 1x,w = 1z? 1x
a117a180du = dx,dv = 1x2 dx? 1y2 dy,dw = 1x2 dx? 1z2 dz
a44a152a144a161a167dw =?w?u du+?w?v dv
a75 1x2 dx? 1z2 dz =?w?u dx+?w?v
parenleftBigg
1
x2 dx?
1
y2 dy
parenrightBigg
a18a110a167a26dz = z2
parenleftBigg
1
x2?
w
u?
1
x2 ·
w
v
parenrightBigg
dx+ z
2
y2 ·
w
v dy
a242a254a170a164a40a189a27?z?x,?z?y a147a92a6a144a167a167a26x2z2?w?u = 0
a113xz negationslash≡ 0a167a75?w?u = 0.
(4) a207u = xy,v = x,w = xz?ya167a75?w?y = x?z?y?1,?w?y =?w?u ·?u?y +?w?v ·?v?y,?u?y =? xy2,?v?y = 0
a117a180?w?y =? xy2?w?u a167a75?z?y = 1x? 1y2?w?u
a117a180y?
2z
y2 + 2
z
y =
2
x+
x
y3
2w
u2 =
2
x
a113xy3 negationslash≡ 0a167a75a6a144a167a67a134a143?
2w
u2 = 0.
224
§4,a152a109a173a130a27a131a130a134a123a178a161
1,a166a101a15a173a130a51a164a171a58a63a27a131a130a134a123a178a161a181
(1) x = asin2t,y = bsint·cost,z = ccos2ta167a51t = pi4 a27a58a63a182
(2) x2 +y2 +z2 = 6,x+y+z = 0a167a51a58(1,?2,1).
a41a181
(1) x0 = a2,y0 = b2,z0 = c2,xprime(t0) = a,yprime(t0) = 0,zprime(t0) =?c
a75a173a130a51t = pi4 a27a58a63a27a131a130a144a167a143
x? a2
a =
z? c2
c
y = b2
a61
x
a +
z
c = 1
y = b2
a123a178a161a144a167a143a
parenleftBigg
x? a2
parenrightBigg
c
parenleftBigg
z? c2
parenrightBigg
= 0a61ax?cz = 12 (a2?c2).
(2) a207D(F1,F2)D(y,z)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingle2y 2z1 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
=?6,D(F1,F2)D(z,x)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingle2z 2x1 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
= 0,
D(F1,F2)
D(x,y)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingle2x 2y1 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
= 6
a75a173a130a51a58(1,?2,1)a27a131a130a144a167a143
braceleftbigg x+z?2 = 0
y =?2
a123a178a161a144a167a143x?z = 0.
2,a51a173a130x = t,y = t2,z = t3a254a166a209a152a58a167a166a100a58a27a131a130a178a49a117a178a161x+ 2y+z = 4.
a41a181a23a164a166a58a143(t0,t20,t30)a167a75xprime(t0) = 1,yprime(t0) = 2t0,zprime(t0) = 3t20
a117a180a173a130a27a131a130a144a149a165a254a143v = {1,2t0,3t20}
a113a178a161a123a165a254n = {1,2,1}a167a75a226a75a191a167a65a107v·n = 1 + 4t0 + 3t20 = 0a167a117a180t0 =?1,t0 =?13
a75a164a166a58a143(?1,1,?1),
parenleftBigg
13,19,? 127
parenrightBigg
.
3,a121a178a173a130x = aet cost,y = aet sint,z = aeta134a73a161x2 +y2 = z2a27a49a130a131a2a164a211a152a14.
a121a178a181a242x,y,za147a92x2 +y2 = z2a167a26a2e2t cos2t+a2e2t sin2t = a2e2t = z2a167a75a173a130a65a51a173a161a254
a11a73x2 +y2 = z2a27a186a58a51a6a58a167a76a11a73a254a63a152a58P(x0,y0,z0)a27a49a130a143a76a6a58
a75a49a130a27a144a149a165a254a143v1 = {x0,y0,z0}
a113a173a130a51a58Pa27a131a149a254a143v2 = {aet0(cost0?sint0),aet0(sint0 + cost0),aet0} = {x0?y0,x0 +y0,z0}
x20 +y20 = z20
a75cos(hatwiderv1,v2) = v1 ·v2|v
1||v2|
= 2√6 =
√6
3 a167a249a134a173a130a254a58(x,y,z)a27a160a152a118a107a39a88
a207a13a173a130a134a73a161a27a49a130a131a2a164a211a152a14.
4,a166a101a15a136a173a130a51a164a171a58a27a131a130a27a144a149a123a117a181
(1) x = t2,y = t3,z = t4a167a51t = 1a27a58a254a182
(2) xyz = 1,y2 = xa167a51a58(1,1,1).
a41a181
(1) a207xprime(t0) = 2,yprime(t0) = 3,zprime(t0) = 4a167a75a131a149a254a143{2,3,4}
a117a180a144a149a123a117a143a181cosα = ± 229 √29,cosβ = ± 329 √29,cosγ = ± 429 √29.
(2) a207D(F1,F2)D(y,z)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingle xz xy?2y 0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
= 2,D(F1,F2)D(z,x)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
=
vextendsinglevextendsingle
vextendsinglevextendsinglexy yz0 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
= 1,
D(F1,F2)
D(x,y)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingleyz xz1?2y
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
=?3a167a75a131a149a254a143{2,1,?3}
a117a180a144a149a123a117a143a181cosα = ±
√14
7,cosβ = ±
√14
14,cosγ = ±
3
14
√14.
225
§5,a173a161a27a131a178a161a134a123a130
1,a166a101a15a173a161a51a164a171a58a27a131a178a161a57a123a130a144a167a181
(1) x = asin?cosθ,y = asin?sinθ,z = acos?a167a51(θ0,?0)a182
(2) exz +eyz = 4a167a51a58(ln2,ln2,1)a182
(3) z = 2x2 + 4y2a167a51a58(2,1,12)a182
(4) ax2 +by2 +cz2 +d = 0a167a51a58(x0,y0,z0).
a41a181
(1) a207D(y,z)D(θ,?)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(θ0,?0)
=
vextendsinglevextendsingle
vextendsinglevextendsingleasin?cosθ acos?sinθ0?asin?
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(θ0,?0)
=?asin2?0 cosθ0,
D(z,x)
D(θ,?)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(θ0,?0)
=?a2 sin2?0 sinθ0,D(x,y)D(θ,?)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(θ0,?0)
=?a2 sin?0 cos?0
a75a131a178a161a144a167a143sin?0 cosθ0x+ sin?0 sinθ0y+ cos?0z = a
a123a130a144a167a143x?asin?0 cosθ0sin?
0 cosθ0
= y?asin?0 sinθ0sin?
0 sinθ0
= z?acos?0cos?
0
.
(2) a207a51(ln2,ln2,1)a58fx = 2,fy = 2,fz =?ln16
a75a131a178a161a144a167a143x+y?2ln2·z = 0a182a123a130a144a167a143x?ln21 = y?ln21 = z?1?2ln2,
(3) a207zx(2,1) = 8,zy(2,1) = 8
a75a131a178a161a144a167a1438x+ 8y?z = 12a182a123a130a144a167a143x?28 = y?18 = z?12?1,
(4) a207a51(x0,y0,z0)a58fx = 2ax0,fy = 2by0,fz = 2cz0
a75a131a178a161a144a167a143ax0x+by0y+cz0z +d = 0a182a123a130a144a167a143x?x0ax
0
= y?y0by
0
= z?z0cz
0
.
2,a51a173a161z = xya254a166a152a58a167a166a249a58a27a123a130a82a134a117a178a161x+ 3y+z + 9 = 0a167a191a21a209a100a123a130a144a167.
a41a181a76a173a161a254a63a152a58M0(x0,y0,z0)a27n1 = {y0,x0,?1}a167a123a130a27a131a149a254a143n2 = {1,3,1}
a135a166a123a130a82a134a117a254a227a178a161a167a75n1 bardbl n2a61?y1 =?x3 = 11
a117a180a164a166a58a143(?3,?1,3)a167a75a123a130a144a167a143x+ 31 = y+ 13 = z?31,
3,a121a178a173a161√x+√y+√z = √a,(a> 0)a254a63a219a152a58a27a131a178a161a51a136a139a73a182a254a27a31a229a131a218a31a117a.
a121a178a181a51a173a161a254a63a18a152a58P0(x0,y0,z0)
a75a173a161a51a84a58a27a131a178a161a144a167a143 12√x
0
(x?x0) + 12√y
0
(y?y0) + 12√z
0
(z?z0) = 0
a61√y0z0(x?x0) +√x0z0(y?y0) +√x0y0(z?z0) = 0
a117a180a131a178a161a51a139a73a182a254a27a31a229a169a143√ax0,√ay0,√az0a167a217a218a143√a(√x0 +√y0 +√z0) = a.
4,a166a252a173a161x2 +y2 = a2,bz = xya27a2a14.
a41a181a23a252a173a161a63a152a2a58M0(x0,y0,z0)
a100a252a173a161a51M0a58a27a123a149a254a143n1 = {2x0,2y0,0},n2 = {y0,x0,?b}
a117a180a2a14?a247a118cos? = n1 ·n2|n
1||n2|
= 2bz0|a|√a2 +b2,
226
§6,a144a149a19a234a218a70a221
1,a166u = x2?xy+y2a51(1,1)a63a247a144a149l = (cosα,sinα)a27a144a149a19a234.a191a63a152a218a166a181
(1) a51a61a135a144a149a254a217a19a234a107a129a140a138a182
(2) a51a61a135a144a149a254a217a19a234a107a129a2a138a182
(3) a51a61a135a144a149a254a217a19a234a1430a182
(4) a166ua27a70a221.
a41a181a207ux = 2x?y,uy =?x+ 2ya167a75ux(1,1) = 1,uy(1,1) = 1
a113?u?l = ux(1,1)cosα+uy(1,1)sinαa167a75?u?l = cosα+ sinα = √2sin
parenleftBigg
α+ pi4
parenrightBigg
a117a180
(1) a8α = pi4 a158a167a51a144a149l =
parenleftBigg√
2
2,
√2
2
parenrightBigg
a254a217a19a234a107a129a140a138√2a182
(2) a8α =?34 pia158a167a51a144a149l =
parenleftBigg
√2
2,?
√2
2
parenrightBigg
a254a217a19a234a107a129a2a138?√2a182
(3) a8α =?pi4,34 pia158a167a51a144a149l =
parenleftBigg√
2
2,?
√2
2
parenrightBigg
a189l =
parenleftBigg
√2
2,
√2
2
parenrightBigg
a254a217a19a234a1430a182
(4) gradu = ux(1,1)i +uy(1,1)j = i + j.
2,a166u = xyza51a58M(1,1,1)a167a247l = (2,?1,3)a27a144a149a19a234a57a70a221.
a41a181a207ux = yz,uy = xz,uz = xya167a75a51(1,1,1)a58ux = uy = uz = 1
a113a149a254la27a144a149a123a117cosα = 2√14,sinβ =? 1√14,cosγ = 3√14
a75?u?l = ux(1,1,1)cosα+uy(1,1,1)cosβ +uz(1,1,1)cosγ = 27 √14 a182gradu = i + j + k.
3,a166a234a254a188a234u = x2 + 2y2 + 3z2 +xy+ 3x?2y?6za51O(0,0,0)a57A(1,1,1)a27a70a221a57a217a140a2.
a41a181a207ux = 2x+y+ 3,uy = 4y+x?2,uz = 6z?6
a75a51O(0,0,0)a58a181ux = 3,uy =?2,uz =?6a167a117a180gradu = 3i?2j + 6k,|gradu| = 7
a51A(1,1,1)a58a181ux = 6,uy = 3,uz = 0a167a117a180gradu = 6i + 3j,|gradu| = 3√5.
4,a121a178a181
(1) grad(αu+βv) = αgradu+βgradva167a217a165α,βa209a180a126a234a182
(2) grad(uv) = ugradv+vgradua182
(3) gradF(u) = F prime(u)gradu
a121a178a181a177a19a3a188a234a143a126a53a121.a45u = u(x,y),v = v(x,y)
(1) a207?(αu+βv)?x = α?u?x +β?v?x,?(αu+βv)?y = α?u?y +β?v?y
a75grad(αu+βv) =
parenleftBigg
(αu+βv)
x,
(αu+βv)
y
parenrightBigg
= α
parenleftBigg
u
x,
v
y
parenrightBigg
+β
parenleftBigg
u
x,
v
y
parenrightBigg
= αgradu+βgradv.
(2) a207?(uv)?x = v?u?x +u?v?x,?(uv)?y = v?u?y +u?v?y
a75grad(uv) =
parenleftBigg
(uv)
x,
(uv)
y
parenrightBigg
= v
parenleftBigg
u
x,
u
y
parenrightBigg
+u
parenleftBigg
v
x,
v
y
parenrightBigg
= ugradv+vgradu.
(3) gradF(u) =
parenleftBigg
F
x,
F
y
parenrightBigg
=
parenleftBigg
F prime(u)?u?x,F prime(u)?u?y
parenrightBigg
= F prime(u)
parenleftBigg
u
x,
u
y
parenrightBigg
= F prime(u)gradu
a245a3a188a234a140a149a19a3a188a234a121a131.
5,a121a178grad1r =? rr3 a167r =radicalbigx2 +y2 +z2,r = xi +yj +zk.
a121a178a181a207?r?x = xr,?r?y = yr,?r?z = zr
a75grad1r = ddr
parenleftBigg
1
r
parenrightBigg
gradr =? 1r2
parenleftBigg
r
x i +
r
y j +
r
z k
parenrightBigg
=? 1r2 · 1r(xi +yj +zk) =? rr3,
227
6,a23a234a254a188a234u = ln 1r,r =radicalbig(x?a)2 + (y?b)2 + (z?c)2a167a51a152a109a165a61a10a58a254a164a225|gradu| = 1a186
a41a181a207?r?x = x?ar,?r?y = y?br,?r?z = z?cr
a75gradu = ddr
parenleftBigg
ln 1r
parenrightBigg
gradr =?1r
parenleftBigg
x?a
r i +
y?b
r j +
z?c
r k
parenrightBigg
=? 1r2 [(x?a)i + (y?b)j + (z?c)k]
a117a180|gradu| = 1r = 1a167a75r = 1a61(x?a)2 + (y?b)2 + (z?c)2 = 1
a249a76a178a51a177(a,b,c)a143a165a37a167a140a187a1431a27a165a161a254a164a225|gradu| = 1.
228
§7,a18a86a250a170
1,a21a209a58(1,?2)a78a67a188a234f(x,y) = 2x2?xy?y2?6x?3y+ 5a27a18a86a250a170.
a41a181a207fx = 4x?y?6,fy =?x?2y?3,fx2 = 4,fxy =?1,fy2 =?2a167a164a107a110a30a160a19a254a1430
a75a51a58(1,?2)a167f = 5,fx = 0,fy = 0,fx2 = 4,fxy =?1,fy2 =?2
a117a180f(x,y) = 5 + 2(x?1)2?(x?1)(y+ 2)?(y+ 2)2.
2,a85xa57ya27a166a152a208a109a188a234f(x,y) = ex ln(1 +y)a20a110a145a143a142.
a41a181a207fx = ex ln(1 +y),fy = e
x
1 +y,fx2 = e
x ln(1 +y),f
y2 =?
ex
(1 +y)2,fxy =
ex
1 +y
fx3 = ex ln(1 +y),fy3 = 2e
x
(1 +y)3,fxy2 =?
ex
(1 +y)2,fyx2 =
ex
1 +y
a75a51a58(0,0)a63a167f = 0,fx = fx2 = fx3 = 0,fy = 1,fxy = 1,fy2 =?1,fxy2 =?1,fyx2 = 1,fy3 = 2
a117a180f(x,y) = y+xy? 12 y2 + 12 x2y? 12 xy2 + 13 y3a34
229
a49a155a202a217 a52a138a218a94a135a52a138
§1,a52a138a218a129a2a19a166a123
1,a166a101a15a188a234a27a52a138a181
(1) z = x2?(y?1)2
(2) z = (x?y+ 1)2
(3) z = 3axy?x3?y3 (a> 0)
(4) z = sinx+ cosy+ cos(x?y)
parenleftbigg x
0 lessorequalslant y lessorequalslant pi2
parenrightbigg
(5) z = xy·
radicalbigg
1? x
2
a2?
y2
b2 (a,b> 0)
a41a181
(1) a45zx = 2x = 0,zy =?2(y?1) = 0
a75a26x = 0,y = 1a167a117a180a58(0,1)a143a140a85a52a138a58
a113zx2 = 2,zxy = 0,zy2 =?2a167a75A = 2,B = 0,C =?2a167a117a180H =?4 < 0a167a108a13a100a188a234a195a52a138.
(2) a45zx = 2(x?y+ 1) = 0,zy =?2(x?y+ 1) = 0
a75a8a58a169a217a51x?y+ 1 = 0a254a158a167a188a234a140a85a107a52a138
a113A = 2,B =?2,C = 2a167a75H = 0a167a25a73a63a152a218a7a228
a207a233a134a130x?y+ 1 = 0a254a27a58a254a107z = 0a167a133z greaterorequalslant 0a240a164a225
a75a188a234za51a134a130x?y+ 1 = 0a254a136a58a18a26a52a2a138z = 0.
(3) a45zx = 3ay?3x2 = 0,zy = 3ax?3y2 = 0
a75a26
braceleftbigg x
1 = 0
y1 = 0
braceleftbigg x
2 = a
y2 = a a61a51a58(0,0),(a,a)a63a140a85a107a52a138
a113zx2 =?6x,zxy = 3a,zy2 =?6y
a75a51a58(0,0)a167A = 0,B = 3a,C = 0a167a117a180H =?9a2 < 0a167a100a158a188a234a195a52a138a182
a51a58(a,a)a167A =?6a< 0,B = 3a,C =?6aa167a117a180H = 27a2 > 0a167a100a158a188a234a107a52a140a138z = a3.
(4) a45zx = cosx?sin(x?y) = 0,zy =?siny+ sin(x?y) = 0
a75a26cosx = sinya167a117a180y = pi2?xa167a25cosx?sin(x?y) = cos?sin
parenleftBigg
2x? pi2
parenrightBigg
= 2cos x2 cos 32 x = 0
a2070 lessorequalslantxlessorequalslant pi2a167a75cos x2 negationslash= 0,cos 32 x = 0a167a117a180x = pi3,y = pi6 a167a61a51a58
parenleftBigg
pi
3,
pi
6
parenrightBigg
a63a140a85a107a52a138
a113zx2 =?sinx?cos(x?y),zxy = cos(x?y),zy2 =?cosy?cos(x?y)
a75A =?√3 < 0,B =
√3
2,C =?
√3a167a117a180H = 9
4 > 0a167a61a51a58
parenleftBigg
pi
3,
pi
6
parenrightBigg
a63a188a234a107a52a140a138z = 32 √3.
(5) a45zx = y
radicalbigg
1? x
2
a2?
y2
b2?
x2y
a2
radicalbigg
1? x
2
a2?
y2
b2
= 0,zy = x
radicalbigg
1? x
2
a2?
y2
b2?
xy2
b2
radicalbigg
1? x
2
a2?
y2
b2
= 0
a75a26
braceleftbigg x
1 = 0
y1 = 0
x2 = a√3
y2 = b√3
x3 =? a√3
y3 =? b√3
x4 = a√3
y4 =? b√3
x5 =? a√3
y5 = b√3
a117a180a51a58P1(0,0),P2
parenleftBigg
a√
3,
b√
3
parenrightBigg
,P3
parenleftBigg
a√3,? b√3
parenrightBigg
,P4
parenleftBigg
a√
3,?
b√
3
parenrightBigg
,P5
parenleftBigg
a√3,b√3
parenrightBigg
a63a140a85a18a52
a138
a113zx2 =?3a
2b2xy+ 2b2x3y+ 3a2xy3
a4b2
parenleftBig
1? x2a2? y2b2
parenrightBig3
2
,zxy = a
4b4?3a2b4x2 + 2b4x4?3a4b2y2 + 3a2b2x2y2 + 2a4y4
a4b4
parenleftBig
1? x2a2? y2b2
parenrightBig3
2
zy2 = 3b
2x3y?3a2b2xy+ 2a2xy3
a2b4
parenleftBig
1? x2a2? y2b2
parenrightBig3
2
230
a51a58P2
parenleftBigg
a√
3,
b√
3
parenrightBigg
,P3
parenleftBigg
a√3,? b√3
parenrightBigg
a63a167A =?4
√3b
3a < 0,B =?
2
3
√3,C =?4√3a
3b
a100a158H = 4 > 0a167a117a180a188a234a107a52a140a138z =
√3
9 aba182
a51a58P4
parenleftBigg
a√
3,?
b√
3
parenrightBigg
,P5
parenleftBigg
a√3,b√3
parenrightBigg
a63a167A = 4
√3b
3a > 0,B =?
2
3
√3,C = 4√3a
3b
a100a158H = 4 > 0a167a117a180a188a234a107a52a2a138z =?
√3
9 aba182
a51a58P1(0,0)a63a167A = 0,B = 1,C = 0a167a100a158H =?1 < 0a167a117a180a188a234a195a52a138.
2,a121a178a188a234z = (1 +ey)cosx?yeya107a195a161a245a135a52a140a138a167a2a195a52a2a138.
a121a178a181a45zx =?(1 +ey)sinx = 0 zy = ey cosx?ey?yey = 0
a2071 +ey negationslash= 0a167a75sinx = 0a167a117a180x = kpi (k ∈Z)
a113ey negationslash= 0a167a75cosx?1?y = 0a61a107y = cosx?1
a8ka143a243a234a158a167y = 0a182a8ka143a219a234a158a167y =?2a167a75a140a85a27a52a138a58a143
braceleftbigg x
1 = 2kpi
y1 = 0
braceleftbigg x
2 = (2k+ 1)pi
y2 =?2 (k =
0,±1,±2,···)
a113zx2 =?(1 +ey)cosx,zxy =?ey sinx,zy2 = ey cosx?2ey?yey
a75a51a58(2kpi,0)a167A =?2 < 0,B = 0,C =?1a167a100a158H = 2 > 0a167a75a100a158a188a234a107a52a140a138z = 2
a51a58((2k+ 1)pi,?2)a167A = 1 + 1e2,B = 0,C =? 1e2 a167a100a158H =? 1e2
parenleftBigg
1 + 1e2
parenrightBigg
< 0a167a75a100a158a188a234a195a52a138
a110a254a140a127a167a188a234z = (1 +ey)cosx?yeya107a195a161a245a135a52a140a138a167a2a195a52a2a138.
3,a51a174a127a177a127a1432pa27a152a131a110a14a47a165a166a209a161a200a129a140a27a110a14a47.
a41a181a23a110a14a47a27a62a127a169a79a143x,y,za167a75C = x+y+z = 2pa167D = x+y+z2 = pa167a117a180z = 2p?x?y
a75S =radicalbigD(D?x)(D?y)(D?z) =radicalbigp(p?x)(p?y)(x+y?p)
a127a196a188a234u = S2 = p(p?x)(p?y)(x+y?p),0 <x,y<p
Sa27a52a138a254a143ua27a52a138a133a8ua51a58(x,y)a18a26a27a52a138a216a1430a158a167Sa143a51a58(x,y)a18a26a52a138
a45ux = p(p?y)(2p?2x?y) = 0,uy = p(p?x)(2p?x?2y) = 0
a207pnegationslash= 0,0 <x,y<pa167a75a41a26x = y = 23 pa167a117a180z = 23 p
a75a8x = y = z = 23 pa158a167ua107a52a138a61Sa107a52a138
a108a13a8x = y = z = 23 pa158a167a161a200a129a140a133a138a143S =
√3
9 p
2.
4,a173a161z = 12 x2?4xy+ 9y2 + 3x?14y+ 12 a51a219a63a107a129a112a58a189a129a36a58a186
a41a181a100
braceleftbigg z
x = x?4y+ 3 = 0
zy =?4x+ 18y?14 = 0 a167a41a26
braceleftbigg x = 1
y = 1 a61a51a58(1,1)a140a85a107a52a138
a113zx2 = 1,zxy =?4,zy2 = 18a167a75A = 1 > 0,B =?4,C = 18a167a117a180H = 2 > 0
a75a100a158a188a234a107a52a2a138z =?5a167a108a13a173a161a107a129a36a58(1,1,?5)
a113a8x2 +y2 → +∞a158a167z → +∞a167a25a173a161a195a129a112a58.
5,a174a127y = ax2 +bx+ca167a121a255a26a152a124a234a226(xi,yi),i = 1,2,···,na167a124a94a129a2a19a166a123a167a166a88a234a,b,ca164a247a118a27a110a3
a152a103a144a167a124.
a41a181a100a174a127a167a26ε =
nsummationdisplay
i=1
(yi?ax2i?bxi?c)2a167a143a166a111a160a11ε(a,b,c)a136a20a129a2a167a100a52a138a27a55a135a94a135a167a107
ε
a =?2
nsummationdisplay
i=1
(yi?ax2i?bxi?c)x2i = 0,?ε?b =?2
nsummationdisplay
i=1
(yi?ax2i?bxi?c)xi = 0,?ε?c =?2
nsummationdisplay
i=1
(yi?ax2i?bxi?c) = 0
a61a,b,ca247a118a101a15a110a3a152a103a144a167a124a181
parenleftBigg nsummationdisplay
i=1
x4i
parenrightBigg
a+
parenleftBigg nsummationdisplay
i=1
x3i
parenrightBigg
b+
parenleftBigg nsummationdisplay
i=1
x2i
parenrightBigg
c =
nsummationdisplay
i=1
x2iyi
parenleftBigg nsummationdisplay
i=1
x3i
parenrightBigg
a+
parenleftBigg nsummationdisplay
i=1
x2i
parenrightBigg
b+
parenleftBigg nsummationdisplay
i=1
xi
parenrightBigg
c =
nsummationdisplay
i=1
xiyi
parenleftBigg nsummationdisplay
i=1
x2i
parenrightBigg
a+
parenleftBigg nsummationdisplay
i=1
xi
parenrightBigg
b+nc =
nsummationdisplay
i=1
yi
6,a173a130y = x2a51[0,1]a254a135a94a159a111a24a27a134a130η = ax+ba53a147a79a167a226a85a166a167a27a178a144a216a11a27a200a169
J(a,b) =
integraldisplay 1
0
(y?η)2 dxa143a52a2a27a191a194a101a143a129a90a67a113a186
231
a41a181J(a,b) =
integraldisplay 1
0
(y?η)2 dx =
integraldisplay 1
0
(x2?ax?b)2 dx = 15 + a
2
3 +b
2? a
2?
2
3 b+ab
a143a10a192a74a,ba166a178a144a216a11a27a200a169J(a,b)a136a20a52a2a167a100a52a138a27a55a135a94a135a167a107
a45?J?a =?12 + 23 a+b = 0,?J?b =?23 +a+ 2b = 0
a75a = 1,b =?16
a117a180a173a130y = x2a94a134a130η = x? 16 a53a147a79a167a140a136a20a129a90a67a113a27a135a166.
232
§2,a94a135a52a138
1,a166a101a15a188a234a51a164a137a94a135a101a52a138a181
(1) f = x+y,a101x2 +y2 = 1a182
(2) f = x?2y+ 2z,a101x2 +y2 +z2 = 1a182
(3) f = xyz,a1011x + 1y + 1z = 1a (x> 0,y> 0,z> 0,a> 0)a182
(4) f = 1x + 1y,a101x+y = 2a182
(5) f = xyz,a101x2 +y2 +z2 = 1,x+y+z = 0.
a41a181
(1) a138a188a234L = x+y+λ(x2 +y2?1)
a41a144a167a124
Lx = 1 + 2λx = 0
Ly = 1 + 2λy = 0
Lλ = x2 +y2?1 = 0
a167a26
x1 =
√2
2
y1 =
√2
2
λ1 =?
√2
2
x2 =?
√2
2
y2 =?
√2
2
λ2 =
√2
2a113L
x2 = 2λ,Lxy = 0,Ly2 = 2λ
a75d2L
parenleftBigg√
2
2,
√2
2
parenrightBigg
=?√2(dx2 + dy2) < 0a167a117a180a188a234a51
parenleftBigg√
2
2,
√2
2
parenrightBigg
a63a18a26a52a140a138√2a182
a211a110a140a26a167a188a234a51
parenleftBigg
√2
2,?
√2
2
parenrightBigg
a63a18a26a52a2a138?√2.
(2) a138a188a234L = x?2y+ 2z +λ(x2 +y2 +z2?1)
a41a144a167a124
Lx = 1 + 2λx = 0
Ly =?2 + 2λy = 0
Lz = 2 + 2λz = 0
Lλ = x2 +y2 +z2?1 = 0
a167a26
x1 = 13
y1 =?23
Z1 = 23
λ1 =?32
x2 =?13
y2 = 23
z2 =?23
λ2 = 32
a113Lx2 = Ly2 = Lz2 = 2λ,Lxy == Lxz = Lyz = 0
a75d2L(x2,y2,z2) = 3(dx2 + dy2 + dz2) > 0a167a117a180a188a234a51
parenleftBigg
13,23,?23
parenrightBigg
a63a18a26a52a2a138?3a182
a211a110a140a26a167a188a234a51
parenleftBigg
1
3,?
2
3,
2
3
parenrightBigg
a63a18a26a52a140a1383.
(3) a138a188a234L = xyz +λ
parenleftBigg
1
x +
1
y +
1
z?
1
a
parenrightBigg
a41a144a167a124
Lx = yz? λx2 = 0
Ly = xz? λy2 = 0
Lz = xy? λz2 = 0
Lλ = 1x + 1y + 1z? 1a = 0
a167a26x = y = z = 3a,λ = 81a4
a113Lx2(3a,3a,3a) = Ly2(3a,3a,3a) = Lz2(3a,3a,3a) = 6a,
Lxy(3a,3a,3a) = Lxz(3a,3a,3a) = Lyz(3a,3a,3a) = 3a
a75d2L(3a,3a,3a) = 3a[(dx+ dy+ dz)2 + dx2 + dy2 + dz2] > 0a167a117a180a188a234a51(3a,3a,3a)a63a18a26a52a2
a13827a3.
(4) a138a188a234L = 1x + 1y+λ(x+y?2)
233
a41a144a167a124
Lx =? 1x2 +λ = 0
Ly =? 1y2 +λ = 0
Lλ = x+y?2 = 0
a167a26x = y = λ = 1
a113Lx2(1,1) = Ly2(1,1) = 2,Lxy(1,1) = 0
a75d2L(1,1) = 2(dx2 + dy2) > 0a167a117a180a188a234a51(1,1)a63a18a26a52a2a1382.
(5) a138a188a234L = xyz +u(x2 +y2 +z2?1) +v(x+y+z)
a41a144a167a124
Lx = yz + 2ux+v = 0
Ly = xz + 2uy+v = 0
Lz = xy+ 2uz +v = 0
Lu = x2 +y2 +z2?1 = 0
Lv = x+y+z = 0
a167a26
x1 =
√6
6
y1 =
√6
6
z1 =?
√6
3
u1 =
√6
12
v1 = 16
x2 =?
√6
6
y2 =?
√6
6
z2 =
√6
3
u2 =?
√6
12
v2 = 16
x3 =?
√6
3
y3 =
√6
6
z3 =
√6
6
u3 =
√6
12
v3 = 16
x4 =
√6
3
y4 =?
√6
6
z4 =?
√6
6
u4 =?
√6
12
v4 = 16
x5 =
√6
6
y5 =?
√6
3
z5 =
√6
6
u5 =
√6
12
v5 = 16
x6 =?
√6
6
y6 =
√6
3
z6 =?
√6
6
u6 =?
√6
12
v6 = 16
a113d2L = 2u(dx2 + dy2 + dz2) + 2(zdxdy+ydxdz +xdydz)
a75a51a58(x1,y1,z1)a63a167d2L =
√6
6 (dx
2 + dy2 + dz2?4dxdy+ 2dxdz + 2dydz)
a100x2 +y2 +z2 = 1a167a262xdx+ 2ydy+ 2zdz = 0a167a75a51a58(x1,y1,z1)a63a167a107dx+ dy = 2dz
a113a100x+y+z = 0a167a26dx+ dy+ dz = 0a167a75dx =?dy,dz = 0a167a117a180d2L(x1,y1,z1) = √6 dx2 > 0a167
a75a188a234a51
parenleftBigg√
6
6,
√6
6,?
√6
3
parenrightBigg
a63a18a26a52a2a138?
√6
18
a211a110a140a26a167a188a234a51(x3,y3,z3),(x5,y5,z5)a63a18a26a52a2a138?
√6
18
a188a234a51(x2,y2,z2),(x4,y4,z4),(x6,y6,z6)a63a18a26a52a140a138
√6
18,
2,a166f = xmynzpa51a94a135x+y+z = a,a> 0,m> 0,n> 0,p> 0,x> 0,y> 0,z> 0a131a101a27a129a140a138.
a41a181a207x> 0,y> 0,z> 0a167a75f = xmynzp a129a140a158a167lnf = mlnx+nlny+plnza143a129a140a167a135a131a189a44a167a25a144a73
a166lnfa27a52a140a58a167a167a143a180fa27a52a140a58
a45L = mlnx+nlny+plnz +λ(x+y+z?a)
a75a41a144a167
Lx = mx +λ = 0
Ly = ny +λ = 0
Lz = pz +λ = 0
Lλ = x+y+z?a = 0
a167a26
x = mam+n+p
y = nam+n+p
z = pam+n+p
λ =?m+n+pa
a75
parenleftBigg
ma
m+n+p,
na
m+n+p,
pa
m+n+p
parenrightBigg
a143a140a85a52a138a58
a113Lx2 =?mx2,Lxy = Lyz = Lxz = 0,Ly2 =? ny2,Lz2 =? pz2,d2L =
parenleftBigg
mx2 dx2? ny2 dy2? pz2 dz2
parenrightBigg
< 0
a25a51
parenleftBigg
ma
m+n+p,
na
m+n+p,
pa
m+n+p
parenrightBigg
a63lnfa107a52a140a138a167a61fa107a52a140a138 m
mnnpp
(m+n+p)m+n+p a
m+n+p
a113f = xmnnzpa8(x,y,z)a170a117a62a46
braceleftbigg x+y = a
z = 0
braceleftbigg x+z = a
y = 0
braceleftbigg y+z = a
x = 0 a158a167f → 0a167a25fa27a141a152a52a140a58
a143a180a167a27a129a140a58.
3,a166a253a11x2 + 3y2 = 12a27a83a26a31a27a110a14a47a167a166a217a46a62a178a49a117a253a11a27a127a182a167a13a161a200a129a140.
a41a181a100a117a75a165a110a14a47a83a26a117a253a11 x
2
(2√3)2 +
y2
4 = 1a180a31a27a110a14a47a167a133a46a62a178a49a117a127a182
234
a25a217a46a62a164a233a186a58a55a180a225a182a254a253a11a27a186a58(0,±2)
a23a110a14a47a27a44a152a135a186a58a139a73a143(x,y) (x,y> 0)a167a75a217a83a26a31a27a110a14a47a46a62a127a1432xa167a112a143y+ 2
a31a27a110a14a47a110a186a58a139a73a143A(0,?2),B(x,y),C(?x,y)a167a100a253a11a27a233a161a53a167a26A(0,2),B(x,?y),C(?x,?y)a143a180
a217a186a58
a75S = x(y+ 2)a167a58(x,y)a51a253a11x2 + 3y2 = 12a254
a113a207a100a175a75a180a166S = x(y+ 2)a51a129a155a94a135x2 + 3y2 = 12a254a27a129a140a138(x,y> 0)
a138a188a234L = x(y+ 2) +λ(x2 + 3y2?12)
a75a41a144a167
Lx = y+ 2 + 2λx = 0
Ly = x+ 6λy = 0
Lλ = x2 + 3y2?12 = 0
a167a26
x = 3
y = 1
λ =?12
a117a180a217a186a58a139a73a143A(0,2),B(3,?1),C(?3,?1)a189A(0,?2),B(3,1),C(?3,1)
a207a100a175a75a143a162a83a175a75a167a129a140a138a55a127a51a167a75a51(0,2),(3,?1),(?3,?1)a189(0,?2),(3,1),(?3,1)a63a217a161a200a129a140a167a217
a138a1439.
4,a193a166a14a212a130y2 = 4xa254a27a58a167a166a167a134a134a130x?y+ 4 = 0a131a229a129a67.
a41a181a23a164a166a58a139a73a143(x,y)a167a75a167a20a134a130a27a229a108a143d = 1√2 |x?y+ 4|a167a217a165y2 = 4x
a134a130x?y+ 4 = 0a242a178a161a169a143a134a33a109a252a220a169a167a134a161x?y+ 4 < 0a167a109a161x?y+ 4 > 0
a13a14a212a130y2 = 4xa51a109a161a220a169a167a207a13a58(x,y)a20a167a27a229a108a143d = 1√2 (x?y+ 4)
a45L = 1√2 (x?y+ 4) +λ(y2?4x)
a75a41a144a167a124
Lx = 1√2?4λ = 0
Ly =? 1√2 + 2λy = 0
Lλ = y2?4x = 0
a167a26
x = 1
y = 2
λ = 14√2
a113Lx2 = 0,Ly2 = 12√2,Lxy = 0,d2L(1,2) = Lx2 dx2 + 2Lxy dxdy+Ly2 dy2 = dy
2
2√2 > 0
a25(1,2)a143a52a2a58a167a61a58(1,2)a20a134a130a27a229a108a129a67.
5,a14a212a161z = x2 +y2a26a178a161x+y+z = 1a31a164a152a253a11a167a166a6a58a20a249a253a11a27a129a127a134a129a225a229a108.
a41a181a226a75a191a167a166a229a108d =radicalbigx2 +y2 +z2a51a129a155a94a135z = x2 +y2,x+y+z = 1a27a129a138
a61a166u = x2 +y2 +z2a51a129a155a94a135a101a27a129a138
a138L = x2 +y2 +z2 +λ(z?x2?y2) +γ(x+y+z?1)
a75a41a144a167a124
Lx = 2x?2λx+γ = 0
Ly = 2y?2λy+γ = 0
Lz = 2z +λ+γ = 0
Lλ = z?x2?y2 = 0
Lγ = x+y+z?1 = 0
a167a26
x1 =?1 +
√3
2
y1 =?1 +
√3
2
z1 = 2?√3
λ1 =?5
√3 + 9
3
γ1 =?7 + 113 √3
,
x2 =?1?
√3
2
y2 =?1?
√3
2
z2 = 2 +√3
λ2 = 5
√3 + 9
3
γ2 =?7? 113 √3
a117a180d(x1,y1,z1) =
radicalbig
9?5√3,d(x2,y2,z2) =
radicalbig
9 + 5√3
a226a175a75a27a162a83a191a194a167a129a127a33a129a225a229a108a127a51
a75a129a127a229a108a143a6a58a20a58
parenleftBigg
1 +
√3
2,?
1 +√3
2,2 +
√3parenrightBigga27a229a108a167a143radicalbig9 + 5√3a182
a129a225a229a108a143a6a58a20a58
parenleftBigg
1 +√3
2,
1 +√3
2,2?
√3parenrightBigga27a229a108a167a143radicalbig9?5√3.
6,a166a152a109a152a58(a,b,c)a20a178a161Ax+By+Cz +D = 0a27a129a225a229a108.
a41a181a23(x,y,z)a143a178a161Ax+By+Cz+D = 0a254a63a152a58a167a75a167a134(a,b,c)a58a27a229a108a143d =radicalbig(x?a)2 + (y?b)2 + (z?c)2a167
a217a165(x,y,z)a247a118Ax+By+Cz +D = 0
a207d> 0a167a25da129a140d2a129a140
a85a75a23a167a65a166d2 = (x?a)2 + (y?b)2 + (z?c)2a51a94a135Ax+By+Cz +D = 0a101a27a52a138
a45L = (x?a)2 + (y?b)2 + (z?c)2 +λ(Ax+By+Cz +D)
235
a75a41a144a167a124
Lx = 2(x?a) +λA = 0
Ly = 2(y?b) +λB = 0
Lz = 2(z?c) +λC = 0
Lλ = Ax+By+Cz?D = 0
a167a26
x = a? 12 λA
y = b? 12 λB
z = c? 12 λC
λ = 2(Aa+Bb+Cc+D)A2 +B2 +C2
a117a180d = |Aa+Bb+Cc+D|√A2 +B2 +C2
a113a8x,y,za165a107a63a152a170a117∞a158a167d→∞a167a207a100a51braceleftbig(x,y,z)vextendsinglevextendsingle(x?a)2 + (y?b)2 + (z?c)2 <dbracerightbiga83a55a18a129a2a138
a75a58(a,b,c)a20a178a161Ax+By+Cz +D = 0a27a129a225a229a108a143d = |Aa+Bb+Cc+D|√A2 +B2 +C2,
236
a49a155a56a217 a219a188a234a127a51a189a110a33a188a234a131a39
§1,a219a188a234a127a51a189a110
1,a101a51a219a188a234a127a51a189a110a165a94a135a85a143a181
(1) a51a171a141D,(x0?alessorequalslantxlessorequalslantx0 + 1,y0?blessorequalslantylessorequalslanty0 +b)a254a235a89a182
(2) F(x0,y0) = 0a182
(3) a8xa27a189a158a167a188a234F(x,y)a180ya27a252a78a188a234a182a75a140a26a20a159a111a24a27a40a216a167a193a121a178a131.
a121a178a181a40a216a57a121a178a181
(1) a51a58(x0,y0)a27a44a152a25a141a83a167a100a144a167F(x,y) = 0a85a141a152a40a189y = f(x)a180xa27a252a78a188a234a133y0 = f(x0).
a100a94a135(3)a127a167a8xa27a189a158a167F(x,y)a180ya27a238a130a252a78a188a234.a216a148a23a167a180ya27a238a130a252a79a188a234
a27a189x0a167a188a234F(x0,y)a180ya27a238a130a79a188a234a167a133F(x0,y0) = 0a167a207a100a107F(x0,y0 +b) > 0,F(x0,y0?b) < 0
a100a94a135(1)a127a167F(x,y)a51a171a141Da254a235a89a167a207a13a127a51η> 0a167a166a8|x?x0|<ηa158a167a189a107
F(x,y0 +b) > 0,F(x,y0?b) < 0
a64a34a233?x∈O(x0,η)a167a100a188a234F(x,y)a51[y0?b,y0 +b]a27a235a89a53a57F(x,y0 +b) > 0,F(x,y0?b) < 0
a226a34a58a127a51a189a110a167a55a127a51y ∈ (y0?b,y0 +b)a167a166F(x,y) = 0
a100a117F(x,y)a51[y0?b,y0 +b]a238a130a252a78a167a108a13a8y>ya158a167F(x,y) > 0a182a8y<ya158a167F(x,y) < 0
a25a254a227ya180a141a152a27
a249a76a178a233?x ∈ O(x0,η)a167a207a76a254a227a144a123a167a107a141a152a27ya134xa233a65a167a133a247a118F(x,y) = 0a167a117a180a40a189a10a189a194
a51O(x0,η)a254a27a252a138a188a234y = f(x)a247a118F(x,f(x)) = 0a167a65a79a107F(x0,y0) = 0a61y0 = f(x0).
(2) f(x)a180a235a89a188a234.
x1 ∈O(x0,a)a167a101a121y = f(x)a51x1a58a235a89.
a233?ε> 0(ε<b)a167a23y1 = f(x1)a167a117a180F(x1,y1) = 0
a113a100a94a135(3)a167F(x,y)a180ya27a238a130a252a79a188a234
a207a100F(x1,y1 +ε) > 0,F(x1,y1?ε) < 0
a75a100Fa27a235a89a53a167a127a127a51a25a141O(x1,δ)?O(x0,a)a167a166a26a8x∈O(x1,δ)a158a167a240a107
F(x,y1 +ε) > 0,F(x,y1?ε) < 0
a117a180a226a34a58a127a51a189a110a167a26a55a107y ∈O(y1,ε)a167a166F(x,y) = 0a61y = f(x)
a61a144a135|x?x1|<δa167a210a107|f(x)?f(x1)| = |y?y1|<εa61y = f(x)a51x1a58a235a89
a100x1 ∈O(x0,a)a27a63a191a53a167a26f(x)a143a235a89a188a234.
2,a188a234F(x,y) ≡y2?x2(1?x2) = 0a51a61a10a58a67a11a140a141a152a47a251a189a252a138a235a89a167a133a107a235a89a19a234a27a188a234y = y(x).
a41a181a19a3a188a234F(x,y) = y2?x2(1?x2)a51a18a135a19a145a152a109a235a89a167a167a27a160a19a234Fx = 4x3?2x,Fy = 2ya143a235a89
a100y2?x2(1?x2) = 0a167a101y = 0a167a75x2(1?x2) = 0a167a41a26x = 0,x = ±1
a113y2 greaterorequalslant 0,x2 greaterorequalslant 0a167a251?x2 greaterorequalslant 0a61?1 lessorequalslantxlessorequalslant 1
a8y negationslash= 0a158a167Fy negationslash= 0
a100a219a188a234a127a51a189a1101a167a51a63a219a247a118{(x,y)vextendsinglevextendsingle|x|< 1,xnegationslash= 0,y2?x2(1?x2) = 0}a67a11a140a141a152a47a251a189a252a138a235a89a133a107
a235a89a19a234a27a188a234y = y(x).
3,a121a178a107a141a152a140a19a27a188a234y = y(x)a247a118a144a167 siny+ sinhy = xa167a191a166a209a19a234yprime(x).
a121a178a181a19a3a188a234F(x,y) = siny+ sinhy?xa51a18a135a19a145a152a109a235a89a167Fx =?1,Fy = cosy+ coshya143a235a89
a113coshy = e
y +e?y
2 greaterorequalslant 1 a133a31a210a144a51y = 0a158a164a225a167a13a100a158cosy = 1a167a51a152a132a156a185a101|cosy|lessorequalslant 1
a75a233a152a131a58(x,y)a167a240a107Fy = cosy+ coshy> 0a167a117a180Fy negationslash= 0
a100a219a188a234a127a51a189a1101a167a51a63a219a247a118a254a227a144a167a27a58(x,y)a167a107a141a152a140a19a27a188a234a247a118a144a167siny+ sinhy = x
a217a19a188a234a143yprime =?FxF
y
= 1cosy+ coshy.
4,a23Da180a58P0,(x0,y0,z0,u0,v0)a27a25a141a167a101
(1) F(x0,y0,z0,u0,v0) = 0,G(x0,y0,z0,u0,v0) = 0a182
(2) F,Ga39a117a152a131a67a254a27a160a19a234a51Da165a235a89a182
(3) J =
vextendsinglevextendsingle
vextendsinglevextendsingleFu FvG
u Gv
vextendsinglevextendsingle
vextendsinglevextendsinglea51P0a58a216a143a34a182
a75a51(x0,y0,z0)a27a25a141Ra83a127a51a141a152a27a152a233a188a234
u = f(x,y,z);v = g(x,y,z)
a247a118a181
237
(1) u0 = f(x0,y0,z0),v0 = g(x0,y0,z0)
(2) F(x,y,z,f,g) ≡ 0,G(x,y,z,f,g) ≡ 0
(3) u = f(x,y,z),v = g(x,y,z)a51a58(x0,y0,z0)a27a25a141Ra83a107a233a152a131a67a254a27a160a19a234a167a133
f
x =?
1
J
D(F,G)
D(x,v),
f
y =?
1
J
D(F,G)
D(y,v),
f
z =?
1
J
D(F,G)
D(z,v)
g
x =?
1
J
D(F,G)
D(u,x),
g
y =?
1
J
D(F,G)
D(u,y),
g
z =?
1
J
D(F,G)
D(u,z)
a121a178a181a100a94a135(3)a127a167Fu,Fva165a150a8a107a152a135a51P0a58a216a31a1170
a216a148a23Fv(P0) negationslash= 0a167a75a100a219a188a234a127a51a189a1102a167a127a51P0a58a27a44a135a25a141a83a140a177a114va108F(x,y,z,u,v) = 0a165a41a209a53.
a23v =?(x,y,z,u)a133v0 =?(x0,y0,z0,u0)a51(x0,y0,z0,u0)a27a44a135a25a141a83a180a141a152a27a167a228a107a39a117x,y,z,ua27a235a89
a160a19a234
a114v =?(x,y,z,u)a147a92G(x,y,z,u,v)a165a26G(x,y,z,u,?(x,y,z,u)) = ψ(x,y,z,u)
a25ψu = Gu +Gv ·vu = Gu +Gv
parenleftBigg
FuF
v
parenrightBigg
=? JF
v
a100a98a23Fv(P0) negationslash= 0a133a51P0a58J negationslash= 0a167a25ψu(x0,y0,z0,u0) negationslash= 0
a75a100a189a1102a167a26a51(x0,y0,z0,u0)a27a44a25a141a83a140a108a144a167G = G(x,y,z,u,?) ≡ψ(x,y,z,u) = 0a165a41a209ua53.
a23u = f(x,y,z)a167a167a51(x0,y0,z0)a27a44a25a141a83a107a235a89a160a19a234a167a133u0 = f(x0,y0,z0)
a114u = f(x,y,z)a147a92?(x,y,z,u)a165a26v =?(x,y,z,f(x,y,z)) = g(x,y,z)
a75a107g(x0,y0,z0) =?(x0,y0,z0,u0) = v0
a25u = f(x,y,z),v = g(x,y,z)a61a143a164a166
a233a144a167a124
braceleftbigg F(x,y,z,u,v) = 0
G(x,y,z,u,v) = 0 a252a224a39a117xa166a19a167a26
F
x +
F
u ·
f
x +
F
v ·
g
x = 0?G
x +
G
u ·
f
x +
G
v ·
g
x = 0
a41a131a167a26?f?x =?1J D(F,G)D(x,v),?g?x =?1J D(F,G)D(x,u)
a211a110a140a26?f?y =?1J D(F,G)D(y,v),?f?z =?1J D(F,G)D(z,v),?g?y =?1J D(F,G)D(u,y),?g?z =?1J D(F,G)D(u,z)
5,a23?i(x)(i = 1,2,···,n)a180xa27a235a89a140a19a188a234a167a133
Gi(x1,···,xn) = Fi(?1(x1),···,?n(xn))
a121a178?(G1,G2,···,Gn)?(x
1,x2,···,xn)
=?(?(x1),···,?n(xn))
nproducttext
i=1
i prime(xi)
a217a165?(x1,x2,···,xn) = D(F1,F2,···,Fn)D(x
1,x2,···,xn)
nproducttext
i=1
i prime(xi) =?1 prime(x1)?2 prime(x2)···?n prime(xn).
a121a178a181a207?i(x)(i = 1,2,···,n)a180xa27a235a89a140a19a188a234a167a133Gi(x1,···,xn) = Fi(?1(x1),···,?n(xn))
a75?Gi?x
j
=?Fi
j
·j?x
j
=?Fi
j
i prime(xj)(i,j = 1,2,···,n)
a117a180?(G1,G2,···,Gn)?(x
1,x2,···,xn)
=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
G1
x1
G1
x2 ···
G1
xn
G2
x1
G2
x2 ···
G2
xn
..........................
Gn
x1
Gn
x2 ···
Gn
xn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
F1
1(x1)?1
prime(x1)?F1
2(x2)?2
prime(x2) ···?F1
n(xn)?n
prime(xn)
F2
1(x1)?1
prime(x1)?F2
2(x2)?2
prime(x2) ···?F2
n(xn)?n
prime(xn)
.............................................................
Fn
1(x1)?1
prime(x1)?Fn
2(x2)?2
prime(x2) ···?Fn
n(xn)?n
prime(xn)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
=
1 prime(x1)?2 prime(x2)···?n prime(xn)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
F1
1(x1)
F1
2(x2) ···
F1
n(xn)
F2
1(x1)
F2
2(x2) ···
F2
n(xn)
......................................
Fn
1(x1)
Fn
2(x2) ···
Fn
n(xn)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
=
nproducttext
i=1
i prime(xi) D(F1,F2,···,Fn)D(?
1(x1),?2(x2),···,?n(xn))
=
(?(x1),···,?n(xn))
nproducttext
i=1
i prime(xi).
6,a23F(x,y,z)a107a19a30a235a89a160a19a234a167a191a100F(x,y,z) = 0a140a40a189z = f(x,y).a63a216z = f(x,y)a27a52a138a27a55a135a218a191a169a94
238
a135.a50a166a100
x2 +y2 +z2?2x+ 2y?4z?10 = 0
a164a40a189a27z = f(x,y)a27a52a138.
a121a178a181a207a188a234z = f(x,y)a18a26a52a138a27a55a135a94a135a143
braceleftbigg z
x = fx(x,y) = 0
zy = fy(x,y) = 0
a113zx =?FxF
z
,zy =?FyF
z
a167a75F(x,y,z)a18a26a52a138a27a55a135a94a135a143
braceleftbigg F
x = 0
Fy = 0
a113a219a188a234a18a52a138a27a27a191a169a94a135a134a119a188a234a97a211a167a144a180a166a19a30a160a19a234a158a135a94a219a188a234a27a112a30a160a19a234a166a123
a45
braceleftbigg F
x = 2x?2 = 0
Fy = 2y+ 2 = 0 a41a26
braceleftbigg x = 1
y =?1 a167a233a65a27za138a143z1 =?2,z2 = 6
a113?z?x = x?12?z,?z?y = 1 +y2?z,?
2z
x2 =
(x?1)2 + (2?z)2
(2?z)3,
2z
y2 =
(1 +y)2 + (2?z)2
(2?z)3,
2z
x?y =
(x?1)(1 +y)
(2?z)3
a117a180a51a58(1,?1,?2),?
2z
x2 =
1
4,
2z
y2 =
1
4,
2z
x?y = 0a167a100
1
4 ·
1
4?0 =
1
16 > 0a57
1
4 > 0a167a75z1 =?2a143a52a2a138a182
a51a58(1,?1,6),?
2z
x2 =?
1
4,
2z
y2 =?
1
4,
2z
x?y = 0a167a100
parenleftBigg
14
parenrightBiggparenleftBigg
14
parenrightBigg
0 = 116 > 0a57?14 < 0a167a75z2 = 6a143a52
a140a138
239
§2,a188a234a49a15a170a27a53a159a33a188a234a131a39
1,a121a178
x = rcosθcos?
y = rcosθsin?
z = rsinθ
a188a234a213a225
a121a178a181a207D(x,y,z)D(r,θ,?) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
cosθcosrsinθcosrcosθsin?
cosθsinrsinθsin? rcosθcos?
sinθ rcosθ 0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle=?r2 cosθ
a75a51r negationslash= 0a133θ negationslash= kpi+ pi2a27a171a141Da83D(x,y,z)D(r,θ,?) negationslash= 0
a117a180a226a189a1105a167a26a6a188a234a124a51a171a141Da83a188a234a213a225.
2,a121a178
y1 = x1 +x2 +···+xn
y2 = x21 +x22 +···+x2n
y3 = x1x2 +x1x3 +···+xn?1xn
a188a234a131a39a167a191a21a209a217a188a234a39a88a170.
a121a178a181a207a127a51a188a234?(t1,t2) = 12(t21?t2)a167a166a26y3 =?(y1,y2) = 12(y21?y2)a51a18a135na145a152a109(x1,x2,···,xn)a83
a164a143a240a31a170
a75a188a234a124a51a18a135na145a152a109a165a188a234a131a39a167a217a188a234a39a88a170a143y3 = 12(y21?y2).
3,a101a15a188a234a180a196a131a39a186
(1) x?yx?z,y?zy?x,z?xz?y
(2) x1?x?y?z,y1?x?y?z,z1?x?y?z
a41a181
(1) a207f1 ·f2 ·f3 =?1a167a75a188a234a131a39.
(2) a45f1(x,y,z) = x1?x?y?z,f2(x,y,z) = y1?x?y?z,f3(x,y,z) = z1?x?y?z
a75Jocobia221a10a143
f1
x
f1
y
f1
z
f2
x
f2
y
f2
z
f3
x
f3
y
f3
z
=
1?y?z
(1?x?y?z)2
x
(1?x?y?z)2
x
(1?x?y?z)2
y
(1?x?y?z)2
1?x?z
(1?x?y?z)2
y
(1?x?y?z)2
z
(1?x?y?z)2
z
(1?x?y?z)2
1?x?y
(1?x?y?z)2
a113a100a221a10a27a157a1433a167a75a226a189a1106a167a26a188a234a124a188a234a213a225.
240
a49a110a220a169 a185a235a67a254a27a200a169a218a50a194a200a169
a49a155a212a217 a185a235a67a254a27a200a169
1,a23F(y) =
integraldisplay y2
y
e?x2y dxa167a79a142F prime(y).
a41a181a207a189a1104a94a135a247a118a167a65a94a189a1104a167a107
F prime(y) =
integraldisplay y2
y
(?x2)e?x2y dx+ 2ye?y5?e?y3 = 52 ye?y5? 32e?y3? 12y F(y).
2,a23F(y) =
integraldisplay y
0
(x+y)f(x)dxa167a217a165f(x)a143a140a135a188a234a167a166F primeprime(y).
a41a181a207f(x)a143a140a135a188a234a167a75f(x)a235a89a167a117a180(x+y)f(x)a235a89a167a75a189a1104a94a135a247a118
a117a180F prime(y) =
integraldisplay y
0
f(x)dx+ 2yf(y)a167F primeprime(y) = 3f(y) + 2yfprime(y).
3,a101F(y) =
integraldisplay 1
0
ln
radicalbig
x2 +y2 dxa167a134a26a79a142a200a169a167a166a209F(y)a167a50a166a209F prime(0)a167a191a117a8a65a94a189a1104a79a142F prime(0)a27a20a40
a53.
a41a181a8y negationslash= 0a158a167a107F(y) =
integraldisplay 1
0
ln
radicalbig
x2 +y2 dx = xln
radicalbig
x2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
integraldisplay 1
0
x2
x2 +y2 dx = ln
radicalbig
1 +y2? 1 +
y
integraldisplay 1
0
dxy
1 +
parenleftBigg
x
y
parenrightBigg2 dx = ln
radicalbig
1 +y2?1 +yarctan 1y,
a207F(0) =
integraldisplay 1
0
lnxdx =?1
a75F+ prime(0) = lim
y→+0
F(y)?F(0)
y =
pi
2,F?
prime(0) = lim
y→?0
F(y)?F(0)
y =?
pi
2
a117a180F prime(0)a216a127a51
a44a152a144a161a167a8x> 0a158a167ylnradicalbigx2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
y=0
= yx2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
y=0
= 0a167a75
integraldisplay 1
0
parenleftBigg
yln
radicalbig
x2 +y2
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
y=0
dx = 0
a113F+ prime(0) = pi2 negationslash= 0 =
integraldisplay 1
0
parenleftBigg
yln
radicalbig
x2 +y2
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
y=0
dx,F? prime(0) =?pi2 negationslash= 0 =
integraldisplay 1
0
parenleftBigg
yln
radicalbig
x2 +y2
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
y=0
dx
a75a8y = 0a158a167a216a85a51a200a169a210a101a166a19a234a167a61a166a166a134a33a109a19a234a143a216a49.
4,a166a188a234
E(k) =
integraldisplay pi
2
0
radicalBig
1?k2 sin2?d?a218F(k) =
integraldisplay pi
2
0
d?radicalbig
1?k2 sin2?
(0 <k< 1)
a27a19a234a133a121a178E(k)a247a118a144a167a181
Eprimeprime(k) + 1k Eprime(k) + E(k)1?k2 = 0
.
a41a181Eprime(k) =?
integraldisplay pi
2
0
ksin2?radicalbig
1?k2 sin2?
d? = 1k
integraldisplay pi
2
0
bracketleftBiggradicalBig
1?k2 sin2 1radicalbig
1?k2 sin2?
bracketrightBigg
d? = 1k[E(k)?F(k)]
F prime(k) =
integraldisplay pi
2
0
ksin2?
(1?k2 sin2?)32
d? =?1k
integraldisplay pi
2
0
bracketleftBigg
1radicalbig
1?k2 sin2?
1
(1?k2 sin2?)32
bracketrightBigg
d? =?1kF(k)+1k
integraldisplay pi
2
0
1
(1?k2 sin2?)32
d?
a207 dd?
parenleftBigg
k2 sin?cos?radicalbig
1?k2 sin2?
parenrightBigg
= k
2(cos2sin2?)(1?k2 sin2?) +k4 sin2?cos2?
(1?k2 sin2?)32
= k
2?1 + (1?k2 sin2?)2
(1?k2 sin2?)32
=
k2?1
(1?k2 sin2?)32
+
radicalbig
1?k2 sin2?
a75 1
(1?k2 sin2?)32
= 11?k2
radicalbig
1?k2 sin2 11?k2 dd?
parenleftBigg
k2 sin?cos?radicalbig
1?k2 sin2?
parenrightBigg
a117a180
integraldisplay pi
2
0
d?
(1?k2 sin2?)32
= 11?k2
integraldisplay pi
2
0
radicalBig
1?k2 sin2?d? = 11?k2 E(k)
241
a75F prime(k) = 1k(1?k2) E(k)? 1k F(k)
a117a180Eprimeprime(k) = (E
prime(k)?F prime(k))k?(E(k)?F(k))
k2 =?
E(k)
k2(1?k2)?
F(k)
k2
a108a13Eprimeprime(k) + 1k Eprime(k) + E(k)1?k2 =? E(k)k2(1?k2) + F(k)k2 + E(k)?F(k)k2 + E(k)1?k2 = 0.
5,a239a196a188a234
F(y) =
integraldisplay 1
0
yf(x)
x2 +y2 dx,(ygreaterorequalslant 0)
a27a235a89a53a167a217a165f(x)a180[0,1]a254a235a89a133a143a20a27a188a234.
a41a181a23d>c> 0a167a18y ∈ [c,d]a167a75a26a200a188a234 yf(x)x2 +y2 a51[0,1]×[c,d]a254a235a89
a100a189a1101a167a26F(y) =
integraldisplay 1
0
yf(x)
x2 +y2 dxa51[c,d]a254a235a89a167a100c,da27a63a191a53a167a26F(y)a51y> 0a235a89
a113f(x)a180[0,1]a254a235a89a133a143a20a27a188a234a167a75f(x)a51[0,1]a254a55a107a129a2a138m> 0
a100a117F(y) greaterorequalslantm
integraldisplay 1
0
y
x2 +y2 dx = marctan
1
ya57 limy→+0arctan
1
y =
pi
2 a167a75 limy→+0F(y) greaterorequalslant
mpi
2 > 0
a113F(0) = 0a167a75F(y)a8y = 0a158a216a235a89.
6,a65a94a233a235a234a166a19a123a79a142a200a169a181
(1)
integraldisplay pi
2
0
ln(a2?sin2x)dx (a> 1)(a216a55a189a126a234a167a101a79a142a158a209a121a195a46a156a185a167a18a52a129a79a142)a182
(2)
integraldisplay pi
0
ln(1?2acosx+a2)dx (|a|< 1)
a41a181
(1) a23I(a) =
integraldisplay pi
2
0
ln(a2?sin2x)dx
a207a26a200a188a234ln(a2?sin2x)a51
bracketleftBigg
0,pi2
bracketrightBigg
×[1,+∞]a254a216a235a89
a75a216a85a94a189a1102a167a143a10a85a94a189a110a167a160a2a137a140
bracketleftBigg
0,pi2
bracketrightBigg
×[b,c](b> 1,c→ +∞)
a249a158f(x,a) = ln(a2?sin2x)a57fa = 2aa2?sin2xa209a51a52a221a47
bracketleftBigg
0,pi2
bracketrightBigg
×[b,c]a254a235a89
a100a189a1102a167a107Iprime(a) =
integraldisplay pi
2
0
2a
a2?sin2x dx =
2√
a2?1
bracketleftBigg
arctan a+ 1√a2?1 + arctan a?1√a2?1
bracketrightBigg
= pi√a2?1
a233aa200a169a167a26I(a) = piln|a+√a2?1|+C
a207a∈ [b,c]a167a100b,ca27a63a191a53a167a75I(a) = piln|a+√a2?1|+C
(2) a23I(a) =
integraldisplay pi
0
ln(1?2acosx+a2)dx
a8|a|< 1a158a167a100a1171?2acosx+a2 greaterorequalslant 1?2|a|+a2 = (1?|a|)2 > 0
a75f(x,a) = ln(1?2acosx+a2)a57fa =?2cosx+ 2a1?2acosx+a2 a209a51a52a221a47[0,pi]×[?b,b]a254a235a89(|a|lessorequalslantb< 1)
a100a189a1102a167a107Iprime(a) =
integraldisplay pi
0
2cosx+ 2a
1?2acosx+a2 dx =
pi
a?
1?a2
a
integraldisplay pi
0
dx
(1 +a2)?2acosx =
pi
a?
1?a2
a(1 +a2)
integraldisplay pi
0
dx
1 +
parenleftBig
2aa2+1
parenrightBig
cosx
a138a147a134t = tan x2
a75
integraldisplay pi
0
dx
1 +
parenleftBig
2aa2+1
parenrightBig
cosx
= 2
integraldisplay +∞
0
1 +a2
(1?a)2 + (1 +a)2t2 dt =
1 +a2
1?a2 pi
a117a180Iprime(a) = 0a167a108a13I(a) = C
a113I(0) = 0a167a75C = 0a167a117a180I(a) = 0
a207a∈ [?bb]a167a100ba27a63a191a53a167a26a8|a|< 1a158a167I(a) = 0.
7,a65a94a200a169a210a101a166a200a169a144a123a79a142a200a169a181
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xb?xa
lnx dx (a> 0,b> 0)
242
(a101a209a121a195a46a156a185a134a99a161a211a24a63a110).
a41a181a216a148a23a<b
a207
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xb?xa
lnx dx =
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
dx
integraldisplay b
a
xy dy =
integraldisplay b
a
dy
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xy dx.
a249a112a167a8x = 0a158a167sin
parenleftBigg
ln 1x
parenrightBigg
xya110a41a1430a167a108a13sin
parenleftBigg
ln 1x
parenrightBigg
xya510 lessorequalslantxlessorequalslant 1,alessorequalslantylessorequalslantba254a235a89
a75a140a65a94a200a169a210a101a27a200a169a123a2a134a200a169a103a83
a138a147a134x = e?ta167a140a26
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xy dx =
integraldisplay +∞
0
e?(y+1)t sintdt = 11 + (1 +y)2
a117a180
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xb?xa
lnx dx =
integraldisplay b
a
dy
1 + (1 +y)2 = arctan(1+b)?arctan(1+a) = arctan
b?a
1 + (1 +b)(1 +a)
8,a121a178
integraldisplay 1
0
dx
integraldisplay 1
0
x2?y2
(x2 +y2)2 dy negationslash=
integraldisplay 1
0
dy
integraldisplay 1
0
x2?y2
(x2 +y2)2 dx.
a121a178a181a207
integraldisplay 1
0
dx
integraldisplay 1
0
x2?y2
(x2 +y2)2 dy =
integraldisplay 1
0
dx
1 +x2 =
pi
4 a167
integraldisplay 1
0
dy
integraldisplay 1
0
x2?y2
(x2 +y2)2 dx =?
integraldisplay 1
0
dy
1 +y2 =?
pi
4
a75
integraldisplay 1
0
dx
integraldisplay 1
0
x2?y2
(x2 +y2)2 dy negationslash=
integraldisplay 1
0
dy
integraldisplay 1
0
x2?y2
(x2 +y2)2 dx.
9,a23a188a234f(x,y)a51D = [a,A;b,B]a107a46a167a216a22Da83a107a129a94a235a89a173a130y =?i(x)a167fa51Da235a89a167a121a178a181
F(x) =
integraldisplay B
b
f(x,y)dy
a51[a,A]a235a89.
a121a178a181a216a148a23a144a107a152a94a235a89a173a130y =?1(x)a167f(x,y)a51a249a94a173a130a254a109a228
a207f(x,y)a107a46a167a80M = sup
[a,A;b,B]
|f(x,y)|
a63a18x0 ∈ [α,β]? [a,A],y0 =?1(x0) ∈ [b,B]
a101a121F(x) =
integraldisplay B
b
f(x,y)dya51x0a58a235a89a167a61a121?ε> 0,?δ> 0a167a8|x?x0|<δa158a167a107
|F(x)?F(x0)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay B
b
f(x,y)dy?
integraldisplay B
b
f(x0,y)dy
vextendsinglevextendsingle
vextendsinglevextendsingle<ε
a100a117y =?1(x)a51x0a58a235a89a167a75a233?ε1 > 0,?δ1 > 0a167a8|x?x0|< 2δ1a158a167a107|y?y0| = |?1(x)1(x0)|<ε1
a117a180a51x0?δ1 lessorequalslantxlessorequalslantx0 +δ1,blessorequalslantylessorequalslantBa27a145a141a83a166f(x,y)a109a228a27a58a144a185a117a177(x0,y0)a143a165a37a27a221a47a141x0?δ1 lessorequalslant
xlessorequalslantx0 +δ1,y0?ε1 <y <y0 +ε1a51a249a145a141a27a254a33a101a252a253(a101y0?ε1a84a208a31a117ba189y0 +ε1a84a208a31a117Ba167a75a144a107
a254a253a189a101a253)a167a52a141a165f(x,y)a143a235a89
a207a13a51a249a252a135(a189a152a135)a52a141a165f(x,y)a143a152a151a235a89a167a65a79a233?ε2 > 0,?δ2 > 0a167a8|x?x0|<δ2a158a167a107
|f(x,y)?f(x0,y)|<ε2
a121a18δ = min(δ1,δ2)a167a8|xprime?x0|<δa158a167a107
|F(xprime)?F(x)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay B
b
f(xprime,y)dy?
integraldisplay B
b
f(x0,y)dy
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay y0?ε1
b
|f(xprime,y)?f(x0,y)|dy+
integraldisplay y0+ε1
y0?ε1
|f(xprime,y)|dy+
integraldisplay y0+ε1
y0?ε1
|f(x0,y)|dy+
integraldisplay B
y0+ε1
|f(xprime,y)?f(x0,y)|dylessorequalslantε2(B?b) + 4ε1M
a101a18ε1 = ε8M,ε1 = ε2(B?b)
a75a8|xprime?x0|<δa158a167a107|F(xprime)?F(x0)|<ε
a8x0 ∈ [a,α]a189x0 ∈ [β,A]a158a167F(x)a51x0a235a89a167a25F(x)a51[a,A]a235a89
a101f(x,y)a107a109a228a27a235a89a173a130a107a65a94a167a75a144a73a114a166f(x,y)a140a85a164a143a109a228a27a58a94a150a245a65a135a2a221a47a133a109a210a49a10
a217a123a216a121a131a211
a100a117f(x,y)a107a46a133a150a245a107a65a94a109a228a130a167a75F(x) =
integraldisplay B
b
f(x,y)dya127a51a133a51[a,A]a235a89.
243
a49a155a108a217 a185a235a67a254a27a50a194a200a169
1,a121a178a181a101a51[a,∞;c,d]a83a164a225|f(x,y)|lessorequalslantF(x,y)a167a191a133a39a117y ∈ [c,d]a200a169
integraldisplay +∞
a
F(x,y)dxa152a151a194a241a167a75
integraldisplay +∞
a
f(x,y)dxa39
a117y ∈ [c,d]a189a152a151a194a241a167a133a253a233a194a241.
a121a178a181a207a200a169
integraldisplay +∞
a
F(x,y)dxa39a117y ∈ [c,d]a152a151a194a241a167a75a100a185a235a67a254a27a50a194a200a169a27a133a220a152a151a194a241a6a110a167a26
a233?ε> 0a167a127a51a134ya195a39a27A0(ε) >aa167a8A,Aprime greaterorequalslantA0a158a167a233a152a131y ∈ [c,d]a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprime
A
F(x,y)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε
a13
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprime
A
f(x,y)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprime
A
|f(x,y)|dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprime
A
F(x,y)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<εa233a152a131y ∈ [c,d]a209a164a225
a100a195a161a129a185a235a67a254a50a194a200a169a27a133a220a152a151a194a241a6a110a167
integraldisplay +∞
a
f(x,y)dxa39a117y ∈ [c,d]a152a151a194a241a167
integraldisplay +∞
a
|f(x,y)|dxa39
a117y ∈ [c,d]a152a151a194a241
a75
integraldisplay +∞
a
f(x,y)dxa39a117y ∈ [c,d]a152a151a194a241a133a253a233a194a241.
2,a121a178a101a15a200a169a51a164a137a189a27a171a109a83a152a151a194a241a181
(1)
integraldisplay +∞
0
cosxy
x2 +y2 dx (ygreaterorequalslanta> 0)
(2)
integraldisplay +∞
0
cosxy
x2 + 1 dx (?∞<y< +∞)
(3)
integraldisplay 1
0
lnxydx
parenleftBigg
1
blessorequalslantylessorequalslantb,b> 1
parenrightBigg
a121a178a181
(1) a207ygreaterorequalslanta> 0a167a75
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosxy
x2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
x2 +a2 a13
integraldisplay +∞
0
dx
x2 +a2 =
pi
2aa194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
0
cosxy
x2 +y2 dxa39a117ya51[a,+∞)(a> 0)a83a152a151a194a241.
(2) a207y ∈ (?∞,+∞),
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosxy
x2 + 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
x2 + 1 a13
integraldisplay +∞
0
dx
x2 + 1 =
pi
2 a194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
0
cosxy
x2 + 1 dxa39a117ya51(?∞,+∞)a83a152a151a194a241.
(3) x = 0a180a219a58a167a81blessorequalslantylessorequalslantb,b> 1,0 <xlessorequalslant 1a158a167|lnxy|lessorequalslant|lnx|+|lny|lessorequalslant?lnx+ lnb = ln bx
a207 lim
x→+0
x14 ln bx = lim
x→+0
ln bx
x?14
= 0a167a75a100a195a46a188a234a50a194a200a169a7a79a123a27a52a129a47a170a167a26
integraldisplay 1
0
ln bx dxa194a241
a108a13a100a159a188a7a79a123a167a26
integraldisplay 1
0
lnxydxa39a117ya51[ 1b,b](b> 1)a254a152a151a194a241.
3,a23f(x,y)a51[a,+∞;c,d]a235a89a167a233[c,d)a254a122a152a135ya167
integraldisplay +∞
a
f(x,y)dxa194a241a167a2a200a169a51y = da117a209.
a121a178a249a200a169a51[c,d]a154a152a151a194a241.
a121a178a181a100
integraldisplay +∞
a
f(x,d)dxa117a209a167a26?ε0 > 0,?A0 >a,?Aprime,Aprimeprime greaterorequalslantA0a167a166
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
f(x,d)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslantε0
a249a76a178a233y = d∈ [c,d]a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
f(x,y)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslantε0a167a96a178
integraldisplay +∞
a
f(x,y)dxa51[c,d]a154a152a151a194a241.
4,a63a216a101a15a200a169a51a141a189a171a109a27a152a151a194a241a53a181
(1)
integraldisplay +∞
1
xαe?x dx (alessorequalslantαlessorequalslantb;a,ba143a63a191a162a234)
(2)
integraldisplay +∞
0
√αe?αx2 dx (0 <α< +∞)
244
(3)
integraldisplay +∞
∞
e?(x?α)2 dx
(i) a<α<b
(ii)?∞<α< +∞
(4)
integraldisplay 1
0
xp?1 ln2xdx
(i) pgreaterorequalslantp0 > 0
(ii) p> 0
(5)
integraldisplay +∞
0
e?αx sinxdx (α> 0)
a41a181
(1) a207α∈ [a,b],x∈ (1,+∞)a167a750 <|xαe?x|lessorequalslantxbe?x
a113 lim
x→+∞
x2 ·xbe?x = 0a167a75a226a195a161a129a50a194a200a169a27a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
xbe?x dxa152a151a194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
1
xαe?x dxa39a117α∈ [a,b](a,ba143a63a191a162a234)a152a151a194a241.
(2)
integraldisplay +∞
0
√αe?αx2 dx = √pi
2 a194a241a167a2a167a51(0,+∞)a39a117αa154a152a151a194a241
a233?A> 0a167a207 lim
α→+0
integraldisplay +∞
A
√αe?αx2 dx = lim
α→+0
integraldisplay +∞
√αA e
t2 dt =
integraldisplay +∞
0
e?t2 dt =
√pi
2
a25a233a1170 < ε0 <
√pi
2 a167a55a127a51α0 > 0a167a166a26
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
A
√α
0e?α0x
2 dx
vextendsinglevextendsingle
vextendsinglevextendsingle =
integraldisplay +∞
A
√α
0e?α0x
2 dx > ε
0a167
a61
integraldisplay +∞
0
√αe?αx2 dxa39a117αa51(0,+∞)a254a216a152a151a194a241.
(3) a233a63a191a27a189a27α∈ (?∞,+∞)a167a200a169
integraldisplay +∞
∞
e?(x?α)2 dxa209a194a241a167a133
integraldisplay +∞
∞
e?(x?α)2 dx = √pi
(i) |x|a191a169a140a158a167a233a152a131a<α<ba167a1070 <e?(x?α)2 < 2e?x
2
4
a207
integraldisplay +∞
∞
e?x
2
4 dx = 2
integraldisplay +∞
0
e?x
2
4 dxa194a241
a75a100a159a188a7a79a123a167a26
integraldisplay +∞
∞
e?(x?α)2 dxa233a<α<ba152a151a194a241.
(ii) a233?A> 0a167a107 lim
α→+∞
integraldisplay +∞
A
e?(x?α)2 dx = lim
α→+∞
integraldisplay +∞
A?α
e?t2 dt = √pi
a75a8αa191a169a140a158a167
integraldisplay +∞
A
e?(x?α)2 dx>
√pi
2
a100a100a167a26
integraldisplay +∞
0
e?(x?α)2 dxa51?∞<α< +∞a254a154a152a151a194a241
a108a13
integraldisplay +∞
∞
e?(x?α)2 dxa51?∞<α< +∞a254a154a152a151a194a241.
(4) (i) |xp?1 ln2x| = xp?1 ln2xlessorequalslantxp0?1 ln2x (pgreaterorequalslantp0 > 0,0 lessorequalslantxlessorequalslant 1)
a200a169
integraldisplay 1
0
xp?1 ln2xdx =
integraldisplay +∞
0
e?p0zz2 dz
lim
z→+∞
z2 ·e?p0zz2 = lim
z→+∞
z4
ep0z = 0 (p0 > 0)
a75a100a133a220a7a79a123a27a52a129a47a170
integraldisplay +∞
0
e?p0zz2 dza194a241a167a117a180
integraldisplay 1
0
xp0?1 ln2xdxa194a241
a108a13a100a159a188a7a79a123a167a26
integraldisplay 1
0
xp?1 ln2xdxa39a117pa51pgreaterorequalslantp0 > 0a254a152a151a194a241.
(ii) a207a8x∈
parenleftBigg
0,1e
parenrightBigg
,ln2xgreaterorequalslant 1
a25a107
integraldisplay 1
0
xp?1 ln2xdx>
integraldisplay 1
e
0
xp?1 ln2xdx>
integraldisplay 1
e
0
xp?1 dx = 1p
parenleftBigg
1
e
parenrightBiggp
→ +∞ (p→ +0)
a117a180
integraldisplay 1
0
xp?1 ln2xdxa51p> 0a158a216a152a151a194a241.
245
(5) a94a135a121a123.
a98a23
integraldisplay +∞
0
e?αx sinxdxa39a117α > 0a152a151a194a241a167a75a233?ε > 0,?M > 0a167a8Aprimeprime > Aprime > Ma158a167a233a152
a131α> 0a164a225
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
e?αx sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε
a108a13a233a117α∈ (0,1)a189a164a225
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
e?αx sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε
a51a216a31a170a252a62a45α→ 0a167a75a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslantεa167a108a13
integraldisplay +∞
0
sinxdxa194a241
a13
integraldisplay A
0
sinxdx = 1?cosAa167a8A→ +∞a158a167cosAa27a52a129a216a127a51a167a117a180
integraldisplay +∞
0
sinxdxa117a209a167a75a103a241a167a25
a98a23a216a164a225
a108a13
integraldisplay +∞
0
e?αx sinxdxa39a117α> 0a216a152a151a194a241.
5,a121a178a181
(1)
integraldisplay +∞
0
α
x2 +α2 dxa51a216a185α = 0a27a63a219a171a109a254a180a235a89a188a234a182
(2) F(p) =
integraldisplay pi
0
sinx
xp(pi?x)2?p dxa51(0,2)a83a235a89.
a121a178a181
(1) a23F(α) =
integraldisplay +∞
0
α
x2 +α2 dx.
a233a63a219α0 negationslash= 0a167a216a148a23α0 > 0a167a56a18δ> 0a167a166a26α0?δ> 0a167a101a121F(α)a51[α0?δ,α0 +δ]a83a152a151a194a241
a175a162a254a167a8α∈ [α0?δ,α0 +δ]a158a167 αx2 +α2 lessorequalslant α0 +δx2 + (α
0?δ)2
a207a200a169
integraldisplay +∞
0
α0 +δ
(α0?δ)2 +x2 dxa194a241a167a75a100a159a188a7a79a123a167a26F(α)a51[α0?δ,α0 +δ]a254a39a117αa152a151a194a241
a117a180a100a235a89a189a110a167a26F(α)a51a84a171a109a254a180αa27a235a89a188a234a167a65a79a51α0a58a235a89
a100a117α0 negationslash= 0a27a63a191a53a167a26
integraldisplay +∞
0
α
x2 +α2 dxa233a63a219αnegationslash= 0a235a89a167a100a100a140a127F(α)a51a63a219a216a185α = 0a27a171a109a254
a209a235a89
a2a100 lim
α→+0
integraldisplay +∞
0
α
α2 +x2 dx =
pi
2,limα→?0
integraldisplay +∞
0
α
α2 +x2 dx =?
pi
2
a26F(α)a51α = 0a63a216a235a89a167a75
integraldisplay +∞
0
α
x2 +α2 dxa51a216a185α = 0a27a63a219a171a109a254a180a235a89a188a234.
(2) a63a18p∈ (0,2)a167a75a127a510 <p1,p2 < 2a167a1660 <p1 lessorequalslantplessorequalslantp2 < 2
a2070a218pia254a140a85a180a219a58a167a242a200a169a169a143a110a227integraldisplay
pi
0
sinx
xp(pi?x)2?p dx =
integraldisplay 1
0
sinx
xp(pi?x)2?p dx+
integraldisplay pi?1
1
sinx
xp(pi?x)2?p dx+
integraldisplay pi
pi?1
sinx
xp(pi?x)2?p dx
a233a117
integraldisplay 1
0
sinx
xp(pi?x)2?p dx
a207 sinxxp(pi?x)2?p lessorequalslant sinxxp
2(pi?x)2?p2
(0 lessorequalslantxlessorequalslant 1,0 <p1 lessorequalslantplessorequalslantp2 < 2)
a133 lim
x→+0
xp2?1 sinxxp
2(pi?x)2?p2
= 1pi2?p
2
a207p2 < 2a167a75p2?1 < 1a167a117a180a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay 1
0
sinx
xp2(pi?x)2?p2 dxa194a241
a108a13a100a159a188a7a79a123a167a26
integraldisplay 1
0
sinx
xp(pi?x)2?p dxa39a117p∈ [p1,p2 ]a152a151a194a241
a113a26a200a188a234 sinxxp(pi?x)2?pa51(0,1]×[p1,p2 ]a254a235a89a167a75a100a235a89a53a189a110a167a26
integraldisplay 1
0
sinx
xp(pi?x)2?p dxa51[p1,p2 ]a235
a89integraldisplay
pi?1
1
sinx
xp(pi?x)2?p dxa180a185a235a67a254a27a126a194a200a169
246
a207a26a200a188a234 sinxxp(pi?x)2?pa51[1,pi?1]×[p1,p2 ]a235a89a167a75a100a235a89a53a189a110a167a26
integraldisplay pi?1
1
sinx
xp(pi?x)2?p dxa51[p1,p2 ]a235
a89
a233a117
integraldisplay pi
pi?1
sinx
xp(pi?x)2?p dx
a207 sinxxp(pi?x)2?p lessorequalslant sin(pi?x)xp
1(pi?x)2?p1
(pi?1 lessorequalslantxlessorequalslantpi,0 <p1 lessorequalslantplessorequalslantp2 < 2)
a133 lim
x→pi?0
(pi?x)1?p1 sin(pi?x)xp
1(pi?x)2?p1
= 1pip
1
a207p1 > 0a167a751?p1 < 1a167a117a180a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay pi
pi?1
sin(pi?x)
xp?1(pi?x)2?p1 dxa194a241
a108a13a100a159a188a7a79a123a167a26
integraldisplay pi
pi?1
sinx
xp(pi?x)2?p dxa39a117p∈ [p1,p2 ]a152a151a194a241
a113a26a200a188a234 sinxxp(pi?x)2?pa51[pi?1,pi)×[p1,p2 ]a254a235a89a167a75a100a235a89a53a189a110a167a26
integraldisplay pi
pi?1
sinx
xp(pi?x)2?p dxa51[p1,p2 ]a235
a89
a110a220a177a254a167a26F(p)a51[p1,p2 ]a235a89a167a108a13a51a217a254a63a152a58pa235a89
a113a100p∈ (0,2)a27a63a191a53a167a26F(p) =
integraldisplay pi
0
sinx
xp(pi?x)2?p dxa51(0,2)a83a235a89..
6,a23f(t)a8t> 0a158a235a89.a88a74
integraldisplay +∞
0
tλf(t)dta8λ = a,λ = ba158a209a194a241a167a64a34
integraldisplay +∞
0
tλf(t)dta39a117λa51[a,b]a254a152a151a194
a241.
a121a178a181a207f(t)a8t> 0a158a235a89a167a75a26a200a188a234tλf(t)a27a219a58a144a140a85a1800
a117a180
integraldisplay +∞
0
tλf(t)dt =
integraldisplay 1
0
tλf(t)dt+
integraldisplay +∞
1
tλf(t)dt
a233a117
integraldisplay 1
0
tλf(t)dt =
integraldisplay 1
0
tλ?ataf(t)dt
a207
integraldisplay +∞
0
taf(t)dta194a241a167a75
integraldisplay 1
0
taf(t)dta194a241a167a108a13a39a117λ ∈ (?∞,+∞)a152a151a194a241a167a13tλ?aa233a117λ greaterorequalslant aa252a78a126
a133|tλ?a|lessorequalslant 1 (0 lessorequalslanttlessorequalslant 1,λgreaterorequalslanta)
a75a100Abela7a79a123a167a26
integraldisplay 1
0
tλf(t)dta39a117λgreaterorequalslantaa152a151a194a241
a233a117
integraldisplay +∞
1
tλf(t)dt =
integraldisplay +∞
1
tλ?btbf(t)dt
a207
integraldisplay +∞
0
tbf(t)dta194a241a167a75
integraldisplay +∞
1
tbf(t)dta194a241a167a108a13a39a117λ ∈ (?∞,+∞)a152a151a194a241a167a13tλ?ba233a117λ lessorequalslant ba252a78a126
a133|tλ?b|lessorequalslant 1 (1 lessorequalslantt< +∞,λlessorequalslantb)
a75a100Abela7a79a123a167a26
integraldisplay +∞
1
tλf(t)dta39a117λlessorequalslantba152a151a194a241
a117a180
integraldisplay +∞
0
tλf(t)dta39a117λa51[a,b]a254a152a151a194a241.
7,a108a31a170e
ax?e?bx
x =
integraldisplay b
a
e?xy dya209a117a167a79a142a200a169
integraldisplay +∞
0
e?ax?e?bx
x dx(b>a> 0)
a41a181a207e
ax?e?bx
x =
integraldisplay b
a
e?xy dya167a75
integraldisplay +∞
0
e?ax?e?bx
x dx =
integraldisplay +∞
0
dx
integraldisplay b
a
e?xy dy
a188a234e?xya51[0,+∞)×[a,b]a254a235a89
a113a233y ∈ [a,b](a> 0)a167a75|e?xy|lessorequalslante?axa133
integraldisplay +∞
0
e?ax dx = 1aa194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
0
e?xy dxa39a117ya51[a,b]a152a151a194a241
a100a200a169a2a134a94a83a189a110a167a26
integraldisplay +∞
0
e?ax?e?bx
x dx =
integraldisplay +∞
0
dx
integraldisplay b
a
e?xy dy =
integraldisplay b
a
dy
integraldisplay +∞
0
e?xy dx =
integraldisplay b
a
dy
y =
ln ba (b>a> 0)
247
8,a193a121a178Γ(s)a27a19a234a127a51a167a166a209Γprime(s)a27a200a169a76a136a170a167a96a178a237a19a76a167a180a220a110a27.
a121a178a181Γ(s) =
integraldisplay +∞
0
xs?1e?x dx
a207xs?1e?xa57s (xs?1e?x) = xs?1e?x lnxa510 <x< +∞,s> 0a254a235a89
integraldisplay +∞
0
xs?1e?x dx =
integraldisplay 1
0
xs?1e?x dx+
integraldisplay +∞
1
xs?1e?x dx
a233a117a63a191a27s> 0a167a111a140a180 <s0 lessorequalslantslessorequalslantS0
xs?1e?x lessorequalslantxs0?1e?x(0 lessorequalslantxlessorequalslant 1)
a207a101s0 < 1a1670a143a219a58a167a100 lim
x→+0
x1?s0xs0?1e?x = 1a57a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay 1
0
xs0?1e?xxdxa194a241a182
a101s0 greaterorequalslant 1a167a75
integraldisplay 1
0
xs0?1e?x dxa143a126a194a200a169a167a25a194a241
a111a131
integraldisplay 1
0
xs0?1e?x dxa194a241a167a108a13a100a159a188a7a79a123a167a26
integraldisplay 1
0
xs?1e?x dxa39a117sa51sgreaterorequalslants0a254a152a151a194a241
a113xs?1e?x lessorequalslantxS0?1e?x(1 lessorequalslantx< +∞)
a207 lim
x→+∞
x2xS0?1e?x = 0a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
xS0?1e?x dxa194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
1
xs?1e?x dxa39a117sa51slessorequalslantS0a254a152a151a194a241
a108a13
integraldisplay +∞
0
xs?1e?x dxa51[s0,S0 ]a254a152a151a194a241a167a25a194a241.
integraldisplay +∞
0
xs?1e?x lnxdx =
integraldisplay 1
0
xs?1e?x lnxdx+
integraldisplay +∞
1
xs?1e?x lnxdx
a233a254a161a270 <s0 lessorequalslantslessorequalslantS0a167|xs?1e?x lnx|lessorequalslantxs0?1|lnx| (0 <xlessorequalslant 1)
a207 lim
x→+0
x1?s02 xs0?1 lnx = lim
x→+0
lnx
x?s02
= 0a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay 1
0
xs0?1e?x|lnx|dx =?
integraldisplay 1
0
xs0?1e?x lnxdxa194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay 1
0
xs?1e?x lnxdxa51sgreaterorequalslants0a254a152a151a194a241
a113xs?1e?x lnx = xse?x lnxx <xS0e?x (1 lessorequalslantx< +∞)
a207 lim
x→+∞
x2xS0e?x = lim
x→+∞
xS0+2
ex = 0a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
xS0e?x dxa194a241a167a117a180a100a159a188
a7a79a123a167a26
integraldisplay +∞
1
xs?1e?x lnxdxa51slessorequalslantS0a254a152a151a194a241
a108a13
integraldisplay +∞
0
xs?1e?x lnxdxa51[s0,S0 ]a254a152a151a194a241
a75a100a200a169a210a101a166a19a189a110a167a26Γ(s)a51[s0,S0 ]a254a140a19a167a8a44a51sa140a19a167a133Γprime(s) =
integraldisplay +∞
0
xs?1e?x lnxdx
a50a100s> 0a27a63a191a53a167a26Γ(s)a51s> 0a140a19a133Γprime(s) =
integraldisplay +∞
0
xs?1e?x lnxdx.
9,(1) a108
integraldisplay +∞
0
e?y2 dy =
√pi
2 a237a209L(c) =
integraldisplay +∞
0
e?y2?c
2
y2 dy =
√pi
2 e
2ca182
(2) a124a94a200a169a210a101a166a19a27a123a75a218a209 dLdc =?2La53a166a26a211a152a40a74a167a191a237a209
integraldisplay +∞
0
e?ay2? by2 dy (a > 0,b > 0)a131
a138.
a121a178a181
(1) L(c) =
integraldisplay +∞
0
e?y2?c
2
y2 dy =
integraldisplay +∞
0
e?
parenleftBig
y?cy
parenrightBig2
2c dy = e?2c
integraldisplay +∞
0
e?
parenleftBig
y?cy
parenrightBig2
dy =
e?2c
integraldisplay +∞
0
e?
parenleftBig
y?cy
parenrightBig2
d
parenleftBigg
y? cy
parenrightBigg
+e?2c
integraldisplay +∞
0
e?
parenleftBig
y?cy
parenrightBig2
dcy
a51a99a152a200a169a165a45u = y? cy a167a51a0a152a200a169a165a45v = cy
a75L(c) = e?2c
integraldisplay +∞
∞
e?u2 du?e?2c
integraldisplay +∞
0
e?(v?cv)2 dv = √pie?2c?L(c)
248
a117a180L(c) =
integraldisplay +∞
0
e?y2?c
2
y2 dy =
√pi
2 e
2c.
(2) L(c) =
integraldisplay +∞
0
e?y2?c
2
y2 dy,
dL
dc = 2
integraldisplay +∞
0
e?y2?c
2
y2
parenleftBigg
cy2
parenrightBigg
dy
a45v = cy a167a75 dLdc =?2
integraldisplay +∞
0
e?v2?c
2
v2 dv =?2L(c)
a117a180lnL =?2c+ lnc0a61ln Lc
0
=?2ca189a61L = c0e?2c
a113L(0) =
integraldisplay +∞
0
e?y2 dy =
√pi
2 a167a75c0 =
√pi
2 a167a117a180L(c) =
√pi
2 e
2c
a75a45u = √aya167a107
integraldisplay +∞
0
e?ay2? by2 dy = 1√a
integraldisplay +∞
0
e?u2?(
√ab)2
u2 du =
1√
a ·
√pi
2 e
2√ab = 1
2
radicalBigg
pi
a e
2√ab (a> 0,b> 0).
249
a49a111a220a169 a245a67a254a200a169a198
a49a155a202a217 a200a169(a19a173a33a110a173a200a169a167a49a152a97
a173a130a33a173a161a200a169)a27a189a194a218a53a159
§2,a200a169a27a53a159
1,a121a178a165a138a189a110a181a101f(M),g(M)a51?a254a235a89a167g(M)a51?a216a67a210a167a75
integraldisplay
f(M)g(M)d? = f(P)
integraldisplay
g(M)d?
a217a165P ∈?.
a121a178a181a23?a180a107a46a52a171a141a133a107a221a254
a207f(M),g(M)a51?a254a235a89a167g(M)a51?a216a67a210
a75f(M),g(M)a51?a254a140a200a167a133a140a23g(M) greaterorequalslant 0a167M = max
M∈?
{f(M)},m = min
M∈?
{f(M)}
a100a53a1594a167a26m
integraldisplay
g(M)d? lessorequalslant
integraldisplay
f(M)g(M)d? lessorequalslantM
integraldisplay
g(M)d?
a101
integraldisplay
g(M)d? = 0a167a100a117g(M) greaterorequalslant 0a133a235a89a167a75a55a107g(M) ≡ 0,M ∈?a167a108a13
integraldisplay
f(M)g(M)d? = 0a61a135a121a216
a31a170a164a225a182
a101
integraldisplay
g(M)d? > 0a167a75mlessorequalslant
integraldisplay
f(M)g(M)d?
integraldisplay
g(M)d?
lessorequalslantM
a100a235a89a188a234a27a48a138a189a110a167a26a55a127a51P ∈?a167a166
integraldisplay
f(M)g(M)d?
integraldisplay
g(M)d?
= f(P)
a61
integraldisplay
f(M)g(M)d? = f(P)
integraldisplay
g(M)d?
a211a110a167a8g(M) lessorequalslant 0a158a167a189a107
integraldisplay
f(M)g(M)d? = f(P)
integraldisplay
g(M)d?.
2,a121a178a181a101f(M)a51?a254a235a89a167f(M) greaterorequalslant 0a167a2f(M) negationslash≡ 0a167a75
integraldisplay
f(M)d? > 0
.
a121a178a181a207f(M) greaterorequalslant 0a167f(M) negationslash≡ 0a167a75a150a8a127a51a152a58M0 ∈?a167a166a26f(M0) > 0
a113f(M)a51?a254a235a89a167a8a44a51M0a235a89a167a75a55a127a51δ> 0a167a8M ∈O(M0,δ)a158a167a107f(M) > 0
a117a180
integraldisplay
f(M)d? =
integraldisplay
\O(M0,δ)
f(M)d? +
integraldisplay
O(M0,δ)
f(M)d? greaterorequalslant
integraldisplay
O(M0,δ)
f(M)d? > 0
3,a121a178a181a101f(M)a51?a254a235a89a167a51?a27a63a219a220a169a171a141?primea254
integraldisplay
prime
f(M)d? = 0
a75f(M) ≡ 0
a100a100a121a178a181a101f(M),g(M)a51?a254a235a89a167a51?a27a63a219a220a169a171a141?primea254a164a225a181
integraldisplay
prime
f(M)d? =
integraldisplay
prime
g(M)d?
a75a51?a254a164a225a181f(M) ≡g(M).
a121a178a181a94a135a121a123.a101a127a51a58Mprime ∈?a167a166f(Mprime) negationslash= 0a167a216a148a23f(Mprime) > 0
a100a117f(M)a51?a254a235a89a167a75a127a51Mprimea27a25a141?prime = O(Mprime,δ)(δ> 0)a167a166a26f(M) > f(M
prime)
2 > 0,?M ∈?
prime
a117a180a107
integraldisplay
prime
f(M)d? greaterorequalslant f(M
prime)
2 ||?
prime||> 0a134a75a23
integraldisplay
prime
f(M)d? = 0a103a241
250
a75a98a23a216a164a225a167a61a107f(M) ≡ 0
a45F(M) = f(M)?g(M)a167a75a51?a27a63a219a220a169a171a141?primea254
integraldisplay
prime
F(M)d? =
integraldisplay
prime
f(M)d
integraldisplay
prime
g(M)d? = 0
a108a13a100a254a161a164a121a40a216a167a107F(M) ≡ 0a167a61f(M)?g(M) ≡ 0a189a61f(M) ≡g(M).
4,a101|f(M)|a51?a254a140a200a167a64a34f(M)a51?a254a180a196a140a200a186a127a9a188a234f(x,y) =?1a167a8xa218ya165a150a8a107a152a135a180a195a110a234
a158a182f(x,y) = 1a167a8xa218ya209a180a107a110a234a158a167a51[0,1;0,1]a254a27a200a169.
a41a181a153a55.
a175a162a254a167f(x,y)a51[0,1;0,1]a254a27a254a218a33a101a218a169a79a143Sprime =
summationdisplay
ik
Mikik = 1,S =
summationdisplay
ik
mikik =?1
a217a165Mik = max
[0,1;0,1]
f(x,y) = 1,mik = min
[0,1;0,1]
f(x,y) =?1
a108a13f(x,y)a51[0,1;0,1]a254a216a140a200
a44a13|f(x,y)|≡ 1a51[0,1;0,1]a254a140a200.
251
a49a19a155a217 a173a200a169a27a79a142a57a65a94
§1,a19a173a200a169a27a79a142
1,a122a19a173a200a169
I =
integraldisplayintegraldisplay
D
f(x,y)dσ
a143a19a103a200a169(a169a79a15a209a233a252a135a67a254a107a0a103a83a216a211a27a19a103a200a169)a167a217a165a200a169a141Da169a79a143a181
(1) Da180a100xa182a134x2 +y2 = r2(y> 0)a164a140a164a27a171a141a182
(2) Da180a100y = 0,y = x2(x> 0)a57x+y = 2a164a140a164a27a171a141a182
(3) Da180a100y = x,x = 2a57y = 1x (x> 0)a164a140a164a27a171a141a182
(4) Da180a11a1301 lessorequalslantx2 +y2 lessorequalslant 4
a41a181
(1) I =
integraldisplay r
r
dx
integraldisplay √r2?x2
0
f(x,y)dy =
integraldisplay r
0
dy
integraldisplay √r2?y2
√
r2?y2
f(x,y)dx
(2) I =
integraldisplay 1
0
dy
integraldisplay 2?y
3√y
f(x,y)dx =
integraldisplay 1
0
dx
integraldisplay x3
0
f(x,y)dy+
integraldisplay 2
1
dx
integraldisplay 2?x
0
f(x,y)dy
(3) I =
integraldisplay 1
1
2
dy
integraldisplay 2
1
y
f(x,y)dx+
integraldisplay 2
1
dy
integraldisplay 2
y
f(x,y)dx =
integraldisplay 2
1
dx
integraldisplay x
1
x
f(x,y)dy
(4) I =
integraldisplay?1
2
dx
integraldisplay √4?x2
√
4?x2
f(x,y)dy+
integraldisplay 1
1
dx
bracketleftBiggintegraldisplay
√
1?x2
√
4?x2
f(x,y)dy+
integraldisplay √4?x2
√
1?x2
f(x,y)dy
bracketrightBigg
+
integraldisplay 2
1
dx
integraldisplay √4?x2
√
4?x2
f(x,y)dy =
integraldisplay?1
2
dy
integraldisplay √4?y2
√
4?y2
f(x,y)dx+
integraldisplay 1
1
dy
bracketleftBiggintegraldisplay
√
1?y2
√
4?y2
f(x,y)dx+
integraldisplay √4?y2
√
1?y2
f(x,y)dx
bracketrightBigg
+
integraldisplay 2
1
dy
integraldisplay √4?y2
√
4?y2
f(x,y)dx
2,a23f(x,y)a51a171a141Da254a235a89a167a217a165Da180a100y = x,y = aa57x = b(b>a)a164a140a164a27a167a121a178
integraldisplay b
a
dx
integraldisplay x
a
f(x,y)dy =
integraldisplay b
a
dy
integraldisplay b
y
f(x,y)dx
a121a178a181a100f(x,y)a51Da254a235a89a167a25a140a200
a45f(x,y) =
braceleftbigg f(x,y),(x,y) ∈D
0,(x,y) ∈ [a,b;a,b]\D a216y = xa9a235a89a167a25a55a140a200
a75
integraldisplayintegraldisplay
D
f(x,y)dxdy =
integraldisplay
[a,b;a,b]
f(x,y)dxdy =
integraldisplay b
a
dx
integraldisplay b
a
f(x,y)dy =
integraldisplay b
a
dx
integraldisplay x
a
f(x,y)dy
integraldisplayintegraldisplay
D
f(x,y)dxdy =
integraldisplay
[a,b;a,b]
f(x,y)dxdy =
integraldisplay b
a
dy
integraldisplay b
a
f(x,y)dx =
integraldisplay b
a
dy
integraldisplay b
y
f(x,y)dx.
3,a51a101a15a200a169a165a85a67a197a103a200a169a27a103a83a181
(1)
integraldisplay 2a
0
dx
integraldisplay √2ax
√
2ax?x2
f(x,y)dya182
(2)
integraldisplay 2pi
0
dx
integraldisplay sinx
0
f(x,y)dya182
(3)
integraldisplay 1
0
dy
integraldisplay 2y
0
f(x,y)dx+
integraldisplay 3
1
dy
integraldisplay 3?y
0
f(x,y)dxa182
(4)
integraldisplay 1
0
dx
integraldisplay x2
0
f(x,y)dy+
integraldisplay 2
1
dx
integraldisplay √1?(x?1)2
0
f(x,y)dy.
a41a181
(1)
integraldisplay 2a
0
dx
integraldisplay √2ax
√
2ax?x2
f(x,y)dy =
integraldisplay a
0
dy
bracketleftBiggintegraldisplay
a?
√
a2?y2
y2
2a
f(x,y)dx+
integraldisplay 2a
a+
√
a2?y2
f(x,y)dx
bracketrightBigg
+
integraldisplay 2a
a
dy
integraldisplay 2a
y2
2a
f(x,y)dx.
252
(2)
integraldisplay 2pi
0
dx
integraldisplay sinx
0
f(x,y)dy =
integraldisplay pi
0
dx
integraldisplay sinx
0
f(x,y)dy+
integraldisplay 2pi
pi
dx
integraldisplay sinx
0
f(x,y)dy =
integraldisplay pi
0
dx
integraldisplay sinx
0
f(x,y)dy?
integraldisplay 2pi
pi
integraldisplay 0
sinx
f(x,y)dy =
integraldisplay 1
0
dy
integraldisplay pi?arcsiny
arcsiny
f(x,y)dx?
integraldisplay 0
1
dy
integraldisplay 2pi+arcsiny
pi?arcsiny
f(x,y)dx.
(3)
integraldisplay 1
0
dy
integraldisplay 2y
0
f(x,y)dx+
integraldisplay 3
1
dy
integraldisplay 3?y
0
f(x,y)dx =
integraldisplay 2
0
dx
integraldisplay 3?x
x
2
f(x,y)dy.
(4)
integraldisplay 1
0
dx
integraldisplay x2
0
f(x,y)dy+
integraldisplay 2
1
dx
integraldisplay √1?(x?1)2
0
f(x,y)dy =
integraldisplay 1
0
dy
integraldisplay 1+√1?y2
√y f(x,y)dx.
4,a79a142a101a15a19a173a200a169a181
(1)
integraldisplayintegraldisplay
[a,b;c,d]
xyex2+y2 dxdya182
(2)
integraldisplayintegraldisplay
xy2 dxdy,?a180a100a14a212a130y2 = 2pxa218a134a130x = ρ2 (ρ> 0)a164a46a27a171a141a182
(3)
integraldisplayintegraldisplay
dxdy√
2a?x (a > 0),?a180a100a11a37a51a58(a,a)a140a187a143aa133a134a139a73a182a131a131a27a11a177a27a22a225a152a227a108a218a139a73a182a164a140
a164a27a171a141a182
(4)
integraldisplayintegraldisplay
(x2 +y2)dxdy,?a180a177y = x,y = x+a,y = aa218y = 3a (a> 0)a143a62a27a171a141.
a41a181
(1)
integraldisplayintegraldisplay
[a,b;c,d]
xyex2+y2 dxdy =
integraldisplay b
a
xex2 dx
integraldisplay d
c
yey2 dy = 14 (eb2?ea2)(ed2?ec2).
(2)
integraldisplayintegraldisplay
xy2 dxdy =
integraldisplay ρ
2
0
xdx
integraldisplay √2px
√2px
y2 dy = pρ
3
21
√pρ.
(3)
integraldisplayintegraldisplay
dxdy√
2a?x =
integraldisplay a
0
dx√
2a?x
integraldisplay a?√2ax?x2
0
dy =
parenleftBigg
2√2? 83
parenrightBigg
a√a.
(4)
integraldisplayintegraldisplay
(x2 +y2)dxdy =
integraldisplay 3a
a
dy
integraldisplay y
y?a
(x2 +y2)dx = 14a4.
5,a121a178
J =
integraldisplay b
a
dx
integraldisplay x
a
f(y)dy =
integraldisplay b
a
f(y)(b?y)dy =
integraldisplay b
a
f(x)(b?x)dx
a121a178a181a242
integraldisplay b
a
dx
integraldisplay x
a
f(y)dya197a145a200a169a167a26
integraldisplayintegraldisplay
f(y)dxdya167a217a165?a180x = b,x = y,y = aa164a140a164a27a171a141
a233a100a200a169a140a122a143a107a233xa0a233ya27a200a169a167a75a26integraldisplay
b
a
dx
integraldisplay x
a
f(y)dy =
integraldisplay b
a
dy
integraldisplay b
y
f(y)dx =
integraldisplay b
a
f(y)(b?y)dy =
integraldisplay b
a
f(x)(b?x)dx.
6,a23a178a161a254a171a141Da51xa182a218ya182a254a27a221a75a127a221a143lx,ly,Da27a161a200a143|D|a167(α,β)a143Da83a63a152a58a167a121a178a181
(1)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplayintegraldisplay
D
(x?α)(y?β)dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslantlxly|D|a182
(2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplayintegraldisplay
D
(x?α)(y?β)dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
l2xl2y
4,
a121a178a181
(1) a100a117(x?α)(y?β)a51Da254a235a89a167a25a100a200a169a165a138a189a110a167a127a51(ξ,η) ∈Da167a166a26vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplayintegraldisplay
D
(x?α)(y?β)dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle(ξ?α)(η?β)
integraldisplayintegraldisplay
D
dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslantlxly|D|
253
(2) a23lx = b?a,ly = d?ca167a75vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplayintegraldisplay
D
(x?α)(y?β)dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplayintegraldisplay
D
|x?α||y?β|dxdylessorequalslant
integraldisplayintegraldisplay
[a,b;c,d]
|x?α||y?β|dxdy =
integraldisplay b
a
|x?α|dx
integraldisplay d
c
|y?β|dy =
parenleftbiggintegraldisplay α
a
(α?x)dx+
integraldisplay b
α
(x?α)dx
parenrightbiggparenleftbiggintegraldisplay β
c
(β?y)dy+
integraldisplay d
β
(y?β)dy
parenrightbigg
=
(b?α)2 + (α?a)2
2 ·
(d?β)2 + (β?c)2
2 lessorequalslant
(b?a)2
2 ·
(d?c)2
2 =
l2xl2y
4
7,a94a52a139a73a79a142
integraldisplayintegraldisplay
f(x,y)dxdya158a167a200a169a129a88a219a27a152(a21a209a101a15a171a141a254a27a252a171a197a103a200a169)a186
(1)?,a140a11x2 +y2 lessorequalslanta2,ygreaterorequalslant 0a182
(2)?,a140a130a2 lessorequalslantx2 +y2 lessorequalslantb2,xgreaterorequalslant 0a182
(3)?,a11x2 +y2 lessorequalslantay (a> 0)a182
(4)?,a20a144a47a1810 lessorequalslantxlessorequalslanta,0 lessorequalslantylessorequalslanta.
a41a181
(1)
integraldisplayintegraldisplay
f(x,y)dxdy =
integraldisplay pi
0
dθ
integraldisplay |a|
0
f(rcosθ,rsinθ)rdr =
integraldisplay |a|
0
rdr
integraldisplay pi
0
f(rcosθ,rsinθ)dθ.
(2)
integraldisplayintegraldisplay
f(x,y)dxdy =
integraldisplay pi
2
pi2
dθ
integraldisplay |b|
|a|
f(rcosθ,rsinθ)rdr =
integraldisplay |b|
|a|
rdr
integraldisplay pi
2
pi2
f(rcosθ,rsinθ)dθ.
(3)
integraldisplayintegraldisplay
f(x,y)dxdy =
integraldisplay pi
0
dθ
integraldisplay asinθ
0
f(rcosθ,rsinθ)rdr =
integraldisplay a
0
rdr
integraldisplay pi?arcsin r
a
arcsin ra
f(rcosθ,rsinθ)dθ.
(4)
integraldisplayintegraldisplay
f(x,y)dxdy =
integraldisplay pi
4
0
dθ
integraldisplay a
cosθ
0
f(rcosθ,rsinθ)rdr+
integraldisplay pi
2
pi
4
dθ
integraldisplay a
sinθ
0
f(rcosθ,rsinθ)rdr
=
integraldisplay a
0
rdr
integraldisplay pi
2
0
f(rcosθ,rsinθ)dθ+
integraldisplay √2 a
a
rdr
integraldisplay arcsin a
r
arccos ar
f(rcosθ,rsinθ)dθ.
8,a51a101a15a200a169a165a218a63a35a67a254u,va167a67a134a101a15a200a169.
(1)
integraldisplay b
a
dx
integraldisplay βx
αx
f(x,y)dy (0 <a<b,0 <α<β)a167a101
u = x,
v = yx ;
(2)
integraldisplay 2
0
dx
integraldisplay 2?x
1?x
f(x,y)dya167a101u = x+y,v = x?ya182
(3)
integraldisplayintegraldisplay
f(x,y)dxdya167a217a165?a180a100a173a130√x+√y = √aa134a139a73a182a164a46a27a171a141.a101
braceleftbigg x = ucos4v
y = usin4v
a41a181
(1) a207
braceleftbigg x = u
y = uv a167a75|J| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
D(x,y)
D(u,v)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= u> 0
a117a180
integraldisplay b
a
dx
integraldisplay βx
αx
f(x,y)dy =
integraldisplay b
a
udu
integraldisplay β
α
f(u,uv)dv
(2) a207
x = u+v2
y = u?v2
a167a75|J| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
D(x,y)
D(u,v)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
2
a117a180
integraldisplay 2
0
dx
integraldisplay 2?x
1?x
f(x,y)dy = 12
integraldisplay 2
1
du
integraldisplay 4?u
u
f
parenleftBigg
u+v
2,
u?v
2
parenrightBigg
dv
(3) a207
braceleftbigg x = ucos4v
y = usin4v a167a75|J| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
D(x,y)
D(u,v)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
usin3 2v
2
a117a180
integraldisplayintegraldisplay
f(x,y)dxdy = 12
integraldisplay a
0
udu
integraldisplay pi
2
0
sin3 2vf(ucos4v,usin4v)dv = 12
integraldisplay pi
2
0
sin3 2vdv
integraldisplay a
0
uf(ucos4v,usin4v)du.
254
9,a65a94a52a139a73a79a142a101a15a19a173a200a169a181
(1)
integraldisplayintegraldisplay
x2+y2lessorequalslantR2
e?(x2+y2) dxdya182
(2)
integraldisplayintegraldisplay
pi2lessorequalslantx2+y2lessorequalslant4pi2
sin
radicalbig
x2 +y2 dxdya182
(3)
integraldisplayintegraldisplay
(x+y)dxdy,(?a180a11x2 +y2 lessorequalslantx+ya27a83a220).
a41a181
(1)
integraldisplayintegraldisplay
x2+y2lessorequalslantR2
e?(x2+y2) dxdy =
integraldisplay 2pi
0
dθ
integraldisplay R
0
re?r2 dr = pi(1?e?R2).
(2)
integraldisplayintegraldisplay
pi2lessorequalslantx2+y2lessorequalslant4pi2
sin
radicalbig
x2 +y2 dxdy =
integraldisplay 2pi
0
dθ
integraldisplay 2pi
pi
rsinrdr =?6pi2
(3) a138a67a134x = 12 +rcosθ,y = 12 +rsinθa167a75|J| = r
a117a180
integraldisplayintegraldisplay
(x+y)dxdy =
integraldisplay 2pi
0
dθ
integraldisplay 1√
2
0
[r+r2(sinθ+ cosθ)]dr = pi2,
10,a166a100a73a161z = hRradicalbigx2 +y2a33a178a161z = 0a57a11a206a161x2 +y2 = R2a164a140a27a225a78a78a200.
a41a181a73a161z = hR radicalbigx2 +y2a33a178a161z = 0a57a11a206a161x2 + y2 = R2a164a140a27a225a78a51XOYa178a161a254a27a19a75a141a180a11
a141? = {(x,y)vextendsinglevextendsinglex2 +y2 lessorequalslantR2}a167a51a49a152a150a129a220a169a80a143?1
a75a124a94a233a161a53a167a26a164a166a225a78a78a200a143
V =
integraldisplayintegraldisplay
zdxdy = 4
integraldisplayintegraldisplay
1
zdxdy = 4hR
integraldisplayintegraldisplay
1
radicalbig
x2 +y2 dxdy = 4hR
integraldisplay pi
2
0
dθ
integraldisplay R
0
r2 dr = 23 piR2h.
11,a166a165a161x2 +y2 +z2 = a2a134a11a206a161x2 +y2 = ax (a> 0)a250a1a220a169a27a78a200.
a41a181a100a233a161a53a167a26V = 2
integraldisplayintegraldisplay
radicalbig
a2?x2?y2 dxdy = 2
integraldisplay pi
2
pi2
dθ
integraldisplay acosθ
0
r
radicalbig
a2?r2 dr = 23 a3
parenleftBigg
pi? 43
parenrightBigg
.
12,a166a100a14a212a130y2 = mx,y2 = nx (0 <m<n)a218a134a130y = αx,y = βx (0 <α<β)a164a140a164a171a141a27a161a200.
a41a181a138a67a134a181u = y
2
x,v =
y
xa167a75|J| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
D(x,y)
D(u,v)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingleD(u,v)D(x,y)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
= 1y2
x3
= uv4
a117a180a164a166a161a200a143D =
integraldisplayintegraldisplay
dxdy =
integraldisplay β
α
dv
v4
integraldisplay n
m
udu = 16 (n2?m2)
parenleftBigg
1
α3?
1
β3
parenrightBigg
.
13,a166a173a130
parenleftBigg
x2
a2 +
y2
b2
parenrightBigg2
= xyc2 a164a140a27a161a200.
a41a181a100a173a130a144a511a333a150a129a133a39a117a6a58a233a161a167a25a144a73a79a142a227a47a51a49a152a150a129a165a27a161a200a167a502a21a61a140
a45x = arcosθ,y = brsinθa167a75|J| = |ab|r,r =
radicalbig|ab|
|c|
√sinθcosθ
a117a180D =
integraldisplayintegraldisplay
D
dxdy = 2
integraldisplay pi
2
0
dθ
integraldisplay √|ab|
|c|
√sinθcosθ
0
|ab|rdr = a
2b2
2c2,
14,a166a152a212a78a27a78a200a167a100a212a78a27a46a161a143a181a178a161z = 0a167a14a212a1612z = x
2
a +
y2
b a167a177a57a177a165a161x
2 +y2 +(z?c)2 = c2a134
a249a135a14a212a161a27a2a130a143a79a130a27a20a206a161(a,b,c> 0).
a41a181a242z = x
2
2a +
y
2b2a147a92a165a144a167a167a26x
2 +y2 +
parenleftBigg
x2
2a +
y2
2b?c
parenrightBigg2
= c2
255
a45x = √a rcosθ,y = √b rsinθa167a75r = 2radicalbigc?(acos2θ+bsin2θ),|J| = √ab r
a117a180V =
integraldisplayintegraldisplay
D
1
2
parenleftBigg
x2
a +
y2
b
parenrightBigg
dxdy = 4
integraldisplay pi
2
0
dθ
integraldisplay 2√c?(acos2 θ+bsin2 θ)
0
√ab
2 r
3 dr = 4√abpi
parenleftBigg
3
8 a
2 + 3
8 b
2 + 1
4 ab?ac?bc+c
2
parenrightBigg
.
15,a166a62a127a143aa27a20a144a47a0a134a27a159a254a167a23a0a134a254a122a152a58a27a151a221a134a84a58a229a20a144a47a44a152a186a58a27a229a108a164a20a39a167a133a51a20a144a47
a27a165a58a63a151a221a143ρ0.
a41a181a23a44a152a186a58a143a6a58(0,0)a167a75ρ = kradicalbigx2 +y2a133a8x = y = a2 a158a167ρ = ρ0a167a117a180k =
√2 ρ
0
a
a75a151a221a188a234a143ρ(x,y) =
√2
a ρ0
radicalbigx2 +y2
a117a180a124a94a497a75(4)a167a26
m =
integraldisplayintegraldisplay
[0,a;0,a]
√2 ρ
0
a
radicalbig
x2 +y2 ddy =
integraldisplay pi
4
0
dθ
integraldisplay a
cosθ
0
√2 ρ
0
a r
2 dr+
integraldisplay pi
2
pi
4
dθ
integraldisplay a
sinθ
0
√2 ρ
0
a r
2 dr
= ρ0a
2
3 [2 +
√2 ln(1 +√2)].
256
§2,a110a173a200a169a27a79a142
1,a79a142a101a15a110a173a200a169a181
(1)
integraldisplayintegraldisplayintegraldisplay
V
xy2z3 dxdydz,Va181a100a173a161z = xy,y = x,z = 0,x = 1a164a140a164a182
(2)
integraldisplayintegraldisplayintegraldisplay
V
xyzdxdydz,Va181a100a173a161x2 +y2 +z2 = 1,xgreaterorequalslant 0,ygreaterorequalslant 0,z greaterorequalslant 0a140a164.
a41a182
(1)
integraldisplayintegraldisplayintegraldisplay
V
xy2z3 dxdydz =
integraldisplay 1
0
xdx
integraldisplay x
0
y2 dy
integraldisplay xy
0
z3 dz = 1364,
(2)
integraldisplayintegraldisplayintegraldisplay
V
xyzdxdydz =
integraldisplay 1
0
xdx
integraldisplay √1?x2
0
ydy
integraldisplay √1?x2?y2
0
zdz = 148,
2,a141a171a101a15a110a173a200a169a27a171a141Va27a47a71a191a85a67a200a169a103a83a181
(1)
integraldisplay 1
0
dx
integraldisplay 1?x
0
dy
integraldisplay x+y
0
f(x,y,z)dza182
(2)
integraldisplay 1
0
dx
integraldisplay x
0
dy
integraldisplay xy
0
f(x,y,z)dza182
(3)
integraldisplay 2
1
dx
integraldisplay 1
0
dy
integraldisplay 0
1?x?y
f(x,y,z)dza182
(4)
integraldisplay 1
1
dx
integraldisplay √1?x2
√
1?x2
dy
integraldisplay 1
√
x2+y2
f(x,y,z)dza182
(5)
integraldisplay 1
0
dx
integraldisplay 1
0
dy
integraldisplay x2+y2
0
f(x,y,z)dz.
a41a182
(1)
integraldisplay 1
0
dx
integraldisplay 1?x
0
dy
integraldisplay x+y
0
f(x,y,z)dz =
integraldisplay 1
0
dy
integraldisplay 1?y
0
dx
integraldisplay x+y
0
f(x,y,z)dz
=
integraldisplay 1
0
dx
integraldisplay x
0
dz
integraldisplay 1?x
0
f(x,y,z)dy+
integraldisplay 1
0
dx
integraldisplay 1
x
dz
integraldisplay 1?x
z?x
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay z
0
dx
integraldisplay 1?x
z?x
f(x,y,z)dy+
integraldisplay 1
0
dz
integraldisplay 1
z
dx
integraldisplay 1?x
0
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay z
0
dy
integraldisplay 1?y
z?y
f(x,y,z)dx+
integraldisplay 1
0
dz
integraldisplay 1
z
dy
integraldisplay 1?y
0
f(x,y,z)dx
=
integraldisplay 1
0
dy
integraldisplay y
0
dz
integraldisplay 1?y
0
f(x,y,z)dx+
integraldisplay 1
0
dy
integraldisplay 1
y
dz
integraldisplay 1?y
z?y
f(x,y,z)dx
(2)
integraldisplay 1
0
dx
integraldisplay x
0
dy
integraldisplay xy
0
f(x,y,z)dz =
integraldisplay 1
0
dy
integraldisplay 1
y
dx
integraldisplay xy
0
f(x,y,z)dz
=
integraldisplay 1
0
dx
integraldisplay x2
0
dz
integraldisplay x
z
x
f(x,y,z)dy =
integraldisplay 1
0
dz
integraldisplay 1
√z dx
integraldisplay x
z
x
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay 1
√z dy
integraldisplay 1
y
f(x,y,z)dx+
integraldisplay 1
0
dz
integraldisplay √z
z
dy
integraldisplay 1
z
y
f(x,y,z)dx
=
integraldisplay 1
0
dy
integraldisplay y2
0
dz
integraldisplay 1
y
f(x,y,z)dx+
integraldisplay 1
0
dy
integraldisplay y
y2
dz
integraldisplay 1
z
y
f(x,y,z)dx
(3)
integraldisplay 2
1
dx
integraldisplay 1
0
dy
integraldisplay 0
1?x?y
f(x,y,z)dz =
integraldisplay 1
0
dy
integraldisplay 2
1
dx
integraldisplay 0
1?x?y
f(x,y,z)dz
=
integraldisplay 1
0
dy
integraldisplay 0
y
dz
integraldisplay 2
1
f(x,y,z)dx+
integraldisplay 1
0
dy
integraldisplay?y
1?y
dz
integraldisplay 2
1?y?z
f(x,y,z)dx
=
integraldisplay?1
2
dz
integraldisplay 1
1?z
dy
integraldisplay 2
1?y?z
f(x,y,z)dx+
integraldisplay 0
1
dz
integraldisplay?z
0
dy
integraldisplay 2
1?y?z
f(x,y,z)dx+
integraldisplay 0
1
dz
integraldisplay 1
z
dy
integraldisplay 2
1
f(x,y,z)dx
257
=
integraldisplay?1
2
dz
integraldisplay 2
z
dx
integraldisplay 1
1?x?z
f(x,y,z)dy+
integraldisplay 0
1
dz
integraldisplay 1?z
1
dx
integraldisplay 1
1?x?z
f(x,y,z)dy+
integraldisplay 0
1
dz
integraldisplay 2
1?z
dx
integraldisplay 1
0
f(x,y,z)dy
=
integraldisplay 2
1
dx
integraldisplay 2
1?x
dz
integraldisplay 1
0
f(x,y,z)dy+
integraldisplay 2
1
dx
integraldisplay 1?x
x
dz
integraldisplay 1
1?x?z
f(x,y,z)dy
(4)
integraldisplay 1
1
dx
integraldisplay √1?x2
√
1?x2
dy
integraldisplay 1
√
x2+y2
f(x,y,z)dz =
integraldisplay 1
1
dy
integraldisplay √1?y2
√
1?y2
dx
integraldisplay 1
√
x2+y2
f(x,y,z)dz
=
integraldisplay 1
1
dx
integraldisplay 1
|x|
dz
integraldisplay √z2?x2
√
z2?x2
f(x,y,z)dy =
integraldisplay 1
0
dz
integraldisplay z
z
dx
integraldisplay √z2?x2
√
z2?x2
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay z
z
dy
integraldisplay √z2?y2
√
z2?y2
f(x,y,z)dx =
integraldisplay 1
1
dy
integraldisplay 1
|y|
dz
integraldisplay √z2?y2
√
z2?y2
f(x,y,z)dx
(5)
integraldisplay 1
0
dx
integraldisplay 1
0
dy
integraldisplay x2+y2
0
f(x,y,z)dz =
integraldisplay 1
0
dy
integraldisplay 1
0
dx
integraldisplay x2+y2
0
f(x,y,z)dz
=
integraldisplay 1
0
dx
integraldisplay x2
0
dz
integraldisplay 1
0
f(x,y,z)dy+
integraldisplay 1
0
dx
integraldisplay x2+1
x2
dz
integraldisplay 1
√
z?x2
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay √z
0
dx
integraldisplay 1
√
z?x2
f(x,y,z)dy+
integraldisplay 1
0
dz
integraldisplay 1
√z dx
integraldisplay 1
0
f(x,y,z)dy+
integraldisplay 2
1
dz
integraldisplay 1
√z?1 dx
integraldisplay 1
√
z?x2
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay √z
0
dy
integraldisplay 1
√
z?y2
f(x,y,z)dx+
integraldisplay 1
0
dz
integraldisplay 1
√z dy
integraldisplay 1
0
f(x,y,z)dx+
integraldisplay 2
1
dz
integraldisplay 1
√z?1 dy
integraldisplay 1
√
z?y2
f(x,y,z)dx
=
integraldisplay 1
0
dy
integraldisplay y2
0
dz
integraldisplay 1
0
f(x,y,z)dx+
integraldisplay 1
0
dy
integraldisplay y2+1
y2
dz
integraldisplay 1
√
z?y2
f(x,y,z)dx
3,a79a142a101a15a110a173a200a169a181
(1)
integraldisplayintegraldisplayintegraldisplay
V
zdxdydza167a217a165a200a169a171a141Va180a100a165a161x2 +y2 +z2 = 4a134a14a212a161z = 13 (x2 +y2)a164a140a164a27a225a78a182
(2)
integraldisplayintegraldisplayintegraldisplay
V
(x2 +y2 +z2)dVa167a217a165Va180x2 +y2 +z2 lessorequalslant 1a182
(3)
integraldisplayintegraldisplayintegraldisplay
V
z2 dxdydz,Va100a252a135a165x2 +y2 +z2 lessorequalslantR2,x2 +y2 +z2 lessorequalslant 2Rza27a250a1a220a169a164a124a164a182
(4)
integraldisplayintegraldisplayintegraldisplay
V
radicalBigg
1? x
2
a2?
y2
b2?
z2
c2 dxdydz,Va143a253a165
x2
a2 +
y2
b2 +
z2
c2 lessorequalslant 1.
a41a182
(1) a124a94a206a161a139a73a167a26
integraldisplayintegraldisplayintegraldisplay
V
zdxdydz =
integraldisplay 2pi
0
dθ
integraldisplay √3
0
dr
integraldisplay √4?r2
r2
3
rzdz = 134 pi
(2) a124a94a165a161a139a73a167a26
integraldisplayintegraldisplayintegraldisplay
V
(x2 +y2 +z2)dV =
integraldisplay 2pi
0
dθ
integraldisplay pi
0
sin?d?
integraldisplay 1
0
ρ4 dρ = 45 pi
(3) a124a94a165a161a139a73a167a26integraldisplayintegraldisplayintegraldisplay
V
z2 dxdydz =
integraldisplay 2pi
0
dθ
integraldisplay pi
3
0
cos2?sin?d?
integraldisplay R
0
ρ4 dρ+
integraldisplay 2pi
0
dθ
integraldisplay pi
2
pi
3
cos2?sin?d?
integraldisplay 2Rcos?
0
ρ4 dρ
= 59480 piR5
(4) a100a50a194a165a161a139a73a167a26
integraldisplayintegraldisplayintegraldisplay
V
radicalBigg
1? x
2
a2?
y2
b2?
z2
c2 dxdydz = abc
integraldisplay 2pi
0
dθ
integraldisplay pi
0
sin?d?
integraldisplay 1
0
ρ2
radicalbig
1?ρ2 dρ = pi
2
4 abc.
4,a124a94a165a161a139a73a189a206a161a139a73a79a142a101a15a173a161a164a46a78a200a181
(1) x2 +y2 +z2 = 4R2a27a83a220a26x2 +y2 = 2Rxa164a121a209a27a220a169a182
(2) (x2 +y2 +z2)3 = 3xyz.
258
a41a182
(1) a124a94a206a161a139a73x = rcosθ,y = rsinθ,z = za133|J| = ra167a51a100a67a134a101a167a173a161a144a167a67a143a181
r2 +z2 = 4R2,r = 2rcosθ
a75V =
integraldisplayintegraldisplayintegraldisplay
V
dxdydz = 2
integraldisplay pi
2
pi2
dθ
integraldisplay 2Rcosθ
0
rdr
integraldisplay √4R2?r2
0
dz = 163 R3
parenleftBigg
pi? 43
parenrightBigg
(2) a100a75a127a225a78a51a49a152a33a49a110a33a49a56a57a49a108a37a129a83a167a233a117a249a10a37a129a169a79a107
x,y,z greaterorequalslant 0;x,y lessorequalslant 0,z greaterorequalslant 0;x,z lessorequalslant 0,y greaterorequalslant 0;xgreaterorequalslant 0,y,z lessorequalslant 0 a207a6a170a134a224a57a109a224a8x,y,za165a63a252a135a211a158a67a210
a158a31a170a69a164a225a167a25a225a78a51a249a111a135a37a129a83a27a136a220a169a167a152a233a152a233a47a233a161a117a139a73a182a131a152.
a100a165a161a139a73x = ρsin?cosθ,y = ρsin?sinθ,z = ρcos?,|J| = ρ2 sin?
a173a161a144a167a67a143a181ρ6 = 3ρ3 sin2?cos?sinθcosθa61ρ3 = 3sin2?cos?sinθcosθa167
a133a51a49a152a37a129a83a167ρgreaterorequalslant 0,0 lessorequalslantθlessorequalslant pi2,0 lessorequalslant?lessorequalslant pi2
a117a180V = 4
integraldisplay pi
2
0
dθ
integraldisplay pi
2
0
sin?d?
integraldisplay 3√3sin2?cos?sinθcosθ
0
ρ2 dρ = 12,
5,a124a94a183a8a27a139a73a67a134a79a142a101a15a173a161a164a140a78a200a181
(1)
parenleftBigg
x2
a2 +
y2
b2 +
z2
c2
parenrightBigg2
= x
2
a2 +
y2
b2
(2)
parenleftBigg
x
a +
y
b
parenrightBigg2
+
parenleftBigg
z
c
parenrightBigg2
= 1,(x> 0,y> 0,z> 0,a,b,c> 0)
(3) z = x2 +y2,z = 2(x2 +y2),xy = a2,xy = 2a2,x = 2y,2x = ya167(a217a165x,y> 0)
a41a182
(1) a100a50a194a165a161a139a73:x = aρsin?cosθ,y = bρsin?sinθ,z = cρcos?a167a217a165ρgreaterorequalslant 0,0 lessorequalslantθ lessorequalslant 2pi,0 lessorequalslant?lessorequalslantpia167
a249a158|J| = abcρ2 sin?
a173a161a144a167a67a143a181ρ = sin?
a75V = abc
integraldisplay 2pi
0
dθ
integraldisplay pi
0
sin?d?
integraldisplay sin?
0
ρ2 dρ = pi
2
4 abc
(2) a138a67a134a181x = arcos2θcos?,y = brsin2θcos?,z = crsin?a167a217a1650 lessorequalslantr lessorequalslant 1,0 lessorequalslantθ lessorequalslant pi2,0 lessorequalslant?lessorequalslant pi2 a167
a249a158|J| = 2abcr2 cosθsinθcos?
a75V =
integraldisplay pi
2
0
dθ
integraldisplay pi
2
0
d?
integraldisplay 1
0
(2abcr2 cosθsinθcos?)dr = abc3
(3) a45z = u(x2 +y2),xy = v,x = wya167a75x = √wv,y =
radicalbigg
v
w,z = u
parenleftBigg
wv+ vw
parenrightBigg
a100a158|J| = v2 + v2w2 a167a1331 lessorequalslantulessorequalslant 2,a2 lessorequalslantvlessorequalslant 2a2,12 lessorequalslantwlessorequalslant 2
a117a180V =
integraldisplay 2
1
du
integraldisplay 2a2
a2
vdv
integraldisplay 2
1
2
parenleftBigg
1
2 +
1
2w2
parenrightBigg
dw = 94 a4
6,a166a228a107a252a160a78a2000 lessorequalslantxlessorequalslant 1,0 lessorequalslantylessorequalslant 1,0 lessorequalslantz lessorequalslant 1a27a212a78a27a159a254a167a101a212a78a51a58M(x,y,z)a27a151a221a143μ = x+y+z.
a41a182m =
integraldisplayintegraldisplayintegraldisplay
V
(x+y+z)dxdydz =
integraldisplay 1
0
dx
integraldisplay 1
0
dy
integraldisplay 1
0
(x+y+z)dz = 32,
259
§3,a200a169a51a212a110a254a27a65a94
1,a166a101a15a173a130a164a46a0a134a27a159a37a139a73a181
(1) ay = x2,x+y = 2a (a> 0)
(2) r = a(1 + cos?) (0 lessorequalslant?lessorequalslantpi)
a41a181
(1) a151a221ρa143a126a234a167a75xG =
integraltextintegraltext
xd?
integraltextintegraltext
d?,yG =
integraltextintegraltext
yd?
integraltextintegraltext
d?
a100
integraldisplayintegraldisplay
d? =
integraldisplay a
2a
dx
integraldisplay 2a?x
x2
a
dy = 92 a2
integraldisplayintegraldisplay
xd? =
integraldisplay a
2a
xdx
integraldisplay 2a?x
x2
a
dy =?94 a3
integraldisplayintegraldisplay
yd? =
integraldisplay a
2a
dx
integraldisplay 2a?x
x2
a
ydy = 365 a3
a75xG =?a2,yG = 85 a
(2) a151a221ρa143a126a234a167a75xG =
integraltextintegraltext
rcos?d?
integraltextintegraltext
d?,yG =
integraltextintegraltext
rsin?d?
integraltextintegraltext
d?
a100
integraldisplayintegraldisplay
d? =
integraldisplay pi
0
d?
integraldisplay a(1+cos?)
0
rdr = 34 a2pi
integraldisplayintegraldisplay
rcos?d? =
integraldisplay pi
0
d?
integraldisplay a(1+cos?)
0
r2 cos?dr = 58 a3pi
integraldisplayintegraldisplay
rsin?d? =
integraldisplay pi
0
d?
integraldisplay a(1+cos?)
0
r2 sin?dr = 43 a3
a75xG = 56 a,yG = 16a9pi,
2,a166a100a101a15a173a161a164a46a27a212a78a27a159a37a181
(1) x
2
a2 +
y2
b2 +
z2
c2 = 1,xgreaterorequalslant 0,ygreaterorequalslant 0,z greaterorequalslant 0
(2) z = x2 +y2,x+y = a,x = 0,y = 0,z = 0
a41a181
(1) a151a221ρa143a126a234a167a75xG =
integraltextintegraltextintegraltext
V
xdxdydz
integraltextintegraltextintegraltext
V
dxdydz,yG =
integraltextintegraltextintegraltext
V
ydxdydz
integraltextintegraltextintegraltext
V
dxdydz,zG =
Dintegraltextintegraltextintegraltext
V
zdxdydz
integraltextintegraltextintegraltext
V
dxdydz
a45x = aρsin?cosθ,y = bρsin?sinθ,z = cρcos?a167a217a1650 lessorequalslant ρ lessorequalslant 1,0 lessorequalslant θ lessorequalslant pi2,0 lessorequalslant? lessorequalslant pi2 a167a100
a158|J| = abcρ2 sin?
a75
integraldisplayintegraldisplayintegraldisplay
V
dxdydz = abc
integraldisplay pi
2
0
dθ
integraldisplay pi
2
0
sin?d?
integraldisplay 1
0
ρ2 dρ = pi6 abc
integraldisplayintegraldisplayintegraldisplay
V
xdxdydz = a2bc
integraldisplay pi
2
0
cosθdθ
integraldisplay pi
2
0
sin2?d?
integraldisplay 1
0
ρ3 dρ = pi16 a2bc
a117a180xG = 38 aa167a100a233a161a53a167a26yG = 38 b,zG = 38 c
(2) a151a221ρa143a126a234a167a75xG =
integraltextintegraltextintegraltext
V
xdxdydz
integraltextintegraltextintegraltext
V
dxdydz,yG =
integraltextintegraltextintegraltext
V
ydxdydz
integraltextintegraltextintegraltext
V
dxdydz,zG =
integraltextintegraltextintegraltext
V
zdxdydz
integraltextintegraltextintegraltext
V
dxdydz
a100
integraldisplayintegraldisplayintegraldisplay
V
dxdydz =
integraldisplay a
0
dx
integraldisplay a?x
0
dy
integraldisplay x2+y2
0
dz = a
4
6
260
integraldisplayintegraldisplayintegraldisplay
V
xdxdydz =
integraldisplay a
0
xdx
integraldisplay a?x
0
dy
integraldisplay x2+y2
0
dz = a
5
15
integraldisplayintegraldisplayintegraldisplay
V
ydxdydz = a
5
15,
integraldisplayintegraldisplayintegraldisplay
V
zdxdydz = 7180 a6
a75xG = 25 a,yG = 25 a,zG = 730 a2.
3,a166a254a33a169a217a117a252a135a11r = 2sinθa57r = 4sinθa131a109a27a171a141a254a27a159a254a27a159a37.
a41a181a100a233a161a53a167a26x = 0
a113y = 13pi
integraldisplay pi
0
dθ
integraldisplay 4sinθ
2sinθ
r2 sinθdr = 73 a167a75a164a166a47a37a143
parenleftBigg
0,73
parenrightBigg
.
4,a51a44a152a41a23a76a167a165a167a135a51a140a11a47a27a134a62a254a86a254a152a135a62a134a134a187a31a127a27a221a47a167a166a18a135a178a161a227a47a27a159a37a225a51a11a37
a254a167a193a166a221a47a27a44a152a62a127.
a41a181a23a151a221ρa143a126a234a167a221a47a27a44a152a62a127a143la167a11a37a51a139a73a6a58(0,0)a167a18a11a160a117xa182a254a144a167a18a221a47a160a117xa182a101a144
a117a180x =
ρ
integraldisplay R
R
xdx
integraldisplay √R2?x2
l
dy
ρparenleftbig12 piR2 + 2Rlparenrightbig = 0
y =
ρ
integraldisplay R
R
dx
integraldisplay √R2?x2
l
ydy
ρparenleftbig12 piR2 + 2Rlparenrightbig ==
2
piR+ 4l
parenleftBigg
2
3 R
2?l2
parenrightBigg
a45y = 0a167a75a26l =
√6
3 R.
5,a166a254a33a169a217a51a100y = x2a134y = 1a164a140a164a27a178a161a227a47a254a27a159a254a39a117a134a130y =?1a27a61a196a46a254.
a41a181Iy=1 =
integraldisplayintegraldisplay
(y+ 1)2 d? =
integraldisplay 1
1
dx
integraldisplay 1
x2
(y+ 1)2 dy = 368105,
6,a166a100a101a15a173a161a164a46a254a33a78a233a117a164a171a182a27a61a196a46a254a181
(1) z = x2 +y2,x+y = ±1,x?y = ±1,z = 0a39a117za182a182
(2) a127a144a78a39a117a167a27a152a99.
a41a181
(1) a173a161a164a46a254a33a212a78a233a117OZa182a27a61a196a46a254a80a143IOZ
a75IOZ =
integraldisplayintegraldisplayintegraldisplay
V
(x2 +y2)dxdydz
=
integraldisplay 1
0
dx
integraldisplay 1?x
x?1
dy
integraldisplay x2+y2
0
(x2 +y2)dz +
integraldisplay 0
1
dx
integraldisplay x+1
(1+x)
dy+
integraldisplay x2+y2
0
(x2 +y2)dz = 1445
(2) a23a127a144a780 lessorequalslantz lessorequalslantc,0 lessorequalslantylessorequalslantb,0 lessorequalslantxlessorequalslanta
a39a117za182a27a61a196a46a254a143IOZ =
integraldisplay a
0
dx
integraldisplay b
0
dy
integraldisplay c
0
(x2 +y2)dz = abc3 (a2 +b2).
7,a166a254a33a0a161x2 +y2 lessorequalslantR2,z = 0a233a117za182a254a152a58(0,0,c) (c> 0)a63a252a160a159a254a27a218a229.
a41a181a218a229a51OX,OYa182a254a27a19a75a1430a167a61Fx = Fy = 0a167a23ρ = ρ0
a75Fz = k
integraldisplayintegraldisplay
ρ0 cd3 d? = kρ0
integraldisplay 2pi
0
dθ
integraldisplay R
0
cr
(r2 +c2)32
dr = 2kρ0pic
bracketleftBigg
1
c?
1√
R2 +c2
bracketrightBigg
.
8,a166a254a33a206a78x2 +y2 lessorequalslanta2,0 lessorequalslantz lessorequalslantha233a117p(0,0,c) (c>h)a58a63a27a252a160a159a254a27a218a229.
a41a181a23ρ = ρ0a167a100a233a161a53a167a26a218a229a51OX,OYa182a254a27a19a75a1430a167a61Fx = Fy = 0
a124a94a206a161a139a73a167a26a218a229a51OZa182a254a27a19a75a143a181
Fz = kρ0
integraldisplayintegraldisplay
dxdy
integraldisplay h
0
z?c
(x2 +y2 + (z?c)2)32
dz = kρ0
integraldisplay 2pi
0
dθ
integraldisplay a
0
rdr
integraldisplay h
0
z?c
[r2 + (z?c)2]32
dz
= 2pikρ0(
radicalbig
a2 +c2?
radicalbig
a2 + (c?h)2?h).
261
§4,a50a194a173a200a169
1,a79a142a101a15a50a194a173a200a169a131a138a181
(1)
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq
(2)
integraldisplayintegraldisplay
x2+y2lessorequalslant1
dxdyradicalbig
1?x2?y2
(3)
integraldisplay +∞
∞
integraldisplay +∞
∞
e?(x2+y2) dxdy.a191a100a100a121a178a86a199a200a169
1√
pi
integraldisplay +∞
∞
e?x2 dx = 1
a41a181
(1) a100a117a26a200a188a234a154a75a167a25I =
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq =
integraldisplay +∞
1
dx
xp
integraldisplay +∞
1
x
dy
yq
a8qlessorequalslant 1a158a167a100xgreaterorequalslant 1a167a1270 < 1xlessorequalslant 1a167a75a26a200a169
integraldisplay +∞
1
x
dy
yq a117a209a133a107
integraldisplay +∞
1
x
dy
yq = +∞a167
a117a180I =
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq = +∞
a8q> 1a158a167
integraldisplay +∞
1
x
dy
yq =
xq?1
q?1
a100a158a167a8p>qa158a167I =
integraldisplay +∞
1
dx
xp
integraldisplay +∞
1
x
dy
yq =
1
q?1
integraldisplay +∞
1
xq?p?1 dx = 1(q?1)(p?q)
a8plessorequalslantqa158a167(p+ 1)?qlessorequalslant 1a167a75a200a169 1q?1
integraldisplay +∞
1
xq?p?1 dx = +∞a167a108a13a26I =
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq = +∞
a110a254a140a127a167a8p>q> 1a158a167I =
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq =
1
(q?1)(p?q) a167a217a123a156a185I = +∞.
(2)
integraldisplayintegraldisplay
x2+y2lessorequalslant1
dxdyradicalbig
1?x2?y2 = limε→1
integraldisplay 2pi
0
dθ
integraldisplay ε
0
r√
1?r2 dr = 2pi.
(3) a138a67a134x = rcosθ,y = rsinθ (0 lessorequalslantθlessorequalslant 2pi,r> 0),|J| = r
a75
integraldisplay +∞
∞
integraldisplay +∞
∞
e?(x2+y2) dxdy =
integraldisplay 2pi
0
dθ
integraldisplay +∞
0
re?r2 dr = pi.
a100
integraldisplay +∞
∞
integraldisplay +∞
∞
e?(x2+y2) dxdy =
integraldisplay +∞
∞
e?x2 dx
integraldisplay +∞
∞
e?y2 dya133
integraldisplay +∞
∞
e?x2 dx =
integraldisplay +∞
∞
e?y2 dya143a44a152
a138
a75
integraldisplay +∞
∞
e?x2 dx = √pia61 1√piintegraltext+∞?∞ e?x2 dx = 1.
2,a63a216a101a161a50a194a173a200a169a27a194a241a53a181
(1)
integraldisplay +∞
∞
integraldisplay +∞
∞
dxdy
(1 +|x|p)(1 +|y|q)
(2)
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
(x,y)
(1 +x2 +y2)p dxdy,0 <mlessorequalslant|?(x,y)|lessorequalslantM
(3)
integraldisplay a
0
integraldisplay a
0
(x,y)
|x?y|p dxdy,0 <mlessorequalslant|?(x,y)|lessorequalslantM
262
(4)
integraldisplayintegraldisplay
x2+y2lessorequalslant1
(x,y)
(x2 +xy+y2)p dxdy,0 <mlessorequalslant|?(x,y)|lessorequalslantM
a41a181
(1) a207a26a200a188a234a143a20a133a39a117OX,OYa182a233a161a167a75
integraldisplay +∞
∞
integraldisplay +∞
∞
dxdy
(1 +|x|p)(1 +|y|q) = 4
integraldisplay +∞
0
integraldisplay +∞
0
dxdy
(1 +xp)(1 +yq)
a113 lim
x→+∞
xp 11 +xp = 1a167a75a100a195a161a129a50a194a200a169a133a220a7a79a123a27a52a129a47a170a167a26
a8p> 1a158a167a200a169
integraldisplay +∞
0
dx
1 +xp a194a241a182a8plessorequalslant 1a158a167a200a169
integraldisplay +∞
0
dx
1 +xp a117a209
a211a110a140a26a167a8q> 1a158a167a200a169
integraldisplay +∞
0
dy
1 +yq a194a241a182a8qlessorequalslant 1a158a167a200a169
integraldisplay +∞
0
dy
1 +yq a117a209
a110a254a140a127a167a8p> 1a133q> 1a158a167a200a169
integraldisplay +∞
∞
integraldisplay +∞
∞
dxdy
(1 +|x|p)(1 +|y|q) a194a241a167a217a123a156a185a254a117a209.
(2) a207 m(1 +x2 +y2)p lessorequalslant?(x,y)(1 +x2 +y2)p lessorequalslant M(1 +x2 +y2)p
a75a100a50a194a173a200a169a27a39a22a7a79a123a57a50a194a173a200a169a53a159a167a26
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
(x,y)
(1 +x2 +y2)p dxdya134
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
dxdy
(1 +x2 +y2)p
a211a241a209
a100a26a200a188a234a27a233a161a53a57a154a75a53a167a26
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
dxdy
(1 +x2 +y2)p = 2
integraldisplay 1
0
dy
integraldisplay +∞
0
dx
(1 +x2 +y2)p
a100a1170 lessorequalslantylessorequalslant 1a167a75
a101pgreaterorequalslant 0a167a75
integraldisplay +∞
0
dx
(2 +x2)p lessorequalslant
integraldisplay +∞
0
dx
(1 +x2 +y2)p lessorequalslant
integraldisplay +∞
0
dx
(1 +x2)p
a101p< 0a167a75
integraldisplay +∞
0
dx
(2 +x2)p greaterorequalslant
integraldisplay +∞
0
dx
(1 +x2 +y2)p greaterorequalslant
integraldisplay +∞
0
dx
(1 +x2)P
a233a117α> 0a167a100a117 lim
x→+∞
x2p 1(α2 +x2)p = 1a167a75a200a169
integraldisplay +∞
0
dx
(α2 +x2)p a8p>
1
2 a158a194a241a182a8plessorequalslant
1
2 a158a117a209
a117a180
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
dxdy
(1 +x2 +y2)p a8p>
1
2 a158a194a241a182a8plessorequalslant
1
2 a158a117a209
a108a13
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
(x,y)
(1 +x2 +y2)p dxdya8p>
1
2 a158a194a241a182a8plessorequalslant
1
2 a158a117a209.
(3) a2070 < m|x?y|p lessorequalslant?(x,y)|x?y|p lessorequalslant M|x?y|p
a75a100a50a194a173a200a169a27a39a22a7a79a123a57a50a194a173a200a169a53a159a167a26
integraldisplay a
0
integraldisplay a
0
(x,y)
|x?y|p dxdya134
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p a211a241a209
a100a26a200a188a234a27a233a161a53a57a154a75a53a167a26
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p dxdy = 2
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p
a8p< 1a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p =
a2?p
(2?p)(1?p)
a75
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p = 2
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p =
2a2?p
(2?p)(1?p)
a8pgreaterorequalslant 1a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p = limε→+0
integraldisplay a
ε
dx
integraldisplay x?ε
0
dy
(x?y)p
a8p = 1a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p = limε→+0
integraldisplay a
ε
dx
integraldisplay x?ε
0
dy
x?y = limε→+0(alna?a+ε?alnε) = +∞
a75
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p a8p = 1a158a117a209a182
a8p = 2a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p = limε→+0
integraldisplay a
ε
dx
integraldisplay x?ε
0
dy
(x?y)2 = limε→+0
parenleftBigg
a+εlnε
ε?1?lna
parenrightBigg
=?∞
a75
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p a8p = 2a158a117a209a182
263
a8p> 1a133pnegationslash= 2a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p = limε→+0
integraldisplay a
ε
dx
integraldisplay x?ε
0
dy
(x?y)p
= lim
ε→+0
bracketleftBigg
1
(p?1)εp?1
parenleftBigg
a? p?1p?2 ε
parenrightBigg
+ 1(p?1)(p?2)ap?2
bracketrightBigg
= +∞
a75
integraldisplay a
0
integraldisplay a
0
dxdy)
|x?y|p a8p> 1a133pnegationslash= 2a158a117a209
a110a254a140a127
integraldisplay a
0
integraldisplay a
0
(x,y)
|x?y|p dxdya8p< 1 a158a194a241a182a8pgreaterorequalslant 1a158a117a209.
(4) (0,0)a180a219a58a167a100a117x2 + +xy+y2 > 0(a8(x,y) negationslash= (0,0))a167a75
m
(x2 +xy+y2)p lessorequalslant
(x,y)
(x2 +xy+y2)p lessorequalslant
M
(x2 +xy+y2)p
a100a50a194a173a200a169a27a39a22a7a79a123a57a50a194a173a200a169a53a159a167a26
integraldisplay a
0
integraldisplay a
0
(x,y)
(x2 +xy+y2)p dxdya134
integraldisplay a
0
integraldisplay a
0
dxdy
(x2 +xy+y2)p
a211a241a209integraldisplayintegraldisplay
x2+y2lessorequalslant1
dxdy
(x2 +xy+y2)p = limε→+0
integraldisplay 1
ε
dr
r2p?1
integraldisplay 2pi
0
dθ
(1 + sinθcosθ)p = N limε→+0
integraldisplay 1
ε
dr
r2p?1
=
N lim
ε→+0
(?lnε) = +∞,p = 1
N lim
ε→+0
1?ε2?2p
2?2p =
N
2?2p,p< 1
∞,p> 1
parenleftBigg
a217a165N =
integraldisplay 2pi
0
dθ
(1 + sinθcosθ)pa143a126a194a200a169a167a143a126a254
parenrightBigg
a111a131a167
integraldisplayintegraldisplay
x2+y2lessorequalslant1
dxdy
(x2 +xy+y2)p a8p< 1a158a194a241a182a8pgreaterorequalslant 1a158a117a209
a108a13
integraldisplayintegraldisplay
x2+y2lessorequalslant1
(x,y)
(x2 +xy+y2)p dxdya8p< 1 a158a194a241a182a8pgreaterorequalslant 1a158a117a209.
3,a121a178a23Da180a100a51a49a152a150a129a27a14a212a130y = x2a167a11a177x2 +y2 = 1a57xa182a164a140a164a27a171a141a167a75
integraldisplayintegraldisplay
D
dxdy
x2 +y2 a127a51.
a121a178a181(0,0)a180a219a58integraldisplayintegraldisplay
D
dxdy
x2 +y2 =
integraldisplay θ0
0
dθ
integraldisplay 1
sinθ
cos2 θ
dr
r =
integraldisplay θ0
0
ln cos
2θ
sinθ dθ,0a180a219a58
parenleftBigg
a217a165θ0a247a118 sinθ0cos2θ
0
= 1a61sinθ0 =
√5?1
2
parenrightBigg
a207 lim
θ→+0
θ12 ln cos
2θ
sinθ = 0a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay θ0
0
ln cos
2θ
sinθ dθ a194a241
a108a13a6a200a169
integraldisplayintegraldisplay
D
dxdy
x2 +y2 a127a51.
4,a166a254a33a20a11a73a78a39a117a51a167a27a186a58a63a27a159a254a143ma27a159a58a27a218a229.
a41a181a218a229a51OX,OYa182a254a27a19a75a1430a167a61Fx = Fy = 0a167
Fz =
integraldisplayintegraldisplayintegraldisplay
V
mz
gr3 dV =
m
g
integraldisplay 2pi
0
dθ
integraldisplay R
0
dρ
integraldisplay h
R ρ
0
ρz
(ρ2 +z2)32
dz = 2mRpigl (l?g).
264
a49a19a155a152a217 a173a130a200a169a218a173a161a200a169a27a79a142
§1,a49a152a97a173a130a200a169a27a79a142
1,a79a142
integraldisplay
l
(x+y)dsa167la180a177O(0,0),A(1,0),B(0,1)a143a186a58a27a110a14a47.
a41a181I =
integraldisplay
l
(x+y)ds =
braceleftbiggintegraldisplay
OA
+
integraldisplay
AB
+
integraldisplay
BO
bracerightbigg
(x+y)ds
a51a134a130a227OAa254a167y = 0,ds = dxa167a75
integraldisplay
OA
(x+y)ds =
integraldisplay 1
0
xdx = 12 a182
a51a134a130a227ABa254a167y = 1?x,ds = √2 dxa167a75
integraldisplay
AB
(x+y)ds =
integraldisplay 1
0
√2dx = √2 a182
a51a134a130a227BOa254a167x = 0,ds = dya167a75
integraldisplay
BO
(x+y)ds =
integraldisplay 1
0
ydy = 12
a117a180I = 1 +√2,
2,a79a142
integraldisplay
l
(x2 +y2)dsa167la180a177a6a58a143a165a37a167a140a187a143Ra27a134a140a11a177.
a41a181a207l,x = Rcosθ,y = Rsinθ,pi2 lessorequalslantθlessorequalslant 32 pia167a75ds =radicalbigx2θ +y2θ dθ = Rdθ
a117a180
integraldisplay
l
(x2 +y2)ds = piR3.
3,a79a142
integraldisplay
l
(x2 +y2 +z2)dsa167la180a11a218a94a130a181x = acost,y = asint,z = bt (0 lessorequalslanttlessorequalslant 2pi).
a41a181a207ds =radicalbigxprime2(t) +yprime2(t) +zprime2(t) dt = √a2 +b2 dt
a75I =
integraldisplay
l
(x2 +y2 +z2)ds = 23 pi(3a2 + 4pi2b2)
radicalbig
a2 +b2.
4,a79a142
integraldisplay
l
x2 dsa167la180a165a161x2 +y2 +z2 = a2a134a178a161x+y+z = 0a131a2a27a11a177.
a41a181a100a233a161a53a167a26
integraldisplay
l
x2 ds =
integraldisplay
l
y2 ds =
integraldisplay
l
x2 dsa167a75
integraldisplay
l
x2 ds = 13
integraldisplay
l
(x2 +y2 +z2)ds = a
2
3
integraldisplay
l
ds = 23 pia3.
5,a79a142
integraldisplay
l
z2
x2 +y2 dsa167la180a218a130a181x = acost,y = asint,z = at,(0 lessorequalslanttlessorequalslant 2pi).
a41a181a207ds =radicalbigxprime2(t) +yprime2(t) +zprime2(t) dt = √2 dta167a75I =
integraldisplay
l
z2
x2 +y2 ds =
8√2
3 pi
3a.
6,a23a152a55a225a106la27a144a167a143a181
x = et cost,y = et sint,z = et,(0 lessorequalslanttlessorequalslantt0)
a167a51a122a152a58a27a151a221a134a84a58a27a165a187a178a144a164a135a39a167a133a51a58(1,0,1)a63a1431a167a166a167a27a159a254.
a41a181a207ρ = kx2 +y2 +z2 a133a51a58(1,0,1)a63ρ = 1a167a75k = 2a167a117a180ρ = 2x2 +y2 +z2 = e?2t
a113ds =radicalbigxprime2(t) +yprime2(t) +zprime2(t) dt = √3et dta167a75m =
integraldisplay
l
ρds = √3(1?e?t0).
7,a166a253a11x = acost,y = bsinta177a46a27a159a254(0 lessorequalslanttlessorequalslant 2pi)a167a101a173a130a51a58M(x,y)a27a130a53a151a221a143ρ = |y|.
a41a181M =
integraldisplay
l
|y|dsa167a217a165la143a253a11x = acost,y = bsint(0 lessorequalslanttlessorequalslant 2pi)
(1) a23a>ba167a75ds =radicalbigxprime2(t) +yprime2(t) dt = aradicalbig1?ε21 cos2t dta167a217a165ε1 =
√a2?b2
a
a117a180M =
integraldisplay
l
|y|ds =
integraldisplay pi
0
absint
radicalbig
1?ε1 cos2t dt+
integraldisplay 2pi
pi
a(?bsint)
radicalBig
1?ε21 cos2t dt =
2ab
radicalBig
1?ε21 + 2abε
1
arcsinε1 = 2b2 + 2abε
1
arcsinε1
(2) a23a<ba167a75ds =radicalbigxprime2(t) +yprime2(t) dt = aradicalbig1 +ε22 cos2t dta167a217a165ε2 =
√b2?a2
a
a117a180M =
integraldisplay
l
|y|ds =
integraldisplay pi
0
absint
radicalBig
1 +ε22 cos2t dt+
integraldisplay 2pi
pi
a(?bsint)
radicalBig
1 +ε22 cos2t dt =
2ab
radicalBig
1 +ε22 + 2abε
2
ln(ε2 +
radicalBig
1 +ε22) = 2b2 + 2abε
2
ln(ε2 +
radicalBig
1 +ε22)
265
(3) a101a = ba167a75ds =radicalbigxprime2(t) +yprime2(t) dt = adta167a117a180M =
integraldisplay
l
|y|ds =
integraldisplay pi
0
a2 sintdt+
integraldisplay 2pi
pi
a(?asint)dt =
4a2
a108a13M =
2b2 + 2abε
1
arcsinε1,a>b
4a2,a = b
2b2 + 2abε
2
ln(ε2 +radicalbig1 +ε22),a<b
266
§2,a49a152a97a173a161a200a169a27a79a142
1,a79a142a101a15a173a161a161a200a181
(1) z = axya157a185a51a11a206x2 +y2 = a2a83a27a220a169a182
(2) a73a161x2 +y2 = 13 z2a134a178a161x+y+z = 2a(a> 0)a164a46a220a169a27a76a161a182
(3) a206a161x2 +y2 = a2a26a19a178a161x+z = 0,x?z = 0(x> 0,y> 0)a164a31a220a169.
a41a181
(1) a100zx = ay,zy = axa167a26radicalbig1 +z2x +z2y =radicalbig1 +a2x2 +a2y2
a100a233a161a53a167a191a124a94a206a161a139a73a167a26
S = 4
integraldisplayintegraldisplay
σxy
radicalbig
1 +a2x2 +a2y2 dxdy = 4
integraldisplay pi
2
0
dθ
integraldisplay a
0
radicalbig
1 +a2r2rdr = 23a2pi
bracketleftBig
(1 +a4)32?1
bracketrightBig
.
(2) a173a161a27a2a130a51xoya178a161a254a27a19a75a1433x2 + 3y2 = (2a?x?y)2a61x2 +y2?xy+ 2a(x+y) = 2a2
a45x = 1√2 (xprime?yprime),y = 1√2(xprime +yprime)a167a75a144a167a67a143(x
prime + 2√2a)2
(2√3a)2 +
yprime2
(2a)2 = 1
a100a100a140a132a167a173a161a164a46a27a212a78a51xoya178a161a254a27a19a75a141a143a1772aa143a225a140a182a1672√3aa143a127a140a182a27a253a11
a212a78a27a76a161a200a100a31a161a218a31a209a27a73a161a252a220a169a124a164
a233a117z = 2a?x?y,z =radicalbig3x2 + 3y2a169a79a107radicalbig1 +z2x +z2y = √3,radicalbig1 +z2x +z2y = 2
a117a180a212a78a27a76a161a200a143S =
integraldisplayintegraldisplay
√3dxdy+integraldisplayintegraldisplay
2dxdy = (√3 + 2)pi·2a·2√3pi = 4pi(3 + 2√3)a2.
(3) a100yx =?xy,yz = 0a167a26radicalbig1 +y2x +y2z =
radicaltpradicalvertex
radicalvertexradicalbt
1 +
parenleftBigg
x
y
parenrightBigg2
= |a|√a2?x2
a75S =
integraldisplayintegraldisplay
σxz
|a|√
a2?x2 dxdz =
integraldisplay |a|
0
dx
integraldisplay x
x
|a|√
a2?x2 dz = 2a
2.
2,a79a142a49a152a97a173a161a200a169a181
(1)
integraldisplayintegraldisplay
S
(x+y+z)dS,S,a134a140a165a161x2 +y2 +z2 = a2,ylessorequalslant 0a182
(2)
integraldisplayintegraldisplay
S
xdS,S,a218a94a161x = ucosv,y = usinv,z = cva254a27a152a220a1690 lessorequalslantulessorequalslanta,0 lessorequalslantvlessorequalslant 2pia182
(3)
integraldisplayintegraldisplay
S
dS,S,a165a161x2 +y2 +z2 = 2cz(c> 0)a89a51a73a161x2 +y2 = z2a83a27a220a169a182
(4)
integraldisplayintegraldisplay
S
(x2 +y2)dS,S,a78a200radicalbigx2 +y2 lessorequalslantz lessorequalslant 1a27a62a46a182
(5)
integraldisplayintegraldisplay
S
dS
r2,Sa143a11a206a161x
2 +y2 = R2a48a117z = 0a218z = Ha131a109a27a220a169a167a217a165ra143a173a161a254a27a58a20a6a58a27a229a108.
a41a181
(1) a242x2 +y2 +z2 = a2a221a75a20xoza178a161a167a100a158a107y =?√a2?x2?z2
a75yx = x√a2?x2?z2,yz = z√a2?x2?z2a167a117a180radicalbig1 +y2x +y2z = a√a2?x2?z2
a117a180
integraldisplayintegraldisplay
S
(x+y+z)dS =
integraldisplay a
a
dx
integraldisplay √a2?x2
√
a2?x2
a√
a2?x2?z2 (x?
radicalbig
a2?x2?z2 +z)dz =?pia3
(2) E = x2u +y2u +z2u = 1,F = xuxv +yuyv +zuzv = 0,G = x2v +y2v +z2v = u2 +c2
a75
integraldisplayintegraldisplay
S
xdS =
integraldisplayintegraldisplay
Σ
ucosv
radicalbig
u2 +c2 dudv =
integraldisplay a
0
u
radicalbig
u2 +c2 du
integraldisplay 2pi
0
cosvdv = 0
267
(3) a207x2 +y2 +z2 = 2cza167a75x2 +y2 + (z?c)2 = c2,z = c+radicalbigc2?x2?y2
a117a180zx =? xradicalbigc2?x2?y2,zy =? yradicalbigc2?x2?y2 a167a75radicalbig1 +z2x +z2y = cradicalbigc2?x2?y2
a117a180
integraldisplayintegraldisplay
S
dS =
integraldisplayintegraldisplay
σ
cradicalbig
c2?x2?y2 dσ =
integraldisplay 2pi
0
dθ
integraldisplay c
0
cr√
c2?r2 dr = 2pic
2.
(4) a169a143a252a220a169a181
a49a152a220a169a181z = 1,radicalbig1 +z2x +z2y = 1a182a49a19a220a169a181z =radicalbigx2 +y2,radicalbig1 +z2x +z2y = √2
a75
integraldisplayintegraldisplay
S
(x2 +y2)dS =
integraldisplay 2pi
0
dθ
integraldisplay 1
0
r3 dr+
integraldisplay 2pi
0
dθ
integraldisplay 1
0
√2r3 dr = pi
2 (1 +
√2).
(5) x = Rcosθ,y = Rsinθ,z = z (0 lessorequalslantθ2pi,0 lessorequalslantz lessorequalslantH)
a75E = x2θ +y2θ +z2θ = R2,F = xθxz +yθyz +zθzz = 0,G = x2z +y2z +z2z = 1
a117a180√EG?F2 = Ra167a108a13
integraldisplayintegraldisplay
S
dS
r2 =
integraldisplay 2pi
0
dθ
integraldisplay H
0
R
R2 +z2 dz = 2piarctan
H
R,
3,a166a14a212a161a138z = 12 (x2 +y2),0 lessorequalslantz lessorequalslant 1a27a159a254.a100a138a27a151a221a143ρ = z.
a41a181a207z = 12 (x2 +y2)a167a75zx = x,zy = ya167a117a180radicalbig1 +z2x +z2y =radicalbig1 +x2 +y2
a75a159a254M =
integraldisplayintegraldisplay
S
ρdS = 12
integraldisplayintegraldisplay
x2+y2lessorequalslant2
(x2+y2)
radicalbig
1 +x2 +y2 dxdy = 12
integraldisplay 2pi
0
dθ
integraldisplay √2
0
r3
radicalbig
1 +r2 dr = 2(1 + 6
√3)
15 pi.
268
§3,a49a19a97a173a130a200a169
1,a79a142a101a15a49a19a97a173a130a200a169a181
(1)
integraldisplay
l
(x2?2xy)dx+ (y2?2xy)dy,la143y = x2a108(1,1)a20(?1,1)a182
(2)
contintegraldisplay
l
(x2 +y2)dx+ (x2?y2)dy,la143a177A(1,0),B(2,0)C(2,1),D(1,1)a143a186a58a27a20a144a47a167a20a149a182
(3)
integraldisplay
l
(2a?y)dx+ dy,la143a94a211a130x = a(t?sint),y = a(1?cost)a167a108(0,0)a20(2pi,0)a182
(4)
integraldisplay
l
ydx?xdy+ (x2 +y2)dz,la143a173a130x = et,y = e?t,z = ata108(1,1,0)a20(e,e?1,a)
a41a181
(1)
integraldisplay
l
(x2?2xy)dx+ (y2?2xy)dy =
integraldisplay?1
1
[x2?2x3 + 2x(x4?2x3)]dx = 1415,
(2) I =
contintegraldisplay
l
(x2 +y2)dx+ (x2?y2)dy =
parenleftbiggintegraldisplay
AB
+
integraldisplay
BC
+
integraldisplay
CD
+
integraldisplay
DA
parenrightbigg
(x2 +y2)dx+ (x2?y2)dy
a247ABa167y = 0a167a25
integraldisplay
AB
(x2?y2)dy = 0
a211a24a167a107
integraldisplay
BC
(x2 +y2)dx =
integraldisplay
CD
(x2?y2)dy =
integraldisplay
DA
(x2 +y2)dx = 0
a75I =
integraldisplay 2
1
x2 dx+
integraldisplay 1
0
(4?y2)dy+
integraldisplay 1
2
(x2 + 1)dx =
integraldisplay 0
1
(1?y2)dy = 2.
(3)
integraldisplay
l
(2a?y)dx+ dy =
integraldisplay 2pi
0
[(a+acost)·a(1?cost) +asint]dt = a2pi.
(4)
integraldisplay
l
ydx?xdy+ (x2 +y2)dz =
integraldisplay 1
0
[e?t ·et?et(?e?t) + (e2t +e?2t)a]dt = 2 + a2 (e2?e?2).
2,a166a200a169
J =
integraldisplay (1,1,1)
(0,0,0)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
1 1 1
x y z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle· dr
a217a165dra143a165a187a144a149a167a200a169a180a187a169a79a143a181
(1) a247a134a130a182
(2) a247a173a130r = isin?+ j(1?cos?) + k 2?pi,
parenleftBigg
0 lessorequalslant?lessorequalslant pi2
parenrightBigg
.
a41a181
(1) a134a130a144a167a143a181x = y = z
a75J =
integraldisplay (1,1,1)
(0,0,0)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
1 1 1
x y z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle· dr =
integraldisplay (1,1,1)
(0,0,0)
(z?y)dx+ (x?z)dy+ (y?x)dz =
integraldisplay 1
0
(x?x)dx+
integraldisplay 1
0
(y?y)dy+
integraldisplay 1
0
(z?z)dz = 0.
(2) J =
integraldisplay (1,1,1)
(0,0,0)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
1 1 1
x y z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle·dr =
integraldisplay pi
2
0
braceleftBiggbracketleftBigg
2?
pi?(1?cos?)
bracketrightBigg
cos?+
parenleftBigg
sin 2?pi
parenrightBigg
sin?+ [(1?cos?)?sin?]· 2pi
bracerightBigg
d? =
1? pi2? 8pi,
3,a23a49a119a52a173a130La51a49a119a173a161Sa254a167Sa27a144a167a143z = f(x,y)a167a173a130La51XYa161a254a27a221a75a173a130a143la167a188a234P(x,y,z)a51La254
a235a89a167a121a178 contintegraldisplay
L
P(x,y,z)dx =
contintegraldisplay
l
P[x,y,f(x,y)]dx
a121a178a181a216a148a23Sa143a173a161a27a254a253a167z = f(x,y),(x,y) ∈D
a75a173a161a27a62a46La51XYa178a161a254a27a221a75a65a180a95a158a2a144a149a27a173a130l,x =?(t),y = ψ(t),a<b,alessorequalslanttlessorequalslantb
a152a109a173a130La27a144a167a145a131a140a76a143L,x =?(t),y = ψ(t),z = ω(t) = f[?(t),ψ(t)],alessorequalslanttlessorequalslantb
a117a180
contintegraldisplay
L
P(x,y,z)dx =
integraldisplay b
a
P (?(t),ψ(t),f[?(t),ψ(t)])?prime(t)dt =
contintegraldisplay
l
P[x,y,f(x,y)]dx.
269
4,a121a178a181a233a117a173a130a200a169a27a15a79a170a143
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
l
P dx+Qdy
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslantLM,(a170a165La143a200a169a173a130a227a127a221)
M = max
(x,y)∈l
radicalbig
P2 +Q2
a124a94a249a135a216a31a170a15a79a181
IR =
contintegraldisplay
x2+y2=R2
ydx?xdy
(x2 +xy+y2)2
a191a121a178 lim
R→∞
IR = 0.
a121a178a181
(1)
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
l
P dx+Qdy
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
l
[P cosα+Qsinα]dS
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
l
|(P,Q)·(cosα,sinα)| dS lessorequalslant
integraldisplay
l
|(P,Q)||(cosα,sinα)|dS =
integraldisplay
l
radicalbig
P2 +Q2 dS =
radicalbig
P2(ξ,η) +Q2(ξ,η)
integraldisplay
l
dS lessorequalslantML.
(2) a207P = y(x2 +xy+y2)2,Q =?x(x2 +xy+y2)2 a167a75radicalbigP2 +Q2 =
radicalbigx2 +y2
(x2 +xy+y2)2 =
R
(R2 +xy)2
a117a180M = max
(x,y)∈l
radicalbigP2 +Q2 = R
(R2 +xy)2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x = ±
√2
2 R
y =?
√2
2
= 4R3
a750 lessorequalslant|IR| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
contintegraldisplay
x2+y2=R2
ydx?xdy
(x2 +xy+y2)2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslantLM =
8pi
R2
a113 lim
R→∞
8pi
R2 = 0a167a75 limR→∞IR = 0.
5,a23a178a161a171a141Da100a152a94a235a89a52a173a130La164a140a164a167a171a141Da27a161a200a23a143Sa167a237a19a94a173a130a200a169a79a142a161a200Sa27a250a170a181
S = 12
contintegraldisplay
L
xdy?ydx
a121a178a181
(1) a196a107a127a196a227a47D = PQRSa167a217a165QR,PS bardblYa182a167PQ,y = y0(x);SR,y = y1(x)a133a51[a,b]a254a235a89
a249a158L,PSRQP
a242Da27a161a200a119a138a252a173a62a70a47abPSa218abQPa27a161a200a131a11(a217a165a,ba169a143SP,RQa134Xa182a27a2a58)
a117a180a107S =
integraldisplay b
a
[y1(x)?y0(x)]dx
a44a152a144a161a167a226IIa46a173a130a79a142a250a170a167a107
integraldisplay
slurabovePQydx =
integraldisplay b
a
y0(x)dx,
integraldisplay
sluraboveSRydx =
integraldisplay b
a
y1(x)dx
a191a53a191a20
integraldisplay
PS
ydx =
integraldisplay
PQ
ydx = 0
a75?
integraldisplay
L
ydx =
integraldisplay
PSRQP
ydx =
parenleftbiggintegraldisplay
PS
+
integraldisplay
sluraboveSR +
integraldisplay
RQ
+
integraldisplay
sluraboveQR
parenrightbigg
ydx =
integraldisplay b
a
y1(x)dx?
integraldisplay b
a
y0(x)dx = S
a61S =?
integraldisplay
L
ydx.
(2) a233a117a171a141D = PQRSa167a217a165PQ,RS bardblXa182a167a211a110a167a107
integraldisplay
L
xdy = S.
(3) a233a117a141a69a44a27a171a141a156a47a140a122a143a254a252a171a156a47a167a211a24a79a142a195a2a172a161a200a167a44a0a131a92a167a53a191a173a69a180a130a131a112a45a158a167
a211a24a140a26a254a252a171a40a74.
a110a254a164a227a167a107S = 12
contintegraldisplay
L
xdy?ydx.
6,a79a142a101a15a173a130a164a140a171a141a27a161a200a181
(1) a253a11a181x = acost,y = bsint,(0 lessorequalslanttlessorequalslant 2pi)a182
(2) a40a47a130a181x = acos3t,y = asin3t,(0 lessorequalslanttlessorequalslant 2pi).
270
a41a181
(1) S = 12
contintegraldisplay
L
xdy?ydx = 12
integraldisplay 2pi
0
abdt = piab.
(2) S = 12
contintegraldisplay
L
xdy?ydx = 32 a2
integraldisplay 2pi
0
sin2tcos2tdt = 38 pia2.
271
§4,a49a19a97a173a161a200a169
1,a79a142
integraldisplayintegraldisplay
S
(x+y)dydz + (y+z)dzdx+ (z +x)dxdy
Sa180a177a6a58a143a165a37a27a20a144a78(a122a62a127a221a1432)a27a62a46a167a141a149a9a253.
a41a181I =
integraldisplayintegraldisplay
S
(x+y)dydz+(y+z)dzdx+(z+x)dxdy =
integraldisplayintegraldisplay
S
(x+y)dydz+
integraldisplayintegraldisplay
S
(y+z)dzdx+
integraldisplayintegraldisplay
S
(z+x)dxdy
a79a142I =
integraldisplayintegraldisplay
S
(x+y)dydz
a207a20a144a78a56a135a161a165a107a111a135a161a82a134a117YOZa178a161a167a75a100a111a135a161a27a161a200a1430
a117a180
integraldisplayintegraldisplay
S
(x+y)dydz =
integraldisplayintegraldisplay
1lessorequalslantylessorequalslant1
1lessorequalslantzlessorequalslant1
(1 +y)dydz?
integraldisplayintegraldisplay
1lessorequalslantylessorequalslant1
1lessorequalslantzlessorequalslant1
(?1 +y)dydz = 2
integraldisplayintegraldisplay
1lessorequalslantylessorequalslant1
1lessorequalslantzlessorequalslant1
dydz = 8
a211a110a140a26
integraldisplayintegraldisplay
S
(y+z)dzdx = 8,
integraldisplayintegraldisplay
S
(z +x)dxdy = 8
a117a180I =
integraldisplayintegraldisplay
S
(x+y)dydz + (y+z)dzdx+ (z +x)dxdy = 24.
2,a79a142
integraldisplayintegraldisplay
S
f(x)dydz +g(y)dxdz +h(z)dxdy
a170a165f,g,ha143a235a89a188a234a167Sa143a178a49a56a161a78(0 lessorequalslantxlessorequalslanta,0 lessorequalslantylessorequalslantb,0 lessorequalslantz lessorequalslantc)a27a62a46a167a141a149a9a253.
a41a181a23S1,x = a;S2,x = 0;S3,y = b;S4,y = 0;S5,z = c;S6,z = 0
a75I =
integraldisplayintegraldisplay
S
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S1
+
integraldisplayintegraldisplay
S2
+
integraldisplayintegraldisplay
S3
+
integraldisplayintegraldisplay
S4
+
integraldisplayintegraldisplay
S5
+
integraldisplayintegraldisplay
S6
f(x)dydz +g(y)dxdz +h(z)dxdy
a207
integraldisplayintegraldisplay
S1
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S1
f(x)dydz = f(a)bc
integraldisplayintegraldisplay
S2
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S2
f(x)dydz =?f(0)bc
integraldisplayintegraldisplay
S3
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S3
g(y))dzdx = g(b)ac
integraldisplayintegraldisplay
S4
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S4
g(y)dzdx =?g(0)ac
integraldisplayintegraldisplay
S5
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S5
h(z)dxdy = h(c)ab
integraldisplayintegraldisplay
S6
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S6
h(z)dxdy =?h(0)ab
a75I =
integraldisplayintegraldisplay
S
f(x)dydz +g(y)dxdz +h(z)dxdy = abc
bracketleftBigg
f(a)?f(0)
a +
g(b)?g(0)
b +
h(c)?h(0)
c
bracketrightBigg
.
3,a79a142
integraldisplayintegraldisplay
S
yzdzdx
S,x
2
a2 +
y2
b2 +
z2
c2 = 1a27a254a140a76a161a27a254a253.
a41a181a242a253a165a161a76a143a235a234(?,θ)a47a170a181x = asin?cosθ,y = bsin?cosθ,z = ccos?
parenleftBigg
0 lessorequalslant?lessorequalslant pi2,0 lessorequalslantθlessorequalslant 2pi
parenrightBigg
I =
integraldisplayintegraldisplay
S
yzdzdx = ±
integraldisplayintegraldisplay
bcsin?cos?sinθ·Bd?dθa167a217a165?a143?θa178a161a254a27a171a1410 lessorequalslant?lessorequalslant pi2,0 lessorequalslantθlessorequalslant 2pi
a133B = z?xθ?x?zθ = acsin2?sinθ
a207a200a169a247a254a253a167a65a18a20a210a167a61a26I = abc2
integraldisplay pi
2
0
sin3?cos?d?
integraldisplay 2pi
0
sin2θdθ = pi4 abc2
272
4,a79a142
integraldisplayintegraldisplay
S
zdxdy+xdydz +ydxdz
Sa143a206a161x2 +y2 = 1a26a178a161z = 0a57z = 3a164a31a220a169a27a9a253.
a41a181a100a117a206a161x2 +y2 = 1a51XOYa178a161a254a27a221a75a143a152a11a177a167a25a217a161a200a1430a167a108a13
integraldisplayintegraldisplay
S
zdxdy = 0
a113
integraldisplayintegraldisplay
S
xdydz =
integraldisplayintegraldisplay
Sa99
+
integraldisplayintegraldisplay
Sa0
xdydz =integraldisplayintegraldisplay
Syz
radicalbig
1?y2 dydz?
integraldisplayintegraldisplay
Syz
(?
radicalbig
1?y2)dydz =
2
integraldisplay 2
0
dz
integraldisplay 1
1
radicalbig
1?y2 dy = 3pi
integraldisplayintegraldisplay
S
ydxdz =
integraldisplayintegraldisplay
Sa109
+
integraldisplayintegraldisplay
Sa134
ydxdz = 2integraldisplay 3
0
dz
integraldisplay 1
1
radicalbig
1?x2 dx = 3pi
a75
integraldisplayintegraldisplay
S
zdxdy+xdydz +ydxdz = 6pi.
5,a79a142
integraldisplayintegraldisplay
S
x3 dydz +y3 dzdx+z3 dxdy
Sa143a165a161x2 +y2 +z2 = a2a27a9a253.
a41a181a226a211a134a233a161a167a144a73a79a142
integraldisplayintegraldisplay
S
x3 dydza167a133
integraldisplayintegraldisplay
S
x3 dydz =
integraldisplayintegraldisplay
S1
x3 dydz +
integraldisplayintegraldisplay
S2
x3 dydz
a217a165S1a57S2a169a79a76a171a101a140a165a161a57a254a140a165a161a167a61S2,x =radicalbiga2?y2?z2a65a18a254a253a182S1,x =?radicalbiga2?y2?z2a65
a18a101a253
a75
integraldisplayintegraldisplay
S
x3 dydz =
integraldisplayintegraldisplay
S2
x3 dydz +
integraldisplayintegraldisplay
S1
x3 dydz = 2
integraldisplayintegraldisplay
y2+z2lessorequalslanta2
(a2?y2?z2)32 dydz =
2
integraldisplay 2pi
0
dθ
integraldisplay a
0
r(a2?r2)32 dr = 45 pia5
a117a180
integraldisplayintegraldisplay
S
x3 dydz +y3 dzdx+z3 dxdy = 125 pia5.
273
a49a19a155a19a217 a136a171a200a169a109a27a233a88a218a124a216a208a218
§1,a136a171a200a169a109a27a233a88
1,a124a94a130a21a250a170a79a142a173a130a200a169a181
(1)
contintegraldisplay
l
xy2 dx?x2ydy,la181a11a177x2 +y2 = a2a182
(2)
contintegraldisplay
l
(x+y)dx?(x?y)dy,la181a253a11a177x
2
a2 +
y2
b2 = 1a182
(3)
contintegraldisplay
l
(x+y)2 dx?(x2 +y2)dy,la181a186a58a143A(1,1),B(3,2),C(2,5)a27a110a14a47a27a62a46a182
(4)
integraldisplay
sluraboveAMO(e
x siny?my)dx+ (ex cosy?m)dya167
a217a165 sluraboveAMOa143a100a58A(a,0)a150a58O(0,0)a178a76a254a140a11a177x2 +y2 = axa27a23a180a182
(5)
contintegraldisplay
l
ex[(1?cosy)dx?(y?siny)dy],la181a171a1410 <x<pi,0 <y< sinxa27a62a46.
a41a181
(1) a100a130a21a250a170a167a100a158P = xy2,Q =?x2y
a75
contintegraldisplay
l
xy2 dx?x2ydy =
integraldisplayintegraldisplay
D
(?2xy?2xy)dxdy =?4
integraldisplay a
a
xdx
integraldisplay √a2?x2
√
a2?x2
ydy = 0
(2) P = x+y,Q =?(x?y)a167a75
contintegraldisplay
l
(x+y)dx?(x?y)dy =?2
integraldisplayintegraldisplay
D
dxdy =?2piab
(3) AB,BC,CAa27a144a167a169a79a143a181AB,x?2y+ 1 = 0;BC,3x+y?11 = 0;CA,4x?y?3 = 0
P = (x+y)2,Q =?(x2 +y2)
a75I =
contintegraldisplay
l
(x+y)2 dx?(x2 +y2)dy =?2
integraldisplayintegraldisplay
D
(2x+y)dxdy
=?2
bracketleftBiggintegraldisplay
2
1
dx
integraldisplay 4x?3
x+1
2
(2x+y)dy+
integraldisplay 3
2
dx
integraldisplay 11?3x
x+1
2
(2x+y)dy
bracketrightBigg
=?4632
(4) a51Oxa182a254a235a26a58O(0,0)a134A(a,0)a167a249a24a66a8a164a181a52a27a140a11a47AMOAa167a133a51a130a227OAa254integraldisplay
OA
(ex siny?my)dx+ (ex cosy?m)dy = 0
a75
integraldisplay
sluraboveAMOA(e
x siny?my)dx+ (ex cosy?m)dy =
integraldisplay
sluraboveAMO(e
x siny?my)dx+ (ex cosy?m)dy
a124a94a130a21a250a170a167a26
integraldisplay
sluraboveAMOA(e
x siny?my)dx+ (ex cosy?m)dy = m
integraldisplayintegraldisplay
D
dxdy = pim8 a2
a117a180
integraldisplay
sluraboveAMO(e
x siny?my)dx+ (ex cosy?m)dy = pim
2
8 a
2
(5) P = ex(1?cosy),Q = (y?siny)(?ex)
a75
contintegraldisplay
l
ex[(1?cosy)dx?(y?siny)dy] =?
integraldisplayintegraldisplay
D
yex dxdy =?
integraldisplay pi
0
ex dx
integraldisplay sinx
0
ydy =?15 (epi?1).
2,a124a94a130a21a250a170a79a142a101a15a173a130a164a140a161a200a181
(1) a40a47a130a181x = acos3t,y = bsin3ta182
(2) a14a212a130a181(x+y)2 = ax(a> 0)a218xa182
a41a181
(1) a100a130a21a250a170a167a161a200Da143D =
integraldisplayintegraldisplay
D
dxdy = 12
contintegraldisplay
l
xdy?ydx
a113x = acos3t,y = bsin3t (0 lessorequalslanttlessorequalslant 2pi)a167a75D = 12
contintegraldisplay
l
xdy?ydx = 38 ab
integraldisplay 2pi
0
sin2 2tdt = 38 piab
274
(2) a138a147a134y = txa167a75a6a144a167a122a143x2(1 +t)2 = ax (a> 0,x> 0)
a117a180a26a173a130a235a234a144a167x = a(1 +t)2,y = at(1 +t)2 (0 lessorequalslantt< +∞)
a167a134Oxa182a27a2a58a143(a,0)a134(0,0)
a51Oxa182a254a108(0,0)a58a20(a,0)a58a27a152a227a254a107xdy?ydx = 0a182a51a14a212a130a254a107xdy?ydx = a
2
(1 +t)4 dt
a117a180a161a200D = 12
contintegraldisplay
l
xdy?ydx = a
2
2
integraldisplay +∞
0
dt
(1 +t)4 =
a2
6,
3,a121a178a101Ca143a178a161a254a181a52a173a130a167la143a63a191a144a149a75
contintegraldisplay
c
cos(l,n)ds = 0
a170a165na143Ca27a9a123a130a144a149.
a121a178a181a216a148a23Ca27a144a149a143a95a158a2a144a149
a207(l,n) = (l,x)?(n,x)a167a75cos(l,n) = cos(l,x)cos(n,x) + sin(l,x)sin(n,x)
a113sin(n,x) =?cos(t,x),cos(n,x) = sin(t,x)a133cos(t,x) = dxds,sin(t,x) = dyds
a75cos(l,n)ds = cos(l,x)dy?sin(l,x)dx
a117a180
contintegraldisplay
C
cos(l,n)ds =
contintegraldisplay
C
[?sin(l,x)dx+ cos(l,x)dy]
a100P =?sin(l,x),Q = cos(l,x)a167a26?P?y = 0 =?Q?x
a117a180
contintegraldisplay
C
cos(l,n)ds =
integraldisplayintegraldisplay
D
parenleftBigg
Q
x?
P
y
parenrightBigg
dxdy = 0
4,a23u(x,y),v(x,y)a180a228a107a19a30a235a89a160a19a234a27a188a234a167a191a23
u≡?
2u
x2 +
2u
y2
a121a178a181
(1)
integraldisplayintegraldisplay
σ
udxdy =
integraldisplay
l
u
n ds
(2)
integraldisplayintegraldisplay
σ
v?udxdy =?
integraldisplayintegraldisplay
σ
parenleftBigg
u
x
v
x +
u
y
v
y
parenrightBigg
dxdy+
contintegraldisplay
l
v?u?n ds
(3)
integraldisplayintegraldisplay
σ
(u?v?v?u)dxdy =?
integraldisplay
l
parenleftBigg
v?u?n?u?v?n
parenrightBigg
ds
a217a165σa143a52a173a130la164a140a27a178a161a171a141a167?u?n,?v?na143a247la9a123a130a144a149a19a234.
a121a178a181
(1)
integraldisplay
l
u
n ds =
integraldisplay
l
parenleftBigg
u
x cos(n,x) +
u
y sin(n,x)
parenrightBigg
ds =
integraldisplay
l
u
x sin(t,x)ds?
integraldisplay
l
u
y cos(t,x)ds
=
integraldisplay
l
u
x dy?
integraldisplay
l
u
y dx =
integraldisplayintegraldisplay
σ
bracketleftBigg
2u
x2 +
2u
y2
bracketrightBigg
dxdy =
integraldisplayintegraldisplay
σ
udxdy
(2) a207
contintegraldisplay
l
v?u?n ds =
contintegraldisplay
l
v
parenleftBigg
u
x cos(n,x) +
u
y sin(n,x)
parenrightBigg
ds =
contintegraldisplay
l
bracketleftBigg
v?u?x sin(t,x)?v?u?y cos(t,x)
bracketrightBigg
ds
=
contintegraldisplay
l
v?u?x dy?v?u?y dx =
integraldisplayintegraldisplay
σ
bracketleftBigg
x
parenleftBigg
v?u?x
parenrightBigg
y
parenleftBigg
v?u?y
parenrightBiggbracketrightBigg
dxdy
=
integraldisplayintegraldisplay
σ
parenleftBigg
u
x
v
x +
u
y
v
y
parenrightBigg
dxdy+
integraldisplayintegraldisplay
σ
v?udxdy
a75
integraldisplayintegraldisplay
σ
v?udxdy =?
integraldisplayintegraldisplay
σ
parenleftBigg
u
x
v
x +
u
y
v
y
parenrightBigg
dxdy+
contintegraldisplay
l
v?u?n ds
275
(3) a100(2)a167a26
integraldisplayintegraldisplay
σ
u?vdxdy =?
integraldisplayintegraldisplay
σ
parenleftBigg
v
x
u
x +
v
y
u
y
parenrightBigg
dxdy+
contintegraldisplay
l
u?v?n ds
a75
integraldisplayintegraldisplay
σ
(u?v?v?u)dxdy =?
integraldisplay
l
parenleftBigg
v?u?n?u?v?n
parenrightBigg
ds
5,a166a177a101a200a169a131a138
I =
contintegraldisplay
l
[xcos(n,x) +ycos(n,y)]ds
la181a157a140a107a46a171a141a27a123a252a181a52a173a130a167na143a167a27a9a123a130a144a149.
a41a181I =
contintegraldisplay
l
[xcos(n,x) +ycos(n,y)]ds =
contintegraldisplay
l
[xsin(t,x)?ycos(t,x)]ds
=
contintegraldisplay
l
xdy?ydx =
integraldisplayintegraldisplay
σ
parenleftBigg
dx
dx+
dy
dy
parenrightBigg
dxdy = 2
integraldisplayintegraldisplay
σ
dxdy = 2S
6,a121a178a181 contintegraldisplay
l
cos(r,n)
r ds = 0
a217a165la180a152a252a235a207a171a141σa27a62a46a13ra180la254a27a152a58a20σa9a44a152a189a58a27a229a108.a101ra76a171la254a152a58a20σa83a44a152a189a58a27a229a108a167
a64a34a249a200a169a131a138a31a1172pi.
a121a178a181a23ra143a58A(x,y)a20la254a27a58M(ξ,η)a27a149a254a167n,ra134Oxa182a27a89a14a169a79a143α,β
a75(r,n) = α?βa167a117a180cos(r,n) = cosαcosβ + sinαsinβ = ξ?xr cosα+ η?yr sinα
a75
contintegraldisplay
l
cos(r,n)
r ds =
contintegraldisplay
l
parenleftBigg
η?y
r2 sinα+
ξ?x
r2 cosα
parenrightBigg
ds =
contintegraldisplay
l
ξ?x
r2 dη?
η?y
r2 dξ
a207P =?η?yr2,Q = ξ?xr2 a167a75?P?η =?(ξ?x)
2 + (η?y)2
r4 =
Q
ξ
a207a13P,Qa27a160a19a234a216a22a58A(a100a63r = 0)a9a167a51a28a178a161a254a180a235a89a27a167a133?Q?ξ =?P?η
a117a180a124a94a130a21a250a170a167a127a8a58Aa51a173a130la131a9a158a167
contintegraldisplay
l
cos(r,n)
r ds = 0
a8a58a51a173a130 l a131a83a158a167P,Q,?P?η,?Q?ξ a254a51(x,y)a216a235a89a167a75a216a85a134a26a166a94a130a21a250a170a167a143a100a51 l a164a157a140a27a171
a141σa83a167a177Aa143a11a37a167Ra143a140a187a138a152a11a167a177a217a11a177a138a143a173a130 lprimea167a191a166a217a157a140a27a171a141σprime? σa167a50a242σa42a140a143σprimeprimea167
a166σ?σprimeprime
a207P,Q,?P?η,?Q?ξ a254a51a216(x,y)a9a27a18a135a178a161a254a235a89a133?P?η =?Q?ξ
a75a51a69a235a207a171a141σprimeprime\(x,y)a165a235a89a133?P?η =?Q?ξ
a249a158
contintegraldisplay
l
ξ?x
r2 dη?
η?y
r2 dξ =
contintegraldisplay
lprime
ξ?x
r2 dη?
η?y
r2 dξ
a13
contintegraldisplay
lprime
ξ?x
r2 dη?
η?y
r2 dξ =
integraldisplay 2pi
0
dθ = 2pi
a75
contintegraldisplay
l
cos(r,n)
r ds = 2pi
(a8a58Aa51la254a158a167
contintegraldisplay
l
cos(r,n)
r ds = pi)
7,a124a94a112a100a250a170a67a134a177a101a200a169a181
(1)
integraldisplayintegraldisplay
S
xydxdy+xzdxdz +yzdydz
(2)
integraldisplayintegraldisplay
S
parenleftBigg
u
x cosα+
u
y cosβ +
u
z cosγ
parenrightBigg
dS
a217a165cosα,cosβ,cosγa180a173a161a27a9a123a130a144a149a123a117.
a41a181
276
(1) a207P = yz,Q = xz,R = xya167a75Px = Qy = Rz = 0
a117a180a100a112a100a250a170a167a26
integraldisplayintegraldisplay
S
xydxdy+xzdxdz +yzdydz = 0
(2) a207P = ux,Q = uy,R = uza167a75a100a112a100a250a170a167a26integraldisplayintegraldisplay
S
parenleftBigg
u
x cosα+
u
y cosβ +
u
z cosγ
parenrightBigg
dS =
integraldisplayintegraldisplayintegraldisplay
V
parenleftBigg
2u
x2 +
2u
y2 +
2u
z2
parenrightBigg
dxdydz
8,a124a94a112a100a250a170a79a142a173a161a200a169a181
(1)
integraldisplayintegraldisplay
S
x2 dydz +y2 dxdz +z2 dxdy,Sa181a225a144a780 lessorequalslantx,y,z lessorequalslantaa27a9a76a161a182
(2)
integraldisplayintegraldisplay
S
x3 dydz +y3 dxdz +z3 dxdy,Sa181a252a160a165a9a76a161a182
(3)
integraldisplayintegraldisplay
S
(x2 cosα+y2 cosβ +z2 cosγ)dS,Sa181x2 +y2 = z2,0 lessorequalslantz lessorequalslanth;cosα,cosβ,cosγa143a100a173a161a9a123a130a144
a149a123a117.
a41a181
(1) a207P = x2,Q = y2,R = z2a167
a75a100a112a100a250a170a167a26
integraldisplayintegraldisplay
S
x2 dydz +y2 dxdz +z2 dxdy = 2
integraldisplay a
0
dx
integraldisplay a
0
dy
integraldisplay a
0
(x+y+z)dz = 3a4
(2) a207P = x3,Q = y3,R = z3
a75a100a112a100a250a170a167a26
integraldisplayintegraldisplay
S
x3 dydz +y3 dxdz +z3 dxdy = 3
integraldisplayintegraldisplayintegraldisplay
V
(x2 +y2 +z2)dxdydz
= 3
integraldisplay 2pi
0
dθ
integraldisplay pi
0
sin?d?
integraldisplay 1
0
r4 dr = 125 pi
(3) a100a112a100a250a170a167a26integraldisplayintegraldisplay
S
(x2 cosα+y2 cosβ +z2 cosγ)dS = 2
integraldisplayintegraldisplayintegraldisplay
V
(x+y+z)dxdydz
= 2
integraldisplay 2pi
0
d?
integraldisplay h
0
rdr
integraldisplay h
r
[r(cos?+ sin?) +z]dz = pih
4
2,
9,a121a178a181a101
u≡?
2u
x2 +
2u
y2 +
2u
z2
Sa180Va27a62a46a173a161a167a75a164a225a101a161a250a170a181
(1)
integraldisplayintegraldisplayintegraldisplay
V
udxdydz =
integraldisplayintegraldisplay
S
u
n dS
(2)
integraldisplayintegraldisplay
S
u?u?n dS =
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBiggparenleftBigg
u
x
parenrightBigg2
+
parenleftBigg
u
y
parenrightBigg2
+
parenleftBigg
u
z
parenrightBigg2bracketrightBigg
dxdydz +
integraldisplayintegraldisplayintegraldisplay
V
u?udxdydz
a170a165ua51V +Sa254a107a235a89a19a30a19a234a167?u?na143a247a173a161Sa9a123a130a144a149a27a19a234.
a121a178a181
(1)
integraldisplayintegraldisplay
S
u
n dS =
integraldisplayintegraldisplay
S
bracketleftBigg
u
x cos(n,x) +
u
y cos(n,y) +
u
z cos(n,z)
bracketrightBigg
dS =
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBigg
2u
x2 +
2u
y2 +
2u
z2
bracketrightBigg
dxdydz =
integraldisplayintegraldisplayintegraldisplay
V
udxdydz
(2)
integraldisplayintegraldisplay
S
u?u?n dS =
integraldisplayintegraldisplay
S
bracketleftBigg
u?u?x cos(n,x) +u?u?y cos(n,y) +u?u?z cos(n,z)
bracketrightBigg
dS
=
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBigg
x
parenleftBigg
u?u?x
parenrightBigg
+y
parenleftBigg
u?u?y
parenrightBigg
+z
parenleftBigg
u?u?z
parenrightBiggbracketrightBigg
dxdydz
277
=
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBiggparenleftBigg
u
x
parenrightBigg2
+
parenleftBigg
u
y
parenrightBigg2
+
parenleftBigg
u
z
parenrightBigg2bracketrightBigg
dxdydz +
integraldisplayintegraldisplayintegraldisplay
V
u
bracketleftBigg
2u
x2 +
2u
y2 +
2u
z2
bracketrightBigg
dxdydz
=
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBiggparenleftBigg
u
x
parenrightBigg2
+
parenleftBigg
u
y
parenrightBigg2
+
parenleftBigg
u
z
parenrightBigg2bracketrightBigg
dxdydz +
integraldisplayintegraldisplayintegraldisplay
V
u?udxdydz
10,a121a178a100a173a161Sa164a157a140a27a78a200a31a117
V = 13
integraldisplayintegraldisplay
S
(xcosα+ycosβ +zcosγ)dS
a170a165cosα,cosβ,cosγa143a173a161Sa27a9a123a130a27a144a149a123a117.
a121a178a181a100a112a100a250a170a167a26
V =
integraldisplayintegraldisplayintegraldisplay
V
dxdydz = 13
integraldisplayintegraldisplay
S
(xdydz +ydxdz +zdxdy) = 13
integraldisplayintegraldisplay
S
(xcosα+ycosβ +zcosγ)dS.
11,a124a94a100a247a142a105a250a170a79a142a173a130a200a169a181
(1)
contintegraldisplay
l
ydx+zdy+xdz,la181a11a177
braceleftbigg x2 +y2 +z2 = a2
x+y+z = 0 a108xa182a20a149a119a22a11a177a180a95a158a2a144a149a27a182
(2)
contintegraldisplay
l
(z?y)dx+ (x?z)dy+ (y?x)dz,la180a108(a,0,0)a178(0,a,0)a218(0,0,a)a163a20(a,0,0)a27a110a14a47.
a41a181
(1) a114a178a161x+y+z = 0a254la164a157a140a27a171a141a80a143σa167a75σa27a123a130a144a149a143(1,1,1)
a75a217a144a149a123a117a143cosα = cosβ = cosγ = 1√3
a117a180
contintegraldisplay
l
ydx+zdy+xdz =
integraldisplayintegraldisplay
S
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
cosα cosβ cosγ
x
y
z
y z x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
=?
integraldisplayintegraldisplay
S
(√3dS =?√3 pia2
(2) a114la164a157a140a27a171a141a80a143σa167a75σa27a123a130a144a149a143(1,1,1)a167a75a217a144a149a123a117a143cosα = cosβ = cosγ = 1√3
a113P = z?y,Q = x?z,R = y?xa167a75?R?yQ?z = 2,?P?zR?x = 2,?Q?xP?y = 2
a117a180
contintegraldisplay
l
(z?y)dx+ (x?z)dy+ (y?x)dz = 2√3
integraldisplayintegraldisplay
S
dS = 3a2
278
§2,a173a130a200a169a218a180a187a27a195a39a53
1,a23a51a44a52a221a47a171a141Da83?P?y =?Q?x a167a193a121
U(x,y) =
integraldisplay x
x0
P(x,y)dx+
integraldisplay y
y0
Q(x0,y)dy+C
a143P dx+Qdya27a6a188a234a167a217a165C = U(x0,y0).
a121a178a181a207U(x,y) =
integraldisplay x
x0
P(x,y)dx+
integraldisplay y
y0
Q(x0,y)dy+C
a75?U?x = P(x,y),?U?y =
integraldisplay x
x0
Py(x,y)dx+Q(x0,y) =
integraldisplay x
x0
Qx(x,y)dx+Q(x0,y) = Q(x,y)
a117a180dU = P(x,y)dx+Q(x,y)dy
a113U(x0,y0) = Ca167a75U(x,y) =
integraldisplay x
x0
P(x,y)dx +
integraldisplay y
y0
Q(x0,y)dy + Ca143P dx + Qdya27a6a188a234a167a217a165C =
U(x0,y0)
2,a79a142a101a15a28a135a169a170a27a130a200a169a181
(1)
integraldisplay (1,1)
(0,0)
(x?y)(dx? dy)
(2)
integraldisplay (a,b)
(0,0)
f(x+y)(dx+ dy)a167a170a165f(u)a180a235a89a188a234a182
(3)
integraldisplay (1,2)
(2,1)
ydx?xdy
x2 a167a247a216a218Oya182a131a2a27a229a187a182
(4)
integraldisplay (0,1,1)
(1,2,3)
yzdx+xzdy+xydz
(5)
integraldisplay (6,8)
(1,0)
xdx+ydyradicalbig
x2 +y2 a167a247a216a207a76a6a58a27a229a187a182
(6)
integraldisplay (1,2)
(2,1)
(x)dx+ψ(y)dya167a217a165?,ψa143a235a89a188a234.
a41a181
(1) a207(x?y)(dx? dy) = d(x?y)
2
2 a167a75
integraldisplay (1,1)
(0,0)
(x?y)(dx? dy) = (x?y)
2
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1)
(0,0)
= 0
(2) a207P +Q = f(x+y)a167a75?P?y = fprime(x+y) =?Q?xa167a117a180a108(0,0)a20(a,b)a200a169a134a180a187a195a39
a18(0,0) → (a,0) → (a,b)a167
a75
integraldisplay (a,b)
(0,0)
f(x+y)(dx+ dy) =
integraldisplay a
0
f(x+ 0)dx+
integraldisplay b
0
f(a+y)dy =
integraldisplay a+b
0
f(u)du
(3) a8xnegationslash= 0a158a167ydx?xdyx2 = d
parenleftBigg
yx
parenrightBigg
a167a75
integraldisplay (1,2)
(2,1)
ydx?xdy
x2 =?
y
x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,2)
(2,1)
=?32
(4) a207yzdx+xzdy+xydz = dxyza167a75
integraldisplay (0,1,1)
(1,2,3)
yzdx+xzdy+xydz = 0
(5) a8(x,y) negationslash= (0,0)a158a167xdx+ydyradicalbigx2 +y2 = dradicalbigx2 +y2a167a75
integraldisplay (6,8)
(1,0)
xdx+ydyradicalbig
x2 +y2 = 9
(6) a100a117?,ψa180a235a89a188a234a167a25a107?(x)dx+ψ(y)dy = d(F(x) +G(x))
a217a165F(x) =
integraldisplay x
2
(u)du,G(y) =
integraldisplay y
1
ψ(v)dv
a117a180a107
integraldisplay (1,2)
(2,1)
(x)dx+ψ(y)dy = (F(x)+G(y))
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,2)
(2,1)
=
integraldisplay 1
2
(u)du+
integraldisplay 2
1
ψ(v)dv =
integraldisplay 2
1
[ψ(x)(x)]dx
3,a166a6a188a234ua181
279
(1) (x2 + 2xy?y2)dx+ (x2?2xy?y2)dy
(2) (2xcosy?y2 sinx)dx+ (2ycosx?x2 siny)dy
(3) az dx+ bz dy+?by?axz2 dz
(4) (x2?2yz)dx+ (y2?2xz)dy+ (z2?2xy)dz
(5) ex[ey(x?y+ 2) +y]dx+ex[ey(x?y) + 1]dy
a41a181
(1) a207?P?y = 2x?2y,?Q?x = 2x?2ya167a75?P?y =?Q?x
a117a180u =
integraldisplay x
0
(x2 + 2xy?y2)dx+
integraldisplay y
0
(?y2)dy+C = x
3
3 +x
2y?xy2? y
3
3 +C
(2) a207(2xcosy?y2 sinx)dx + (2ycosx?x2 siny)dy = cosydx2 + x2 dcosy + y2 dcosx + cosxdy2 =
d(x2 cosy+y2 cosx)
a75u = x2 cosy+y2 cosx+C
(3) a207az dx+ bz dy+?by?axz2 dz = azdx?xdzz2 +bzdy?ydzz2 = dax+byz
a75u = 1z (ax+by) +C
(4) a207(x2?2yz)dx+ (y2?2xz)dy+ (z2?2xy)dz = 13 (dx3 + dy3 + dz3)?2(yzdx+xzdy+xydz) =
d
parenleftBigg
x3 +y3 +z3
3?2xyz
parenrightBigg
a75u = 13 (x3 +y3 +z3)?2xyz +C
(5) a207ex[ey(x?y + 2) + y]dx + ex[ey(x?y) + 1]dy = (x?y)ex+y(dx + dy) + 2ex+y dx + d(yex) =
d((x?y)ex+y) +ex+y d(x+y) + d(yex) = d((x?y+ 1)ex+y) + d(yex)
a75u = (x?y+ 1)ex+y +yex +C
4,a8a121a181
P dx+Qdy = 12 xdy?ydxAx2 + 2Bxy+Cy2
(A,B,Ca143a126a234a167a133AC?B2 > 0)a183a220a94a135a181
P
y =
Q
x
a166a181a39a117a219a58(0,0)a27a204a130a126a234.
a121a178a181a207P =? y2(Ax2 + 2Bxy+Cy2),Q = x2(Ax2 + 2Bxy+Cy2)
a75?P?y = Cy
2?Ax2
2(Ax2 + 2Bxy+Cy2)2 =
Q
x
a117a180ω =
contintegraldisplay
x2+y2=1
P dx+Qdy =
integraldisplay pi
2
pi2
dt
2(Acos2t+ 2Bsintcost+Csin2t) =
1√
AC?B2 arctan
Cparenleftbigtant+ BCparenrightbig√
AC?B2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
pi
2
pi2
=
pi√
AC?B2,C > 0
pi√AC?B2,C < 0
5,a121a178a181 integraldisplay
xdx+ydy
x2 +y2
a39a117a219a58(0,0)a27a204a130a126a234a1430a167a108a13xdx+ydyx2 +y2 a27a200a169a134a180a187a195a39.
a121a178a181a207P = xx2 +y2,Q = yx2 +y2 a167a75?P?y =? 2xy(x2 +y2)2 =?Q?x
a117a180ω =
contintegraldisplay
x2+y2=1
P dx+Qdy = 0
280
(1) a101a52a180la216a157a140(0,0)a58a167a140a242a219a58(0,0)a134a171a141Da27a62a46a94a152a94a173a130Ca235a26a229a53a167a117a180a69a235a207a171a141a67a164a10
a252a235a207a171a141
a113?P?y =?Q?x a167a75a100a31a100a94a135a167a26
contintegraldisplay
l
xdx+ydy
x2 +y2 = 0
(2) a101a52a180la157a140a219a58(0,0)a167a207a247a130a55a219a58a27a63a152a52a180a27a200a169a31a117a204a130a126a234a167a75
contintegraldisplay
l
xdx+ydy
x2 +y2 = 0
a111a131xdx+ydyx2 +y2 a27a200a169a134a180a187a195a39.
281
§3,a124a216a208a218
1,a23H(t) = eta +e?tba167a217a165a,ba143a126a149a254a167ta143a235a234a167
(1) a166 dHdt
(2) a121a178 d
2H
dt2 = H
a41a181a207H(t) = eta +e?tba167a,ba143a126a149a254a167a75
(1) dHdt = eta?e?tb
(2) d
2H
dt2 =
d
dt
parenleftBigg
dH
dt
parenrightBigg
= eta +e?tb = H
2,a121a178a181 ddt[A·(B×C)] = dAdt ·(B×C) + A·
parenleftBigg
dB
dt ×C
parenrightBigg
+ A·
parenleftBigg
B× dCdt
parenrightBigg
a121a178a181a23A = Axi +Ayj +Azk,B = Bxi +Byj +Bzk,C = Cxi +Cyj +Czk
a75A·(B×C) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
Ax Ay Az
Bx By Bz
Cx Cy Cz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglea167a233a31a170a252a224a166a19a167a109a224a94a233a49a15a170a166a19a123a75a167a26
d
dt[A·(B×C)] =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
Axt Ayt Azt
Bx By Bz
Cx Cy Cz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle+
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
Ax Ay Az
Bxt Byt Bzt
Cx Cy Cz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle+
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
Ax Ay Az
Bx By Bz
Cxt Cyt Czt
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
= dAdt ·(B×C) + A·
parenleftBigg
dB
dt ×C
parenrightBigg
+ A·
parenleftBigg
B× dCdt
parenrightBigg
3,a23a = 3i + 20j?15ka167a233a101a15a234a254a124φa169a79a166a209gradφa57div(φa).
(1) φ = (x2 +y2 +z2)?12
(2) φ = x2 +y2 +z2
(3) φ = ln(x2 +y2 +z2)
a41a181
(1) gradφ = φxi +φyj +φzk =?x
(x2 +xy+y2)32
i +?y
(x2 +xy+y2)32
j +?z
(x2 +xy+y2)32
k
div(φa) = φdiva + gradφ·a = gradφ·a =?3x?20y+ 15z
(x2 +xy+y2)32
(2) gradφ = 2xi + 2yj + 2zk,div(φa) = 6x+ 40y?30z
(3) gradφ = 2xx2 +y2 +z2 i + 2yx2 +y2 +z2 j + 2zx2 +y2 +z2 k,div(φa) = 6x+ 40y?30zx2 +y2 +z2
4,a23U(x,y,z) = xyz
(1) a166U(x,y,z)a51a58P1(0,0,0),P2(1,1,1)a57P3(2,1,1)a63a247b = 2i + 3j?4ka27a144a149a19a234a182
(2) a51a254a227a110a58a63a167U(M)a27a129a140a144a149a19a234a143a219a138a186
(3) a51a254a227a110a58a63a167a166divgradU(M)a57rotgradU(M).
a41a181
(1) a207ba27a144a149a123a117a143cosα = 2√29,cosβ = 3√29,cosγ =? 4√29
a75?U?b = yzcosα+xzcosβ +xycosγ = 1√29 (2yz + 3xz?4xy)
a117a180a51P1(0,0,0)a58?U?b = 0a182a51P2(1,1,1)a58?U?b =
√29
29 a182a51P3(2,1,1)a58
U
b = 0
(2) a207?U?b = gradU ·b0 = |gradU|cos(gradU,b0)a167a217a165b0a180ba144a149a27a252a160a149a254
a75U(M)a27a129a140a144a149a19a234a143|gradU| =radicalbigy2z2 +x2z2 +x2y2
a117a180a51P1(0,0,0)a58|gradU| = 0a182a51P2(1,1,1)a58|gradU| = √3a182a51P3(2,1,1)a58|gradU| = 3
282
(3) a207gradU = yzi +xzj +xyk
a75divgradU =?(yz)?x +?(xz)?y +?(xy)?z = 0
rotgradU =
parenleftBigg
(xy)
y?
(xz)
z
parenrightBigg
i +
parenleftBigg
(yz)
z?
(xy)
x
parenrightBigg
j +
parenleftBigg
(xz)
x?
(yz)
y
parenrightBigg
k = 0
a117a180a51a254a227a110a58a63a167divgradU(M) = 0,rotgradU(M) = 0.
5,a166a149a254a = x2i +y2j +z2ka66a76a165x2 +y2 +z2 = 1,x> 0,y> 0,z> 0a27a54a254.
a41a181Φ =
integraldisplayintegraldisplay
S
x2 dydz +y2 dxdz +z2 dxdya167
integraldisplayintegraldisplay
S
z2 dxdy =
integraldisplay pi
2
0
dθ
integraldisplay 1
0
(1?r2)rdr = pi8
a97a113a47a167a169a79a149XOZ,YOZa178a161a221a75a167a140a26
integraldisplayintegraldisplay
S
y2 dxdz =
integraldisplayintegraldisplay
S
x2 dydz = pi8 a167a117a180Φ = 38 pi.
6,a166a = yzi +xzj +xyka207a76Sa27a54a254a167a23
(1) Sa143a11a206a78x2 +y2 lessorequalslanta2,0 lessorequalslantz lessorequalslantha27a253a161a182
(2) Sa143(1)a165a11a206a78a27a254a46a161a182
(3) Sa143(1)a165a11a206a78a27a76a161.
a41a181
(3)
integraldisplayintegraldisplay
S
an dS =
integraldisplayintegraldisplayintegraldisplay
V
divadV =
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBigg
(yz)
x +
(xz)
y +
(xy)
z
bracketrightBigg
dV = 0
a117a180a149a254aa66a76a11a206a78a76a161a27a54a254a1430
(2)a207a51a11a206a78a27a254a33a101a46a161an = xya167a75
integraldisplayintegraldisplay
Sa254
an dS =
integraldisplayintegraldisplay
x2+y2lessorequalslanta2
xydxdy =
integraldisplay 2pi
0
dθ
integraldisplay a
0
r3 sinθcosθdr = 0
a211a110
integraldisplayintegraldisplay
Sa101
an dS = 0a167a117a180a149a254aa66a76a11a206a78a254a46a161a27a54a254a1430
(1)a207
integraldisplayintegraldisplay
S
an dS =
integraldisplayintegraldisplay
Sa253
an dS +
integraldisplayintegraldisplay
Sa254
an dS +
integraldisplayintegraldisplay
Sa101
an dSa167a75
integraldisplayintegraldisplay
Sa253
an dS = 0.
7,a166a = grad
parenleftBigg
arctan yx
parenrightBigg
a247a173a130la27a130a54a254a181
(1) la143(x?2)2 + (y?2)2 = 1,z = 0a182
(2) la143x2 +y2 = 4,z = 1.
a41a181
(1) a100a174a127a167a107a = grad
parenleftBigg
arctan yx
parenrightBigg
=? yx2 +y2 i + xx2 +y2 j
a75rota =
parenleftBig
x
x2+y2
parenrightBig
x?
parenleftBig
yx2+y2
parenrightBig
y
k = 0(a216x = y = 0a61Oza182a254a27a58)
a207l,(x?2)2 + (y?2)2 = 1,z = 0a180a216a140a55za182a27a173a130a167a25a140a117la254a220a152a173a161Sa167a166Sa134Oza182a216a131a2
a75a226a100a247a142a105a250a170a167a107a130a54a254
contintegraldisplay
l
adl =
integraldisplayintegraldisplay
S
n·rotadS = 0
(2) a207l,x2 + y2 = 4,z = 1a167a100a158la20a208a140a55Oza182a94a61a152a177a167a18a126a234c > 0a191a169a2(c < 2)a167a166la160a117a178
a161z = ca27a254a144a167a51a178a161z = ca254a140a55Oza182a18a152a11a177lr,x2 + y2 = r2,z = ca167ra191a169a2a167a166ra2a1172a167
a177la134lra143a62a46a220a254a152a173a161Sa167a166Sa134Oza182a216a131a2
a100a100a247a142a105a250a170a167a26
contintegraldisplay
l
a· dr +
contintegraldisplay
lr
a· dr =
integraldisplayintegraldisplay
S
n·rotadS = 0a167a217a165?lra76a171a247a94a158a2a144a149
a117a180a130a54a254
contintegraldisplay
l
a· dr =
contintegraldisplay
lr
a· dr
a113a18lra27a235a234a144a167x = rcosθ,y = rsinθ,z = ca167a26
contintegraldisplay
lr
a· dr =
integraldisplay 2pi
0
dθ = 2pi
a108a13
contintegraldisplay
l
a· dr = 2pi.
283
8,a166a149a254a =?yi +xj +ck(ca143a126a234)a27a130a54a254a181
(1) a247a11a177x2 +y2 = 1,z = 0a182
(2) a247a11a177(x?2)2 +y2 = 1,z = 0.
a41a181
(1) a207l,x2 +y2 = 1,z = 0a167a75l = costi + sintj + 0k(0 lessorequalslanttlessorequalslant 2pi)
a117a180a·dl = dta167a108a13a164a166a130a54a254a143
contintegraldisplay
l
a· dl =
integraldisplay 2pi
0
dt = 2pi
(2) a207l,(x?2)2 +y2 = 1,z = 0a167a75l = (2 + cost)i + sintj + 0k(0 lessorequalslanttlessorequalslant 2pi)
a117a180a·dl = (2cost+ 1)dta167a108a13a164a166a130a54a254a143
contintegraldisplay
l
a· dl =
integraldisplay 2pi
0
(2cost+ 1)dt = 2pi
9,a121a178a181
(1) rot(uA) = urotA + gradu×Aa182
(2) div(φa) = φdiva + gradφ·a
(3) graddiva?rotrota =?a
a121a178a181
(1) a207rotx(uA) = u
parenleftBigg
Az
y?
Ay
z
parenrightBigg
+
parenleftBigg
Az?u?y?Ay?u?z
parenrightBigg
= urotxA + (gradu×A)x
a211a123a140a26a167roty(uA) = urotyA + (gradu×A)y,rotz(uA) = urotzA + (gradu×A)z
a117a180rot(uA) = urotA + gradu×A
(2) a207?(φax)?x = φ?ax?x +ax?φ?x,?(φay)?y = φ?ay?y +ay?φ?y,?(φaz)?z = φ?az?z +az?φ?z
a75div(φa) = φdiva + gradφ·a
(3) a207graddiva = grad
parenleftBigg
ax
x +
ay
y +
az
z
parenrightBigg
=
parenleftBigg
2ax
x2 +
2ay
x?y +
2az
x?z
parenrightBigg
i +
parenleftBigg
2ax
x?y +
2ay
y2 +
2az
y?z
parenrightBigg
j +
parenleftBigg
2ax
x?z +
2ay
y?z +
2az
z2
parenrightBigg
k
rotrota = rot
bracketleftBiggparenleftBigg
az
y?
ay
z
parenrightBigg
i +
parenleftBigg
ax
z?
az
x
parenrightBigg
j +
parenleftBigg
ay
x?
ax
y
parenrightBigg
k
bracketrightBigg
=
parenleftBigg
2ay
x?y +
2az
x?z?
2ax
y2?
2ax
z2
parenrightBigg
i+
parenleftBigg
2ax
x?y +
2az
y?z?
2ay
x2?
2ay
z2
parenrightBigg
j+
parenleftBigg
2ax
x?z +
2ay
y?z?
2az
x2?
2az
y2
parenrightBigg
k
a75graddiva?rotrota =?axi +?ayj +?azk =?a
10,a166rot(w×r)a167wa143a126a165a254a167ra143a165a187a149a254.
a41a181a23w = (w1,w2,w3),r = (x,y,z)
a117a180w×r =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
w1 w2 w3
x y z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= (w2z?w3y)i + (w3x?w1z)j + (w1y?w2x)k
rot(w×r) = 2w1i + 2w2j + 2w2k = 2w
11,a121a178a = yz(2x+y+z)i +xz(x+ 2y+z)j +xy(x+y+ 2z)ka143a2a197a124a167a191a166a217a179a188a234.
a121a178a181a233a152a109a63a152a58(x,y,z)a167a107
rota =
braceleftBigg
y [xy(x+y+ 2z)]?
z [xz(x+ 2y+z)]
bracerightBigg
i
+
braceleftBigg
z [yz(2x+y+z)]?
x [xy(x+y+ 2z)]
bracerightBigg
j
+
braceleftBigg
x [xz(x+ 2y+z)]?
y [yz(2x+y+z)]
bracerightBigg
k
= 0
a75aa143a2a197a124
a100a117a179φa247a118dφ = a·dl = ax dx+ay dy+az dz = d[(xyz(x+y+z)]
a75a217a179a188a234a143u(x,y,z) = xyz(x+y+z) +Ca167a217a165Ca143a63a191a126a234.
284
12,a166a149a254a = ra247a218a130r = acosti +asintj +btk(0 lessorequalslanttlessorequalslant 2pi)a27a152a227a164a138a27a245.
a41a181a207a = xi +yj +zk,l,x = acost,y = asint,z = bt(0 lessorequalslant 2pi)
a75a164a166a27a245a143W =
integraldisplay
l
xdx+ydy+zdz = 2b2pi2
13,a23φa143a140a135a188a234a167a79a142a181gradφ(r),div(φ(r)r)a57rot(φ(r)r).
a41a181gradφ(r) = φprime(r)gradr = φprime(r)· rr
div(φ(r)r) = φ(r)divr + r·gradφ(r) = 3φ(r) +rφprime(r)
rot(φ(r)r) = φ(r)rotr + gradφ(r)×r = 0
14,a166a247a118a94a135div(φ(r)r) = 0a27a188a234φ(r).
a41a181a100a254a75a167a26div(φ(r)r) = 3φ(r) +rφprime(r)
a135a166div(φ(r)r) = 0a167a144a1353φ(r) +rφprime(r) = 0a61a135φ
prime(r)
φ(r) =?
3
r
a75a26φ(r) = cr3 (ca143a126a234)
15,a166a177a101a136a149a254a27a209a221a57a94a221(a,ba143a126a149a254)a181
(1) (a·r)b
(2) a×r
(3) φ(r)(a×r)
(4) r×(a×r)
a41a181
(1) div[(a·r)b] = (a·r)divb + grad(a·r)·b = grad(a·r)·b = a·b
rot[(a·r)b] = (a·r)rotb + grad(a·r)×b = grad(a·r)×b = a×b
(2) a207(a×r)x = ayz?azy,(a×r)y = azx?axz,(a×r)z = axy?ayx
a75div(a×r) =x (ayz?azy) +y (azx?axz) +z(axy?ayx) = 0
rot(a×r) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
x
y
z
ayz?azy azx?axz axy?ayx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
= 2a
(3) div[φ(r)(a × r)] = φ(r)div(a × r) + grad(φ(r))(a × r) = φ(r)(r·rota? a·rotr) + (rφprime(r) · (a × r) =
φprime(r)rr ·(a×r) = φ
prime(r)
r [r·(a×r)] = 0rot[φ(r)(a×r)] = φ(r)rot(a×r) + grad(φ(r))×(a×r) =
φ(r)[?(axi +ayj +azk) + 3a] +
parenleftBigg
r
r φ
prime(r)
parenrightBigg
×(a×r) = 2φ(r)a + φ
prime(r)
r [r
2a?(a·r)r]
(4) a207r×(a×r) = |r|2a?(a·r)r
a75div[r×(a×r)] = |r|2diva+grad|r|2·a?[(r·a)divr+r·grad(a·r)] = (ax+ay+az)?4(xax+yay+zaz)
rot[r×(a×r)] = |r|2rotr + grad|r|2 ×a?(r·a)rotr?grad(r·a)×r = 1r (r×a)?a×r.
16,a121a178a177a101a31a170a181
(1) grad(a·b) = a×rotb + b×rota + (b·?)a + (a·?)b
(2) rot(a×b) = (b·?)a?(a·?)b + (divb)a?(diva)b
(3) c·grad(a·b) = a·(c·?) + b·(c·?)a
(4) (c·?)(a×b) = a×(c·?)b?b×(c·?)a
a121a178a181
(1) grad(a·b) = grad(axbx +ayby +azbz) =parenleftBigg
ax?bx?x +bx?ax?x +ay?by?x +by?ay?x +az?bz?x +bz?az?x
parenrightBigg
i +
parenleftBigg
ax?bx?y +bx?ax?y +ay?by?y +by?ay?y +az?bz?y +bz?az?y
parenrightBigg
j +
285
parenleftBigg
ax?bx?z +bx?ax?z +ay?by?z +by?ay?z +az?bz?z +bz?az?z
parenrightBigg
k
= a×rotb + b×rota + (b·?)a + (a·?)b
(2) rot(a×b) =
parenleftBigg
bxx +byy +bzz
parenrightBigg
a?
parenleftBigg
axx +ayy +azz
parenrightBigg
b + (divb)a?(diva)b
= (b·?)a?(a·?)b + (divb)a?(diva)b
(3) c·grad(a·b) =
cx
parenleftBigg
ax?bx?x +bx?ax?x +ay?by?x +by?ay?x +az?bz?x +bz?az?x
parenrightBigg
+
cy
parenleftBigg
ax?bx?y +bx?ax?y +ay?by?y +by?ay?y +az?bz?y +bz?az?y
parenrightBigg
+
cz
parenleftBigg
ax?bx?z +bx?ax?z +ay?by?z +by?ay?z +az?bz?z +bz?az?z
parenrightBigg
= a·(c·?) + b·(c·?)a
(4) (c·?)(a×b) =
parenleftBigg
cxx +cyy +czz
parenrightBigg
[(aybz?azby)i + (azbx?axbz)j + (axby?aybx)k]
=
parenleftBigg
bzcx?ay?x +aycx?bz?x?bycx?az?x?azcx?by?x +bzcy?ay?y +aycy?bz?y?bycy?az?y?azcy?by?y +
bzcz?ay?z +aycz?bz?z?bycz?az?z?azcz?by?z
parenrightBigg
i +
parenleftBigg
bxcx?az?x +azcx?bx?x?bzcx?ax?x?axcx?bz?x +bxcy?az?y +azcy?bx?y?bzcy?ax?y?axcy?bz?y +
bxcz?az?z +azcz?bx?z?bzcz?ax?z?axcz?bz?z
parenrightBigg
j +
parenleftBigg
bycx?ax?x +axcx?by?x?bxcx?ay?x?aycx?bx?x +bycy?ax?y +axcy?by?y?bxcy?ay?y?aycy?bx?y +
bycz?ax?z +axcz?by?z?bxcz?ay?z?aycz?bx?z
parenrightBigg
k
= a×(c·?)b?b×(c·?)a
17,a193a121divgradsin2ra140a76a171a164F(r)a27a47a170a167a191a21a209F(x).
a121a178a181divgradsin2r =?sin2r =?
2 sin2r
x2 +
2 sin2r
y2 +
2 sin2r
z2 =
x
parenleftBigg
sin2r· xr
parenrightBigg
+y
parenleftBigg
sin2r· yr
parenrightBigg
+z
parenleftBigg
sin2r· zr
parenrightBigg
= 2cos2r+ 2sin2r· 1r = 2r (sin2r+rcos2r)
a61F(r) = 2r (sin2r+rcos2r)
a75F(x) = divgradsin2xr = d
2 sin2x
dx2 = 2cos2x
18,a121a178a181a8|a|2 =a126a234a158a167a107(a·?)a =?a×rota.
a121a178a181a207|a|2 =a126a234a167a75grad|a|2 = 0
a10016a75(1)a167a26grad(a·a) = 2[a×rota + (a·?)a] = 0
a75(a·?)a =?a×rota.
1 4
1?ü? 4?D?
1?ù Ctê
§1,?êVg
1,)e? a§?x?xμ
(1)?2 < 1x+ 2
(2) (x?1)(x+ 2)(x?3) < 0
(3) 1x?1 <a
(4) 0 lessorequalslant cosxlessorequalslant 12
(5)
braceleftbigg x2?16 < 0
x2?2xgreaterorequalslant 0
)μ
(1) x<?52?x>?32
a45
a27a24
0 x-1-2-3 a98a98
(2) 1 <x< 3?x<?2
a45
a27 a24a24
0 x-1-2 31 2a99 a99a99
(3) a> 0?§x< 1?x> 1 + 1a?
a45
a24a27
0 x1 1 + 1
a
a99 a99
a< 0?§1 + 1a <x< 1
a45
a27 a24
0 x11 + 1
a
a99a99
a = 0?§x< 1
a45
a24
0 x1a99
2
(4) 2kpi+ pi3 lessorequalslantxlessorequalslant 2kpi+ pi2?2kpi? pi2 lessorequalslantxlessorequalslant 2kpi? pi3(k∈Z)
a45a3a0a3a0a3a0a3a0 0 x
(5)?4 <xlessorequalslant 0?2 lessorequalslantx< 4
a45a11 a8 a11 a80 x-4 2 4a99 a99
2,y2eyé aμ
(1) |x?y|greaterorequalslant||x|?|y||
(2) |x1 +x2 +x3 +···+xn|lessorequalslant|x1|+|x2|+···+|xn|
(3) |x+x1 +···+xn|greaterorequalslant|x|?(|x1|+···+|xn|)
y2μ
(1)?|x||y|greaterorequalslantxy§K(x?y)2 greaterorequalslant (|x|?|y|)2§u′|x?y|greaterorequalslant||x|?|y||
(2) ^ê?8B{y2.
(i) n = 2?§d|x1 +x2|lessorequalslant|x1|+|x2|§(?¤á.
(ii) bn = k?(?¤á§=k|x1 +x2 +x3 +···+xk|lessorequalslant|x1|+|x2|+···+|xk|.
Kn = k+ 1?§|x1 +x2 +x3 +···+xk+1|lessorequalslant|x1 +x2 +x3 +···+xk|+|xk+1|lessorequalslant|x1|+|x2|+
···+|xk|+|xk+1|
nt?§ég,ên§|x1 +x2 +x3 +···+xn|lessorequalslant|x1|+|x2|+···+|xn|t¤á.
(3) |x+x1 +···+xn|greaterorequalslant|x|?|x1 +x2 +x3 +···+xn|greaterorequalslant|x|?(|x1|+···+|xn|)
3,)eyé a§?x?xμ
(1) |x|>|x+ 1|
(2) 2 < 1|x| < 4
(3) |x|>A
(4) |x?a|<η,η?~ê§η> 0
(5)
vextendsinglevextendsingle
vextendsinglevextendsinglex?2
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle> x?2
x+ 1
(6) 2 < 1|x+ 2| < 3
)μ
(1) x<?12
a45
a24
0 x-1 a98
(2)?12 <x<?14?14 <x< 12
a45
a11a8a11a8
0 x1
2-
1
2
a101a101 a101 a101
3
(3) Agreaterorequalslant 0?§x<?A?x>A
a45
a24 a27
0 xA-A a101a101
A< 0?§x∈R
(4) a?η<x<a+η
a45
a27 a24
a0 xa+ηa?η a101a101
(5) a dux?2x+ 1 < 0§K?1 <x< 2
a45
a27 a24
0 x-1 1 2a98 a98
(6)?53 <x<?3252 <x<?73
a45
a7a4 a7a4
0 x-1-2-3 a101a101 a101a101
4,|e?ê9§3:t?ê?μ
(1) y = f(x) =?x+ 1x9f(?1),f(1)úf(2)?
(2) y = f(x) = √a2?x29f(0),f(a)úf
parenleftBig
a2
parenrightBig
(3) s = s(t) = 1te?t9s(1),s(2)?
(4) y = g(α) = α2 tanα9g(0),g
parenleftBigpi
4
parenrightBig
,g
parenleftBig
pi4
parenrightBig
(5) x = x(θ) = sinθ+ cosθ9x
parenleftBig
pi2
parenrightBig
,x(?pi)
(6) y = f(x) = 1(x?1)(x+ 2)9f(0),f(?1)
4
)μ
(1)?êX = (?∞,0)uniontext(0,∞)§f(?1) = 0,f(1) = 0,f(2) =?32
(2)?êX = [?|a|,|a|]§f(0) = |a|,f(a) = 0,f
parenleftBig
a2
parenrightBig
=
√3
2 |a|
(3)?ê(?∞,0)uniontext(0,∞)§s(1) = 1e,s(2) = 12e2
(4)?ê
braceleftBig
x
vextendsinglevextendsingle
vextendsinglex∈R,xnegationslash= kpi+ pi2,k∈Z
bracerightBig
§g(0) = 0,g
parenleftBigpi
4
parenrightBig
= pi
2
16,g
parenleftBig
pi4
parenrightBig
=?pi
2
16
(5)?êX = (?∞,∞)§x
parenleftBig
pi2
parenrightBig
=?1,x(?pi) =?1
(6)?êX = (?∞,?2)uniontext(?2,1)uniontext(1,+∞)§f(0) =?12,f(?1) =?12
5,|e?ê9μ
(1) y = √2 +x?x2
(2) y = √cosx
(3) y = ln
parenleftBig
sin pix
parenrightBig
(4) y = 1sinpix
)μ
(1)?êX = [?1,2]§
bracketleftbigg
0,32
bracketrightbigg
(2)?ê
bracketleftBig
2kpi? pi2,2kpi+ pi2
bracketrightBig
(k∈Z)§[0,1]
(3)?ê
parenleftbigg 1
2k+ 1,
1
2k
parenrightbigg
(k∈Z)§(?∞,0]
(4)?ê(n?1,n)(n = 0,±1,±2,···)§(?∞,?1]uniontext[1,+∞)
6,f(x) = x+ 1,?(x) = x?2§á)?§|f(x) +?(x)| = |f(x) +|?(x)|
)μd?§f(x)?(x) greaterorequalslant 0=(x+ 1)(x?2) greaterorequalslant 0§Kxgreaterorequalslant 2?xlessorequalslant?1.
7,f(x) = (|x|+x)(1?x)§|÷ve?ax?μ
(1) f(0) = 0
(2) f(x) < 0
)μ
(1)?f(x) = 0§K|x|+x = 0?1?x = 0§=xlessorequalslant 0?x = 1
(2)?|x|+xgreaterorequalslant 0§K?f(x) < 0§1?x< 0=?§=x> 1
8,?1-5L?>3|V!?>{R0ú?C>{R|¤>′.3?mS§A,Bü:m>?V?±w
¤~t.|?>6Iú?C>{R?êa.
)μd?9?n?￡§V = I(R0 +R).
9,3
/NìS?,?M?§T
/Nì.′a§p?h§?M?pY′x￡?1-6¤,T
M?NèVúx?m?ê'XV = V(x)§??§ú.
)μd?§V = pia2x§§[0,h]§[1,pia2h]
10.,/Y± ?è′F/§X?1-7§.°2?§>??45o§CDL?Y?§| ?ABCD?
èS?Y h?ê'X.
)μd?9?§S = h(h+ 2).
11,k? ?H?3§X^Rò?±z|¨ωlY?Yl?3S?L-?§|-?./?
lsú?mt?ê'X￡?1-8¤.
)μd?9?§s = H?ωRt
parenleftbigg
t∈
bracketleftbigg
0,Hωt
bracketrightbiggparenrightbigg
12,y = f(x) =
braceleftbigg 1 +x2,x< 0
x?1,xgreaterorequalslant 0 §|f(?2),f(?1),f(0),f(1)úf
parenleftbigg1
2
parenrightbigg
.
)μd?§f(?2) = 5,f(?1) = 2,f(0) =?1,f(1) = 0,f
parenleftbigg1
2
parenrightbigg
=?12.
5
13,x(t) =
0,0 lessorequalslantt< 10
1 +t2,10 lessorequalslanttlessorequalslant 20
t?10,20 <tlessorequalslant 30
§|x(0),x(5),x(10),x(15),x(20),x(25),x(30)§?x?ùê?/.
)μd?§x(0) = 0,x(5) = 0,x(10) = 101,x(15) = 226,x(20) = 401,x(25) = 15,x(30) = 20
14,e]y′&?-tx?ê.Uìe?5?§éuIS ×2&§U&?-t§z-20?AGe]8?§?
v20±20?O?.&?-t360?±S?§á?ùêL?a§?x?§?/.
)μd?§y = f(x) =
8,0 <xlessorequalslant 20
16,20 <xlessorequalslant 40
24,40 <xlessorequalslant 60
15,óàu)ì)n?§ù?/X?1-9§ê'Xu = u(t)(0 lessorequalslanttlessorequalslant 20).
)μd?9?§u = u(t) =
braceleftbigg 1.5t,0 lessorequalslanttlessorequalslant 10
30?1.5t,10 <tlessorequalslant 20
16,e?êfú?′ §oo
(1) f(x) = xx,?(x) = 1
(2) f(x) = x,?(x) = √x2
(3) f(x) = 1,?(x) = sin2x+ cos2x
)μ
(1)?f(?∞,0)uniontext(0,+∞)§?(?∞,+∞)§ùüê .
(2)?f(x) = x,?(x) = |x|§ùüê?êL?a§Kùüê .
(3)(x) = sin2x+ cos2x = 1e¤á§ùüê? .
17,y2éuêf(x) = ax + b§egCê?x = xn(n = 1,2,···)|¤?
ê§KéA?ê
yn = f(xn)(n = 1,2,···)?|¤?
ê.
y2μxm?1,xm,xm+1′xn￥3ê(2 lessorequalslantmlessorequalslantn)
aK?§2xm = xm?1 +xm+1
qyn = f(xn) = axn + b§Kym?1 = axm?1 + b,ym = axm + b,ym+1 = axm+1 + b§u′2ym =
2axm + 2b,ym+1 +ym?1 = axm+1 +b+axm?1 +b = 2axm + 2b§l
2ym = ym?1 +ym+1
qxm?1,xm,xm+1′xn￥3ê§Kym?1,ym,ym+1′yn￥3ê§u′yn = f(xn)(n =
1,2,···)?|¤?
ê.
18,XJ-?y = f(x)t^u?pu§¤?l￡?1-10¤§y2? af(x1) +f(x2)2 >f
parenleftBigx1 +x2
2
parenrightBig
é
u¤kx1,x2(x1 negationslash= x2)¤á￡kt?A5?ê?à?ê¤.
y2μ3-?t?ü:A(x1,f(x1)),B(x2,f(x2))§?AB§ù￥:C(xC,yC)§Kf(x1) + f(x2) =
2yC,x1 +x2 = 2xC
q-?txD = x1 +x22 ¤é:p?I?yD = f
parenleftBigx1 +x2
2
parenrightBig
§KxC = xD
q-?y = f(x)t^u?pu§¤?l?x1,x2?u?l:§KyC > yD=f(x1) +f(x2)2 >
f
parenleftBigx1 +x2
2
parenrightBig
éu¤kx1,x2(x1 negationslash= x2)¤á.
a45
a54
a28
a28
a28
a28
a28
a28
a28a28
0 x1 x2 x
A
C
B
xD
y
f(x)
19,y2eê3¤mS′üNO\?êμ
(1) y = x2(0 lessorequalslantx< +∞)
(2) y = sinx
parenleftBig
pi2 lessorequalslantxlessorequalslant pi2
parenrightBig
y2μ
6
(1) 0 lessorequalslantx1 <x2
Ky2?y1 = x22?x21 = (x2 +x1)(x2?x1) > 0§u′?êy = x20 lessorequalslantxüNO\.
(2) ?pi2 lessorequalslantx1 <x2 lessorequalslant pi2
Ky2?y1 = sinx2?sinx1 = 2cos x2 +x12 sin x2?x12
q?pi2 lessorequalslant x1 < x2 lessorequalslant pi2§K?pi2 < x1 +x22 < pi2,0 < x2x
1
2 lessorequalslant pi2§u′cos x1 +x22 > 0,sin x2?x12 >
0§l
y2?y1 > 0=?êy = sinx?pi2 lessorequalslantxlessorequalslant pi2üNO\.
20,y2e?ê3¤mS′üN~?êμ
(1) y = x2(?∞<xlessorequalslant 0)
(2) y = cosx(0 lessorequalslantxlessorequalslantpi)
y2μ
(1) 0 lessorequalslantx1 <x2
Ky2?y1 = x22?x21 = (x2 +x1)(x2?x1) < 0§u′?êy = x2xlessorequalslant 0üN~.
(2) 0 lessorequalslantx1 <x2 lessorequalslantpi
Ky2?y1 = cosx2?cosx1 =?2sin x2 +x12 sin x2?x12
q0 lessorequalslant x1 < x2 lessorequalslant pi§K0 < x1 +x22 < pi,0 < x2x
1
2 lessorequalslant pi2§u′sin x1 +x22 > 0,sin x2?x12 > 0§l
y2?y1 < 0=?êy = cosx0 lessorequalslantxlessorequalslantpiüN~.
21,e?ê?ó5μ
(1) y = x+x2?x5
(2) y = a+bcosx
(3) y = x+ sinx+ex
(4) y = xsin 1x
(5) y = sgnx =
1,x> 0?
0,x = 0?
1 x< 0?
(6) y =
2
x2,
1
2 <x< +∞?
sinx2,? 12 lessorequalslantxlessorequalslant 12?
1
2x
2,?∞<x<?1
2?
)μ
(1)?y = f(x) = x+x2?x5§Kf(?x) =?x+x2 +x5§f(?x) negationslash= f(x),f(?x) negationslash=?f(x)§u′d?ê
′ó?ê.
(2)?y = f(x) = a+bcosx§Kf(?x) = a+bcos(?x) = a+bcosx = f(x)§u′d?ê′ó?ê.
(3)?y = f(x) = x+sinx+ex§Kf(?x) =?x?sinx+e?x§f(?x) negationslash= f(x),f(?x) negationslash=?f(x)§u′d
ê′ó?ê.
(4)?y = f(x) = xsin 1x§Kf(?x) =?xsin 1?x = xsin 1x = f(x)§u′d?ê′ó?ê.
(5)?y = f(x) =
1,x> 0?
0,x = 0?
1 x< 0?
§
Kf(?x) =
1,?x> 0?
0,?x = 0?
1 ?x< 0?
=
1,x> 0?
0,x = 0?
1 x< 0?
=?f(x)§u′d?ê′ê.
7
(6)?y = f(x) =
2
x2,
1
2 <x< +∞?
sinx2,? 12 lessorequalslantxlessorequalslant 12?
1
2x
2,?∞<x<?1
2?
§
Kf(?x) =
2
(?x)2,
1
2 <?x< +∞?
sin(?x)2,? 12 lessorequalslant?xlessorequalslant 12?
1
2(?x)
2,?∞<?x<?1
2?
=
1
2x
2,1
2 <x< +∞?
sinx2,? 12 lessorequalslantxlessorequalslant 12?
2
x2,?∞<x<?
1
2?
§
f(?x) negationslash= f(x),f(?x) negationslash=?f(x)§u′d?ê′ó?ê.
22,áyü?ó?ê|è′ó?ê§üê|è′ê§êó?ê|è′ê.
y2μf1(x),f2(x)3(?a,a)(a > 0)Só?ê§g1(x),g2(x)3(?a,a)(a > 0)S
ê§F1(x) = f1(x)f2(x),F2(x) = g1(x)g2(x),F3(x) = f1(x)f2(x)
Kf1(?x) = f1(x),f2(?x) = f2(x),g1(x) =?g1(x),g2(?x) =?g2(x)§u′
F1(?x) = f1(?x)f2(?x) = f1(x)f2(x) = F1(x)
F2(?x) = g1(?x)g2(?x) = (?g1(x))(?g2(x)) = g1(x)g2(x) = F2(x)
F3(?x) = f1(?x)g1(?x) = f1(x)(?g1(x)) =?f1(x)g1(x) =?F3(x)
l
F1(x)′ó?ê?F2(x)′ó?ê?F3(x)′ê.
23,f(x)3(?∞,+∞)Sê§y2F1(x) ≡f(x) +f(?x)′ó?ê§F2(x) ≡f(x)?f(?x)′?
ê.?éAue?êF1(x),F2(x)μ
(1) y = ax
(2) y = (1 +x)n
y2μ?F1(?x) = f(?x) +f(x) = F1(x)§KF1(x) = f(x) +f(?x)′ó?ê
qF2(?x) = f(?x)?f(x) =?F2(x)§KF2(x) = f(x)?f(?x)′ê.
(1) F1(x) = f(x) +f(?x) = ax +a?x,F2(x) = f(x)?f(?x) = ax?a?x
(2) F1(x) = f(x) +f(?x) = (1 +x)n + (1?x)n,F2(x) = f(x)?f(?x) = (1 +x)n?(1?x)n
24,`2e?ê=
′±ê§?|?±?μ
(1) y = sin2x
(2) y = sinx2
(3) y = sinx+ 12 sin2x
(4) y = cos pi4x
(5) y = |sinx|+|cosx|
(6) y = √tanx
(7) y = x?[x]
(8) y = sinnpix
)μ
(1)?y = sin2x = 12? 12 cos2x§KT = 2pi2 = pi
(2) by = sinx2±ê?T = ω> 0
a±ê§éx ∈ (?∞,+∞)§ksin(x + ω)2 = sinx2§AOéx = 0?AT¤á§
Ksinω2 = 0§u′ω2 = kpi,ω = √kpi(k∈Z+)
qéx = √2ω = √2kpi?¤á§sin(√2ω + ω)2 = sinω2 = 0§K(√2 + 1)2kpi = npi(n ∈ Z+)§u
′(√2 + 1)2 = kn(k,n∈Z+)
q(√2 + 1)2 = 3 + 2√2 ∈Q?§
kn ∈Q+§Kb?¤á§=?êy = sinx2?′±ê.
(3)?y1 = sinxT = 2pi?y2 = 12 sin2xT = pi§Ky = sinx+ 12 sin2xT = 2pi.
(4) T = 2pipi
4
= 8
8
(5)?f(x) = |sinx|+|cosx|,f
parenleftBig
x+ pi2
parenrightBig
=
vextendsinglevextendsingle
vextendsinglesin
parenleftBig
x+ pi2
parenrightBigvextendsinglevextendsingle
vextendsingle+
vextendsinglevextendsingle
vextendsinglecos
parenleftBig
x+ pi2
parenrightBigvextendsinglevextendsingle
vextendsingle= |cosx|+|sinx| = f(x)
a2§y = |sinx|+|cosx|T = pi2.
(6)?f(x) = tanxT = pi§Ky = √tanxT = pi.
(7)?y = x?[x] = (x)§Ky = x?[x]T = 1.
(8) T = 2pinpi = 2n
9
§2,Eü?êúê
1,e?êU?¤Eü?êy = f(?(x))§XJU
¤KdEü?êúμ
(1) y = f(u) = 2u,u =?(x) = x2
(2) y = f(u) = lnu,u =?(x) = 1?x2
(3) y = f(u) = u2 +u3,u =?(x) =
braceleftbigg 1,x?knê?
1,xnê?
(4) y = f(u) = 2§U1§u =?(x)§X§U2
(5) y = f(u) = √u,u =?(x) = cosx
)μ
(1)?y = f(u) = 2u(?∞,+∞)§u =?(x) = x2[0,+∞)
Kd?êU¤Eü?êy = 2x2§§(?∞,+∞)§[1,+∞)
(2)?y = f(u) = lnu(0,+∞)§u =?(x) = 1?x2(?∞,1]
Kd?êU¤Eü?êy = ln(1?x2)§§(?1,1)§(?∞,0]
(3)?y = f(u) = u2 +u3(?∞,+∞)§
u =?(x) =
braceleftbigg 1,x?knê?
1,xnê? {?1,1}
Kd?êU¤Eü?êy =
braceleftbigg 2,x?knê?
0,xnê? §§(?∞,+∞)§{0,2}
(4)?y = f(u) = 2U1§u =?(x)U2
U1intersectiontextU2 negationslash= φ?§d?êU¤Eü?êy = 2§§à?N?ê
§{2}?
U1intersectiontextU2 = φ?§d?ê?U¤Eü?ê
(5)?y = f(u) = √u[0,+∞)§u =?(x) = cosx[?1,1]
Kd?êU¤Eü?êy = √cosx§§
bracketleftBig
2kpi? pi2,2kpi+ pi2
bracketrightBig
(k = 0,±1,±2,···)§
[0,1]
2,f(x) = ax2 +bx+c§y2f(x+ 3)?3f(x+ 2) + 3f(x+ 1)?f(x) ≡ 0
y2μd?§
f(x+3)?3f(x+2)+3f(x+1)?f(x) = a(x+3)2 +b(x+3)+c?3[a(x+2)2 +b(x+2)+c]+3[a(x+1)2 +
b(x+1)+c]?(ax2 +bx+c) = a[(x+3)2?x2]+b(x+3?x)?3a[(x+2)2?(x+1)2]?3b[x+2?(x+1)] =
6ax+ 9a+ 3b?3a(2x+ 3)?3b≡ 0
3,(1) y = f(x) = a+bx+ cx§|f
parenleftbigg2
x
parenrightbigg
(2) y = f(x) = x2 ln(1 +x)§|f(e?x)
(3) y = f(x) = √1 +x+x2§|f(x2)9f(?x2)
(4) y = f(t) = 1√a2 +x2§|f(atanx)
)μ
(1)?y = f(x) = a+bx+ cx§Kf
parenleftbigg2
x
parenrightbigg
= a+ 2bx + c2
x
= a+ 2bx + cx2 = cx
2 + 2ax+ 4b
2x
(2)?y = f(x) = x2 ln(1 +x)§Kf(e?x) = (e?x)2 ln(1 +e?x) = ln(e
x + 1)?x
e2x
(3)?y = f(x) = √1 +x+x2§Kf(x2) = √1 +x2 +x4,f(?x2) = √1?x2 +x4
(4)?y = f(t) = 1√a2 +x2§Kf(atanx) = 1radicalbiga2 + (atanx)2 = 1√a2 sec2x = 1|asecx|
4,ef(x) = x2,?(x) = 2x§|f(?(x))9?(f(x)).
)μ?f(x) = x2,?(x) = 2x§Kf(?(x)) = (2x)2 = 22x = 4x,?(f(x)) = 2x2
5,e?(x) = x3 + 1§|?(x2),(?(x))29?(?(x)).
)μ(x) = x3 + 1§K
(x2) = (x2)3 +1 = x6 +1,(?(x))2 = (x3 +1)2 = x6 +2x3 +1,?(?(x)) = (x3 +1)3 +1 = x9 +3x6 +3x3 +2
10
6,f(x) = 11?x§|f(f(x)),f(f(f(x))),f
parenleftbigg 1
f(x)
parenrightbigg
.
)μ?f(x) = 11?x§K
f(f(x)) = 1
1? 11?x
= x?1x,f(f(f(x))) = 1
1? 1
1? 11?x
= 1
1? x?1x
= x,f
parenleftbigg 1
f(x)
parenrightbigg
= 11?(1?x) =
1
x
7,|e?êê9êμ
(1) y = x2(?∞<xlessorequalslant 0)
(2) y = √1?x2(?1 lessorequalslantxlessorequalslant 0)
(3) y = sinx
parenleftbiggpi
2 lessorequalslantxlessorequalslant
3
2pi
parenrightbigg
(4) y =
x,?∞<x< 1?
x2,1 lessorequalslantxlessorequalslant 4?
2x,4 <x< +∞?
)μ
(1)?y = x2(?∞<xlessorequalslant 0)§Kx =?√y(0 lessorequalslanty< +∞)§l
d?êê?y =?√x(0 lessorequalslanty< +∞)
(2)?y = √1?x2(?1 lessorequalslantxlessorequalslant 0)§Kx =?radicalbig1?y2(0 lessorequalslanty lessorequalslant 1)§l
d?êê?y =?√1?x2(0 lessorequalslant
xlessorequalslant 1)
(3)?y = sinx
parenleftbiggpi
2 lessorequalslantxlessorequalslant
3
2pi
parenrightbigg
§Kx = pi? arcsiny(?1 lessorequalslant y lessorequalslant 1)§l
d?êê?y = pi?
arcsinx(?1 lessorequalslantxlessorequalslant 1)
(4)?y =
x,?∞<x< 1?
x2,1 lessorequalslantxlessorequalslant 4?
2x,4 <x< +∞?
§Kx =
y,?∞<y< 1?√
y,1 lessorequalslanty lessorequalslant 16?
log2y,16 <x< +∞?
§l
d?êê
y =
x,?∞<x< 1?√
x,1 lessorequalslantxlessorequalslant 16?
log2x,16 <x< +∞?
.
11
§3,? D ?ê
1,re3[0,1)t?êò??￠?t§|§¤?±1?±??êμ
(1) y = x2
(2) y = sinx
(3) y = ex
)μ
(1) ò? ?ê?y = (x?n)2(nlessorequalslantx<n+ 1,n∈Z)
(2) ò? ?ê?y = sin(x?n)(nlessorequalslantx<n+ 1,n∈Z)
(3) ò? ?ê?y = ex?n(nlessorequalslantx<n+ 1,n∈Z)
2,re3[0,+∞)t?êò??￠?t§(a)|§?¤ê?(b)|§?¤?ó?êμ
(1) y = x2
(2) y = sinx
)μ
(1) ò? ?ê?μ
(a) f(x) =
braceleftbigg x2,xgreaterorequalslant 0
x2,x< 0
(b) f(x) = x2
(2) ò? ?ê?μ
(a) f(x) = sinx
(b) f(x) = sin|x|
3,?e?ê?/μ
(1) y = sgncosx
(2) y = [x]?2
bracketleftBigx
2
bracketrightBig
)μ
(1)
a45
a54
0 pi x-pi
y
1
-1
a113a113 a113a113
a98a98
a98 a98
a98
a98
a98
(2)
a45
a54
0 1 2 3-1-2-3 x
y
1a99 a98
a98 a98a98a98 a98
4,êy = (x)?/.
)μ
a45
a54
a0
a0a0
a0
a0a0
a0
a0a0
a0a0
a0a0
a0
a0a0
a0
a0a0a0
0 1 2 3-1-2-3 x
y
1a99 a98a98 a98a98a98a98
5,êy = [x]?x?/.
)μ
a45a54
a64
a64a64
a64
a64a64
a64
a64a64
a64a64a64
a64a64
a64
a64a64
a64
a64a64a64a64
0 1 2 3-1-2-3 x
y
-1a99 a98a98 a98a98a98a98
12
6,ê′^e{μ3z?mn lessorequalslant x < n + 1(ù￥n?ê)Sf(x)′?5?f(n) =
1,f
parenleftbigg
n+ 12
parenrightbigg
= 0§á?d?ê?/.
)μ
a45
a54
a1
a1
a1
a1a1
a1
a1
a1
a1a1
a1
a1
a1
a1a1
a1a1a1
a1
a1
a1a1
a1
a1
a1
a1a1
a1
a1
a1
a1a1a1a1
0 1 2 3-1-2-3 x
y
1
-1
a99 a98a98 a98a98a98a98
7,êy = |sinx+ 2cosx|?/.
)μ
a45
a54
0 pi
2
pi?pi
2
pi x
y
√5
-1
8,e??êf(x) = tanx§?e?ê?/μ
(1) y = f(2x)
(2) y = f(kx+b)(knegationslash= 0)
(3) y = f
parenleftBigx
2
parenrightBig
1
)μ
(1)
a45
a54
0 pi
4
pi
2
3pi
4-
pi
4-
pi
2-
3pi
4
x
y
(2) (k,b> 0)
13
a45
a54
0 pi?2b2k
pi?b
k 3pi?2b
2k
-bk
-pi+2b2k-pi+bk x
y
(3)
a45
a54
0 2pipi
-pi
-2pi x
y
9,e??êy = f(x)?/§êy1 = |f(x)|,y2 = f(?x),y3 =?f(?x)?/§?`2y1,y2,y3?/
y?/'X.
)μy = f(x)?/Xeμ
a45
a54
0 x
y
Ky1?/?μ
a45
a54
0 x
y
Ky2?/?μ
a45
a54
0 x
y
Ky3?/?μ
a45
a54
0 x
y
y1?/f(x) < 0y?/'ux?é?§f(x) > 0y?/??
y2?/?y?/'uy?é?§
y3?/?y?/'u:é?§
14
10,e?f(x),g(x)?/§áêy = 12{f(x)+g(x)+|f(x)?g(x)|}?/§?`2y?/?f(x),g(x)?
/'X.
)μy = max{f(x),g(x)}
a45
a54
0 x
y
g(x) f(x)
11,éu3[0,pi]t?êy = x§kr§ò?[0,2pi]|§'ux = pi?é?§, 2r?ò?[0,2pi]t?
êò??￠?t|?ê?±2pi?±??ê.
)μ¤|?ê?:f(x) =
x,x∈ [0,pi]
2pi?x,x∈ [pi,2pi]
x?2npi,x∈ [2npi,(2n+ 1)pi](n = ±1,±2,···)
2npi?x,x∈ [(2n?1)pi,2npi](n = 0,?1,±2,···)
= pi
vextendsinglevextendsingle
vextendsinglexpi?2
bracketleftBigx+pi
2pi
bracketrightBigvextendsinglevextendsingle
vextendsingle
a45
a54
a0
a0
a0
a0a0a64a64
a64
a64a64
a64
a64
a64
a64a64a0a0
a0
a0a0
0 pi?pi 2pi
-pi
-2pi x
y
pi
15
1ù 4Y
§1,ê4?út
1,?eêco?μ
(1) xn = 13n sinn3
(2) xn = m(m?1)···(m?n+ 1)n! xn
(3) xn = 1√n2 + 1 + 1√n2 + 2 +···+ 1√n2 +n
(4) x1 = a> 0,y1 = b> 0,xn+1 = √xnyn,yn+1 = xn +yn2
(5) x2n = 1 + 12 +···+ 1n (n = 1,2,3,···)
x2n+1 = 1n (n = 1,2,···)
):
(1) x1 = 13 sin1,x2 = 16 sin8,x3 = 19 sin27,x4 = 112 sin64
(2) x1 = mx,x2 = m(m?1)2 x2,x3 = m(m?1)(m?2)6 x3,
x4 = m(m?1)(m?2)(m?3)24 x4
(3) x1 = 1√2,x2 = 1√5 + 1√6,x3 = 1√10 + 1√11 + 1√12,
x4 = 1√17 + 1√18 + 1√19 + 1√20
(4) x1 = a,x2 = √ab,x3 =
radicalbigg√
aba+b2,
x4 = 8√ab· 4
radicalbigga+b
2 ·
√a+√b
2
y1 = b,y2 = a+b2,y3 = (
√a+√b)2
4,
y4 = (
√a+√b)2
4 +
4√abradicalbig2(a+b)
16
(5) x2 = 1,x3 = 1,x4 = 32,x5 = 12
2,Uy2±eêtμ
(1) n+ 1n2 + 1
(2) sinnn
(3) n+ (?1)
n
n2?1
(4) 1n!
(5) 1n? 12n + 13n?···+ (?1)n+1 1n2
(6) (?1)n(0.999)n
(7) 1n +e?n
16
(8) e
n
n
(9) √n+ 1?√n
(10) 1 + 2 + 3 +···+nn3
y2:
(1) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglen+ 1
n2 + 1?0
vextendsinglevextendsingle
vextendsinglevextendsingle = n+ 1
n2 + 1 <
2n
n2 =
2
n§?|
vextendsinglevextendsingle
vextendsinglevextendsinglen+ 1
n2 + 1?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§2
n < ε=?"
N =
bracketleftbigg2
ε
bracketrightbigg
+ 1,Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsinglen+ 1
n2 + 1?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±n+ 1
n2 + 1 → 0(n→∞)
(2) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglesinn
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsinglesinn
n
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsinglesinn
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle<ε§1
n <ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§
Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsinglesinn
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±sinn
n → 0(n→∞)
(3) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglen+ (?1)n
n2?1?0
vextendsinglevextendsingle
vextendsinglevextendsingle = n+ (?1)n
n2?1 <
n+ 1
n2?1 =
1
n?1§?|
vextendsinglevextendsingle
vextendsinglevextendsinglen+ (?1)n
n2?1?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§?
1n?1 < ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§Kn > N?§
vextendsinglevextendsingle
vextendsinglevextendsinglen+ (?1)n
n2?1?0
vextendsinglevextendsingle
vextendsinglevextendsingle< εo¤á§¤±n+ (?1)n
n2?1 →
0(n→∞)
(4) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle 1
n!?0
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
n! <
1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle 1
n!?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§1
n < ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§K
n>N?§
vextendsinglevextendsingle
vextendsinglevextendsingle 1
n!?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤± 1
n! → 0(n→∞)
(5) Sn = 1n? 12n + 13n?···+ (?1)n+1 1n2
é?ε> 0§duSn = 1n(1? 12 + 13?···+ (?1)n+1 1n)
δn = 1?12+13?···+(?1)n+1 1n§KSn = δnnn = 2k+1?§k0 <δn = 1?(12?13)?(14?15)?···?
( 12k? 12k+ 1) < 1?n = 2k?§k0 <δn = 1?(12?13)?(14?15)?···?( 12k?2? 12k?1)? 12k < 1"
o?§k0 <δn < 1l
|Sn?0| = Sn = δnn < 1n?||Sn?0|<ε§1n <ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+1§
Kn>N?§|Sn?0|<εo¤á§¤±1n? 12n + 13n?···+ (?1)6n+ 1 1n2 → 0(n→∞)
(6) é?ε > 0§dun > lnn§Ken > n§u′e?n < 1n§l
vextendsinglevextendsingle
vextendsinglevextendsingle1
n +e
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
n + e?n <
2
n§?
||(?1)n(0.999)n?0|<ε§(0.999)n <ε=?"N =
bracketleftbigg
2500ln 1ε
bracketrightbigg
+1§Kn>N?§|(?1)n(0.999)n?
0|<εo¤á§¤±(?1)n(0.999)n → 0(n→∞)
(7) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle1
n +e
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
n! <
1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle1
n +e
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle< ε§2
n < ε=?"N =
bracketleftbigg2
ε
bracketrightbigg
+
1§Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsingle1
n +e
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±1
n +e
n → 0(n→∞)
(8) é?ε > 0§due?n < e0 = 1§K
vextendsinglevextendsingle
vextendsinglevextendsinglee?n
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle = e?n
n <
1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsinglee?n
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§1
n < ε=?"
N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsinglee?n
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±e?n
n → 0(n→∞)
(9) é?ε> 0§du|√n+ 1?√n?0| = 1√n+ 1 +√n < 12√n§?||√n+ 1?√n?0|<ε§ 12√n <
ε=?"N =
bracketleftbigg 1
4ε2
bracketrightbigg
+1§Kn>N?§|√n+ 1?√n?0|<εo¤á§¤±√n+ 1?√n→ 0(n→
∞)
(10) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 2 + 3 +···+n
n3?0
vextendsinglevextendsingle
vextendsinglevextendsingle = n+ 1
2n2 <
2n
2n2 =
1
n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle1
n?0
vextendsinglevextendsingle
vextendsinglevextendsingle< ε§1
n < ε=?"
N =
bracketleftbigg1
ε
bracketrightbigg
+ 1§Kn > N?§
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 2 + 3 +···+n
n3?0
vextendsinglevextendsingle
vextendsinglevextendsingle < εo¤á§¤±1 + 2 + 3 +···+n
n3 →
0(n→∞)
17
3,T~`2e'ut′μ
(1) éε> 0§3N§n>N?§¤áxn <ε?
(2) éε> 0§3xn§||xn|<ε.
):
(1) ~Xμê{?1 + (?1)n+1}(?{?n})={0,?2,0,?2,···} (?{?1,?2,?3,···})÷vt?^?§?′
t?
(2) ~Xμê{1,12,1,13,···,1,1n,···}÷vt?^?§?′t"
4,Uy2μ
(1) lim
n→∞
3n2 +n
2n2?1 =
3
2
(2) lim
n→∞
(0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9) = 1
(3) lim
n→∞
√n2 +n
n = 1
(4) xn = 11·2 + 12·3 +···+ 1(n?1)·n → 1(n→∞)
(5) lim
n→∞
rn = 1§d?rn =
n?1
n n?óên+ 1
n nê
(6) lim
n→∞
rn = 1§d?rn =
3 n = 3k(k = 1,2,3,···)
3n+ 1
n n = 3k+ 1
2 + 1 +n3?√n+n n = 3k+ 2
y2μ
(1) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle3n2 +n
2n2?1?
3
2
vextendsinglevextendsingle
vextendsinglevextendsingle = 2n+ 3
4n2?2 <
4(n+ 1)
4(n+ 1)(n?1) =
1
n?1(n greaterorequalslant 2)§?|
vextendsinglevextendsingle
vextendsinglevextendsingle3n2 +n
2n2?1?
3
2
vextendsinglevextendsingle
vextendsinglevextendsingle<
ε§ 1n?1 < ε=?"N = max(
bracketleftbigg1
ε
bracketrightbigg
+ 1,2),Kn > N?§
vextendsinglevextendsingle
vextendsinglevextendsingle3n2 +n
2n2?1?
3
2
vextendsinglevextendsingle
vextendsinglevextendsingle < εo¤á§¤
±3n
2 +n
2n2?1 →
3
2(n→∞)
(2) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle = (0.1)n =
1
10n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle< ε§
1
10n < ε=?"N =
bracketleftbigg
lg 1ε
bracketrightbigg
+ 1,Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±0.
nbracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright
99···9 → 1(n→∞)
(3) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle
√n2 +n
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle =
√n2 +n?n
n =<
1√
n2 +n+n <
1
2n§?|
vextendsinglevextendsingle
vextendsinglevextendsingle
√n2 +n
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle<ε§?
12n <ε=?"N =
bracketleftbigg 1
2ε
bracketrightbigg
+ 1,Kn>N?§
vextendsinglevextendsingle
vextendsinglevextendsingle
√n2 +n
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§¤±
√n2 +n
n → 1(n→
∞)
(4) é?ε> 0§duxn = 1? 12 + 12? 13 +···+ 1n?1? 1n = 1? 1n§K|xn?1| = 1n§?||xn?1|<ε§
1n <ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1,Kn>N?§|xn?1|<εo¤á§¤±xn → 1(n→∞)
(5) é?ε> 0§du|rn?1| =
vextendsinglevextendsingle
vextendsinglevextendsinglen±1
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
n§?||rn?1| <ε§
1
n <ε=?"N =
bracketleftbigg1
ε
bracketrightbigg
+ 1,K
n>N?§|rn?1|<εo¤á§¤±rn → 1(n→∞)
(6) é?ε > 0§du|r3k?3| = 0,|r3k+1?3| = 1n,|r3k+2?3| =
√n?2
3?√n+n =
n?4
n√n+n+√n+ 6 <
n
n√n =
1√
n§?||rn?3| < ε§
1
n < ε?
1√
n < ε=?"N = max
parenleftbiggbracketleftbigg1
ε
bracketrightbigg
+ 1,
bracketleftbigg 1
ε2
bracketrightbigg
+ 1
parenrightbigg
,K
n>N?§|rn?3|<εo¤á§¤±rn → 3(n→∞)
18
5,(1) Uy2§ean →a(n→∞)§Kég,êk§an+k →a(n→∞)
(2) Uy2§ean →a(n→∞)§K|an|→|a|.q′?¤áo
(3) e|an|→ 0§áˉan →a′¤áooo
y2μ
(1) duan → a(n → ∞)§é?ε > 0,?N ∈ Z+§n > N?§|an?a| < ε§Ké?k ∈ Z+,n + k >
N?§|an+k?a| < ε§u′é?ε > 0,?N ∈ Z+§n + k > N?§|an+k?a| < ε§l
an+k →
a(n→∞)
triangled(?`2μKêc?k§K?ù5"
(2) (i) duan → a§é?ε > 0,?N ∈ Z+§n > N?§|an?a| < ε.qvextendsinglevextendsingle|an|?|a|vextendsinglevextendsingle< |an?a|§u′
é?ε> 0,?N ∈Z+§n>N?§vextendsinglevextendsingle|an|?|a|vextendsinglevextendsingle<ε¤á§=|an|→|a|(n→∞
(ii)¤á"
~μ
(a)?¤áμan = (?1)n§K|an|→ 1§
an?4
(b) ¤áμan = 1n§K|an|→ 0,an → 0
(3) du|an| → 0§é?ε > 0,?N ∈ Z+§n > N?§vextendsinglevextendsingle|an|? 0vextendsinglevextendsingle < ε§q|an? 0| = vextendsinglevextendsingle|an|? 0vextendsinglevextendsingle§u
′é?ε > 0,?N ∈ Z+§n > N?§|an? 0| < ε¤á§=an → 0(n → ∞)"l
e|an| → 0§
Kan → 0¤á"
6,Uy2§exn →a§?a>b§K3N§n>N?§¤áxn >b.
y2μduxn → a§é?ε > 0,?N ∈ Z+§n > N?§|xn? 0| < ε§=a?ε < xn < a+ε.qa > b§
a?b> 0§Kε = a?b> 0§l
N ∈Z+§n>N?§kxn >a?ε = a?(a?b) = b.=3N§
n>N?§¤áxn >b.
7,e{xnyn}§U{xn},{yn}.
)μ?U"
~μxn = (?1)n,yn = (?1)n(n = 1,2,···),xnyn ≡ 1(n = 1,2,···)§K{xnyn}§{xn},{yn}t
"e{xnyn}§?U{xn},{yn}.
8,|^4?5?9O?y2μ
(1) lim
n→∞
parenleftbigg 1
n2 +
1
(n+ 1)2 +···+
1
(2n)2
parenrightbigg
= 0
(2) lim
n→∞
parenleftbigg 1
√n2 + 1 + 1√n2 + 2 +···+ 1√n2 +n
parenrightbigg
= 1
(3) |^(1 +h)n =
nsummationtext
k=0
Cknhk = 1 +nh+ n(n?1)2 h2 +···+hn
y2μ
(i) lim
n→∞
n
an = 0(a> 1)
(ii) lim
n→∞
n5
en = 0(e≈ 2.7)
y2μ
(1) é?n∈Z+§k0 lessorequalslant 1n2 + 1(n+ 1)2 +···+ 1(2n)2 lessorequalslant n+ 1n2 §? lim
n→∞
n+ 1
n2 = 0§K limn→∞
parenleftbigg 1
n2 +
1
(n+ 1)2 +···+
1
(2n)2
parenrightbigg
=
0
(2) é?n ∈ Z+§k nn+ 1 < 1√n2 + 1 + 1√n2 + 2 + ··· + 1√n2 +n < nn = 1? lim
n→∞
n
n+ 1 = 1§
K lim
n→∞
parenleftbigg 1
√n2 + 1 + 1√n2 + 2 +···+ 1√n2 +n
parenrightbigg
= 1
(3) (i) a = 1 +h(h> 0)§du0 < nan = n(1 +h)n = n
1 +nh+ n(n1)2 h2 +···+hn
< nn(n?1)
2 h
2
=
2
(n?1)h2§q
2
h2§
1
n?1 → 0(n→∞)§K
2
(n?1)h2 → 0.l
limn→∞
n
an = 0
19
(ii) e = 1 + h(h ≈ 1.7)§du0 < n
5
en =
n5
(1 +h)n =
n5
1 +nh+C2nh2 +···+hn <
n5
C6nh6 <
720n5
(n?5)6h6§q
720
h6§
n5
(n?5)6 → 0(n→∞)§K
720n5
(n?5)6h6 → 0(n→∞)§l
limn→∞
n5
en =
0
9,|e4?μ
(1) lim
n→∞
3n3 + 2n2?n+ 1
2n3?3n2 + 2
(2) lim
n→∞
6n2?n+ 1
n3 +n2 + 2
(3) lim
n→∞
parenleftbigg
1? 1n√2
parenrightbigg
cosn
(4) lim
n→∞
1 + 12 +···+ 12n
1 + 14 +···+ 14n
(5) lim
n→∞
bracketleftBigg
(sinn!)
parenleftbiggn?1
n2 + 1
parenrightbigg10
parenleftbigg 1
1·2 +
1
2·3 +···+
1
(n?1)·n
parenrightbigg2n2 + 1
n2?1
bracketrightBigg
(6) lim
n→∞
(?2)n + 3n
(?2)n+1 + 3n+1
)μ
(1) lim
n→∞
3n3 + 2n2?n+ 1
2n3?3n2 + 2 =
3
2
(2) lim
n→∞
6n2?n+ 1
n3 +n2 + 2 = 0
(3) dun√2 → 1(n→∞)§1? n√2 → 0(n→∞)§q|cosn|lessorequalslant 1§l
lim
n→∞
parenleftbigg
1? 1n√2
parenrightbigg
cosn = 0
(4) lim
n→∞
1 + 12 +···+ 12n
1 + 14 +···+ 14n
= lim
n→∞
1?(12)n+1
1? 12
1?(14)n+1
1? 14
= 24
3
= 32
(5) du{sinn!}?k.ê§
parenleftbiggn?1
n2 + 1
parenrightbigg10
→ 0,1? 1n → 1,2n
2 + 1
n2 + 1 → 2(n→∞)§
lim
n→∞
bracketleftBigg
(sinn!)
parenleftbiggn?1
n2 + 1
parenrightbigg10
parenleftbigg 1
1·2 +
1
2·3 +···+
1
(n?1)·n
parenrightbigg2n2 + 1
n2?1
bracketrightBigg
=?2
(6) lim
n→∞
(?2)n + 3n
(?2)n+1 + 3n+1 = limn→∞
(?23 )n + 1
(?2)(?23 )n + 3
= 13
10,exn →a> 0§áyμ
(1) √xn →√a
(2)
radicalbig
a0xmn +a1xm?1n +···+am?1xn +am →radicalbiga0am +a1am?1 +···+am?1a+am
(ù￥a0am +a1am?1 +···+am?1a+am > 0)
y2μ
(1) duxn → a > 0§é?ε > 0,?N ∈ Z+§n > N?§|xn? a| < √aε§?|√xn? √a| =vextendsingle
vextendsinglevextendsingle
vextendsingle
xn?a√
xn +√a
vextendsinglevextendsingle
vextendsinglevextendsingle < |xn?a|√
a < ε§=ét?ε > 0,?N ∈ Z
+§n > N?§|√xn? √a| < ε§l
√xn →√a(n→∞)
20
(2) duxn →a(n→∞)§a0xmn +a1xm?1n +···+am?1xn+am →a0am+a1am?1+···+am?1a+am > 0§
Ka(1)radicalbig
a0xmn +a1xm?1n +···+am?1xn +am →radicalbiga0am +a1am?1 +···+am?1a+am
11,éê{xn}§ex2k →a(k→∞),x2k+1 →a(k→∞)§y2μxn →a(n→∞)
y2μ?ε> 0§?x2k →a(n→∞)§?K1 ∈Z+§|k>K1?§|x2k?a|<ε¤á"
q?x2k+1 →a(n→∞)§?K2 ∈Z+§|k>K2?§|x2k+1?a|<ε¤á"
N = max{2K1,2K2 + 1}§Kn>N?§en?óê§n = 2k>N greaterorequalslant 2K1,k>K1,|xn?a| = |x2k?a|<
ε§
enê§n = 2k+ 1 >N greaterorequalslant 2K2 + 1,k>K2,|xn?a| = |x2k+1?a|<ε§
dxn →a(n→∞)
12,|^üNk.7k4?§y2 lim
n→∞
xn3§?|?§μ
(1) x1 = √2,···,xn = √2xn?1
(2) x0 = 1,x1 = 1 + x01 +x
0
,···,xn+1 = 1 + xn1 +x
n
y2μ
(1) w,x1 < x2§bxn?1 < xn§Kxn = √2xn?1 < √2xn§d8B{§{xn}′üNO\§qxn =√
2xn?1§x2n = 2xn1 lessorequalslant 2xn§u′xn lessorequalslant 2§={xn}dt."l
lim
n→∞
xn3§P lim
n→∞
xn = l§
3x2n = 2xn?1ü>-n→∞§l2 = 2l§)?l = 2§= lim
n→∞
xn = 2"
(2) w,xn greaterorequalslant 1§k^?xn = 1 + xn?11 +x
n?1
= 2? 11 +x
n?1
< 2§{xn}k."qx1 = 1 + x01 +x
0
=
1 + 11 + 1 = 32 > 1 = x0§bxn1 < xn§Kxn = 2? 11 +x
n?1
< 2? 11 +x
n
= xn+1§d
8B{§{xn}′üNO\"l
lim
n→∞
xn3§P lim
n→∞
xn = l§3xn = 2? 11 +x
n?1
ü>-
n → ∞§l = 2? 11 +l§=l2 = 1 + l§)l1 = 1 +
√5
2,l2 =
1?√5
2 ￡?üK?§¤§
= lim
n→∞
xn = 1 +
√5
2 "
13,ex1 = a> 0,y1 = b> 0(a<b),xn+1 = √xnyn,yn+1 = xn +yn2 §y2μ lim
n→∞
xn = lim
n→∞
yn.
y2μdu√xnyn lessorequalslant xn +yn2?da? ?=xn = yn§xn+1 lessorequalslant yn+1 ò¤á?=xn =
yn.q0 <a<b§x1 <y1§Kd4íúa§xn+1 <yn+1?xn > 0,yn > 0(n∈Z+).
xn+1 = √xnyn >
√x
nxn = xn,yn+1 =
xn +yn
2 <
yn +yn
2 = yn§Kxn <xn+1 <yn+1 <yn.qdx1 = a> 0,y1 = b> 0§
a < xn < xn+1 < yn+1 < yn < b§`2{xn}?{yn}?′üNk.ê§l
{xn},{yn}tk4?§
lim
n→∞
xn = α,lim
n→∞
yn = β§qdxn+1 = √xnyn§x2n+1 = xnyn§3 aü>-n → ∞§α2 = αβ q
d0 <a<xn <xn+1§?k0 <alessorequalslantα§l
α = β=k lim
n→∞
xn = lim
n→∞
yn.
14,|^üNk.7k4?y2±eê7k4?μ
(1) xn = 1 + 122 +···+ 1n2
(2) xn = 13 + 1 + 132 + 1 +···+ 13n + 1
(3) xn = n
k
an(a> 1,k?ê)
(4) xn = n√a (0 <a< 1)
y2μ
(1) duxn+1?xn = 1(n+ 1)2 > 0§xn+1 > xn§K{xn}?üNO\,q1 < xn < 1 + 11˙2 + ··· +
1
n˙(n+ 1) = 1 +
parenleftbigg
1? 12 +···+ 1n
1
1n
parenrightbigg
= 2? 1n < 2§{xn}k.§u′{xn}34?"
(2) duxn+1?xn = 13n+1 + 1 > 0§xn+1 >xn§K{xn}?üNO\,q14 <xn < 14+ 132 +···+ 13n <
1
3 +
1
32 +···+
1
3n =
1
3
1? 13
= 12§{xn}k.§u′{xn}34?"
21
(3) dua > 1,k?ê§xn = n
k
an > 0§K{xn}ke."q
xn+1
xn =
parenleftbigg
1 + 1n
parenrightbiggk
a =
1
a
parenleftbigg
1 + 1n
parenrightbiggk
→
1
a(n → ∞) < 1§?N ∈ Z
+§n > N?§kxn+1
xn < 1§KlN + 1?mkxn+1 < xn§u
′{xn}?üN~(n>N)§l
{xn}34?"
(4) dulnxn = 1n lna = yn,0 <a< 1§{yn}′üNO\§l
dxn = n√a = eyn{xn}′üNO\
"q0 <xn = n√a< n√1 = 1§{xn}k.§u′{xn}34?"
15,y2μexnt,§yneü§
xn?ynt§Kxnúyn7kó?4?"
y2μdxnt,§x1 lessorequalslant x2 lessorequalslant ··· lessorequalslant xn lessorequalslant ···§qyneü§y1 greaterorequalslant y2 greaterorequalslant ··· greaterorequalslant yn greaterorequalslant ···§qxn?yn
t§{xn?yn}k.§|xn?yn|lessorequalslantC(n = 1,2,···)￡ù￥C?,~ꤧK?C lessorequalslantxn?yn lessorequalslantC=xn lessorequalslant
yn +C lessorequalslant y1 +C§u′{xn}kt.§l
{xn}34?"qyn greaterorequalslant xn?C greaterorequalslant x1?C§u′{yn}ke.§l
{yn}34?§K lim
n→∞
xn? lim
n→∞
yn = lim
n→∞
(xn?yn) = 0§u′ lim
n→∞
xn = lim
n→∞
yn.
16,x￠ê§qyn(x) = sinsin···sinbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
n
x§y2{yn(x)}4?3§?|d4?.
y2μk0 lessorequalslant x lessorequalslant pi§K0 lessorequalslant sinx lessorequalslant x§l
kyn+1(x) = sinyn(x) lessorequalslant yn(x)§{yn(x)}′±0?e.ü
Neü?ê§7k4?§Ké?x0 ∈ [0,pi]§k0 lessorequalslant u0 = lim
n→∞
yn(x0) = sin
parenleftBig
lim
n→∞
fn?1(x0)
parenrightBig
= sinu0§
Ku0 = 0§l
é?x∈ [0,pi],lim
n→∞
yn(x) = 0.
ón?yx∈ [?pi,0]k lim
n→∞
yn(x) = 0.
2d±?5? lim
n→∞
yn(x) = 0
17,e lim
n→∞
xn = a§áyμ lim
n→∞
x1 +x2 +···+x+n
n = a
y2μd lim
n→∞
xn = a§é?ε> 0,?N1 ∈Z+§n>N1?§k|xn?a|< ε2§Kk
vextendsinglevextendsingle
vextendsinglex1 +x2 +···+xnn?a
vextendsinglevextendsingle
vextendsingle=vextendsingle
vextendsinglevextendsingle
vextendsingle
(x1?a) + (x2?a) +···+ (xn?a)
n
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant |x1?a|+|x2?a|+···+|xN1?a|+|xN1+1?a|+···+|xn?a|
n <
|x1?a|+|x2?a|+···+|xN1?a|
n +
n?N1
n ·
ε
2 <|x
1?a|+|x2?a|+···+|xN1?a|
n +
ε
2(∵
n?N1
n < 1)
M = max(|x1?a|,|x2?a|,···,|xn1?a|)§K|x1?a|+|x2?a|+···+|xN1?a|n lessorequalslant N1 ·Mn §qN1·M?
§KN1 ·Mn → 0(n→∞)§
u′ét?ε> 0,?N2 =
bracketleftbigg2N
1 ·M
ε
bracketrightbigg
∈Z+§n>N2?§k
|x1?a|+|x2?a|+···+|xN1?a|
n <
ε
2
N = max(N1,N2)§Kn>N?§k
vextendsinglevextendsingle
vextendsinglex1 +x2 +···+xnn?a
vextendsinglevextendsingle
vextendsingle< ε2 + ε2 = ε§
=k lim
n→∞
x1 +x2 +···+x+n
n = a
5μe lim
n→∞
x1 +x2 +···+xn
n = anotdblarrowright limn→∞xn3"
~μxn = (?1)n?1(n = 1,2,···)§Kw,lim
n→∞
x1 +x2 +···+xn
n = 0§ limn→∞xn?3"
18,y2μe lim
n→∞
an = a,lim
n→∞
bn = b§K lim
n→∞
a1bn +a2bn?1 +···+anb1
n = aby2μ
(1) a = 0§y lim
n→∞
a1bn +a2bn?1 +···+anb1
n = 0
d lim
n→∞
bn = b§Ka?n4(P38)§?M > 0§||bn|lessorequalslantM(n∈Z+)
d lim
n→∞
an = 0§Ké?ε> 0,?N1 ∈Z+§n>N1?§k|an|< ε2M.N = max
braceleftbiggbracketleftbigg2(|a
1|+···+|an|)M
ε
bracketrightbigg
+ 1,N1
bracerightbigg
§
u′ngreaterorequalslantN(greaterorequalslantN1)?§k
vextendsinglevextendsingle
vextendsinglevextendsinglea1bn +a2bn?1 +···+anb1
n
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsinglea1bn +a2bn?1 +···+aN1bn?N1+1 +aN1+1bn?N1 +···+anb1
n
vextendsinglevextendsingle
vextendsinglevextendsingle
lessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsinglea1bn +a2bn?1 +···+aN1bn?N1+1
n
vextendsinglevextendsingle
vextendsinglevextendsingle+
vextendsinglevextendsingle
vextendsinglevextendsingleaN1+1bn?N1 +···+anb1
n
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant (|a1|+···+|aN1|)M
n +
(n?N1)· ε2M ·M
n <
22
ε
2 +
ε
2 = ε§l
limn→∞
a1bn +a2bn?1 +···+anb1
n = 0
(2) anegationslash= 0,bnegationslash= 0?§d lim
n→∞
bn = b§ lim
n→∞
bn +bn?1 +···+b1
n = bnegationslash= 0§q limn→∞an = a§ limn→∞(an?a) = 0
d(1) lim
n→∞
(a1?a)bn +···+ (an?a)b1
n = 0§
u′ lim
n→∞
a1 ·
bn
n +···+an ·
b1
n
bn +···+b1
n
a
= lim
n→∞
(a1?a)· bnn +···+ (an?a)· bnn
bn +···+b1
n
= 0b = 0§
= lim
n→∞
a1 · bnn +···+an · b1n
bn +···+b1
n
= a§
l
lim
n→∞
a1bn +a2bn?1 +···+anb1
n = limn→∞
a1 ·
bn
n +···+an ·
b1
n
bn +···+b1
n
· bn +···+b1n
= ab
19,Uy2eêtμ
(1) √n
(2) n!
(3) lnn
(4) n
2 + 1
2n+ 1
(5) n
2 + 1
2n?1
(6) 1 + 12 + 13 +···+ 1n
y2μ
(1) é?G > 0§?||√n| > G§n > G2=?.N = [G2]§Kn > N?§|√n| > Go¤á§
{√n}′t"
(2) é?G > 0§du|n!| > n§?||n!| > G§n > G=?.N = [G]§Kn > N?§|n!| > Go¤
á§{n!}′t"
(3) é?G > 0§?||lnn| > G§n > eG=?.N = [eG]§Kn > N?§|lnn| > Go¤á§
{lnn}′t"
(4) é?G > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle > n2
3n =
n
3§?|
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle > G§n
3 > G=?.N = [3G]§Kn >
N?§
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle>Go¤á§{n2 + 1
2n+ 1}′t"
(5) é?G > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n?1
vextendsinglevextendsingle
vextendsinglevextendsingle > n2
2n =
n
2§?|
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n?1
vextendsinglevextendsingle
vextendsinglevextendsingle > G§n
2 > G=?.N = [2G]§Kn >
N?§
vextendsinglevextendsingle
vextendsinglevextendsinglen2 + 1
2n?1
vextendsinglevextendsingle
vextendsinglevextendsingle>Go¤á§{n2 + 1
2n?1}′t"
(6) é?G > 0§du lim
n→∞
parenleftbigg
1 + 1n
parenrightbiggn
= e?
parenleftbigg
1 + 1n
parenrightbiggn
üNO\§K
parenleftbigg
1 + 1n
parenrightbiggn
< e§u′ln
parenleftbigg
1 + 1n
parenrightbigg
<
1
n§l
1+
1
2+
1
3+···+
1
n > ln2+ln
3
2+···+ln
parenleftbigg
1 + 1n
parenrightbigg
= ln(n+1) > lnn§K?|
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 1
2 +
1
3 +···+
1
n
vextendsinglevextendsingle
vextendsinglevextendsingle>
G§lnn > G=?.N = [eG]§Kn > N?§
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 1
2 +
1
3 +···+
1
n
vextendsinglevextendsingle
vextendsinglevextendsingle > Go¤á§{1 + 1
2 +
1
3 +···+
1
n}′t"
20,y2μe{xn}′t§xn negationslash= 0(n = 1,2,3,···)§K
braceleftbigg 1
xn
bracerightbigg
′t"
y2μdu{xn}′t§é?ε> 0,?N ∈Z+§n>N?§k|xn|<ε
23
qxn negationslash= 0(n = 1,2,3,···)§ 1x
n
3?
vextendsinglevextendsingle
vextendsinglevextendsingle 1
xn
vextendsinglevextendsingle
vextendsinglevextendsingle> 1
ε
qε′§1ε?′§l
braceleftbigg 1
xn
bracerightbigg
′t"
21,y2μe{xn}t§{yn}?k.Ct§K{xn ±yn}t"
ddO?e4?μ
(1) lim
n→∞
parenleftbigg
sinn+ n
2
√n2 + 1
parenrightbigg
(2) lim
n→∞
(n?arctann)
(3) lim
n→∞
bracketleftbigg
n+ (?1)n
parenleftbigg 1
1·3 +
1
3·5 +···+
1
(2n?1)(2n+ 1)
parenrightbiggbracketrightbigg
qμütú4?NoáU?/"
i)y2μdu{yn}?k.Ct§73êM§||yn| lessorequalslant M§q{xn}t§é?G > M >
0,?N ∈ Z+§n > N?§k|xn| > G§Kn > N?§k|xn ±yn| greaterorequalslant |xn|?|yn| > G?M.dG5
9G>M > 0§?G?M > 0?G?M′§l
{xn ±yn}t"
ii))μ
(1) du lim
n→∞
n2√
n2 + 1 = ∞?|sinn|lessorequalslant 1§K limn→∞
parenleftbigg
sinn+ n
2
√n2 + 1
parenrightbigg
= ∞
(2) du lim
n→∞
n = ∞?|arctann|lessorequalslant pi2§K lim
n→∞
(n?arctann) = ∞
(3) xn = (?1)n
parenleftbigg 1
1·3 +
1
3·5 +···+
1
(2n?1)(2n+ 1)
parenrightbigg
§Kxn = (?1)
n
2
bracketleftbigg
1? 13 + 13? 15 +···+ 12n?1? 12n+ 1
bracketrightbigg
=
(?1)n
2
parenleftbigg
1? 12n+ 1
parenrightbigg
= (?1)
n
2 ·
2n
2n+ 1 =
(?1)n
2 + 1n
§k13 <|xn|< 12.qd lim
n→∞
n = ∞§l
lim
n→∞
bracketleftbigg
n+ (?1)n
parenleftbigg 1
1·3 +
1
3·5 +···+
1
(2n?1)(2n+ 1)
parenrightbiggbracketrightbigg
= ∞
iii))μ
(1) xn = n→ +∞,yn = 2n→ +∞;xn +yn = 3n→ +∞
(2) xn =?n→?∞,yn =?2n→?∞;xn +yn =?3n→?∞
(3) xn =?n→?∞,yn = 2n→ +∞;xn +yn = n→ +∞
(4) xn = n→ +∞,yn =?2n→?∞;xn +yn =?n→?∞
(5) xn = n+a→ +∞,yn =?n→?∞;xn +yn = a￡~t¤
(6) xn = n+ (?1)n → +∞,yn =?n→ +∞;xn +yn = (?1)n?4?
22,tútú!
!?4??/"
)μ
(1) ú!
μ?yn → 0(n → ∞)§{yn}k."qxn → ∞(n → ∞)§KdtK(?§k{xn ±yn}
t"
(2)?μxn negationslash= 0,yn negationslash= 0?§duxn →∞,yn → 0(n→∞)§Kkyn · 1x
n
→ 0§=ynx
n
→ 0,xny
n
→∞
23,T~`2tút|è?Uu)/"
)μ
(1) xn = n→ +∞,yn = 1n2 → 0(n→∞);xn ·yn = 1n → 0(n→∞)
(2) xn = n2 → +∞,yn = 1n → 0(n→∞);xn ·yn = n→ +∞(n→∞)
(3) xn = n→ +∞,yn = an → 0(n→∞);xn ·yn = a￡~t¤
(4) xn = n(?1)n →∞,yn = 1n → 0(n→∞);xn ·yn = (?1)n?4?k.
(5) xn = n2n(?1)n → ∞,yn = 1n → 0(n→ ∞);xn ·yn = n·n(?1)n = n1+(?1)n?4?§?.￡′?
t¤
24
24,exn →∞,yn →anegationslash= 0§y2xnyn →∞
y2μduxn → ∞(n → ∞)§ 1x
n
→ 0(n → ∞)?qyn → a negationslash= 0(n → ∞)§ 1y
n
→ 1a(n → ∞)§u
′ 1x
n
· 1y
n
→ 0(n→∞)§l
xnyn →∞(n→∞)
25,exn → +∞,yn →?∞§y2xnyn →?∞.
y2μ?xn → +∞§Ké?G1 > 0,?N1 ∈ Z+§n > N1?§kxn > G1?qyn →?∞§Ké?G2 >
0,?N2 ∈ Z+§n > N2?§k?yn > G2 > 0,N = max(N1,N2)§Kn > N?§k?xnyn > G1G2§
=xnyn <?G1G2.dG1,G25§G1G2′?G1G2 > 0§Kxnyn →?∞.
26,exn → +∞§y2x1 +x2 +···+xnn → +∞
y2μ?xn → +∞§Ké?G > 0,?N1 ∈ Z+§n > N1?§kxn > 3G§u′x1 +x2 +···+xnn =
x1 +···+xN1
n +x
N1+1 +···+xn
n >
x1 +···+xN1
n +
n?N1
n ·3G§
M = max(|x1|,···,|xN1|)§K
vextendsinglevextendsingle
vextendsinglex1 +·+xN1n
vextendsinglevextendsingle
vextendsingle lessorequalslant |x1|+···+|xN1|n lessorequalslant N1 ·Mn §u′ét?G > 0§
N2 =
bracketleftbigg2N
1 ·M
G
bracketrightbigg
§Kn>N2?§k
vextendsinglevextendsingle
vextendsinglex1 +···+xN1n
vextendsinglevextendsingle
vextendsingle< G2§l
x1 +···+xN1n >?G2"q limn→∞ n?N1n =
1§éuε = 12,?N3 ∈Z+§n>N3?§k
vextendsinglevextendsingle
vextendsinglevextendsinglen?N1
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle< 1
2§l
n?N1
n >
1
2§N = max{N1,N2,N3}§
Kn>N?§kx1 +x2 +···+xnn >?G2 + 3G2 = G§ddx1 +x2 +···+xnn → +∞(n→∞).
25
§2,?ê4?
1,^y2μ
(1) lim
x→?1
x?3
x2?9 =
1
2
(2) lim
x→3
x?3
x2?9 =
1
6
(3) lim
x→1
x?1√
x?1 = 2
(4) lim
x→1
(x?2)(x?1)
x?3 = 0
(5) lim
t→1
t(t?1)
t2?1 =
1
2
(6) lim
x→∞
x?1
x+ 2 = 1
(7) lim
x→3
x
x2?9 = ∞
(8) lim
x→∞
x2 +x
x+ 1 = ∞
y2μ
(1) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x+ 3?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle x+ 1
2x+ 6
vextendsinglevextendsingle
vextendsinglevextendsingle§?x→?1§|x+ 1|< 1§K?2 <x<
0§l
2 < |2x + 6| < 6§u′
vextendsinglevextendsingle
vextendsinglevextendsingle x+ 1
2x+ 6
vextendsinglevextendsingle
vextendsinglevextendsingle < |x+ 1|
2 §?|
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§|x+ 1|
2 < ε=?"
δ = min{2ε,1}> 0§K0 <|x?(?1)|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§ lim
x→?1
x?3
x2?9 =
1
2
(2) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
6
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x+ 3?
1
6
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle x?3
6x+ 18
vextendsinglevextendsingle
vextendsinglevextendsingle§?x → 3§|x? 3| < 1§K2 < x <
4§l
30 < |6x + 18| < 42§u′
vextendsinglevextendsingle
vextendsinglevextendsingle x?3
6x+ 18
vextendsinglevextendsingle
vextendsinglevextendsingle < |x?3|
30 §?|
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
6
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§|x?3|
30 < ε=
"δ = min{30ε,1}> 0§K0 <|x?3|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsinglex?3
x2?9?
1
6
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§lim
x→3
x?3
x2?9 =
1
6
(3) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x?1?2
vextendsinglevextendsingle
vextendsinglevextendsingle = |√x+ 1?2| = |√x?1| =
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle§?x→ 1§|x?1|< 1§
K0 < x < 2§l
1 < |√x + 1| < √2 + 1§u′
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle < |x? 1|§?|
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x?1?2
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§?
|x? 1| < ε=?"δ = min{ε,1} > 0§K0 < |x? 1| < δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsingle x?1√
x?1?2
vextendsinglevextendsingle
vextendsinglevextendsingle < εo¤á§
lim
x→1
x?1√
x?1 = 2
(4) é?ε > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle(x?2)(x?1)
x?3?0
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle
parenleftbigg
1 + 1x?3
parenrightbigg
(x?1)
vextendsinglevextendsingle
vextendsinglevextendsingle§?x → 1§|x? 1| < 1§K0 <
x < 2§l
0 <
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle < 2
3§u′
vextendsinglevextendsingle
vextendsinglevextendsingle1 + 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle < 2
3|x? 1|§?|
vextendsinglevextendsingle
vextendsinglevextendsingle(x?2)(x?1)
x?3?0
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§?
23|x?1|<ε=?"δ = min
braceleftbigg3
2ε,1
bracerightbigg
> 0§K0 <|x?1|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsingle(x?2)(x?1)
x?3?0
vextendsinglevextendsingle
vextendsinglevextendsingle<εo
¤á§lim
x→1
(x?2)(x?1)
x?3 = 0
(5) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglet(t?1)
t2?1?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle t
t+ 1?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle t?1
2t+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle§?t→ 1§|t?1|< 1§K0 <t< 2§
l
2 < |2t + 2| < 6§u′
vextendsinglevextendsingle
vextendsinglevextendsingle t?1
2t+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle < |t?1|
2 §?|
vextendsinglevextendsingle
vextendsinglevextendsinglet(t?1)
t2?1?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§|t?1|
2 < ε=?"
δ = min{2ε,1}> 0§K0 <|x?(?1)|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsinglet(t?1)
t2?1?
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§lim
t→1
t(t?1)
t2?1 =
1
2
26
(6) é?ε> 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglex?1
x+ 2?1
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle 3
x+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle§?x→∞§|x|> 2§K|x+2|>|x|?2§u′
vextendsinglevextendsingle
vextendsinglevextendsingle 3
x+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle<
3
|x|?2§?|
vextendsinglevextendsingle
vextendsinglevextendsingle 3
x+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle < ε§ 3
|x|?2 < ε=?§=|x| >
3
ε"X =
3
ε + 2§K|x| > X?§ò
k
vextendsinglevextendsingle
vextendsinglevextendsinglex?1
x+ 2?1
vextendsinglevextendsingle
vextendsinglevextendsingle<εo¤á§ lim
x→∞
x?1
x+ 2 = 1
(7) é?G > 0§du
vextendsinglevextendsingle
vextendsinglevextendsingle x
x2?9
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle x
x+ 3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle§?x → 3§|x? 3| < 1§K2 < x < 4§l
27 <
vextendsinglevextendsingle
vextendsinglevextendsingle x
x+ 3
vextendsinglevextendsingle
vextendsinglevextendsingle < 4
5§u′
vextendsinglevextendsingle
vextendsinglevextendsingle x
x+ 3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle > 2
7
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle§?|
vextendsinglevextendsingle
vextendsinglevextendsingle x
x2?9
vextendsinglevextendsingle
vextendsinglevextendsingle > G§2
7
vextendsinglevextendsingle
vextendsinglevextendsingle 1
x?3
vextendsinglevextendsingle
vextendsinglevextendsingle > G=
"δ = min
braceleftbigg 2
7G,1
bracerightbigg
> 0§K0 <|x?3|<δ?§òk
vextendsinglevextendsingle
vextendsinglevextendsingle x
x2?9
vextendsinglevextendsingle
vextendsinglevextendsingle>Go¤á§lim
x→3
x
x2?9 = ∞
(8) é?G > 0§du
vextendsinglevextendsingle
vextendsinglevextendsinglex2 +x
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle = |x|§?x → ∞§X = G > 0§K|x| > X?§òk
vextendsinglevextendsingle
vextendsinglevextendsinglex2 +x
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle> Go¤
᧠lim
x→∞
x2 +x
x+ 1 = ∞
2,|4?μ
(1) lim
x→0
x2?1
2x2?x?1
(2) lim
x→1
x2?1
2x2?x?1
(3) lim
x→∞
x2?1
2x2?x?1
(4) lim
x→0
(1 +x)(1 + 2x)(1 + 3x)?1
x
(5) lim
t→1
t2(t?1)
t2?1
(6) lim
t→1
t2?√t√
t?1
(7) lim
x→3
√1 +x?2
x?3
(8) lim
x→0
(1 +x)5?(1 + 5x)
x2 +x5
(9) lim
x→0
(1 +mx)n?(1 +nx)m
x2 ￡m,n?g,ê¤
(10) lim
x→3
x2?5 + 6
x2?8x+ 15
(11) lim
x→∞
x2 + 3x
x2
(12) lim
x→∞
5x?7
2x+√x
)μ
(1) lim
x→0
x2?1
2x2?x?1 = 1
(2) lim
x→1
x2?1
2x2?x?1 = limx→1
(x?1)(x+ 1)
(2x+ 1)(x?1) = limx→1
x+ 1
2x+ 1 =
2
3
(3) lim
x→∞
x2?1
2x2?x?1 =
1
2
(4) lim
x→0
(1 +x)(1 + 2x)(1 + 3x)?1
x = limx→0(6 + 11x+ 6x
2) = 6
(5) lim
t→1
t2(t?1)
t2?1 = limt→1
t2
t+ 1 =
1
2
(6) lim
t→1
t2?√t√
t?1 = limt→1
√t(√t?1)(t+√t+ 1
√t?1 =
lim
t→1
√t(t+√t+ 1) = 3
27
(7) lim
x→3
√1 +x?2
x?3 = limx→3
1√
1 +x+ 2 =
1
4
(8) lim
x→0
(1 +x)5?(1 + 5x)
x2 +x5 = limx→0
10x2 + 10x3 + 5x4 +x5
x2 +x5 = 10
(9) lim
x→0
(1 +mx)n?(1 +nx)m
x2 =
lim
x→0
(C2nm2?C2mn2)x2 + (C3nm3?C3mn3)x3 +···+mnxn?nmxm
x2 = C
2
nm
2?C2
mn
2 = n2m?m2n
2
(10) lim
x→3
x2?5 + 6
x2?8x+ 15 = limx→3
(x?2)(x?3)
(x?3)(x?5) = limx→3
x?2
x?5 =?
1
2
(11) lim
x→∞
x2 + 3x
x2 = 1
(12) lim
x→∞
5x?7
2x+√x =
5
2
3, R(x) = P(x)Q(x)
a￥P(x)úQ(x)?xa§P(a) = Q(a) = 0§ˉlim
x→a
k=
U?o
)μduP(x)úQ(x)?xa?P(a) = Q(a) = 0§
KP(x) = (x?a)mP1(x),Q(x) = (x?a)nQ1(x)(P1(a) negationslash= 0,Q1(x) negationslash= 0)§u′lim
x→a
R(x) = lim
x→a
P(x)
Q(x) =
lim
x→a
(x?a)mP1(x)
(x?a)nQ1(x)
μ
(1) n = m?§lim
x→a
R(x) = P1(a)Q
1(a)
(2) n>m?§lim
x→a
(x?a)m?n = ∞?lim
x→a
P1(x)
Q1(x) =
P1(a)
Q1(a) negationslash= 0§limx→aR(x) = ∞
(3) n<m?§lim
x→a
(x?a)m?n = 0?lim
x→a
P1(x)
Q1(x) =
P1(a)
Q1(a)§limx→aR(x) = 0
4,|e4?μ
(1) lim
x→0
sin2x?sin3x
x
(2) lim
h→0
cos(x+h)?cosx
h
(3) lim
x→+∞
(√x2 + 1?x)
(4) lim
x→?∞
(√x2 + 1?x)
(5) lim
x→0
x2
1?cosx
(6) lim
x→+∞
radicalBig
x+radicalbigx+√x
x+ 1
(7) lim
x→0
cosx?cos3x
x2
(8) lim
x→0
sin5x?sin3x
sin2x
(9) lim
x→1
(1?x)tan pix2
(10) lim
x→a
sinx?sina
x?a
(11) lim
x→0
(√1 +x2 +x)n?(√1 +x2?x)n
x
(12) lim
x→0
x
bracketleftbigg1
x
bracketrightbigg
)μ
28
(1) lim
x→0
sin2x?sin3x
x = limx→0
sin2x
x? limx→0
sin3x
x = 2?3 =?1
(2) lim
h→0
cos(x+h)?cosx
h = limh→0
2sin 2x+h2 sin h2
h = limh→0
sin h2
h sin
2x+h
2 =?sinx
(3) lim
x→+∞
(√x2 + 1?x) = lim
x→+∞
1√
x2 + 1 +x = 0
(4) lim
x→?∞
(√x2 + 1?x) = +∞
(5) lim
x→0
x2
1?cosx = limx→0
x2
x2
2
= 2
(6) lim
x→+∞
radicalBig
x+radicalbigx+√x
x+ 1 = 0
(7) lim
x→0
cosx?cos3x
x2 = limx→0
2sinxsin2x
x2 = 4
(8) lim
x→0
sin5x?sin3x
sin2x = limx→0
2sinxcos4x
2x = 1
(9) -y = x? 1§Klim
x→1
(1?x)tan pix2 = lim
y→0
ytan
parenleftBigpi
2(1 +y)
parenrightBig
= lim
y→0
ycot pi2y = lim
y→0
ycos pi2y
sin pi2y
=
lim
y→0
y
pi
2y
= 2pi
(10) lim
x→a
sinx?sina
x?a = limx→a
2cos x+a2 sin x?a2
x?a = cosa
(11) lim
x→0
(√1 +x2 +x)n?(√1 +x2?x)n
x =
lim
x→0
2C1n(1 +x2)n?12 x+ 2C3n(1 +x2)n?32 x2 +···
x =
lim
x→0
bracketleftBig
2n(1 +x2)n?12 + 2C3n(1 +x2)n?32 x+···
bracketrightBig
= 2n
(12) du
bracketleftbigg1
x
bracketrightbigg
= 1x?
parenleftbigg1
x
parenrightbigg
0 lessorequalslant
parenleftbigg1
x
parenrightbigg
< 1§
Klim
x→0
x
bracketleftbigg1
x
bracketrightbigg
= lim
x→0
braceleftbigg
1?x
parenleftbigg1
x
parenrightbiggbracerightbigg
= 1? lim
x→0
x
parenleftbigg1
x
parenrightbigg
= 1
5,e lim
x→x0
f(x) = A,lim
x→x0
g(x) = B§3δ> 0§0 <|x?x0|<δ?kf(x) greaterorequalslantg(x)§y2AgreaterorequalslantB.
qe0 <|x?x0|<δ?f(x) >g(x)§′¤áA>B
y2μ
(1) ^?y{"bA<B§Kd lim
x→x0
f(x) = A,lim
x→x0
g(x) = B95?1§?δ0 > 0§|0 < |x?x0| <
δ0?§kg(x) > f(x)"ùμ?δ > 0§0 < |x?x0| < δ?§kf(x) greaterorequalslant g(x)g?§b?¤
á§=AgreaterorequalslantB¤á"
(2)"~μ
(i) ¤á"f(x) = 2(x
2 + 3x4)
x2,g(x) = x
2 + 3x4x2,?δ > 0§0 < |x| < δ?§kf(x) > g(x)"
qA = lim
x→x0
f(x) = 2,B = lim
x→x0
g(x) = 1§A>B¤á"
(ii)?¤á"f(x) = x
2 + 3x4
x2,g(x) = x
2 +x4x2,?δ > 0§0 < |x| < δ?§kf(x) > g(x)"qA =
lim
x→x0
f(x) = 1,B = lim
x→x0
g(x) = 1§kA = B"
6,e3:x0?Skg(x) lessorequalslant f(x) lessorequalslant h(x)§g(x)úh(x)3x04?3 uA§y2 lim
x→x0
f(x) =
A.
y2μXJéxn,xn →x0,xn negationslash= x0 §bxn ∈O(x0,δ)?{x0}§kg(xn) lessorequalslantf(xn) lessorequalslanth(xn)
±9g(xn) → A,h(xn) → A(n → ∞) §dê4?5?μf(xn) → A(n → ∞) §ùòy2
f(x) →
A(x→x0).
29
7,e lim
x→x0
f(x) = A,lim
x→x0
g(x) = B negationslash= 0§y2 lim
x→x0
f(x)
g(x) =
A
B.
y2μ
vextendsinglevextendsingle
vextendsinglevextendsinglef(x)
g(x)?
A
B
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingleBf(x)?Ag(x)
Bg(x)
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingleBf(x)?AB +AB?Ag(x)
BG(x)
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant |B||f(x)?A|+|A||g(x)?B|
|B||g(x)| §
du lim
x→x0
f(x) = A,lim
x→x0
g(x) = B§é?ε> 0,?δ1 > 0§0 <|x?x0|<δ1?§k|f(x)?A|<ε?ét
ε> 0,?δ2 > 0§0 <|x?x0|<δ2?§k|g(x)?B|<ε
qa|{$?μ lim
x→x0
Bg(x) = B2 > B
2
2 §Ka5?3§?δ3 > 0§0 <|x?x0|<δ3?§kBg(x) >
B2
2
δ = min{δ1,δ2,δ3}§0 <|x?x0|<δ?§k
vextendsinglevextendsingle
vextendsinglevextendsinglef(x)
g(x)?
A
B
vextendsinglevextendsingle
vextendsinglevextendsingle< (|A|+|B|)ε
B2
2
= 2(|A|+|B|)B2 ε
u′§é?ε> 0,?δ> 0§0 <|x?x0|<δ?§k
vextendsinglevextendsingle
vextendsinglevextendsinglef(x)
g(x)?
A
B
vextendsinglevextendsingle
vextendsinglevextendsingle< 2(|A|+|B|)
B2 ε§l
limx→x0
f(x)
g(x) =
A
B.
8,(1) f(x) =
0 x> 1
1 x = 1
x2 + 2 x< 1
|f(x)3x = 1?m4?"
(2) f(x) =
xsin 1x x> 0
1 +x2 x< 0
|f(x)3x = 0?m4?"
)μ
(1) lim
x→1?0
f(x) = lim
x→1?0
(x2 + 2) = 3,lim
x→1+0
f(x) = 0
(2) lim
x→?0
f(x) = lim
x→?0
(1 +x2) = 1,
lim
x→+0
f(x) = lim
x→+0
(xsin 1x) = 0
9,`2e?ê3¤?:?m4/μ
(1) y =
1
2x 0 <xlessorequalslant 1
x2 1 <x< 2 (3x = 1.5,2,1n:)
2x 2 <x< 3
(2) y = x·sin 1x￡3x = 0:¤
(3) y = 2
1
x + 1
21x?1
￡3x = 0:¤
(4) y = 1x?
bracketleftbigg1
x
bracketrightbigg
￡3x = 1n:¤
(5) D(x) =
braceleftbigg 1 x?knê
0 xnê (3:)
(6) y = (x?1)(?1)
[x]
x2?1 (3x =?1¤
)μ
(1) lim
x→1.5?0
y = lim
x→1.5+0
y = 2.25,
lim
x→2?0
y = lim
x→2?0
x2 = 4,lim
x→2+0
y = lim
x→2+0
(2x) = 4
lim
x→1?0
y = lim
x→1?0
1
2x =
1
2,limx→1+0y = limx→1+0x
2 = 1
(2) lim
x→+0
y = lim
x→+0
y = 0
(3) du lim
x→+0
1
x = +∞,limx→?0
1
x =?∞§
K lim
x→+0
21x = +∞,lim
x→?0
21x = 0§
u′ lim
x→+0
y = lim
x→=0
21x + 1
21x?1
= lim
x→+0
parenleftbigg
1 + 2
21x?1
parenrightbigg
= 1,lim
x→+?0
y = lim
x→?0
21x + 1
21x?1
=?1
30
(4) lim
x→1n+0
y = lim
x→1n+0
parenleftbigg1
x?
bracketleftbigg1
x
bracketrightbiggparenrightbigg
= n?(n?1) = 1
lim
x→1n?0
y = lim
x→1n?0
parenleftbigg1
x?
bracketleftbigg1
x
bracketrightbiggparenrightbigg
= n?n = 0
(5) d?ê3:?m43"
x0?Rt:§dknêú?nê3ê?tè?5§?knS{x(1)n } → x0 + 0§?nS
{x(2)n }→x0 + 0§
lim
x(1)n →x0+0
D
parenleftBig
x(1)
parenrightBig
= 1,lim
x(2)n →x0+0
D
parenleftBig
x(2)
parenrightBig
= 0§l
d?ê3:m43
ón§d?ê3:?43
l
d?ê3:?m43"
(6) y = (x?1)(?1)
[x]
x2?1 =
(?1)[x]
x+ 1? limx→?1+0[x] =?1,limx→?1?0[x] =?2
K lim
x→?1+0
y =?∞,lim
x→?1+0
y =?∞
10,e4?μ
(1) lim
x→∞
sinx
x
(2) lim
x→∞
ex sinx
(3) lim
x→∞
xarctanx
(4) lim
x→∞
xtanx(xnegationslash= npi+ pi2)
)μ
(1) du lim
x→∞
1
x = 0?sinx′k.t§ limx→∞
sinx
x = 0
(2) du lim
x→+∞
ex = +∞§exn = 2npi → +∞(n → ∞)§Kexn sinxn = e2npi sin2npi = 0 → 0(n →
∞)?exn = pi2 + 2npi → +∞(n → ∞)§Kexn sinxn = epi2 +2npi sin
parenleftBigpi
2 + 2npi
parenrightBig
= epi2 +2npi →
+∞(n→∞)§ lim
x→+∞
ex sinx?3§l
lim
x→∞
ex sinx?3.
(3) du lim
x→?∞
arctanx =?pi2,lim
x→+∞
xarctanx = pi2§
K lim
x→?∞
xarctanx = +∞,lim
x→+∞
xarctanx = +∞§l
lim
x→∞
xarctanx = +∞
(4) xn = npi → ∞(n → ∞)§k lim
n→∞
xn tanxn = lim
n→∞
npitannpi = 0?,xn = pi4 +npi → ∞(n →
∞)§k lim
n→∞
xn tanxn = lim
n→∞
parenleftBigpi
4 +npi
parenrightBig
tan
parenleftBigpi
4 +npi
parenrightBig
= lim
n→∞
parenleftBigpi
4 +npi
parenrightBig
= +∞§ lim
x→∞
xtanx(xnegationslash=
npi+ pi2)?3.
11,l^? lim
x→∞
parenleftbiggx2 + 1
x+ 1?ax?b
parenrightbigg
= 0§|~êaúb.
)μdu lim
x→∞
parenleftbiggx2 + 1
x+ 1?ax?b
parenrightbigg
= lim
x→∞
(x2 + 1)?ax(x+ 1)?b(x+ 1)
x+ 1 = limx→∞
(1?a)x2?(a+b)x?b+ 1
x+ 1 =
0§Kk
braceleftbigg 1?a = 0
a+b = 0 §l
braceleftbigg a = 1
b =?1
12,l^? lim
x→?∞
(√x2?x+ 1?a1x?b1) = 0,lim
x→?∞
(√x2?x+ 1?a2x?b2) = 0§|~êa1,b1,a2,b2.
)μdu lim
x→?∞
(√x2?x+ 1?a1x?b1) = lim
x→?∞
(1?a21)x2?(1 + 2a1b1)x+ 1?b21√
x2?x+ 1 +a1x+b1 = 0§K
braceleftbigg 1?a2
1 = 0
1 + 2a1b1 = 0 §
u′
braceleftBigg a
1 = ±1
b1 =?12,
qa^μea1 = 1§K lim
x→?∞
(√x2?x+ 1?a1x?b1) = +∞§l
braceleftBigg a
1 =?1
b1 = 12 §ón
braceleftBigg a
2 = 1
b2 =?12,
13,e lim
x→+∞
[f(x)?(kx+b)] = 0§Ky = kx+b′-?y = f(x)x→ +∞ìC?.|^ù§í?ì
C?37^?.
y2μe-?3ìC?§Kk
lim
x→+∞
[f(x)?(kx+b)] = 0,(1)
31
f(x)x = 1x[f(x)?kx?b] +k+ bx§-x→ +∞üà45?(1)a§
lim
x→+∞
f(x)
x = k (2)
.
Q|?
k§2l(1)a|
b = lim
x→+∞
[f(x)?kx] (3)
.
§e(2)!(3)üa¤á§á=?w?^?(1)¤á.
-?y = f(x)x → +∞?3ìC?y = kx+b7?^?′4? lim
x→+∞
f(x)
x = k! limx→+∞[f(x)?
kx] = bt¤á.
14,e lim
x→?∞
f(x) = A> 0§y23X > 0§|x<?X¤áμA2 <f(x) < 32A.
y2μdu lim
x→?∞
f(x) = A > 0§éε = A2 > 0,?X > 0 §x <?X?§k|f(x)?A| < A2§
=A2 <f(x) < 32A.
15,e lim
x→+∞
f(x) = A,lim
x→+∞
g(x) = B§y2 lim
x→+∞
f(x)g(x) = AB.
y2μdu lim
x→+∞
f(x) = A§é?ε > 0,?X1 > 0§x > X1?§k|f(x)?A| < εX2 > 0,M > 0§
x>X2?§k|f(x)|<A.
q lim
x→+∞
g(x) = B§ét?ε> 0,?X3 > 0§x>X3?§k|g(x)?B|<ε.
X = max{X1,X2,X3}§ét?ε> 0§x>X?§
k|f(x)g(x)?AB| = |f(x)g(x)?f(x)B+f(x)B?AB|lessorequalslant|f(x)||g(x)?B|+|B||f(x)?A|lessorequalslantMε+|B|ε =
(M +|B|)ε§= lim
x→+∞
f(x)g(x) = AB.
16,y2 lim
x→+∞
f(x) = A^?′μéêxn → +∞,f(xn) →A.
y2μ
du lim
x→+∞
f(x) = A§é?ε> 0,?X > 0§x>X?§k|f(x)?A|<ε.
qxn → +∞(n → ∞)§ét?X > 0,?N ∈ Z+§n > N?§kxn > X§l
|f(xn)?A| < ε§u
′ lim
n→∞
f(xn) = A.
^?y{"b lim
x→+∞
f(x) negationslash= A§K?ε0 > 0§é?X > 0§?kxprime§xprime >X?§k|f(xprime)?A| greaterorequalslant
ε0.
AO/§X?1,2,3,···§?xprime1,xprime2,xprime3,···§|
xprime1 > 1?§k|f(xprime1)?A|greaterorequalslantε0?xprime2 > 2?§k|f(xprime2)?A|greaterorequalslantε0?xprime3 > 3?§k|f(xprime3)?A|greaterorequalslantε0?···
l?>?±w?xprimen → +∞(n → ∞)§
lm>w? lim
n→∞
f(xprimen) negationslash= A§g?§Kb?¤á§
lim
x→+∞
f(x) = A
17,y2 lim
x→x0+0
f(x) = +∞^?′μéêxn,xn >x0,xn →x0§kf(xn) → +∞.
y2μ
du lim
x→x0+0
f(x) = +∞§é?G> 0,?δ> 0§0 <x?x0 <δ?§kf(x) >G.
qxn > x0,xn → x0(n → ∞)§ét?δ > 0,?N ∈ Z+§n > N?§k0 < xn?x0 < δ§l
f(xn) >
G§u′ lim
n→∞
f(xn) = +∞.
^?y{"b lim
x→x0+0
f(x) negationslash= +∞§K?G0 > 0§é?δ > 0§?kxprime§0 < xprime?x0 < δ?§
kf(xprime) lessorequalslantG0.
AO/§δ?1,12,13,···§?xprime1,xprime2,xprime3,···§|
0 < xprime1? x0 < 1?§kf(xprime1) lessorequalslant G0?0 < xprime2? x0 < 12?§kf(xprime2) lessorequalslant G0?0 < xprime3? x0 < 13?§
kf(xprime3) lessorequalslantG0?···
l?>?±w?xprimen > x0,xprimen → x0§
lm>w? lim
x→x0+0
f(x) negationslash= +∞§g?§Kb?¤á§
lim
x→x0+0
f(x) = +∞
18,Tüe?|f(x)
(1) f(+0) = 0,f(?0) = 1
32
(2) f(+0)?3§∞,f(?0) = 0
(3) f(+∞) = 0,f(?∞)?3
(4) f(+∞) = f(?∞) = A￡~ê¤
(5) f(x0 + 0)úf(x0?0)3
(6) f(x0 + 0) = +∞,f(x0?0) =?∞
(7) f(x0 + 0) = 1,f(x0?0) = +∞
(8) f(+∞)?3§∞,f(?∞) =?∞
)μ
(1) f(x) =
braceleftbigg 0 x> 0
1 xlessorequalslant 0
(2) f(x) =
braceleftBigg
sin 1x x> 0
0 xlessorequalslant 0
(3) f(x) = e?x
(4) f(x) = Ax+ 1x
(5) f(x) = sin 1x?x
0
(6) f(x) = 1x?x
0
(7) f(x) = 1 +e?
1
x?x0
(8) f(x) =
braceleftbigg sinx xgreaterorequalslant 0
x x< 0
33
§3,?Y?ê
1,Uy2e?ê3S?Yμ
(1) y = √x
(2) y = 1x
(3) y = |x|
(4) y = sin 1x
y2μ
(1) x0?(0,+∞)S:§|√x?√x0|< |x?x0|√x+√x
0
lessorequalslant |x?x0|√x
0
é?ε> 0§δ = √x0ε§|x?x0|<δ?§k|√x?√x0|< |x?x0|√x
0
<ε§y = √x3x0:?Y.
qdx03(0,+∞)￥5§Ky = √x3(0,+∞)S?Y.
x0 = 0?§ét?ε> 0§δ = ε2§0 <x?x0 <δ?§k|√x?√x0|<√x<ε§f(+0) = 0 =
f(0)§
l
y = √x3[0,+∞)S?Y.
(2) x0?(0,+∞)S:§|x?x0| < x02 §Kx> x02,xx0 > x
2
0
2 §u′|
1
x?
1
x0 | =
|x?x0|
xx0 <
|x?x0|
x20
2
ex0?(?∞,0)S:§|x?x0|<?x02 §Kx< x02,xx0 > x
2
0
2 §u′|
1
x?
1
x0 | =
|x?x0|
xx0 <
|x?x0|
x20
2
x0?(?∞,0)uniontext(0,+∞)S:§
é?ε > 0§δ = min
braceleftbigg|x
0|
2,
x20
2 ε
bracerightbigg
> 0§|x?x0| < δ?§k
vextendsinglevextendsingle
vextendsinglevextendsingle1
x?
1
x0
vextendsinglevextendsingle
vextendsinglevextendsingle = |x?x0|
xx0 > ε§y =
1
x3x0:?Y
qdx03(?∞,0)uniontext(0,+∞)S5§y = 1x3(?∞,0)uniontext(0,+∞)S?Y.
(3) x0?(?∞,+∞)S:§||x|?|x0||lessorequalslant|x?x0|.
é?ε> 0§δ = ε> 0§|x?x0|<δ?§k||x|?|x0||lessorequalslant|x?x0|<ε§y = |x|3x0:?Y
qdx03(?∞,+∞)S5§y = |x|3(?∞,+∞)S?Y.
(4) x0?(0,+∞)S:§|x? x0| < x02 §Kx > x02,xx0 > x
2
0
2 §u′
vextendsinglevextendsingle
vextendsinglevextendsinglesin 1
x?sin
1
x0
vextendsinglevextendsingle
vextendsinglevextendsingle =
2
vextendsinglevextendsingle
vextendsinglevextendsinglesin x+x0
2xx0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglecos x?x0
2xx0
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant |x?x0|
xx0 <
|x?x0|
x20
2
ex0?(?∞,0)S:§|x?x0| <?x02 §Kx < x02,xx0 > x
2
0
2 §u′
vextendsinglevextendsingle
vextendsinglevextendsinglesin 1
x? sin
1
x0
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
|x?x0|
xx0 <
|x?x0|
x20
2
x0?(?∞,0)uniontext(0,+∞)S:§
é?ε > 0§δ = min
braceleftbigg|x
0|
2,
x20
2 ε
bracerightbigg
> 0§|x?x0| < δ?§k
vextendsinglevextendsingle
vextendsinglevextendsinglesin 1
x?sin
1
x0
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant |x?x0|
xx0 < ε§
y = sin 1x3x0:?Y
qdx03(?∞,0)uniontext(0,+∞)S5§y = sin 1x3(?∞,0)uniontext(0,+∞)S?Y.
2,|^?Y?ê$?§|e?ê?Yμ
(1) y = tanx
34
(2) y = 1xn
(3) y = secx+ cscx
(4) y = 1√cosx
(5) y = ln(1 +x)x2?2x
(6) y = [x]tanx1 + sinx
)μ
(1)?y = tanx = sinxcosx§Kcosxnegationslash= 0?§y = tanx?Y§y = tanx?Y
parenleftBig
pi2 +kpi,pi2 +kpi
parenrightBig
(k∈
Z).
(2) en > 0§Ky = 1xn?Y(?∞,0)uniontext(0,+∞)?en lessorequalslant 0§Ky = 1xn?Y§=§?Y
(?∞,+∞).
(3)?secx?Y
parenleftbigg
k? 12
parenrightbigg
pi < x <
parenleftbigg
k+ 12
parenrightbigg
pi(k = 0,±1,±2,···)§cscx?Ykpi < x <
(k+ 1)pi(k = 0,±1,±2,···)§
y = secx+ cscx?Y
parenleftBig
kpi? pi2
parenrightBiguniontextparenleftBig
kpi,kpi+ pi2
parenrightBig
((k = 0,±1,±2,···).
(4) cosx> 0?§y = 1√cosx?Y§y = 1√cosx?Y
parenleftBig
pi2 + 2kpi,pi2 + 2kpi
parenrightBig
.
(5)?ln(1+x)x>?1Y§ 1x2?2xxnegationslash= 0,xnegationslash= 2Y§y = ln(1 +x)x2?2x ?Y(?1,0)uniontext(0,2)uniontext(2,+∞).
(6)?y = [x]tanx1 + sinx = [x]sinx(1 + sinx)cosx§Ksinx negationslash= 1,cosx negationslash= 0,x /∈ Z/{0}?§y = [x]tanx1 + sinx?Y§
y = [x]tanx1 + sinx?Yx∈
parenleftBig
kpi? pi2,kpi+ pi2
parenrightBig
x /∈Z/{0}(k∈Z).
3,e?ê?Y5§?x?ù?/.
(1) y =
x2?4
x?2,exnegationslash= 2
4,x = 2
(2) y =
vextendsinglevextendsingle
vextendsinglevextendsinglesinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle,xnegationslash= 0
1,x = 0
(3) y ==
sinx
|x|,xnegationslash= 0
1,x = 0
(4) y=[x]
)μ
(1)?lim
x→2
y = lim
x→2
x2?4
x?2 = limx→2(x+ 2) = 4§?x = 2?§y = 4§?ê3x = 2?Y
xnegationslash= 2?§y = x
2?4
x?2 = x+ 2w,?Y§
y =
x2?4
x?2,exnegationslash= 2
4,x = 2
3(?∞,+∞)S?Y.
(2) x negationslash= 0?§y =
vextendsinglevextendsingle
vextendsinglevextendsinglesinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle = sinx
x?y =?
sinx
x w,?Y"qlimx→0
vextendsinglevextendsingle
vextendsinglevextendsinglesinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle = 1 = f(0)§?ê3x = 0?
Y§
u′y =
vextendsinglevextendsingle
vextendsinglevextendsinglesinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle,xnegationslash= 0
1,x = 0
3(?∞,+∞)S?Y.
35
(3)? lim
x→+0
y = lim
x→+0
sinx
|x| = 1,limx→?0y = limx→?0
sinx
|x| =?1 §limx→0y?3"qx> 0?§y =
sinx
|x| =
sinx
x §x< 0?§y =
sinx
|x| =?
sinx
x §w,?Y§d?ê3?0 ?Y§=3(?∞,0)
uniontext(0,+∞)S
Y.
(4)? lim
x→k+0
y = lim
x→k+0
[x] = k,lim
x→k?0
y = lim
x→k?0
[x] = k? 1(k ∈ Z)§Klim
x→k
y?3§x = k(k ∈
Z)?y = [x]m?:§3m?:?m?Y
k<x<k+ 1(k∈Z)?§y = [x]w,?Y§d?ê3?k(k∈Z) ?Y.
4,ef(x)?Y§|f(x)|úf2(x)′Yoqe|f(x)|?f2(x)?Y§f(x)′Yo
)μ
(1) f(x)3ùIt?Y§x0?It:
f(x)3x0:?Y§é?ε> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|<ε
||f(x)|?|f(x0)||lessorequalslant|f(x)?f(x0)|<ε§=é?ε> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|<ε§
|f(x)|3x0:?Y
qdx03It5§|f(x)|3ItY
ó|f2(x)? f2(x0)| = |f(x)? f(x0)||f(x) + f(x0)| = |f(x)? f(x0)||f(x)? f(x0) + 2f(x0)| lessorequalslant
|f(x)?f(x0)|(|f(x)?f(x0)|+ 2f(x0)) <ε(ε+ 2f(x0))§f2(x)3x0:?Y
qdx03It5§f2(x)3ItY
(2)?L5§e|f(x)|?f2(x)?Y§f(x)Y.
(i)Y"~μf(x) =
braceleftbigg 1,xgreaterorequalslant 0
1,x< 0 §|f(x)| = 1úf
2(x) = 1t3(?∞,+∞)S?Y§f(x)3x =
0:Y?
(ii)?Y"~μf(x) = x,Kf(x)!|f(x)|!f2(x)3(?∞,+∞)St?Y"
5,(1)?êf(x)x = x0Y§
êg(x)x = x0Y§ˉd?êú3x0:′Yo
(2) x = x0êf(x)úg(x)Y§ˉd?êúf(x) +g(x)3?:x0′?7Yo
):
(1) ^?y{"bf(x) +g(x)3x0:?Y"
f(x)x = x0Y§Kd?Y?ê5?§g(x) = [f(x) +g(x)]?f(x)x0Yg?"
b?¤á§=f(x) +g(x)3x0:?Y.
(2)"
(i)?Yμ~μf(x) =
braceleftbigg 1,xgreaterorequalslant 0
1,x< 0,g(x) =
braceleftbigg?1,xgreaterorequalslant 0
1,x< 0 3x = 0Y§f(x) +
g(x) = 03x = 0?Y.
(ii)Yμ~μf(x) = g(x) = 1x3x = 0Y§f(x) +g(x) = 2x3x = 0Y.
6,(1)?êf(x)3x0?Y§
êg(x)3x0Y?
(2) x = x0êf(x)úg(x)Y§ˉd?ê|èf(x)g(x)3?:x0′?7Yo
):
(1)"
(i)?Yμ~μf(x) = 03x = 0?Y§g(x) =
braceleftbigg 1,xgreaterorequalslant 0
0,x< 0 3x = 0Y§f(x)g(x) = 03x =
0?Y.
(ii)Yμ~μf(x) = x3x = 0?Y§g(x) = 1x23x = 0Y§f(x)g(x) = 1x3x = 0Y.
(2)"
(i)?Yμ~μf(x) =
braceleftbigg 1,xgreaterorequalslant 0
1,x< 0,g(x) =
braceleftbigg?1,xgreaterorequalslant 0
1,x< 0 3x = 0Y§f(x)g(x) =
13x = 0?Y.
(ii)Yμ~μf(x) = g(x) = 1x3x = 0Y§f(x)g(x) = 1x23x = 0Y.
7,ef(x)3[a,∞)?Y§ lim
x→∞
f(x)3§y2f(x)3[a,∞)k..
y2μdu lim
x→∞
f(x)3§ lim
x→∞
f(x) = A
Kéε = 1,?X > 0§x>X?§k|f(x)?A|<ε = 1¤á§l
|f(x)| = |f(x)?A+A|lessorequalslant|f(x)?A|+
36
|A|< 1 +|A|
X1 = max{X,a+ 1}§Kf(x)3(X1,∞)Sk.§?|f(x)|<|A|+ 1,x∈ (X1,∞)
qduf(x)3[a,X1]t?Y§f(x)3[a,X1]tk.§ù.?M > 0§=?x∈ [a,X1]§k|f(x)|lessorequalslantM
G = max{|A|+ 1,M}§K?x∈ [a,∞),f(x) lessorequalslantG§
=f(x)3[a,∞)k..
8,eéε> 0§f(x)3[a+ε,b?ε]?Y§ˉμ
(1) f(x)′?(a,b)3?Yo
(2) f(x)′?3[a,b]?Yo
)μ
(1)?x0 ∈ (a,b)§ε = min
braceleftbiggx
0?a
2,
b?x0
2
bracerightbigg
§Kx0 ∈ [a+ε,b?ε]
éε> 0§f(x)3[a+ε,b?ε]?Y§f(x)3x0:?Y
dx0 ∈ (a,b)5§f(x)3(a,b)S?Y.
(2)Y"
(i)Y"~μf(x)3[0 +ε,1?ε](ε> 0)S?Y§f(x)3[0,1]tY§3x = 0:?m.
(ii)?Y"~μf(x)3[1 +ε,2?ε](ε> 0)S?Y§?f(x)3[1,2]t?Y.
9,ef(x)3x0:?Y§f(x0) > 0§y23x0δ?O(x0,δ)§x ∈ O(x0,δ)?§f(x) greaterorequalslant c> 0§c?,
~ê.
y2μduf(x)3x0:?Y§?f(x0) > 0§Kf(x0) >c> 0
éε = f(x0)?c> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|<ε = f(x0)?c§Kf(x0)?[f(x0)?
c] lessorequalslantf(x)§=f(x) greaterorequalslantc> 0.
10,y2e?Y?ê3kn:?ê0§Kd?êe?0.
y2μf(x)?￠?t?Y?ê§x0n:.
dkn:3ê?tè?5§?±?nê{xn}§|xn →x0(n→∞).
f(x)3x0?Y§Kf(x0) = lim
n→∞
f(xn) = 0§
dx0:5§f(x)3¤k?n:?ê0.
qf(x)3kn:?ê0§Kd?êe?0.
11,ef(x)3[a,b]?Y§e§Uy2 1f(x)3[a,b]?Y.
y2μduf(x)3[a,b]?Y§e§Kf(x)3(a,b)?Y§f(x) > 0§ 1f(x)3§x∈ [a,b]
x0?(a,b)S:§Ké?ε> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|<ε.
qf(x)3[a,b]?Y§Kd4?m?Y?ê5?2§?f(x)3[a,b]t?m > 0§=f(x) greaterorequalslant m,x ∈
[a,b]§u′
vextendsinglevextendsingle
vextendsinglevextendsingle 1
f(x)?
1
f(x0)
vextendsinglevextendsingle
vextendsinglevextendsingle= |f(x)?f(x0)|
f(x)f(x0) <
ε
m2§ limx→x0
1
f(x) =
1
f(x0)§l
1
f(x)3x0?Y.
dx03(a,b)S5§f(x)3(a,b)S?Y.
qf(a+ 0) = f(a) > 0§K 1f(a+ 0) = 1f(a)§f(x)3[a,b)?Y?
qf(b?0) = f(b) > 0§K 1f(b?0) = 1f(b)§f(x)3[a,b]?Y.
12,ef(x)úg(x)?3[a,b]?Y§áy2max(f(x),g(x))±9min(f(x),g(x))?3[a,b]?Y.
y2μduf(x)úg(x)?3[a,b]?Y§f(x)?g(x)úf(x) +g(x)?3[a,b]?Y.
d14K(?§k|f(x)?g(x)|3[a,b]?Y.
-?(x) = max(f(x),g(x)) = 12(f(x) +g(x) +|f(x)?g(x)|),
ψ(x) = min(f(x),g(x)) = 12(f(x) +g(x)?|f(x)?g(x)|)§
?(x),ψ(x)?3[a,b]?Y.
13,ef(x)′?Y§y2éc> 0§?êg(x) =
c,ef(x) <?c
f(x),e|f(x)|lessorequalslantc
c,ef(x) >c
′?Y.
y2μdug(x) = max(?c,min(f(x),c))
qduf(x)?Y§?éc> 0§?(x) = c?Y§ψ(x) =?c?Y§
KdtK(?§min(f(x),c)?Y§l
2dtK(?§g(x)?Y.
14,e?êY:5?￡=Y:¤μ
(1) y = x(1 +x)2
37
(2) y = 1 +x1 +x3
(3) y = x
2?1
x3?3x+ 2
(4) y = xsinx
(5) y = cos2 1x
(6) y = [x] + [?x]
(7) y = 1lnx
(8) y = x
2?x
|x|(x2?1)
(9) y =
1
q,x =
p
q(q> 0,q,p?p?ê)
0,xnê
(10) y =
braceleftbigg x,|x|lessorequalslant 1
1,|x|> 1
(11) y =
braceleftBigg
cos pix2,|x|lessorequalslant 1
|x?1|,|x|> 1
(12) y =
braceleftbigg sinpix,x?knê
0,xnê
)μ
(1)? lim
x→?1?0
x
(1 +x)2 =?∞§x =?1?1aY:￡m?:¤.
(2)? lim
x→?1
1 +x
1 +x3 =
1
3§y3x =?1:vk§x =?1￡Y:.
(3)?y = x
2?1
x3?3x+ 2 =
(x?1)(x+ 1)
(x?1)(x2 +x+ 1)?3(x?1) =
(x?1)(x+ 1)
(x?1)(x2 +x?2) =
(x?1)(x+ 1)
(x?1)2(x+ 2)§
q lim
x→1?0
y =?∞,lim
x→?2?0
y =?∞§x =?2,x = 1?1aY:.
(4)?lim
x→0
x
sinx = 1y3x = 0§x = 0￡Y:?
q lim
x→kpi
k∈Z,knegationslash=0
x
sinx = ∞§x = kpi(k∈Z,knegationslash= 0)?1aY:.
(5)?lim
x→0
cos2 1x3[0,1]m
§?
.4?§d43§u′x = 0?1aY:.
(6)?x→k+ 0?§?x→?k?0§ lim
x→k+0
y = lim
x→k+0
([x] + [?x]) = k+ (?k?1) =?1?
q?x→k?0?§?x→?k+ 0§ lim
x→k?0
y = lim
x→k?0
([x] + [?x]) = k?1 + (?k) =?1(k∈Z)
qx = k?§y = [x] + [?x] = [k] + [?k] = 0(k∈Z)§ê:t￡Y:.
(7)? lim
x→1+0
1
lnx = +∞§x =?1?1aY:?
lim
x→?0
1
lnx?3§x = 0?1aY:.
(8) y = x(x?1)|x|(x?1)(x+ 1)
lim
x→1
y = 12y3x = 1§x = 1￡Y:?
lim
x→+0
y = 1,lim
x→?0
y =?1§x = 0?1?aY:￡am?:¤?
lim
x→?1+0
y =?∞§x =?1?1am?:.
(9)?d?ê′±1?±??ê§?3?m[0,1]§ù§?m?/?daq.
3[0,1]t§?1?1knêkü?μ01,111?2knêkμ12?
1?3knêkü?μ13,231?4knêkü?μ14,34?
1?5knêko?μ15,25,35,451?6knêkü?μ16,56?···
38
o?§?1Lkknê?êllessorequalslant 2 + 1 + 2 + 3 +···+ (k?1) = k(k?1)2 + 2§=?1Lkk
nê?kk"
e?5y§3:x0 ∈ [0,1]§x→x0?§y → 0.
é?ε> 0§k =
bracketleftbigg1
ε
bracketrightbigg
§3[0,1]t§?1Lkknê?r1,r2,···,rl.
δ = min
limits1lessorequalslantilessorequalslantl|ri?x0|§K0 <|x?x0|<δ§=x /∈{r1,r2,···,rn}§?ò′xnê§k
nêpq§?q greaterorequalslantk+ 1 >k?§òk|y?0| =
1
q lessorequalslant
1
k+ 1,x?knêx =
p
q,q>k
0 <ε,xnê
.
lim
x→x0
y = 0 §u′μn:?′d?ê?Y:§kn:?′d?ê?￡Y:.
(10)? lim
x→?1+0
y =?1,lim
x→?1?0
y = 1§x =?1?1?aY:.
(11)? lim
x→?1+0
y = 0,lim
x→?1?0
y = 2§x =?1?1?aY:.
(12) (i) x0 negationslash= n,n∈Z§
kn:rn →x0?rn >x0§K lim
rn→x0+0
f(rn) = sinpix0 negationslash= 0?
?n:xn →x0?xn >x0§K lim
xn→x0+0
f(xn) = 0"
f(x0 + 0)?3§l
xnegationslash= n(n∈Z)ê1aY:.
(ii) x0 = n,n∈Z§
xnê?§|f(x)?f(n)| = 0?
x?knê?§|f(x)?f(n)| lessorequalslant pi|x?n|§é?ε > 0,?δ = εpi > 0§||x?n| < δ?§k|f(x)?
f(n)|<ε§f(x)3x = n(n∈Z)?Y.
15,x = 0?e?êf(x)§áf(0)ê?§|f(x)3x = 0?Yμ
(1) f(x) =
√1 +x?1
3√1 +x?1
(2) f(x) = tan2xx
(3) f(x) = sinx·sin 1x
(4) f(x) = (1 +x)1x
)μ
(1)?lim
x→0
f(x) = lim
x→0
√1 +x?1
3√1 +x?1 = limx→0
3radicalbig(1 +x)2 + 3√1 +x+ 1
√1 +x+ 1 = 32§
f(0) = 32.
(2)?lim
x→0
f(x) = lim
x→0
tan2x
x = 2§
f(0) = 2.
(3)?lim
x→0
f(x) = lim
x→0
sinx·sin 1x = 0§
f(0) = 0.
(4)?lim
x→0
f(x) = lim
x→0
(1 +x)1x = e§
f(0) = e.
16,ef(x)3[a,b]?Y§a<x1 <x2 <···<xn <b§K3[x1,xn]￥7kξ§|f(ξ) = f(x1) +f(x2) +···+f(xn)n,
y2μM = max
1lessorequalslantilessorequalslantn
f(xi),m = min
1lessorequalslantilessorequalslantn
f(xi)
Kf(x1) +f(x2) +···+f(xn)n lessorequalslantM?
ónf(x1) +f(x2) +···+f(xn)n greaterorequalslantm.
duf(x)3[x1,xn]? [a,b]t?Y§d0n§7?ξ ∈ [x1,xn]? [a,b]§|f(ξ) = f(x1) +f(x2) +···+f(xn)n,
17,^Yyμ
39
(1) f(x) = 3√x3[0,1]t′Y?
(2) f(x) = sinx3(?∞,+∞)t′Y?
(3) f(x) = sinx23(?∞,+∞)tY.
y2μ
(1) éx1,x2 ∈ [0,1]§| 3√x1? 3√x2| = |x1?x2|3radicalbigx2
1 + 3
√x
1x2 + 3
radicalbigx2
2
= |x1?x2|3
4(
3√x1 + 3√x2)2 + 1
4(
3√x1? 3√x2)2 lessorequalslant
|x1?x2|
1
4(
3√x1? 3√x2)2§
=14| 3√x1? 3√x2|3 lessorequalslant|x1?x2|§?=| 3√x1? 3√x2|lessorequalslant 3radicalbig4|x1?x2|
é?ε> 0,?δ = ε
3
4 > 0§|é?x1,x2 ∈ [0,1]§|x1?x2|<δ?§ok|
3√x1? 3√x2|lessorequalslant 3
radicalbig4|x
1?x2|<
ε
l
f(x) = 3√x3[0,1]t′Y.
(2) éx1,x2 ∈ (?∞,+∞)§|sinx1?sinx2| = 2
vextendsinglevextendsingle
vextendsinglecos x1 +x22 sin x1?x22
vextendsinglevextendsingle
vextendsinglelessorequalslant 2
vextendsinglevextendsingle
vextendsinglex1?x22
vextendsinglevextendsingle
vextendsingle= |x1?x2|§
é?ε > 0,?δ = ε > 0§|é?x1,x2 ∈ (?∞,+∞)§|x1?x2| < δ?§ok|sinx1? sinx2| lessorequalslant
|x1?x2|<ε
l
f(x) = sinx3(?∞,+∞)t′Y.
(3) ε0 = 1§éδ> 0§xprimen =radicalbig2npi+ pi2,xprimeprimen =radicalbig2npi? pi2§|xprimen?xprimeprimen| = |radicalbig2npi+ pi2?radicalbig2npi? pi2| =vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle piradicalbig2npi+ pi
2 +
radicalbig2npi? pi
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle→ 0(n→∞)
n§k|xprimen?xprimeprimen|<δ§
|sin(xprimen)2?sin2(xprimeprimen)2| = |1?(?1)| = 2 > 1 = ε0
l
f(x) = sinx23(?∞,+∞)tY.
40
§4,tút
1,|etx→ 0? úì?ü?μ
(1) x3 +x6
(2) 4x2 + 6x3?x5
(3) √x·sinx
(4) radicalbigx2 + 3√x
(5) √1 +x?√1?x
(6) tanx?sinx
(7) ln(1 +x)
)μ
(1) dulim
x→0
x3 +x6
x3 = limx→0(1 +x
3) = 1§§′3 t§§ì?üx3.
(2) dulim
x→0
4x2 + 6x3?x5
4x2 = limx→0(1 +
3
2x?
x3
4 ) = 1§§′2 t§§ì?ü4x
2.
(3) dulim
x→0
√x·sinx
|x| = limx→0
radicalbigg
sinx
x = 1§§′1 t§§ì?ü|x|.
(4) dulim
x→0
radicalbigx
2 + 3√x
6√x = limx→0
radicalBig
x53 + 1 = 1§§′16 t§§ì?ü 6√x.
(5) dulim
x→0
√1 +x?√1?x
x = limx→0
2x
x(√1 +x+√1?x) = 1§§′1 t§§ì?ü?
x.
(6) dulim
x→0
tanx?sinx
x3
2
= lim
x→0
2tanx?sinxx3 = lim
x→0
2(1?cosx)
cosx·x2 = limx→0
x2
x2 = 1§§′3
t§§ì?üx
3
2,
(7) dulim
x→0
ln(1 +x)
x = 1§§′1 t§§ì?üx.
2,x→∞?§|eCt úì?ü?μ
(1) x2 +x6
(2) 4x2 + 6x4?x5
(3) 3
radicalbigg
x2 sin 1x
(4)
radicalBig
1 +radicalbig1 +√x
(5) 2x
5
x3?3x+ 1
)μ
(1) du lim
x→∞
x2 +x6
x6 = 1§§′6 t§§ì?üx
6.
(2) du lim
x→∞
4x2 + 6x4?x5
x5 = 1§§′5 t§§ì?üx
5.
(3) du lim
x→∞
3
radicalbigg
x2 sin 1x
3√x = limx→∞
3√x
3√x = 1§§′
1
3 t§§ì?ü
3√x.
(4) du lim
x→∞
radicalBig
1 +radicalbig1 +√x
8√x = limx→∞
radicaltpradicalvertex
radicalvertexradicalbtparenleftbigg1
x
parenrightbigg1
4 +
radicalBiggparenleftbigg
1
x
parenrightbigg1
2 + 1 = 1§§′1
8 t§§ì?
ü 8√x.
41
(5) du lim
x→∞
2x5
x3?3x+ 1
2x2 = limx→∞
x3
x3?3x+ 1 = 1§§′2 t§§ì?ü2x
2.
3,áyμ?x→ 0?
(1) o(?xm) +o(?xn) = o(?xn)(m>n> 0)
(2) o(?xm)o(?xn) = o(?xm+n)(m,n> 0)
(3) |f(x)|lessorequalslantM§Kf(x)o(?x) = o(?x)
(4)?xm ·o(1) = o(?xm)
y2μ
(1) du?x→ 0§?xm → 0,?xn → 0§u′o(?x
m)
xm → 0,
o(?xn)
xn → 0
qm>n> 0§?x
m
xn =?x
m?n → 0 §u′o(?xm) +o(?xn)
xn =
o(?xm)
xm ·
xm
xn +
o(?xn)
xn → 0§
l
o(?xm) +o(?xn) = o(?xn)
(2) du?x→ 0§?xm → 0,?xn → 0§u′o(?x
m)
xm → 0,
o(?xn)
xn → 0
u′o(?x
m)o(?xn)
xm+n =
o(?xm)
xm ·
o(?xn)
xn → 0§
l
o(?xm)o(?xn) = o(?xm+n)
(3)?x → 0§o(?x)?x → 0§q|f(x)| lessorequalslant M§f(x)k.§u′f(x)o(?x)?x = f(x)o(?x)?x → 0§l
f(x)o(?x) = o(?x).
(4) do(1)u′t§Ko(1) → 0§u′?x
m ·o(1)
xm =
xm
xmo(1) = o(1) → 0§l
x
m ·o(1) =
o(?xm).
42
1ü? 4?Y?
1nù 'u￠ê? ?n9
4?mt?Y?ê5?y2
§1,'u￠ê? ?n
1,luy2e(.5.
y2μα,αprime?′ê8Ee(.§u′?x∈E§?kxgreaterorequalslantα§=α′Ee.?xgreaterorequalslantαprime§=αprime′Ee..
duα′Ee(.§′e.￥§l
kα greaterorequalslant αprime?ódαprime′Ee(.§kαprime greaterorequalslant α.dd
α = αprime.
2,β = supE,β /∈E§áygE￥?àê{xn}§ù4β?qeβ ∈E§K?/X?o
y2μ
(1) duβ = supE,β /∈E§Kdt(.§
(i) é?x∈E§?kx<β?
(ii) é?ε> 0§?3êx0 ∈E§|x0 >β?ε.
εn = 1n§éz?εn?kxn ∈ E§|β > xn > β?εn§=0 < β?xn < εn§u′?àê
{xn}?E.
q lim
n→∞
(β?εn) = β? lim
n→∞
εn = β?β greaterorequalslant lim
n→∞
xn greaterorequalslant lim
n→∞
(β?εn) = β§ lim
n→∞
xn = β.
(2) β ∈ E?§·K¤á"~μ?¤á"E = (1,12,13,···,1n,···),β = supE = 1,1 ∈ E.
q1n → 0(n→∞)§KE￥f4?t?0§β ∈E?§·K?¤á"
¤á"E =
braceleftbigg
sin pi8,sin 2pi8,···,sin npi8,···
bracerightbigg
,β = supE = 1,1 ∈E§xn = sin 16n+ 48 pi§K lim
n→∞
xn =
1§β ∈E?§·K¤á"
3,T~μ
(1) kt(.?e(.ê?
(2) 1kt(.?1ke(.ê?
(3) Q1kt(.q1ke(.ê?
(4) Q?1kt(.§q?1ke(.ê§ù￥t!e(.?k?.
)μ
(1) {xn} = {?n},sup{xn} =?1
(2) {xn} = {1n},sup{xn} = 1 ∈{xn},inf{xn} = 0 /∈{xn}
(3) {xn} = {1 + (?1)n},sup{xn} = 2 ∈{xn},inf{xn} = 0 ∈{xn}
(4) E =
parenleftbigg
1,12,1 + 12,13,1 + 23,···,1n,1 + n?1n
parenrightbigg
,supE = 2 /∈E,infE = 0 /∈E
4,áyê7kt(.úe(.§au+∞ê7ke(.§au?∞ê7kt(..
y2μ
(1) éue?~êê§w,t!e(.t.
éu?e?~êê§?{xn}§={xn}k4?§Kd1ù§1?n4§ê{xn}′k.ê.
l
d ù?nn§ê{xn}kt!e(.§=ê7kt!e(..
5μy2μt!e(.β,α￥?káu{xn}.
ˉ￠t§eα = β§Kα = β = xn,n = 1,2,···
eαnegationslash= β§?α /∈{xn}§KdSK2§3f
braceleftBig
x(1)nk
bracerightBig
u�3f
braceleftBig
x(2)nk
bracerightBig
uβ§
{xn}§ù{xn}g?§α,β￥?káu{xn}.
(2)?{xn}′au+∞ê§K?N ∈Z+§n>N?§ekxn >x1§u′x1,x2,···,xN￥??§=
{xn}e(."
43
(3)?{xn}′au?∞ê§K?N ∈Z+§n>N?§ekxn <x1§u′x1,x2,···,xN￥§=
{xn}t(."
5,|ê{xn}t!e(.:
(1) xn = 1? 1n
(2) xn =?n[2 + (?2)n]
(3) x2k = k,x2k+1 = 1 + 1k(k = 1,2,3,···)
)μ
(1) α = 0￡¤§β = 1￡¤
(2)? lim
k→∞
bracketleftbig?2kparenleftbig2 + (?2)2kparenrightbigbracketrightbig=?∞,lim
k→∞
bracketleftbig?(2k+ 1)parenleftbig2 + (?2)2k+1parenrightbigbracketrightbig= +∞§{x
n}?t!e(..
(3)? lim
k→∞
x2k = lim
x→∞
k = +∞§{xn}?t(.?
q?x2k greaterorequalslant 1,k = 1,2,3,··· ;x2k+1 > 1?min{x2k} = 1§inf{xn} = 1￡¤.
6,y2μüN~ke.ê7k4?.
y2μdu{yn}ke.§{yn}7ke(..
de(.kμ(i)yn greaterorequalslantα(n = 1,2,3,···)?(ii)é?ε> 0§?kyN ∈{yn}§|yN <α+ε.
du{yn}′üN~ê§n > N?§kyn < α+ε§=n > N?§k0 lessorequalslant yn?α < ε§u′yn →
α(n→∞).
l
üN~ke.ê7k4?.
7,ám@?n^?μeò4?mU?m?m§(JX?oeò^?[a1,b1]? [a2,b2]? ···K?ò^
bn?an → 0K§(JNoáT~`2.
)μ
(1) 3?m@?n￥§eò4?mU?m?m§=
(i) (an+1,bn+1)? (an,bn)?
(ii) lim
n→∞
(bn?an) = 0
K?±y2{an},{bn}Euó?4?ξ§= lim
n→∞
an = lim
n→∞
bn = ξ§d?ξ?U? ?áuù
m
m§=ξ /∈ (an,bn)(n∈Z+)§?=ξ?U(an,bn)ú:.
~μm?m
braceleftbigg
(0,1n)
bracerightbigg
§
(i)
parenleftbigg
0,1n+ 1
parenrightbigg
parenleftbigg
0,1n
parenrightbigg
(ii) lim
n→∞
parenleftbigg1
n?0
parenrightbigg
= lim
n→∞
1
n = 0?
an = 0 → 0(n→∞);bn = 1n → 0(n→∞)§Kξ = 0 /∈
parenleftbigg
0,1n
parenrightbigg
§=(¤á.
(2) eò^?[an+1,bn+1]? [an,bn]K§=?k^?bn?an → 0¤á§K?Uy{an}?{bn}.
~μ4?m
bracketleftbigg
n? 1n,n+ 1n
bracketrightbigg
′@,lim
n→∞
bracketleftbigg
n+ 1n?
parenleftbigg
n? 1n
parenrightbiggbracketrightbigg
= lim
n→∞
2
n = 0§
limn→∞
parenleftbigg
n+ 1n
parenrightbigg
lim
n→∞
parenleftbigg
n? 1n
parenrightbigg
.
?3ξ?{an},{bn}ú4?§=(¤á.
(3) eò^?bn?an → 0K§=?k^?[an+1,bn+1]? [an,bn]¤á.K?±y2{an},{bn}￡m
@?ny2?¤§?Uy lim
n→∞
bn = lim
n→∞
an¤á§l
[an,bn]ú:§$yú
?m.
~μ4?m
bracketleftbigg
1? 1n+ 1,2 + 1n+ 1
bracketrightbigg
bracketleftbigg
1? 1n,2 + 1n
bracketrightbigg
,n∈Z+§? lim
n→∞
bracketleftbigg
2 + 1n?
parenleftbigg
1? 1n
parenrightbiggbracketrightbigg
= 1.
d lim
n→∞
an = 1,lim
n→∞
bn = 2§[1,2]?
bracketleftbigg
1? 1n,2 + 1n
bracketrightbigg
,n∈Z+§=(¤á.
8,e{xn}?.§t§K73ü?fx(1)nk →∞,x(2)nk →a(a?,k?ê).
y2μky
braceleftBig
x(1)nk
bracerightBig
′t.
du{xn}?.§é￠êM > 0§?knprime ∈Z+§||xnprime|>M.
M = 1§K73n1§|
vextendsinglevextendsingle
vextendsinglex(1)n1
vextendsinglevextendsingle
vextendsingle > 1?M = 2§K73n2§|
vextendsinglevextendsingle
vextendsinglex(1)n2
vextendsinglevextendsingle
vextendsingle > 2?···?M = K§K7
44
3nK >nK?1§|
vextendsinglevextendsingle
vextendsinglex(1)nK
vextendsinglevextendsingle
vextendsingle>K§···.
K??f
braceleftBig
x(1)nk
bracerightBig
§é?M ∈Z+§K = M§Kk>K?§òk
vextendsinglevextendsingle
vextendsinglex(1)nk
vextendsinglevextendsingle
vextendsingle>M§k limk→∞x(1)nk = ∞.
d?{xn}?′t§Kd§?M0 > 0§é?N ∈ Z+§?km ∈ Z+§m > N?§
k|xm|<M0.
yN = m0parenleftbigm0 ∈Z+parenrightbig§K?km1 >m0§||xm1|lessorequalslantM0
2N = m1§K?km2 >m1§||xm2|lessorequalslantM0§···
Xd?1e§K??mtμm1 < m2 < ··· < mt < ···§||xmt| lessorequalslant M0§=f{xmt}?|xmt| lessorequalslant
M0(mt ∈Z+)§ù`2f{xmt}k.§d5?n§k.f{xmt}7kf.
Pùf?{x(2)nk}§§?′{xn}f?§ua.= lim
k→∞
x(2)nk = a￡a?,k?ê¤.
9,k.ê{xn}e§K73ü?fx(1)nk →a,x(2)nk →b(anegationslash= b).
y2μdu{xn}k.§Kd5?n§7kfx(1)nk →a.
du{xn}§3ε0 > 0§3(a?ε0,a+ε0) k{xn}§¤{xn}f§P?
braceleftBig
x(2)n
bracerightBig
.
du
braceleftBig
x(2)n
bracerightBig
k.§3fx(2)nk →b§w,anegationslash= b.
10,e3?m[a,b]￥ü?ê
braceleftBig
x(1)n
bracerightBig
9
braceleftBig
x(2)n
bracerightBig
÷vx(1)n?x(2)n → 0(n→∞)§K3düê￥Ué?k?óv
Inkf§|x(1)nk →x0,x(1)nk →x0(k→∞).
y2μ?
braceleftBig
x(1)n
bracerightBig
[a,b]§K
braceleftBig
x(1)n
bracerightBig
k.ê§Kd5?n§
braceleftBig
x(1)n
bracerightBig
7kf§P?
braceleftBig
x(1)nk
bracerightBig
§
lim
k→∞
x(1)n = x0.
3
braceleftBig
x(2)n
bracerightBig
￥
braceleftBig
x(1)nk
bracerightBig
k?óvIf
braceleftBig
x(2)nk
bracerightBig
.
x(1)n?x(2)n → 0(n→∞)§K lim
k→∞
parenleftBig
x(1)nk?x(2)nk
parenrightBig
= 0§
u′ lim
k→∞
x(2)nk = lim
k→∞
bracketleftBig
x(1)nk?
parenleftBig
x(1)nk?x(2)nk
parenrightBigbracketrightBig
= lim
k→∞
x(1)nk? lim
k→∞
parenleftBig
x(1)nk?x(2)nk
parenrightBig
= x0?0 = x0.
11,|^?üneê5μ
(1) xn = a0 +a1q+a2q2 +···+anqn(|q|< 1,|ak|lessorequalslantM)
(2) xn = 1 + sin12 + sin222 +···+ sinn2n
(3) xn = 1? 12 + 13?···+ (?1)n+1 1n
y2μ
(1) n>m§K|xn?xm| =vextendsinglevextendsingleam+1qm+1 +am+1qm+1 +···+anqnvextendsinglevextendsinglelessorequalslantMparenleftbig|q|m+1 +|q|m+2 +···+|q|nparenrightbig=
M|q|m+1 1?|q|
n?m
1?|q| <M|q|
m+1 1
1?|q| → 0(m→∞)
é?ε> 0,?N ∈Z+§n>m>N?§kM|q|m+1 11?|q| <ε§l
k|xn?xm|<ε.
d?ün§{xn}7.
(2) m > n§é?ε > 0￡ε < 12¤§du|xm?xn| =
vextendsinglevextendsingle
vextendsinglevextendsinglesin(n+ 1)
2n+1 +
sin(n+ 2)
2n+2 +···+
sinm
2m
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
1
2n+1 +
1
2n+2 +···+
1
2m =
1
2n+1
parenleftbigg
1 + 12 +···+ 12m?n?1
parenrightbigg
= 12n+1
1?
parenleftbigg1
2
parenrightbiggm?n
1? 12
< 12n§e?|xm?
xn|<ε§ 12n <ε=?.
N =
lnε
ln 12
∈Z+§m>n>N?§k|x
m?xn|<ε.
d?ün§{xn}7.
￡?μ3￡1¤￥-a0 = 1,ak = sink,q = 12§Kd￡1¤=￡2¤¤.
(3) é?ε> 0§é?k∈Z+§du|xn+k?xn| =
vextendsinglevextendsingle
vextendsinglevextendsingle(?1)n+2
n+ 1 +
(?1)n+3
n+ 2 +···+
(?1)n+k?1
n+k
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle 1
n+ 1?
1
n+ 2 +···+
(?1)k?1
n+k
vextendsinglevextendsingle
vextendsinglevextendsingle=
1
n+ 1?
parenleftbigg 1
n+ 2?
1
n+ 3 +···+
(?1)k
n+k
parenrightbigg
< 1n+ 1 < 1n§e?|xn+k?xn|<ε§1n <ε=?.
45
N =
bracketleftbigg1
ε
bracketrightbigg
§Kn+k>n>N?§k|xn+k?xn|<ε.
d?ün§{xn}7.
12,|^k?CX?ny2?dA.d?n.
y2μ{xn}?k.ê§K73a,b§|alessorequalslantxn lessorequalslantb.
^?y{"b{xn}f§Kéx0 ∈ [a,b]§?kε0 > 0§|3O(x0,ε0)￥?1
k{xn}k.
Ké?ε> 0§3O(x0,ε)￥1k{xn}.
εn = 1n§w,3O(x0,εn)￥?1k{xn}§K3{xn}￥??μxn1 ∈ O(x0,1)§q?
xn2 ∈O
parenleftbigg
x0,12
parenrightbigg
(n2 >n1)§Xd?1e§?{xn}f{xnk},|xnk?x0|< 1k§é?M ∈Z+§
K = M§Kk>K?§òk|xnk?x0|< 1k < 1K < 1M§Kxnk →x0(k→∞)ù?bg?.
dx0 ∈ [a,b]5§é[a,b]￥z?:?kù?§|d1{xn}k§¤kù
?
¤[a,b]mCX.
dk?CX?n§K3kCX[a,b]§?
[a,b]1k{xn}k§ùxn ∈ [a,b]g
§b?¤á§K{xn}7kf.
13,|^?dA.d?ny2üNk.ê7k4?.
y2μ{xn}?üNO\k.ê§x1 lessorequalslantx2 lessorequalslant···lessorequalslantxn lessorequalslant···lessorequalslantM
a?dA.d?n§3f{xnk},lim
k→∞
xnk = a.
eyμ lim
n→∞
xn = a.
kyxn lessorequalslanta,n = 1,2,···.e?,§?N ∈Z+§|xN >a.
dunk →∞(k→∞)§k§7knk >N§l
xnk greaterorequalslantxN >a§u′a = lim
k→∞
xnk greaterorequalslantxN >ag?.
2y lim
n→∞
xn = a.
é?ε> 0,?k0§|
vextendsinglevextendsingle
vextendsinglexnk0?a
vextendsinglevextendsingle
vextendsingle= a?xnk0 <ε.
N = nk0§Kn>N?§kxn greaterorequalslantxnk0 = xN§l
k|a?xn| = a?xn lessorequalslanta?xnk0 <ε§ lim
n→∞
xn = a.
=üNO\k.ê7k4?.
ón?§üN~k.ê7k4?§l
üNk.ê7k4?.
14,(1) y2üNk.?ê3?!m4
(2) y2üNk.?êY:1?aY:.
y2μ
(1) d??f(x)3(a,b)tüNO\k.§?x0 ∈ (a,b)§β(x0) = sup
a<x<x0
f(x)§
dt(.§é?ε> 0§?kxprime ∈ (a,x0)§|f(xprime) >β(x0)?ε=f(xprime) +ε>β(x0)
δ = x0?xprime > 0§?f(x)3(a,b)tüNO\§δ>x0?x> 0=xprime <x?§kf(xprime) <f(x)§u′
kf(x) +ε>β(x0)=0 lessorequalslantβ(x0)?f(x) <ε§l
|β(x0)?f(x)|<ε
`2 lim
x→x0?0
f(x) = β(x0).=f(x)3x03?4?.
ón?§f(x)3(a,b)tüN~k.?§f(x)3x03?4?§l
üNk.?ê3?4?.
ón?§üNk.?ê3m4?.
(2) x0?f(x)Y:§Kd￡1¤(?f (x0?0)úf (x0 + 0)3§d?f (x0?0) negationslash= f (x0 + 0)"
K§f (x0?0) = f (x0 + 0)§df(x)üN5§7kf (x0) = f (x0?0) = f (x0 + 0).
ù`2x0′?Y:§g?§f (x0?0) negationslash= f (x0 + 0)§l
x0′f(x)1?aY:.
15,y2 lim
x→+∞
f(x)37?^?′μéε > 0§3X > 0§xprime,xprimeprime > X?ek|f(xprime)?
f(xprimeprime)|<ε.
y2μ lim
x→+∞
f(x)3§ lim
x→+∞
f(x) = A.
é?ε> 0,?X > 0§x>X?§k|f(x)?A|< ε2
xprime,xprimeprime > X?§k|f(xprime)?A| < ε2,|f(xprimeprime)?A| < ε2§K|f(xprime)?f(xprimeprime)| = |f(xprime)?A? (f(xprimeprime)?A)| lessorequalslant
|f(xprime)?A|+|f(xprimeprime)?A|<ε§l
éε> 0§3X > 0§xprime,xprimeprime >X?ek|f(xprime)?f(xprimeprime)|<ε.
3f(x)S§àê{xn}§|xn → +∞(n→∞)
d?§é?ε> 0,?X > 0§xprime,xprimeprime >X?§ek|f(xprime)?f(xprimeprime)|<ε.
q?xn → +∞§u′ét?X > 0§N ∈ Z+§n > N?§kxn > X§l
n,m > N?§ò
kxn >X,xm >X§?
k|f(xn)?f(xm)|<ε.
46
d?ün§ lim
n→∞
f(xn)3§ lim
n→∞
f(xn) = A
dxn59?ê4ê4?'X§ lim
x→+∞
f(x) = A= lim
x→+∞
f(x)3.
16,y2 lim
x→x0
f(x)37?^?′μéε> 0§3δ > 0§0 < |xprime?x0| <δ,0 < |xprimeprime?x0| <
δ?§ek|f(xprime)?f(xprimeprime)|<ε.
y2μ lim
x→x0
f(x)3§ lim
x→x0
f(x) = A.
é?ε> 0,?δ> 0§0 <|x?x0|<δ?§k|f(x)?A|< ε2
0 < |xprime?x0| < δ,0 < |xprimeprime?x0| < δ?§k|f(xprime)?A| < ε2,|f(xprimeprime)?A| < ε2§K|f(xprime)?f(xprimeprime)| =
|f(xprime)?A? (f(xprimeprime)?A)| lessorequalslant |f(xprime)?A| + |f(xprimeprime)?A| < ε§l
éε > 0§3δ > 0§0 <
|xprime?x0|<δ,0 <|xprimeprime?x0|<δ?§ek|f(xprime)?f(xprimeprime)|<ε.
3f(x)S§àê{xn}§|xn →x0?xn negationslash= x0(n→∞)
d?§é?ε > 0,?δ > 0§xprime,xprimeprime ∈ D(f)§?0 < |xprime?x0| < δ,0 < |xprimeprime?x0| < δ?§òk|f(xprime)?
f(xprimeprime)|<ε.
q?xn →x0,xn negationslash= x0(n→ ∞)§u′ét?δ > 0§N ∈Z+§n>N?§k0 <|xn?x0|<δ§l
n,m>N?§òk0 <|xn?x0|<δ,0 <|xm?x0|<δ§?
k|f(xn)?f(xm)|<ε.
dê?ün§ lim
n→∞
f(xn)3§ lim
n→∞
f(xn) = A
d{xn}′±x0?4?ê?xn negationslash= x0 9?ê4ê4?'X§ lim
x→x0
f(x) = A= lim
x→x0
f(x)
3.
17,y2f(x)3x0:?Y7?^?′μéε> 0§3δ> 0§|xprime?x0|<δ,|xprimeprime?x0|<δ?§e
k|f(xprime)?f(xprimeprime)|<ε.
y2μf(x)3x0:?Y§Ké?ε> 0,?δ> 0§|x?x0|<δ?§k|f(x)?f(x0)|< ε2
|xprime? x0| < δ,|xprimeprime? x0| < δ?§k|f(xprime)? f(x0)| < ε2,|f(xprimeprime)? f(x0)| < ε2§K|f(xprime)? f(xprimeprime)| =
|f(xprime)?f(x0)? (f(xprimeprime)?f(x0))| lessorequalslant |f(xprime)?f(x0)| + |f(xprimeprime)?f(x0)| < ε§l
éε > 0§
3δ> 0§|xprime?x0|<δ,|xprimeprime?x0|<δ?§ek|f(xprime)?f(xprimeprime)|<ε.
xprime = x0,xprimeprime = x§Kd?§é?ε> 0,?δ> 0§|x?x0|<δ?§òk|f(x)?f(x0)|<ε.
l
f(x)3x0:?Y.
47
§2,4?mt?Y?ê5?y2
1,y2μeüNk.?êf(x)?f(a),f(b)?m§Kf(x)3[a,b]?Y.
y2μf(x)?üNO\k.?ê.
d ù§1,14K￡1¤§f(x)3[a,b]à:a(b)?m￡?¤4?3§d?f(a) = f(a+ 0)(f(b) = f(b?
0))§
e?,§7kf(a) <f(a+ 0) = inf
a<x<b
f(x)(f(b) >f(b?0) = sup
a<x<b
f(x))§u′df(x)?f(a)?f(b)?
m§éf(a) < y < f(a + 0)(f(b? 0) < y < f(b))§7kx ∈ (a,b)§|f(x) = y§d
f(a+ 0) = inf
a<x<b
f(x)(f(b?0) = sup
a<x<b
f(x))g?.
dd?f(x)3a(b)m￡?¤?Y.
ekx0 ∈ (a,b)§|f(x)3x0:Y"d§1,14(2)(?§x07?1?am?:§=f(x0 + 0)úf(x0?
0)3§f(x0 + 0) negationslash= f(x0?0).
q?f(x)?üNO?ê§f(x0?0) lessorequalslantf(x0) <f(x0 + 0)?f(x0?0) <f(x0) lessorequalslantf(x0 + 0)§ù?f(x)?
(f(x0?0),f(x0 + 0))?méuf(x0)?§ùg?§b?¤á.
u′f(x)3[a,b]?Y.
ón§f(x)?üN~k.?ê?§f(x)3[a,b]?Y.
l
f(x)3[a,b]?Y.
2,y2μ?êf(x)3(a,b)?Y§f(a+ 0),f(b?0)3§Kf(x)?f(a+ 0)úf(b?0)?m(?U?
uf(a+ 0),f(b?0)).
y2μduf(a+ 0),f(b?0)3§Kf(a) = f(a+ 0),f(b) = f(b?0).
qf(x)3(a,b)?Y§Kf(x)3[a,b]?Y§?
f(x)3[a,b]t7kMú??m.
2d0n§f(x)?±Múmm.
eM = f(a+ 0)(?f(b?0))§m = f(b?0)(?f(b?0))§ù?f(x)?(f(a+ 0),f(b?0))￥(
U? uf(a+ 0),f(b?0)).
eM >f(a+ 0)(?f(b?0))§m<f(b?0)(?f(b?0))§ù?f(x)?(f(a+ 0),f(b?0))￥(?
U uf(a+ 0),f(b?0)),f(x)?f(a+ 0)úf(b?0)?m(?U? uf(a+ 0),f(b?0)).
3,y2(a,b)t?Y?êY7?^?′μf(a+ 0),f(b?0)3.
y2μ?f(x)?(a,b)t?Y?ê
f(a+ 0),f(b?0)3§Kf(a) = f(a+ 0),f(b) = f(b?0)§u′f(x)3[a,b]?Y§Kdx÷?
n§f(x)3[a,b]tY§l
f(x)3(a,b)tY.
f(x)3(a,b)tY§Kd§é?ε > 0,?δ(ε) > 0§x1,x2 ∈ (a,b)?|x1?x2| < δ(ε)?§
k|f(x1)?f(x2)|<ε.
éa§0 <x1?a< δ(ε)2,0 <x2?a< δ(ε)2?§|x1?x2| = |(x1?a)?(x2?a)|lessorequalslant|x1?a|+|x2?a|<δ(ε)§
Kk|f(x1)?f(x2)|<ε.
d?ün§ lim
x→a+0
f(x)3§=f(a+ 0)3?k?.
ón§f(b?0)3?k?.
4,e?êf(x)3(?∞,+∞)tk?4?mt?Y§K§3(?∞,+∞)tk?m?mtY.
y2μ(a,b)?(?∞,+∞)tk?m?m§K[a,b]?(?∞,+∞)tk?4?m.
f(x)3[a,b]t?Y§Kdx÷?n§f(x)3[a,b]tY§?
f(x)3(a,b)tY.
d(a,b)5§f(x)3(?∞,+∞)tk?m?mtY.
5,?êf(x) = x23(?∞,+∞)9(?l,l)t(l> 0)′Yo
)μ
(1) f(x) = x23(?∞,+∞)tY.
x1 >x2 > 0§?x1,x2 ∈ (?∞,+∞),|f(x1)?f(x2)| = |x21?x22| = |x1+x2||x1?x2| = (x1+x2)(x1?
x2) > 2x2(x1?x2)§3ε0 > 0§é?η> 0§x2 = 2ε0η,x1 = x2 + η2§
w,kx1 >x2 > 0?|x1?x2| = η2 <η§|f(x1)?f(x2)|> 2x2(x1?x2) = 2· 2ε0η · η2 = 2ε0 >ε0§
l
f(x) = x23(?∞,+∞)tY.
(2) f(x) = x23(?l,l)(l> 0)tY.
f(x)3[?l,l](l> 0)t′?Y§Kdx÷?n§f(x)3[?l,l]tY§l
f(x) = x23(?l,l)t
Y.
6,ef(x)3(a,b)Sk§é(a,b)Sx§3x,??Ox§|f(x)3OxSk..ˉμf(x)3(a,b)S
′?k.oqeò(a,b)U?[a,b]§X?o
y2μ
48
(1) f(x)3(a,b)k..
~μ?.μf(x) = 1x3(0,1)Sk§?é?x∈ (a,b)?Y§7?ük.§=3x?Ox(O(x,δx))§
|§3Ox(O(x,δx))Sk.§§3(0,1)S?..
k.μf(x) = sinx3
parenleftBig
0,pi2
parenrightBig
k§é
parenleftBig
0,pi2
parenrightBig
Sx§3x,??Ox§|f(x)3OxSk
.?f(x)3
parenleftBig
0,pi2
parenrightBig
tk.§?0 <f(x) < 1.
(2) f(x)3[a,b]k..
f(x)3[a,b]Sk§Kμf(x)3(a?δ,a)f(a)§f(x)3(b,b+δ)f(b).
d?é[a,b]Sx§3x,??Ox§|f(x)3OxSk.§=?M > 0§||f(x)| lessorequalslant M§?
d3[a,b]tz?:?ù?￡?=m?m¤§ù
m?m N¤m?m8§§CX
[a,b].
dk?CX?n§3ù
m?m8￥7kkm?mCX
[a,b]§Pùkm?m?(x1?
δ1,x1 +δ1),(x2?δ2,x2 +δ2),···,(xk?δk,xk +δk)§?AM?OP?M1,M2,···,Mk§X8
M? = max{M1,M?2,···,Mk}.
é[a,b]t:x§d?mCXVg§3ùk?m?mO(xi,δi)(i = 1,2,···,k)￥?k1x§
P§?O(xi,δi)§?3ù?m?mt§k|f(x)|lessorequalslantMi§|f(x)|lessorequalslantMi lessorequalslantM?.
dux?[a,b]t:§K3[a,b]to¤á|f(x)|lessorequalslantM?§l
y2
f(x)3[a,b]tk..
7,y2(a,b)tY?ê7k..
y2:?f(x)?(a,b)tY?ê§KdSK3§f(x)3(a,b)t?Y?f(a+0),f(b?0)3§u′
μf(a) = f(a+ 0),f(b) = f(b?0)§Kf(x)3[a,b]t?Y§u′f(x)3[a,b]tk.§l
f(x)3(a,b)t
k..
8,Uy2§üY?êúEY.kˉμüY?êèX?o
y2μ
(1) f(x)?g(x)3mXtY.
f(x)3?mXtY§Kdé?ε> 0,?δ1 > 0§é?mXSü:xprime,xprimeprime§|xprime?xprimeprime|<
δ1§òk|f(xprime)?f(xprimeprime)|< ε2.
q?g(x)3?mXtY§Kdét?ε > 0,?δ2 > 0§é?mXSü:xprime,xprimeprime§|xprime?
xprimeprime|<δ2§òk|g(xprime)?g(xprimeprime)|< ε2.
δ = min{δ1,δ2}§K|xprime?xprimeprime|<δ?§k|f(xprime)+g(xprime)?(f(xprimeprime)+g(xprimeprime))| = |f(xprime)?f(xprimeprime)+(g(xprime)?
g(xprimeprime))|lessorequalslant|f(xprime)?f(xprimeprime)|+|g(xprime)?g(xprimeprime)|< ε2 + ε2 = ε.
l
f(x)3?mXtY.
(2) (i) e?mX?km§K(?¤á.
f(x),g(x)3?mXtY§KdtK(?§3~êL> 0,M > 0§||f(x)|<L,g(x) <
M(x∈X).
qd§?ε > 0,?δ1 > 0§é?mXSü:xprime,xprimeprime§|xprime?xprimeprime| < δ1§ò
k|f(xprime)?f(xprimeprime)|< ε2M.
ó§ét?ε > 0,?δ2 > 0§é?mXSü:xprime,xprimeprime§|xprime? xprimeprime| < δ2§òk|g(xprime)?
g(xprimeprime)|< ε2L.
δ = min{δ1,δ2}§K|xprime?xprimeprime|<δ?§òk|f(xprime)?f(xprimeprime)|< ε2M,|g(xprime)?g(xprimeprime)|< ε2Ló?¤
á.
dd?§|f(xprime)g(xprime)?f(xprimeprime)g(xprimeprime)| =
|[f(xprime)?f(xprimeprime)]g(xprime) +f(xprimeprime)[g(xprime)?g(xprimeprime)]| lessorequalslant |f(xprime)?f(xprimeprime)||g(xprime)| + |f(xprimeprime)||g(xprime)?g(xprimeprime)| <ε
2M ·M +L·
ε
2L =
ε
2 +
ε
2 = ε.
l
f(x)g(x)3?mXtY.
(ii) f(x),g(x)3(?∞,+∞)Y?§f(x)g(x)3(?∞,+∞)tY.
~μ
(a)Y.
f(x) = g(x) = x§?é?ε > 0§9x1,x2 ∈ (?∞,+∞)§δ = ε§|x1? x2| < δ?§
k|x1?x2|<ε§f(x) = g(x) = x3(?∞,+∞)tY.
f(x)g(x) = x2§d15K?f(x)g(x)3(?∞,+∞)tY.
(b)Y.
f(x) = 1§?é?ε> 0§éx1,x2 ∈ (?∞,+∞)§δ = ε§|x1?x2|<δ?§k|f(x1)?
f(x2)|<ε§f(x) = 13(?∞,+∞)tY.
49
g(x) = x§Kd?g(x) = x3(?∞,+∞)tY§?f(x)g(x) = x3(?∞,+∞)t
Y.
50
1? üCt?è
1?ü? üCt
1où ê
§1,êú
1,L-?y = x2tü:A(2,4)úB(2+?x,2+?y)??§?O|??x = 19?x = 0.1???§?|
-?3A:?.
)μkAB = (2 +?x)
2?22
x = 4 +?x
?x = 1?§kAB = 5??x = 0.1?§kAB = 4.1
-?3A:k = lim
x→0
kAB = lim
x→0
(4 +?x) = 4.
2,|y = x23A(1,1):ú3B(?2,4):§ú{§.
)μ?yprime = 2x§3:A(1,1)μk1 = 2§§?μy? 1 = 2(x? 1)=2x?y? 1 = 0?{§
y?1 =?12(x?1)=x+ 2y?3 = 0
3:B(?2,4)μk2 =?4§§?μy? 4 =?4(x + 2)=4x + y + 4 = 0?{§?y? 4 = 14(x +
2)=x?4y+ 18 = 0
3,ey = f(x) = x3§|
(1) L-?t:x0,x0 +?x???￡x0 = 2,?x?O?0.1§0.01§0.001¤?
(2) 3x = x0?-?.
)μ
(1)?k = f(x0 +?x)?f(x0)?x = (x0 +?x)
3?x3
x = 3x
2
0 + 3x0?x+ (?x)
2§
μ?x = 0.1?§k = 12.61??x = 0.01?§k = 12.0601??x = 0.001?§k = 12.006001.
(2) fprime(x) = lim
x→0
f(x0 +?x)?f(x0)
x = 3x
2§
u′fprime(x0) = 3x20
4,es = vt? 12gt2§|
(1) 3t = 1,t = 1 +?t?m2t?Y￡?t = 1,0.1,0.01¤?
(2) 3t = 1]Y.
)μ
(1)?ˉv =
v(1 +?t)? 12g(1 +?t)2?
parenleftbigg
vt? 12gt2
parenrightbigg
t = v?g?
1
2g?t
2§
μ?t = 1?§ˉv = v? 32g??t = 0.1?§ˉv = v? 2120g??t = 0.01?§ˉv = v? 201200g.
(2) 3t = 1]Yv = lim
t→0
ˉv = v?g.
5,y = x23=?:21uy = 4x?5o3=?:R?u2x?6y+ 5 = 0o
)μy = 4x? 5k = 4§Kdfprime(x) = 2x = k§x = 2§=(2,4):21uy =
4x?5?
2x?6y + 5 = 0k = 13§Kdfprime(x) = 2x =?1k =?3§x =?32§=(?32,94):R?
u2x?6y+ 5 = 0.
6,|e?ê3¤?:?y?xμ
(1) y = √x(x = 2,?x = 0.01)
(2) y = 1x(x = 4,?x = 0.04)
51
)μ
(1)?y?x =
√x+?x?√x
x =
√2.01?√2
0.01 = 100
parenleftBig√
2.01?√2
parenrightBig
= 1√2.01 +√2
(2)?y?x =
1
x+?x?
1
x
x =?
1
x(x+?x) =?
1
4(4 + 0.04) =?
25
404
7,y2μ
(1)?(f(x)±g(x)) =?f(x)±?g(x)
(2)?[f(x)·g(x)] = g(x+?x)·?f(x) +f(x)·?g(x)
y2μ
(1)?(f(x)±g(x)) = [f(x+?x)±g(x+?)]?[f(x)±g(x)] = [f(x+?x)?f(x)]±[g(x+?x)?g(x)] =
f(x)±?g(x)
(2)?[f(x)·g(x)] = f(x+?x)·g(x+?x)?f(x)·g(x) = f(x+?x)·g(x+?x)?f(x)·g(x+?x)+f(x)·g(x+
x)?f(x)·g(x) = [f(x+?x)?f(x)]·g(x+?x)+f(x)[g(x+?x)?g(x)] = g(x+?x)·?f(x)+f(x)·?g(x)
52
§2,{ü?êê
1,dê|y = cosxê.
)μyprime = lim
x→0
cos(x+?x)?cosx
x = lim?x→0
2sin 2x+?x2 sin?x2
x =? lim?x→0sin
parenleftbigg
x+?x2
parenrightbiggsin?x
2
x
2
=
sinx§=(cosx)prime =?sinx.
2,dê|y = 3√xê.
)μyprime = lim
x→0
3√x+?x? 3√x
x = lim?x→0
x13
bracketleftBiggparenleftbigg
1 +?xx
parenrightbigg1
3?1
bracketrightBigg
x = lim?x→0
x?23
bracketleftBiggparenleftbigg
1 +?xx
parenrightbigg1
3?1
bracketrightBigg
x
x
= x
23
3 =
1
3 3√x2§=(
3√x)prime = 1
3 3√x2
3,Uy2μ?ó?êù?ê′ê§?êù?ê′ó?ê.
y2μf(x)ó?ê§Kf(?x) = f(x)?g(x)ê§Kg(?x) =?g(x)
u′fprime(?x) = lim
x→0
f(?x+?x)?f(?x)
x = lim?x→0
f(xx)?f(x)
x = limx→0
[f(xx)?f(x)]
x =
fprime(x)=?ó?êù?ê′ê?
gprime(?x) = lim
x→0
g(?x+?x)?g(?x)
x = lim?x→0
g(xx) +g(x)
x = limx→0
g(xx)?g(x)]
x = g
prime(x)=?
êù?ê′ó?ê.
4,Uy2μ?±ê§ù?êE?±ù?ê.
y2μf(x)±T?ê§Kf(x+T) = f(x)§
u′fprime(x+T) = lim
x→0
f(x+T +?x)?f(x+T)
x = lim?x→0
f(x+?x)?f(x)
x = f
prime(x)=?±ê§
ù?êE?±ù?ê.
53
§3,|{K
1,|^?2êúa§|e?êêμ
(1) y = x5
(2) y = x11
(3) y = x6
(4) y = 2x
(5) y = log10x
(6) y = 10x
)μ
(1) yprime = (x5)prime = 5x4
(2) yprime = (x11)prime = 11x10
(3) yprime = (x6)prime = 6x5
(4) yprime = (2x)prime = 2x ln2
(5) yprime = (log10x)prime = 1xln10
(6) yprime = (10x)prime = 10x ln10
2,|e?êêμ
(1) f(x) = 2x2?3x+ 1§?|fprime(0),fprime(1)
(2) f(x) = x5 + 3sinx§?|fprime(0),fprime
parenleftBigpi
2
parenrightBig
(3) f(x) = ex + 2cosx?7x§?|fprime(0),fprime(pi)
(4) f(x) = 4sinx?lnx+x2
(5) f(x) = anxn +an?1xn?1 +···+a1x+a0§?|fprime(0),fprime(1)
)μ
(1) fprime(x) = 4x?3§fprime(0) =?3,fprime(1) = 1
(2) fprime(x) = 5x4 + 3cosx§?|fprime(0) = 3,fprime
parenleftBigpi
2
parenrightBig
= 5pi
4
16
(3) fprime(x) = ex?2sinx?7§?|fprime(0) =?6,fprime(pi) = epi?7
(4) fprime(x) = 4cosx? 1x + 2x
(5) f(x) = nanxn?1 + (n1)an?1xn?2 +···+a1§?|fprime(0) = a1,fprime(1) =
nsummationtext
i=1
iai
3,|e?êêμ
(1) y = x2 sinx§?|fprime(0),fprime
parenleftBigpi
2
parenrightBig
(2) y = xcosx+ 3x2§?|fprime(?pi)úfprime(pi)
(3) y = xtanx+ 7x?6
(4) y = ex sinx?7cosx+ 5x2
(5) y = 4√x+ 1x?2x3
(6) y = (3x2 + 2x?1)sinx
)μ
(1) yprime = 2xsinx+x2 cosx§fprime(0) = 0,fprime
parenleftBigpi
2
parenrightBig
= pi
(2) yprime = cosx?xsinx+ 6x§fprime(?pi) =?1?6pi,fprime(pi) =?1 + 6pi
(3) yprime = tanx+xsec2x+ 7
(4) yprime = ex sinx+ex cosx+ 7sinx+ 10x = ex(sinx+ cosx) + 7sinx+ 10x
54
(5) yprime = 2√x? 1x2?6x2
(6) yprime = (3x2 + 2x?1)cosx+ (6x+ 2)sinx
4,|e?êêμ
(1) y = 2 + sinxx
(2) y = cotx
(3) y = 3x
2 + 7x?1
√x
(4) y = (1 +x
2)sinx
2x
(5) y = xlnx1 +x
(6) y = xe
x?1
sinx
)μ
(1) yprime = x(2 + sinx)
prime?(x+ sinx)
x2 =
xcosx?sinx?2
x2
(2) yprime =
parenleftBigcosx
sinx
parenrightBigprime
= sinx(cosx)
prime?cosx(sinx)prime
sin2x =?
1
sin2x =?csc
2x
(3) yprime =
√x(3x2 + 7x?1)prime?(√x)prime(3x2 + 7x?1)
x =
√x(6x+ 7)? 3x2 + 7x?1
2√x
x =
9x2 + 7x+ 1
2x√x =
9
2x
1
2 + 72x?
1
2 + 12x?
3
2
(4) yprime = 2x[(1 +x
2)sinx]prime?2(1 +x2)sinx
4x2 =
2x[2xsinx+ (1 +x2)cosx]?2(1 +x2)sinx
4x2 =
(x2?1)sinx+x(1 +x2)cosx
2x2
(5) yprime = (1 +x)(xlnx)
prime?xlnx
(1 +x)2 =
(1 +x)(lnx+ 1)?xlnx
(1 +x)2 =
x+ lnx+ 1
(1 +x)2
(6) yprime = sinx(xe
x?1)prime?(sinx)prime(xex?1)
sin2x =
ex sinx(x+ 1)?cosx(xex?1)
sin2x
5,|e?êêμ
(1) y =
√x+ cosx
x?1?7x
2
(2) y = xsinx+ cosxxsinx?cosx
(3) y = x2ex sinx+ 3 +x
2
√x?xlnx+ 8x2
(4) y = sinx1 + tanx
(5) y = xcosx?lnxx+ 1
(6) y = 1x+ cosx
)μ
(1) yprime =
(x?1)( 12√x?sinx)?(√x+ cosx)
(x?1)2? 14x =
(x?1)(1?2√xsinx)?(2x+ 2√xcosx)
2√x(x?1)2? 14x
(2) yprime = (xsinx?cosx)(sinx+xcosx?sinx)?(xsinx+ cosx)(sinx+xcosx+ sinx(xsinx?cosx)2 =?2(sinxcosx+x)(xsinx?cosx)2 =
2x+ sin2x(xsinx?cosx)2
55
(3) yprime = 2xex sinx+x2ex sinx+x2ex cosx+
2x√x? 3 +x
2
2√x
x? lnx? 1 + 16x = xe
x(2sinx+xsinx+
xcosx) + 3x
2?1
2x√x?lnx?1 + 16x
(4) yprime = cosx(1 + tanx)?sinx·sec
2x
(1 + tanx)2
(5) yprime =
(x+ 1)(cosx?xsinx? 1x)?(xcosx?lnx)
(x+ 1)2 =
xcosx?(x2 sinx+ 1)(x+ 1) +xlnx
x(x+ 1)2
(6) yprime =? 1?sinx(x+ cosx)2 = sinx?1(x+ cosx)2
6,|-?y+ 1 = (x?2)33:A(3,0)?§9{§.
)μ?y+ 1 = (x?2)3§Ky = (x?2)3?1§u′yprime = 3(x?2)2§K¤|k = y prime|x=3 = 3§
l
¤|§?μy = 3(x?3)=3x?y?9 = 0?¤|{§?μy =?13(x?3)=x+ 3y?3 = 0.
7,|-?y = lnx3:(1,0)?§ú{§.
)μ?y = lnx§Kyprime = 1x§u′¤|k = y prime|x=1 = 1§
l
¤|§?μy = x?1=x?y?1 = 0?¤|{§?μy =?(x?1)=x+y?1 = 0.
8,y = x2?2x+ 43=?:21ux?o3=?:x??45oo
)μ?y = x2?2x+ 4§yprime = 2x?2.
q21ux?k = 0§K2x?2 = 0§u′x = 1§=¤|:?(1,3)?
q?x??45ok = 1§K2x?2 = 1§u′x = 32§=¤|:?
parenleftbigg3
2,
13
4
parenrightbigg
.
9,÷$??N§ù$§?s = 3t4? 20t3 + 36t2§|ù?Y§?ˉ?Nc$?o $
o
)μ?s = 3t4?20t3 + 36t2§v = sprime = 12t3?60t2 + 72t.
v> 0=0 <t< 2?t> 3?§?N?c$v< 0=2 <t< 3?§?N? $?.
10,du ^§?￥÷XtE§D?Y?5§$§?s = 5t?t2§áˉd￥meEo
)μ?s = 5t?t2§v = sprime = 5?2t§v = 0=t = 52?§￥meE.
11,3x = 2?§?-?y = 0.1x3§áˉ: §d-3o
)μ?y = 0.1x3§yprime = 0.3x2§u′3x = 2?§k = y |x = 2 = 1.2§l
d-?3?
:(2,0.8)?§?y?0.8 = 1.2(x?2)§=6x?5y?8 = 0?d
braceleftbigg y = 0.1x3
6x?5y?8 = 0 §x
3?12x+16 =
0§K(x?2)2(x+ 4) = 0§)x1 = x2 = 2,x3 =?4§Kd-3:(?4,?6.4).
12,-?y = xn￡n?ê¤t:(1,1)?x?u:(ξn,0)§| lim
n→∞
y(ξn).
)μ?y = xn§Kyprime = nxn?1§Kd-?3x = 1?k = y prime|x=1 = n§u′d-?3:(1,1)?
§?y?1 = n(x?1)=y = nx?n+ 1.
y = 0?§x = n?1n =ξn = n?1n §K lim
n→∞
y(ξn) = lim
n→∞
parenleftbiggn?1
n
parenrightbiggn
= lim
n→∞
parenleftbigg
1? 1n
parenrightbiggn
= 1e.
13,§?y = x2 +ax+b§áˉ:(x0,y0)?u§?±l:(x0,y0)édü^
^§o
)μ(x0,y0)?2?t:§(x,y)?L(x0,y0):.
d?§?k = yprime = 2x+a§K¤|y?y0 = (2x+a)(x?x0)=y0?y =
(2x?a)(x0?x)§
qy = x2+ax+b§Ky0?(x2+ax+b) = (2x+a)(x0?x)§x2?2x0x+y0?ax0§K? = 4x20?4(y0?b?ax0)
? > 0=y0 < x20 + ax0 + b?§ü^? = 0=y0 = x20 + ax0 + b?§^
? < 0=y0 >x20 +ax0 +b?§.
14,ˉ.êao§y = xaU?éê-?y = logaxo3o
)μdK?§xprime = (logax)prime§=1 = 1xlna§Kx = 1lna§u′y = 1lna.
qdu3?:§ùp?I7L? §K logax = 1lna§u′x = e§K?lna = 1e§=a = e1e=.
êa = e1e?§y = xaU?éê-?y = logax§3:(e,e).
56
§4,Eü?ê|{
1,|e?êêμ
(1) y = 2sin3x
(2) y = 4cos(3t?1)
(3) y = 3e2x + 5cos2x
(4) y = (x+ 1)2
(5) y = (1?x+x2)3
(6) y = 3e?2t + 1
(7) y = ln(x+ 1)
(8) y = (3x+ 1)4
(9) y = √1 +x2
(10) y =
parenleftbigg
1? 1x
parenrightbigg2
(11) y = tan x2 + sin3x
(12) y = lnsinx
(13) y = x√1 +x2
(14) y = 1√2pie?3t2
)μ
(1) yprime = 6cos3x
(2) yprime =?12sin(3t?1)
(3) yprime = 6e2x?10sin2x
(4) yprime = 2(x+ 1)
(5) yprime = 3(1?x+x2)2(2x?1)
(6) yprime =?6e?2t
(7) yprime = 1x+ 1
(8) yprime = 12(3x+ 1)3
(9) yprime = x√1 +x2
(10) yprime = 2
parenleftbigg
1? 1x
parenrightbigg
·
parenleftbigg
1x2
parenrightbigg
= 2(x?1)x3
(11) yprime = 12 sec2 x2 + 3cos3x
(12) yprime = cosxsinx = cotx
(13) yprime =
√1 +x2? x2√
1 +x2
1 +x2 =
1
(1 +x2)32
(14) yprime =?3
√2t
√pi e?3t2
2,|e?êêμ
(1) y = sin3 2x
(2) y = (at+b)e?2t(a,b?~ê)
(3) y = e2t sin3t+ t
2
2
57
(4) y = ln 1?x
2
1 +x2
(5) y = e
kt sinωt
1 +t (k,ω?~ê)
(6) y = 4(x+ cos2x)2
(7) y = e?t(cost+ sint)
(8) y = x√1 + cos2x
(9) y = (x?1)√x2 + 1
(10) y = (2 + 3t)sin2t+ 7t2?7
)μ
(1) yprime = 6sin2 2xcosx = 3sin4xsin2x
(2) yprime = ae?2t?2(at+b)e?2t =?(2at+ 2b?a)e?2t
(3) yprime = 2e2t sin3t+ 3e2t cos3t+t = e2t(2sin3t+ 3cos3t) +t
(4) yprime = 1 +x
2
1?x2 ·
2x(1 +x2)?2x(1?x2)
(1 +x2)2 =
4x
x4?1
(5) yprime = (1 +t)e
kt(?ksinωt+ωcosωt)?(e?kt sinωt
(1 +t)2 =
(kt+k+ 1)e?kt sinωt+ω(1 +t)e?kt cosωt
(1 +t)2
(6) yprime =?4[(x+ cos2x)
2]prime
(x+ cos2x)4 =?
8(1?2sin2x)
(x+ cos2x)2
(7) yprime =?e?t(cost+ sint) +e?t(?sint+ cost) =?2e?t sint
(8) yprime =
√1 + cos2x?x?2sinxcosx
2√1 + cos2x
1 + cos2x =
1 + cos2x+xsinxcosx
(1 + cos2x)32
(9) yprime = √x2 + 1 + (x?1) 2x2√x2 + 1 = 2x
2?x+ 1
√x2 + 1
(10) yprime = 3sin2t+ 2(2 + 3t)cos2t+ 14t
3,|e?êêμ
(1) y = e?kt(3cosωt+ 4sinωt)(k,ω?~ê)
(2) y = xarctanx
(3) y = (2x2 + 1)2e?x sin3x
(4) y = e
t sin3t
√1 +t2
(5) y = (3t+ 1)et(cos3t?7sin3t)
(6) y = tarcsin3t+ 7e?2t lnt+ 8t
(7) y = x√a2?x2 + x√a2?x2 (a?~ê)
)μ
(1) yprime =?ke?kt(3cosωt+4sinωt)+e?kt(?3ωsinωt+4ωcosωt) = e?kt[(4ω?3k)cosωt?(3ω+4k)sinωt]
(2) yprime = arctanx+ x1 +x2
(3) yprime = 4x(2x2 + 1)e?x sin3x? (2x2 + 1)2e?xsin3x+ 3(2x2 + 1)2e?xcos3x = e?x(2x2 + 1)[(?2x2 +
8x?1)sin3x+ 3(2x2 + 1)cos3x]
(4) yprime =
e?t(?sin3t+ 3cos3t)√1 +t2?e?t sin3t t√1 +t2
1 +t2 =
e?t[3(1 +t2)cos3t?(t2 +t+ 1)sin3t]
(1 +t2)32
58
(5) yprime = 3et(cos3t?7sin3t) + (3t+ 1)et(cos3t?7sin3t) + (3t+ 1)et(?3sin3t?21cos3t) =?et[(60t+
17)cos3t+ (30t+ 31)sin3t]
(6) yprime = arcsin3t+ 3t√1?9t2?14e?2t lnt+ 7e
2t
t + 8
(7) yprime = √a2?x2? x
2
√a2?x2 +
√a2?x2 + x2√
a2?x2
a2?x2 =
(a2?2x2)(a2?x2) +a2
(a2?x2)32
4,|e?êêμ
(1) y = sinnxcosnx
(2) y = sinhnxcoshnx
(3) y = e?x2+2x
(4) y = (sinx+ cosx)n
(5) y = arcsin(sinx·cosx)
(6) y = ln
radicalbigg(x+ 2)(x+ 3)
x+ 1
(7) y = arctan 2x1?x2
(8) y = xa2√a2 +x2
)μ
(1) yprime = nsinn?1xcosxcosnx?nsinnxsinnx =
nsinn?1xcos(n+ 1)x
(2) yprime = nsinhn?1xcoshxcoshnx+nsinhnxsinhnx =
nsinhnxcosh(n+ 1)x
(3) yprime =?2(x?1)e?x2+2x
(4) yprime = n(sinx+ cosx)n?1(cosx?sinx) = n(sinx+ cosx)n?2 cos2x
(5) yprime = cos2xradicalbig1?(sinx·cosx)2 = 2cos2x√4?sin22x
(6)?y = ln
radicalbigg(x+ 2)(x+ 3)
x+ 1 =
1
2[ln(x+2)+ln(x+3)?ln(x+1)]§y
prime = 1
2
bracketleftbigg 1
x+ 2 +
1
x+ 3?
1
x+ 1
bracketrightbigg
=
x2 + 2x?1
2(x+ 1)(x+ 2)(x+ 3).
(7) yprime = 1
1 +
parenleftbigg 2x
1?x2
parenrightbigg2 · 2(1?x
2) + 4x2
(1?x2)2 =
2
1 +x2
(8) yprime =
√a2 +x2? x2√
a2 +x2
a2(a2 +x2) =
1
(a2 +x2)32
5,|^éê2|?{§|e?êêμ
(1) y = x
radicalbigg1?x
1 +x
(2) y = x
2
1?x
radicalbigg x+ 1
1 +x+x2
(3) y = (x?α1)α1(x?α2)α2 ···(x?αn)αn
(4) y = (x+√1 +x2)n
(5) y = xmmx
)μ
59
(1)?y = x
radicalbigg1?x
1 +x§K lny = lnx+
1
2 ln(1?x)?
1
2 ln(1 +x)§ü>éx|§
1
yy
prime = 1
x +
1
2(1?x)?
1
2(1 +x)§
Kyprime = 1?x?x
2
(1 +x)√1?x2 (0 <|x|< 1)
(2)?y = x
2
1?x
radicalbigg x+ 1
1 +x+x2§K lny = 2lnx? ln(1?x) +
1
2 ln(x+ 1)?
1
2 ln(1 +x+x
2)§ü>éx|
§1yyprime = 2x + + 11?x + 12(1 +x)? 1 + 2x2(1 +x+x2)§
Kyprime = x
2
1?x
radicalbigg x+ 1
1 +x+x2
parenleftbigg2
x + 11?x+ 12(x+ 1)?
2x+ 1
2(1 +x+x2)
parenrightbigg
(3)?y = (x?α1)α1(x?α2)α2 ···(x?αn)αn =
nproducttext
i=1
(x?αi)αi9y3éê?òS§Aproducttext nlim
i=1
(x?αi)αi >
0§Klny =
nsummationtext
i=1
αi ln|x?αi|§ü>éx|ê§1yyprime =
nsummationdisplay
i=1
αi
x?αi§
Kyprime =
nsummationtext
i=1
αi
x?αi
nproductdisplay
i=1
(x?αi)αi(x∈D)ù￥D =
braceleftbigg nproducttext
i=1
(x?αi)αi > 0
bracerightbigg
(4)?y = (x + √1 +x2)n§K lny = nln(x + √1 +x2)§ü>éx|§1yyprime = n
1 + x√1 +x2
x+√1 +x2 =n
√1 +x2§Kyprime = n√1 +x2 (x+
radicalbig
1 +x2)n
(5)?y = xmmx§Klny = mln|x| +xlnm§ü>éx|§1yyprime = mx + lnm§Kyprime = xm?1mx+1 +
xmmx lnm
6,f(x)′éx?|?ê§|dydx.
(1) y = f(x2)
(2) y = f(ex)·ef(x)
(3) y = f(f(f(x)))
)μ
(1) dydx = 2xfprime(x2)
(2) dydx = exfprime(ex)·ef(x) +fprime(x)f(ex)ef(x) = ef(x)(exfprime(ex) +f(ex)fprime(x))
(3) dydx = fprime(f(f(x)))fprime(f(x))fprime(x)
7,?(x),ψ(x)?éx?|?ê§|dydx.
(1) y =radicalbig?2(x) +ψ2(x)
(2) y = arctan?(x)ψ(x)(ψ(x) negationslash= 0)
(3) y =?(x)radicalbigψ(x)(?(x) negationslash= 0,ψ(x) > 0)
(4) y = log?(x)ψ(x)(?(x) > 0,ψ(x) negationslash= 0)
)μ
(1) dydx =?(x)?
prime(x) +ψ(x)ψprime(x)
radicalbig?2(x) +ψ2(x)
(2) dydx =?
prime(x)ψ(x)?ψprime(x)?(x)
2(x) +ψ2(x)
(3) dydx =?(x)
radicalbig
ψ(x)
parenleftbigg ψprime(x)
(x)ψ(x)?
prime(x)lnψ(x)
2(x)
parenrightbigg
60
(4) dydx =
ψprime(x)
ψ(x) ln?(x)?
prime(x)
(x) lnψ(x)
(ln?(x))2 =
ψprime(x)
ψ(x)ln?(x)?
prime(x)lnψ(x)
(x)(ln?(x))2 =
log?(x)ψ(x)
bracketleftbigg ψprime(x)
ψ(x)lnψ(x)?
prime(x)
(x)ln?(x)
bracketrightbigg
8,|?4-7¤?-Y?\?w?$??Y.
)μ?s =
radicalbig
l2?r2 sin2ωt?rcosωt§v = sprime = rωsinωt? r
2ωsin2ωt
2
radicalbig
l2?r2 sin2ωt
.
9,|-?y = √1?x23x = 12?§ú{§.
)μ?yprime =? x√1?x2§K3x = 12?k =?
√3
3 §
u′¤|§?μy?
√3
2 =?
√3
3
parenleftbigg
x? 12
parenrightbigg
=x+√3y?2 = 0?
¤|{§?μy?
√3
2 =
√3parenleftbiggx? 1
2
parenrightbigg
=√3x?y = 0.
10,|-?y = e?xt?:§|LT:y =?ex21§??T:{§.
)μ?k = yprime =?e?x =?e§Kx =?1§KL(?1,e):y =?ex21§LT:{§
y?e = 1e(x+ 1)=x?ey+e2 + 1 = 0.
11,|-?y = √1?x2tY2.
)μ?k = yprime =? x√1?x2 = 0§Kx = 0§u′d-?3(0,1)?Y2§§?y = 1.
12,|-?y = 12(1 + 2x2 ±
radicalbig
1 + 4x2)tIx = U:?§.ù-?uo
)μ?yprime = 2x ± 2x√1 + 4x2§K-?3x = U?k = 2U ± 2U√1 + 4U2§u′d-?3?
:(U,12(1 + 2U2 ±
radicalbig
1 + 4U2))?§?y? 12(1 + 2U2 ±
radicalbig
1 + 4U2)) = (2U ± 2U√1 + 4U2 )(x?U)§
=2U(√1 + 4U2 ±1)x?√1 + 4U2y± 12 + 12(1?2U2)
radicalbig
1 + 4U2 = 0§d-?u?
U(
√1 + 4U2 ±1)
√1 + 4U2,12
1 + 2U2(
√1 + 4U2 ±1)2
1 + 4U2 ±
radicalBigg
1 + 4U
2(√1 + 4U2 ±1)2
1 + 4U2
.
13,y = f(x)3x0?§P?(t) = f(x0 +at)§a?~ê§|?prime(0).
)μea = 0§K?(t) = f(x0)§K?prime(0) = 0
eanegationslash= 0§K?prime(x) = lim
t→0
(x)(0)
t = limt→0
f(x0 +at)?f(x0)
t = alimt→0
f(x0 +at)?f(x0)
at = af
prime(x
0).
61
§5,9ù$?
1,|e?ê3:μ
(1) y = anxn +an?1xn?1 +···+a0§|dy(0),dy(1)
(2) y = secx+ tanx§|dy(0),dy
parenleftBigpi
4
parenrightBig
,dy(pi)
(3) y = 1a arctan xa§|dy(0),dy(a)
(4) y = 1x + 1x2§|dy(0.1),dy(0.01)
)μ
(1)?dy = [nanxn?1 + (n?1)an?1xn?2 +···+a1]dx§Kdy(0) = a1dx,dy(1) =
nsummationtext
i=1
iaidx
(2)?dy = (tanxsecx+ sec2x)dx§Kdy(0) = dx,dy
parenleftBigpi
4
parenrightBig
= (√2 + 2)dx,dy(pi) = dx
(3)?dy = dxa2 +x2dx§Kdy(0) = dxa2dx,dy(a) = dx2a2dx
(4)?y =?x+ 2x3 dx§Kdy(0.1) =?2100dx,dy(0.01) =?2010000dx
2,|e?êy = y(x)μ
(1) y = x? 12x2 + 13x3? 14x4
(2) y = x2 sinx
(3) y = x1?x2
(4) y = xlnx?x
(5) y = (1?x2)n
(6) y = √x+ lnx? 1√x
(7) y = lntanx
(8) y = sinaxcosbx
(9) y = eax cosbx
(10) y = arcsin√1?x2
)μ
(1) dy = (1?x+x2?x3)dx
(2) dy = (2xsinx+x2 cosx)dx
(3) dy = 1 +x
2
(1?x2)2dx
(4) dy = lnxdx
(5) dy =?2nx(1?x2)n?1dx
(6) dy = x+ 2
√x+ 1
x32
dx
(7) dy = 2sin2xdx
(8) dy = (acosaxcosbx?bsinaxsinbx)dx
(9) dy = eax(acosbx?bsinbx)dx
(10) dy =? x|x|√1?x2dx
3,|e?êyμ
(1) y = sin2t,t = ln(3x+ 1)
62
(2) y = ln(3t+ 1),t = sin2x
(3) y = e3u,u = 12 lnt,t = x2?2x+ 5
(4) y = arctanu,u = (lnt)2,t = 1 +x2?cotx
)μ
(1) dy = 3sin(2ln(3x+ 1))3x+ 1 dx
(2) y = 3sin2x3sin2x+ 1dx
(3) y = 3(3x
2?2)
2(x3?2x+ 5)e
3
2 ln(x
2?2x+5)dx
(4) y = 2ln(1 +x
2?cotx)(2x+ csc2x)
[1 + (ln(1 +x2?cotx))4](1 +x2?cotx)dx
4,eu,v,w?xê§|?êydyμ
(1) y = u·v·w
(2) y = u·wv2
(3) y = 1√u2 +v2
(4) y = ln√u2 +v2
(5) y = arctan uv
)μ
(1) dy = (uprime ·v·w+u·vprime ·w+u·v·wprime)dx
(2) dy = v
2(uprimew+uwprime)?2uvvprimew
v4 dx
(3) dy =? uu
prime +vvprime
(u2 +v2)32
dx(u2 +v2 > 0)
(4) dy = uu
prime +vvprime
u2 +v2 dx
(5) dy = u
primev?uvprime
u2 +v2 dx(v negationslash= 0)
63
§6,ê9?ê?§¤Lê|{
1,|eêêdydxμ
(1) x
2
a2 +
y2
b2 = 1§ù￥a,b?~ê
(2) y2 = 2px§ù￥p?~ê
(3) x2 +xy+y2 = a2§ù￥a?~ê
(4) x3 +y3?xy = 0
(5) y = x+ 12 siny
(6) x23 +y23 = a23§ù￥a?~ê
(7) y?cos(x+y) = 0
(8) y = x+ arctany
(9) y = 1?ln(x+y) +ey
(10) arctan yx = ln
radicalbig
x2 +y2
)μ
(1) 3?§üàéx|ê§?5?y′x?ê§òk2xa2 + 2yy
prime
b2 = 0§Ky
prime =?b
2x
a2y(y negationslash= 0).
(2) 3?§üàéx|ê§?5?y′x?ê§òk2yyprime = 2p§Kyprime = py(y negationslash= 0).
(3) 3?§üàéx|ê§?5?y′x?ê§òk2x+xyprime +y+ 2yyprime = 0§Kyprime =?2x+yx+ 2y.
(4) 3?§üàéx|ê§?5?y′x?ê§òk3x2 + 3y2yprime?xyprime?y = 0§Kyprime = 3x
2?y
x?3y2,
(5) 3?§üàéx|ê§?5?y′x?ê§òkyprime = 1 + y
prime
2 cosy§Ky
prime = 2
2?cosy.
(6) 3?§üàéx|ê§?5?y′x?ê§òk23x?13 + 23y?13yprime = 0§Kyprime =? 3
radicalbiggx
y.
(7) 3?§üàéx|ê§?5?y′x?ê§òkyprime+(1+yprime)sin(x+y) = 0§Kyprime =? sin(x+y)1 + sin(x+y).
(8) 3?§üàéx|ê§?5?y′x?ê§òkyprime = 1 + y
prime
1 +y2§Ky
prime = 1 +y
2
y2,
(9) 3?§üàéx|ê§?5?y′x?ê§òkyprime =?1 +y
prime
x+y +y
primeey§Kyprime = 1
(x+y)ey?x?y?1.
(10) 3?§üàéx|ê§?5?y′x?ê§òkxy
prime?y
x2 +y2 =
x+yyprime
x2 +y2§Ky
prime = x+y
x?y.
2,|eê3:êdydxμ
(1) y = cosx+ 12 siny§:
parenleftBigpi
2,0
parenrightBig
(2) yex + lny = 1§:(0,1)
)μ
(1) 3?§üàéx|ê§?5?y′x?ê§òkyprime =?sinx+ y
prime
2 cosy§Ky
prime = 2sinx
cosy?2§u′3
:
parenleftBigpi
2,0
parenrightBig
§yprime =?2.
(2) 3?§üàéx|ê§?5?y′x?ê§òkex(y +yprime) + y
prime
y = 0§Ky
prime =? y
2ex
yex + 1§u′3
:(0,1)?§yprime =?12.
64
3,|-?x32 +y32 = 163:(4,4)§ú{§.
)μ3?§üàéx|ê§?5?y′x?ê§òk32x12 + 32y12yprime = 0§Kyprime =?
radicalbiggx
y§u′y
prime|x=4
y=4
=
1§l
§?y?4 =?(x?4)§=x+y?8 = 0
{§?y?4 = x?4§=x = y.
4,|e?ê?§3¤?:êμ
(1)
braceleftbigg x = acost
y = bsint 3t =
pi
3ú
pi
4?
(2)
braceleftbigg x = t?sint
y = 1?cost §3t =
pi
2,pi?
(3)
braceleftbigg x = 1?t2
y = t?t3 §3t =
√2
2,
√3
3?
(4)
braceleftbigg x = a(t?sint)
y = a(1?cost) (a′~ê)§3t = 0,
pi
2?
)μ
(1)?xprime(t) =?asint,yprime(t) = bcost§Kdydx = y
prime(t)
xprime(t) =?
b
a cott§u′§t =
pi
3?§y
prime =?
√3b
3a?
t = pi4?§yprime =?ba
(2)?xprime(t) = 1? cost,yprime(t) = sint§Kdydx = y
prime(t)
xprime(t) =
sint
1?cost§u′§t =
pi
2?§y
prime = 1?t =
pi?§yprime = 0
(3)?xprime(t) =?2t,yprime(t) = 1? 3t2§Kdydx = y
prime(t)
xprime(t) =
3t2?1
2t §u′§t =
√2
2?§y
prime =
√2
4?t =√
3
3?§y
prime = 0
(4)?xprime(t) = a(1? cost),yprime(t) = asint§Kdydx = y
prime(t)
xprime(t) = cot
t
2§u′§t = 0?§y
primet =
pi
2?§y = 1
5,|e?ê?§êμ
(1)
braceleftbigg x = acosht
y = bsinht
(2)
braceleftbigg x = sin2t
y = cos2t
(3)
braceleftbigg x = acos3t
y = asin3t
(4)
braceleftbigg x = e2t cos2t
y = e2t sin2t
)μ
(1) dydx = y
prime(t)
xprime(t) =
asinht
bcosht =
a
b cotht
(2) dydx = y
prime(t)
xprime(t) =
2costsint
2sintcost =?1
(3) dydx = y
prime(t)
xprime(t) =
3sin2tcost
3cos2tsint =?!tant
(4) dydx = y
prime(t)
xprime(t) =
e2t(2sin2t+ 2sintcost)
e2t(2cos2t?2costsint) = tant·
sint+ cost
cost?sint
6,?
I/Nì§ 10o§to
4o￡?4-11¤μ
(1) /\Y?§|YNèVéY?pYhCz
(2) |NèVéNì ?
RCz?.
65
)μ?NèV?Nì ?
R§Y?pYh'X?V = 13piR2h§?d?§R4 = h10=h = 52R§u
′
(1) V = 13pi
parenleftbigg2
5h
parenrightbigg2
h = 475pih3§l
dVdh = 425pih2?
(2) V = 13piR2 · 52R = 56piR3§l
dVdR = 52piR2.
7,?
I/Nì.t?X§§o?2arctan 34§8?p??,N§
(1) ?Nr?3§O\?Ydrdt?14?§NèO\?YdVdt′?o
(2) ?N6§NèO\?Y?24?§O\?Y′?o
)μ?NèVNr'X?V = 49pir3§V,r?′?mt?ê§ü>ét|§dVdt = 49pi(3r2)drdt=dVdt =
4
3pir
2dr
dt§K
(1) r = 3,drdt = 14?§dVdt = 3pi?
(2) ddrdt = 34pir2 dVdt§r = 6,dVdt = 24?§drdt = 12pi.
8,Ylp?18f?!.6f?
I/|ì6\5f?
/ùS.?|ì￥Y ?12f§
|ì￥Y?eü?Y?1f?/?§|d?
ù￥Y?t,?Y.
)μlm?|Y?2t?¨ §
I/|ì￥M? Y?xf?§
/ù￥Y?,p
yf?"d?§
|ì￥|?M?Nè?13pi·62 ·18? 13pi
parenleftBigx
18 ·6
parenrightBig2
·x = 216pi? pi27x3￡á?f?¤§
/ù￥5\
M?Nè?pi·52 ·y = 25piy￡á?f?¤"aK?§25piy = 216pi? pi27x3§y = 125
parenleftbigg
216? x
3
27
parenrightbigg
§u
′dydt = dydx · dxdt =? 1675 ·3x2 · dxdt =? 1225x2 · dxdt"x = 12(f?)?§dxdt =?1(f?/?)§u′d?
ù
￥Y?t,?Y?dydx =? 1225 ·122(?1) = 1625 = 0.64(f?/?).
9,?4-12¤?>′￥§P = i2R§ù￥>6i = Ur+R.|N?C>{R?§PCz?dPdR.
)μ?P = i2R,i = Ur+R§KdPdR = 2iR didR +i2 =?2U
2R
(r+R)3 +
U2
(r+R)2 =
U2(r?R)
(r+R)3
66
§7,?êT~
1,|e?ê3¤?:x0?êfprime?(x0)úmêfprime+(x0)μ
(1) y =
braceleftbigg x2,xlessorequalslant 0,
xex,x> 0,x0 = 0
(2) y =
braceleftBigg x
1 +e1x
,xnegationslash= 0,
0,x = 0,
x0 = 0
(3) y =
braceleftBigg
x2 sin 1x,xnegationslash= 0,
0,x = 0,
x0 = 0
)μ
(1) fprime+(x0) = lim
x→+0
xex?0
x = 1?f
prime
(x0) = limx→?0
x2?0
x = 0.
(2) fprime+(x0) = lim
x→+0
x
1 +e1x
0
x = limx→+0
1
1 +e1x
= 0?
fprime?(x0) = lim
x→?0
1
1 +e1x
= 1.
(3) fprime+(x0) = lim
x→+0
x2 sin 1x?0
x = 0?f
prime
(x0) = limx→?0
x2 sin 1x?0
x = 0.
2,|e?ê3ê?3:?!mêμ
(1) y = |ln|x||
(2) y = |tanx|
(3) y = √1?cosx
)μ
(1) y = |ln|x|| =
ln(?x),xlessorequalslant?1
ln(?x),?1 <x< 0
lnx,0 <x< 1
lnx,xgreaterorequalslant 1
dd?§?ê3x = 0,x = ±1?ê?3"
fprime+(?1) = lim
x→+0
ln[?(?1 +?x)]?ln(?(?1))
x = lim?x→+0ln(1x)
1?x = 1?
fprime?(?1) = lim
x→?0
ln[?(?1 +?x)]?ln(?(?1))
x = lim?x→?0ln(1x)
1?x =?1?
ê3x = 0:§fprime+(0)úfprime?(0)
fprime+(1) = lim
x→+0
ln(1 +?x)?ln1
x = lim?x→+0ln(1 +?x)
1
x = 1?
fprime?(1) = lim
x→?0
ln(1 +?x)?ln(1)
x = lim?x→?0?ln(1 +?x)
1
x =?1.
(2) y = |tanx| =
tanx,x∈
parenleftBig
kpi? pi2,kpi
bracketrightBig
tanx,x∈
parenleftBig
kpi,kpi+ pi2
parenrightBig k∈Z
ù￥x = kpi+ pi2(k∈Z)ê§m?:§?!mê
x = kpi(k∈Z)?ê?3:.
fprime+(kpi) = lim
x→+0
tan(kpi+?x)?(?tankpi)
x = lim?x→+0
tan?x
x = 1?f
prime
(kpi) = lim?x→?0
tan(kpi+?x)?(?tankpi)
x =
lim
x→?0
tan?x?x =?1.
(3)?yprime = sinx√1?cosxxnegationslash= 2kpi(k∈Z)?ak§x = 2kpi(k∈Z)?y = √1?cosx:.
fprime+(2kpi) = lim
x→+0
radicalbig1 + cos(2kpi+?x)?√1 + 2kpi
x = lim?x→+0
√1?cos?x
x =? lim?x→+0
radicalbigg
1?cos?x
x2 =
67
√2
2?
fprime?(2kpi) = lim
x→?0
radicalbig1 + cos(2kpi+?x)?√1 + 2kpi
x = lim?x→?0
√1?cos?x
x =? lim?x→?0?
radicalbigg
1?cos?x
x2 =
√2
2,
3,e
(1) f(x)3x0:?§g(x)3x0:§y2?êF(x) = f(x) +g(x)3x0:?
(2) f(x)úg(x)3x0:§U|?ú?êF(x) = f(x) +g(x)3x0:o
y2μ
(1) bF(x) = f(x) +g(x)3x0:?§qf(x)3x0:?§Kg(x) = F(x)?f(x)3x0:?§ù
g?§b?¤á"l
êF(x) = f(x) +g(x)3x0:.
(2)?U"~μ
(i)?μf(x) = |x|+x2,g(x) = x?|x|2 3x = 0:§§?ú?êF(x) = f(x) +g(x) =
x3x = 0:??Fprime(0) = 1?
(ii)μf(x) = |x|2,g(x) = |x|2 3x = 0:§§?ú?êF(x) = f(x) +g(x) = |x|3x =
0:.
4,3tK^?e§§?èG(x) = f(x)·g(x)??1No
)μ
(1) §?èG(x) = f(x)·g(x)3x0:?U?"
~μ
(i)?μf(x) = x3x = 0:??fprime(0) = 1?g(x) = |x|3x = 0:§§?èG(x) =
f(x)·g(x) = x|x|3x = 0??Gprime(0) = lim
x→0
G(x)
x = lim?x→0
x|?x|
x = lim?x→0|?x| = 0
(ii)μf(x) = 13x = 0:??fprime(0) = 0?g(x) = |x|3x = 0:§§?èG(x) =
f(x)·g(x) = |x|3x = 0:.
(2) §?èG(x) = f(x)·g(x)3x0:?U?"
~μ
(i)?μf(x) = |x|,g(x) = |x|3x = 0:§§?èG(x) = f(x) ·g(x) = x23x = 0?
Gprime(0) = 0
(ii)μf(x) = x23,g(x) = |x13|3x = 0:§§?èG(x) = f(x)·g(x) = |x|3x = 0:
.
5,e?êf(x)3km(a,b)￥kê§?lim
x→a
f(x) = ∞§′?7klim
x→a
fprime(x) = ∞o±~ff(x) = 1x +
cos 1x`2?.
§ef(x)3km(a,b)￥kê§?lim
x→a
fprime(x) = ∞§′?7klim
x→a
f(x) = ∞o±~ff(x) = 3√x`
2?.
)μ
(1)/`§?Uyklim
x→a
fprime(x) = ∞.
~μéu
parenleftBig
0,pi2
parenrightBig
S?êf(x) = 1x + cos 1x§w,klim
x→0
f(x) = ∞.
qfprime(x) =? 1x2? 1x2
parenleftbigg
sin 1x
parenrightbigg
= 1x2
parenleftbigg
sin 1x?1
parenrightbigg
§
éuA??Gêxn = 1
2npi+ pi2
(n = 1,2,···)§kfprime(xn) = 0§ lim
n→∞
fprime(xn) = 0?
éuxprimen = 1npi(n = 1,2,···)§kfprime(xprimen) =?n2pi2§ lim
n→∞
fprime(xprimen) =?∞§fprime(x)3x = 0:4
3§§=lim
x→0
fprime(x) = ∞?¤á.
(2)?Uy7klim
x→a
f(x) = ∞.
~μf(x) = 3√x§§3(0,b)(b> 0)tkê§?fprime(x) = 13 3√x2§lim
x→0
fprime(x) = ∞§lim
x→0
f(x) = 0.
68
6,e
(1) f(x)3x = g(x0)kê§
g(x)3x0:vkê?
(2) f(x)3x = g(x0)vkê§
g(x)3x0:kê?
(3) f(x)3x = g(x0)vkê§
g(x)3x0:?vkê?
KEü?êF(x) = f(g(x))3x0:′o
)μ
(1) Eü?êF(x) = f(g(x))3x0:?U?.
~μ
(i)?μf(u) = u2,g(x) = |x|,x0 = 0§f(u) = u23u0 = 0 = g(x0)??fprime(0) = 0§g(x) =
|x|3x0 = 0?F(x) = f(g(x)) = |x|2 = x23x0 = 0??Fprime(0) = 0?
(ii)?μf(u) = u,g(x) = |x|,x0 = 0§f(u) = u3u0 = 0 = g(x0)??fprime(0) = 1§g(x) =
|x|3x0 = 0?F(x) = f(g(x)) = |x|3x0 = 0.
(2) Eü?êF(x) = f(g(x))3x0:?U?.
~μ
(i)?μf(u) = |u|,g(x) = x2,x0 = 0§f(u) = |u|3u0 = 0 = g(x0)§g(x) = x23x0 = 0?
gprime(0) = 0?F(x) = f(g(x)) = |x2| = x23x0 = 0??Fprime(0) = 0?
(ii)?μf(u) = |u|,g(x) = x,x0 = 0§f(u) = |u|3u0 = 0 = g(x0)§g(x) = x3x0 = 0?
gprime(0) = 1?F(x) = f(g(x)) = |x|3x0 = 0.
(3) Eü?êF(x) = f(g(x))3x0:?U?.
~μ
(i)?μf(u) = 2u+|u|,g(x) = 23x?|x|3,x0 = 0§f(u) = 2u+|u|3u0 = 0 = g(x0)§g(x) =
2
3x?
|x|
3 3x0 = 0?F(x) = f(g(x)) = 2
parenleftbigg2
3x?
|x|
3
parenrightbigg
+
vextendsinglevextendsingle
vextendsinglevextendsingle2
3x?
|x|
3
vextendsinglevextendsingle
vextendsinglevextendsingle =
braceleftbigg x,xgreaterorequalslant 0
x,x< 0 =
x=F(x) = x(?x∈ (?∞,+∞)§F(x)3x0 = 0??Fprime(0) = 1?
(ii)?μf(u) = |u|,g(x) = |x|,x0 = 0§f(u) = |u|3u0 = 0 = g(x0)§g(x) = |x|3x0 = 0?
?F(x) = f(g(x)) = |x|3x0 = 0.
69
§8,p ê?p
1,y = 2x3 +x2 +x+ 1§|yprime,yprimeprime,y(3)úy(4).
)μyprime = 6x2 + 2x+ 1,yprimeprime = 12x+ 2,y(3) = 12,y(4) = 0
2,y = eαt(α?~ê)§|yprimeprime,y(3),y(n).
)μyprime = αeαt,yprimeprime = α2eαt,y(3) = α3eαt,y(n) = αneαt
3,|e?êp êμ
(1) y = x√1?x2§|yprimeprime
(2) y = xlnx§|yprimeprime
(3) y = e?x2§|yprimeprime
(4) y = arcsinx√1?x2§|yprimeprime
(5) y = x2 ·e2x§|yprimeprimeprime
(6) y = a3x§|yprimeprimeprime
(7) y = x3 sinhx§|y(30)
(8) y = x3 cosx§|y(50)
)μ
(1) yprime =
√1?x2 + x2√
1?x2
1?x2 =
1
(1?x2)32
= (1?x2)?32,yprimeprime = 3x(1?x2)?52
(2) yprime = 1 + lnx,yprimeprime = 1x
(3) yprime =?2xe?x2,yprimeprime =?2e?x2(1?2x2) = 2e?x2(2x2?1)
(4) yprime =
1 + xarcsinx√1?x2
1?x2 =
1
1?x2 +
xarcsinx
(1?x2)32
,
yprimeprime = 2x(1?x2)2 +
parenleftbigg
arcsinx+ x√1?x2
parenrightbigg
(1?x2)32 + 3x(1?x2)12 ·xarcsinx
(1?x2)3 =
3x
(1?x2)2 +
(2x2 + 1)arcsinx
(1?x2)52
(5) yprimeprimeprime = (x2e2x)primeprimeprime = x2(e2x)primeprimeprime + 3(x2)prime(e2x)primeprime + 3(x2)primeprime(e2x)prime + (x2)primeprimeprimee2x = 4e2x(2x2 + 6x+ 3)
(6) yprime = 3a3x lna,yprimeprime = 9ln2a·a3x,yprimeprimeprime = 27ln3a·a3x
(7)?(x3)prime = 3x2,(x3)primeprime = 6x,(x3)primeprimeprime = 6,(x3)(4) = ··· = (x3)(30) = 0;(sinhx)(30) = sinhx,(sinhx)(29) =
coshx,(sinhx)(28) = sinhx,(sinhx)(27) = coshx§y(30) = (x3 sinhx)(30) = x3(sinhx)(30)+30(x3)prime(sinhx)(29)+
435(x3)primeprime(sinhx)(28) + 4060(x3)primeprimeprime(sinh)(27) = xsinhx(x2 + 2610) + 30coshx(3x2 + 812)
(8)?(x3)prime = 3x2,(x3)primeprime = 6x,(x3)primeprimeprime = 6,(x3)(4) = ··· = (x3)(50) = 0;(cosx)(50) =?cosx,(cosx)(49) =
sinx,(cosx)(48) = cosx,(cosx)(47) = sinx§y(50) = (x3 cosx)(50) = x3(cosx)(50)+50(x3)prime(cosx)(49)+
1225(x3)primeprime(cosx)(48) + 19600(x3)primeprimeprime(cosx)(47) = xcosx(7350?x2) + 150sinx(784?x2)
4,|^ê?8B{y2e?úaμ
(1) (ax)(n) = ax ·(lna)n(a> 0)
(2) (cosx)(n) = cos
parenleftBig
x+n· pi2
parenrightBig
(3) (lnx)(n) = (?1)
n?1 ·(n?1)!
xn
y2μ
(1) (i) n = 1?§(ax)prime = ax lna = ax(lna)1§Kn = 1?úa¤á.
(ii) bn = k?úa¤á§=(ax)(k) = ax(lna)k¤á§
Kn = k+ 1?§(ax)(k+1) =
bracketleftBig
(ax)(lna)(k)
bracketrightBigprime
= (lna)k(ax)prime = (lna)k ·ax lna = ax(lna)k+1§u
′n = k+ 1?úa?¤á.
nüt§ng,ê?§úa(ax)(n) = ax ·(lna)n(a> 0)?¤á.
70
(2) (i) n = 1?§(cosx)prime =?sinx = cos
parenleftBig
x+ pi2
parenrightBig
§Kn = 1?úa¤á.
(ii) bn = k?úa¤á§=(cosx)(k) = cos
parenleftBig
x+k· pi2
parenrightBig
¤á§
Kn = k + 1?§(cosx)(k+1) =
bracketleftBig
(cosx)(k)
bracketrightBigprime
=
bracketleftBig
cos
parenleftBig
x+k· pi2
parenrightBigbracketrightBigprime
=?sin
parenleftBig
x+k· pi2
parenrightBig
=
cos
parenleftBig
x+ (k+ 1)· pi2
parenrightBig
§u′n = k+ 1?úa?¤á.
nüt§ng,ê?§úa(cosx)(n) = cos
parenleftBig
x+n· pi2
parenrightBig
¤á.
(3) (i) n = 1?§(lnx)prime = 1x = (?1)
1?1(1?1)!
x1 §Kn = 1?úa¤á.
(ii) bn = k?úa¤á§=(lnx)(k) = (?1)
k?1 ·(k?1)!
xn ¤á§
Kn = k + 1?§(lnx)(k+1) =
bracketleftBig
(lnx)(k)
bracketrightBigprime
=
bracketleftbigg(?1)k?1 ·(k?1)!
xk
bracketrightbiggprime
=?k· (?1)
k?1 ·(k?1)!
xk+1 =
(?1)k ·k!
xk+1 =
(?1)k+1?1 ·(k+ 1?1)!
xk+1 §u′n = k+ 1?úa?¤á.
nüt§ng,ê?§úa(lnx)(n) = (?1)
n?1 ·(n?1)!
xn?¤á.
5,|n êμ
(1) y = 1x(1?x)
(2) y = 1x2?2x?8
(3) y = amxm +am?1xm?1 +···+a0
(4) y = cos2ωx
(5) y = e
x
x
(6) y = 2x ·lnx
(7) y = eaxpn(x)§ù￥pn(x)?nga.
)μ
(1)?y = 1x(1?x) = 1x + 11?x§Ky(n) =
parenleftbigg1
x?
1
1?x
parenrightbigg(n)
=
parenleftbigg1
x
parenrightbigg(n)
+
parenleftbigg 1
1?x
parenrightbigg(n)
= (x?1)(n) +[(1?
x)?1](n) = (?1)n ·n!·x?n?1 + (?1)2n ·n!·(1?x)?n?1 = n![(?1)nx?n?1 + (1?x)?n?1]
(2)?y = 1x2?2x?8 = 1(x+ 2)(x?4) = 16
parenleftbigg 1
x?4?
1
x+ 2
parenrightbigg
§Ky(n) = 16
parenleftbigg 1
x?4?
1
x+ 2
parenrightbigg(n)
=
1
6[((x?4)
1)(n)?((x+2)?1)(n)] = 1
6[(?1)
n·n!(x?4)?n?1?(?1)n·n!(n+2)?n?1] = (?1)n
6 n!
parenleftbigg 1
(x?4)n+1?
1
(x+ 2)n+1
parenrightbigg
(3) n>m?§x(n) = (x2)(n) = ··· = (xm)(n) = 0§Ky(n) = 0?
n = m?§x(n) = (x2)(n) = ··· = (xm?1)(n) = 0,(xm)(n) = (xm)(m) = m!§Ky(n) = am ·m!?
n < m?§x(n) = (x2)(n) = ··· = (xn?1)(n) = 0,(xn)(n) = n!,···,(xm)(n) = m!(m?n)!xm?n§
Ky(n) = am· m!(m?n)!xm?n+am?1· (m?1)!(m?n?1)!xm?n?1+···+ann! =
m?nsummationdisplay
i=0
am?i (m?i)!(m?n?i)!xm?n?i.
(4)?yprime =?2ωcosωxsinωx =?ωsin2ωx§Ky(n) = (yprime)(n?1) = (?ωsin2ωx)(n?1) =?ωsin
parenleftbigg
2ωx+ n?12 pi
parenrightbigg
·
(2ω)n?1 =?2n?1ωn sin
parenleftbigg
2ωx+ n?12 pi
parenrightbigg
= 2n?1ωn cos
parenleftBig
2ωx+ n2pi
parenrightBig
(5) y(n) =
parenleftbiggex
x
parenrightbigg(n)
=
parenleftbigg
ex · 1x
parenrightbigg(n)
=
nsummationtext
k=0
Cknex
parenleftbigg1
x
parenrightbigg(k)
=
ex
bracketleftBigg
1
x +
nsummationdisplay
k=1
(?1)kn(n?1)···(n?k+ 1)xk+1
bracketrightBigg
71
(6) y(n) = (2x ·lnx)(n) =
nsummationtext
k=0
Ckn(2x)(n?k)(lnx)(k) =
nsummationtext
k=1
Ckn(ln2)n?k ·2x · (?1)
k?1(k?1)!
xk + 2
x(ln2)n lnx =
2x[(ln2)n lnx+n(ln2)n?1x?1 +···+ (?1)n?2(n?2)!·nln2·x?(n?1) + (?1)n?1(n?1)!·x?n]
(7) y(n) = (eaxpn(x))(n) = aneaxpn(x)+C1nan?1eaxpprimen(x)+···+eaxp(n)n (x) = eax[anpn(x)+C1nan?1pprimen(x)+
···+p(n)n (x)]
6,ef(x) =
braceleftBigg
e? 1x2,xnegationslash= 0
0,x = 0 y2f
(n)(0) = 0.
y2μx negationslash= 0?§fprime(x) = 2x3e? 1x2,fprimeprime(x) = e? 1x2
parenleftbigg
6x4 + 4x6
parenrightbigg
§ddí?f(n)(x) = e? 1x2Pn
parenleftbigg1
x
parenrightbigg
(x negationslash=
0)§ù￥Pn(t)′'uta"
e?y2μéên§tk·Kf(n)(x) = e? 1x2Pn
parenleftbigg1
x
parenrightbigg
(xnegationslash= 0)
n = 1?§·Kw,¤á.
bn = k?§·K¤á§=kf(k)(x) = e? 1x2Pk
parenleftbigg1
x
parenrightbigg
(xnegationslash= 0),Pk(t)′'uta§
Kn = k+ 1?§f(k+1)(x) = [f(k)(x)]prime =
bracketleftbigg
e? 1x2Pk
parenleftbigg1
x
parenrightbiggbracketrightbiggprime
=
e? 1x2
bracketleftbigg 2
x3Pk
parenleftbigg1
x
parenrightbigg
1x2Pprimek
parenleftbigg1
x
parenrightbiggbracketrightbigg
= e? 1x2
bracketleftBigg
2
parenleftbigg1
x
parenrightbigg3
Pk
parenleftbigg1
x
parenrightbigg
parenleftbigg1
x
parenrightbigg2
Pprimek
parenleftbigg1
x
parenrightbiggbracketrightBigg
= e? 1x2Pk+1
parenleftbigg1
x
parenrightbigg
§
ù￥Pk+1(t)′'ut,a.
aê?8B{?§·Kég,ênt¤á.
n = 1?§fprime(0) = lim
x→0
f(0 +?x)?f(0)
x = lim?x→0
e?( 1?x)2
x = lim?x→0
1
x
e( 1?x)2
= 0
bn = k?§f(k)(0) = 0§Kf(k+1) = lim
x→0
f(k)(0 +?x)?f(k)(0)
x = lim?x→0
e?( 1?x)2Pk
parenleftbigg 1
x
parenrightbigg
x =
lim
x→0
1
xPk
parenleftbigg 1
x
parenrightbigg
e( 1?x)2
= 0
aê?8B{?§f(n)(0) = 0.
7,f(x)? ê3§|yprimeprime9yprimeprimeprimeμ
(1) y = f(x2)
(2) y = f
parenleftbigg1
x
parenrightbigg
(3) y = f(e?x)
(4) y = f(lnx)
)μ
(1) yprime = 2xfprime(x2),yprimeprime = 2fprime(x2) + 4x2fprimeprime(x2),
yprimeprimeprime = 12xfprimeprime(x2) + 8x3fprimeprimeprime(x2)
(2) yprime =? 1x2fprime
parenleftbigg1
x
parenrightbigg
,yprimeprime = 2x3fprime
parenleftbigg1
x
parenrightbigg
+ 1x4fprimeprime
parenleftbigg1
x
parenrightbigg
,
yprimeprimeprime =? 6x4fprime
parenleftbigg1
x
parenrightbigg
6x5fprimeprime
parenleftbigg1
x
parenrightbigg
1x6fprimeprimeprime
parenleftbigg1
x
parenrightbigg
(3) yprime =?e?xfprime(e?x),yprimeprime = e?xfprime(e?x)+e?2xfprimeprime(e?x),yprimeprimeprime =?e?xfprime(e?x)?3e?2xfprimeprime(e?x)?e?3xfprimeprimeprime(e?x)
(4) yprime = 1xfprime(lnx),yprimeprime = 1x2fprimeprime(lnx)? 1x2fprime(lnx) = 1x2 [fprimeprime(lnx)?fprime(lnx)],yprimeprimeprime = 1x3 [2fprime(lnx)?3fprimeprime(lnx)+
fprimeprimeprime(lnx)]
8,y = ex sinx,z = ex cosx§y2§?÷v?§yprimeprime = 2z,zprimeprime =?2y.
y2μ?y = ex sinx,z = ex cosx§Kyprime = ex(sinx + cosx),yprimeprime = 2ex cosx;zprime = ex(cosx? sinx),zprimeprime =
2ex sinx§u′yprimeprime = 2z,zprimeprime =?2y.
72
9,y = C1eλ1x +C2eλ2x§C1,C2,λ1,λ2′~ê§y2§÷v?§yprimeprime?(λ1 +λ2)yprime +λ1λ2y = 0.
y2μ?y = C1eλ1x + C2eλ2x§C1,C2,λ1,λ2′~ê§Kyprime = C1λ1eλ1x + C2λ2eλ2x,yprimeprime = C1λ21eλ1x +
C2λ22eλ2x§
u′yprimeprime?(λ1 +λ2)yprime +λ1λ2y = C1λ21eλ1x +C2λ22eλ2x?(λ1 +λ2)(C1λ1eλ1x +C2λ2eλ2x)+λ1λ2(C1eλ1x +
C2eλ2x) = 0=yprimeprime?(λ1 +λ2)yprime +λ1λ2y = 0.
10,y = C1 sinx+C2 cosx§y2y÷v?§yprimeprime +y = 0.
y2μ?y = C1 sinx + C2 cosx§Kyprime = C1 cosx?C2 sinx,yprimeprime =?C1 sinx?C2 cosx =?(C1 sinx +
C2 cosx) =?y=yprimeprime +y = 0.
11,e?ê(x) = f(x)?f(a)fprime(a)
bracketleftbigg
1 + f(x)?f(a)fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
§|?prime(a)9?primeprime(a).
)μ(x) = f(x)?f(a)fprime(a)
bracketleftbigg
1 + f(x)?f(a)fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
§
K?prime(x) = f
prime(x)
fprime(a)
bracketleftbigg
1 + f(x)?f(a)fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
+
f(x)?f(a)
fprime(a)
bracketleftbiggfprime(x)
fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
,
primeprime(x) = f
primeprime(x)
fprime(a)
bracketleftbigg
1 + f(x)?f(a)fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
+ 2f
prime(x)
fprime(a)
bracketleftbiggfprime(x)
fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
+
f(x)?f(a)
fprime(a)
bracketleftbiggfprimeprime(x)
fprime(a)2
parenleftbigg
fprime(a)? 12fprimeprime(a)
parenrightbiggbracketrightbigg
§
K?prime(a) = 1,?primeprime(a) = 2
12,x =?(y)′y = f(x)ê§áˉX?dfprime,fprimeprime,fprimeprimeprimeprimeprimeprime(y)o
)μprime(y) = 1fprime(x)§K?primeprime(y)fprime(x) =? f
primeprime(x)
[fprime(x)]2§u′?
primeprime(y) =? f
primeprime(x)
[fprime(x)]3§
K?primeprimeprime(y)fprime(x) =?f
primeprimeprime(x)[fprime(x)]3?3[fprime(x)]2[fprimeprime(x)]2
[fprime(x)]6 §l
primeprimeprime(y) = 3[f
primeprime(x)]2?fprimeprimeprime(x)fprime(x)
[fprime(x)]5,
13,á|{Z?s = ae?λt sinωt3t?Yú\?Y§?|Y?=:.
)μ?Yv = sprime = ae?λt(?λsinωt+ωcosωt)§\?Ya = vprime = sprimeprime = ae?λt[(λ2?ω2)sinωt?2λωcosωt]?
Y?=:=v = 0§K?λsinωt+ωcosωt = 0§u′tanωt = ωλ(λnegationslash= 0).
14,|e?ê?§ êd
2y
dx2μ
(1)
braceleftbigg x = 2t?t2
y = 3t?t3
(2)
braceleftbigg x = acost
y = asint
(3)
braceleftbigg x = a(t?sint)
y = a(1?cost)
(4)
braceleftbigg x = et cost
y = et sint
(5)
braceleftbigg x = acos3t
y = asin3t
(6)
braceleftbigg x = fprime(t)
y = tfprime(t)?f(t)
)μ
(1) dydx = 3?3t
2
2?2t =
3
2(1 +t)
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
= 34(1?t)
(2) dydx = acost?asint =?cott
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
=? 1asin3t
73
(3) dydx = asinta(1?cost) = cot t2
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
=? 1
4asin4 t2
(4) dydx = e
t(sint+ cost)
et(cost?sint) =
sint+ cost
cost?sint
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
= 2et(cost?sint)3
(5) dydx = 3asin
2tcost
3acos2tsint =?tant
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
= 13acos4tsint
(6) dydx = tf
primeprime(t)
fprimeprime(t) = t
d2y
dx2 =
d
dt
parenleftbiggdy
dx
parenrightbigg
dx
dt
= 1fprimeprime(t)
15,|dê¤(? êμ
(1) ex+y?xy = 0
(2) x3 +y3?3axy = 0
(3) y2 + 2lny?x4 = 0
)μ
(1) é?§ex+y?xy = 0üà'ux|§
(1 +yprime)ex+y?y?xyprime = 0 (4)
u′yprime = y?e
x+y
ex+y?x§
2é(4)üà'ux|§yprimeprimeex+y + (1 +yprime)2ex+y?2yprime?xyprimeprime = 0§Kyprimeprime = 2y
prime?(1 +yprime)2ex+y
ex+y?x §
òyprime = y?e
x+y
ex+y?x?\ta§=y
primeprime = 2(y?e
x+y)
(ex+y?x)2?
(x?y)2ex+y
(ex+y?x)3,
(2) é?§x3 +y3?3axy = 0üà'ux|§
x2 +y2yprime?axyprime?ay = 0 (1)
u′yprime = ay?x
2
y2?ax§
2é(1)üà'ux|§2x+ 2y(yprime)2 +y2yprimeprime?2ayprime?axyprimeprime = 0§Kyprimeprime = 2ay
prime?2y(yprime)2?2x
y2?ax §
òyprime = ay?x
2
y2?ax?\ta§=y
primeprime = 2a(ay?x
2)
(y2?ax)2?
2y(ay?x2)2
(y2?ax)3?
2x
y2?ax.
(3) é?§y2 + 2lny?x4 = 0üà'ux|§
yyprime + 1yyprime?2x3 = 0 (1)
u′yprime = 2x
3y
y2 + 1§
2é(1)üà'ux|§(yprime)2 +yyprimeprime + yy
primeprime?(yprime)2
y2?6x
2 = 0§Kyprimeprime = 6x2y2 + (yprime)2(1?y2)
y(y2 + 1) §
òyprime = 2x
3y
y2 + 1?\ta§=y
primeprime = 2x
2y
(y2 + 1)3 [3(y
2 + 1)2 + 2x4(1?y2)].
74
16,|p (x′gCt)μ
(1) y = √1 +x2§|d2y
(2) y = xx§|d2y
(3) y = xcos2x§|d3y
(4) y = 1√x§|d3y
(5) y = xn ·ex§|dny
(6) y = lnxx §|dny
)μ
(1) dy = x√1 +x2dx,d2y = (1 +x2)?32dx2
(2) dy = xx(lnx+ 1)dx,d2y = xx
bracketleftbigg
(lnx+ 1)2 + 1x
bracketrightbigg
dx2
(3) d3y = (xcos2x)(3)dx3 = (x(cos2x)(3) + 3(cos2x)(2))dx3 = (8xsin2x?12cos2x)dx3
(4) d3y =
parenleftbigg 1
√x
parenrightbigg(3)
dx3 =?158 x?72dx3
(5) dny = (xn ·ex)(n)dxn =
parenleftbigg
ex
nsummationtext
k=0
Ckn n!(n?k)!xn?k
parenrightbigg
dxn
(6) dny =
parenleftbigglnx
x
parenrightbigg(n)
dxn =
parenleftbigg1
x lnx
parenrightbigg(n)
dxn =
bracketleftBigg
(?1)nn!lnxxn+1 +
nsummationdisplay
k=1
Ckn(?1)n?1 (n?k)!(k?1)!xn+1
bracketrightBigg
dxn =
(?1)n n!xn+1
bracketleftBigg
lnx?
nsummationdisplay
k=1
1
k
bracketrightBigg
dxn
17,éy = ex|d2y§?e?ü/μ
(1) x′gCt
(2) x′￥mCt?.
)μ
(1) dy = exdx,d2y = exdx2
(2) dy = exdx,d2y = ex(dx2 +d2x)
18,eu,v?x?ê§v
g§|p μ
(1) y = u(x)·v(x)§|d2y
(2) y = u(x)v(x)§|d2y
(3) y = um(x)vn(x)(m,n?~ê)§|d2y
(4) y = au(x)(a> 0)§|d2y
(5) y = lnu(x)§|d3y
(6) y = sin(u(x))§|d3y
)μ
(1) dy = (uprime(x)v(x) +u(x)vprime(x))dx,
d2y = [uprimeprime(x)v(x) + 2uprime(x)vprime(x) +u(x)vprimeprime(x)]dx2
(2) dy = u
prime(x)v(x)?u(x)vprime(x)
v2(x) dx,
d2y =
bracketleftbigguprimeprime(x)
v(x)?
u(x)vprimeprime(x) + 2uprime(x)vprime(x)
v2(x) +
2u(x)(vprime(x))2
v3(x)
bracketrightbigg
dx2
75
(3) dy = [mum?1(x)vn(x)uprime(x) +num(x)vn?1(x)vprime(x)]dx,
d2y = [m(m? 1)um?2(x)vn(x)(uprime(x))2 + 2mnum?1(x)vn?1(x)uprime(x)vprime(x) + mum?1(x)vn(x)uprimeprime(x) +
n(n?1)um(x)vn?2(x)(vprime(x))2 +num(x)vn?1(x)vprimeprime(x)]dx2
(4) dy = au(x) lna·uprime(x)dx,
d2y = au(x) lna[lna(uprime(x))2 +uprimeprime(x)]dx2
(5) dy = u
prime(x)
u(x) dx,d
2y =
bracketleftbigguprimeprime(x)
u(x)?
(uprime(x))2
u2(x)
bracketrightbigg
dx2,
d3y =
bracketleftbigguprimeprimeprime(x)
u(x)?
3uprime(x)uprimeprime(x)
u2(x) +
2(uprime(x))3
u3(x)
bracketrightbigg
dx3
(6) dy = cos(u(x))uprime(x)dx,d2y = [cos(u(x))uprimeprime(x)?sin(u(x))(uprime(x))2]dx2,
d3y = [cos(u(x))uprimeprimeprime(x)?3sin(u(x))uprime(x)uprimeprime(x)?cos(u(x))(uprime(x))3]dx3.
76
1êù? ?n9ùA^
§1,￥n
1,3¤ê?n￥§ex0mà:§áT~`2(¤á.
)μ~μ?êy = x3?m[?1,1]tk§§3à:x0 = 1?§=?x∈ [?1,1]§ekf(x) lessorequalslant
f(x0) = 1§,
yprime|x=1 = 1 negationslash= 0.
2,éux0 ∈ (a,b)§efprime(x0) > 0§K3§?!m?O?(x0,δ),O+(x0,δ)|x∈O?(x0,δ)f(x0) >
f(x)§x∈O+(x0,δ)f(x0) <f(x).
y2μ?fprime(x0) = lim
x→x0
f(x)?f(x0)
x?x0 > 0§a4?5?§3x0δ(δ > 0)?O(x0,δ)? (a,b)§
|x ∈ O(x0,δ)?§kf(x)?f(x0)x?x
0
> 0§l
x ∈ O?(x0,δ)=x? x0 < 0?§kf(x0) > f(x)§
x∈O+(x0,δ)=x?x0 > 0?§kf(x0) <f(x).
3,y2μefprime+(x0) > 0,fprime?(x0) < 0§K3x0?§|3d?Sf(x) greaterorequalslantf(x0).
y2μ?fprime+(x0) = lim
x→x0+0
f(x)?f(x0)
x?x0 > 0§Kdm4?5?§73x0δ1(δ1 > 0)m?O+(x0,δ1)§
|x∈O+(x0,δ1)=0 <x?x0 <δ1?§kf(x)?f(x0)x?x
0
> 0§l
kf(x0) <f(x)?
qfprime?(x0) = lim
x→x0?0
f(x)?f(x0)
x?x0 < 0§Kd?4?5?§73x0δ2(δ2 > 0)??O?(x0,δ2)§|
x∈O?(x0,δ2)=0 <x0?x<δ2?§kf(x)?f(x0)x?x
0
< 0§l
kf(x0) <f(x)?
δ = min(δ1,δ2)§x∈O(x0,δ)?§okf(x) greaterorequalslantf(x0).
4,ef(x)3[a,b]?Y§f(a) = f(b) = 0,fprime(a)·fprime(b) > 0§Kf(x)3(a,b)S?k":.
y2μ?fprime(a)·fprime(b) > 0§fprime(a) > 0,fprime(b) > 0(fprime(a) < 0,fprime(b) < 0?1ón?y)
qfprime(a) = fprime+(a) = lim
x→a+0
f(x)?f(a)
x?a > 0§Kdm4?5?§73aδ1(δ1 > 0)m?O+(a,δ1)§|
x∈O+(a,δ1)=0 <x?a<δ1?§kf(x)?f(a)x?a > 0§l
kf(a) <f(x)?
?x1 ∈O+(a,δ1)§Kkf(x1) >f(a)
qf(a) = 0§Kf(x1) > 0
qfprime(b) = fprime?(b) = lim
x→b?0
f(x)?f(b)
x?b > 0§Kd?4?5?§73bδ2(δ2 > 0)??O?(b,δ2)§|
x∈O?(b,δ2)=0 <b?x<δ2?§kf(x)?f(b)x?x
0
> 0§l
kf(b) >f(x)?
?x2 ∈O?(b,δ1)§Kkf(x2) <f(b)
qf(b) = 0§Kf(x2) < 0
f(x)3[a,b]?Y§3[x1,x2]Y§qf(x1) > 0,f(x2) < 0§Kd":3?n?§3[x1,x2]S?
k":§
q[x1,x2]? [a,b]§l
f(x)3[a,b]S?k":.
ón§fprime(a) < 0,fprime(b) < 0?§f(x)3[a,b]S?k":.
5,df(x+?x)?f(x) = fprime(x+θ?x)?x(0 <θ< 1)§|?êθ = θ(x,?x)§
(1) f(x) = ax2 +bx+c(anegationslash= 0)
(2) f(x) = 1x
(3) f(x) = ex
)μ
(1) fprime(x) = 2ax+b,fprime(x+θ?x) = 2a(x+θ?x) +b§
K[2a(x + θ?x) + b]?x = f(x +?x)?f(x) = a(x +?x)2 + b(x +?x) + c? (ax2 + bx + c) =
2ax·?x+a(?x)2 +b?x =
bracketleftbigg
2a
parenleftbigg
x+ 12?x
parenrightbigg
+b
bracketrightbigg
x§u′θ = 12
(2) fprime(x) =? 1x2,fprime(x+θ?x) =? 1(x+θ?x)2§
Kx(x+θ?x)2 = f(x+?x)?f(x) = 1x+?x? 1x =xx(x+?x)§l
x2θ2 + 2x·?xθ = 0§
77
u′θ =?x±
√x2 +x?x
x §d?Kò?à(θ ∈ (0,1)
§?Ak
x
x >?1(x negationslash= 0)(dx
2 +
x?x> 0§K?xx >?1)
(3) fprime(x) = ex,fprime(x+θ?x) = ex+θ?x§
Kex+θ?x?x = f(x +?x)?f(x) = ex+?x?ex = ex(e?x? 1)§l
eθ?x?x = e?x? 1§u′θ =
1
x ln
e?x?1
x §?±yθ∈ (0,1)
6,f(x)3?m[a,b]S?Y§3(a,b)?§|^?ê
Φ(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
x f(x) 1
b f(b) 1
a f(a) 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
y2.?KFúa§?QêΦ(x)A.
y2μ?Φ(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
x f(x) 1
b f(b) 1
a f(a) 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= (a?b)f(x) + (f(b)?f(a))x+bf(a)?af(b)§
qf(x)3?m[a,b]S?Y§Kd?Y?êoK$?{K§Φ(x)3[a,b]?Y?
qf(x)3(a,b)?§Kd??êoK$?{K§Φ(x)3(a,b)?.
qΦ(a) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
a f(a) 1
b f(b) 1
a f(a) 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle = 0,Φ(b) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
b f(b) 1
b f(b) 1
a f(a) 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle = 0§Kda?n§3(a,b)S?k?:ξ§
|Φprime(ξ) = 0.
Φprime(x) = (a?b)fprime(x) +f(b)?f(a)§K0 = Φprime(ξ) = (a?b)fprime(ξ) +f(b)?f(a)=fprime(ξ) = f(b)?f(a)b?a,
Φ(x)Aμn/?èúaS? = 12
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
x1 y1 1
x2 y2 1
x3 y3 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle§ù￥(x1,y1),(x2,y2),(x3,y3)L?o:?I§
KΦ(x)L?±A(x,f(x)),B(a,f(a)),C(b,f(b))?o:n/?èü.
7,áée?ê?.?KFúaf(b)?f(a) = fprime(c)(b?a)§?|c.
(1) f(x) = x3,x∈ [0,1]
(2) f(x) = arctanx,x∈ [0,1]
)μ
(1)?fprime(x) = 3x2§K3c2(1?0) = 13?03=3c2 = 1§qc∈ (0,1)§c =
√3
3,
(2)?fprime(x) = 11 +x2§K 11 +c2 (1?0) = arctan1?arctan0= 11 +c2 = pi4§qc∈ (0,1)§c =
radicalbigg4
pi?1.
8,áée?êüúaf(b)?f(a)g(b)?g(a) = f
prime(c)
gprime(c)§?|c.
(1) f(x) = sinx,g(x) = cosx,x∈
bracketleftBig
0,pi2
bracketrightBig
(2) f(x) = x2,g(x) = √x,x∈ [1,4]
)μ
(1)?fprime(x) = cosx,gprime(x) =?sinx§K
f
parenleftBigpi
2
parenrightBig
f(0)
g
parenleftBigpi
2
parenrightBig
g(0)
= f
prime(c)
gprime(c)=
1?0
0?1 =
cosc
sinc§?=cotc = 1§qc ∈
parenleftBig
0,pi2
parenrightBig
§c = pi4.
(2)?fprime(x) = 2x,gprime(x) = 12√x§Kf(4)?f(1)g(4)?g(1) = f
prime(c)
gprime(c)=
16?1
2?1 =
2c
1
2√c
§?=4c32 = 15§qc∈ (1,4)§
c =
parenleftbigg15
4
parenrightbigg2
3.
78
9,áêy = |x?1|3?m[0,3]t?/§ùpovk21uu§.?KF?n￥=?^¤
áo
)μ?ê3:x = 1§=ù?/ACBò?§dò?3C(0,1):3§.?KF?n￥
1?^?=3(0,3)S?ù?^÷v.
a45
a54
a0
a0
a0
a0
a0
a0
a0
a64
a64
a64
a64
a64
a64
a64a64
0 3 x
y
10,|^.?KFúay2? aμ
(1) |sinx?siny|lessorequalslant|x?y|
(2) x∈
parenleftBig
pi2,pi2
parenrightBig
§|x|lessorequalslant|tanx|( ò?k3x = 0?¤á)
(3) n·yn?1(y?x) <xn?yn <n·xn?1(x?y)(n> 1,x>y)
(4) x1 +x < ln(1 +x) <x(x> 0)
(5) exnegationslash= 0,ex > 1 +x(?x> 0,x< 0ü1y2)
y2μ
(1)x>y,f(t) = sint3[y,x]t?Y§3(y,x)S?§.?KF?n¤á§?
ksinx?siny =
cosξ(x? y)(ξ ∈ (y,x))§K|sinx? siny| = |cosξ(x? y)| = |cosξ||(x? y)| lessorequalslant |x? y|(?(x,y ∈
(?∞,+∞))¤á.
(2)x ∈
parenleftBig
0,pi2
parenrightBig
,f(t) = tant3[0,x]t?Y§3(0,x)S?§.?KF?n¤á§?
ktanx?
tan0 = sec2ξ(x?0)
parenleftbigg
ξ ∈ (0,x),x∈
parenleftbigg
0,pi2
parenrightbiggparenrightbigg
§Kx = cos2ξ·tanx< tanx
ón?y§x∈
parenleftBig
pi2,0
parenrightBig
§?x<?tanx.
x = 0?§|tanx| = |x|.
o?§x∈
parenleftBig
pi2,pi2
parenrightBig
§|x|lessorequalslant|tanx|¤á.
x = 0?§ ò¤á?0 <|ξ|<|x|< pi2?§0 < cos2ξ< 1§?U¤á|x|<|tanx|.
(3) f(t) = tn3[y,x]t?Y§3(y,x)S?§.?KF?n¤á§?
kxn?yn = n·ξn?1(x?y)(0 <
y<ξ<x)§
qn> 1§Kyn?1 <ξn?1 <xn?1§n·yn?1(x?y) <n·ξn?1(x?y) <n·xn?1(x?y)=n·yn?1(x?y) <
xn?yn <n·xn?1(x?y)¤á.
(4) f(t) = ln(1 +t)3[0,x]t?Y§3(0,x)S?§.?KF?n¤á§?
kln(1 +x) = ln(1 +x)?
ln1 = 11 +ξ(1 + x? 1) = x1 +ξ(0 < ξ < x)§q1 < 1 + ξ < 1 + x§K 11 +x < 11 +ξ < 1§l
k x1 +x < x1 +ξ <x(x> 0)= x1 +x < ln(1 +x) <x(x> 0)¤á.
(5) f(t) = etw,÷v.?KF?n^?.
x> 0?§éf(t) = et3[0,x]A^.?KFúa§kex?e0 = eξ(x?0)=ex?1 = xeξ(0 <ξ<x)§
0 <ξ<x§Keξ > 1§l
ex?1 = xeξ >x=ex > 1 +x?
x< 0?§éf(t) = et3[x,0]A^.?KFúa§ke0?ex = eξ(0?x)=1?ex =?xeξ(x<ξ< 0)§
x<ξ< 0§K0 <eξ < 1§l
1?ex =?xeξ <?x=ex > 1 +x.
o?§exnegationslash= 0§okex > 1 +x.
11,efprime(x) ≡k§áyf(x) = kx+b.
y2μ?F(x) = f(x)?kx
duFprime(x) = fprime(x)? k ≡ 0§a.?KF?ní?1§F(x) = f(x)? kx = b(?x ∈ (?∞,+∞)§
f(x) = kx+b.
79
12,y2?§x3?3x+c = 03[0,1]S?1küó?.
y2μ-f(x) = x3?3x+c
^?y{.f(x)3[0,1]Sküó?0 <x1 <x2 < 1.
d?f(x1) = f(x2) = 0§aa?n§73ξ ∈ (x1,x2)§|fprime(ξ) = 0=3ξ2? 3 = 0§)ξ = ±1§ù
ξ ∈ (x1,x2)? (0,1)g?.
b?¤á.=?§x3?3x+c = 03[0,1]S?1küó?.
13,e3[a,b]t|fprime(x)|greaterorequalslant|?prime(x)|,fprime(x) negationslash= 0§K|?f(x)|greaterorequalslant|(x)|.?y3
bracketleftbigg1
2,x
bracketrightbigg
t?arctanxlessorequalslant?ln(1 +x2)§d
dy23
bracketleftbigg1
2,1
bracketrightbigg
t±e? a¤áμarctanx?ln(1 +x2) greaterorequalslant pi4?ln2.
y2μ?3[a,b]t|fprime(x)|greaterorequalslant|?prime(x)|,fprime(x) negationslash= 0§f(x),?(x)3[a,b]t?§l
3[a,b]t?Y.
x,x+?x∈ [a,b],?x> 0§Kf(x),?(x)3[x,x+?x]t?Y??fprime(x) negationslash= 0.
d?ü?n§73ξ ∈ (x,x+?x)§|?(x+?x)(x)f(x+?x)?f(x) =?
prime(ξ)
fprime(ξ)=
(x)
f(x) =
prime(ξ)
fprime(ξ)§u′
vextendsinglevextendsingle
vextendsinglevextendsingle(x)
f(x)
vextendsinglevextendsingle
vextendsinglevextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle?prime(ξ)
fprime(ξ)
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 1=|?f(x)|greaterorequalslant|(x)|.
(arctanx)prime = 11 +x2,(ln(1 + x2))prime = 2x1 +x2§?3
bracketleftbigg1
2,x
bracketrightbigg
t§k2x > 1§K(ln(1 + x2))prime = 2x1 +x2 >
1
1 +x2 = (arctanx)
prime > 0§f(x) = ln(1+x2),?(x) = arctanx§Kdt?(?§3
bracketleftbigg1
2,x
bracketrightbigg
t§?arctanx =
|?arctanx|lessorequalslant|?ln(1 +x2)| =?ln(1 +x2).
3
bracketleftbigg1
2,1
bracketrightbigg
t?x§3[x,1]tkarctan1? arctanx =?arctanx lessorequalslant?ln(1 +x2) = ln(1 + 12)? ln(1 +
x2)=pi4?arctanxlessorequalslant ln2?ln(1 +x2)§l
arctanx?ln(1 +x2) greaterorequalslant pi4?ln2.
14,ef(x)3?mX(d)￥?kk.ê§=|fprime(x)|lessorequalslantM§Kf(x)3X￥Y.
y2μ?ef(x)3?mXt?§l
3Xt?Y§?|fprime(x)|lessorequalslantM,M > 0
x1,x2 ∈X§x1 <x2§Kf(x)3[x1,x2]t?Y?.
d.?KF￥n§?ξ ∈ (x1,x2)§|f(x2)?f(x1) = fprime(ξ)(x2?x1)§K|f(x2)?f(x1)| = |fprime(ξ)|(x2?
x1) lessorequalslantM(x2?x1)§u′é?ε> 0§δ = εM§K|x2?x1|<δ = εM?§|f(x2)?f(x1)|lessorequalslantM(x2?x1) <
ε¤á§u′f(x)3X￥Y.
80
§2,Vúa
1,|x|?§íeCqúaμ
tanx≈x;cosx·sinx≈x; n√1±x≈ 1± xn;ex ≈ 1 +x.
y2μ
(1) -f(x) = tanx§?|x|§^Cqúax0 = 0§u′f(x0) = 0,fprime(0) = sec2xvextendsinglevextendsinglex=0 = 1§l
f(x) ≈f(0) +fprime(0)(x?0)=?tanx≈x.
(2) -f(x) = cosx·sinx§?|x|§^Cqúax0 = 0§u′f(x0) = 0,fprime(0) = (?sin2x+ cos2x)vextendsinglevextendsinglex=0 =
1§l
f(x) ≈f(0) +fprime(0)(x?0)=?cosx·sinx≈x.
(3) -f(x) = n√1±x§?|x|§^Cqúax0 = 0§u′f(x0) = 1,fprime(0) = ±1n(1±x) 1n?1
vextendsinglevextendsingle
vextendsinglevextendsingle
x=0
=
±1n§l
f(x) ≈f(0) +fprime(0)(x?0)=?n√1±x≈ 1± xn.
(4) -f(x) = ex§?|x|§^Cqúax0 = 0§u′f(x0) = 1,fprime(0) = ex|x=0 = 1§l
f(x) ≈f(0) +fprime(0)(x?0)=?ex ≈ 1 +x.
2,|tan4oCq?.
)μdtK§tanx≈x§tan4o = tan pi45 ≈ pi45 ≈ 0.0698.
3,|√37Cq?.
)μ?√37 = √36 + 1 = 6
radicalbigg
1 + 136§a11K§√37 = 6
radicalbigg
1 + 136 ≈ 6
parenleftbigg
1 + 172
parenrightbigg
≈ 6.083.
4,?5-5¤à?o§?oàR§2H§H'R.
(1) y2μ?otYD≈ H
2
2R?
(2) 2H = 50§R = 100§|D.
)μ
(1)?D = R?√R2?H2§KD = R
1?
radicalBigg
1?
parenleftbiggH
R
parenrightbigg2?
.
qH'R§
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
parenleftbiggH
R
parenrightbigg2vextendsinglevextendsinglevextendsingle
vextendsinglevextendsingle§K
radicalBigg
1?
parenleftbiggH
R
parenrightbigg2
≈ 1?12
parenleftbiggH
R
parenrightbigg2
= 1?H
2
2R2§l
D≈R
bracketleftbigg
1?
parenleftbigg
1? H
2
2R2
parenrightbiggbracketrightbigg
=
H2
2R.
(2) D = R?√R2?H2 = 100?√1002?252 ≈ 3.175;D≈ H
2
2R =
252
200 = 3.125
5,?
g30.12§?ù?
0.05.|
g ?èyé?
ú?é?
.
)μ?
èS = pi4D2§K|^êO?
§Skyé?
|?S| ≈
vextendsinglevextendsingle
vextendsinglepi2D?D
vextendsinglevextendsingle
vextendsingle = pi2 × 30.12 × 0.05 ≈
2.3656(2)é?
vextendsinglevextendsingle
vextendsinglevextendsingle?S
S
vextendsinglevextendsingle
vextendsinglevextendsingle≈
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
2D?Dpi
4D
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle2?D
D
vextendsinglevextendsingle
vextendsinglevextendsingle≈ 0.33%.
6,?7á￥ND = 10.12§?
D = 0.05.O?￥NNè9ùyé?
§?é?
.
)μ?￥NèV = pi6D3§￥NNèV = pi6(10.12)3 ≈ 542.675(3)?
|^ê?
O§Vkyé?
|?V|≈
vextendsinglevextendsingle
vextendsinglepi2D2?D
vextendsinglevextendsingle
vextendsingle= pi2 ×10.122 ×0.05 ≈ 8.044(3)?
é?
vextendsinglevextendsingle
vextendsinglevextendsingle?V
V
vextendsinglevextendsingle
vextendsinglevextendsingle≈
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
2D
2?D
pi
6D
3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle3?D
D
vextendsinglevextendsingle
vextendsinglevextendsingle≈ 1.48%.
7,|e?ê3x = 0:VDmaμ
(1) f(x) = √1 +x
(2) f(x) = 11 +x
(3) f(x) = esinx(Dm?1kx3?)
81
(4) f(x) = cosx
(5) f(x) = lncosx(Dm?1kx6?)
(6) f(x) = ln(1 +x)
)μ
(1) f(x) = √1 +x,fprime(x) = 12(1+x)?12,fprimeprime(x) =?14(1+x)?32,···,f(n)(x) = (?1)n?1 (2n?3)!!2n (1+x)12?n
rx = 0?g?\t?a§kf(0) = 1,fprime(0) = 12,fprimeprime(0) =?14,···,f(n)(0) = (?1)n?1 (2n?3)!!2n
u′?êf(x) = √1 +x3x = 0VDmaμf(x) = 1 + x2? x
2
8 +···+
(?1)n?1 (2n?3)!!2n
n! x
n +
o(xn) = 1 + x2? x
2
8 +···+
(?1)n?1(2n?3)!!
n!·2n +o(x
n)
(2) f(x) = 11 +x = (1+x)?1,fprime(x) =?(1+x)?2,fprimeprime(x) = 2(1+x)?3,···,f(n) = (?1)n·n!(1+x)?(n+1)
rx = 0?g?\t?a§kf(0) = 1,fprime(0) =?1,fprimeprime(0) = 2,···,f(n)(0) = (?1)n ·n!
u′?êf(x) = 11 +x3x = 0VDmaμf(x) = 1?x+x2?···+ (?1)nxn +o(xn)
(3) 5?sinx?x d.
Kesinx = 1+sinx+ 12! sin2x+ 13! sin3x+o1(sin3x) = 1+(x? x
3
3! +o(x
3))+ 1
2(x+o(x
2))2 + 1
6(x+
o(x2))3 +o1(sin3x) = 1 +x? x
3
6 +
x2
2 +
x3
6 +o(x
3) = 1 +x+ x2
2 +o(x
3).
(4) f(x) = cosx,fprime(x) =?sinx,fprimeprime(x) =?cosx,···,f(k)(x) = cos
parenleftbigg
x+ k2pi
parenrightbigg
rx = 0?g?\t?a§kf(0) = 1,fprime(0) = 0,fprimeprime(0) =?1,···,f(2m)(0) = (?1)m,f(2m+1)(0) =
0,···(m∈Z+)
u′?êf(x) = cosx3x = 0VDmaμf(x) = 1? x
2
2 +···+ (?1)
n x2n
(2n)! +o(x
2n)
(5) f(x) = lncosx = 12 ln(1?sin2x) =?12
parenleftbigg
sin2x+ sin
4x
2 +
sin6x
3 +o(sin
6x)
parenrightbigg
=
12
bracketleftBiggparenleftbigg
x? x
3
3! +
x5
5! +o1(x
5)
parenrightbigg2
+ 12
parenleftbigg
x? x
3
3! +o2(x
3)
parenrightbigg4
+ 13parenleftbigx+o3(x2))parenrightbig6 +o(x6)
bracketrightBigg
=?x
2
2?
x4
12?
x6
45 +o(x
6)
(6) f(x) = ln(1 +x),fprime(x) = 11 +x = (1 +x)?1,fprimeprime(x) =?(1 +x)?2,···,f(n)(x) = (?1)n?1!(1 +x)?n
rx = 0?g?\t?a§kf(0) = 0,fprime(0) = 1,fprimeprime(0) =?1,···,f(n)(0) = (?1)(n?1)!
u′?êf(x) = ln(1 +x)3x = 0VDmaμf(x) = x? x
2
2 +···+ (?1)
n?1xn
n +o(x
n)
8,|?êlnx3x = 1VDma.
)μdtK(?§lnx = ln(1 +x?1) = (x?1)? (x?1)
2
2 +···+ (?1)
n?1 (x?1)n
n +o((x?1)
n).
9,|?ê√x3x = 1VDma￡Dmx3?¤.
):f(x) = √x,fprime(x) = 12(1 +x)?12,fprimeprime(x) =?14(1 +x)?32,fprimeprimeprime(x) = 38(1 +x)?52
rx = 1?g?\t?a§kf(1) = 1,fprime(1) = 12,fprimeprime(1) =?14,fprimeprimeprime(1) = 38
u′?êf(x) = √x3x = 1VDmaμf(x) = 1 + 12(x?1)? 18(x?1)2 + 116(x?1)3 +o((x?1)3).
10,òaP3(x) = 1 + 3x+ 5x2?2x3L¤x+ 1ê?a.
)μ?P3(x) = 1 + 3x+ 5x2?2x3,Pprime3(x) = 3 + 10x?6x2,Pprimeprime3 (x) = 10?12x,Pprimeprimeprime3 (x) =?12,P(4)3 = ··· =
P(n)3 = 0
rx =?1?g?\t?a§kP3(?1) = 5,Pprime3(?1) =?13,Pprimeprime3 (?1) = 22,Pprimeprimeprime3 (?1) =?12,P(4)3 = ··· =
P(n)3 = 0
u′P3(x) = 5?13(x+ 1) + 11(x+ 1)2?2(x+ 1)3.
82
11,|^VúaO? 3√7?o?ê.
)μ 3√7 = 2
parenleftbigg
1? 18
parenrightbigg1
3 ≈ 2
bracketleftBigg
1 + 13
parenleftbigg
18
parenrightbigg
+ 12! · 13
parenleftbigg1
3?1
parenrightbiggparenleftbigg
18
parenrightbigg2
+ 13! · 13
parenleftbigg1
3?1
parenrightbiggparenleftbigg1
3?2
parenrightbiggparenleftbigg
18
parenrightbigg3bracketrightBigg
≈
1.9130
< 2· 14! · 13 · 23 · 53 · 83
parenleftbigg1
8
parenrightbigg4
≈ 2.01×10?5.
12,|^Vúa|e4?μ
(1) lim
x→0
cosx?e?x
2
2 +
1
12x
4
x6
(2) lim
x→0
ex sinx?x(1 +x)
x3
(3) lim
x→∞
bracketleftbigg
x?x2 ln
parenleftbigg
1 + 1x
parenrightbiggbracketrightbigg
(4) lim
x→0
parenleftbigg1
x?
1
sinx
parenrightbigg
(5) lim
x→+∞
( 6√x6 +x5? 6√x6?x5)
(6) lim
x→0
1?cos(sinx)
2ln(1 +x2)
)μ
(1) |^Vúa§kcosx = 1? x
2
2 +
x4
4!?
x6
6! +o(x
6),e?x22 = 1? x2
2 +
x4
8?
x6
48 +ox
6§
Kcosx?e?x
2
2 +
1
12x
4 = 7
360x
6 +o(x6)§u′lim
x→0
cosx?e?x
2
2 +
1
12x
4
x6 =
7
360.
(2) |^Vúa§kex = 1+x+x
2
2 +o(x
2),sinx = x?x3
3! +o(x
3)§Kex sinx?x(1+x) = x3
3 +o(x
3)§
u′lim
x→0
ex sinx?x(1 +x)
x3 =
1
3.
(3) |^Vúa§kln
parenleftbigg
1 + 1x
parenrightbigg
= 1x? 12x2 + 13x3 +o
parenleftbigg 1
x3
parenrightbigg
Kx?x2 ln
parenleftbigg
1 + 1x
parenrightbigg
= 12? 13x +o
parenleftbigg1
x
parenrightbigg
§
u′ lim
x→∞
bracketleftbigg
x?x2 ln
parenleftbigg
1 + 1x
parenrightbiggbracketrightbigg
= 12.
(4) |^Vúa§ksinx = x? x
3
3! + o(x
3)§K1
x?
1
sinx =
sinx?x
xsinx =
x
3
3! +o(x
3)
x
parenleftbigg
x? x
3
3! +o(x
3)
parenrightbigg =
x6 +o(x)
1? x
2
6 +o(x
2)
§u′lim
x→0
parenleftbigg1
x?
1
sinx
parenrightbigg
= 0
(5)? 6√x6 +x5? 6√x6?x5 = x
parenleftbigg
1 + 1x
parenrightbigg1
6?x
parenleftbigg
1? 1x
parenrightbigg1
6
|^Vúa§k
parenleftbigg
1 + 1x
parenrightbigg1
6 = 1 + 1
6x?
5
72x2 +o
parenleftbigg 1
x2
parenrightbigg
,
parenleftbigg
1? 1x
parenrightbigg1
6 = 1? 1
6x?
5
72x2 +o
parenleftbigg 1
x2
parenrightbigg
§
K 6√x6 +x5? 6√x6?x5 = 13 +o
parenleftbigg1
x
parenrightbigg
§u′ lim
x→+∞
( 6√x6 +x5? 6√x6?x5) = 13.
(6) |^Vúa§kcos(sinx) = 1? sin
2x
2 +
sin4x
4! + o(sin
4x),ln(1 + x2) = x2? x4
2 + o(x
4)§
K1?cos(sinx)2ln(1 +x2) =
sin2x
bracketleftbigg
1? 112 sin2x+o(sin2x)
bracketrightbigg
4x2
parenleftbigg
1? 12x2 +o(x2)
parenrightbigg §l
lim
x→0
1?cos(sinx)
2ln(1 +x2) =
1
4
83
13,α,β§| lim
x→+∞
( 4√16x4?8x3 + 10x?7?αx?β) = 0.
)μ? 4√16x4?8x3 + 10x?7 = 2x· 4
radicalBigg
1 +
parenleftbigg
12x + 58x3? 716x4
parenrightbigg
= 2x?14+ 516x2? 732x3 +ε( lim
x→+∞
ε = 0)
4√16x4?8x3 + 10x?7?αx?β = (2?α)x?
parenleftbigg1
4 +β
parenrightbigg
+ 516x2? 732x3 +ε
dd?§?| lim
x→+∞
( 4√16x4?8x3 + 10x?7?αx?β) = lim
x→+∞
bracketleftbigg
(2?α)x?
parenleftbigg1
4 +β
parenrightbigg
+ 516x2? 732x3 +ε
bracketrightbigg
=
0§7Lα = 2,β =?14.
14,A§|4?lim
x→0
nradicalbigQ(x)?A
x 3§ù￥Q(x) = a0 +a1x+···+amx
m,a0 negationslash= 0,m?g,ê.
)μlim
x→0
nradicalbigQ(x)?A
x = limx→0
n√a0 +a1x+···+amxm?A
x = limx→0
n√a0
parenleftbigg
n
radicalbigg
1 + a1a
0
x+···+ ama
0
xm?A
parenrightbigg
x =
lim
x→0
n√a0
parenleftbigg
1 + 1n
parenleftbigga
1
a0x+···+
am
a0 x
m
parenrightbigg
+o(x)?A
parenrightbigg
x 3
n√a0?A = 0=A = n√a0§d?a= a1 ·
n√a0
na0,
84
§3,?ê,ü!à5?4?
1,y2e?êüN5μ
(1) y = x?sinx
(2) y =
parenleftbigg
1 + 1x
parenrightbiggx
(x> 0)
y2μ
(1)?y = f(x)3(?∞,+∞)S?Y?§fprime(x) = 1? cosx?q?1 lessorequalslant cosx lessorequalslant 1§fprime(x) greaterorequalslant 0§u
′y = x?sinx3(?∞,+∞)üNt,.
(2)?y =
parenleftbigg
1 + 1x
parenrightbiggx
§yprime =
parenleftbigg
1 + 1x
parenrightbiggxbracketleftbigg
ln(1 +x)?lnx? 11 +x
bracketrightbigg
qx> 0§
parenleftbigg
1 + 1x
parenrightbiggx
> 0§K?I?)ò￥af?ò.
-f(x) = lnx3[x,1+x](é?x> 0)tA^.?KF?n§kln(1+x)?lnx = 1ξ(1+x?x) = 1ξ(x<ξ<
1+x)§u′1x > 1ξ > 11 +x§ln(1+x)?lnx = 1ξ > 11 +x=ln(1+x)?lnx? 11 +x > 0(?x> 0)§
dd?yprime > 0§l
y =
parenleftbigg
1 + 1x
parenrightbiggx
3(0,+∞)tüNO\.
2,üN?êê′?7?üNo
)μ.
~μy = x33(?∞,+∞)tüNt,§yprime = 3x2%?üN.
3,y2e? aμ
(1) x> sinx> 2pix
parenleftBig
0 <x< pi2
parenrightBig
(2) x? x
3
6 > sinx>x(x< 0)
(3) x? x
2
2 < ln(1 +x) <x(x> 0)
(4) tanx>x+ x
3
3
parenleftBig
0 <x< pi2
parenrightBig
(5) 2√x> 3? 1x(x> 1)
(6) 12p?1 lessorequalslantxp + (1?x)p lessorequalslant 1(0 lessorequalslantxlessorequalslant 1,p> 1)
y2μ
(1) f(x) = x?sinx§d11K§f(x)3
parenleftBig
0,pi2
parenrightBig
SüNt,§qf(0) = 0§é?x∈
parenleftBig
0,pi2
parenrightBig
§kf(x) >
f(0) = 0=x?sinx> 0§l
x> sinx
parenleftBig
0 <x< pi2
parenrightBig
g(x) = sinxx,g
parenleftBigpi
2
parenrightBig
= 2pi,gprime(x) = xcosx?sinxx2
parenleftBig
0 <x< pi2
parenrightBig
5?u(x) = xcosx? sinx
parenleftBig
0 <x< pi2
parenrightBig
u(0) = 0§duuprime(x) =?xsinx < 0
parenleftBig
0 <x< pi2
parenrightBig
§
x ∈
parenleftBig
0,pi2
parenrightBig
§u(x)üNeü=u(x) < u(0) = 0
parenleftBig
0 <x< pi2
parenrightBig
§dd§gprime(x) < 0
parenleftBig
0 <x< pi2
parenrightBig
§
g(x)3
parenleftBig
0,pi2
parenrightBig
tüNeü§u′g(x) >g
parenleftBigpi
2
parenrightBig
= 2pi=sinxx > 2pi,x∈
parenleftBig
0 <x< pi2
parenrightBig
§Ksinx> 2pix§
l
x> sinx> 2pix
parenleftBig
0 <x< pi2
parenrightBig
(2) f(x) = x? sinx§d11K§f(x)3(?∞,0)SüNt,§qf(0) = 0§é?x ∈ (?∞,0)§
kf(x) <f(0) = 0=x?sinx> 0§l
x< sinx(x< 0)?
g(x) = x? x
3
6?sinx,g(0) = 0,g
prime(x) = 1? x2
2?cosx
2h(x) = 1?x
2
2?cosx(x< 0)?h(0) = 0§duh
prime(x) =?x+sinx> 0§x∈ (?∞,0)?§h(x)ü
Nt,=h(x) < h(0) = 0(x < 0)§dd§gprime(x) < 0(x < 0)§g(x)3(?∞,0)tüNeü§u
′g(x) >g(0) = 0=x? x
3
6?sinx> 0(x< 0)§Kx?
x3
6 > sinx§l
x?
x3
6 > sinx>x(x< 0)
85
(3) f(x) = ln(1+x)?x,g(x) = ln(1+x)?x+x
2
2 (x> 0)§f
prime(x) = 1
1 +x?1 =?
x
1 +x < 0(x> 0)§
Kf(x)3(0,+∞)SüNeü§qf(0) = 0§é?x> 0§kf(x) <f(0) = 0=ln(1 +x) <x(x> 0)?
gprime(x) = 11 +x?1 +x = x
2
1 +x > 0(x> 0)
g(x)3(0,+∞)tüNt,§qg(0) = 0§u′g(x) > g(0) = 0=ln(1 + x) > x? x
2
2 (x > 0)§l
x? x
2
2 < ln(1 +x) <x(x> 0)
(4) f(x) = tanx?x?x
3
3
parenleftBig
0 <x< pi2
parenrightBig
§fprime(x) = sec2x?1?x2 = tan2x?x2 = (tanx+x)(tanx?x)§
q?(tanx?x)prime = sec2x? 1 = tan2x lessorequalslant 0
parenleftBig
x∈
parenleftBig
0,pi2
parenrightBigparenrightBig
§Ktanx?x3
parenleftBig
0,pi2
parenrightBig
SüNt,§
é?x∈
parenleftBig
0,pi2
parenrightBig
§ktanx?x> 0
parenleftBig
0 <x< pi2
parenrightBig
u′fprime(x) = (tanx+x)(tanx?x) > 0
parenleftBig
0 <x< pi2
parenrightBig
§dd?§f(x)3
parenleftBig
0,pi2
parenrightBig
tüNt,§qf(0) =
0§u′f(x) >f(0) = 0=tanx?x? x
3
3 > 0
parenleftBig
0 <x< pi2
parenrightBig
§l
tanx>x+ x
3
3
parenleftBig
0 <x< pi2
parenrightBig
(5) f(x) = 2√x?3 + 1x(x> 1)§fprime(x) = 1√x? 1x2 = x
3
2?1
x2 > 0(x> 1)§u′f(x)3(1,+∞)tü
Nt,§qf(1) = 0§u′f(x) >f(1) = 0=2√x?3 + 1x > 0(x> 1)§l
12p?1 lessorequalslant xp + (1?x)p lessorequalslant
1(0 lessorequalslantxlessorequalslant 1,p> 1)
(6) f(x) = xp + (1?x)p(0 lessorequalslantxlessorequalslant 1,p> 1)§fprime(x) = pxp?1?p(1?x)p?1§
-fprime(x) = pxp?1?p(1?x)p?1 = 0§)x = 12§'f(0) = 1,f(1) = 1,f
parenleftbigg1
2
parenrightbigg
= 12p?1§dd
min
0lessorequalslantxlessorequalslant1
f(x) = 12p?1,max
0lessorequalslantxlessorequalslant1
f(x) = 1 §l
12p?1 lessorequalslantxp + (1?x)p lessorequalslant 1(0 lessorequalslantxlessorequalslant 1,p> 1)
4,(?e?êt,!eü?mμ
(1) y = x3?6x
(2) y = 2x3?3x2?12x+ 1
(3) y = x4?2x3
(4) y = x+ sinx
(5) y = 2x1 +x2
(6) y = 2x2?sinx
(7) y = xne?x(n> 0,xlessorequalslant 0)
)μ
(1)?yprime = 3x2?6 = 3(x2?2)§7:x = ±√2
x<?√2?x>√2?§yprime > 0§?êt,??√2 <x<√2?§yprime < 0§?êeü.
l
3?m(?∞,?√2)uniontext(√2,+∞)t?êt,?3?m(?√2,√2)t?êeü.
(2)?yprime = 6x2?6x?12 = 6(x2?x?2) = 6(x?2)(x+ 1)§7:x =?1,x = 2
x<?1?x> 2?§yprime > 0§?êt,??1 <x< 2?§yprime < 0§?êeü.
l
3?m(?∞,?1)uniontext(2,+∞)t?êt,?3?m(?1,2)t?êeü.
(3)?yprime = 4x3?6x2 = 2x2(2x?3)§7:x = 0,x = 32
x> 32?§yprime > 0§?êt,?x< 32?§yprime lessorequalslant 0?=3x = 0?yprime = 0§?êeü.
l
3?m
parenleftbigg3
2,+∞
parenrightbigg
t?êt,?3?m
parenleftbigg
∞,32
parenrightbigg
t?êeü.
(4)?yprime = 1 + cosxlessorequalslant 0§?ê3(?∞,+∞)t?êt,.
(5)?yprime = 2(1?x
2)
(1 +x2)2§7:x = ±1
x<?1?x> 1?§yprime < 0§?êeü??1 <x< 1?§yprime > 0§?êt,.
l
3?m(?∞,?1)uniontext(1,+∞)t?êeü?3?m(?1,1)t?êt,.
86
(6)?yprime = 4x?cosx,yprimeprime = 4 + sinx> 0§Kyprime3(?∞,+∞)tüNt,.
qyprime(0) =?1,yprime
parenleftBigpi
2
parenrightBig
= 2pi§K3
parenleftBig
0,pi2
parenrightBig
Sk:x0÷vyprime(x0) = 0=4x0 = cosx0
x>x0?§yprime > 0§?êt,?x<x0?§yprime < 0§?êeü.
l
3?m(x0,+∞)t?êt,?3?m(?∞,x0)t?êeü.
(7)?yprime = nxn?1e?x?xne?x = xn?1e?x(n?x)
n> 0,x> 0§xn?1 > 0,e?x > 0§Kxn?1e?x > 0
0 <x<n?§yprime > 0§?êt,?x>n?§yprime < 0§?êeü.
l
3?m(0,n)t?êt,?3?m(n,+∞)t?êeü.
5,|e?ê4?μ
(1) y = x?ln(1 +x)
(2) y = √xlnx
(3) y = x+ 1x
(4) y = sin3x+ cos3x
(5) y = cosx+ coshx
)μ
(1)?yprime = 1? 11 +x = x1 +x,yprimeprime = 1(1 +x)2 > 0
d?ê(?1,+∞)§K7:?x = 0§?ê?U3ù:k4?§u′x = 0′?ê4:§4
y = 0.
(2)?yprime = 1√x + 12√x lnx = 12√x(lnx+ 2),yprimeprime =? 1
2x32
+ 1
2x32
1
4x32
lnx =?lnx
232
7:?x = e?2§?ê?U3ù:k4?§qyprimeprime|x=e?2 > 0§u′x = e?2′?ê4:§4?
y =?2e.
(3)?yprime = 1? 1x2,yprimeprime =? 1x3 > 0
d?ê(?∞,0)uniontext(0,+∞)§K7:?x = ±1§?ê?U3ùü:k4?§qyprimeprime|x=1 = 1 >
0,yprimeprime|x=?1 =?1 < 0§u′x = 1′?ê4:§4y = 2?x =?1′?ê4?:§4
y = 2.
(4)?yprime = 3sinxcosx(sinx?cosx) = 32 sin2x(sinx?cosx),yprimeprime = 3cos2x(sinx?cosx)+ 32 sin2x(cosx+
sinx)
7:?x = kpi + pi4,x = kpi2 (k ∈ Z)§qyprimeprime|x=2kpi =?3 < 0,yprimeprime|x=2kpi+pi4 = 32√2 > 0,yprimeprime|x=2kpi+pi2 =
3 < 0,yprimeprime|2kpi+pi = 3 > 0,yprimeprime|x=2kpi+5pi
4
=?32√2 < 0,yprimeprime|x=2kpi+3pi
2
= 3 > 0§
u′x = 2kpi?§k4y = 1?x = 2kpi + pi2?§k4y = 1?x = 2kpi + 5pi4?§k4?
y =?
√2
2?
x = 2kpi + pi4?§k4?y =
√2
2?x = 2kpi + pi?§k4?y =?1?x = 2kpi +
3pi
2?§k4
y =?1.
(5)?yprime = 1?sinx+ sinhx§?′|7:§d?sinx+ e
x?e?x
2 = 0′?x = 0′7:
d?sinx,sinhx3
parenleftBig
pi2,pi2
parenrightBig
üN5§ù7:′.
yprimeprime =?cosx+coshx,yprimeprime(0) = 0;yprimeprimeprime = sinx+sinhx,yprimeprimeprime(0) = 0;y(4) = cosx+coshx,y(4)(0) = 2 > 0§
u′x = 0′?ê4:§4y = 2.
6,ef(x)3:x0?k?n ?Yê§fprime(x0) = fprimeprime(x0) = ··· = f(n?1)(x0) = 0,f(n)(x0) negationslash= 0§@
onê?§f(x0)?4n?óê
f(n)(x0) > 0?§f(x0)?4n?óê
f(n)(x0) <
0?§f(x0)?4.
y2μòf(x)3x = x0:^VúaDmμf(x) = f(x0) + fprime(x0)(x? x0) + f
primeprime(x
0)
2 (x? x0)
2 + ··· +
f(n)(x0)
n! (x?x0)
n +o((x?x
0)
n)
fprime(x0) = fprimeprime(x0) = ··· = f(n?1)(x0) = 0§f(x) = f(x0) + f
(n)(x
0)
n! (x?x0)
n +o((x?x
0)
n)
87
x→x0?§o((x?x0)n) → 0§xCx0?§=|x?x0|?§f(x)?f(x0)?f
(n)(x
0)
n! (x?
x0)nk?ó?ò
ef(n)(x0) > 0§
(1) nê?§ex>x0§K(x?x0)n > 0§u′f
(n)(x
0)
n! (x?x0)
n > 0§l
f(x)?f(x
0) > 0=f(x) >
f(x0)?
ex<x0§K(x?x0)n < 0§u′f
(n)(x
0)
n! (x?x0)
n < 0§l
f(x)?f(x
0) < 0=f(x) <f(x0)
df(x0)?′4?.
(2) n?óê?§xCx0§x > x0§?′x < x0§?k(x?x0)n > 0§d?f
(n)(x
0)
n! (x?
x0)n > 0(xnegationslash= x0)§l
f(x)?f(x0) > 0§=3x0,?S§ekf(x) >f(x0)§ùL2f(x0)′
4?.
ef(n)(x0) < 0§
(1) nê?§ex>x0§K(x?x0)n > 0§u′f
(n)(x
0)
n! (x?x0)
n < 0§l
f(x)?f(x
0) < 0=f(x) <
f(x0)?
ex<x0§K(x?x0)n < 0§u′f
(n)(x
0)
n! (x?x0)
n > 0§l
f(x)?f(x
0) > 0=f(x) >f(x0)
df(x0)?′4?.
(2) n?óê?§xCx0§x > x0§?′x < x0§?k(x?x0)n > 0§d?f
(n)(x
0)
n! (x?
x0)n < 0(xnegationslash= x0)§l
f(x)?f(x0) < 0§=3x0,?S§ekf(x) <f(x0)§ùL2f(x0)′
4.
7,|e?ê3mtú??μ
(1) y = |x2?3x+ 2|,[?10,10]
(2) y = e|x?3|,[?5,5]
)μ
(1) y =
(x?2)(x?1),xlessorequalslant 1
(x?2)(x?2),1 <xlessorequalslant 2
(x?2)(x?1) x> 2
§
|§yprime =
2x?3,x< 1
3,x = 1
2x+ 3,?1 <x< 2
3,x = 2
2x?3,x> 2
§K7:x = 32§ê?3:x = 1,x = 2
qy(?10) = 132,y(1) = 0,y
parenleftbigg3
2
parenrightbigg
= 54,y(2) = 0,y(10) = 72§?ê′132§??′0.
(2) y =
braceleftbigg ex?3,xgreaterorequalslant 3
e3?x,x< 3 §|§y
prime =
ex?3,x> 3
3,x = 3
e3?x,x< 3
§w,?7:
qy(?5) = e8,y(3) = 1,y(5) = e2§?êe8§?1.
8,c′tAB??l?100úp§ó?C?A40úp§ACR?uAB.8?3AB￥m?:D?ó?C^
ú′￡?5-21¤§|l?A?B$àó?C¤^$¤.ˉD:AT3o?z?úpc′$
¤?ú′$¤?'′3:5.
)μ|AD| = xúp§K|DB| = 100?xúp?zúpc′$¤?3t§Kzúpú′$¤?5t§o$¤
yt
Kyt = √x2 + 1600(5t)+(100?x)(3t)=y = 5√x2 + 1600+3(100?x)§u′yprime = 5x?3
√x2 + 1600
√x2 + 1600,yprimeprime =
8000
(x2 + 1600)32
> 0§7:?x = 30§?x = 30?4:§D:A3?A30úp?.
9,r
7?¤Y/7^.ˉY/ú°§ ?èo
)μ
7 R§Y/!°?O?x,y§Kradicalbigx2 +y2 = 2R§u′y = √4R2?x2§l
S =
88
xy = x√4R2?x2
KSprime = 4R
2?2x2
√4R2?x2,Sprimeprime = 2x
3?12R2x
(4R2?x2)32
§7:?x = √2R§d?Sprimeprime < 0§Kx = √2R?4?:§d
x = y = √2R§Y/!°t√2R?§ ?è.
10,S = (x?a1)2 + (x?a2)2 +···+ (x?an)2.ˉx§S?o
)μSprime = 2[nx? (a1 + a2 + ··· + an)],Sprimeprime = 2n > 0§7:?x = a1 +a2 +···+ann §?x?4:§=
x = a1 +a2 +···+ann?§S?.
11,
/G?§?ùNè?V§üà?ázüèda§y?ázü?db§
ˉG?úp' u??§Edo
)μd
/G?D§p?H§KV = 14piD2H§u′H = 4VpiD2
EdG = 2a
parenleftBigpi
4D
2
parenrightBig
+bpiDH = pi2aD2+b4VD §KGprime = piaD?4bVD2 §7:?D = 3
radicalbigg4bV
api,D<
3
radicalbigg4bV
api?§G
prime <
0?D> 3
radicalbigg4bV
api?§G
prime > 0§KD = 3
radicalbigg4bV
api ′4:§l
′?:.
u′DH = D4V
piD2
= piD
3
4V =
b
a=G?p'?
b
a?§Ed.
12,^R
/c?§}
%?α
÷/?¤|ì.ˉα§|ìNèo
)μdK§{eü?
%?x = 2pi?α§|ì.±?Rx = R(2pi?α)§.Rx2pi§ùp
h =
radicalBigg
R2?
parenleftbiggRx
2pi
parenrightbigg2
= R2pi
radicalbig
4pi2?x2(x > 0)§u′|ìNè?V = 13pi
parenleftbiggRx
2pi
parenrightbigg2
· R2pi
radicalbig
4pi2?x2 =
R3
24pi2x
2radicalbig4pi2?x2(x> 0)
UK§?I?x§?êf(x) = x4(4pi2?x2).
fprime(x) = 16pi2x3?6x5,fprimeprime(x) = 48pi2x2?30x4§7:?x = 2pi
radicalbigg2
3§?f
primeprime
parenleftBigg
2pi
radicalbigg2
3
parenrightBigg
< 0§x = 2pi
radicalbigg2
3?
4?:§?
}
%A?α = 2pi
parenleftBigg
1?
radicalbigg2
3
parenrightBigg
§¤?|ìNè.
13.,?a§p?hn/§á|ùSY/?è.
)μùSY/!°?O?b,c
Kd?§ba = h?ch =b = h?ch a§u′S = bc = ach?ch = ahc?ac
2
h §KS
prime = ah?2ac
h,S
primeprime =
2ah < 0§7:?c = h2§u′c = h2?4?:§d?b = a2§l
è?S = bc = ah4,
14,?l§ár§ü?§|±ùü>¤?¤Y/?è.
)μdY/?x§K°?l?x
S = x(l?x) = lx?x2§KSprime = l?2x,Sprimeprime =?2 < 0§7:?x = l2§?x = l2?4?:§?dx = l2?§
Y/?�S = l
2
4"
15,Suy
x
2
a2 +
y2
b2 = 1§
>21u?Y/.
)μd?¤|Y/?xu(x,0)§Kù?yu
parenleftbigg
0,ba
radicalbig
a2?x2
parenrightbigg
dY/?è?S§K14S = x· ba
radicalbig
a2?x2§l
S = 4ba
radicalbig
a2?x2§
Ksprime = 4ba · a
2?2x2
√a2?x2,Sprimeprime = 4ba · 2x
3?3a2x
(a2?x2)32
§7:?x =
√2
2 a§d?S
primeprime < 0§Kx =
√2
2 a?S4:§
u′x =
√2
2 a?Y/?è§è?S = 2ab.
16,|:M(p,p)y2 = 2px?á?l.
)μ:M(p,p)y2 = 2pxt:(x,y)?l?d =radicalbig(x?p)2 + (y?p)2 =
radicalBiggparenleftbigg
y2
2p?p
parenrightbigg2
+ (y?p)2 =
radicalBigg
y4
4p2 + 2p
2?2py
89
f(y) = y
4
4p2 + 2p
2?2py§Kfprime(y) = 1
p2 (y
3?2p3),fprimeprime(y) = 3y2
p2 > 0§7:?y =
3√2p§?§ò′f(y)4
?:§?d¤|?á?l?d =
radicalBig
f( 3√2p) = |p|
radicalBig
2 + 2?23?234,
17,`E±u = 20?/??Y?àê1§ì?3ù?h = 82kˉE±v = 16?/??Y?H
ê1§ˉüE?l?Co
):x? üE?l?C§üES?§KS =radicalbig(82?16x)2 + (20x)2 = √656x2?2624x+ 6724
-f(x) = 656x2?2624x+ 6724§|ù??"Kfprime(x) = 1312x?2624,fprimeprime(x) = 1312 > 0§7:?x = 2?
§?f(x)4?:§K2? üE?l?C§d?S = 10√41.
18,2/t-?§-t?Pú6.??N?/? TXê?μ"y\F§|?Nm?￡?.ˉdY2
Y§^o￡?5-22¤o
)μaK§kF cos? = μ(PG?F sin?)=F = μPGcos?+μsin?
-y = cos?+μsin?§?|F?§|y
dyprime =?sin?+μcos?,yprimeprime =?cosμsin?§7: = arctanμ§d?yprimeprime < 0§L2? = arctanμ?§y
§l
F??§=^.
19,X?5-23¤?§k`!ˉü)èü^?C?ì§ˉC?ìMA3§¤^?>o
)μMA3?`Rl?xúp?§¤^?>?l
d?§l = √1 +x2 + radicalbig2.252 + (3?x)2§Klprime = x√1 +x2 + x?3radicalbig2.252 + (3?x)2,lprimeprime = 1
(1 +x2)32
+
2.25
(2.252 + (3?x)2)32
> 0§7:?x = 1.2§?:§=x = 1.2úp?§¤^?>.
20,y
x
2
a2 +
y2
b2 = 1ü?IOuA,Bü:§
(1) |ABü:m?l?
(2) |?OAB??è.
)μ?:?(x,y)§Kk =?b
2x
a2y§u′§?Y?y =?
b2x
a2y(X?x)§
5§?:M(x.y)31§3ü?I?t O?a
2
x,
b2
y§K
(1) ¤|ABü:m?l?d =
radicalBigg
a4
x2 +
b4
y2 = a
radicalbigg
a2
x2 +
b2
a2?x2
-f(x) = a
2
x2 +
b2
a2?x2§?|d??§?I|f(x)??.
dfprime(x) =?2a
2
x3 +
2b2x
(a2?x2)2,f
primeprime(x) = 6a2
x4 +
2a2b2 + 6b2x2
(a2?x2)3 > 0§?dux ∈ [0,a],x
2 lessorequalslant a2§K7:
÷vx2 = a
3
a+b?d?f(x)??§=d??§?á?l?d = a
radicalbigf(x) = a+b.
(2) UK§kS = 12 · a
2
x ·
ab√
a2?x2 =
a3b
2x√a2?x2§êg(x) = x
2(a2?x2)
|S??§|g(x)
dgprime(x) = 2a2x?4x3,gprimeprime(x) = 2a2?12x2§7:?x = a√2?d?gprimeprime(x) < 0§=x = a√2?g(x)?
§l
S??§??è?S = ab.
21,êxα(α> 190 <α< 1),ex,lnx,xlnx3(0,+∞)Sà5.
)μf(x) = xα,fprime(x) = αxα?1,fprimeprime(x) = α(α?1)xα?2
α> 1?§fprimeprime(x) > 0§Kxα3(0,+∞)Seà?0 <α< 1?§fprimeprime(x) < 0§Kxα3(0,+∞)Stà.
f(x) = ex,fprime(x) = ex,fprimeprime(x) = ex > 0(x> 0)§Kex3(0,+∞)Seà
f(x) = lnx,fprime(x) = 1x,fprimeprime(x) =? 1x2 < 0(x> 0)§Klnx3(0,+∞)Stà
f(x) = xlnx,fprime(x) = 1 + lnx,fprimeprime(x) = 1x > 0(x> 0)§Kxlnx3(0,+∞)Seà
22,e?êà5ú$:μ
(1) y = 3x2?x3
(2) y = a
2
a2 +x2 (a> 0)
90
(3) y = x+ sinx
(4) y = √1 +x2
)μ
(1) yprime = 6x?3x2,yprimeprime = 6?6x§yprimeprime = 0x = 1§LXeμ
x (?∞,1) (1,+∞)
yprimeprime?ò + -
y eà tà
§$:
I?(1,2)
(2) yprime =? 2ax(a2 +x2)2,yprimeprime = 2a
2(3x2?a2)
(a2 +x2)3 §y
primeprime = 0x = ±
√3
3 a§LXeμ
x
parenleftbigg
∞,?
√3
3 a
parenrightbigg parenleftbigg
√3
3 a,
√3
3 a
parenrightbigg parenleftbigg√3
3 a,+∞
parenrightbigg
yprimeprime?ò + - +
y eà tà eà
§$:?I?
parenleftbigg
√3
3 a,
3
4
parenrightbigg
,
parenleftbigg√3
3 a,
3
4
parenrightbigg
(3) yprime = 1 + cosx,yprimeprime =?sinx§yprimeprime = 0x = kpi(k∈Z)§LXeμ
x ((2k?1)pi,2kpi) (2kpi,(2k+ 1)pi) ((2k+ 1)pi,2(k+ 1)pi)
yprimeprime?ò + - +
y eà tà eà
§$:?I?(kpi,kpi)
(4) yprime = x√1 +x2,yprimeprime = 1
(1 +x2)32
§Kyprimeprime > 0§?ê′eà§l
$:.
23,y2-?y = x+ 1x2 + 1k?uótn?$:.
y2μyprime = 1?2x?x
2
(x2 + 1)2,y
primeprime = 2(x?1)(x+ 2?
√3)(x+ 2 +√3)
(x2 + 1)3
-yprimeprime = 0§x1 = 1,x2 =?2 +√3,x3 =?2?√3
x <?2?√3?§yprimeprime < 0??2?√3 < x <?2 +√3?§yprimeprime > 0??2 +√3 < x <?1?§yprimeprime < 0?
x>?1?§yprimeprime > 0
u′-?3x1,x2,x3?kn?$:A(1,1),B
parenleftbigg
2 +√3,
√3 + 1
4
parenrightbigg
,C
parenleftbigg
(2 +√3),1?
√3
4
parenrightbigg
LA,B¤?y = 14x+ 34§òC:?I?\t§§1?
√3
4 =
2?√3
4 +
3
4 =
1?√3
4 =C÷v
d?§§K-?y = x+ 1x2 + 1k?uótn?$:.
24,ef(x)′eà?ê￡eà?ꤧfprime(x0)3§K
f(x) greaterorequalslantf(x0) +fprime(x0)(x?x0)f(x) >f(x
0) +f
prime(x
0)(x?x0)
bracerightbigg
(xnegationslash= x0).
y2μx?f(x)S:§xnegationslash= x0￡x>x0¤
-x1 = x+x02 §df(x)?eà?ê§Kf(x)?f(x0)x?x
0
greaterorequalslant f(x1)?f(x0)x
1?x0
x2 = x1 +x02 §df(x)?eà?
ê§Kf(x1)?f(x0)x?x
0
greaterorequalslant f(x2)?f(x0)x
2?x0
x3 = x2 +x02 §df(x)?eà?ê§Kf(x2)?f(x0)x?x
0
greaterorequalslant f(x3)?f(x0)x
3?x0
Xd?1e§?ê{xn}§|xn?x0| = |x?x0|2n → 0(n→∞)§Kxn →x0(n→∞)§?f(xn)?f(x0)x
n?x0
greaterorequalslant
f(xn+1)?f(x0)
xn+1?x0
qfprime(x0)3§K lim
n→∞
f(xn)?f(x0)
xn?x0 = limxn→x0
f(xn)?f(x0)
xn?x0 = f
prime(x
0)
qf(x)?f(x0)x?x
0
greaterorequalslant f(xn)?f(x0)x
n?x0
§Kd4?5?§f(x)?f(x0)x?x
0
greaterorequalslantfprime(x0)§l
f(x) greaterorequalslantf(x0)+fprime(x0)(x?
x0)
ón?y§ef(x)′eà?ê§Kf(x) >f(x0) +fprime(x0)(x?x0).
25,ef(x)′eà?ê§K?f(x)′tà?ê.
y2μ?f(x)′eà?ê§Kf(x)3[a,b]t?Y§é[a,b]￥ü:x1,x2§ekf
parenleftBigx1 +x2
2
parenrightBig
lessorequalslant f(x1) +f(x2)2 §
u′?f
parenleftBigx1 +x2
2
parenrightBig
greaterorequalslant?f(x1) +f(x2)2 = (?f(x1)) + (?f(x2))2 §l
f(x)′tà?ê.
26,(1) efn(x)′eà?ê§ˉF(x) = min
n
{fn(x)}′?′eà?êo
91
(2) ef(x),g(x)′eà?ê§ˉf(x) +g(x)′?′eà?êo
(3) `2ng?ê?′eà?ê.
)μ
(1).
~μ
f1(x) = 1x,f2(x) = x2(x> 0)?§f1(x),f2(x)?′eà?ê§F(x) = min
braceleftbigg1
x,x
2
bracerightbigg
3(1,1):?÷
veà?ê§=F(x)?′eà?ê.
f1(x) = x2,f2(x) = x
2
2?§f1(x),f2(x)?′eà?ê§?F(x) = min
braceleftbigg
x2,x
2
2
bracerightbigg
= x
2
2 ′eà?ê.
(2) f(x) +g(x)′eà?ê.
f(x),g(x)′eà?ê§Kf
parenleftBigx1 +x2
2
parenrightBig
lessorequalslant f(x1) +f(x2)2,
g
parenleftBigx1 +x2
2
parenrightBig
lessorequalslant g(x1) +g(x2)2 §u′(f+g)
parenleftBigx1 +x2
2
parenrightBig
= f
parenleftBigx1 +x2
2
parenrightBig
+g
parenleftBigx1 +x2
2
parenrightBig
lessorequalslant f(x1) +f(x2)2 +
g(x1) +g(x2)
2 =
1
2[(f +g)(x1) + (f +g)(x2)]=f(x) +g(x)′eà?ê.
(3) f(x) = ax3 +bx2 +cx+d(anegationslash= 0)§Kfprime(x) = 3ax2 + 2bx+c,fprimeprime(x) = 6ax+ 2b
u′§
a> 0?§x>? b3a?§fprimeprime(x) > 0§f(x)′eà?ê?x<? b3a?§fprimeprime(x) < 0§f(x)′tà?ê
a< 0?§x>? b3a?§fprimeprime(x) < 0§f(x)′tà?ê?x<? b3a?§fprimeprime(x) > 0§f(x)′eà?ê
Kf(x)?′eà?ê§3x =? b3a?k$:.
27,X?àJ?êh> 0§?U|-?y = h√pie?h2x23x = ±σ(σ> 0,σ~ê)?k$:.
)μyprime =?2h
3
√pixe?h2x2,yprimeprime = 2h
3
√pie?h2x2(2h2x2?1)
-yprimeprime = 0§Kx = ± 1√2h
x <? 1√2h?§yprimeprime > 0§-?eà?? 1√2h < x < 1√2h?§yprimeprime < 0§-?tà?x > 1√2h?§yprimeprime >
0§-?eà
K3x = ± 1√2h?kü?$:§u′± 1√2h = ±σ§qh,σ> 0§Kh = 1√2σ.
28,|y = x
2
x2 + 14?9$:§?|$:?§.
)μyprime = 2x(1 +x2)2,yprimeprime = 2?6x
2
(1 +x2)3
7:?x = 0§?eLμ
x (?∞,0) 0 (0,+∞)
yprime?ò - 0 +
y arrowsoutheast 4?0 arrownortheast
Kx = 0§yk4?y = 0.
-yprimeprime = 0§Kx = ±
√3
3 §?eLμ
x
parenleftbigg
∞,?
√3
3
parenrightbigg parenleftbigg
√3
3,
√3
3
parenrightbigg parenleftbigg√3
3,+∞
parenrightbigg
yprimeprime?ò - + -
y tà eà tà
$:?
parenleftbigg
√3
3,
1
4
parenrightbigg
,
parenleftbigg√3
3,
1
4
parenrightbigg
.
3$:
parenleftbigg
√3
3,
1
4
parenrightbigg
§?y? 14 =
2
√3
3parenleftbigg
1
3 + 1
parenrightbigg2
parenleftbigg
x+
√3
3
parenrightbigg
=3√3x+ 8y+ 1 = 0?
92
3$:
parenleftbigg√3
3,
1
4
parenrightbigg
§?y? 14 =
2√3
3parenleftbigg
1
3 + 1
parenrightbigg2
parenleftbigg
x?
√3
3
parenrightbigg
=3√3x?8y?1 = 0.
29,e?ê?/μ
(1) y = x3?6x
(2) y = 3x1 +x2
(3) y = 5e?x2
(4) y = e
x +e?x
2
(5) y = 1x2?1
(6) y = ln 1 +x1?x
(7) y = (x?1)2(x+ 2)3
(8) y = (x?1)
3
(x+ 1)3
(9) y = x
2?2x?3
x2 + 1
(10) y = x+ arctanx
)μ
(1) (i)(?∞,+∞)§′ê§-?'u:é?.
(ii) yprime = 3x2?6,yprimeprime = 6x§x = ±√2?§yprime = 0?x = 0?§yprimeprime = 0.
(iii) LXeμ
x (?∞,?√2)?√2 (?√2,0) 0 (0,√2) √2 (√2,+∞)
yprime + 0 - - - 0 +
yprimeprime - - - 0 + + +
y tàarrownortheast 44√2 tàarrowsoutheast 0 eàarrowsoutheast 44√2 eàarrownortheast
a45
a54
-√6-√2 0
√6√2
x
y
(2) (i)(?∞,+∞)§′ê§-?'u:é?.
(ii) yprime = 3(1?x
2)
(1 +x2)2,y
primeprime = 6x(x2?3)
(1 +x2)3 §x = ±1?§y
prime = 0?x = 0,x = ±√3?§yprimeprime = 0.
(iii) LXeμ
x (?∞,?√3)?√3 (?√3,?1) -1 (?1,0) 0 (0,1) 1 (1,√3) √3 (√3,+∞)
yprime - - - 0 + + + 0 - - -
yprimeprime - 0 + + + 0 - - - 0 +
y tàarrowsoutheast?34√3 eàarrowsoutheast 4? eàarrownortheast 0 tàarrownortheast 4 tàarrowsoutheast 34√3 eàarrowsoutheast
32 32
93
(iv) x→∞?§y → 0§y = 0′-??^Y2ìC?.
a45
a54
-1-
√3
0 1 √3 x
y
(3) (i)(?∞,+∞)§′ó?ê§-?'uy?é?.
(ii) yprime =?10xe?x2,yprimeprime = 10e?x2(2x2?1)§x = 0?§yprime = 0?x = ± 1√2?§yprimeprime = 0.
(iii) LXeμ
x (?∞,? 1√2)?DF1√2 (? 1√2,0) 0 (0,1sqrt2) 1√2 ( 1√2,+∞)
yprime + + + 0 - - -
yprimeprime + 0 - - - 0 +
y eàarrownortheast 5√e tàarrownortheast 4 tàarrowsoutheast 5√e eàarrowsoutheast
5
(iv) x→∞?§y → 0§y = 0′-??^Y2ìC?.
a45
a54
- 1√2 0
5
1√
2
x
y
(4) (i)(?∞,+∞)§′ó?ê§-?'uy?é?￡ù′V-{u?êcoshx = e
x +e?x
2 ¤.
(ii) yprime = sinhx,yprimeprime = coshx§x = 0?§yprime = 0?
duyprimeprime > 0(x∈ (?∞,+∞)§y3(?∞,+∞)t′eà.?yprimeprime|x=0 = 1 > 0§ymin = 1.
a45
a54
1
x
y
(5) (i)(?∞,?1)uniontext(?1,1)uniontext(1,+∞)§′ó?ê§-?'uy?é?.
(ii) yprime =? 2x(x2?1)2,yprimeprime = 2(x
2 + 1)
(x2?1)3§x = 0?§y
prime = 0?x = ±1?§yprime?3?x =
±1?§yprimeprime?3.
94
(iii) LXeμ
x (?∞,?1) -1 (?1,0) 0 (0,1) 1 (1,+∞)
yprime +?3 + 0 -?3 -
yprimeprime +?3 - - -?3 +
y eàarrownortheast tàarrownortheast 4 tàarrowsoutheast eàarrowsoutheast
-1
(iv) x→∞?§y → 0§y = 0′-??^Y2ìCx→±1?§y →∞§x = ±1′-?
?^R?ìC?.
a45
a54
-1 10
-1
x
y
(6) (i)(?1,1)§′ê§-?'u:é?.
(ii) yprime = 21?x2,yprimeprime = 4x(1?x2)2§yprime = 0?)?x = 0?§yprimeprime = 0.
(iii) LXeμ
x (?1,0) 0 (0,1)
yprime + + +
yprimeprime - 0 +
y tàarrownortheast 0 eàarrownortheast
(iv) x→ 1§y → +∞§x = 1′-??^R?ìC
x→?1+?§y →?∞§x =?1′-??^R?ìC?.
a45
a54
-1 10 x
y
(7) (i)(?∞,+∞).
(ii) yprime = (x?1)(x?2)2(5x?7),yprimeprime = 2(x?2)(10x2?28x+19)§x = 1,x = 2,x = 75 = 1.4?§yprime =
0?x = 2,x = 14±
√6
10?§y
primeprime = 0.
(iii) LXeμ
x (?∞,1) 1
parenleftBigg
1,?14?
√6
10
parenrightBigg
14?√6
10
parenleftBigg
14?
√6
,1.4
parenrightBigg
1.4
yprime + 0 - - - 0
yprimeprime - - - 0 + +
y tàarrownortheast 4 tàarrowsoutheast -0.0154 eàarrowsoutheast 4?
0 -0.0346
95
x
parenleftBigg
1.4,14 +
√6
10
parenrightBigg
14 +√6
10
parenleftBigg
14 +√6
,2
parenrightBigg
2 (2,+∞)
yprime + + + 0 +
yprimeprime + 0 - 0 +
y eàarrownortheast -0.0186 tàarrownortheast 0 eàarrownortheast
a45
a54
10 2
-8
x
y
(8) (i)(?∞,?1)uniontext(?1,+∞).
(ii) yprime = 6(x?1)
2
(x+ 1)4,y
primeprime =?12(x?1)(x?3)
(x+ 1)5 §x = 1?§y
prime = 0?x = 1,x = 3?§yprimeprime = 0?
x =?1?§yprime,yprimeprimet?3.
(iii) LXeμ
x (?∞,?1) -1 (?1,1) 1 (1,3) 3 (3,+∞)
yprime +?3 + 0 + + +
yprimeprime -?3 - 0 + 0 -
y tàarrownortheast tàarrownortheast 0 eàarrownortheast 18 tàarrownortheast
(iv) x→?1§y → +∞§x =?1′-??^R?ìC
x→∞?§y → 1§y = 1′-??^Y2ìC?.
a45
a54
-1 10 3 x
y
(9) (i)(?∞,+∞).
96
(ii) yprime = 2(x
2 + 4x?1)
(x2 + 1)2,y
primeprime =?4(x3 + 6x2?3x?2)
(x2 + 1)3 §x =?2 ±
√5?§yprime = 0?yprimeprime = 0?
x1,x2,x3§ù￥x1 ∈ (?7,?6),x2 ∈ (?1,0),x3 ∈
parenleftbigg1
2,1
parenrightbigg
.
(iii) LXeμ
x (?∞,x1) x1 (x1,?2?√5)?2?√5 (?2?√5,x2) x2
yprime + + + 0 - -
yprimeprime + 0 - - - 0
y eàarrownortheast $,tàarrownortheast 4 tàarrowsoutheast $:√
5?1
x (x2,?2 +√5)?2 +√5 (?2 +√5,x3) x3 (x3,+∞)
yprime - 0 + + +
yprimeprime + + + 0 -
y eàarrowsoutheast 4? eàarrownortheast $,tàarrownortheast
√5?1
(iv) x→∞?§y → 1§x = 1?-??^Y2ìC?.
a45
a54
-1 10 3-7 -6 x
y
1
(10) (i)(?∞,+∞)§′ê§-?'u:éx = 0?§y = 0.
(ii) yprime = 1 + 11 +x2 > 0§-?üNt,§?4?:.
yprimeprime =? 2x(1 +x2)2§x = 0?§yprimeprime = 0?x > 0?§yprimeprime < 0?x < 0?§yprimeprime > 0§K(0,0)?$
:.
(iii) k = lim
x→∞
y
x = 1,b1 = limx→?∞(y?kx) =?
pi
2,b2 = limx→+∞(y?kx) =
pi
2§-?kü^ìC
μy = x+ pi2,y = x? pi2.
a45
a54
0 x
y
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
30,á?e?ê?/μy =
9x+x4
x?x3,xnegationslash= 0
9,x = 0
)μ
(1)(?∞,?1)uniontext(?1,1)uniontext(1,+∞).
(2) yprime =
x4 + 3x2 + 18x
(1?x2)2,xnegationslash= 0
0,x = 0
,yprimeprime =
2(x
3 + 27x2 + 3x+ 9
(x2?1)2,xnegationslash= 0
18,x = 0
§x = 0,x = 3?§yprime = 0?yprimeprime = 0x1§ù￥x1 ∈ (?27,?26)?x = ±1?§yprime,yprimeprimet?3.
97
(3) LXeμ
x (?∞,x1) x1 (x1,?1) -1 (?1,0) 0
yprime - - -?3 - 0
yprimeprime + 0 - - -
y eàarrowsoutheast $,tàarrowsoutheast tàarrowsoutheast 4?
9
x (0,1) 1 (1,3) 3 (3,+∞)
yprime +?3 + 0 -
yprimeprime -?3 - - -
y tàarrownortheast tàarrownortheast 4 tàarrowsoutheast
92
(4) x→±1?§y →∞§x = ±1′-?R?ìC?.
a45
a54
-27 0 x
y
98
§4,2?-?-?
1,|-?y = 4x?x2-?±93:(2,4)-.
)μ?y = 4x?x2§yprime = 4? 2x,yprimeprime =?2§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
yprimeprime
(1 +yprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
2
[1 + 4(2?x)2]32
§u′-
ρ = 1K = 12[1 + 4(x?2)2]32§l
3:(2,4)-ρ = 12.
2,|e-?--μ
(1) ]ó?y = acosh xa(a> 0)
(2) y2 = 2px(p> 0)
(3) ^ó?x = a(t?sint),y = a(1?cost)(a> 0)
(4) %9?ρ = a(1 + cosθ)(a> 0)
(5) VY?ρ2 = 2a2 cos2θ(a> 0)
(6) éêú?ρ = aeλθ(λ> 0)
)μ
(1)?y = acosh xa§yprime = sinh xa,yprimeprime = 1a cosh xa§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
yprimeprime
(1 +yprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
1
acosh2 xa
§u′-
ρ = 1Kacosh2 xa.
(2)?y2 = 2px§K2yyprime = 2p=yprime = py§yprimeprime =? py2yprime =?p
2
y3§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
yprimeprime
(1 +yprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
p2
(y2 +p2)32
=
p2
(2px+p2)32
= 1
p
parenleftbigg
1 + 2xp
parenrightbigg3
2
§u′-ρ = 1K = p
parenleftbigg
1 + 2xp
parenrightbigg3
2
parenleftBigg
(y
2 +p2)32
p2
parenrightBigg
.
(3)?x = a(t? sint),y = a(1? cost)§xprime = a(1? cost),xprimeprime = asint;yprime = asintyprimeprime = acost§K-
K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
xprimeyprimeprime?xprimeprimeyprime
(xprime2 +yprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
4a
vextendsinglevextendsingle
vextendsinglevextendsinglesin t
2
vextendsinglevextendsingle
vextendsinglevextendsingle
§u′-ρ = 1K = 4a
vextendsinglevextendsingle
vextendsinglevextendsinglesin t
2
vextendsinglevextendsingle
vextendsinglevextendsingle(= 2radicalbig2ay).
(4)?ρ = a(1 + cosθ)§ρprime =?asinθ,ρprimeprime =?acosθ§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ρ2 + 2ρprime2?ρρprimeprime
(ρ2 +ρprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
3
2√2aρ§u′-
R = 2
√2aρ
3,
(5)?ρ2 = 2a2 cos2θ§K2ρρprime =?4a2 sin2θ§ρprime =?2a
2 sin2θ
ρ,ρ
primeprime =?4a4 +ρ4
ρ3 §K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ρ2 + 2ρprime2?ρρprimeprime
(ρ2 +ρprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
3ρ
2a2§u′-R =
2a2
3ρ,
(6)?ρ = aeλθ§ρprime = λaeλθ = λρ,ρprimeprime = aλ2eλθ = λ2ρ§K-?K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ρ2 + 2ρprime2?ρρprimeprime
(ρ2 +ρprime2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
|ρ|(1 +λ2)12
§
u′-R = |ρ|√1 +λ2.
3,|-?y = 2(x?1)2?-.
)μ?y = 2(x?1)2§yprime = 4(x?1),yprimeprime = 4§K-R = 1K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1 +yprime2)32
yprimeprime
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
[1 + 16(x?1)2]32
4
|R?§K7k[1 + 16(x?1)2]32?§=x = 1?§Rmin = 14.
4,÷′?y = x
2
4000￡ü¤?:à?1§3?I:O?Yv = 140?/|§?1
N
-G = 70ú6.|dé?1
.
)μd?n?￡§?!?
±$??N¤é?%F = mv
2
R §ù￥mN?t§v?§?
99
Y§R?
.
¤|é?1
A?F = Gg+ mv
2
R §ùA
%.
aK?§k|-§yprime = x2000,yprimeprime = 12000§K-R = 1K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(20002 +x2)32
20002
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle§u′3?I
:OR = 2000￡?¤§q3?I:O?Yv = 140?/|§l
F = 1372(N).
5,-t′P§± ?v¨L?x￡?5-32¤§x?ACB′?§ùo?X.|e?LC:?éx?
.
)μ±O?:§AB?x?§CO?yá?IX§K§y =?4δl2 x2 +δ
d?n?§e?LC:?éx?F = mv
2
R cosθ+mg
aK?§k|-§yprime =?8δl2 x,yprimeprime =?8δl2 §K-R = 1K =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(l2 + 8δx)32
8lδ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle§u′3:CR =
l2
8δ§q3:Cθ = pi§l
F = Pg+
Pv2
R cosθ =
gl2?8δv2
l2 P.
100
§5,.
1,|^a7?{K|e4?μ
(1) lim
x→0
tanax
sinbx
(2) lim
x→0
1?cosx2
x3 sinx
(3) lim
x→∞
pi
2?arctanx
sin 1x
(4) lim
x→∞
xb
eax
(5) lim
x→1
parenleftbigg 1
lnx?
1
x?1
parenrightbigg
(6) lim
x→pi
(pi?x)tan x2
(7) lim
x→0
ln(cosax)
ln(cosbx)
(8) lim
x→0
cos(sinx)?cosx
x4
(9) lim
x→0
ax?bx
x
(10) lim
x→1
x?1
lnx
(11) lim
x→a
ax?xa
x?a (a> 0)
(12) lim
x→pi6
1?2sinx
cos3x
(13) lim
x→0
lnx
cotx
(14) lim
x→+∞
lncx
xb (b,c> 0)
(15) lim
x→0
(1 +x)1x?e
x
(16) lim
x→0
xb lncx(b,c> 0)
(17) lim
x→0
xsinx
(18) lim
x→1
x 11?x
(19) lim
x→0
parenleftbigg1
x?
1
ex?1
parenrightbigg
(20) lim
x→+0
parenleftbigg
ln 1x
parenrightbiggx
)μ
(1) lim
x→0
tanax
sinbx = limx→0
asec2ax
bcosbx =
a
b
(2) lim
x→0
1?cosx2
x3 sinx = limx→0
1?cosx2
x4 = limx→0
2xsinx2
4x3 limx→0
sinx2
2x2 =
1
2
(3) lim
x→∞
pi
2?arctanx
sin 1x
= lim
x→∞
11 +x2
1x2 cos 1x
= lim
x→∞
x2
(1 +x2)cos 1x
= 1
101
(4) b?ê§ lim
x→∞
xb
eax = limx→∞
bxb?1
aeax = ··· = limx→∞
b!
abeax = 0
bê§K[b] lessorequalslantb< [b]+1§u′|x|
[b]
eax lessorequalslant
|x|b
eax <
|x|[b]+1
eax (|x|> 1)§
!müàx→∞?§
ty2§?40§?d§￥m40.
l
§éa,b§tk lim
x→∞
xb
eax = 0
(5) lim
x→1
parenleftbigg 1
lnx?
1
x?1
parenrightbigg
= lim
x→1
x?1?lnx
(x?1)lnx = limx→1
1? 1x
lnx+ x?1x
= lim
x→1
x?1
xlnx+x?1 = limx→1
1
lnx+ 1 + 1 =
1
2
(6) lim
x→pi
(pi?x)tan x2 = lim
x→pi
pi?x
cot x2
= lim
x→pi
1
12 csc2 x2
= 2
(7) lim
x→0
ln(cosax)
ln(cosbx) = limx→0
atanax
btanbx =
a
b limx→0
tanax
tanbx =
a
b limx→0
asec2ax
bsec2bx =
a2
b2 (bnegationslash= 0)
(8) lim
x→0
cos(sinx)?cosx
x4 = limx→0
sin(sinx)cosx+ sinx
4x3
= lim
x→0
cos(sinx)cos2x+ sin(sinx)sinx+ cosx
12x2 =
lim
x→0
sin(sinx)cos3x+ 32 cos(sinx)sin2x+ sin(sinx)cosx?sinx
24x =
lim
x→0
bracketleftbiggcos(sinx)cos4x?3sin(sinx)sin2xcosx+ 3cos(sinx)cos2x
24 +
cos(sinx)cos2x?sin(sinx)sinx?cosx
24
bracketrightbigg
=
1
6
(9) lim
x→0
ax?bx
x = limx→0
ax lna?bx lnb
1 = lna?lnb = ln
a
b(anegationslash= 0,bnegationslash= 0)
(10) lim
x→1
x?1
lnx = limx→1
1
1
x
= 1
(11) lim
x→a
ax?xa
x?a = limx→a
ax lna?axa?1
1 = a
a(lna?1)
(12) lim
x→pi6
1?2sinx
cos3x = limx→pi6
2cosx
3sin3x =
√3
3
(13) lim
x→0
lnx
cotx = limx→0
1
x
csc2x =? limx→0
sin2x
x = 0
(14) -y = lnx,Kx = ey§u′ lim
x→+∞
lncx
xb = limy→+∞
yc
eby = 0(d(4))
(15) lim
x→0
(1 +x)1x?e
x = limx→0(1+x)
1
x
bracketleftbigg 1
x(1 +x)?
1
x2 ln(1 +x)
bracketrightbigg
= e lim
x→0
x?(1 +x)ln(1 +x)
x2 = e limx→0
1?1?ln(1 +x)
2x =
e2
(16) -y = lnx,Kx = ey§u′lim
x→0
xb lncx = lim
y→?∞
ebyyc = lim
y→?∞
yc
e?by = 0(d(4))
(17) lim
x→0
xsinx = elimx→0sinxlnx§
lim
x→0
sinxlnx = lim
x→0
lnx
1
x
= lim
x→0
1
x
1x2
=? lim
x→0
x = 0§u′lim
x→0
xsinx = 1
(18) lim
x→1
x 11?x = elimx→1
lnx
1?x§
lim
x→1
lnx
1?x =? limx→1
1
x =?1§u′limx→1x
1
1?x = 1e
(19) lim
x→0
parenleftbigg1
x?
1
ex?1
parenrightbigg
= lim
x→0
ex?x?1
x(ex?1) = limx→0 =
ex?1
ex?1 +xex = limx→0
ex
2ex +xex =
1
2
102
(20) lim
x→+0
parenleftbigg
ln 1x
parenrightbiggx
= e
limx→+0xln
ln 1x
-y = 1x§K lim
x→+0
xln(ln 1x) = lim
y→+∞
ln(lny)
y = limy→+∞
1
ylny = 0§l
limx→+0
parenleftbigg
ln 1x
parenrightbiggx
= 1
2,á`2e?ê?U^a7?{K|4?μ
(1) lim
x→0
x2 sin 1x
sinx
(2) lim
x→∞
x+ sinx
x?cosx
(3) lim
x→∞
2x+ sin2x
(2x+ sinx)esinx
(4) lim
x→1
(x2?1)sinx
ln
parenleftBig
1 + sin pi2x
parenrightBig
)μ
(1)?
x2 sin 1x
sinx ?f!?1ó?éx|ê§
2xsin 1x?cos 1x
cosx §
cos 1x
cosxx→ 0?43§?da
7?{K?U·^§′4?′3"ˉ￠t§klim
x→0
x2 sin 1x
sinx = limx→0
x
sinx ·xsin
1
x = 0
(2)?x+ sinxx?cosx?f!?1ó?éx|ê§1 + cosx1 + sinx§x→∞?d?ê43§?da7?{
K?U·^§′4?′3"ˉ￠t§k lim
x→∞
x+ sinx
x?cosx = limx→∞
1 + sinxx
1? cosxx
= 1
(3) éu?óSμxprimen = 2npi+ pi29xprimeprimen = 2npi(n = 1,2,···)§n→∞?§K?ó4?1e91§l
43.
^a7?{K|)§k lim
x→∞
2x+ sin2x
(2x+ sinx)esinx =
lim
x→∞
2 + 2cos2x
(2 + cosx+ 2xcosx+ sinxcosx)esinx =
lim
x→∞
4cos2x
[2 + cosx(1 + 2x+ sinx)]esinx =
lim
x→∞
4bracketleftbigg
2
cos2x +
1
cosx(1 + 2x+ sinx)
bracketrightbigg
esinx
§?esinx greaterorequalslante?1,1+2x+sinxgreaterorequalslant 2x§K
vextendsinglevextendsingle
vextendsinglevextendsingle[ 2
cos2x +
1
cosx(1 + 2x+ sinx)]e
sinx
vextendsinglevextendsingle
vextendsinglevextendsinglegreaterorequalslant
e?1(?2 + 2|x|) → +∞(x→∞)§K lim
x→∞
2x+ sin2x
(2x+ sinx)esinx = 0.
(4)?|4lim
x→1
(x2?1)sinx
ln
parenleftBig
1 + sin pi2x
parenrightBig = 0§d4ü^a7?{K|4?^?.
103
§6,?§Cq)
1,|?§x3?x?4 = 0?§|?
L0.0001.
)μf(x) = x3?x? 4§3[1,2]m§f(1) =?4 < 0,f(2) = 2 > 0=f(1)f(2) < 0?fprime(x) = 3x2? 1 >
0,fprimeprime(x) = 6x> 0
f(2)fprimeprime(2) = 24 > 0§Kl:(2,f(2))=:(2,2)m§x0 = 2?D?.
u′x1 = 2? f(2)fprime(2) ≈ 1.81818,x2 = x1? f(x1)fprime(x
1)
≈ 1.79663,x3 = x2? f(x2)fprime(x
2)
≈ 1.79632,x4 = x3? f(x3)fprime(x
3)
≈
1.79632
x3?x4c5?ê?ó§ùLCu?°(?"?
`2°(Y§^1.7963á?e§kf(1.7963) ≈
0.00019 < 0§
f(1.79632) ≈ 0.00002 > 0§e1.7963Cq?§K?
L0.0001.
2,|?§x3?x?4 = 0?§|?
L0.0001.
)μf(x) = x3? 5x2 + 6x? 1§f(0) =?1 < 0,f(1) = 1 > 0,fprime(x) = 3x2? 10x+ 6§d?fprime(0) = 6 >
0,fprime(1) =?1 < 0§3(0,1)Sfprime(x)k":5?
√7
3 §d?f
prime(x)3
parenleftbigg
0,5?
√7
3
parenrightbigg
S??fprime(x)3
parenleftbigg5?√7
3,1
parenrightbigg
S
K.
y?O?f(x)3(0,0.7)?(0.7,1)￥?
f(0.7) = 1.093 > 0§3(0,0.7)￥7k￠?ξ§3(0.7,1)￥.
y|ξ,fprimeprime(x) = 6x?10 < 0(?x∈ (0,0.7))§?f(0) =?1,fprimeprime(0) =?10§x0 = 0?D?.
u′x1 = 0? f(0)fprime(0) ≈ 0.16667,x2 = x1? f(x1)fprime(x
1)
≈ 0.19706,x3 = x2? f(x2)fprime(x
2)
≈ 0.19806,x4 = x3? f(x3)fprime(x
3)
≈
0.19806
x3?x4c5?ê?ó§ùLCu?ξ°(?"?
`2°(Y§^0.1980á?e§kf(0.1980) ≈
0.00026 < 0§
f(0.1981) ≈ 0.01397 > 0§e0.1980Cq?§K?
L0.0001.
104
a49a19a220a169 a252a67a254a200a169a198
a49a56a217 a216a189a200a169
§1,a216a189a200a169a27a86a103a57a36a142a123a75
1,a121a178a181a101integraltext f(t)dt = F(t) +Ca167a75integraltext f(ax+b)dx = 1aF(ax+b) +C.
a121a178a181a207integraltext f(t)dt = F(t) +Ca167a25[F(t) +C]prime = f(t)a167a75
bracketleftbigg1
aT(ax+b)
bracketrightbiggprime
= 1a[F(ax+b)]prime = f(ax+b)a167a117
a180integraltext f(ax+b)dx = 1aF(ax+b) +C.
2,a166a101a15a216a189a200a169a181
(1)
integraldisplay
(2?sec2x)dx
(2)
integraldisplay parenleftbigg
x4?2x3 +
√x
2
parenrightbigg
dx
(3)
integraldisplay parenleftbigg√
x+ 3√x+ 2√x + 23√x?2
parenrightbigg
dx
(4)
integraldisplay parenleftbigg
ex + 1x + 1x2 + 1x3
parenrightbigg
dx
(5)
integraldisplay parenleftbigg
2cosx+ 12 sinx
parenrightbigg
dx
(6)
integraldisplay parenleftbigg
cosx? 21 +x2 + 14√1?x2
parenrightbigg
dx
(7)
integraldisplay parenleftbigg1
2 cosx+ sinx+ 1
parenrightbigg
dx
(8)
integraldisplay parenleftbigg
2x +
parenleftbigg1
3
parenrightbiggx
e
x
5
parenrightbigg
dx
(9)
integraldisplay
(3?x2)3 dx
(10)
integraldisplay parenleftbigg
1? 1x2
parenrightbiggradicalBig
x√xdx
a41a181
(1)
integraldisplay
(2?sec2x)dx = 2x?tanx+C
(2)
integraldisplay parenleftbigg
x4?2x3 +
√x
2
parenrightbigg
dx = 15x5? 12x4 + 13x32 +C
(3)
integraldisplay parenleftbigg√
x+ 3√x+ 2√x + 23√x?2
parenrightbigg
dx = 23x32 + 34x43?2x+ 3x23 + 4x12 +C
(4)
integraldisplay parenleftbigg
ex + 1x + 1x2 + 1x3
parenrightbigg
dx = ex + ln|x|? 1x? 12x2 +C
(5)
integraldisplay parenleftbigg
2cosx+ 12 sinx
parenrightbigg
dx = 2sinx? 12 cosx+C
(6)
integraldisplay parenleftbigg
cosx? 21 +x2 + 14√1?x2
parenrightbigg
dx = sinx?2arctanx+ 14 arcsinx+C
(7)
integraldisplay parenleftbigg1
2 cosx+ sinx+ 1
parenrightbigg
dx = 12 sinx?cosx+x+C
(8)
integraldisplay parenleftbigg
2x +
parenleftbigg1
3
parenrightbiggx
e
x
5
parenrightbigg
dx = 1ln22x? 1ln3
parenleftbigg1
3
parenrightbiggx
e
x
5 +C
(9)
integraldisplay
(3?x2)3 dx =
integraldisplay
(27?27x2 + 9x4?x6)dx = 27x?9x3 + 95x5? 17x7 +C
(10)
integraldisplay parenleftbigg
1? 1x2
parenrightbiggradicalBig
x√xdx =
integraldisplay
(x34?x?54 )dx = 47x74 + 4x?14 +C
105
§2,a216a189a200a169a27a79a142
1,a166a101a15a216a189a200a169a181
(1)
integraldisplay dx
5x?7
(2)
integraldisplay
cos(ωt)dt
(3)
integraldisplay dx
radicalbigg
1?
parenleftBigx
2 + 3
parenrightBig2
(4)
integraldisplay dx
√1?2x2
(5)
integraldisplay
tan10xsec2xdx
(6)
integraldisplay
eαx ·2xdx
(7)
integraldisplay
(2x + 3x)2 dx
(8)
integraldisplay
tanxdx
(9)
integraldisplay
tan
radicalbig
1 +x2 · xdx√1 +x2
(10)
integraldisplay
(αx2 +β)μxdx(μnegationslash=?1)
(11)
integraldisplay dx
1?cosx
(12)
integraldisplay dx
A2 sin2x+B2 cos2x
(13)
integraldisplay sinx·cosx
1 + sin4x dx
(14)
integraldisplay dx
sin2
parenleftBig
x+ pi4
parenrightBig
(15)
integraldisplay
x2 8
radicalbig
1 +x3 dx
(16)
integraldisplay sin2xcosx
1 + sin3x dx
(17)
integraldisplay 1?2sinx
cos2x dx
(18)
integraldisplay dx
ex +e?x
(19)
integraldisplay sinx+ cosx
3√sinx?cosx dx
(20)
integraldisplay 1 + sin2x
sin2x dx
(21)
integraldisplay radicalBiggln(x+√1 +x2)
1 +x2 dx
(22)
integraldisplay dx
√1 +e2x
(23)
integraldisplay dx
x2?2x+ 2
(24)
integraldisplay dx
(arcsinx)2√1?x2
106
(25)
integraldisplay x2 + 7
x2?2x?3 dx
(26)
integraldisplay x2?1
x4 + 1 dx
a41a181
(1)
integraldisplay dx
5x?7 =
1
5
integraldisplay d(5x?7)
5x?7 =
1
5 ln|5x?7|+C
(2)
integraldisplay
cos(ωt)dt = 1ω
integraldisplay
cos(ωt)d(ωt) = 1ω sin(ωt) +C
(3)
integraldisplay dx
radicalbigg
1?
parenleftBigx
2 + 3
parenrightBig2 = 2
integraldisplay dparenleftBigx
2 + 3
parenrightBig
radicalbigg
1?
parenleftBigx
2 + 3
parenrightBig2 = 2arcsin
parenleftBigx
2 + 3
parenrightBig
+C
(4)
integraldisplay dx
√1?2x2 =
√2
2
integraldisplay d(√x)
radicalBig
1?(√2x)2
=
√2
2 arcsin(
√2x) +C
(5)
integraldisplay
tan10xsec2xdx =
integraldisplay
tan10xd(tanx) =? 111 tan11x+C
(6)
integraldisplay
eαx ·2xdx =
integraldisplay
(2eα)xdx = (2e
α)x
ln(2eα) +C
(7)
integraldisplay
(2x + 3x)2 dx =
integraldisplay
(4x + 2·6x + 9x)dx = 4
x
ln4 +
2
ln66
x + 9x
ln9 +C
(8)
integraldisplay
tanxdx =
integraldisplay sinx
cosx dx =?
integraldisplay d(cosx)
cosx =?ln|cosx|+C = ln|secx|+C
(9)
integraldisplay
tan
radicalbig
1 +x2 · xdx√1 +x2 =
integraldisplay
tan
radicalbig
1 +x2 d(
radicalbig
1 +x2) = ln|sec
radicalbig
1 +x2|+C
(10)
integraldisplay
(αx2 +β)μxdx = 12α
integraldisplay
(αx2 +β)μd(αx2 +β) = (αx
2 +β)μ+1
2α(μ+ 1) +C
(11)
integraldisplay dx
1?cosx =
integraldisplay
csc2 x2 d
parenleftBigx
2
parenrightBig
=?cot x2 +C
(12)
integraldisplay dx
A2 sin2x+B2 cos2x =
1
AB
integraldisplay 1
1 +
parenleftbiggA
B
parenrightbigg2
tan2x
dAtanxB = 1AB arctan
parenleftbiggA
B tanx
parenrightbigg
+C
(13)
integraldisplay sinx·cosx
1 + sin4x dx =
1
2
integraldisplay 1
1 + (sin2x)2 d(sin
2x) = 1
2 arctan(sin
2x) +C
(14)
integraldisplay dx
sin2
parenleftBig
x+ pi4
parenrightBig =
integraldisplay
csc2
parenleftBig
x+ pi4
parenrightBig
d
parenleftBig
x+ pi4
parenrightBig
=?cot
parenleftBig
x+ pi4
parenrightBig
+C
(15)
integraldisplay
x2 8
radicalbig
1 +x3 dx = 13
integraldisplay
8radicalbig1 +x3 d(1 +x3) = 8
27(1 +x
3)98 +C
(16)
integraldisplay sin2xcosx
1 + sin3x dx =
1
3
integraldisplay d(1 + sin3x)
1 + sin3x =
1
3 ln(1 + sin
3x) +C
(17)
integraldisplay 1?2sinx
cos2x dx =
integraldisplay
sec2xdx+ 2
integraldisplay dcosx
cos2x = tanx?2secx+C
(18)
integraldisplay dx
ex +e?x =
integraldisplay dex
e2x + 1 = arctan(e
x) +C
(19)
integraldisplay sinx+ cosx
3√sinx?cosx dx =
integraldisplay d(sinx?cosx)
3√sinx?cosx =
3
2(sinx?cosx)
2
3 +C
(20)
integraldisplay 1 + sin2x
sin2x dx =
integraldisplay
csc2xdx+
integraldisplay d(sin2x)
sin2x =?cotx+ ln(sin
2x) +C =?cotx+ 2ln|sinx|+C
(21)
integraldisplay radicalBiggln(x+√1 +x2)
1 +x2 dx =integraldisplay radicalBig
ln(x+
radicalbig
1 +x2)d(ln(x+
radicalbig
1 +x2) = 23[ln(x+
radicalbig
1 +x2]32 +C
107
(22)
integraldisplay dx
√1 +e2x =?
integraldisplay de?x
√1 +e?2x =?ln(e?x +
radicalbig
1 +e?2x) +C
(23)
integraldisplay dx
x2?2x+ 2 =
integraldisplay d(x?1)
(x?1)2 + 1 = arctan(x?1) +C
(24)
integraldisplay dx
(arcsinx)2√1?x2 =
integraldisplay d(arcsinx)
(arcsinx)2 =?
1
arcsinx +C
(25)
integraldisplay x2 + 7
x2?2x?3 dx =
integraldisplay parenleftbigg
1 + 2x+ 10(x+ 1)(x?3)
parenrightbigg
dx =
integraldisplay parenleftbigg
1? 2x+ 1 + 4x?3
parenrightbigg
dx = x?2ln|x+ 1|+ 4ln|x?3|+C = x+ 2ln (x?3)
2
|x+ 1| +C
(26)
integraldisplay x2?1
x4 + 1 dx =
integraldisplay 1?x?2
x2 +x?2 dx =
integraldisplay d
parenleftbigg
x+ 1x
parenrightbigg
parenleftbigg
x+ 1x
parenrightbigg2
2
=
√2
4 ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x+ 1x?√2
x+ 1x +√2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+C =
√2
4 ln
vextendsinglevextendsingle
vextendsinglevextendsinglex2?
√2x+ 1
x2 +√2x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle+C
2,a166a101a15a216a189a200a169a181
(1)
integraldisplay sin√x
√x dx
(2)
integraldisplay (2√u+ 1)2
u2 du
(3)
integraldisplay
e
√x+1
dx
(4)
integraldisplay x2
√4?x2 dx
(5)
integraldisplay radicalbig
x2 +a2 dx
(6)
integraldisplay radicalbig
x2?a2 dx
(7)
integraldisplay dx
radicalbig(x?a)(b?x)
(8)
integraldisplay dx
x2radicalbigαx2 +β
(9)
integraldisplay xdx
√5 +x?x2
(10)
integraldisplay radicalbig
2 +x?x2 dx
a41a181
(1)
integraldisplay sin√x
√x dx =?2
integraldisplay
sin√xd(√x) =?2cos√x+C
(2)
integraldisplay (2√u+ 1)2
u2 du =
integraldisplay parenleftbigg4
u +
4
u32
+ 1u2
parenrightbigg
du = 4ln|u|?8u?12? 1u +C
(3) a45√1 +x = ta167a75x = t2?1,dx = 2tdta167a117a180
integraldisplay
e
√x+1
dx = 2
integraldisplay
tetdt = 2(t?1)et+C = 2(√1 +x?
1)e
√1+x
+C
(4)
integraldisplay x2
√4?x2 dx =?
integraldisplay 4?x2?4
√4?x2 dx =?
integraldisplay radicalbig
4?x2 dx+ 4
integraldisplay dx
√4?x2 =?x2
radicalbig
4?x2?2arcsin x2 +
4arcsin x2 +C = 2arcsin x2? x2
radicalbig
4?x2 +C
108
(5) I =
integraldisplay radicalbig
x2 +a2 dx = x
radicalbig
x2 +a2?
integraldisplay x2
√x2 +a2 dx = x
radicalbig
x2 +a2?
integraldisplay radicalbig
x2 +a2 dx+
integraldisplay a2
√x2 +a2 dx =
x
radicalbig
x2 +a2?I +a2 ln|x+
radicalbig
x2 +a2|+C1a167
a117a1802I = x√x2 +a2+a2 ln|x+√x2 +a2|+C1a167a108a13I = x2
radicalbig
x2 +a2+a
2
2 ln(x+
radicalbig
x2 +a2)+C(C =
C1
2 )
(6) I =
integraldisplay radicalbig
x2?a2 dx = x
radicalbig
x2?a2?
integraldisplay x2
√x2?a2 dx = x
radicalbig
x2?a2?
integraldisplay radicalbig
x2?a2 dx?
integraldisplay a2
√x2?a2 dx =
x
radicalbig
x2?a2?I?a2 ln|x+
radicalbig
x2?a2|+C1a167
a117a1802I = x√x2?a2?a2 ln|x+√x2?a2|+C1a167a108a13I = x2
radicalbig
x2?a2?a
2
2 ln(x+
radicalbig
x2?a2)+C(C =
C1
2 )
(7)
integraldisplay dx
radicalbig(x?a)(b?x) =
integraldisplay dx
radicalbig?[x2?(a+b)x]?ab =
integraldisplay d
parenleftbigg
x? a+b2
parenrightbigg
radicalBigg
parenleftbigg
x? a+b2
parenrightbigg2
+
parenleftbigga?b
2
parenrightbigg2 = arcsin
x? a+b2
a?b
2
+C = arcsin 2x?a?ba?b +Ca163a217a165a<ba164
(8)
integraldisplay dx
x2radicalbigαx2 +β =
integraldisplay dx
x3
radicalbigg
α+ βx2
=?12
integraldisplay d 1
x2radicalbigg
α+ βx2
=?1β
radicalbigg
α+ βx2 +C
(9)
integraldisplay xdx
√5 +x?x2 =
integraldisplay xdx
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 =
integraldisplay x? 1
2radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 +
1
2
integraldisplay dx
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 =?
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2
+
1
2 arcsin
x? 12
√21
2
+C =?
radicalbig
5 +x?x2 + 12 arcsin 2x?1√21 +C
(10)
integraldisplay radicalbig
2 +x?x2 dx =
integraldisplay radicalBigg9
4?
parenleftbigg
x? 12
parenrightbigg2
dx =
x? 12
2
radicalBigg
9
4?
parenleftbigg
x? 12
parenrightbigg2
+98 arcsin
x? 12
3
2
+C = 2x?14
radicalbig
2 +x?x2+
9
8 arcsin
2x?1
3 +C
3,a166a101a15a216a189a200a169a181
(1)
integraldisplay
x2 cosxdx
(2)
integraldisplay
x3 lnxdx
(3)
integraldisplay
lnxdx
(4)
integraldisplay
xnlnxdx(na143a20a18a234)
(5)
integraldisplay arcsinx
√1?x dx
(6)
integraldisplay
cscxdx
(7)
integraldisplay
cos(lnx)dx
(8)
integraldisplay xdx
sin2x
(9)
integraldisplay
xcos2xdx
109
(10)
integraldisplay
xsin2xdx
(11)
integraldisplay
arccosxdx
(12)
integraldisplay
(arcsinx)2 dx
(13)
integraldisplay
eaxcosbxdx
(14)
integraldisplay
ln(x+
radicalbig
1 +x2)dx
a41a181
(1)
integraldisplay
x2 cosxdx = x2 sinx? 2
integraldisplay
xsinxdx = x2 sinx + 2xcosx? 2
integraldisplay
cosxdx = x2 sinx + 2xcosx?
2sinx+C
(2)
integraldisplay
x3 lnxdx = 14x4 lnx? 14
integraldisplay
x3 dx = 14x4 lnx? x
4
16 +C
(3)
integraldisplay
lnxdx = xlnx?
integraldisplay
dx = xlnx?x+C
(4)
integraldisplay
xnlnxdx = x
n+1
x+ 1 lnx?
1
n+ 1
integraldisplay
xndx = x
n+1
n+ 1 lnx?
xn+1
(n+ 1)2 +C
(5)
integraldisplay arcsinx
√1?x dx =?2arcsinx·√1?x+ 2
integraldisplay 1
√1 +x dx =?2√1?xarcsinx+ 4√1 +x+C
(6)
integraldisplay
cscxdx =
integraldisplay dx
sinx =
integraldisplay 12cos( x
2)
tan x2
dx =
integraldisplay dparenleftBigtan x
2
parenrightBig
tan x2
= ln
vextendsinglevextendsingle
vextendsingletan x2
vextendsinglevextendsingle
vextendsingle+C
(7) a207I =
integraldisplay
cos(lnx)dx = xcoslnx +
integraldisplay
sin(lnx)dx = xcos(lnx) + xsin(lnx)?
integraldisplay
cos(lnx)dx =
xcos(lnx) +xsin(lnx)?I +C1a167a25I = x2[cos(lnx) + sin(lnx)] +C
parenleftbigg
C = C12
parenrightbigg
(8)
integraldisplay xdx
sin2x =
integraldisplay
xcsc2xdx =?xcotx+
integraldisplay
cotxdx =?xcotx+ ln|sinx|+C
(9)
integraldisplay
xcos2xdx = 12
integraldisplay
x(1 + cos2x)dx = x
2
4 +
1
2
integraldisplay
xcos2xdx = x
2
4 +
x
4 sin2x?
1
4
integraldisplay
sin2xdx =
x2
4 +
x
4 sin2x+
1
8 cos2x+C
(10)
integraldisplay
xsin2xdx =
integraldisplay
x(1? cos2x)dx = x
2
2?
integraldisplay
xcos2xdx = x
2
2?
parenleftbiggx2
4 +
x
4 sin2x+
1
8 cos2x
parenrightbigg
+C =
x2
4?
x
4 sin2x?
1
8 cos2x
(11)
integraldisplay
arccosxdx = xarccosx+
integraldisplay x
√1?x2 dx = xarccosx?
radicalbig
1?x2 +C
(12)
integraldisplay
(arcsinx)2 dx = x(arcsinx)2?
integraldisplay 2xarcsinx
√1?x2 dx = x(arcsinx)2 + 2arcsinx·
radicalbig
1?x2? 2
integraldisplay
dx =
x(arcsinx)2 + 2
radicalbig
1?x2 arcsinx?2x+C
(13) I =
integraldisplay
eaxcosbxdx = 1aeaxcosbx+ba
integraldisplay
eaxsinbxdx = 1aeaxcosbx+ ba2eaxsinbx?b
2
a2
integraldisplay
eaxcosbxdx =
1
ae
axcosbx+ b
a2e
axsinbx? b2
a2I +C1a167a75I =
acosbx+bsinbx
a2 +b2 e
ax +C
parenleftbigg
C = a
2
a2 +b2C1
parenrightbigg
(14)
integraldisplay
ln(x+
radicalbig
1 +x2)dx = xln(x+
radicalbig
1 +x2)?
integraldisplay x
√1 +x2x = xln(x+
radicalbig
1 +x2)?
radicalbig
1 +x2 +C
4,a166a101a15a216a189a200a169a181
(1)
integraldisplay x3 + 1
x3?5x2 + 6x dx
110
(2)
integraldisplay dx
(x+ 1)(x+ 2)2
(3)
integraldisplay dx
(x+ 1)(x+ 2)2(x+ 3)3
(4)
integraldisplay x2 + 5x+ 4
x4 + 5x2 + 4 dx
(5)
integraldisplay dx
(x2?4x+ 4)(x2?4x+ 5)
(6)
integraldisplay dx
x4 +x2 + 1
(7)
integraldisplay dx
(x+ 1)(x2 + 1)
(8)
integraldisplay dx
x3 + 1
(9)
integraldisplay x2 dx
1?x4
(10)
integraldisplay x6 +x4?4x2?2
x3(x2 + 1)2 dx
a41a181
(1) a207 x
3 + 1
x3?5x2 + 6x = 1 +
1
6x?
9
2(x?2) +
28
3(x?3)a167a25
integraldisplay x3 + 1
x3?5x2 + 6x dx = x+
1
6 ln|x|?
9
2 ln|x?
2|+ 283 ln|x?3|+C
(2) a207 1(x+ 1)(x+ 2)2 = 1x+ 1? 1x+ 2? 1(x+ 2)2a167a25
integraldisplay dx
(x+ 1)(x+ 2)2 = ln|x+1|?ln|x+2|+
1
x+ 2 +
C = ln
vextendsinglevextendsingle
vextendsinglevextendsinglex+ 1
x+ 2
vextendsinglevextendsingle
vextendsinglevextendsingle+ 1
x+ 2 +C
(3) a207 1(x+ 1)(x+ 2)2(x+ 3)3 = 18(x+ 1) + 2x+ 2? 1(x+ 2)2? 178(x+ 3)? 54(x+ 3)2? 12(x+ 3)3a167
a25
integraldisplay dx
(x+ 1)(x+ 2)2(x+ 3)3 =
1
8 ln|x+1|+2ln|x+2|+
1
x+ 2?
17
8 ln|x+3|+
5
4(x+ 3) +
1
4(x+ 3)2 +
C = 18 ln|x+ 1|+ 2ln|x+ 2|? 178 ln|x+ 3|+ 9x
2 + 50x+ 68
4(x+ 2)(x+ 3)2 +C
(4) a207 x
2 + 5x+ 4
x4 + 5x2 + 4 =
5
3x+ 1
x2 + 1 +
53x
x2 + 4a167a25
integraldisplay x2 + 5x+ 4
x4 + 5x2 + 4 dx =
5
6 ln(x
2+1)+arctanx?5
6 ln(x
2+4)+C =
5
6 ln
parenleftbiggx2 + 1
x2 + 4
parenrightbigg
+ arctanx+C
(5) a207 1(x2?4x+ 4)(x2?4x+ 5) = 1x2?4x+ 4? 1x2?4x+ 5 = 1(x?2)2? 1(x?2)2 + 1a167a25
integraldisplay dx
(x2?4x+ 4)(x2?4x+ 5) =
1x?2?arctan(x?2) +C
(6) a207 1x4 +x2 + 1 = x+ 12(x2 +x+ 1)? x?12(x2?x+ 1)a167a25
integraldisplay dx
x4 +x2 + 1 =
1
4 ln(x
2+x+1)+
√3
6 arctan
parenleftbigg√3
3 (2x+ 1)
parenrightbigg
1
4 ln(x
2?x+ 1) +
√3
6 arctan
parenleftbigg√3
3 (2x?1)
parenrightbigg
(7) a207 1(x+ 1)(x2 + 1) = 12(x+ 1)? x?12(x2 + 1)a167a25
integraldisplay dx
(x+ 1)(x2 + 1) =
1
2 ln|x + 1|?
1
4 ln(x
2 + 1) +
1
2 arctanx+C =
1
4 ln
(x+ 1)2
x2 + 1 +
1
2 arctanx+C
(8) a207 1x3 + 1 = 13(x+ 1) +?x+ 23(x2?x+ 1)a167
a25
integraldisplay dx
x3 + 1 =
1
3 ln|x + 1|?
1
6 ln|x
2? x + 1| +
√3
3 arctan
√3(2x?1)
3 + C =
1
6 ln
(x+ 1)2
x2?x+ 1 +√
3
3 arctan
√3(2x?1)
3 +C
111
(9) a207 x
2
1?x4 =
1
2(1?x2)?
1
2(1 +x2) =
1
4(1?x) +
1
4(1 +x)?
1
2(x2 + 1)a167a25
integraldisplay x2 dx
1?x4 =?
1
4 ln|1?x|+
1
4 ln|1 +x|?
1
2 arctanx+C =
1
4 ln
vextendsinglevextendsingle
vextendsinglevextendsingle1 +x
1?x
vextendsinglevextendsingle
vextendsinglevextendsingle+ 1
2 arctanx+C
(10) a207x
6 +x4?4x2?2
x3(x2 + 1)2 =
x4(x2 + 1)
x3(x2 + 1)2?
4x2 + 2
x3(x2 + 1)2 =
x
x2 + 1? 2
(x2 + 1)2?x4
x3(x2 + 1)2 =
x
x2 + 1?
2
x3 +
2x
(x2 + 1)2a167a25
integraldisplay x6 +x4?4x2?2
x3(x2 + 1)2 dx =
1
x2 +
1
2 ln(x
2 + 1)? 1
x2 + 1 +C
5,a166a101a15a216a189a200a169a181
(1)
integraldisplay dx
4 + 5cosx
(2)
integraldisplay dx
sinx+ tanx
(3)
integraldisplay xdx
√5 +x?x2
(4)
integraldisplay 1
x 4√1 +x4 dx
(5)
integraldisplay xdx
√2 + 4x
(6)
integraldisplay cosx
1 + sinx dx
(7)
integraldisplay dx
x(1 + 2√x+ 3√x)
(8)
integraldisplay √x+ 1?√x?1
√x+ 1 +√x?1 dx
(9)
integraldisplay dx
3radicalbig(x+ 1)2(x?1)4
(10)
integraldisplay dx
√x(1 + 4√x)3
(11)
integraldisplay dx
√ax2 +bx+c(a> 0)
(12)
integraldisplay xdx
4radicalbigx3(a?x)
(13)
integraldisplay
x
radicalbig
x4 + 2x2?1dx
(14)
integraldisplay radicalbig
2 +x?x2 dx
(15)
integraldisplay x2 dx
√1 +x?x2
(16)
integraldisplay x2 + 1
x√x4 + 1 dx
(17)
integraldisplay
sin6xdx
(18)
integraldisplay
sin2xcos4xdx
(19)
integraldisplay
sin4xcos4xdx
(20)
integraldisplay cos4x
sin3x dx
(21)
integraldisplay dx
sin3xcos5x
(22)
integraldisplay
tanx·tan(x+a)dx
112
(23)
integraldisplay
sin5xcosxdx
(24)
integraldisplay sin2x
1 + sin2x dx
(25)
integraldisplay dx
sin(x+a)sin(x+b)
(26)
integraldisplay
xexcosxdx
(27)
integraldisplay dx
(2 + cosx)sinx
(28)
integraldisplay
ln(x+
radicalbig
1 +x2)2 dx
(29)
integraldisplay sinxcosx
sinx+ cosx dx
(30)
integraldisplay lnx
(1 +x2)32
dx
(31)
integraldisplay
xexsinxdx
(32)
integraldisplay x3 arccosx
√1?x2 dx
(33)
integraldisplay
(x+|x|)2 dx
(34)
integraldisplay
x2excosxdx
(35)
integraldisplay xex
(1 +x)2 dx
(36)
integraldisplay √
xln2xdx
(37)
integraldisplay dx
radicalbig(x?a)(b?x)
(38)
integraldisplay
xln 1 +x1?x dx
(39)
integraldisplay
xarctanx·ln(1 +x2)dx
(40)
integraldisplay
sinh2xcosh2xdx
a41a181
(1) a45tan x2 = ta167a75cosx = 1?t
2
1 +t2,dx =
2dt
1 +t2a167
a117a180
integraldisplay dx
4 + 5cosx =
integraldisplay 2
(3?t)(3 +t) dt =
1
3 ln
vextendsinglevextendsingle
vextendsinglevextendsingle3 +t
3?t
vextendsinglevextendsingle
vextendsinglevextendsingle+C = 1
3 ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
3 + tan x2
3?tan x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle+C
(2) a45tan x2 = ta167a75sinx = 2t1 +t2,tanx = 2t1?t2,dx = 2dt1 +t2a167
a117a180
integraldisplay dx
sinx+ tanx =
integraldisplay 1?t2
2t dt =
1
2 ln|t|?
t2
4 +C =
1
2 ln
vextendsinglevextendsingle
vextendsingletan x2
vextendsinglevextendsingle
vextendsingle? 14
parenleftBig
tan x2
parenrightBig2
+C
(3)
integraldisplay xdx
√5 +x?x2 =
integraldisplay xdx
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 =
integraldisplay x? 1
2radicalBigg
21
4 dx?
parenleftbigg
x? 12
parenrightbigg2 +
1
2
integraldisplay dx
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2 =?
radicalBigg
21
4?
parenleftbigg
x? 12
parenrightbigg2
+
1
2 arcsin
x? 12
√21
2
+C =?
radicalbig
5 +x?x2 + 12 arcsin 2x?1√21 +C
113
(4) a45t = 4√1 +x4a167a75x = 4√t4?1,dx = t3(t4?1)?34 dt
a117a180
integraldisplay 1
x 4√1 +x4 dx =
integraldisplay t2
t4?1 =
1
4
integraldisplay parenleftbigg 1
t?1?
1
t+ 1
parenrightbigg
dt+12
integraldisplay 1
1 +t2 dt =
1
4 ln
vextendsinglevextendsingle
vextendsinglevextendsinglet?1
t+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle+1
2 arctant+
C = 14 ln
vextendsinglevextendsingle
vextendsinglevextendsingle 4
√1 +x4?1
4√1 +x4 + 1
vextendsinglevextendsingle
vextendsinglevextendsingle+ 1
2 arctan(
4radicalbig1 +x4) +C
(5)
integraldisplay xdx
√2 + 4x = 12
integraldisplay
xd√4x+ 2 = 12x√2 + 4x? 12
integraldisplay
(2 + 4x)12 dx = x2√2 + 4x? 112(2 + 4x)32 +C
(6)
integraldisplay cosx
1 + sinx dx =
integraldisplay dsinx
1 + sinx = ln(1 + sinx) +C
(7) a45 6√x = ta167a75x = t6,dx = 6t5 dt
a117a180
integraldisplay dx
x(1 + 2√x+ 3√x) =
6
integraldisplay dt
t(1 + 2t3 +t2) = 6
integraldisplay bracketleftbigg1
t?
1
4(t+ 1)?
6t?1
4(2t2?t+ 1)
bracketrightbigg
dt
a113
integraldisplay 6t?1
4(2t2?t+ 1) dt =
3
8
integraldisplay d(2t2?t+ 1)
2t2?t+ 1 +
1
8
integraldisplay dt
2t2?t+ 1 =
3
8 ln|2t
2?t+1|+ 1
4√7 arctan
4t?1√
7 +
C1a167
a108a13
integraldisplay dx
x(1 + 2√x+ 3√x) = 6ln|t|?
3
2 ln|t+1|?
9
4 ln|2t
2?t+1|? 3
2√7 arctan
4t?1√
7 +C = 6ln|
6√x|?
3
2 ln|
6√x+1|? 9
4 ln|2
3√x? 6√x+1|? 3
2√7 arctan
4 6√x?1√
7 +C =
3
4 ln
x 3√x
(1 + 6√x)2(2 3√x? 6√x+ 1)3?
3
2√7 arctan
4 6√x?1√
7 +C
(8)
integraldisplay √x+ 1?√x?1
√x+ 1 +√x?1 dx =
integraldisplay (√x+ 1?√x?1)2
(√x+ 1 +√x?1)(√x+ 1?√x?1) dx =
integraldisplay
(x?
radicalbig
x2?1)dx = x
2
2?
x
2
radicalbig
x2?1 + 12 ln|x+
radicalbig
x2?1|+C
(9) a45 3
radicalbiggx+ 1
x?1 = ta167a75x =
t3 + 1
t3?1,dx =?
6t2
(t3?1)2 dt
a117a180
integraldisplay dx
3radicalbig(x+ 1)2(x?1)4 =?
3
2
integraldisplay
dt =?32t+C =?32 3
radicalbiggx+ 1
x?1 +C
(10) a45 4√x = ta167a75x = t4,dx = 4t3 dt
a117a180
integraldisplay dx
√x(1 + 4√x)3 = 4
integraldisplay t
(1 +t)3 dt = 4
integraldisplay bracketleftbigg 1
(1 +t)2?
1
(1 +t)3
bracketrightbigg
dt =? 41 +t + 2(1 +t)2 + C =
41 + 4√x + 2(1 + 4√x)2 +C
(11)
integraldisplay dx
√ax2 +bx+c = 1√a
integraldisplay d
parenleftbigg√
a
parenleftbigg
x+ b2a
parenrightbiggparenrightbigg
bracketleftbigg√
a
parenleftbigg
x+ b2a
parenrightbiggbracketrightbigg2
+ 4ac?b
2
4a
=
1√
a ln
vextendsinglevextendsingle
vextendsinglevextendsingle√a
parenleftbigg
x+ b2a
parenrightbigg
+
radicalbig
ax2 +bx+c
vextendsinglevextendsingle
vextendsinglevextendsingle+C
(12) a45 4
radicalbigga?x
x = t
a117a180
integraldisplay xdx
4radicalbigx3(a?x) =?
integraldisplay 4at2
(1 +t4)2 dt =
4a
integraldisplay bracketleftbigg t
(t2 +√2t+ 1)(t2?√2t+ 1)
bracketrightbigg2
dt =?a2
integraldisplay dt
(t2?√2t+ 1)2?
a
2
integraldisplay dt
(t2 +√2t+ 1)2 +a
integraldisplay dt
t4 + 1
a113
integraldisplay dt
(t2?√2t+ 1)2 =
integraldisplay d
parenleftbigg
t?
√2
2
parenrightbigg
bracketleftBiggparenleftbigg
t?
√2
2
parenrightbigg2
+ 12
bracketrightBigg2 = 2t?
√2
2(t2?√2t+ 1) +
√2arctan(√2t?1) +C
1
integraldisplay dt
(t2 +√2t+ 1)2 =
2t+√2
2(t2 +√2t+ 1) +
√2arctan(√2t+ 1) +C
2
integraldisplay 1
t4 + 1 dt =
1
2
integraldisplay t2 + 1
t4 + 1 dt?
1
2
integraldisplay t2?1
t4 + 1 dt
114
a13
integraldisplay t2 + 1
t4 + 1 dt =
integraldisplay 1 + 1
t2
t2 + 1t2
dt =
integraldisplay d
parenleftbigg
t? 1t
parenrightbigg
parenleftbigg
t? 1t
parenrightbigg2
+ 2
= 1√2 arctan t
2?1
√2t +C3,
integraldisplay t2?1
t4 + 1 dt =
1
2√2 ln
t2?√2t+ 1
t2 +√2t+ 1 +C4
a108a13
integraldisplay xdx
4radicalbigx3(a?x) =?
at3
1 +t4?
a√2
2 arctan
√2t
1?t2 +
a
4√2 ln
vextendsinglevextendsingle
vextendsinglevextendsinglet2 +
√2t+ 1
t2?√2t+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle+ a
2√2 arctan
t2?1√
2t +Ca167
a217a165t = 4
radicalbigga?x
x,
(13)
integraldisplay
x
radicalbig
x4 + 2x2?1dx = 12
integraldisplay radicalbig
(x2 + 1)2?2dx2 = x
2 + 1
4
radicalbig
x4 + 2x2?1?12 ln(x2+1+
radicalbig
x4 + 2x2?1)+
C
(14)
integraldisplay radicalbig
2 +x?x2 dx =
integraldisplay radicalBigg9
4?
parenleftbigg
x? 12
parenrightbigg2
dx = 2x?14
radicalbig
2 +x?x2 + 98 arcsin 2x?13 +C
(15)
integraldisplay x2 dx
√1 +x?x2 =?
integraldisplay radicalbig
1 +x?x2 dx+
integraldisplay x+ 1
√1 +x?x2 dx =?2x?14
radicalbig
1 +x?x2?58 arcsin 2x?1√5?
radicalbig
1 +x?x2 + 32 arcsin 2x?1√5 +C =?2x+ 34
radicalbig
1 +x?x2 + 78 arcsin 2x?1√5 +C
(16)
integraldisplay x2 + 1
x√x4 + 1 dx =
integraldisplay 1 + 1
x2radicalbigg
x2 + 1x2
dx =
integraldisplay d
parenleftbigg
x? 1x
parenrightbigg
radicalBiggparenleftbigg
x? 1x
parenrightbigg2
+ 2
= ln
parenleftBigg
x? 1x +
radicalbigg
x2 + 1x2
parenrightBigg
+C = ln
vextendsinglevextendsingle
vextendsinglevextendsinglex2?1 +
√x4 + 1
x
vextendsinglevextendsingle
vextendsinglevextendsingle+
C
(17)
integraldisplay
sin6xdx =
integraldisplay parenleftbigg1?cos2x
2
parenrightbigg3
dx = 18
integraldisplay
(1? 3cos2x+ 3cos2 2x? cos3 2x)dx = 18x? 316 sin2x+
3
16
integraldisplay
(1+cos4x)dx? 116
integraldisplay
cos2 2xdsin2x = 18x? 316 sin2x+ 316x+ 364 sin4x? 116 sin2x+ 148 sin3 2x+
C = 516x? 14 sin2x+ 364 sin4x+ 148 sin3 2x+C
(18)
integraldisplay
sin2xcos4xdx =?15
integraldisplay
sinxdcos5x =?15 sinxcos5x+
1
5
integraldisplay
cos6x =?15 sinxcos5x+ 15
bracketleftbigg 5
16x?
1
4 sin2x+
3
64 sin4x+
1
48 sin
3 2x
bracketrightbigg
+C = 116x? 120 sin2x+
3
320 sin4x+
1
240 sin
3 2x? 1
5 sinxcos
5x+C
(19)
integraldisplay
sin4xcos4xdx =
integraldisplay parenleftbiggsin2x
2
parenrightbigg4
dx = 116
integraldisplay parenleftbigg1?cos4x
2
parenrightbigg2
dx = 164
integraldisplay
(1?2cos4x+ cos2 4x)dx =
3
128x?
sin4x
128 +
1
1024 sin8x+C
(20)
integraldisplay cos4x
sin3x dx =?
1
2
integraldisplay
cos3xd 1sin2x =?12 · cos
3x
sin2x?
3
2
integraldisplay cos2x
sinx dx =?
cos3x
2sin2x?
3
2
integraldisplay dx
sinx +
3
2
integraldisplay
sinxdx =? cos
3x
2sin2x?
3
2
integraldisplay sec2 x
2
tan x2
dx2? 32 cosx =? cos
3x
2sin2x?
3
2 ln
vextendsinglevextendsingle
vextendsingletan x2
vextendsinglevextendsingle
vextendsingle? 32 cosx+C
(21)
integraldisplay dx
sin3xcos5x =
integraldisplay sin2x+ cos2x
sin3xcos5x dx =
integraldisplay dx
sinxcos5x +
integraldisplay dx
sin3xcos3x =
integraldisplay sin2x+ cos2x
sinxcos5x dx +integraldisplay
sin2x+ cos2x
sin3xcos3x dx =
integraldisplay sinx
cos5x dx+2
integraldisplay dx
sinxcos3x+
integraldisplay dx
sin3xcosx =
1
4 cos
4x+2
integraldisplay sin2x+ cos2x
sinxcos3x dx+integraldisplay
sin2x+ cos2x
sin3xcosx dx =
1
4 sec
4x+ 2
integraldisplay sinx
cos3x dx+ 3
integraldisplay dx
sinxcosx +
integraldisplay cosx
sin3x dx =
1
4 sec
4x+ sec2x+
3
integraldisplay dtanx
tanx?
1
2 csc
2x = 1
4 sec
4x+sec2x+3ln|tanx|?1
2 csc
2x+C1 = 1
4 tan
4x+ 3
2 tan
2x?1
2 cot
2x+
3ln|tanx|+C
(22)
integraldisplay
tanx·tan(x+a)dx =
integraldisplay
tanx· tanx+ tana1?tanxtana dx =
integraldisplay tan2x+ tanxtana+ 1?1
1?tanxtana dx =
integraldisplay 1 + tan2x
1?tanxtana dx?integraldisplay
dx =
integraldisplay dtanx
1?tanxtana?x =?cotaln|1?tanxtana|?x+C1 = cotaln
vextendsinglevextendsingle
vextendsinglevextendsingle cosx
cos(x+a)
vextendsinglevextendsingle
vextendsinglevextendsingle?x+C
115
(23)
integraldisplay
sin5xcosxdx = 12
integraldisplay
(sin6x+ sin4x)dx =? 112 cos6x? 18 cos4x+C
(24)
integraldisplay sin2x
1 + sin2x dx =
integraldisplay 1
csc2x+ 1 dx =
integraldisplay parenleftbigg
1? csc
2x
1 + csc2x
parenrightbigg
dx = x+
integraldisplay dcotx
2 + cot2x = x+
√2
2 arctan
parenleftbigg√2
2 cotx
parenrightbigg
+
C
(25) a23sin(a?b) negationslash= 0a167
a75
integraldisplay dx
sin(x+a)sin(x+b) =
1
sin(a?b)
integraldisplay sin[(x+a)?(x+b)]
sin(x+a)sin(x+b) dx =
1
sin(a?b)
integraldisplay bracketleftbiggcos(x+b)
sin(x+b)?
cos(x+a)
sin(x+a)
bracketrightbigg
dx =
1
sin(a?b) ln
vextendsinglevextendsingle
vextendsinglevextendsinglesin(x+b)
sin(x+a)
vextendsinglevextendsingle
vextendsinglevextendsingle+C
(26) I =
integraldisplay
xexcosxdx = xexcosx?
integraldisplay
ex(cosx?xsinx)dx = xexcosx?
integraldisplay
excosxdx+
integraldisplay
xexsinxdx =
xexcosx?sinx+ cosx2 ex+xexsinx?
integraldisplay
ex(sinx+xcosx)dx = xexcosx?sinx+ cosx2 ex+xexsinx?
sinx?cosx
2 e
x?
integraldisplay
xexcosxdx+C1 = ex(xcosx+xsinx?sinx)?I +C1a167
a75I =
integraldisplay
xexcosxdx = e
x
2 (xcosx+xsinx?sinx) +C
(27) a45tanx2 = ta167a75sinx = 2t1 +t2,cosx = 1?t
2
1 +t2,dx =
2dt
1 +t2
a117a180
integraldisplay dx
(2 + cosx)sinx =
integraldisplay 1 +t2
t(3 +t2) dt =
integraldisplay parenleftbigg 1
3t +
2t
3(3 +t2)
parenrightbigg
dt = 13 ln|t(t2+3)|+C = 13 ln
vextendsinglevextendsingle
vextendsingletan x2
parenleftBig
tan2 x2 + 3
parenrightBigvextendsinglevextendsingle
vextendsingle+
C =
1
6 ln
(1?cosx)(2 + cosx)2
(1 + cosx)3 +C
(28)
integraldisplay
ln(x+
radicalbig
1 +x2)2 dx = xln(x+
radicalbig
1 +x2)2?
integraldisplay
x· 1(x+√1 +x2)2 · 2(x+
radicalbig
1 +x2) ·
parenleftbigg
1 + x√1 +x2
parenrightbigg
dx = xln(x+
radicalbig
1 +x2)2?
integraldisplay d(1 +x2)
√1 +x2 =
xln(x+
radicalbig
1 +x2)2?2
radicalbig
1 +x2 +C
(29)
integraldisplay sinxcosx
sinx+ cosx dx =
integraldisplay sin2parenleftBigx+ pi
4
parenrightBig
12
√2sinparenleftBigx+ pi
4
parenrightBig dx =
√2
2
integraldisplay
sin
parenleftBig
x+ pi4
parenrightBig
dx? 12√2
integraldisplay dx
sin
parenleftBig
x+ pi4
parenrightBig =?
√2
2 cos
parenleftBig
x+ pi4
parenrightBig
1
2√2
integraldisplay dparenleftBigtanparenleftBigx
2 +
pi
8
parenrightBigparenrightBig
tan
parenleftBigx
2 +
pi
8
parenrightBig = 12(sinx?cosx)?
√2
4 ln
vextendsinglevextendsingle
vextendsingletan
parenleftBigx
2 +
pi
8
parenrightBigvextendsinglevextendsingle
vextendsingle+C
(30)
integraldisplay lnx
(1 +x2)32
dx =
integraldisplay
lnxd
parenleftbigg x
√1 +x2
parenrightbigg
= xlnx√1 +x2?
integraldisplay dx
√1 +x2 = xlnx√1 +x2?ln(x+
radicalbig
1 +x2) +C
(31)
integraldisplay
xexsinxdx = xexsinx?
integraldisplay
ex(sinx + xcosx)dx = xexsinx?
integraldisplay
exsinxdx?
integraldisplay
xexcosxdx =
xexsinx? sinx?cosx2 ex? e
x
2 (xcosx+xsinx?sinx) +C =
ex
2 (xsinx?xcosx+ cosx) +C
(32)
integraldisplay x3 arccosx
√1?x2 dx =?x2 arccosx
radicalbig
1?x2+2
integraldisplay
xarccosx
radicalbig
1?x2 dx?
integraldisplay
x2 dx =?x2
radicalbig
1?x2 arccosx?
x3
3?
2
3(1?x
2)32 arccosx? 2
3
integraldisplay
(1?x2)dx =?x2
radicalbig
1?x2 arccosx? x
3
3?
2
3(1?x
2)32 arccosx?
2
3
parenleftbigg
x? x
3
3
parenrightbigg
+C =?6x+x
3
9?
2 +x2
3
radicalbig
1?x2 arccosx+C
(33)
integraldisplay
(x+|x|)2 dx =
integraldisplay
(2x2+2x|x|)dx = 23x3+2
integraldisplay
x·sgnx·xdx = 23x3+23x3sgnx+C = 23x3+23x2|x|+C
(34)
integraldisplay
x2excosxdx = x2excosx?
integraldisplay
ex(2xcosx?x2 sinx)dx = x2excosx?ex(xcosx+xsinx?sinx)+
x2exsinx?integraldisplay
ex(2xsinx+x2 cosx)dx = x2ex(cosx+ sinx)?ex(xcosx+xsinx?sinx)?ex(xsinx?xcosx+
cosx)?
integraldisplay
x2excosxdx+C1 = ex(x2 cosx+x2 sinx?2xsinx+ sinx?cosx)?I +C1a167
a75I =
integraldisplay
x2excosxdx = e
x
2 (x
2 cosx+x2 sinx?2xsinx+ sinx?cosx) +C
116
(35)
integraldisplay xex
(1 +x)2 dx =?
integraldisplay
xexd 11 +x =? xe
x
1 +x +
integraldisplay
exdx =? xe
x
1 +x +e
x +C = ex
x+ 1 +C
(36)
integraldisplay √
xln2xdx = 23 ln2x·x32? 43
integraldisplay
x12 lnxdx = 23 ln2x·x32? 89x32 lnx+ 89
integraldisplay √
xdx = 23 ln2x·x32?
8
9x
3
2 lnx+ 1627x
3
2 +C = 227x
3
2 (9ln2x?12lnx+ 8) +C
(37) a45t =
radicalbiggb?x
x?aa167a75x =
b+at2
1 +t2,x?a =
b?a
1 +t2,dx =?
2(b?a)t
(1 +t2)2 dta167
a117a180
integraldisplay dx
radicalbig(x?a)(b?x) =?2
integraldisplay dt
1 +t2 =?2arctant+C =?2arctan
radicalbiggb?x
x?a +C
(38)
integraldisplay
xln 1 +x1?x dx = x
2
2 ln
1 +x
1?x?
integraldisplay x2
1?x2 dx =
x2
2 ln
1 +x
1?x?
integraldisplay dx
1?x2 +
integraldisplay
dx = 12x2 ln 1 +x1?x?
1
2 ln
1 +x
1?x +x+C
(39)
integraldisplay
xarctanx·ln(1 +x2)dx = x
2
2 arctanxln(1 +x
2)?
1
2
integraldisplay
x2
bracketleftbiggln(1 +x2)
1 +x2 +
2xarctanx
1 +x2
bracketrightbigg
dx = x
2
2 arctanxln(1+x
2)?1
2
integraldisplay
ln(1+x2)dx+12
integraldisplay ln(1 +x2)
1 +x2 dx?integraldisplay
xarctanxdx+
integraldisplay xarctanx
1 +x2 dx =
x2
2 arctanxln(1+x
2)?x
2 ln(1+x
2)+
integraldisplay x2
1 +x2 dx+
1
2 arctanxln(1+
x2)?
integraldisplay xarctanx
1 +x2 dx +
integraldisplay xarctanx
1 +x2 dx?
x2
2 arctanx +
1
2
integraldisplay x2
1 +x2 dx =
x2
2 arctanxln(1 + x
2)?
x
2 ln(1+x
2)+ 3
2x?
3
2 arctanx+
1
2 arctanxln(1+x
2)? x2
2 arctanx+C =
1
2 arctanx[x
2 ln(1+x2)+
ln(1 +x2)?x2?3]? x2 ln(1 +x2) + 32x+C
(40)
integraldisplay
sinh2xcosh2xdx = 14
integraldisplay
sinh2 2xdx = 18
integraldisplay
(cosh4x?1)dx = 132 sinh4x? x8 +C
117
a49a212a217 a189a200a169
§1 a189a200a169a27a86a103
a124a94a189a200a169a27a189a194a79a142a200a169a181
(1)
integraldisplay l
0
f(x)dxa167a217a165f(x) = ax+b,a,ba180a126a234
(2)
integraldisplay 2
1
x2 dx
(3)
integraldisplay 1
0
axdx
a41a181
(1) a207f(x)a51[0,l]a254a235a89a167a25a189a200a169a55a127a51a167a226a189a200a169a189a194a167a242a171a109[0,l]na31a169a167a75a122a152a102a171a109a27a127a143?xi = lna167
a18ξia143a122a135a102a171a109[xi?1,xi]a27a109a224a58a167a61ξi = inl(i = 1,2,···,n).
a138a200a169a218
nsummationdisplay
i=1
f(ξi)?xi =
nsummationdisplay
i=1
(aξi +b)?xi =
nsummationdisplay
i=1
parenleftbiggia
nl+b
parenrightbigg l
n =
nsummationdisplay
i=1
(nb+ia) ln2 = bl+ n+ 12n al2a167
a117a180
integraldisplay l
0
f(x)dx = lim
||x||= 1n→0
nsummationdisplay
i=1
(aξi +b)?xi = lim
n→∞
parenleftbigg
bl+ n+ 12n al2
parenrightbigg
= bl+ a2l2
(2) a207x2a51[?1,2]a254a235a89a167a25a189a200a169a55a127a51a167a226a189a200a169a189a194a167a242a171a109[?1,2]na31a169a167a75a122a152a102a171a109a27a127a143?xi =
3
na167a18ξia143a122a135a102a171a109[xi?1,xi]a27a109a224a58a167a61ξi =?1 +
3i
n =
3i?n
n (i = 1,2,···,n).
a138a200a169a218
nsummationdisplay
i=1
f(ξi)?xi =
nsummationdisplay
i=1
ξ2i?xi =
nsummationdisplay
i=1
parenleftbigg3i?n
n
parenrightbigg2 3
n =
nsummationdisplay
i=1
3
n3 (9i
2?6ni+n2) = 9
2
parenleftbigg
1 + 1n
parenrightbiggparenleftbigg
2 + 1n
parenrightbigg
9
parenleftbigg
1 + 1n
parenrightbigg
+ 3a167
a117a180
integraldisplay 2
1
x2 dx = lim
||x||= 1n→0
nsummationdisplay
i=1
ξ2i?xi = lim
n→∞
bracketleftbigg9
2
parenleftbigg
1 + 1n
parenrightbiggparenleftbigg
2 + 1n
parenrightbigg
9
parenleftbigg
1 + 1n
parenrightbigg
+ 3
bracketrightbigg
= 3
(3) a207axa51[0,1]a254a235a89a167a25a189a200a169a55a127a51a167a226a189a200a169a189a194a167a242a171a109[0,1]na31a169a167a75a122a152a102a171a109a27a127a143?xi = 1na167
a18ξia143a122a135a102a171a109[xi?1,xi]a27a109a224a58a167a61ξi = in(i = 1,2,···,n).
a138a200a169a218
nsummationdisplay
i=1
f(ξi)?xi =
nsummationdisplay
i=1
aξi?xi =
nsummationdisplay
i=1
1
na
i
n =
a1n(1?a)
n(1?a1n)
,anegationslash= 1
1,a = 1
a117a180
integraldisplay 1
0
axdx = lim
||x||= 1n→0
nsummationdisplay
i=1
aξi?xi =
lim
n→∞
a1n(1?a)
n(1?a1n)
= a?1lna,anegationslash= 1
1,a = 1
118
§2 a189a200a169a127a51a27a94a135
1,a7a228a101a15a188a234a27a140a200a53a181
(1) f(x)a51[?2,2]a254a107a46a167a167a27a216a235a89a58a1801n(n = 1,2,3,···)
(2) f(x) = sgn
parenleftBig
sin pix
parenrightBig
a167a51[0,1].
a41
(1) f(x)a51[?2,2]a254a180a140a200a27.
a207f(x)a51[?2,2]a254a107a46a167a25?M > 0a167a166|f(x)|lessorequalslantM,?x∈ [?2,2]a167a108a13a217a8a204ω(f) lessorequalslant 2M.
ε> 0a167a18a103a44a234Na247a118N =
bracketleftbigg2M
ε
bracketrightbigg
+1a167a117a180a51
bracketleftbigg 1
N,2
bracketrightbigg
a254f(x)a144a107a107a129a245a135a216a235a89a58a167a207a13f(x)a51
bracketleftbigg 1
N,2
bracketrightbigg
a254
a140a200.
a51
bracketleftbigg
0,1N
bracketrightbigg
a254a167a242a217a169a127a143a220a169a171a109?xia167a49ia135a2a171a109?xia254a27a8a204a23a143ωi(f) lessorequalslantω(f) lessorequalslant 2Ma167a75
nsummationdisplay
i=1
ωi(f)?xi lessorequalslant
2M
nsummationdisplay
i=1
xi lessorequalslant 2MN < 2M · ε2M = εa167a25f(x)a51
bracketleftbigg
0,1N
bracketrightbigg
a254a143a180a140a200a27.
a113a100a117f(x)a51[?2,0]a254a235a89a167a8a44a140a200.
a226a200a169a39a117a171a109a140a92a53a167a26f(x)a51[?2,2]a254a140a200.
(2) a214a191a189a194f(0) = 0.
a100a117f(x)a51[0,1]a254a107a46a167a113sgn
parenleftBig
sin pix
parenrightBig
a144a51x = 0,1n(n = 1,2,···)a109a228a167a25a100a29a75a1631a164a167a26f(x)a51[0,1]a254
a140a200.
2,a101a188a234f(x)a51[a,b]a254a140a200a167a217a200a169a180Ia167a56a51[a,b]a83a107a129a135a58a254a85a67f(x)a27a138a166a167a164a143a44a152a135a188a234f?(x)a167a121
a178f?(x)a143a51[a,b]a254a140a200a167a191a133a217a200a169a69a143I.
a121a178a181a45F(x) = f(x)?f?(x)a167a75F(x)a51[a,b]a254a216a85a67a10f(x)a27a188a234a138a27a107a129a135a58a9a254a1430a167a61a216a249a107a129a135
a58a9a167a188a234a235a89a167a108a13F(x)a51[a,b]a254a140a200a167a133a200a169a1430.
a113f?(x) = f(x)?F(x)a167a226a140a200a188a234a27a11a69a140a200a167
a107
integraldisplay b
a
f?(x)dx =
integraldisplay b
a
f(x)dx?
integraldisplay b
a
F(x)dx =
integraldisplay b
a
f(x)dx = I.
3,a63a216f,f2,|f|a110a246a109a140a200a53a27a39a88.
a41a181
(1) a101f(x)a51[a,b]a254a140a200a167a75f2(x)a51[a,b]a254a143a140a200.
a207f(x)a51[a,b]a254a140a200a167a25f(x)a51[a,b]a254a55a107a46,a23f(x) lessorequalslantM,Ma143a126a234(x∈ [a,b])
a51a171a109[xi?1,xi]a254a63a18a252a58xprime,xprimeprimea167a127a196f2(xprimeprime)?f2(xprime) = [f(xprimeprime)?f(xprime)][f(xprimeprime) +f(xprime)]
a18ωia76a171f(x)a51[xi?1,xi]a254a27a204a221a167a75|f2(xprimeprime)?f2(xprime)|lessorequalslant 2ωiM
a101f2(x)a51[xi?1,xi]a254a27a204a221a143?ia167a210a107?i lessorequalslant 2ωiMa167a108a13a107
summationdisplay
i
i?xi lessorequalslant 2M
summationdisplay
i
ωi?xi
a100a117f(x)a140a200a167a107
summationdisplay
i
ωi?xi → 0(λ(?) → 0)a167a75
summationdisplay
i
i?xi → 0(λ(?) → 0)a167a249a210a96a178a10f2(x)a51[a,b]a254
a27a140a200a53.
(2) a101f(x)a51[a,b]a254a140a200a167a75|f(x)|a51[a,b]a254a143a140a200.
a169a79a114a188a234f(x)a134|f(x)|a51a171a109[xi?1,xi]a254a27a204a221a80a143ωi,ω?i.
a207a233a225a117[xi?1,xi]a27a63a191a252a58xprime,xprimeprimea167a107||f(xprime)|?|f(xprimeprime)||lessorequalslant|f(xprime)?f(xprimeprime)|a167a25a107ω?i lessorequalslantωia167a117a180
summationdisplay
i
ω?i?xi lessorequalslant
summationdisplay
i
ωi?xi
a100a117
summationdisplay
i
ωi?xi → 0(λ(?) → 0)a167a210a140a177a237a26λ(?) → 0a158a167a143a107
summationdisplay
i
ω?i?xi → 0a167a249a210a96a178a10|f(x)a51[a,b]a254
a27a140a200a53.
(3) a101|f(x)|a51[a,b]a254a140a200a167a216a85a146a189f(x)a51[a,b]a254a143a140a200.
a126a181f(x) =
braceleftbigg 1,xa143a107a110a234
1,xa143a195a110a234
|f(x)| = 1a51a63a219a52a171a109a254a140a200a167a2f(x)a37a51a63a219a52a171a109a254a209a216a140a200.
119
(4) a101f2(x)a51[a,b]a254a140a200a167a216a85a146a189f(x)a51[a,b]a254a143a140a200.
a126a181f(x) =
braceleftbigg 1,xa143a107a110a234
1,xa143a195a110a234
f2(x) = 1a51a63a219a52a171a109a254a140a200a167a2f(x)a37a51a63a219a52a171a109a254a209a216a140a200.
(5) a101|f(x)|a51[a,b]a254a140a200a167a75a100(1)a167a26f2(x) = |f(x)|2a51[a,b]a254a152a189a140a200.
(6) a101f2(x)a51[a,b]a254a140a200a167a75a100|f(x)| =radicalbigf2(x)a167a9a0a188a234a235a89a167a112a0a188a234a140a200a167a26a100a188a234a55a140a200.
4,a101a188a234f(x)a51[a,b]a254a140a200a167a121a178a127a51a242a130a188a234a15?n(x)(n = 1,2,3,···)a166a26
integraldisplay b
a
f(x)dx = lim
n→∞
integraldisplay b
a
n(x)dx
a121a178a181a242[a,b]na31a169a167a23a169a58a143a = x(n)0 <x(n)1 <···<x(n)n?1 <x(n)n = ba61x(n)i = a+ in(b?a),i = 0,1,···,n
a51[x(n)i?1,x(n)i ]a254a45?n(x)a143a76a58(x(n)i?1,f(x(n)i?1))a57(x(n)i,f(x(n)i ))a27a134a130a167a61a8x ∈ (x(n)i?1,x(n)i )a158a167a45?n(x) =
f(x(n)i?1) + x?x
(n)
i?1
x(n)i?x(n)i?1
(f(x(n)i )?f(x(n)i?1))
a75?n(x)a180[a,b]a254a27a152a135a242a130a188a234a15a167a8a44a180a235a89a188a234a15a167a207a100
integraldisplay b
a
n(x)dxa107a189a194.
a101a45m(n)i,M(n)i a57ω(n)i a169a79a76a171a188a234f(x)a51[x(n)i?1,x(n)i ]a254a27a101a40a46a33a254a40a46a57a8a204a167a75a8x∈ [x(n)i?1,x(n)i ]a158a167m(n)i lessorequalslant
n(x) lessorequalslantM(n)i,m(n)i lessorequalslantf(x) lessorequalslantM(n)i a167a108a13|?n(x)?f(x)|lessorequalslantω(n)i
a117a180a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay b
a
f(x)dx?
integraldisplay b
a
n(x)dx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay b
a
|f(x)n(x)|dx =
nsummationdisplay
i=1
integraldisplay x(n)
i
x(n)i?1
|f(x)n(x)|dxlessorequalslant
nsummationdisplay
i=1
ω(n)i?x(n)i
a100a117f(x)a51[a,b]a254a140a200a167a25a8max|?x(n)i | = b?an → 0a158a167a55a107
nsummationdisplay
i=1
ω(n)i?x(n)i → 0a167a207a13
integraldisplay b
a
f(x)dx =
lim
n→∞
integraldisplay b
a
n(x)dx.
5,a101a188a234f(x)a51[A,B]a140a200a167a121a178lim
h→0
integraldisplay b
a
|f(x+h)?f(x)|dx = 0a167a217a165A<a<b<B(a249a152a53a159a161a143a200a169a27
a235a89a53)
a121a178a181a207f(x)a51[A,B]a140a200a167a100a254a75a40a216a167a233?ε> 0a167a127a51[A,B]a254a27a235a89a188a234?(x)a167a166
integraldisplay B
A
|f(x)(x)|dx<
ε
4
a207?(x)a51[A,B]a235a89a167a108a13a152a151a235a89a167a75a233a254a227ε > 0a167a127a51δ > 0a167a233[A,B]a165a63a191a252a58xprime,xprimeprimea167a144a135|xprime?
xprimeprime|<δa167a210a107|?(xprime)(xprimeprime)|< ε2(b?a)
a117a180a8|h|<δa158a167a107
integraldisplay b
a
|f(x+h)?f(x)|dx =
integraldisplay b
a
|f(x+h)(x+h)+?(x+h)(x)+?(x)?f(x)|dxlessorequalslant
integraldisplay b
a
|f(x+h)(x+h)|dx+
integraldisplay b
a
|?(x+h)(x)|dx+
integraldisplay b
a
|f(x)(x)|dxlessorequalslant
integraldisplay B
A
|f(x+h)(x+h)|dx+
integraldisplay b
a
|?(x+h)(x)|dx+
integraldisplay B
A
|f(x)(x)|dx< ε4 +
integraldisplay b
a
ε
2(b?a) dx+
ε
4 = ε
a108a13lim
h→0
integraldisplay b
a
|f(x+h)?f(x)|dx = 0
120
§3 a189a200a169a27a53a159
1,a101f(x),g(x)a51[a,b]a140a200a167a121a178f(x)+g(x)a143a51[a,b]a140a200a167a191a133
integraldisplay b
a
(f(x)+g(x))dx =
integraldisplay b
a
f(x)dx+
integraldisplay b
a
g(x)dx.
a121a178a181a207f(x),g(x)a51[a,b]a140a200a167a61
integraldisplay b
a
f(x)dx,
integraldisplay b
a
g(x)dxa127a51a167a25a233a63a191a169a123?,a = x0 < x1 < ··· <
xn = ba177a57[xi?1,xi]a165a63a191ξia167a107 lim
λ(?)→0
nsummationdisplay
i=1
f(ξi)?xi =
integraldisplay b
a
f(x)dx
a100a169a123?a57ξia27a63a191a53a167a26g(x)a51a100a63a191a169a123a101a167a233a254a227[xi?1,xi]a165a27ξia167a143a107 lim
λ(?)→0
nsummationdisplay
i=1
g(ξi)?xi =
integraldisplay b
a
g(x)dxa167
a117a180
integraldisplay b
a
f(x)dx+
integraldisplay b
a
g(x)dx = lim
λ(?)→0
nsummationdisplay
i=1
f(ξi)?xi+ lim
λ(?)→0
nsummationdisplay
i=1
g(ξi)?xi = lim
λ(?)→0
nsummationdisplay
i=1
(f(ξi)+g(ξi))?xi =
integraldisplay b
a
(f(x) +g(x))dx
a108a13f(x) +g(x)a143a51[a,b]a140a200a167a191a133
integraldisplay b
a
(f(x) +g(x))dx =
integraldisplay b
a
f(x)dx+
integraldisplay b
a
g(x)dx.
2,a23f(x) =
braceleftbigg 1,a8xa143a107a110a234
1,a8xa143a195a110a234
a121a178a181|f(x)|a51a63a219a171a109[a,b]a254a140a200a167a2f(x)a51[a,b]a216a140a200.
a121a178a181a207f(x) =
braceleftbigg 1,a8xa143a107a110a234
1,a8xa143a195a110a234 a167a25|f(x)| = 1a167
a75|f(x)|a51[a,b]a254a235a89a167a108a13|f(x)|a51[a,b]a254a140a200
a233a117a188a234f(x)a167a51[a,b]a27a63a152a220a169a171a109[xi?1,xi](i = 1,2,···,n)a254ωi = 2a167a25
nsummationdisplay
i=1
ωi?xi = 2
nsummationdisplay
i=1
xi =
2(b?a)notarrowright0(n→∞)a167a117a180a188a234f(x)a51[a,b]a254a216a140a200.
3,a23f(x)a51[a,b]a235a89a167f(x) greaterorequalslant 0a167f(x)a216a240a143a34a167a121a178
integraldisplay b
a
f(x)dx> 0.
a121a178a181a207f(x) greaterorequalslant 0a133a216a240a143a34a167a75a55a127a51x0 ∈ [a,b]a167a166a26f(x0) > 0
a100a235a89a188a234a27a219a220a2a210a53a167a127a510 <δlessorequalslant min
parenleftbiggx
0?a
2,
b?x0
2
parenrightbigg
a167a166a8x∈ [x0?δ,x0+δ]a158a167f(x) > f(x0)2 >
0a167a117a180a107
integraldisplay b
a
f(x)dxgreaterorequalslant
integraldisplay x0+δ
x0?δ
f(x)dxgreaterorequalslant
integraldisplay x0+δ
x0?δ
f(x0)
2 dx = f(x0)δ> 0.
4,a39a22a101a15a136a75a165a200a169a27a140a2a181
(1)
integraldisplay 1
0
xdx,
integraldisplay 1
0
x2 dx
(2)
integraldisplay pi
2
0
xdx,
integraldisplay pi
2
0
sinxdx
(3)
integraldisplay?1
2
parenleftbigg1
3
parenrightbiggx
dx,
integraldisplay 1
0
3xdx
a41a181
(1) a207x∈ (0,1)a158a167x>x2a167a75
integraldisplay 1
0
xdx>
integraldisplay 1
0
x2 dx
(2) a207x∈
parenleftBig
0,pi2
parenrightBig
a158a167x> sinxa167a75
integraldisplay pi
2
0
xdx>
integraldisplay pi
2
0
sinxdx
(3) a207
integraldisplay?1
2
parenleftbigg1
3
parenrightbiggx
dx =
integraldisplay 1
0
parenleftbigg1
3
parenrightbiggx?2
dx =
integraldisplay 1
0
32?xdxa133a8x∈ (0,1)a158a1672?x>xa167a2532?x > 3xa167
a108a13
integraldisplay?1
2
parenleftbigg1
3
parenrightbiggx
dx>
integraldisplay 1
0
3xdx
5,a23f(x)a51[a,b]a235a89a167
integraldisplay b
a
f2(x)dx = 0a167a121a178f(x)a51[a,b]a254a240a143a34.
a121a178a181a94a135a121a123.a98a23f(x)a51[a,b]a254a216a240a143a34a167a75f2(x) greaterorequalslant 0a133a216a240a143a34.
121
a113f(x)a51[a,b]a235a89a167a25f2(x)a51[a,b]a235a89a167
a75a226a493a75a140a127
integraldisplay b
a
f2(x)dx> 0a167a249a134a174a127
integraldisplay b
a
f2(x)dx = 0a103a241.
a117a180a98a23a134a216a167a108a13f(x)a51[a,b]a254a240a143a34.
6,a222a126a96a178a181f2(x)a51[a,b]a140a200a167a2f(x)a51[a,b]a216a140a200.
a41a181a126a181f(x) =
braceleftbigg 1,a8xa143a107a110a234
1,a8xa143a195a110a234 a167a25f
2(x) = 1a167
a75f2(x)a51[a,b]a254a235a89a167a108a13f2(x)a51[a,b]a254a140a200
a113a100a49a19a75a140a127a167a188a234f(x)a51[a,b]a254a216a140a200.
7,a23f(x),g(x)a51[a,b]a235a89a167a121a178 lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(θi)?xi =
integraldisplay b
a
f(x)g(x)dxa167a217a165xi?1 lessorequalslant ξi lessorequalslant xi,xi?1 lessorequalslant
θi lessorequalslantxi(i = 1,2,···,n),?xi = xi?xi?1(x0 = a,xn = b).
a121a178a181a207f(x),g(x)a51[a,b]a235a89a167a75f(x)·g(x)a51[a,b]a235a89a167a25f(x)·g(x)a51[a,b]a140a200a167a61 lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(ξi)?xi =
integraldisplay b
a
f(x)g(x)dx
a13 lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(θi)?xi = lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(ξi)?xi +
lim
max(?xi)→0
bracketleftBigg nsummationdisplay
i=1
f(ξi)g(θi)?xi?
nsummationdisplay
i=1
f(ξi)g(ξi)?xi
bracketrightBigg
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
i=1
f(ξi)g(θi)?xi?
nsummationdisplay
i=1
f(ξi)g(ξi)?xi
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
i=1
f(ξi)(g(θi)?g(ξi))?xi
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
i=1
|f(ξi)||g(θi)?g(ξi)|?xi lessorequalslant
nsummationdisplay
i=1
M(f)ωi(g)?xi = M(f)
nsummationdisplay
i=1
ωi(g)?xia167a217a165M(f)a76a171|f|a51[a,b]a254a27a254a46a167ωi(g)a76a171ga51[xi?1,xi]a254a27a8
a204.
a100fa27a235a89a53a218ga27a140a200a53a167a8max(?xi) → 0a158a167a254a161a216a31a170a109a224M(f)
nsummationdisplay
i=1
ωi(g)?xi → 0a167a108a13 lim
max(?xi)→0
nsummationdisplay
i=1
f(ξi)g(θi)?xi =
integraldisplay b
a
f(x)g(x)dx.
8,a23y =?(x)(x greaterorequalslant 0)a180a238a130a252a78a79a92a27a235a89a188a234a167?(0) = 0,x = ψ(x)a180a167a27a135a188a234a167a121a178
integraldisplay a
0
(x)dx +
integraldisplay b
a
ψ(y)dygreaterorequalslantab(agreaterorequalslant 0,bgreaterorequalslant 0).
a121a178a181a100y =?(x)a143a180a238a130a252a78a79a92a27a235a89a188a234a167?(0) = 0a127a217a135a188a234x = ψ(y)a180a238a130a252a78a79a92a27a235a89a188
a234a167a133ψ(0) = 0a167a207a13
integraldisplay a
0
(x)dx,
integraldisplay b
0
ψ(y)dya107a189a194
a45g(x) = bx?
integraldisplay x
0
(t)dta167a65a79a47a167a107
g(a) = ab?
integraldisplay a
0
(t)dt (1)
a13a133gprime(x) = b(x)
a100?(x)a180a238a130a252a78a79a92a27a235a89a188a234a167a207a100a80 < x < ψ(b)a158a167a107gprime(x) > 0a182a8x > ψ(b)a158a167a107gprime(x) < 0a182
a8x = ψ(b)a158a167a107gprime(x) = 0a167a207a100a8x = ψ(b)a158a167g(x)a18a129a140a138a167a61a107
g(a) lessorequalslant maxg(x) = g(ψ(b)) (2)
a169a220a200a169a167a26
integraldisplay ψ(b)
0
x?prime(x)dx = bψ(b)?
integraldisplay ψ(b)
0
(x)dx = g(ψ(b))a167a94a67a254a147a134y =?(x)a167a75x = ψ(y)a167a117a180
g(ψ(b)) =
integraldisplay ψ(b)
0
x?prime(x)dx =
integraldisplay b
0
ψ(y)dy (3)
a242()a33()a147a92()a210a26a20
integraldisplay a
0
(x)dx+
integraldisplay b
a
ψ(y)dygreaterorequalslantab(agreaterorequalslant 0,bgreaterorequalslant 0).
122
§4 a189a200a169a27a79a142
1,a79a142a101a15a189a200a169a181
(1)
integraldisplay 2
1
(x+ 1)(x2?3)
3x2 dx
(2)
integraldisplay pi
2
1
(asinx+bcosx)dx
(3)
integraldisplay 1
0
parenleftbiggx?1
x+ 1
parenrightbigg4
dx
(4)
integraldisplay 1
0
x2 + 1
x4 + 1 dx
(5)
integraldisplay 1√
5
0
x3(1?5x2)10 dx
(6)
integraldisplay 1
0
x2(2?3x2)2 dx
(7)
integraldisplay 1
5
15
x√2?5xdx
(8)
integraldisplay pi
2
0
sinmxcosnxdx
(9)
integraldisplay 1
3
13
18x?4√
9x2 + 6x+ 5 dx
(10)
integraldisplay √ln2
0
x3e?x2 dx
(11)
integraldisplay 1
0
xarctanxdx
(12)
integraldisplay 2pi
0
xcos2xdx
(13)
integraldisplay pi
pi
x2 cosxdx
(14)
integraldisplay 2pi
ω
0
sinωtsin(ωt+?)dt
(15)
integraldisplay 3
0
xdx
1 +√1 +x
(16)
integraldisplay 4
0
x(x+√x)dx
(17)
integraldisplay pi
pi
sinmxsinnxdx
(18)
integraldisplay pi
pi
sinmxcosnxdx
(19)
integraldisplay 0
1
(x+ 1)
radicalbig
1?x?x2 dx
(20)
integraldisplay 0.75
0
dx
(x+ 1)√x2 + 1
a41a181
(1)
integraldisplay 2
1
(x+ 1)(x2?3)
3x2 dx =
integraldisplay 2
1
x3 +x2?3x+ 3
3x2 dx =
1
3
integraldisplay 2
1
parenleftbigg
x+ 1? 3x? 3x2
parenrightbigg
dx = 13
parenleftbiggx2
2 +x?3lnx+
3
x
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
2
1
= 13?ln2
123
(2)
integraldisplay pi
2
1
(asinx+bcosx)dx = (?acosx+bsinx)|pi20 = a+b
(3) a45y = x+ 1a167a75
integraldisplay 1
0
parenleftbiggx?1
x+ 1
parenrightbigg4
dx =
integraldisplay 2
1
parenleftbiggy?2
y
parenrightbigg4
dy =
integraldisplay 2
1
parenleftbigg
1? 8y + 24y2? 32y3 + 16y4
parenrightbigg
dy =
parenleftbigg
y?8lny? 24y + 16y2? 163y3
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
2
1
= 173?8ln2
(4)
integraldisplay 1
0
x2 + 1
x4 + 1 dx =
1
2
integraldisplay 1
0
bracketleftbigg 1
x2 +√2x+ 1 +
1
x2?√2x+ 1
bracketrightbigg
dx =
1
2 [
√2arctan(√2x+ 1) +√2arctan(√2x?1)]vextendsinglevextendsingle
vextendsingle
1
0
=
√2
2 (arctan(
√2 + 1) + arctan(√2?1)) = √2
4 pi
(5) a45x = 1√5 sinu,dx = 1√5 cosudua167
a75
integraldisplay 1√
5
0
x3(1?5x2)10 dx = 125
integraldisplay pi
2
0
sin3ucos21udu =? 125
integraldisplay pi
2
0
(1?cos2u)cos21udcosu =? 125
parenleftbiggcos22u
22?
cos24u
24
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
pi
2
0
=
1
6600
(6)
integraldisplay 1
0
x2(2?3x2)2 dx =
integraldisplay 1
0
(4x2?12x4 + 9x6)dx = 23105
(7) a45u = √2?5x
a75
integraldisplay 1
5
15
x√2?5xdx = 225
integraldisplay √3
1
(2u2?u3)du = 2375(3√3?7)
(8) a8mnegationslash= ±na158a167
integraldisplay pi
2
0
sinmxcosnxdx = 12
integraldisplay pi
2
0
[sin(m+n)x+sin(m?n)x]dx = mm2?n2?
cos m+n2 pi
2(m+n)?
cos m?n2 pi
2(m?n) a182
a8m = ±na133m negationslash= 0a158a167
integraldisplay pi
2
0
sinmxcosnxdx = ±12
integraldisplay pi
2
0
sin2nxdx =? 14n cos2nx
vextendsinglevextendsingle
vextendsingle
pi
2
0
= ± 14n(1?
cosnpi) = ± 14n[1?(?1)n](n∈Za133nnegationslash= 0)a182
a8m = 0a158a167
integraldisplay pi
2
0
sinmxcosnxd = 0
(9)
integraldisplay 1
3
13
18x?4√
9x2 + 6x+ 5 dx =
integraldisplay 1
3
13
d(9x2 + 6x+ 5)√
9x2 + 6x+ 5?
10
3
integraldisplay 1
3
13
dxradicalBig
1 +parenleftbig3x+12 parenrightbig2
=
2radicalbig9x2 + 6x+ 5? 103 ln
3x+ 12 +
radicalBigg
1 +
parenleftbigg3x+ 1
2
parenrightbigg2?
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1
3
13
= 4(√2?1)? 103 ln(√2 + 1)
(10) a45u = x2integraldisplay √
ln2
0
x3e?x2 dx = 12
integraldisplay ln2
0
ue?udu = 12
parenleftbigg
ue?u
vextendsinglevextendsingle
vextendsingle
ln2
0
+
integraldisplay ln2
0
e?udu
parenrightbigg
= 14(1?ln2)
(11)
integraldisplay 1
0
xarctanxdx = x
2
2 arctanx
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
12
integraldisplay 1
0
x2
1 +x2 dx =
pi
8?
1
2 +
1
2 arctanx
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
= pi?24
(12)
integraldisplay 2pi
0
xcos2xdx = 12
integraldisplay 2pi
0
x(1 + cos2x)dx = x
2
4
vextendsinglevextendsingle
vextendsinglevextendsingle
2pi
0
+ 14xsin2x
vextendsinglevextendsingle
vextendsinglevextendsingle
2pi
0
14
integraldisplay 2pi
0
sin2xdx = pi2
(13)
integraldisplay pi
pi
x2 cosxdx = 2
integraldisplay pi
0
x2 cosxdx = 2x2 sinx
vextendsinglevextendsingle
vextendsingle
pi
0
4
integraldisplay pi
0
xsinxdx = 4xcosx
vextendsinglevextendsingle
vextendsingle
pi
0
4
integraldisplay pi
0
cosxdx =?4pi
(14)
integraldisplay 2pi
ω
0
sinωtsin(ωt+?)dt = 12
integraldisplay 2pi
ω
0
[coscos(2ωt+?)]dt = piω cos 14ω sin(2ωt+?)
vextendsinglevextendsingle
vextendsinglevextendsingle
2pi
ω
0
= piω cos?
(15)
integraldisplay 3
0
xdx
1 +√1 +x =
integraldisplay 3
0
x(1?√1 +x)
x dx =
integraldisplay 3
0
(√1 +x?1)dx = 23(1 +x)32
vextendsinglevextendsingle
vextendsinglevextendsingle
3
0
3 = 53
(16)
integraldisplay 4
0
x(x+√x)dx =
integraldisplay 4
0
(x2 +x32 )dx = 51215
124
(17) a8mnegationslash= ±n(m,n∈Z)a158a167
integraldisplay pi
pi
sinmxsinnxdx = 12
integraldisplay pi
pi
[cos(m?n)x?cos(m+n)x]dx = sin(m?n)x2(m?n)
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
pi
sin(m+n)x
2(m+n)
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
pi
= 0a182
a8m = ±n(m,n∈Z)a133mnegationslash= 0a158a167
integraldisplay pi
pi
sinmxsinnxdx = ±
integraldisplay pi
pi
sin2mxdx = ±
integraldisplay pi
0
(1?cos2mx)dx =
±pia182
a8m = 0a158a167
integraldisplay pi
pi
sinmxsinnxdx = 0
(18)
integraldisplay pi
pi
sinmxcosnxdx = 0
(19)
integraldisplay 0
1
(x+1)
radicalbig
1?x?x2 dx =?12
integraldisplay 0
1
(1?x?x2)12 d(1?x?x2)+12
integraldisplay 0
1
radicalbig
1?x?x2 dx = 12
integraldisplay 0
1
radicalbig
1?x?x2 dx =
1
4 +
5
8 arcsin
√5
5
(20) a45x = tant
a75
integraldisplay 0.75
0
dx
(x+ 1)√x2 + 1 =
integraldisplay arctan0.75
0
dt
sint+ cost =
integraldisplay arctan0.75
0
d
parenleftBig
t+ pi4
parenrightBig
√2sinparenleftBigt+ pi
4
parenrightBig =
√2
2 lntan
parenleftbiggt
2 +
pi
8
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
arctan0.75
0
= 1√2 ln
tan
parenleftbiggarctan0.75
2 +
pi
8
parenrightbigg
tan pi8
= 1√2 ln 9 + 4
√2
7
2,a79a142a101a15a200a169a181
(1)
integraldisplay pi
2
0
sin7xdx
(2)
integraldisplay pi
2
0
cos4xdx
(3)
integraldisplay pi
0
sin5xdx
(4)
integraldisplay 2pi
0
cos6xdx
(5)
integraldisplay a
0
(a2?x2)ndx
(6)
integraldisplay 1
0
(1?x2)6 dx
a41a181
(1)
integraldisplay pi
2
0
sin7xdx = 6!!7!!
(2)
integraldisplay pi
2
0
cos4xdx = 3!!4!! · pi2 = 3pi16
(3)
integraldisplay pi
0
sin5xdx =
integraldisplay pi
2
0
sin5xdx+
integraldisplay pi
pi
2
sin5xdx
a51a0a152a200a169a165a167a45x = pi? ya167a75sinx = siny,dx =?dya167a117a180
integraldisplay pi
pi
2
sin5xdx =?
integraldisplay 0
pi
2
sin5ydy =
integraldisplay pi
2
0
sin5ydy = I5
a108a13
integraldisplay pi
0
sin5xdx = 2I5 = 2· 4!!5!! = 1615
(4)
integraldisplay 2pi
0
cos6xdx =
integraldisplay pi
pi
cos6xdx = 2
integraldisplay pi
0
cos6xdx = 4
integraldisplay pi
2
0
cos6xdx = 4· 5!!6!! · pi2 = 1524pi
(5) a45x = acost,dx =?asintdta167
a75
integraldisplay a
0
(a2?x2)ndx =?a2n+1
integraldisplay 0
pi
2
sin2n+1tdt = a2n+1
integraldisplay pi
2
0
sin2n+1tdt = (2n)!!(2n+ 1)!!a2n+1
125
(6) a51a254a75a165a167a45a = 1,n = 6a167a75
integraldisplay 1
0
(1?x2)6 dx = 12!!13!!
3,a23f(x)a180a177a207a188a234a167a177a207a180Ta167a121a178
integraldisplay a+nT
a
f(x)dx = n
integraldisplay T
0
f(x)dxa167a100a63na180a20a18a234.
a121a178a181
integraldisplay a+nT
a
f(x)dx =
integraldisplay 0
a
f(x)dx+
integraldisplay T
0
f(x)dx+···+
integraldisplay nT
(n?1)T
f(x)dx+
integraldisplay a+nT
nT
f(x)dx
a233a254a227a31a170a27a129a0a152a135a200a169a167a23x?nT = ta167a75
integraldisplay a+nT
nT
f(x)dx =
integraldisplay a
0
f(t+nT)dt =
integraldisplay a
0
f(t)dt
a2331 <i<na167a127a196a200a169
integraldisplay iT
(i?1)T
f(x)dxa167a23x?(i?1)T = ta167a75
integraldisplay iT
(i?1)T
f(x)dx =
integraldisplay T
0
f(t+ (n?1)T)dt =
integraldisplay T
0
f(t)dt
a108a13
integraldisplay a+nT
a
f(x)dx = n
integraldisplay T
0
f(x)dx
4,a121a178a181(m,na143a20a18a234)
(1)
integraldisplay pi
pi
sin2mxdx = pi,
integraldisplay pi
pi
cos2mxdx = pi
(2)
integraldisplay pi
pi
cosmxcosnxdx = 0(mnegationslash= n)
a121a178a181
(1)
integraldisplay pi
pi
sin2mxdx = 1m
integraldisplay pi
pi
sin2mxdmx = 1m
integraldisplay pi
0
(1?cos2mx)dmx = pi? 12m sin2mx
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
0
= pi
a211a110a140a26a167
integraldisplay pi
pi
cos2mxdx = pi
(2)
integraldisplay pi
pi
cosmxcosnxdx = 12
integraldisplay pi
pi
[cos(m+n)x+cos(m?n)x]dx = sin(m+n)x(m+n)
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
0
+ sin(m?n)x(m?n)
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
0
= 0
5,a121a178a101a188a234f(x)a51a52a171a109[0,1]a235a89a167a75a181
(1)
integraldisplay pi
2
0
f(sinx)dx =
integraldisplay pi
2
0
f(cosx)dx
(2)
integraldisplay pi
0
xf(sinx)dx = pi2
integraldisplay pi
0
f(sinx)dx
a100a100a79a142
integraldisplay pi
0
xsinx
1 + cos2x dx
a121a178a181
(1) a45pi2?t = xa167a75dx =?dt,f(sinx) = f(cosx)a167
a117a180
integraldisplay pi
2
0
f(sinx)dx =?
integraldisplay 0
pi
2
f(cost)dt =
integraldisplay pi
2
0
f(cost)dt
a61
integraldisplay pi
2
0
f(sinx)dx =
integraldisplay pi
2
0
f(cosx)dx
(2) a23t = pi?xa167a75dx =?dta133xf(sinx) = (pi?t)f(sint)
a117a180
integraldisplay pi
0
xf(sinx)dx =?
integraldisplay 0
pi
(pi?t)f(sint)dt = pi
integraldisplay pi
0
f(sint)dt?
integraldisplay pi
0
tf(sint)dta167a752
integraldisplay pi
0
xf(sinx)dx =
pi
integraldisplay pi
0
f(sinx)dxa167a108a13
integraldisplay pi
0
xf(sinx)dx = pi2
integraldisplay pi
0
f(sinx)dx
integraldisplay pi
0
xsinx
1 + cos2x dx =
pi
2
integraldisplay pi
0
sinx
1 + cos2x dx =?
pi
2
integraldisplay pi
0
dcosx
1 + cos2x =?
pi
2 arctan(cosx)
vextendsinglevextendsingle
vextendsingle
pi
0
= pi
2
4
6,a121a178a219a188a234a27a152a131a6a188a234a27a143a243a188a234a167a243a188a234a27a6a188a234a165a107a152a143a219a188a234.
a121a178a181a23f(x)a51[?l,l]a254a107a189a194a167a133F(x)a180f(x) a27a152a135a6a188a234
a8f(x)a143a219a188a234a61a8f(?x) =?f(x)a158a167a100a117f(x) = ddxF(x),
f(?x) =? ddxF(?x)a167a25a107 ddx[F(x)?F(?x)] = 0a167a108a13a140a26F(x)?F(?x) = C1a133C1 = 0a167a117a180F(x) =
F(?x)a167a75f(x)a27a152a135a6a188a234F(x)a143a243a188a234a167a108a13f(x)a27a63a152a135a6a188a234F(x) +C(Ca143a63a191a126a234)a143a143a243a188a234
a8f(x)a143a243a188a234a61a8f(?x) = f(x)a158a167a97a113a140a26F(x) + F(?x) = C2a133C2 = 2F(0)a167a117a180f(x)a107a152a135a6a188
a234F(x)?F(0)a180a219a188a234.
126
7,a101f(x)a39a117x = Ta233a161a167a133a<T <ba167a75
integraldisplay b
a
f(x)dx = 2
integraldisplay b
T
f(x)dx+
integraldisplay 2T?b
a
f(x)dx
a121a178a181
integraldisplay b
a
f(x)dx =
integraldisplay 2T?b
a
f(x)dx+
integraldisplay T
2T?b
f(x)dx+
integraldisplay b
T
f(x)dx
a233a254a227a31a170a109a224a27a49a19a135a200a169a167a23x = 2T?ta167a75
integraldisplay T
2T?b
f(x)dx =?
integraldisplay T
b
f(2T?t)dt
a113f(x)a39a117x = Ta233a161a167a75f(2T?t) = f(t)
a117a180
integraldisplay T
2T?b
f(x)dx =?
integraldisplay T
b
f(2T?t)dt =?
integraldisplay T
b
f(t)dt =
integraldisplay b
T
f(t)dta167a108a13
integraldisplay b
a
f(x)dx = 2
integraldisplay b
T
f(x)dx +
integraldisplay 2T?b
a
f(x)dx
8,a121a178a181
integraldisplay a
0
x3f(x2)dx = 12
integraldisplay a2
0
xf(x)dx(a> 0)
a121a178a181a45t = x2a167a752xdx = dta167a117a180
integraldisplay a
0
x3f(x2)dx = 12
integraldisplay a2
0
tf(t)dt = 12
integraldisplay a2
0
xf(x)dx
9,a124a94a169a220a200a169a121a178a181
integraldisplay x
0
f(u)(x?u)du =
integraldisplay x
0
braceleftbiggintegraldisplay u
0
f(x)dx
bracerightbigg
du
a121a178a181
integraldisplay x
0
braceleftbiggintegraldisplay u
0
f(x)dx
bracerightbigg
du = u
integraldisplay u
0
f(x)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
x
0
integraldisplay x
0
uf(u)du = x
integraldisplay x
0
f(t)dt?
integraldisplay x
0
uf(u)du =
integraldisplay x
0
xf(u)du?
integraldisplay x
0
uf(u)du =
integraldisplay x
0
f(u)(x?u)du
10,a152a127a221a143la27a238a249a167a164a201a49a214a85a53a198p(x) = a+bx+cx2a169a217a167a193a100a101a227a94a135a251a189a88a234a,b,ca182a111a49a214a180P =integraldisplay
l
0
p(x)dxa167a52a140a49a214a160a11723la63a167a133a51a52a140a58a27a134a109a252a62a136a171a201a111a49a214a27a152a140.
a41a181a100a174a127a167a26
(1) P =
integraldisplay l
0
p(x)dx = al+ b2l2 + c3l3
(2) a45P(x) =
integraldisplay x
0
p(t)dta167a75Pprime
parenleftbigg2
3l
parenrightbigg
= p
parenleftbigg2
3l
parenrightbigg
= a+ 23bl+ 49cl2 = 0
(3)
integraldisplay 2
3l
0
p(x)dx = 23al+ 29bl+ 881cl3 = P2
a233a225a144a167a124
al+ b2l2 + c3l3 = P
2
3al+
2
9bl+
8
81cl
3 = P
2
a+ 23bl+ 49cl2 = 0
a166a41a167a26
a = 4lP
b =? 694l2P
c = 1358l3 P
11,a101f(x)a235a89a167a166a181
(1) ddx
parenleftbiggintegraldisplay b
x
f(t)dt
parenrightbigg
(2) ddx
parenleftBiggintegraldisplay
x2
0
f(t)dt
parenrightBigg
a41a181
(1) ddx
parenleftbiggintegraldisplay b
x
f(t)dt
parenrightbigg
= ddx
parenleftbigg
integraldisplay x
b
f(t)dt
parenrightbigg
=? ddx
parenleftbiggintegraldisplay x
b
f(t)dt
parenrightbigg
=?f(x)
(2) a207 ddx2
parenleftBiggintegraldisplay
x2
0
f(t)dt
parenrightBigg
= f(x2)a167a25 ddx
parenleftBiggintegraldisplay
x2
0
f(t)dt
parenrightBigg
= ddx2
parenleftBiggintegraldisplay
x2
0
f(t)dt
parenrightBigg
· dx
2
dx = 2xf(x
2)
12,a166a52a129a181
127
(1) lim
n→∞
parenleftbigg 1
n2 +
2
n2 +···+
n?1
n2
parenrightbigg
(2) lim
n→∞
1p + 2p +···+np
np+1 (p> 0)
(3) lim
n→∞
1
n
parenleftBiggradicalbigg
1 + 1n +
radicalbigg
1 + 2n +···+
radicalbigg
1 + n?1n
parenrightBigg
(4) lim
n→∞
n√n!
n
a41a181
(1) a188a234f(x) = xa51[0,1]a235a89a167a207a13a140a200.a242[0,1]na31a169a167a169a58a143in,?xi = 1n(i = 0,1,···,n?1)a167a51a122a135a2
a171a109[xi,xi+1] =
bracketleftbiggi
n,
i+ 1
n
bracketrightbigg
a167a18ξi = ina167a75f(ξi) = ina167
a117a180 lim
n→∞
parenleftbigg 1
n2 +
2
n2 +···+
n?1
n2
parenrightbigg
= lim
n→∞
1
n
parenleftbigg1
n +
2
n +···+
n?1
n
parenrightbigg
= lim
n→∞
n?1summationdisplay
i=0
i
n·
1
n =
integraldisplay 1
0
xdx =
1
2
(2) a188a234f(x) = xpa51[0,1]a235a89a167a207a13a140a200.a242[0,1]na31a169a167a169a58a143in,?xi = 1n(i = 1,2,···,n)a167a51a122a135a2a171
a109[xi?1,xi] =
bracketleftbiggi?1
n,
i
n
bracketrightbigg
a167a18ξi = ina167a75f(ξi) =
parenleftbiggi
n
parenrightbiggp
a167
a117a180 lim
n→∞
1p + 2p +···+np
np+1 = limn→∞
nsummationdisplay
i=1
parenleftbiggi
n
parenrightbiggp
· 1n =
integraldisplay 1
0
xpdx = 1p+ 1
(3) a188a234f(x) = √1 +xa51[0,1]a235a89a167a207a13a140a200.a242[0,1]na31a169a167a169a58a143in,?xi = 1n(i = 0,1,···,n? 1)a167a51
a122a135a2a171a109[xi,xi+1] =
bracketleftbiggi
n,
i+ 1
n
bracketrightbigg
a167a18ξi = ina167a75f(ξi) =
radicalbigg
1 + ina167
a117a180 lim
n→∞
1
n
parenleftBiggradicalbigg
1 + 1n +
radicalbigg
1 + 2n +···+
radicalbigg
1 + n?1n
parenrightBigg
= lim
n→∞
n?1summationdisplay
i=0
radicalbigg
1 + in·1n? lim
n→∞
1
n =
integraldisplay 1
0
√x+ 1dx?
0 = 23(2√2?1)
(4) a207
n√n!
n =
n
radicalbigg
n!
nn =
parenleftbigg1
n
parenrightbigg1
n
parenleftbigg2
n
parenrightbigg1
n ···parenleftBign
n
parenrightBig1
na167a25ln
n√n!
n =
1
n
parenleftbigg
ln 1n + ln 2n +···+ ln nn
parenrightbigg
a113a188a234f(x) = lnxa51(0,1]a235a89a167a127a196 lim
ξ→+0
integraldisplay 1
ξ
lnxdx.a242(0,1]na31a169a167a169a58a143in,?xi = 1n(i = 1,2,···,n)a167
a51a122a135a2a171a109[xi?1,xi] =
bracketleftbiggi?1
n,
i
n
bracketrightbigg
a167a18ξi = ina167a75f(ξi) =
parenleftbiggi
n
parenrightbiggp
a167
a117a180 lim
n→∞
ln
n√n!
n = limn→∞
1
n
parenleftbigg
ln 1n + ln 2n +···+ ln nn
parenrightbigg
= lim
n→∞
nsummationdisplay
i=1
ln in · 1n = lim
ξ→+0
integraldisplay 1
ξ
lnxdx =
lim
ξ→+0
(xlnx?x)|1ξ =?1
a108a13 lim
n→∞
n√n!
n = e
1 = 1
e
13,a138a226a1267a107 lim
n→∞
I2n
I2n+1 = 1a167a100a100a237a121
pi
2 =
2
1 ·
2
3 ·
4
3 ·
4
5 ···
2n
2n?1 ·
2n
2n+ 1 ···
a121a178a181a207I2n =
integraldisplay pi
2
0
sin2nxdx = (2n?1)(2n?3)···3·12n(2n?2)···4·2 ·pi2,I2n+1 =
integraldisplay pi
2
0
sin2n+1xdx = 2n(2n?2)···4·2(2n+ 1)(2n?1)···5·3a167
a75I2n+1I
2n
= 21 · 23 · 43 · 45 ··· 2n2n?1 · 2n2n+ 1 · 2pi
a80 lessorequalslant x lessorequalslant pi2a158a1670 lessorequalslant sinx lessorequalslant 1,sin2n+1x lessorequalslant sin2nx lessorequalslant sin2n?1xa167a75
integraldisplay pi
2
0
sin2n+1xdx lessorequalslant
integraldisplay pi
2
0
sin2nxdx lessorequalslant
integraldisplay pi
2
0
sin2n?1xdxa61I2n+1 lessorequalslantI2n lessorequalslantI2n+1a167a117a1801 lessorequalslant I2nI
2n+1
lessorequalslant I2n?1I
2n+1
a113a100a52a237a250a170In = n?1n In?2,I2n+1 = 2n2n+ 1I2n?1a61I2n+1I
2n?1
= 2n2n+ 1a167a25 lim
n→∞
I2n+1
I2n?1 = limn→∞
2n
2n+ 1 =
1a167a117a180 lim
n→∞
I2n
I2n+1 = 1a167a108a13
pi
2 =
2
1 ·
2
3 ·
4
3 ·
4
5 ···
2n
2n?1 ·
2n
2n+ 1 ···
128
14,a23f(x)a134g(x)a209a51[a,b]a140a200a167a121a178
bracketleftbiggintegraldisplay b
a
f(x)g(x)dx
bracketrightbigg2
lessorequalslant
integraldisplay b
a
f2(x)dx·
integraldisplay b
a
g2(x)dx
a113a31a170a51a219a158a164a225a186
a121a178a181a233a63a219a162a234ha167a207[hf(x)?g(x)]2 = h2f2(x)?2hf(x)g(x) +g2(x) greaterorequalslant 0
a100a200a169a27a53a159a167a26
integraldisplay b
a
(h2f2(x)?2hf(x)g(x)+g2(x))dxgreaterorequalslant 0a61h2
integraldisplay b
a
f2(x)dx?2h
integraldisplay b
a
f(x)g(x)dx+
integraldisplay b
a
g2(x)dxgreaterorequalslant
0
a100a19a103a110a145a170a154a75a27a94a135a167a26
parenleftbigg
2
integraldisplay b
a
f(x)g(x)dx
parenrightbigg2
4
integraldisplay b
a
f2(x)dx·
integraldisplay b
a
g2(x)dxlessorequalslant 0a61
bracketleftbiggintegraldisplay b
a
f(x)g(x)dx
bracketrightbigg2
lessorequalslant
integraldisplay b
a
f2(x)dx·
integraldisplay b
a
g2(x)dx
a135a166a31a210a164a225a167a144a135
parenleftbigg
2
integraldisplay b
a
f(x)g(x)dx
parenrightbigg2
4
integraldisplay b
a
f2(x)dx·
integraldisplay b
a
g2(x)dx = 0a61h2
integraldisplay b
a
f2(x)dx?2h
integraldisplay b
a
f(x)g(x)dx+
integraldisplay b
a
g2(x)dx = 0a107a173a138.
a216a148a23h0a143a144a167a27a173a138a167a75h20
integraldisplay b
a
f2(x)dx?2h0
integraldisplay b
a
f(x)g(x)dx+
integraldisplay b
a
g2(x)dx = 0a61
integraldisplay b
a
[h0f(x)?g(x)]2 dx =
0
a13a8g(x) = h0f(x)a158a167
integraldisplay b
a
[h0f(x)? g(x)]2 dx = 0(a217a165h0a143a144a167h2
integraldisplay b
a
f2(x)dx? 2h
integraldisplay b
a
f(x)g(x)dx +
integraldisplay b
a
g2(x)dx = 0a27a173a138)
129
a49a108a217 a189a200a169a27a65a94a218a67a113a79a142
§1 a178a161a227a47a27a161a200
1,a166a100a101a15a136a173a130a164a140a164a27a227a47a161a200a181
(1) y2 = 4(x+ 1),y2 = 4(1?x)
(2) y = |lnx|,y = 0,(0.1 lessorequalslantxlessorequalslant 10)
(3) y = x,y = x+ sin2x,(0 lessorequalslantxlessorequalslantpi)
(4) y2 = 1?x,2y = x+ 2
(5) a93a130r = acosθ+b(bgreaterorequalslanta)a167a8b = aa158a61a143a37a57a130
(6) r = 3cosθ,r = 1 + cosθ
(7) a94a211a130x = a(t?sint),y = a(1?cost)(0 lessorequalslanttlessorequalslant 2pi)a177a57xa182
(8) a40a47a130x23 +y23 = a23
a41a181
(1) a252a94a173a130x = y
2?4
4 a134x =
y2 + 4
4 =?
y2?4
4 a27a2a58a27a112a139a73a169a79a143?2a572a167
a117a180A =
integraldisplay 2
2
bracketleftbigg
y
2?4
4?
y2?4
4
bracketrightbigg
dy =?
integraldisplay 2
0
(y2?4)dy = 163
(2) a252a94a173a130y = |lnx|a134y = 0a27a2a58a27a238a139a73a1431a167
a117a180A =
integraldisplay 10
0.1
[ln|x|? 0]dx =
integraldisplay 1
0.1
(?lnx)dx +
integraldisplay 10
1
lnxdx =?(xlnx?x)
vextendsinglevextendsingle
vextendsingle
1
0.1
+ (xlnx?x)
vextendsinglevextendsingle
vextendsingle
10
1
=
9.9ln10?8.1 ≈ 14.69559
(3) A =
integraldisplay pi
0
(x+ sin2x?x)dx =
integraldisplay pi
0
sin2xdx = 12
integraldisplay pi
0
(1?cos2x)dx = pi2? sin2x4
vextendsinglevextendsingle
vextendsinglevextendsingle
pi
0
= pi2
(4) a252a94a173a130a27a2a58a169a79a143(0,1),(?8,?3)a167
a117a180A =
integraldisplay 1
3
[1?y2?(2y?2)]dy =
integraldisplay 1
3
(3?y2?2y)dy = 323
(5) a164a166a161a200a143a181A = 12
integraldisplay 2pi
0
r2 dθ = 12
integraldisplay 2pi
0
(acosθ+b)2 dθ = pi2a2 +pib2
(6) a164a166a161a200a143a181A = pi
parenleftbigg3
2
parenrightbigg2
A1 = 94pi?A1
a217a165A1 = 12
integraldisplay pi
3
pi3
[9cos2θ?(1 + cosθ)2]dθ =
integraldisplay pi
3
0
[8cos2θ?1?2cosθ)dθ = pia167
a108a13A = 54pi
(7) a164a166a161a200a143a181A =
integraldisplay 2pi
0
ydx =
integraldisplay 2pi
0
a(1?cost)·a(1?cost)dt = a2
integraldisplay 2pi
0
(1?cost)2 dt = 3pia2
(8) a23x = acos3t,y = asin3ta167a217a1650 lessorequalslanttlessorequalslant pi3a167a167a233a65a117a111a169a131a152a27a161a200a167a164a166a161a200a143a217a111a21
a61A = 4
integraldisplay a
0
ydx = 4
integraldisplay 0
pi
2
(?3a2 sin4tcos2t)dt = 12a2
integraldisplay pi
2
0
(sin4x?sin6x)dx = 3pi8 a2
2,a134a130y = xa114a253a11x2 + 3y2 = 6ya27a161a200a169a164a252a220a169A(a2a27a152a172)a218B(a140a27a152a172)a167a166ABa131a138.
a41a181a100a174a127a167a26a253a11a144a167a143x
2
3 + (y?1)
2 = 1a167a75a253a11a161a200a143S = piab = √3pi
a113y = xa134a253a11x2 + 3y2 = 6ya27a2a58a27a112a139a73a1430,32
a117a180A =
integraldisplay 3
2
0
(
radicalbig
6y?3y2? y)dy = √3
integraldisplay 3
2
0
radicalbig
1?(y?1)2 dy? y
2
2
vextendsinglevextendsingle
vextendsinglevextendsingle
3
2
0
=
√3
3 pi?
3
4a167a75B = S? A =
2
3
√3pi+ 3
4a167
a108a13AB = 4
√3pi?9
8√3pi+ 9.
130
3,a166a173a130y = √1?x2 + arccosxa134xa182a57x =?1a164a140a27a161a200.
a41a181a207y1 = √1?x2a27a189a194a141a143[?1,1]a167a138a141a143[0,1]a182
y2 = arccosxa27a189a194a141a143[?1,1]a167a138a141a143[0,pi]
a75a161a200A =
integraldisplay 1
1
y1 dx+
integraldisplay 1
1
y2 dx =
integraldisplay 1
1
radicalbig
1?x2 dx+
integraldisplay 1
1
arccosxdx = 32pi
131
§2 a173a130a27a108a127
a166a101a15a173a130a27a108a127a181
1,y = x32 (0 lessorequalslantxlessorequalslant 4)
2,x = 14y2? 12 lny(1 lessorequalslantylessorequalslante)
3,a40a47a130x23 +y23 = a23 (a> 0)
4,a94a211a130x = a(t?sint),y = a(1?cost)(0 lessorequalslanttlessorequalslant 2pi)
5,a11a27a236a109a130x = a(cost+tsint),y = a(sint?tcost)(0 lessorequalslanttlessorequalslant 2pi)
6,a37a57a130r = a(1 + cosθ)(0 lessorequalslantθlessorequalslant 2pi)
a41a181
1,a164a166a108a127a143s =
integraldisplay 4
0
radicalBig
1 + [(x32 )prime]2 dx =
integraldisplay 4
0
radicalbigg
1 + 94xdx = 827(10√10?1)
2,a164a166a108a127a143s =
integraldisplay e
1
radicalbig
1 + (xprime)2 dy =
integraldisplay e
1
radicalBigg
1 +
parenleftbiggy
2?
1
2y
parenrightbigg2
dy =
integraldisplay e
1
radicalBiggparenleftbigg
y
2 +
1
2y
parenrightbigg2
dy =
integraldisplay e
1
parenleftbiggy
2 +
1
2y
parenrightbigg
dy =
e2 + 1
4
3,a100a174a127a140a23x = acos3t,y = asin3t(0 lessorequalslanttlessorequalslant 2pi)
a75a164a166a108a127a143s = 4
integraldisplay pi
2
0
radicalBig
[(acos3t)prime]2 + [(asin3t)prime]2 dt =
4a
integraldisplay pi
2
0
radicalBig
(?3cos2tsint)2 + (3sin2tcost)2 dt = 12a
integraldisplay pi
2
0
sintcostdt = 6a
4,s =
integraldisplay 2pi
0
radicalbig
[(a(t?sint))prime]2 + [(a(1?cost))prime]2 dt =
|a|
integraldisplay 2pi
0
radicalBig
(1?cost)2 + sin2tdt = 2|a|
integraldisplay 2pi
0
radicalbigg
1?cost
2 dt = 2|a|
integraldisplay 2pi
0
sin t2 dt = 8|a|
5,s =
integraldisplay 2pi
0
radicalbig
[(a(cost+tsint))prime]2 + [(a(sint?tcost))prime]2 dt =
|a|
integraldisplay 2pi
0
radicalbig
(tcost)2 + (tsint)2 dt = |a|
integraldisplay 2pi
0
tdt = 2pi2|a|
6,s =
integraldisplay 2pi
0
radicalbig
r2 + (rprime)2 dθ =
integraldisplay 2pi
0
radicalBig
a2(1 + cosθ)2 +a2 sin2θdθ = 4|a|
integraldisplay pi
0
radicalbigg
1 + cosθ
2 dθ = 4|a|
integraldisplay pi
0
cos θ2 dθ =
8|a|
132
§3 a78a200
1,a166a209a100a101a15a136a173a161a164a140a164a27a65a219a78a78a200a181
(1) a166a31a73a78a78a200a167a217a254a101a46a27a143a253a11a167a253a11a27a182a127a169a79a31a117A,Ba218a,ba167a13a112a143ha182
(2) a166a253a165a78a78a200a181x
2
a2 +
y2
b2 +
z2
c2 = 1a182
(3) a166a100a101a15a252a173a161a181x2 +y2 +z2 = a2,x2 +y2 = axa164a140a164a27a78a200a182
(4) a166a94a207a76a46a161a134a187a27a178a161a108a134a11a206a254a131a101a27a252a47a78a78a200a167a23a11a206a27a46a140a187a143aa167a46a161a144a167a143x2 +y2 lessorequalslant
a2a167a31a161a207a76xa182a254a27a134a187a133a134a46a161a164αa14.
a41a181
(1) a138a152a178a49a117a254a33a101a46a133a229a108a101a46a143xa27a31a161a167a100a31a161a143a253a11a167a217a140a182a169a79a143a181aprime = A+
parenleftBig
1? xh
parenrightBig
(a?A),
bprime = B +
parenleftBig
1? xh
parenrightBig
(b?B)
a117a180a100a31a161a161a200a143a181A(x) = piaprimebprime =
pi
bracketleftbigg
AB + (a?A)(b?B)
parenleftBig
1? xh
parenrightBig2
+ (A(b?B) +B(a?A))
parenleftBig
1? xh
parenrightBigbracketrightbigg
a108a13a164a166a78a200a143V =
integraldisplay h
0
A(x)dx = pi6[(2a+A)b+ (a+ 2A)B]
(2) a94a82a134a117Oxa182a27a178a161a31a253a165a26a31a232a143a152a253a11a167a167a51yOza178a161a254a27a221a75a143 y
2
b2
parenleftbigg
1? x
2
a2
parenrightbigg+ z
2
c2
parenleftbigg
1? x
2
a2
parenrightbigg =
1
a100a100a140a132a217a140a182a169a79a143b
radicalbigg
1? x
2
a2a57c
radicalbigg
1? x
2
a2a167a108a13a26a100a253a11a161a200a143A(x) = pibc
parenleftbigg
1? x
2
a2
parenrightbigg
a117a180a167a164a166a27a253a165a78a78a200a143a181
V =
integraldisplay a
a
A(x)dx = 2
integraldisplay a
0
pibc
parenleftbigg
1? x
2
a2
parenrightbigg
dx = 43abc
(3) z =radicalbiga2?x2?y2(a254a140a161)a167a217a67a122a137a140a143?√ax?x2 lessorequalslantylessorequalslant√ax?x2
a217a31a161a200a143A(x) = 2
integraldisplay √ax?x2
0
radicalbig
a2?x2?y2 dy = a32x12?a12x32 + (a2?x2)arcsin
radicalbigg x
a+x
a117a180a167a164a166a78a200a143a181
V = 2
integraldisplay a
0
A(x)dx =
2
integraldisplay a
0
bracketleftbigg
a32x12?a12x32 + (a2?x2)arcsin
radicalbigg x
a+x
bracketrightbigg
dx = 23a3
parenleftbigg
pi? 43
parenrightbigg
(4) y = √a2?x2,z = √a2?x2 tanαa167
a75A(x) = 12
radicalbig
a2?x2 ·
radicalbig
a2?x2 tanα = 12(a2?x2)tanα
a108a13a164a166a78a200a143a181
V =
integraldisplay a
a
A(x)dx =
integraldisplay a
0
(a2?x2)tanαdx = 23a3 tanα
2,a166a94a61a78a27a78a200a181
(1) x
2
a2 +
y2
b2 = 1a55Xa182a182
(2) y = sinx,y = 0(0 lessorequalslantxlessorequalslantpi)
(i) a55xa182
(ii) a55ya182
(3) x = asin3t,y = bcos3t(0 lessorequalslanttlessorequalslant 2pi)
(i) a55xa182
(ii) a55ya182
(4) a121a178a100alessorequalslantxlessorequalslantb,0 lessorequalslantylessorequalslanty(x)(a217a165y(x)a180a235a89a188a234)a164a140a164a27a161a200a55ya182a94a61a164a164a27a94a61a78a27a78a200a143a181
V =
integraldisplay b
a
2pixy(x)dx
133
(5) x = a(t?sint),y = a(1?cost)(0 lessorequalslanttlessorequalslant 2pi,y = 0)
(i) a55xa182
(ii) a55ya182
(iii) a55a134a130y = 2a
a41a181
(1) V = pi
integraldisplay a
a
y2 dx = pi
integraldisplay a
a
bracketleftbigg
b2
parenleftbigg
1? x
2
a2
parenrightbiggbracketrightbigg
dx = 43piab2
(2) (i) V = pi
integraldisplay pi
0
sin2xdx = pi
2
2
(ii) V = 2pi
integraldisplay pi
0
xsinxdx = 2pi2
(3) (i) V = 2pi
integraldisplay pi
2
0
y2 dx = 2pi
integraldisplay pi
2
0
b2 cos6t·3asin2tcostdt = 6a
integraldisplay pi
2
0
ab2 sin2tcos7tdt = 6piab2
integraldisplay pi
2
0
(cos7t?
cos9t)dt = 32105piab2
(ii) a124a94a233a161a53a167a144a73a242a254a170a137a89a165a,ba233a78a167a61a26
V = 32105pia2b
(4) a121a178a181a138[a,b]a27a63a191a169a123a181a = x0 <x1 <···<xn = b
a51[xi?1,xi]a165a63a18a152a58ξia167a233a65a27a188a234a138a143y(ξi)a182Ai ≈y(ξi)?xi,?Vi ≈ 2piξiy(ξi)?xia167a75V = lim
λ→0
nsummationdisplay
i=1
2piξiy(ξi)?xia167
a108a13V =
integraldisplay b
a
2pixy(x)dx
(5) (i) V = pi
integraldisplay 2pi
0
y2 dx = pi
integraldisplay 2pi
0
a3(1?cost)3 dt = 5pi2a3
(ii) V = 2pi
integraldisplay 2pi
0
a3(1?cost)2(t?sint)dt = 6pi3a3
(iii) a138a178a163y = y+ 2a,x = xa167a75a173a130a144a167a143x = a(t?sint),y =?a(1 + cost)a57y =?2a
a117a180V = pi
integraldisplay 2pi
0
[4a2?a2(1+cost)2]a(1?cost)dt = pia3
integraldisplay 2pi
0
(3?2cost?cos2t)(1?cost)dt = 7pi2a3
3,a121a178a114a161a2000 lessorequalslant α lessorequalslant θ lessorequalslant β lessorequalslant pi,0 lessorequalslant r lessorequalslant r(θ)(r(θ)a51[α,β]a254a235a89)a55a52a182a94a61a164a164a27a78a200a31a117a181V =
2pi
3
integraldisplay β
α
r3(θ)sinθdθa167a191a166a209r = a(1 + cosθ)a55a52a182a94a61a164a164a27a78a200.
(1) a121a178a181a94a135a3a123.
a207a177ra143a140a187a167a134a52a130a164θa14a27a247a47a55a52a182a94a61a152a177a164a26a27a78a200a143a181
V = pi3(rsinθ)2rcosθ+pi
integraldisplay r
rcosθ
(r2?x2)dx = 23pir3(1?cosθ)
a138[α,β]a27a63a191a169a123a181α = θ0 <θ1 <···<θn = β,?θi = θi?θi?1,λ = max
i
{?θi}
a51a122a135[θi?1,θi]a209a127a51θprimeia167a166cosθi?1?cosi =?sinθprimei(θi?1?θi) = sinθprimei?θi
a177r(θprimei)a138a2a247a47Aia27a140a187a167a75a247a47a55a52a182a94a61a152a177a0a164a26a27a78a200a143a181
Vi = 23pir3(θprimei)(1?cosθi)? 23pir3(θprimei)(1?cosθi?1) =
2
3pir
3(θprime
i)(cosθi?1?cosθi) =
2
3pir
3(θprime
i)sinθ
prime
i?θia167a75a18a135a173a62a247a47a55a52a182a94a61a26
nsummationdisplay
i=1
2
3pir
3(θprime
i)sinθ
prime
i?θi
a108a13V = lim
λ→0
nsummationdisplay
i=1
2
3pir
3(θprime
i)sinθ
prime
i?θi =
2pi
3
integraldisplay β
α
r3(θ)sinθdθ.
(2) a41a181V = 23pi
integraldisplay pi
0
a3(1 + cosθ)3 sinθdθ = 83pia3
4,a114a14a212a130y = x(x?a)a51a238a139a730a134c(c>a> 0)a131a109a27a108a227a55xa182a94a61a167a175ca143a219a138a158a167a84a94a61a78a27a78a200Va31a117
a177a117OPa55xa182a94a61a164a164a27a73a78a78a200a186(a2278-14)
a41a181a207a14a212a130y = x(x?a),xP = ca167a25P(c,c(c?a))
a75a177a117OPa55xa182a94a61a164a164a27a73a78a78a200a143a181V1 = 13pic[c(c?a)]2 = pi3c3(c?a)2
134
a164a166a27a94a61a78a78a200a143a181V2 = pi
integraldisplay c
0
[x(x?a)]2 dx = pi
parenleftbiggc5
5?
a
2c
4 + a2
3 c
3
parenrightbigg
a113V1 = V2a167a25pi3c3(c?a)2 = pi
parenleftbiggc5
5?
a
2c
4 + a2
3 c
3
parenrightbigg
a167
a108a13c = 54a
5,a114a173a130y =
√x
1 +x2a55xa182a94a61a26a152a94a61a78a167a167a51a58x = 0a134x = ξa131a109a27a78a200a80a138V(ξ)a167a166aa31a117a219a138a158a167a85
a166V(a) = 12 lim
ξ→∞
V(ξ).
a41a181a207V(ξ) = pi
integraldisplay ξ
0
parenleftbigg √x
1 +x2
parenrightbigg2
dx = ξ
2
2(1 +ξ2)pia167a75V(a) =
a2
2(1 +a2)pi
a113V(a) = 12 lim
ξ→∞
V(ξ) = 12 lim
ξ→∞
ξ2
2(1 +ξ2)pi =
pi
4a167a117a180a
2 = 1
a113a> 0a167a25a = 1.
6,a253a11b2x2 +a2y2 = a2b2a55xa182a94a61a26a152a94a61a253a165a78a167a114a167a247xa182a144a149a139a152a66a37a27a11a154a167a166a144a101a27a130a47a78a78a200a31
a117a253a165a78a78a200a27a152a140a167a251a189a125a152a27a140a187ρ(a2278-15).
a41a181a23a75a165a144a101a27a130a47a78a78a200a143Va167a253a165a78a78a200a143V1
a207b2x2 +a2y2 = a2b2a167a75y =
√a2b2?b2x2
a
a75V1 = pi
integraldisplay a
a
y2 dx = pi
integraldisplay a
a
a2b2?b2x2
a2 dx =
4
3piab
2
V = V1?2piρ2
radicalbiga2b2?a2ρ2
b?2pi
integraldisplay a
√
a2b2?a2ρ2
b
parenleftbigg√a2b2?b2x2
a
parenrightbigg2
dx =
4
3piab
radicalbig
b2?ρ2? 4pia3b ρ2
radicalbig
b2?ρ2 = 43piab(b2?ρ2)32
a100a75a191a167a26V = 12V1a6143piab(b2?ρ2)32 = 23piab2a167a41a100a144a167a167a26ρ = b
radicalBig
1?2?23
135
§4 a94a61a173a161a27a161a200
1,a166a101a15a94a61a173a161a27a161a200a181
(1) x2 = 2py+a(0 lessorequalslantxlessorequalslanta,a> 1)a55xa182a57ya182
(2) y = sinx(0 lessorequalslantxlessorequalslantpi)a55xa182
(3) a253a11x
2
a2 +
y2
b2 = 1a55ya182
(4) a94a211a130x = a(t?sint),y = a(1?cost)(0 lessorequalslanttlessorequalslant 2pi)a55xa182
(5) a86a221a130r2 = 2a2 cos2θ
(i) a55a52a130
(ii) a55a182θ = pi2
(iii) a55a182θ = pi4
a41a181
(1) y = x
2?a
2p,?
a
2p lessorequalslantylessorequalslant
a2?a
2p (p> 0)
(i) Fx = 2pi
integraldisplay a
0
x2?a
2p
radicaltpradicalvertex
radicalvertexradicalbt
1 +
bracketleftBiggparenleftbigg
x2?a
2p
parenrightbiggprimebracketrightBigg2
dx = pip2
integraldisplay a
0
(x2?a2)
radicalbig
x2 +p2 dx =
bracketleftBigg
a(2a2?4a+p2)
8p2
radicalbig
p2 +a2? p
2 + 4a
8 ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a+radicalbiga2 +p2
p
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
bracketrightBigg
pi
(ii) radicalbig1 + (xprime)2 =
radicalbigp2 + 2py+a
√2py+a
a75Fy = 2pi
integraldisplay a2?a
2p
a2p
radicalbig2py+aradicalbigp2 + 2py+a√
2py+a dy =
pi
p
integraldisplay a2?a
2p
a2p
radicalbig
2py2 +p2 +ad(2py+p2 +a) = 2pi3p(p2 +a2)32? 23p2pi
(2) F = 2pi
integraldisplay pi
0
sinx·
radicalbig
1 + [(sinx)prime]2 dx = 2pi
integraldisplay pi
0
sinx
radicalbig
1 + cos2xdx = 2√2pi+ 2piln(√2 + 1)
(3) F = 2pi
integraldisplay b
b
x
radicalbig
1 + (xprime)2 dy = 2pi
integraldisplay b
b
a
b
radicalbig
b2?y2·
radicaltpradicalvertex
radicalvertexradicalbt
1 +
parenleftBigg
a(?y)
bradicalbigb2?y2
parenrightBigg2
dy = 2piab
integraldisplay b
b
radicalbigg
b2 + a
2?b2
b2 y
2 dy =
4api
b
integraldisplay b
0
radicalbigg
b2 + c
2
b2y
2 dy = 2api
parenleftbigg
a+ b
2
c ln
a+c
b
parenrightbigg
.
(4) a207dS = radicalbig(xprime)2 + (yprime)2 dt = 2asin t2 dta167a75F = 2pi
integraldisplay 2pi
0
yds = 2pi
integraldisplay 2pi
0
a(1? cost) · 2asin t2 dt =
16a2pi
integraldisplay 2pi
0
sin3 t2 dt2 = 643 pia2.
(5) (i) y = √2a√cos2θsinθ,dS =
√2a
√cos2θ dθ
parenleftBig
pi4 lessorequalslantθlessorequalslant pi4
parenrightBig
a100a233a161a53a167a26F = 2×2pi
integraldisplay pi
4
0
2a2 sinθdθ = 4pia2(2?√2)
(ii) x = √2a√cos2θcosθ,dS =
√2a
√cos2θ dθ
parenleftBig
pi4 lessorequalslantθlessorequalslant pi4
parenrightBig
a75F = 2pi
integraldisplay pi
4
pi4
2a2 cosθdθ = 4√2pia2
(iii) x = √2a√cos2θcosθ,y = √2a√cos2θsinθ,
dS =
√2a
√cos2θ dθ
parenleftBig
pi4 lessorequalslantθlessorequalslant pi4
parenrightBig
a53a191a20a51?pi4 lessorequalslantθlessorequalslant pi4a83a167a240a107x?ygreaterorequalslant 0a167
a117a180a167a164a166a27a76a161a200a143F = 2×2pi
integraldisplay pi
4
pi4
x?y√
2 ·
√2a
√cos2θ dθ = 4√2pia2
integraldisplay pi
4
pi4
(cosθ?sinθ)dθ = 8pia2
136
2,a121a178a100x =?(t),y = ψ(t),z = χ(t)(t0 lessorequalslanttlessorequalslantT)a134Oxya178a161a109a164a129a27a206a161a161a200a31a117S =
integraldisplay T
t0
χ(t)
radicalbig
prime2(t) +ψprime2(t)dt
a121a178a181a23a173a130CDa51Oxya178a161a27a221a75a143ABa167a75ABa27a144a167a143
braceleftbigg x =?(t)
y = ψ(t) (t0 lessorequalslanttlessorequalslantT)
a51ABa254a18a169a58a181A = M0,M1,···,Mi?1,Mi,···,Mn = B
a51CDa233a65a27a169a58a181C = N0,N1,···,Ni?1,Ni,···,Nn = D
a233a65a27a235a234a181t0,t1,···,ti?1,ti,···,tn
a23Mia27a139a73a143χi =?(ti),yi = ψ(ti)a167a75MiNi = χ(ti)
a134a14a70a47MiMi?1Ni?1Nia27a161a200a181
Si = MiNi +Mi?1Ni?12 ·Mi?1Mi = χ(ti) +χ(ti?1)2
radicalbig
[?(ti)(ti?1)]2 + [ψ(ti)?ψ(ti?1)]2 =
χ(ti)
radicalbig
[?(ti)(ti?1)]2 + [ψ(ti)?ψ(ti?1)]2?
χ(ti)?χ(ti?1)
2
radicalbig
[?(ti)(ti?1)]2 + [ψ(ti)?ψ(ti?1)]2
a100a135a169a165a138a189a110a181?(ti)(ti?1) =?prime(ξi)?ti,ψ(ti)?ψ(ti?1) = ψprime(ηi)?ti,χ(ti)?χ(ti?1) = χprime(ζi)?ti
a147a92a167a26Si = χ(ti)radicalbig?prime2(ξi) +ψprime2i (ηi)?ti? 12χprime(ζi)
radicalBig
prime2(ξi) +ψprime2i (ηi)(?ti)2
a108a13a206a161a161a200a143a181
S = lim
λ→0
nsummationdisplay
i=1
χ(ti)
radicalBig
prime2(ξi) +ψprime2i (ηi)?ti? lim
λ→0
1
2
nsummationdisplay
i=1
χprime(ζi)
radicalBig
prime2(ξi) +ψprime2i (ηi)(?ti)2
=
integraldisplay T
t0
χ(t)
radicalbig
prime2(t) +ψprime2(t)dt?0 =
integraldisplay T
t0
χ(t)
radicalbig
prime2(t) +ψprime2(t)dt
137
§5 a159a37
1,a166a101a15a173a130a227a27a159a37a139a73a181
(1) a140a187a143aa167a108a127a14312aα(αlessorequalslantpi)a27a254a33a11a108a182
(2) a177A(0,0),B(0,1),C(2,1),D(2,0)a143a186a58a27a221a47a177a46a167a173a130a254a63a152a58a27a151a221a31a117a84a58a20a6a58a229a108a27a19
a21a182
(3) a233a234a218a130r = aekθ(a> 0,k> 0)a254a100a58(0,a)a20a58(θ,r)a27a254a33a108a227.
a41a181
(1) a177a6a58a143a11a37a167a108a140a178a27a229a169a62a164a51a134a130a143xa182a239a225a134a14a139a73a88a167a75a11a108a144a167a143x = acosα,y =
asinαa167a117a180xprime =?asinα,yprime = acosαa167
a108a13x =
integraldisplay α
2
0
acosα
radicalbig
xprime2 +yprime2 dα
s =
a2
integraldisplay α
2
0
cosαdα
s =
a2 sin α2
1
2aα
=
2asin α2
α
y =
integraldisplay α
2
0
asinα
radicalbig
xprime2 +yprime2 dα
s =
a2
integraldisplay α
2
0
sinαdα
s =
2a
α
parenleftBig
1?cos α2
parenrightBig
(2) a107a166a209a151a221a188a234.
ABa27a144a167a143
braceleftbigg x = 0
y = y y ∈ [0,1]a167a217a254a63a152a58a27a151a221a143
ρ = 2y
BCa27a144a167a143
braceleftbigg x = x
y = 1 x∈ [0,2]a167a217a254a63a152a58a27a151a221a143
ρ = 2√1 +x2
CDa27a144a167a143
braceleftbigg x = 2
y = y y ∈ [0,1]a167a217a254a63a152a58a27a151a221a143
ρ = 2radicalbig4 +y2
DAa27a144a167a143
braceleftbigg x = x
y = 0 x∈ [0,2]a167a217a254a63a152a58a27a151a221a143
ρ = 2x
a117a180mAB =
integraldisplay 1
0
2y
radicalbig
02 + 12 dy = 1,
xAB = 0,yAB =
integraldisplay 1
0
y·2y
radicalbig
02 + 12 dy
mAB =
2
3a182
mBC =
integraldisplay 2
0
2
radicalbig
1 +x2
radicalbig
12 + 02 dx = 2√5 + ln(2 +√5),
xBC =
integraldisplay 2
0
x·2
radicalbig
1 +x2
radicalbig
12 + 02 dx
mBC =
2(5√5?1)
6√5 + 3ln(2 +√5),
yBC = 1a182
mCD =
integraldisplay 1
0
2
radicalbig
4 +y2
radicalbig
02 + 12 dy = √5 + 4ln 1 +
√5
2,
xCD = 2,yCD =
integraldisplay 1
0
y·2
radicalbig
4 +y2
radicalbig
02 + 12 dy
mCD =
2(5√5?8)
3
parenleftbigg√
5 + 4ln 1 +
√5
2
parenrightbigga182
mDA =
integraldisplay 2
0
2x
radicalbig
12 + 02 dx = 4,
xDA =
integraldisplay 2
0
x·2x
radicalbig
12 + 02 dx
mDA =
4
3,yDA = 0
a25a100a221a47a177a46a27a159a37a139a73a143a181
x = mABxAB +mBCxBC +mCDxCD +mDAxDAm
AB +mBC +mCD +mDA
=
138
16√5 + 14 + 24ln 1 +
√5
2
9√5 + 15 + 3ln(2 +√5) + 12ln 1 +
√5
2
y = mAByAB +mBCyBC +mCDyCD +mDAyDAm
AB +mBC +mCD +mDA
=
16√5?14 + 3ln(2 +√5)
9√5 + 15 + 3ln(2 +√5) + 12ln 1 +
√5
2
(3) a173a37a27a134a14a139a73a143a181
x =
integraldisplay θ
0
x
radicalbig
xprime2 +yprime2 dx
s =
integraldisplay θ
0
rcosθ
radicalbig
[(rcosθ)prime]2 + [(rsinθ)prime]2 dθ
integraldisplay θ
0
radicalbig
[(rcosθ)prime]2 + [(rsinθ)prime]2 dθ
=
integraldisplay θ
0
rcosθ
radicalbig
a2(1 +k2)ekθ dθ
integraldisplay θ
0
radicalbig
a2(1 +k2)ekθ dθ
=
a
integraldisplay θ
0
e2kθ cosθdθ
integraldisplay θ
0
ekθ dθ
= ake
2kθ(sinθ+ 2kcosθ)?2ak2
(4k2 + 1)(ekθ?1)
a211a123a140a26y = ake
2kθ(2ksinθ?cosθ) +ak
(4k2 + 1)(ekθ?1)
a117a180a167a173a37a27a52a139a73a143a181
r =
radicalbig
x2 +y2 = ak(4k2 + 1)(ekθ?1)
radicalbig
(e4kθ + 1?2e2kθ cosθ)(4k2 + 1)
tanθ0 = yx = e
2kθ(sinθ+ 2kcosθ)?2k
e2kθ(2ksinθ?cosθ) + 1
a133a159a37a139a73a143(r,θ0)
2,a94a152a138a151a221a254a33a27a55a225a106a11a164a140a187a143aa27a140a11a108a167a51a252a224a94a211a24a27a55a225a106a26a254a252a94a131a130a163a2278-19a164a167a175a131a130
a127ba143a245a8a158a167a144a85a166a55a225a106MABNa27a159a37a20a208a51a11a37Oa186
a41a181
a45
a54
a39
a38
x
y
0
A
B
a
M
N
a23a55a225a106a27a151a221a143μa167a140a11a108a27a159a254a143a181m =
integraldisplay 3pi
2
pi
2
μds = piaμ
a140a11a108a181x = acosθ,y = asinθ
parenleftbiggpi
2 lessorequalslantθlessorequalslant
3
2pi
parenrightbigg
,ds =radicalbigxprime2 +yprime2 dθ = adθ
a75a140a11a108a27a159a37a139a73a143
x =
integraldisplay 3
2pi
pi
2
xμds
m =
μa2
integraldisplay 3
2pi
pi
2
cosθdθ
piaμ =?
2a
pi ;
y =
integraldisplay 3
2pi
pi
2
yμds
m =
μa2
integraldisplay 3
2pi
pi
2
sinθdθ
piaμ = 0
a113a252a94a131a130a27a159a37a139a73a143a181x = b2,y = 0a167a159a254a143a1812bμ
a117a180a100a174a127a167a26a159a58a88a159a37a139a73a143a181x =
2api ·piaμ+ b2 ·2bμ
piaμ+ 2bμ = 0a167a108a13b =
√2a
3,a182a12710a146a167a151a221a169a217a143ρ = ρ(x) = (6 + 0.3x)a90a142/a146a167a217a165xa143a229a182a27a152a135a224a58a27a229a108a167a166a182a27a159a254.
a41a181m =
integraldisplay 10
0
ρ(x)dx =
integraldisplay 10
0
(6 + 0.3x)dx = 75(a90a142)
4,a174a127a152a14a212a130a227y = x2(?1 lessorequalslant x lessorequalslant 1)a167a173a130a227a254a63a152a58a63a27a151a221a134a84a58a20ya182a27a229a108a164a20a39a167x = 1a63a151a221
a1435a167a166a100a173a130a227a27a159a254.
139
a41a181a100a174a127a167a26ρ(x) = k|x|
a207x = 1a158a167ρ(1) = 5a167a75k = 5a167a117a180ρ(x) = 5|x|
a113ds =radicalbig1 + (yprime)2 dx = √1 + 4x2 dxa167
a75m =
integraldisplay 1
1
ρ(x)ds = 2
integraldisplay 1
0
5x
radicalbig
1 + 4x2 dx = 256 √5? 56
140
§6 a178a254a138a33a245
1,a174a127a18a54a62a180a165a62a123Ra252a224a27a62a216a129a140a138a143Uma167a11a170a199a143ωa167a79a142a158a209a51Ra254a27a178a254a245a199(a169a140a197a18a54a218a28a197
a18a54a252a171a156a185a63a216).
a41a181a140a197a18a54a158a167a158a209a51Ra254a27a178a254a245a199a143a181P1 = 2T
integraldisplay T
2
0
P(t)dt = 2ω2pi
integraldisplay pi
ω
0
U2m
R cos
2ωtdt = U2m
2R
a28a197a18a54a158a167a158a209a51Ra254a27a178a254a245a199a143a181P2 = 1T
integraldisplay T
0
P(t)dt = ω2pi
integraldisplay 2pi
ω
0
U2m
R cos
2ωtdt = U2m
2R
2,a79a142a2a54a62a216u = Umcosωta51
bracketleftBig
0,piω
bracketrightBig
a218
bracketleftBig
pi2ω,pi2ω
bracketrightBig
a83a27a178a254a138.
a41a181a51
bracketleftBig
0,piω
bracketrightBig
a83a27a178a254a138a143a181u = ωpi
integraldisplay pi
ω
0
Umcosωtdt = 0a182
a51
bracketleftBig
pi2ω,pi2ω
bracketrightBig
a83a27a178a254a138a143a181u = ωpi
integraldisplay pi
2ω
pi2ω
Umcosωtdt = 2piUm.
3,a166a101a15a188a234a51a137a189a171a109a83a27a178a254a138a181
(1) y = sinx,[0,pi]
(2) y = xex,[0,1]
a41a181
(1) y = 1pi
integraldisplay pi
0
sinxdx = 2pi
(2) y =
integraldisplay 1
0
xexdx = 1
4,a114a6a148a46a127a164a73a27a229a134a6a148a27a27a127a164a20a39.a174a127a152a250a54a27a229a85a166a6a148a27a1271a102a146a167a175a114a6a148a46a12710a102a146a135a138a245
a8a245a186
a41a181a100a7a142a189a110a127a167a6a53a161a69a229Fa134a27a127a254xa164a20a39a61F = kx.
a100a94a135a167a127k = 1a167a207a13F = xa167a117a180a164a166a27a245a143W =
integraldisplay 10
0
F dx =
integraldisplay 10
0
xdx=50(a90a142·a102a146a164=5J
5,a63a239a140a120a120a234a158a135a107a101a140a51.a23a152a11a206a47a140a51a27a134a187a14320a146a167a89a2927a146a167a140a51a112a209a89a1613a146a167a135a114a89a196a166a167a79
a142a142a209a173a229a164a138a27a245.
a41a181a207?W = pir2 ·?x·x·103g = 105gpix?x
a75W = 105g
integraldisplay 30
3
pixdx = 4.37×108pi(J)
6,a44a89a165a27a185a128a180a152a70a47a167a254a466a146a167a101a462a146a167a11210a146a167a166a89a47a247a158a185a128a164a201a27a229.a23a89a27a39a173a1431a235/a1462.
a41a181a207?F = 2xyg?x = 2gx
parenleftBig
3? x5
parenrightBig
x
a75F = 2g
integraldisplay 10
0
x
parenleftBig
3? x5
parenrightBig
dx = 1.63×106(N)
7,a212a78a85a53a198x = ct3(c> 0)a138a134a130a36a196a167xa76a171a51a158a109ta83a212a78a163a196a27a229a108a167a23a48a159a27a123a229a134a132a221a178a144a164a20a39a167
a166a212a78a108x = 0a20x = aa158a123a229a164a138a27a245.
a41a181a207x = ct3(c> 0)a167a25v = xprime = 3ct2
a113a48a159a123a229a134a132a221a27a178a144a164a20a39a167a75a23f = kv2(ka143a126a234)a167a117a180f = 9kc2t4
a113a8xa108x = 0a20x = aa158a167ta108t = 0a20t =
parenleftBiga
c
parenrightBig1
3a167
a75W =
integraldisplay (a
c)
1
3
0
9kc2t4 ·3ct2 dt = 27kc3
integraldisplay (a
c)
1
3
0
t6 dt = 277 kc23a73
8,a140a187a143ra27a165a156a92a89a165a167a167a134a89a161a131a26a167a165a27a39a173a1431a167a121a242a165a108a89a165a18a209a167a135a138a245a8a245a186
a41a181a207?W = 1·pi(radicalbigr2?(x?r)2)2?x(2r?x) = pi(4r2x?4rx2 +x3)?x
a75W = pi
integraldisplay 2r
0
(4r2x?4rx2 +x3)dx = 43pir4
141
§7 a189a200a169a27a67a113a79a142
1,a94a14a212a130a47a250a170a166
integraldisplay 1
0
dx
1 +x2a27a67a113a138(a18n = 3).
a41a181a18n = 3a167a79a142a204a160a2a234a167a140a26a181
x0 = 0,y0 = 1.0000;x1 = 16,4y1 = 3.8919;x2 = 13,2y2 = 1.8000;x3 = 12,4y3 = 3.2000;x4 = 23,2y4 =
1.3846;x5 = 56,4y5 = 2.3607;x6 = 1,y6 = 0.5000
a124a94a14a212a130a47a250a170a167a107integraldisplay
1
0
dx
1 +x2 ≈
1
18[y0 + 2(y2 +y4) + 4(y1 +y3 +y5) +y6] = 0.7854
2,a166a44a202a46a27a161a200.a202a46a88a2278-24a164a171a167xa182a180a167a27a233a161a182a167OAa1272a146a16710a31a169a167a255a26a234a226a88a101a163a252a160a181a102
a146a164a181
x 0 20 40 60 80 100 120 140 160 180 200
y 0 8.5 11.0 11.5 10.5 10.0 8.0 6.5 4.5 2.5 0
a41a181a124a94a14a212a130a47a250a170a167a107
A≈ 2006×5[0 + 0 + 4(8.5 + 11.5 + 10.0 + 6.5 + 2.5) + 2(11.0 + 10.5 + 8.0 + 4.5)] = 203 ×224 ≈ 1493.3(cm2)
3,a51a176a14320a146a27a224a161a254a167a255a254a224a54a238a31a161a27a161a200.a88a74a108a224a27a152a87a149a233a87a122a1332a146a167a255a26a224a89a29a221a88a101a76a164a15a181
x 0 2 4 6 8 10 12 14 16 18 20
y(a89a29) 0.4 0.8 1.4 2.0 2.4 2.1 1.9 1.6 1.3 0.8 0.4
(a89a29a252a160a181a146)a166a100a224a54a238a31a161a27a161a200a163a2278-25a164
a41a181a124a94a14a212a130a47a250a170a167a107
A≈ 206×5[0.4 + 0.4 + 4(0.8 + 2.0 + 2.1 + 1.6 + 0.8) + 2(1.4 + 2.4 + 1.9 + 1.3)] = 23 ×44 ≈ 29.3(m2)
142
a49a110a159 a63a234a216
a49a152a220a169 a234a145a63a234a218a50a194a200a169
a49a202a217 a234a145a63a234
§1,a253a23a127a163a181a254a52a129a218a101a52a129
1,a121a178a181
(1) lim
n→∞
(xn +yn) lessorequalslant lim
n→∞
xn + lim
n→∞
yn
(2) lim
n→∞
(xn +yn) greaterorequalslant lim
n→∞
xn + lim
n→∞
yn
a121a178a181
(1) a23 lim
n→∞
xn,lim
n→∞
yna254a143a107a129a234a167a75{xn},{yn}a254a107a46.
a45αk = sup
n>k
{xn},βk = sup
n>k
{yn}a34a117a180a167a8n>ka158a167a107xn +yn lessorequalslantαk +βk
a108a13sup
n>k
{xn +yn}lessorequalslantαk +βk
a25 lim
n→∞
(xn +yn) = lim
k→∞
sup
n>k
{xn +yn}lessorequalslant lim
k→∞
αk + lim
k→∞
βk = lim
n→∞
xn + lim
n→∞
yn
a53a181a101 lim
n→∞
xn,lim
n→∞
yna131a152a143+∞.a126a88a181 lim
n→∞
xn = +∞a167a143a166a39a88a170a109a62a92a123a36a142a107a191a194a167a75 lim
n→∞
yna216
a26a143?∞.a249a158{xn}a195a254a46a167{xn+yn}a189a195a254a46a167a254a227a39a88a170a119a44a164a225a182a101 lim
n→∞
xn =?∞a167a143a166a39a88a170
a109a62a92a123a36a142a107a191a194a167a75 lim
n→∞
yna216a26a143+∞.a117a180{yn}a254a107a46a167a108a13 lim
n→∞
(xn +yn) =?∞a167a254a227a39a88a170
a119a44a164a225.
(2) a207xn greaterorequalslant inf{xn},yn greaterorequalslant inf{yn}a167a25xn +yn greaterorequalslant inf{xn}+ inf{yn}
a226a101a40a46a143a101a46a165a129a140a27a167a75inf{xn +yn}greaterorequalslant inf{xn}+ inf{yn}a167
a108a13inf
n>k
{xn +yn}greaterorequalslant inf
n>k
{xn}+ inf
n>k
{yn}
a75 lim
k→∞
inf
n>k
{xn +yn}greaterorequalslant lim
k→∞
parenleftbigg
inf
n>k
{xn}+ inf
n>k
{yn}
parenrightbigg
= lim
k→∞
inf
n>k
{xn}+ lim
k→∞
inf
n>k
{yn}
a61 lim
n→∞
(xn +yn) greaterorequalslant lim
n→∞
xn + lim
n→∞
yn.
2,a23xn greaterorequalslant 0,yn greaterorequalslant 0a167a121a178a181
(1) lim
n→∞
xnyn lessorequalslant lim
n→∞
xn · lim
n→∞
yn
(2) lim
n→∞
xnyn greaterorequalslant lim
n→∞
xn · lim
n→∞
yn
a121a178a181
(1) a2070 lessorequalslantxn lessorequalslant sup{xn},0 lessorequalslantyn lessorequalslant sup{yn}a167a750 lessorequalslantxnyn lessorequalslant sup{xn}·sup{yn}
a226a254a40a46a180a254a46a165a129a2a27a167a75a1070 lessorequalslant sup{xn ·yn}lessorequalslant sup{xn}·sup{yn}
a108a130 lessorequalslant sup
n>k
{xn ·yn}lessorequalslant sup
n>k
{xn}· sup
n>k
{yn}
a75 lim
k→∞
sup
n>k
{xn ·yn}lessorequalslant lim
k→∞
parenleftbigg
sup
n>k
{xn}· sup
n>k
{yn}
parenrightbigg
= lim
k→∞
sup
n>k
{xn}· lim
k→∞
sup
n>k
{yn}
a61 lim
n→∞
xnyn lessorequalslant lim
n→∞
xn · lim
n→∞
yn.
(2) a207xn greaterorequalslant inf{xn}greaterorequalslant 0,yn greaterorequalslant inf{yn}greaterorequalslant 0a167a75xnyn greaterorequalslant inf{xn}·inf{yn}greaterorequalslant 0
a226a101a40a46a180a101a46a165a129a140a27a167a75a107inf{xn ·yn}greaterorequalslant inf{xn}·inf{yn}greaterorequalslant 0
a108a13inf
n>k
{xn ·yn}greaterorequalslant inf
n>k
{xn}· inf
n>k
{yn}greaterorequalslant 0
a75 lim
k→∞
inf
n>k
{xn ·yn}greaterorequalslant lim
k→∞
parenleftbigg
inf
n>k
{xn}· inf
n>k
{yn}
parenrightbigg
= lim
k→∞
inf
n>k
{xn}· lim
k→∞
inf
n>k
{yn}
a61 lim
n→∞
xnyn greaterorequalslant lim
n→∞
xn · lim
n→∞
yn
3,a101 lim
n→∞
xna127a51a167a75a233a63a219a234a15{yn}a164a225a181
(1) lim
n→∞
(xn +yn) = lim
n→∞
xn + lim
n→∞
yn
143
(2) lim
n→∞
(xn ·yn) = lim
n→∞
xn · lim
n→∞
yna167a101 lim
n→∞
xn > 0
a121a178a181a23 lim
n→∞
xn = α
a101 lim
n→∞
yn = +∞(a189?∞)a167a75(1)a119a44a164a225,a207α> 0a167a75(2)a119a44a164a225.
a25a216a148a23 lim
n→∞
yn = βa143a107a129a234
a207 lim
n→∞
yn = βa167a25a127a51{yn}a27a102a15{ynk}a167a166 lim
k→∞
ynk = βa133βa143a164a107a194a241a102a15a27a52a129a165a27a129a140a246.
a113 lim
n→∞
xn = αa167a75 lim
k→∞
xnk = αa167a25 lim
k→∞
(xnk +ynk) = α+β,lim
k→∞
(xnk ·ynk) = αβ
a101a121α+βa143{xn +yn}a131a152a131a194a241a102a15a27a52a129a165a27a129a140a246(a94a135a121a123)
a98a23{xn +yn}a27a152a135a194a241a102a15{xnkprime +ynkprime}a167a166 lim
kprime→∞
parenleftbigx
nkprime +ynkprime
parenrightbig= γ >α+β
a75 lim
kprime→∞
ynkprime = lim
kprime→∞
parenleftbigx
nkprime +ynkprime
parenrightbig? lim
kprime→∞
xnkprime = γ?α>β
a249a134βa143{yn}a27a164a107a194a241a102a15a27a52a129a165a27a129a140a138a103a241.
a117a180α+βa210a180{xn +yn}a164a107a194a241a102a15a52a129a27a129a140a138.
a211a110a140a121a167a8α> 0a158a167α+βa143{xn +yn}a27a152a131a194a241a102a15a27a52a129a165a27a129a140a138.
a108a13 lim
n→∞
(xn +yn) = α+β = lim
n→∞
xn + lim
n→∞
yn
lim
n→∞
(xn ·yn) = αβ = lim
n→∞
xn · lim
n→∞
yna167a101 lim
n→∞
xn > 0
4,a166a101a15a234a15a27a254a52a129a134a101a52a129a181
(1) an = 12?n + (?1)n (n = 1,2,···)
(2) an = (?1)n
parenleftBigg
1 + 1n
parenrightBigg
(n = 1,2,···)
(3) an = (?1)
n
n (n = 1,2···)
(4) an = sin npi5 (n = 1,2,···)
a41a181
(1) a167a144a107a252a135a228a52a129a27a102a234a15a181a2k → 1,a2k+1 →?1 (k →∞) (k = 1,2,3···)
a117a180 lim
n→∞
an = 1,lim
n→∞
an =?1.
(2) a167a144a107a252a135a228a52a129a27a102a234a15a181a2k → 1,a2k+1 →?1 (k →∞) (k = 1,2,3···)
a117a180 lim
n→∞
an = 1,lim
n→∞
an =?1.
(3) a207 lim
n→∞
an = 0a167a25 lim
n→∞
an = 0,lim
n→∞
an = 0.
(4)?sin 25pilessorequalslant sin npi5 lessorequalslant sin 25pi
a8n = 10k+ 2 (k = 1,2,···)a158a167a10k+2 → sin 25pi (k →∞)
a8n = 10k?2 (k = 1,2,···)a158a167a10k?2 →?sin 25pi (k →∞)
a117a180 lim
n→∞
an = sin 25pi,lim
n→∞
an =?sin 25pi.
5,a101 lim
n→∞
nradicalbig|an| = αa167a75 lim
n→∞
nradicalbig|ak
0+n| = α
a100a63k0a180a63a191a27a189a27a18a234.
a121a178
(1) a207|ak0+n|1n = |ak0+n| 1k0+n
parenleftBig
|ak0+n| 1k0+n
parenrightBigk0n
a113 lim
n→∞
k0+nradicalbig|ak
0+n| = αa167a133a8α> 0a158a167 limn→∞
k0
n ln|ak0+n|
1
k0+n = 0a167a25 lim
n→∞
parenleftBig
|ak0+n| 1k0+n
parenrightBigk0n
= 1
a100a492a75(1)a167a26 lim
n→∞
nradicalbig|ak
0+n|lessorequalslantα
(2) a207 lim
n→∞
nradicalbig|an| = αa167a25a127a51a102a15{an
k}a167a166a26 limk→∞|ank|
1
nk = αa167
a133a8α> 0a158a167a107 lim
k→∞
|ank| 1nk?k0 = lim
k→∞
|ank| 1nk · lim
k→∞
parenleftBig
|ank| 1nk
parenrightBig k0n
k?k0 = α
144
a108a13 lim
n→∞
nradicalbig|ak
0+n|greaterorequalslantα
a110a220(1)(2)a167a26a8α> 0a158a167a40a216a164a225.
(3) a101α = 0a167a75a119a44a107 lim
n→∞
nradicalbig|an| = 0a167a108a13 lim
n→∞
nradicalbig|ak
0+n| = limn→∞
parenleftBig
|ak0+n| 1k0+n
parenrightBigk0+nn
= 0
a117a180a26a100a40a216a20a40.
6,a101 lim
n→∞
an = a<ba167a121a178a181a55a127a51Na167a8n>Na158a167a107an <b,a113a101 lim
n→∞
an = a<b.a156a185a88a219a186
a121a178a181
(1) a18ε = b?a2 a167a100§1a189a1101a167a26{an}a165a150a245a144a107a107a129a145a225a117(a+ε,+∞) =
parenleftBigg
a+b
2,+∞
parenrightBigg
a45a249a107a129a145a27a118a73a129a140a246a143Na167a75a8n>Na158a167a107an <a+ε = a+b2 < b+b2 = b
(2) a101 lim
n→∞
an = a<ba167a40a216a153a55a164a225.
a126a181an = 1 + (?1)n,n = 1,2,···a167a249a135a234a15a1430,2,0,2,···a167a119a44 lim
n→∞
an = 2,lim
n→∞
an = 0a167
a13 lim
n→∞
an = 0 < 1a167a2a107a195a161a245a145a2n = 2 > 1 (n = 1,2,···)
145
§2,a63a234a27a194a241a53a57a217a196a29a53a159
1,a63a216a101a15a63a234a27a241a209a53a181
(1) 11·6 + 16·11 +···+ 1(5n?4)(5n+ 1) +···
(2) 1 + 23 + 35 +···+ n2n?1 +···
(3)
parenleftBigg
1
2 +
1
3
parenrightBigg
+
parenleftBigg
1
22 +
1
32
parenrightBigg
+···+
parenleftBigg
1
2n +
1
3n
parenrightBigg
+···
(4) 11·4 + 14·7 +···+ 1(3n?2)(3n+ 1) +···
(5) cos pi3 + cos pi4 + cos pi5 +···
a41a181
(1) a207Sn = 11·6+ 16·11+···+ 1(5n?4)(5n+ 1) = 15
bracketleftBigg
1? 16 + 16? 111 +···+ 15n?4? 15n+ 1
bracketrightBigg
= 15
parenleftBigg
1? 15n+ 1
parenrightBigg
a75 lim
n→∞
Sn = lim
n→∞
1
5
parenleftBigg
1? 15n+ 1
parenrightBigg
= 15
a117a180a226a189a194a127a167a63a234 11·6 + 16·11 +···+ 1(5n?4)(5n+ 1) +···a194a241.
(2) a207 lim
n→∞
n
2n?1 =
1
2 negationslash= 0a167a25a63a234a117a209.
(3) a100a117
∞summationdisplay
n=1
1
2na134
∞summationdisplay
n=1
1
3na254a143a194a241a27a65a219a63a234a167
a25a100a234a15a63a234a53a1592a167a127
∞summationdisplay
n=1
parenleftBigg
1
2n +
1
3n
parenrightBigg
=
∞summationdisplay
n=1
1
2n +
∞summationdisplay
n=1
1
2n = 1 +
1
2 =
3
2
(4) a207Sn = 11·4+ 14·7+···+ 1(3n?2)(3n+ 1) = 13
bracketleftBigg
1? 14 + 14? 17 +···+ 13n?2? 13n+ 1
bracketrightBigg
= 13
parenleftBigg
1? 13n+ 1
parenrightBigg
a75 lim
n→∞
Sn = lim
n→∞
1
3
parenleftBigg
1? 13n+ 1
parenrightBigg
= 13
a117a180a226a189a194a127a167a63a234 11·4 + 14·7 +···+ 1(3n?2)(3n+ 1) +···a194a241.
(5) a207 lim
n→∞
cos pin+ 2 = 1 negationslash= 0a167a25a63a234a117a209.
2,a124a94a133a220a194a241a6a110a7a79a101a15a63a234a180a194a241a132a180a117a209.
(1) a0 +a1q+a2q2 +···+anqn +···,|q|< 1,|an|lessorequalslantA,(n = 0,1,2,···)
(2) 1 + 12? 13 + 14 + 15? 16 +···
a121a178a181
(1) a207a233a63a219a103a44a234pa167
|Sn+p?Sn| =vextendsinglevextendsingleanqn +an+1qn+1 +···+an+p?1qn+p?1vextendsinglevextendsinglelessorequalslantvextendsinglevextendsingleanvextendsinglevextendsingle|qn|+vextendsinglevextendsinglean+1vextendsinglevextendsinglevextendsinglevextendsingleqn+1vextendsinglevextendsingle+···+vextendsinglevextendsinglean+p?1vextendsinglevextendsinglevextendsinglevextendsingleqn+p?1vextendsinglevextendsinglelessorequalslant
A|q|n1?|q|
p
1?|q|
a113|q|< 1a167a750 < 1?|q|p < 1a167a117a180|Sn+p?Sn|<A· |q|
n
1?|q|
a108a13a233?ε> 0a167a18N =
bracketleftBigg
ln (1?|q|)εA /ln|q|
bracketrightBigg
a167a8n>Na158a167a233a63a219p = 1,2,3,···a167
a111a164a225|Sn+p?Sn|<ε
a85a194a241a6a110a167a63a234a0 +a1q+a2q2 +···+anqn +···a194a241.
146
(2) a100a63a234a143
∞summationdisplay
n=0
parenleftBigg
1
3n+ 1 +
1
3n+ 2?
1
3n+ 3
parenrightBigg
a180 <ε0 < 16a167a216a216na245a140a167a101a45p = na167a75a107
|Sn+p?Sn| = |S2n?Sn| = 13n+ 1+ 13n+ 2? 13n+ 3+···+ 16n?2+ 16n?1? 16n> 13n+ 3+ 13n+ 3?
1
3n+ 3 +···+
1
6n+
1
6n?
1
6n =
1
3
parenleftBigg
1
n+ 1 +
1
n+ 2 +···+
1
2n
parenrightBigg
> 13
1
2n+
1
2n+···+
1
2nbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
na145
=
1
6 >ε0
a207a100a63a234
∞summationdisplay
n=0
parenleftBigg
1
3n+ 1 +
1
3n+ 2?
1
3n+ 3
parenrightBigg
a117a209.
3,a23a107a20a145a63a234
∞summationdisplay
n=1
an(a61a122a152a145an > 0)a167a193a121a178a101a233a217a145a92a41a210a0a164a124a164a27a63a234a194a241a167a75
∞summationdisplay
n=1
ana189a194a241.
a121a178a181a23
∞summationdisplay
n=1
ana220a169a218a234a15a143{Sn}a167a92a41a210a0a164a124a164a27a63a234a143
∞summationdisplay
n=1
An
a217a165An = ain?1+1 +ain?1+2 +···+aina167a119a44
∞summationdisplay
n=1
Ana69a143a20a145a63a234.
a23a217a220a169a218a234a15a143{Snprime}a167a217a165Snprime = (a1 +a2 +···+ai1)+(ai1+1 +···+ai2)+···+(ain?1+1 +···+ain)
a119a44Snprime greaterorequalslantSn
a113
∞summationdisplay
n=1
Ana194a241a167a100a196a29a189a110a167a26{Snprime}a107a254a46a167a61a127a51M > 0a167a166Snprime lessorequalslant Ma167a108a13Sn lessorequalslant Snprime lessorequalslant Ma167a96
a178{Sn}a107a254a46
a75a100a196a29a189a110a167a26
∞summationdisplay
n=1
ana194a241.
4,a40a189a166a101a15a63a234a194a241a27xa27a137a140.
(1)
∞summationdisplay
n=0
1
(1 +x)n
(2)
∞summationdisplay
n=1
(lnx)n
a41a181
(1) a100a63a234a180a250a39a143 11 +xa27a31a39a63a234a167a25a8
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
1 +x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle< 1a158a63a234a194a241
a108a13a194a241a141a143x<?2a189x> 0.
(2) a100a63a234a180a250a39a143lnxa27a31a39a63a234a167a25a8|lnx|< 1a158a63a234a194a241
a108a13a194a241a141a1431e<x<e.
147
§3,a20a145a63a234
1,a7a228a101a15a63a234a27a194a241a218a117a209.
(1)
∞summationdisplay
n=1
1√
n2 +n
(2)
∞summationdisplay
n=1
1
(2n?1)·22n?1
(3)
∞summationdisplay
n=1
n?√n
2n?1
(4)
∞summationdisplay
n=1
sin pi2n
(5)
∞summationdisplay
n=1
1
1 +an,(a> 1)
(6)
∞summationdisplay
n=1
1
n· n√n
(7)
∞summationdisplay
n=1
parenleftBigg
1
2n+ 1
parenrightBiggn
(8)
∞summationdisplay
n=1
1
[ln(n+ 1)]n
(9)
∞summationdisplay
n=1
2 + (?1)n
2n
(10)
∞summationdisplay
n=1
2nsin pi3n
(11)
∞summationdisplay
n=1
nn
n!
(12)
∞summationdisplay
n=1
xn
(1 +x)(1 +x2)···(1 +xn),(xgreaterorequalslant 0)
(13)
∞summationdisplay
n=1
parenleftBigg
b
an
parenrightBiggn
a167a217a165an →a,an,b,aa27a20a234a167anegationslash= 0
a41a181
(1) a207 lim
n→∞
1√
n2 +n
1
n
= lim
n→∞
n√
n2 +n = 1a167a13a63a234
∞summationdisplay
n=1
1
na180a117a209a27
a75a100a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
1√
n2 +na189a117a209.
(2) a207 lim
n→∞
un+1
un = limn→∞
1
(2n+ 1)22n+1
1
(2n?1)22n?1
= lim
n→∞
2n?1
4(2n+ 1) =
1
4 < 1
a75a100a136a75a19a16a7a79a123a167a26a63a234
∞summationdisplay
n=1
1
(2n?1)·22n?1a194a241.
(3) a207 lim
n→∞
n?√n
2n?1 =
1
2notarrowright0a167a25a63a234
∞summationdisplay
n=1
n?√n
2n?1a117a209.
(4) a207sin pi2n lessorequalslant pi2na167a13
∞summationdisplay
n=1
pi
2na194a241a167a25a63a234
∞summationdisplay
n=1
sin pi2na194a241.
148
(5) a207 11 +an lessorequalslant
parenleftBigg
1
a
parenrightBiggn
a167a13
∞summationdisplay
n=1
parenleftBigg
1
a
parenrightBiggn
a194a241a167a25a63a234
∞summationdisplay
n=1
1
1 +ana194a241.
(6) a207 lim
x→+0
xx = lim
x→+0
elnxx = lim
x→+0
exlnx = e limx→+0xlnx = 1a167a25 lim
n→∞
1
n√n = limn→∞
parenleftBigg
1
n
parenrightBigg1
n
= 1
a113 lim
n→∞
1
n· n√n
1
n
= lim
n→∞
1
n√n = 1a167a13a63a234
∞summationdisplay
n=1
1
na180a117a209a27
a75a100a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
1
n· n√na117a209.
(7) a207 lim
n→∞
n
radicaltpradicalvertex
radicalvertexradicalbtparenleftBigg 1
2n+ 1
parenrightBiggn
= lim
n→∞
1
2n+ 1 = 0 < 1a167a25a63a234
∞summationdisplay
n=1
parenleftBigg
1
2n+ 1
parenrightBiggn
a194a241.
(8) a207 lim
n→∞
n
radicalBigg
1
[ln(n+ 1)]n = limn→∞
1
ln(n+ 1) = 0 < 1a167a25a63a234
∞summationdisplay
n=1
1
[ln(n+ 1)]na194a241.
(9) a207 2 + (?1)
n
2n lessorequalslant
3
2n a133a63a234
∞summationdisplay
n=1
3
2n a194a241
a75a226a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
2 + (?1)n
2n a194a241.
(10) a207 0 < 2nsin pi3n lessorequalslantpi
parenleftBigg
2
3
parenrightBiggn
a133a63a234
∞summationdisplay
n=1
parenleftBigg
2
3
parenrightBiggn
a194a241
a75a226a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
2nsin pi3n a194a241.
(11) a207 lim
n→∞
un+1
un = limn→∞
(n+ 1)n+1
(n+ 1)!
nn
n!
= lim
n→∞
parenleftBigg
1 + 1n
parenrightBiggn
= e> 1a167a25a63a234
∞summationdisplay
n=1
nn
n!a117a209.
(12) a207 lim
n→∞
un+1
un = limn→∞
xn+1/[(1 +x)(1 +x2)···(1 +xn)(1 +xn+1)]
xn/[(1 +x)(1 +x2)···(1 +xn)] = limn→∞
x
1 +xn+1 =
0 < 1,x> 1a189x = 0
1
2 < 1,x = 1x< 1,0 <x< 1
a75a226a136a75a19a16a7a79a123a167a26a63a234
∞summationdisplay
n=1
xn
(1 +x)(1 +x2)···(1 +xn)a194a241.
(13) a207 lim
n→∞
n
radicaltpradicalvertex
radicalvertexradicalbtparenleftBigg b
an
parenrightBiggn
= lim
n→∞
b
an =
b
a
a75a8ba< 1a61b<aa158a167a63a234
∞summationdisplay
n=1
parenleftBigg
b
an
parenrightBiggn
a194a241a182
a8ba> 1a61b>aa158a167a63a234
∞summationdisplay
n=1
parenleftBigg
b
an
parenrightBiggn
a117a209a182
a8ba = 1a61b = aa158a167a73a63a152a218a7a228a34a126a88a181a63a234
∞summationdisplay
n=1
parenleftBigg
1
n√n
parenrightBiggn
=
∞summationdisplay
n=1
1
na117a209a182a13a63a234
∞summationdisplay
n=1
parenleftBigg
1
n√n2
parenrightBiggn
=
∞summationdisplay
n=1
1
n2a194a241.
2,a101a20a145a63a234
∞summationdisplay
n=1
una194a241a167a121a178
∞summationdisplay
n=1
u2na143a194a241a167a217a95a88a219a186
a121a178a181a207
∞summationdisplay
n=1
una194a241a167a75 lim
n→∞
un = 0
a18ε0 = 1a167a75a127a51a20a18a234Na167a8n>Na158a167a107|un|<ε0 = 1a610 lessorequalslantun < 1a167a117a1800 lessorequalslantu2n <un(n>N)a167
149
a108a13a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
u2na194a241.
a217a95a216a253.a126a181
∞summationdisplay
n=1
1
n2a194a241a167a2
∞summationdisplay
n=1
1
na117a209a182
∞summationdisplay
n=1
1
n4a194a241a167
∞summationdisplay
n=1
1
n2a194a241.
3,a23
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna143a252a20a145a63a234a167 lim
n→∞
un
vn = 0a167a121a178a181a8
∞summationdisplay
n=1
vna194a241a158a167
∞summationdisplay
n=1
una143a194a241.a113a101
∞summationdisplay
n=1
vna117a209a158a167
∞summationdisplay
n=1
una88
a219a186a101 lim
n→∞
un
vn = ∞a167a64a111
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna27a241a209a53a131a109a107a159a111a39a88a186
a121a178a181
(1) a207 lim
n→∞
un
vn = 0a167
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna143a252a20a145a63a234
a18ε0 = 1a167a75a127a51a20a18a234Na167a8n>Na158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
un
vn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε0 = 1a610 lessorequalslant
un
vn < 1a167a117a180un <vn(n>N)
a113
∞summationdisplay
n=1
vna194a241a167a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
una194a241
a101
∞summationdisplay
n=1
vna117a209a167a75
∞summationdisplay
n=1
una140a85a194a241a167a143a140a85a117a209
a126a181
∞summationdisplay
n=1
1
na117a209a167 limn→∞
1
n2
1
n
= 0a167a2
∞summationdisplay
n=1
1
n2a194a241a182
∞summationdisplay
n=1
1√
na117a209a167 limn→∞
1
n
1√
n
= 0a167a133
∞summationdisplay
n=1
1
na117a209.
(2) a207 lim
n→∞
un
vn = ∞a167
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna143a252a20a145a63a234
a18G0 = 1a167a75a127a51a20a18a234Na167a8n>Na158a167a107unv
n
>G0 = 1a167a117a180un >vn(n>N)
a101
∞summationdisplay
n=1
una194a241a167a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
vna194a241a182a101
∞summationdisplay
n=1
vna117a209a167a75
∞summationdisplay
n=1
una117a209a182a101
∞summationdisplay
n=1
una117a209a167a75
∞summationdisplay
n=1
vna241
a209a53a216a189.
4,a101a252a20a145a63a234
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna117a209a167
∞summationdisplay
n=1
max(un,vn)a167
∞summationdisplay
n=1
min(un,vn)a252a63a234a88a219a186
a41a181a207a252a20a145a63a234
∞summationdisplay
n=1
una218
∞summationdisplay
n=1
vna117a209a167un lessorequalslant max(un,vn)
a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
max(un,vn)a117a209.
a233a117
∞summationdisplay
n=1
min(un,vn)a241a209a53a216a189.
a126a181
∞summationdisplay
n=1
1
na167
∞summationdisplay
n=1
1
2na209a117a209a167
∞summationdisplay
n=1
min
parenleftBigg
1
n,
1
2n
parenrightBigg
=
∞summationdisplay
n=1
1
2na143a117a209a182
∞summationdisplay
n=1
1 + (?1)n
2 a167
∞summationdisplay
n=1
1?(?1)n
2 a209a117a209a167a2
∞summationdisplay
n=1
min
parenleftBigg
1 + (?1)n
2,
1?(?1)n
2
parenrightBigg
= 0 + 0 +···+ 0 +···a37a194a241.
5,a124a94a63a234a194a241a27a55a135a94a135a121a178a181
(1) lim
n→∞
nn
(n!)2 = 0
(2) lim
n→∞
(2n)!
an! = 0(a> 1)
a121a178a181
150
(1)
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
nn
(n!)2
a207 lim
n→∞
un+1
un = limn→∞
(n+ 1)n+1
[(n+ 1)!]2
nn
(n!)2
= lim
n→∞
1
n+ 1
parenleftBigg
1 + 1n
parenrightBiggn
= 0 < 1
a75a226a136a75a19a16a7a79a123a27a52a129a47a170a167a26
∞summationdisplay
n=1
nn
(n!)2a194a241a167a108a13a100a63a234a194a241a27a55a135a94a135a167a26 limn→∞
nn
(n!)2 = 0
(2)
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
(2n)!
an!
a2070 < un+1u
n
=
[2(n+ 1)]!
a(n+1)!
(2n)!
an!
= (2n+ 2)(2n+ 1)an2(n?1)! < 4(n+ 1)
2
an+1 (a> 1)
a13 lim
n→∞
nk
an = 0(a> 1,k ∈N)a167a117a180 limn→∞
4(n+ 1)2
an+1 = 0a167a108a13 limn→∞
un+1
un = 0
a75a226a136a75a19a16a7a79a123a27a52a129a47a170a167a26
∞summationdisplay
n=1
(2n)!
an! a194a241a167a108a13a100a63a234a194a241a27a55a135a94a135a167a26 limn→∞
(2n)!
an! = 0
6,a63a216a101a15a63a234a27a194a241a53a181
(1)
∞summationdisplay
n=1
1
n·(lnn)p
(2)
∞summationdisplay
n=1
1
n·lnn·lnlnn
(3)
∞summationdisplay
n=1
1
n·(lnn)1+σ lnlnn
(4)
∞summationdisplay
n=1
1
n·(lnn)p(lnlnn)q
a41a181
(1) a100a117a216a216pa143a219a234a167a8xa191a169a140a158a167a188a234f(x) = 1x(lnx)p a209a180a154a75a52a126a27a167a133
lim
n→∞
integraldisplay n
2
dx
x(lnx)p =
1
p?1(ln2)
1?p,p> 1
∞,plessorequalslant 1
a25a8p> 1a158a167a63a234a194a241a182a8plessorequalslant 1a158a167a63a234a117a209.
(2) a23f(x) = 1xlnxlnlnxa167f(x)a8xgreaterorequalslant 3a180a20a138a52a126a188a234.
lim
n→∞
integraldisplay n
3
dx
xlnxlnlnx = limn→∞(lnlnlnn?lnlnln2) = ∞a167a75a63a234
∞summationdisplay
n=1
1
n·lnn·lnlnna117a209.
(3) a207 lim
n→∞
integraldisplay n
2
dx
x(lnx)1+σ = limn→∞
1
σ
parenleftBigg
1
(ln2)σ?
1
(lnn)σ
parenrightBigg
= 1σ(ln2)σ (σ> 0)
a25a63a234
∞summationdisplay
n=2
1
n(lnn)1+σa194a241.
a113 1n·(lnn)1+σ lnlnnlessorequalslant 1n(lnn)1+σ a167a75a100a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
1
n·(lnn)1+σ lnlnna194a241.
(4) a45f(x) = 1x(lnx)p(lnlnx)q a167a8nlessorequalslant 3a158a180a20a138a52a126a188a234.
a113a207a143
integraldisplay +∞
3
dx
x(lnx)p(lnlnx)q =
integraldisplay +∞
lnln3
dt
e(p?1)ttq
a233a63a219qa167a8p? 1 > 0a158a167a200a169a194a241a167a8p? 1 < 0a158a167a200a169a117a209a182a8p = 1a158a167a101q > 1a167a200a169a194a241a167
a101qlessorequalslant 1a167a200a169a117a209.
151
a100a133a220a200a169a7a79a123a127a167a6a63a234a241a209a53a134a200a169a241a209a53a94a135a152a151
a75a6a63a234a8p> 1a158a194a241a182a8p< 1a158a117a209a182a8p = 1a158a167q> 1a158a63a234a194a241a182qlessorequalslant 1a158a63a234a117a209.
7,a101
∞summationdisplay
n=1
una180a194a241a27a20a145a63a234a167a191a133a234a15{un}a252a78a101a252a167a121a178 lim
n→∞
nun = 0.
a121a178a181a207
∞summationdisplay
n=1
una194a241a167a23S =
∞summationdisplay
n=1
una167Sn =
nsummationdisplay
k=1
uk
a75 lim
n→∞
Sn = S = lim
n→∞
S2na167a117a180 lim
n→∞
(S2n?Sn) = 0
a113{un}a252a78a101a252a167a75S2n?Sn = un+1 +un+2 +···+u2n greaterorequalslantu2n +u2n +···+u2n = nu2n
a113un greaterorequalslant 0a167a750 lessorequalslantnu2n lessorequalslantS2n?Sna167a117a180a26 lim
n→∞
nu2n = 0a167a108a13 lim
n→∞
(2n)u2n = 0
a113a207u2n+1 lessorequalslantu2n,un greaterorequalslant 0a167a750 lessorequalslant (2n+ 1)u2n+1 lessorequalslant (2n+ 1)u2n = 2n+ 12n (2nu2n) → 0(n→∞)
a117a180 lim
n→∞
(2n+ 1)u2n+1 = 0a167a108a13 lim
n→∞
nun = 0
8,a121a178a136a75a19a16a7a79a123a57a217a52a129a47a170.
a121a178a181
(1) a136a75a19a16a7a79a123a181
a207n>Na158a167a107un+1u
n
lessorequalslantq< 1a167a75uN+2u
N+1
lessorequalslantq,uN+2 lessorequalslantquN+1; uN+3u
N+2
lessorequalslantq,uN+3 lessorequalslantquN+2;··· ; uN+k+1u
N+k
lessorequalslant
q,uN+k+1 lessorequalslantquN+k lessorequalslant···lessorequalslantqKuN+1
a207q < 1a167a75
∞summationdisplay
k=1
qka194a241a167a117a180a100a194a241a63a234a27a53a1591a127a167
∞summationdisplay
k=1
qkuN+1a143a194a241a167a108a13a100a39a22a7a79a123a167
a26
∞summationdisplay
k=2
uN+ka143a194a241
a50a100a194a241a63a234a27a53a1595a127a167a86a92a107a129a145u1,u2,···,uN+1a0a26a20a27a35a63a234
∞summationdisplay
n=1
una143a194a241.
a101n>Na158a167un+1u
n
greaterorequalslant 1a167a75uN+1u
N
greaterorequalslant 1,uN+1 greaterorequalslantuNa167a249a96a178{uN}a180a252a78a79a92a27
a113un greaterorequalslant 0a167a75unnotarrowright0(n→∞)a167a117a180
∞summationdisplay
n=1
una117a209.
(2) a136a75a19a16a7a79a123a27a52a129a47a170a181
(i) a101 lim
n→∞
un
un?1 = ˉr< 1
a100a162a234a27a200a151a53a127a55a127a51ε0 > 0a167a166a26r< ˉr+ε0 < 1
a100a254a52a129a27a189a1101a27a121a178a165a167a127
braceleftBigg
un+1
un
bracerightBigg
a144a107a107a129a145a140a117ˉr +ε0a167a117a180a189a127a51a152a135a20a18a234N(a144a135a18
a107a129a145a165a101a73a129a140a27a137Na61a140)a167a166a26a8n> Na158a167a107un+1u
n
< ˉr +ε0 < 1a167a25a100a136a75a19a16a7a79a123a127
a63a234a194a241.
(ii) a101 lim
n→∞
un
un?1 = r> 1
a100a162a234a27a200a151a53a127a55a127a51ε0 > 0a167a166a26ˉr>r?ε0 > 1
a100a254a52a129a27a189a1102a27a121a178a165a167a127
braceleftBigg
un+1
un
bracerightBigg
a144a107a107a129a145a2a117r +ε0a167a117a180a189a127a51a152a135a20a18a234N(a144a135a18
a107a129a145a165a101a73a129a140a27a137Na61a140)a167a166a26a8n> Na158a167a107un+1u
n
> r +ε0 > 1a167a25a100a136a75a19a16a7a79a123a127
a63a234a117a209.
(iii) a222a126a96a178a181
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
1
na167 limn→∞
un+1
un = limn→∞
un+1
un = 1a167a2
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
1
na117a209a182
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
1
n2 a167 limn→∞
un+1
un = limn→∞
un+1
un = 1a167a2
∞summationdisplay
n=1
un =
∞summationdisplay
n=1
1
n2 a194a241.
152
§4,a63a191a145a63a234
1,a63a216a101a15a63a234a27a194a241a53a163a157a41a94a135a194a241a189a253a233a194a241a164a181
(1) 12? 310 + 122? 3103 + 123? 3105 +···
(2) 1? 12 + 13!? 14 + 15!?···
(3)
∞summationdisplay
n=2
(?1)n?1 lnnn
(4)
∞summationdisplay
n=1
(?1)n?1n
3
2n
(5)
∞summationdisplay
n=1
(?1)n+1 n(n+ 1)2
(6)
∞summationdisplay
n=1
(?1)nsin xn (xnegationslash= 0)
(7) 1√2?1? 1√2 + 1 + 1√3?1? 1√3 + 1 +···+ 1√n?1? 1√n+ 1 +···
a41a181
(1) a207
∞summationdisplay
n=1
1
2na194a241a167
∞summationdisplay
n=1
1
102n?1a194a241a167a75
∞summationdisplay
n=1
3
102n?1a194a241
a117a180
∞summationdisplay
n=1
1
2n +
∞summationdisplay
n=1
1
102n?1a194a241a167a61
1
2 +
3
10 +
1
22 +
3
103 +
1
23 +
3
105 +···a194a241a167a108a13a6a63a234a253a233a194a241.
(2) a207
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
parenleftBigg
12n
parenrightBigg
a117a209
a113a233a63a234
∞summationdisplay
n=1
1
(2n?1)!
a207 lim
n→∞
1
(2n+1)!
1
(2n?1)!
= lim
n→∞
1
2n(2n+ 1) = 0 < 1a167a75a100a136a75a19a16a7a79a123a27a52a129a47a170a167a26a63a234
∞summationdisplay
n=1
1
(2n1)!a194a241
a117a180a6a63a234a117a209.
(3) a207
∞summationdisplay
n=2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle(?1)
n?1 lnn
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=2
lnn
n
a113 lim
n→∞
lnn
n
1
n
= lim
n→∞
lnn = +∞a133
∞summationdisplay
n=1
1
na117a209a167a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=2
lnn
n a117a209
a113a23f(x) = lnxx (xgreaterorequalslant 3)a167a75fprime(x) =
parenleftBigg
lnx
x
parenrightBiggprime
= 1?lnxx2 < 0 (xgreaterorequalslant 3)a167a117a180f(x) = lnxx a252a78a101a252a167a108
a13
braceleftBigg
lnn
n
bracerightBigg
a51ngreaterorequalslant 3a158a252a78a101a252
a113 lim
x→+∞
lnx
x = limx→+∞
1
x = 0a167a75 limn→∞
lnn
n = 0
a117a180a226a52a217a90a91a189a110a167a26
∞summationdisplay
n=2
(?1)n?1 lnnn a94a135a194a241.
(4)
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle(?1)
n?1n
3
2n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
n3
2n
a207 lim
n→∞
(n+1)3
2n+1
n3
2n
= lim
n→∞
1
2
parenleftBigg
n+ 1
n
parenrightBigg3
= 12 < 1a167a75a226a136a75a19a16a7a79a123a167a26
∞summationdisplay
n=1
n3
2na194a241
a108a13
∞summationdisplay
n=1
(?1)n?1n
3
2na253a233a194a241.
153
(5)
∞summationdisplay
n=1
(?1)n+1 n(n+ 1)2 =
∞summationdisplay
n=1
n
(n+ 1)2
a207 lim
n→∞
n
(n+1)2
1
n
= lim
n→∞
n2
(n+ 1)2 = 1a133
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
n
(n+ 1)2a117a209
a23f(x) = x(x+ 1)2 (x greaterorequalslant 2)a167a75fprime(x) = 1?x(x+ 1)3 < 0(x greaterorequalslant 2)a167a117a180a8x greaterorequalslant 2a158a167f(x)a252a78a101a252a167a108
a13
braceleftBigg
n
(n+ 1)2
bracerightBigg
a8ngreaterorequalslant 2a158a252a78a101a252
a113 lim
n→∞
n
(n+ 1)2 = 0a167a75a226a52a217a90a91a189a110a167a26
∞summationdisplay
n=1
(?1)n+1 n(n+ 1)2a194a241
a108a13
∞summationdisplay
n=1
(?1)n+1 n(n+ 1)2a94a135a194a241.
(6)
∞summationdisplay
n=1
(?1)nsin xn =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin xn
1
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle→|x|negationslash= 0(n→∞)a133
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea117a209
a113a233?x ∈ R,x negationslash= 0a167a207xn → 0(n → ∞)a167a75a127a51N ∈ Z+a167a8n > Na158a167a1070 <
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<
pi
2a167a117a180a8n >
Na158a167sin xna134xa107a131a211a27a206a210a133
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea145na79a140a13a126a2a200a167a75a100a52a217a90a91a7a79a123a167a26
∞summationdisplay
n=1
(?1)nsin xna194
a241
a108a13
∞summationdisplay
n=1
(?1)nsin xna94a135a194a241.
(7) a23a220a169a218a234a15a143{Sn}a167a75S2n =
n+1summationdisplay
k=2
parenleftBigg
1√
k?1?
1√
k+ 1
parenrightBigg
=
n+1summationdisplay
k=2
2
k?1 = 2
nsummationdisplay
k=1
1
k
a117a180 lim
n→∞
S2n = +∞a167a75a100a63a234a92a41a210a0a117a209a167a108a13a6a63a234a117a209.
2,a121a178a181a101a63a234a27a145a92a41a210a0a164a138a164a27a63a234a194a241a167a191a133a51a211a152a135a41a210a83a145a27a206a210a131a211a167a64a34a22a75a41a210a0a167a100a63
a234a189a194a241a182a191a100a100a127a9a63a234
∞summationdisplay
n=1
(?1)[
√n]
n a27a194a241a53.
a121a178:
(1) a174a127a35a63a234
∞summationdisplay
n=1
uprimen = (u1 +···+un1) + (un1+1 +···+un2) +···+ (unk?1+1 +···+unk) +···a194a241a133
a51a211a152a41a210a83a27a206a210a131a211
a23
nsummationdisplay
k=1
uk = Sn,
nsummationdisplay
k=1
uprimek = Sprimena167a75S1 prime = Sn1,S2 prime = Sn2,···,Skprime = Snk,···
a8a6a63a234a27a101a73na108nk?1a20nka158a167
∞summationdisplay
n=1
una27a220a169a218a252a78a67a122a167a61
a8unk?1+1,···,unka254a143a20a158a167a107Sk?1 prime = Snk?1 <Sn <Snk = Skprime
a8unk?1+1,···,unka254a143a75a158a167a107Sk?1 prime = Snk?1 >Sn >Snk = Skprime
a174a127
∞summationdisplay
n=1
uprimena194a241a167a61 lim
k→∞
Skprime = lim
k→∞
Sk?1 prime = Sprimea167a75 lim
n→∞
Sn = Sprimea167a117a180
∞summationdisplay
n=1
una194a241.
(2) a127a196
∞summationdisplay
n=1
(?1)[
√n]
n
a8n = k2,k2+1,···,k2+2k(k = 1,2,···)a158a167a195ana211a210.a80Ak = 1k2 + 1k2 + 1+···+ 1k2 + 2ka167
∞summationdisplay
k=1
(?1)kAka180
a2a134a63a234
a207
integraldisplay k2+2k
k2?1
dx
x lessorequalslant
1
k2 +
1
k2 + 1 + ··· +
1
k2 + 2k lessorequalslant
integraldisplay (k+1)2
k2
dx
x a61ln
k2 + 2k
k2?1 lessorequalslant Ak lessorequalslant ln
(k+ 1)2
k2 a167a108a13
a8k →∞a158a167Ak → 0
154
a113Ak?Ak+1 greaterorequalslant ln k
2 + 2k
k2?1? ln
(k+ 2)2
(k+ 1)2 = ln
k2 +k
k2 +k?2 > 0a167a75a100a52a217a90a91a7a79a123a127
∞summationdisplay
k=1
(?1)kAka194
a241a167a108a13a6a63a234a194a241.
3,a63a216a101a15a63a234a180a196a253a233a194a241a189a94a135a194a241a181
(1)
∞summationdisplay
n=1
(?1)n
n+x
(2)
∞summationdisplay
n=1
sin(2nx)
n!
(3)
∞summationdisplay
n=1
sinnx
n,(0 <x<pi)
(4)
∞summationdisplay
n=1
cosnx
np,(0 <x<pi)
a41a181
(1)
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(?1)n
n+x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
1
|n+x|
a207 lim
n→∞
1
|n+x|
1
n
= lim
n→∞
n
|n+x| = 1a167a75a100a39a22a7a79a123a167a26
∞summationdisplay
n=1
1
|n+x|a117a209
a8xgreaterorequalslant 0a158a167 1n+xa252a78a126a8a167a133 lim
n→∞
1
n+x = 0a167a75a100a52a217a90a91a189a110a167a26
∞summationdisplay
n=1
(?1)n
n+xa194a241
a8x < 0a133a216a143a75a18a234a158a167a207xa143a189a234a167a75a8na191a169a140a158a167a61a127a51N ∈ Z+a167a8n > Na158a167a107n + x >
0a167a117a180
∞summationdisplay
n=N+1
(?1)n
n+xa180a2a134a63a234a167a133a100
1
n+xa252a78a126a8a57 limn→∞
1
n+x = 0a167a75
∞summationdisplay
n=N+1
(?1)n
n+xa194a241a167a108
a13
∞summationdisplay
n=1
(?1)n
n+xa194a241
a75a8xa216a143a75a18a234a158a167a100a63a234a143a94a135a194a241.
(2) a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(2nx)
n!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n!a167a133 limn→∞
1
(n+1)!
1
n!
= lim
n→∞
1
n+ 1 = 0 < 1a167a75a100a136a75a19a16a7a79a123a167a26
∞summationdisplay
n=1
1
n!a194a241
a50a226a39a22a7a79a123a167a26
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(2nx)
n!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea194a241a167a108a13
∞summationdisplay
n=1
sin(2nx)
n! a253a233a194a241.
(3) a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
sinkx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cos x2?cos 2n+12 x
2sin x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1vextendsingle
vextendsinglesin x2vextendsinglevextendsinglea133a234a15
braceleftBigg
1
n
bracerightBigg
a252a78a170a1170
a75a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
n=1
sinnx
n a194a241.
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinnx
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
sin2nx
n =
1
2n?
cos2nx
2n a133
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
cos2kx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinx?sin(2n+ 1)x
2sinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
|sinx|a57a234a15
braceleftBigg
1
2n
bracerightBigg
a252
a78a170a1170
a75a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
n=1
cos2nx
n a194a241.
a113
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
1
2na117a209a167a117a180
∞summationdisplay
n=1
parenleftBigg
1
2n?
cos2nx
2n
parenrightBigg
a117a209a167a108a13
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinnx
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea117a209
a75a63a234
∞summationdisplay
n=1
sinnx
n (0 <x<pi)a94a135a194a241.
(4) (i) a8p> 1a158a167a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosnx
np
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
npa133
∞summationdisplay
n=1
1
npa8p> 1a158a194a241a167a75a63a234
∞summationdisplay
n=1
cosnx
np (0 <x<pi)a253a233a194a241.
(ii) a80 <plessorequalslant 1a158a167a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
coskx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin x2?sin 2n+12 x
2sin x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1vextendsingle
vextendsinglesin x2vextendsinglevextendsinglea133a234a15
braceleftBigg
1
np
bracerightBigg
a252a78a170a1170
a75a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
n=1
cosnx
np a194a241.
155
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosnx
np
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
cos2nx
np =
1
2np +
cos2nx
2np a133a100a102a226a121a178a140a26
∞summationdisplay
n=1
cos2nx
(2n)p a194a241.
a75
∞summationdisplay
n=1
cos2nx
(2n)p ·2
p?1a194a241a167a61
∞summationdisplay
n=1
cos2nx
2np a194a241
a113a80 <plessorequalslant 1a158a167
∞summationdisplay
n=1
1
npa117a209a167a75
∞summationdisplay
n=1
1
2npa117a209a167a117a180
∞summationdisplay
n=1
parenleftBigg
1
2np +
cos2nx
2np
parenrightBigg
a117a209
a108a13
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosnx
np
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglea117a209
a75a63a234
∞summationdisplay
n=1
cosnx
np (0 <x<pi)a80 <plessorequalslant 1a158a94a135a194a241.
(iii) a8plessorequalslant 0a158a167a207 cosnxnp notarrowright0a167a75a63a234
∞summationdisplay
n=1
cosnx
np (0 <x<pi)a8plessorequalslant 0a158a117a209.
4,a101a63a234
∞summationdisplay
n=1
ana194a241a167a191a133 lim
n→∞
an
bn = 1a167a85a196a228a189
∞summationdisplay
n=1
bna143a194a241a186
a121a178a181
(1) a101a63a234
∞summationdisplay
n=1
ana167
∞summationdisplay
n=1
bna209a180a20a145a63a234
a100a63a234
∞summationdisplay
n=1
ana194a241a167 lim
n→∞
an
bn = 1a167a75a226a20a145a63a234a39a22a7a79a123a167a26a63a234
∞summationdisplay
n=1
bna194a241
(2) a101a63a234
∞summationdisplay
n=1
ana167
∞summationdisplay
n=1
bna216a152a189a209a180a20a145a63a234
a100a63a234
∞summationdisplay
n=1
ana194a241a167a216a140a228a189
∞summationdisplay
n=1
bna194a241
a126a181a63a234
∞summationdisplay
n=1
(?1)n√
n a143a52a217a90a91a46a63a234a167a75a217a194a241a133 limn→∞
(?1)n√
n
(?1)n√
n +
1
n
= 1
a100a117
∞summationdisplay
n=1
(?1)n√
n a194a241a167
∞summationdisplay
n=1
1
na117a209a167a75
∞summationdisplay
n=1
parenleftBigg
(?1)n√
n +
1
n
parenrightBigg
a117a209.
5,a121a178a181a101
∞summationdisplay
n=1
an
nx0a194a241a167a64a34a8x>x0a158
∞summationdisplay
n=1
an
nxa143a194a241.
a121a178a181a207x>x0a167a75
1
(n+1)x?x0
1
nx?x0
=
parenleftBigg
n
n+ 1
parenrightBiggx?x0
=
parenleftBigg
1? 1n+ 1
parenrightBiggx?x0
< 1a167a75 1(n+ 1)x?x
0
< 1nx?x
0
a133 1nx?x
0
lessorequalslant 1
a117a180a234a15
braceleftBigg
1
nx?x0
bracerightBigg
a252a78a107a46a167a133 1nx?x
0
lessorequalslant 1
a113a63a234
∞summationdisplay
n=1
an
nx0 a194a241a167a75a100a67a19a16a7a79a123a167a26
∞summationdisplay
n=1
an
nxa194a241.
6,a23{nan}a194a241a167
∞summationdisplay
n=1
n(an?an?1)a194a241a167a75
∞summationdisplay
n=1
ana143a194a241.
a121a178a181a207{nan}a194a241a167a23a217a52a129a143a
a113
∞summationdisplay
n=1
n(an?an?1)a194a241a167a75a217a220a169a218a234a15
braceleftBigg nsummationdisplay
k=1
k(ak?ak?1)
bracerightBigg
a107a52a129a167a23a217a52a129a143S
a113
nsummationdisplay
k=1
k(ak?ak?1) = (a1?a0) + 2(a2?a1) +···+n(an?an?1) = nan?
n?1summationdisplay
k=0
ak
a61
n?1summationdisplay
k=0
ak = nan?
nsummationdisplay
k=1
k(ak?ak?1)a167a75 lim
n→∞
n?1summationdisplay
k=0
ak = lim
n→∞
nan? lim
n→∞
nsummationdisplay
k=1
k(ak?ak?1) = a?S
a117a180
∞summationdisplay
n=0
ana194a241a167a108a13
∞summationdisplay
n=1
an =
∞summationdisplay
n=0
an?a0a194a241.
156
7,a101
∞summationdisplay
v=1
(av?av?1)a253a233a194a241a167
∞summationdisplay
v=1
bva194a241a167a64a34
∞summationdisplay
v=1
avbva194a241.
a121a178a181a45Bn+mn =
n+msummationdisplay
v=n+1
bv
a100Abel a67a134a167a26
n+psummationdisplay
v=n+1
avbv = an+pBn+pn +
p?1summationdisplay
i=1
Bn+in (an+i?an+i+1)
a25
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
n+psummationdisplay
v=n+1
avbv
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant|an+p|
vextendsinglevextendsingleBn+p
n
vextendsinglevextendsingle+p?1summationdisplay
i=1
vextendsinglevextendsingle
vextendsingleBn+in
vextendsinglevextendsingle
vextendsingle|an+i?an+i+1|
a45Hpn = maxbraceleftbigvextendsinglevextendsingleBn+1n vextendsinglevextendsingle,vextendsinglevextendsingleBn+2n vextendsinglevextendsingle,···,vextendsinglevextendsingleBn+pn vextendsinglevextendsinglebracerightbiga167a75a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
n+psummationdisplay
v=n+1
avbv
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslantH
p
n
bracketleftBigg
|an+p|+
p?1summationdisplay
i=1
|an+i?an+i+1|
bracketrightBigg
a207
∞summationdisplay
v=1
|av?av?1|a194a241a167a25
∞summationdisplay
v=1
(av?av?1)a194a241a133
∞summationdisplay
v=1
(av?av?1) =?a0 +ana167a25 lim
n→∞
ana127a51
a207a13a127a51M > 0a167a166a233a152a131na167a107
p?1summationdisplay
i=1
|an+i?an+i+1|+|an+p|<M (4)
a113
∞summationdisplay
v=1
bva194a241a167a108a13a233?ε> 0,?N ∈Z+a167a8n>Na158a167a233a152a131p∈Z+a167a107
Hpn < εM (5)
a100(),()a127a167a8n>Na158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
n+psummationdisplay
v=n+1
avbv
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<εa167a249a76a178a63a234
∞summationdisplay
v=1
avbva194a241
8,a124a94a133a220a194a241a6a110a121a178a2a134a63a234a27a52a217a90a91a189a110.
a121a178a181a233a63a219a103a44a234pa167a107
|Sn+p?Sn| =vextendsinglevextendsingle(?1)n+2un+1 + (?1)n+3un+2 +···+ (?1)n+p+1un+pvextendsinglevextendsingle=vextendsinglevextendsingle(?1)n+2(un+1?un+2 +···+ (?1)p?1un+p)vextendsinglevextendsingle=vextendsingle
vextendsingleun+1?un+2 +···+ (?1)p?1un+pvextendsinglevextendsingle
a8pa143a243a234a158a167(un+1?un+2) +···+ (un+p?1?un+p) greaterorequalslant 0
a8pa143a219a234a158a167(un+1?un+2) +···+ (un+p?2?un+p?1) +un+p greaterorequalslant 0
a111a131vextendsinglevextendsingleun+1?un+2 +···+ (?1)p?1un+pvextendsinglevextendsingle= un+1?un+2 +···+ (?1)p?1un+p
a113a8pa143a243a234a158a167un+1?(un+2?un+3)?···?(un+p?2?un+p?1)?un+p lessorequalslantun+1
a8pa143a219a234a158a167un+1?(un+2?un+3)?···?(un+p?1?un+p) lessorequalslantun+1
a111a131un+1?un+2 +···+ (?1)p?1un+p lessorequalslantun+1
a233a63a191ε> 0a167a207 lim
n→∞
un = 0a167a75 lim
n→∞
un+1 = 0
a117a180a55a127a51N ∈Z+a167a8n>Na158a167a107|un+1?0|<εa167a75un+1 <ε
a100a100a8n>Na158a167a233a63a219a103a44a234pa209a107|Sn+p?Sn| = un+1?un+2 +···+ (?1)pun+p lessorequalslantun+1 <ε
a108a13a100a133a220a194a241a6a110a167a26
∞summationdisplay
n=1
(?1)n+1una194a241.
157
§5,a253a233a194a241a63a234a218a94a135a194a241a63a234a27a53a159
1,a23|x|< 1,|y|< 1a167a121a178
∞summationdisplay
v=1
(xv?1 +xv?2y+···+yv?1) = 1(1?x)(1?y)
a121a178a181a207|x|< 1,|y|< 1a167a75
∞summationdisplay
v=1
xv?1 = 1 +x+x2 +···+xv +··· = 11?xa253a233a194a241 (6)
∞summationdisplay
v=1
yv?1 = 1 +y+y2 +···+yv +··· = 11?y a253a233a194a241 (7)
()·()a167a26
∞summationdisplay
v=1
xv?1
∞summationdisplay
v=1
yv?1 = 1(1?x)(1?y)
a113
∞summationdisplay
v=1
xv?1
∞summationdisplay
v=1
yv?1 = (1+x+x2+···+xv+···)(1+y+y2+···+yv+···) =
∞summationdisplay
v=1
(xv?1+xv?2y+···+yv?1)a167
a75
∞summationdisplay
v=1
(xv?1 +xv?2y+···+yv?1) = 1(1?x)(1?y).
2,a121a178a181
∞summationdisplay
n=0
xn
n!
∞summationdisplay
n=0
yn
n! =
∞summationdisplay
n=0
(x+y)n
n!
a121a178a181a207 lim
n→∞
|x|n+1
(n+1)!
|x|n
n!
= lim
n→∞
|x|
n+ 1 = 0 < 1a167a75a226a136a75a19a16a7a79a123a27a52a129a47a170a167a26a63a234
∞summationdisplay
n=0
|x|n
n! a194a241
a117a180a63a234
∞summationdisplay
n=0
|x|n
n! a253a233a194a241
a211a110a167a63a234
∞summationdisplay
n=0
|y|n
n! a253a233a194a241
a140a21a164
∞summationdisplay
n=0
xn
n!
∞summationdisplay
n=0
yn
n! =
∞summationdisplay
n=0
Cn
a217a165Cn =
nsummationdisplay
i=0
xi
i! ·
yn?i
(n?i)! =
yn
n! +
x
1! ·
yn?1
(n?1)! +···+
xn
n! =
1
n!(C
0
ny
n+C1
nxy
n?1 +···+Cn
nx
n) = (x+y)
n
n!
a75
∞summationdisplay
n=0
xn
n!
∞summationdisplay
n=0
yn
n! =
∞summationdisplay
n=0
(x+y)n
n!
3,a121a178a181a140a177a138a209a94a135a194a241a63a234a27a141a83a63a234a167a166a217a117a209a20+∞.
a121a178a181a23
∞summationdisplay
n=1
una94a135a194a241
a100a189a1101a167a26
∞summationdisplay
n=1
vna218
∞summationdisplay
n=1
wna209a117a209a167a133
∞summationdisplay
n=1
vna117a209a20+∞a167
∞summationdisplay
n=1
(?wn)a117a209a20?∞
a192a18a117a209a20+∞a27a234a15{βn}a167a61 lim
n→∞
βn = +∞
a114
∞summationdisplay
n=1
vna85a94a83a152a145a152a145a92a229a53
a18m1a167a166v1 +v2 +···+vm1 >β1 +w1
a44a0a18m2a167a166v1 +v2 +···+vm1 +vm1+1 +···+vm2 >β2 +w1 +w2
a152a132a47a167a140a18a191a169a140a27mk >mk?1a167a166a26v1+v2+···+vm1+···+vm2+···+vmk >βk+w1+w2+···+wk (k =
3,4,···)
a249a24a2a134a47a152a92a152a124a20a145a218a152a135a75a145a181
(v1 +···+vm1?w1) + (vm1+1 +···+vm2?w2) +···+ (vmk?1+1 +···+vmk?wk) +··· (?)
a100a63a234a119a44a143a6a63a234a27a141a83a63a234
a207(?)a92a41a210a0a27a63a234
∞summationdisplay
k=1
(vmk?1+1 +···+vmk?wk)a27ka103a220a169a218
(v1 +···+vm1?w1) + (vm1+1 +···+vm2?w2) +···+ (vmk?1+1 +···+vmk?wk) >βk
158
a13 lim
k→∞
βk = +∞
a75
∞summationdisplay
k=1
(vmk?1+1 +···+vmk?wk)a117a209a20+∞
a100a117a209a63a234a140a63a191a22a41a210a167a75a140a177a138a209a94a135a194a241a63a234a27a141a83a63a234a167a166a217a117a209a20+∞.
159
§6,a195a161a166a200
1,a63a216a195a161a166a200a27a194a241a53a181
(1)
∞productdisplay
n=3
n2?4
n2?1
(2)
∞productdisplay
n=1
a(?1)
n
n (a> 0)
(3)
∞productdisplay
n=0
radicalBigg
n+ 1
n+ 2
a41a181
(1) a207n
2?4
n2?1 = 1?
3
n2?1,ngreaterorequalslant 3a167a133?
3
n2?1 < 0
a113 lim
n→∞
3
n2?1
1
n2
= 3a133
∞summationdisplay
n=3
1
n2a194a241a167a75
∞summationdisplay
n=3
3
n2?1a194a241a167a117a180
∞summationdisplay
n=3
parenleftBigg
3n2?1
parenrightBigg
a194a241
a108a13a226a189a1102a167a26
∞productdisplay
n=3
n2?4
n2?1a194a241.
(2)
∞summationdisplay
n=1
lna(?1)
n
n =
∞summationdisplay
n=1
(?1)n
n lna = lna
∞summationdisplay
n=1
(?1)n
n
a207a63a234
∞summationdisplay
n=1
(?1)n
n a143a52a217a90a91a46a63a234a167a75a217a194a241a167a117a180a63a234
∞summationdisplay
n=1
lna(?1)
n
n a194a241a167a108a13a195a161a166a200
∞productdisplay
n=1
a(?1)
n
n a194
a241.
(3) a100a117a220a169a166a200Pn =
radicalBigg
1
2 ·
2
3 ···
n
n+ 1 ·
n+ 1
n+ 2 =
radicalBigg
1
n+ 2 → 0(n→∞)
a25a195a161a166a200
∞productdisplay
n=0
radicalBigg
n+ 1
n+ 2a117a209a1170.
2,a121a178a181a101
∞summationdisplay
n=1
x2na194a241a167a75
∞productdisplay
n=1
cosxna194a241.
a121a178a181a207
∞productdisplay
n=1
cosxn =
∞productdisplay
n=1
parenleftBigg
1?2sin2 xn2
parenrightBigg
a1330 lessorequalslant 2sin2 xn2 lessorequalslant 2·
parenleftBigg
sinxn
2
parenrightBigg2
= x
2
n
2
a113
∞summationdisplay
n=1
x2na194a241a167a75
∞summationdisplay
n=1
2sin2 xn2 a194a241
a117a180a226a189a1102a167a26
∞productdisplay
n=1
cosxna194a241.
3,a121a178a181a101
∞summationdisplay
n=1
αna253a233a194a241a167a75
∞productdisplay
n=1
tan
parenleftBigg
pi
4 +αn
parenrightBigg
a194a241
parenleftBigg
a217a165|αn|< pi4
parenrightBigg
.
a121a178a181
∞productdisplay
n=1
tan
parenleftBigg
pi
4 +αn
parenrightBigg
=
∞productdisplay
n=1
1 + tanαn
1?tanαn =
∞productdisplay
n=1
parenleftBigg
1 + 2tanαn1?tanα
n
parenrightBigg
a207
∞summationdisplay
n=1
αna253a233a194a241a167a75 lim
n→∞
αn = 0a167a117a180 lim
n→∞
vextendsinglevextendsingle
vextendsingle 2tanαn1?tanαn
vextendsinglevextendsingle
vextendsingle
|αn| = limn→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
2
1?tanαn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
tanαn
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 2
a100
∞summationdisplay
n=1
αna253a233a194a241a167a26
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle 2tanαn
1?tanαn
vextendsinglevextendsingle
vextendsinglevextendsinglea194a241a167a117a180
∞summationdisplay
n=1
2tanαn
1?tanαna253a233a194a241
a108a13
∞productdisplay
n=1
tan
parenleftBigg
pi
4 +αn
parenrightBigg
a253a233a194a241.
160
a49a155a217 a50a194a200a169
§1,a195a161a129a27a50a194a200a169
1,a166a101a15a50a194a200a169a27a138a181
(1)
integraldisplay +∞
2
1
x2?1 dx
(2)
integraldisplay +∞
0
1
(x2 +p)(x2 +q)dx,(p,q> 0)
(3)
integraldisplay +∞
0
e?ax2xdx(a> 0)
(4)
integraldisplay +∞
0
e?axsinbxdx,(a> 0)
a41a181
(1)
integraldisplay +∞
2
1
x2?1 dx =
1
2ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x?1
x+ 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+∞
2
= 12ln3 = ln√3
(2)
integraldisplay +∞
0
1
(x2 +p)(x2 +q)dx =
1
q?p
parenleftBigg
1√
parctan
x√
p?
1√
qarctan
x√
q
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+∞
0
=
pi
2√
pq(√p+√q) =
pi
2(q√p+p√q)
(3)
integraldisplay +∞
0
e?ax2xdx =?e
ax2
2a
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+∞
0
= 12a
(4)
integraldisplay +∞
0
e?axsinbxdx =?asinbx?bcosbxa2 +b2 e?ax
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
+∞
0
= ba2 +b2
2,a63a216a101a15a200a169a27a194a241a53a181
(1)
integraldisplay +∞
0
dx
3√x4 + 1
(2)
integraldisplay +∞
1
xarctanx
1 +x2 dx
(3)
integraldisplay +∞
1
sin 1x2 dx
(4)
integraldisplay +∞
0
dx
1 +x|sinx|
(5)
integraldisplay +∞
0
x
1 +x2 sin2x dx
(6)
integraldisplay +∞
0
xm
1 +xn dx,(n> 0,m> 0)
a41a181
(1)
integraldisplay +∞
0
dx
3√x4 + 1 =
integraldisplay 1
0
dx
3√x4 + 1 +
integraldisplay +∞
1
dx
3√x4 + 1
a207
integraldisplay 1
0
dx
3√x4 + 1a143a20a126a200a169a167a75a217a55a194a241
a233
integraldisplay +∞
1
dx
3√x4 + 1a167a207
integraldisplay +∞
1
dx
3√x4a194a241a167a75
integraldisplay +∞
1
dx
3√x4 + 1a194a241a167a108a13
integraldisplay +∞
0
dx
3√x4 + 1a194a241.
(2) a207 lim
x→+∞
x
1+x3 arctanx
1
x2
= lim
x→+∞
x3
1 +x3 arctanx =
pi
2a167a133
integraldisplay +∞
1
1
x2a194a241
a75a100a39a22a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
xarctanx
1 +x2 dxa194a241.
161
(3) a207 lim
x→+∞
sin 1x2
1
x2
= 1a133
integraldisplay +∞
1
dx
x2 a194a241a167a108a13
integraldisplay +∞
1
sin 1x2 dxa194a241.
(4) 11 +x|sinx|greaterorequalslant 11 +x
a207 lim
x→+∞
1
1+x
1
x
= 1a133
integraldisplay +∞
1
dx
x a117a209a167a108a13a100a39a22a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
dx
1 +xa117a209
a113
integraldisplay 1
0
dx
1 +xa143a20a126a200a169a75a194a241a167a117a180
integraldisplay +∞
0
dx
1 +xa117a209
a108a13a100a39a22a7a79a123a167a26
integraldisplay +∞
0
dx
1 +x|sinx|a117a209.
(5) a207x∈ [0,+∞)a158a167a107 x1 +x2 sin2xgreaterorequalslant x1 +x2a133
integraldisplay +∞
0
x
1 +x2 dx =
1
2ln(1 +x
2)
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
0
= +∞
a75a100a39a22a7a79a123a167a26
integraldisplay +∞
0
x
1 +x2 sin2x dxa117a209.
(6)
integraldisplay +∞
0
xm
1 +xn dx =
integraldisplay 1
0
xm
1 +xn dx+
integraldisplay +∞
1
xm
1 +xn dxa133
integraldisplay 1
0
xm
1 +xn dxa143a126a194a200a169
(i) a8n?m> 1a158a167a107 1xn?m · 11 + 1
xn
< 1xn?ma133a200a169
integraldisplay +∞
1
xm
1 +xn dxa194a241a167a25a6a200a169a194a241a182
(ii) a8n?mlessorequalslant 1a133xgreaterorequalslant 1a158a167a107 x
m
1 +xn greaterorequalslant
1
2xn?ma167a133
integraldisplay +∞
1
1
xn?m dxa117a209a167a25a6a200a169a117a209
a75a8n?m> 1a158a167
integraldisplay +∞
0
xm
1 +xn dxa194a241a182a8n?mlessorequalslant 1a158a167
integraldisplay +∞
0
xm
1 +xn dxa117a209.
3,a121a178a253a233a194a241a27a50a194a200a169a55a194a241a167a2a135a131a216a44.
a121a178a181a174a127
integraldisplay +∞
a
|f(x)|dxa194a241a167a100a133a220a7a79a6a110a167a26a233?ε> 0,?A> 0a167a8Aprimeprime >Aprime >Aa158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
|f(x)|dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<εa167a75
integraldisplay Aprimeprime
Aprime
|f(x)|dx<εa167a117a180
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
f(x)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
integraldisplay Aprimeprime
Aprime
|f(x)|dx<εa167
a108a13
integraldisplay +∞
a
f(x)dxa194a241.
a194a241a27a50a194a200a169a153a55a253a233a194a241.
a126a181
integraldisplay +∞
1
sinx
x dxa194a241a182a13
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinx
x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dxa117a209(a132a214a25455a144).
4,a121a178a233a117a195a161a129a200a169a167a169a220a200a169a250a170a164a225a163a8a250a170a165a136a220a169a107a191a194a158a164
integraldisplay +∞
a
f(x)gprime(x)dx = f(x)g(x)
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
a
integraldisplay +∞
a
g(x)f prime(x)dx
a121a178a181a233a117a63a191A>aa167a164a225
integraldisplay A
a
f(x)gprime(x)dx = f(x)g(x)
vextendsinglevextendsingle
vextendsinglevextendsingle
A
a
integraldisplay A
a
g(x)f prime(x)dx
a252a62a18a52a129a167a26 lim
A→+∞
integraldisplay A
a
f(x)gprime(x)dx = lim
A→+∞
parenleftBigg
f(x)g(x)
vextendsinglevextendsingle
vextendsinglevextendsingle
A
a
parenrightBigg
lim
A→+∞
parenleftbiggintegraldisplay A
a
g(x)f prime(x)dx
parenrightbigg
a75
integraldisplay +∞
a
f(x)gprime(x)dx = f(x)g(x)
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
a
integraldisplay +∞
a
g(x)f prime(x)dx
5,a121a178a181
(1) a23f(x)a143[0,+∞)a254a27a152a151a235a89a188a234a167a191a133a200a169
integraldisplay +∞
0
f(x)dxa194a241a167a75 lim
x→+∞
f(x) = 0a182a88a74a61a61a200
a169
integraldisplay +∞
0
f(x)dxa194a241a167a177a57f(x)a51[0,+∞)a235a89a167f(x) greaterorequalslant 0a167a180a196a69a206a164a225 lim
x→+∞
f(x) = 0a186
a121a178a181a94a135a121a123.a23 lim
x→+∞
f(x) negationslash= 0a167a75?ε> 0a167a233a63a191a140a27A> 0a167a209a127a51xA >Aa167a166a26|f(xA)|greaterorequalslant 2ε.
a18a83a15An → +∞(n→∞)a167a107a83a15xn → +∞a133xn >An(n = 1,2,···)a167a166|f(xn)|greaterorequalslant 2ε
a44a152a144a161a167a100f(x)a27a152a151a194a241a53a167a233a254a227ε> 0,?δ> 0a167a166a26a8|xprime?xprimeprime|<δa158a167a107|f(xprime)?f(xprimeprime)|<ε
a207a100a167a233a152a131na167a8x∈
parenleftBigg
xn? δ2,xn + δ2
parenrightBigg
a158a167a107|f(x)?f(xn)|<εa167a61f(xn)?ε<f(x) <f(xn) +ε
162
a8f(xn) > 0a158a167|f(xn)| = f(xn) greaterorequalslant 2εa167a100a134a224a216a31a170a167a26f(x) > 2ε?ε = ε
a8f(xn) < 0a158a167|f(xn)| =?f(xn) greaterorequalslant 2εa167a100a109a224a216a31a170a167a26f(x) <?2ε+ε =?ε
a108a13a167
integraldisplay xn+δ
2
xn?δ2
f(x)dx>εδa163a8f(xn) > 0a158a164a189
integraldisplay xn+δ
2
xn?δ2
f(x)dx<?εδa163a8f(xn) < 0a158a164
a100a134
integraldisplay +∞
0
f(x)dxa194a241a103a241a167a75a98a23a216a164a225a167a117a180 lim
x→+∞
f(x) = 0.
(2) a200a169
integraldisplay +∞
0
f(x)dxa194a241a167a177a57f(x)a51[0,+∞)a235a89a167f(x) greaterorequalslant 0a167a191a216a85a2a121 lim
x→+∞
f(x) = 0.
a126a181
integraldisplay +∞
0
x
1 +x6 sin2x dx.
a167a180a253a233a194a241a27.
a207
integraldisplay +∞
0
x
1 +x6 sin2x dx =
∞summationdisplay
n=0
integraldisplay (n+1)pi
npi
x
1 +x6 sin2x dx =
integraldisplay pi
0
x
1 +x6 sin2x dx+
∞summationdisplay
n=1
parenleftbigI1
n +I
2
n
parenrightbig
a217a165I1n =
integraldisplay (n+1
2)pi
npi
x
1 +x6 sin2x dx =
integraldisplay pi
2
0
npi+z
1 + (npi+z)6 sin2z dz,
I2n =
integraldisplay (n+1)pi
(n+12)pi
x
1 +x6 sin2x dx =
integraldisplay pi
2
0
npi+pi?z
1 + (npi+pi?z)6 sin2z dz
a53a191a20a80 <z< pi2a158a1672pi< sinzz lessorequalslant 1a167a117a180(npi+z)6 sin2z greaterorequalslant (npi)6
parenleftBigg
2z
pi
parenrightBigg2
= (2pi2n3z)2,
(npi+pi?z)6 sin2z greaterorequalslant (2pi2n3z)2
a25a107I1n lessorequalslant
integraldisplay pi
2
0
(n+ 1)pi
1 + (2pi2n3z)2 dz =
n+ 1
2n3pi
integraldisplay (npi)3
0
dy
1 +y2 lessorequalslant
n+ 1
4n3
a211a110a167a107I2n lessorequalslant n+ 14n3
a207
integraldisplay pi
0
x
1 +x6 sin2x dxa143a20a126a200a169a167a75a55a194a241
a113n+ 12n3 < 1n2a133
∞summationdisplay
n=1
1
n2a194a241a167a75
∞summationdisplay
n=1
n+ 1
2n3 a194a241
a117a180
integraldisplay +∞
0
x
1 +x6 sin2x dxlessorequalslant
integraldisplay pi
0
x
1 +x6 sin2x dx+
∞summationdisplay
n=1
n+ 1
2n3 a253a233a194a241
a119a44f(x) = x1 +x6 sin2xa51[0,+∞)a254a154a75a235a89
a2a101a18xn = 2npi(n = 0,1,2,···)a167a107f(xn) = f(2npi) = 2npi → +∞(n→∞)
a75 lim
x→∞
f(x) negationslash= 0.
6,a121a178a181a101f(x),g(x)a51a63a219a171a109[a,A]a140a200a167a113a23f2(x),g2(x)a51[a,+∞)a200a169a194a241a167a64a34[f(x) +g(x)]2a218
|f(x)·g(x)|a51[a,+∞)a254a27a140a200.
a121a178a181a207f(x),g(x)a51a63a219a171a109[a,A]a140a200a167a75
integraldisplay A
a
|f(x)·g(x)|dxa127a51a167
integraldisplay A
a
[f(x) +g(x)]2 dxa127a51
a113
integraldisplay +∞
a
f2(x)dxa218
integraldisplay +∞
a
g2(x)dxa209a194a241a167a75
integraldisplay +∞
a
[f2(x) +g2(x)]dxa194a241a167
a117a180
integraldisplay +∞
a
2[f2(x) +g2(x)]dxa218
integraldisplay +∞
a
1
2[f
2(x) +g2(x)]dxa209a194a241
a113[|f(x)|?|g(x)|]2 = f2(x) +g2(x)?2|f(x)·g(x)|greaterorequalslant 0a61|f(x)·g(x)|lessorequalslant 12[f2(x) +g2(x)]
a75a100a39a22a7a79a123a167a26|f(x)·g(x)|a51[a,+∞)a254a140a200
a113[f(x) +g(x)]2 = f2(x) +g2(x) + 2f(x)·g(x) lessorequalslantf2(x) +g2(x) + 2|f(x)·g(x)|lessorequalslant 2[f2(x) +g2(x)]
a75a100a39a22a7a79a123a167a26[f(x) +g(x)]2a51[a,+∞)a254a140a200.
7,a233a195a161a129a50a194a200a169a167a63a216a178a144a140a200a218a253a233a140a200a27a39a88.a127a9a126a102a181
integraldisplay +∞
1
dx
x3/2a218
integraldisplay +∞
1
f(x)dxa167a217a165
f(x) = n2
parenleftBigg
a8nlessorequalslantx<n+ 1n4
parenrightBigg
a167f(x) = 0
parenleftBigg
a8n+ 1n4 lessorequalslantx<n+ 1
parenrightBigg
.
a121a178a181a178a144a140a200notdblarrowrighta253a233a140a200
a126a181
integraldisplay +∞
1
1
x3/2 dxa194a241a167a2
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
x3/4
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay +∞
1
dx
x3/4a117a209a182
163
integraldisplay +∞
1
1
x3 dxa194a241a167a133
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
x3/2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay +∞
1
1
x3/2 dxa194a241
a253a233a140a200notdblarrowrighta178a144a140a200
a126a181
integraldisplay +∞
1
f(x)dxa167a217a165f(x) = n2
parenleftBigg
a8nlessorequalslantx<n+ 1n4
parenrightBigg
a167f(x) = 0
parenleftBigg
a8n+ 1n4 lessorequalslantx<n+ 1
parenrightBigg
integraldisplay +∞
1
|f(x)|dx =
integraldisplay +∞
1
f(x)dx = 114 ·12 + 124 ·22 +···+ 1n4 ·n2 +··· = 1 + 122 +···+ 1n2 +··· =
∞summationdisplay
n=1
1
n2a194a241
integraldisplay +∞
1
f(x)dxa167a217a165f(x) = n4
parenleftBigg
a8nlessorequalslantx<n+ 1n4
parenrightBigg
a167f(x) = 0
parenleftBigg
a8n+ 1n4 lessorequalslantx<n+ 1
parenrightBigg
integraldisplay +∞
1
f2(x)dx = 114 ·14 + 124 ·24 +···+ 1n4 ·n4 +··· = 1 + 1 +···+ 1 +··· =
∞summationdisplay
n=1
1a117a209a182
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
x3/2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay +∞
1
1
x3/2 dxa194a241a167a133
integraldisplay +∞
1
1
x3 dxa194a241
8,a63a216a101a15a200a169a27a253a233a194a241a53a57a94a135a194a241a53a181
(1)
integraldisplay +∞
0
√xcosx
x+ 100 dx
(2)
integraldisplay +∞
1
cosx
xλ dx,
integraldisplay +∞
1
sinx
xλ dx
(3)
integraldisplay +∞
a
Pm(x)
Qn(x) sinxdx,Pm(x),Qn(x)a136a143m,na103a245a145a170a133a8xgreaterorequalslantaa158a167Qn(x) negationslash= 0
(4)
integraldisplay +∞
2
lnlnx
lnx sinxdx
a41a181
(1) a233A> 0a167a100a117
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
0
cos dx
vextendsinglevextendsingle
vextendsinglevextendsingle= |sinA?sin0|lessorequalslant 1
a113 lim
x→+∞
√x
x+ 100 = 0,
parenleftBigg √
x
x+ 100
parenrightBiggprime
= 100?x2√x(x+ 100)2 a167a8x> 100a158a167
parenleftBigg √
x
x+ 100
parenrightBiggprime
< 0a167a75
√x
x+ 100a252
a78a126a8
a117a180a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
0
√xcosx
x+ 100 dxa194a241.
a2a167a216a143a253a233a194a241.
a100a117
√x|cosx|
x+ 100 greaterorequalslant
√xcos2x
x+ 100 =
1
2
parenleftBigg √
x
x+ 100 +
√xcos2x
x+ 100
parenrightBigg
a207 lim
x→+∞
parenleftBigg
x12 ·
√x
x+ 100
parenrightBigg
= 1a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
√x
x+ 100 dxa117a209
a113
integraldisplay 1
0
√x
x+ 100 dxa143a20a126a200a169a167a75
integraldisplay +∞
0
√x
x+ 100 dx =
integraldisplay 1
0
√x
x+ 100 dx+
integraldisplay +∞
1
√x
x+ 100 dxa117a209
a157a99a140a227a27a121a178a167a140a127
integraldisplay +∞
0
√xcos2x
x+ 100 dxa194a241a167a108a13a200a169
integraldisplay +∞
0
√xcos2x
x+ 100 dxa117a209
a75
integraldisplay +∞
0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
√xcosx
x+ 100
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay +∞
0
√x|cosx|
x+ 100 dxa117a209a167a108a13a200a169
integraldisplay +∞
0
√xcosx
x+ 100 dxa94a135a194a241.
(2) (i) a8λ> 1a158a167a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosx
xλ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
|cosx|
xλ lessorequalslant
1
xλa133a8λ> 1a158a167
integraldisplay +∞
1
dx
xλa194a241a167a108a13
integraldisplay +∞
1
cosx
xλ dxa253a233a194a241
a211a110
integraldisplay +∞
1
sinx
xλ dxa253a233a194a241
(ii) a80 <λlessorequalslant 1a158
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
cosxdx
vextendsinglevextendsingle
vextendsinglevextendsingle= |sinA?sin1|lessorequalslant 2a133 1
xλa80 <λlessorequalslant 1a158a252a78a126a8a167a8x→ +∞a158a170a1170a167
164
a75a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
1
cosx
xλ dxa194a241
a2|cosx|xλ greaterorequalslant cos
2x
xλ =
1
2
parenleftBigg
1
xλ +
cos2x
xλ
parenrightBigg
a167a100a99a161a121a178a167a140a127
integraldisplay +∞
1
cos2x
xλ dxa194a241
a113
integraldisplay +∞
1
dx
xλ (0 <λlessorequalslant 1)a117a209a167a75
integraldisplay +∞
1
|cosx|
xλ dxa117a209a167a108a13
integraldisplay +∞
1
cosx
xλ dxa94a135a194a241
a211a110a167
integraldisplay +∞
1
sinx
xλ dxa94a135a194a241
(iii) a8λlessorequalslant 0a158
a207n→ +∞,2npi → +∞a167a117a180a233a63a191A> 0a167a150a8a140a177a233a20(2n+ 1)pi> 2npi>A
a18ε0 = 2a167a8(2n+1)pi> 2npi>Aa158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay (2n+1)pi
2npi
sinx
xλ dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
integraldisplay (2n+1)pi
2npi
sinx
xλ dxgreaterorequalslant
integraldisplay (2n+1)pi
2npi
sinxdx =
2 = ε0
a75a8λlessorequalslant 0a158a167
integraldisplay +∞
1
sinx
xλ dxa117a209
a211a110a167
integraldisplay +∞
1
cosx
xλ dxa117a209.
a110a220a127a167λ> 1a158a167
integraldisplay +∞
1
cosx
xλ dxa167
integraldisplay +∞
1
sinx
xλ dxa253a233a194a241a182
0 <λlessorequalslant 1a158a167
integraldisplay +∞
1
cosx
xλ dxa167
integraldisplay +∞
1
sinx
xλ dxa94a135a194a241a182
λ< 0a158a167
integraldisplay +∞
1
cosx
xλ dxa167
integraldisplay +∞
1
sinx
xλ dxa117a209.
(3) (i) a23m<n.a100a158a167a253a169a170Pm(x)Q
n(x)
a8xa118a10a140a158a167a145x→ +∞a13a252a78a101a252a170a1170
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
a
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 2(a233?A>a)a167a75a226a41a225a142a52a7a79a123a167a6a200a169a194a241
(ii) a23Qn(x) ≡ 1.a100a158a245a145a170a143Pm(x) = amxm +···+a0a167a216a148a23am > 0
a100a117 lim
x→+∞
Pm(x)
xm = am > 0a167a25a127a51bpi+pi> 0a167a166a8x>bpi+pia158a167Pm(x) =
am
2 x
m
a117a180a107
integraldisplay +∞
a
Pm(x)sinxdx =
integraldisplay bpi+pi
a
Pm(x)sinxdx+
∞summationdisplay
n=b+1
Ina167a217a165In =
integraldisplay (n+1)pi
npi
Pm(x)sinxdx
a100a158a107|In| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay (n+1)pi
npi
Pm(x)sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay pi
0
Pm(npi+z)(?1)nsinzdz
vextendsinglevextendsingle
vextendsinglevextendsinglegreaterorequalslant am
2 (npi)
m
integraldisplay pi
0
sinzdz =
am(npi)ma167a75In →∞(n→∞)
a113
integraldisplay bpi+pi
a
Pm(x)sinxdxa143a20a126a200a169a167a75a55a194a241a167a117a180
integraldisplay +∞
a
Pm(x)sinxdxa117a209
(iii) a8mgreaterorequalslantna158a167Pm(x)Q
n(x)
= R(x) +S(x)a167a217a165R(x)a143a253a169a170a167S(x)a143a18a170
a100(ii)a127a167
integraldisplay +∞
a
S(x)sinxdxa117a209a182a100(i)a127a167
integraldisplay +∞
a
R(x)sinxdxa194a241a167a25
integraldisplay +∞
a
Pm(x)
Qn(x) sinxdxa117
a209
(iv) a23Qn(x) = bnxn +···+b1x+b0
a100a117 lim
x→+∞
vextendsinglevextendsingle
vextendsinglePn(x)Qm(x) sinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingleam
bn x
m?nsinx
vextendsinglevextendsingle
vextendsingle
= 1a167a75a1008(2)a127a167a8λ = n?m> 1a158a167a200a169a253a233a194a241
a110a220a127a181mgreaterorequalslantna158a167a200a169a117a209a182m = n?1a158a167a200a169a94a135a194a241a182m<n?1a158a167a200a169a253a233a194a241.
(4) a233A> 2a167
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
2
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 2a167 lim
x→+∞
lnlnx
lnx = limx→+∞
1
lnx ·
1
x
1
x
= lim
x→+∞
1
lnx = 0a167parenleftBigg
lnlnx
lnx
parenrightBiggprime
= 1?lnlnxx(lnx)2 a167a8x>eea158a167
parenleftBigg
lnlnx
lnx
parenrightBiggprime
< 0a167a100a158a100a188a234a252a78a126a170a1170
a75a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
ee
lnlnx
lnx sinxdxa194a241
a113
integraldisplay ee
2
lnlnx
lnx sinxdxa143a20a126a200a169a167a75a55a194a241a167a117a180
integraldisplay +∞
2
lnlnx
lnx sinxdxa194a241
165
a113
integraldisplay +∞
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lnlnx
lnx sinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx =
integraldisplay n0pi
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lnlnx
lnx sinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx+
∞summationdisplay
n=n0
Ina167a217a165n0 > e
e
pia143a20a18a234
In =
integraldisplay (n+1)pi
npi
lnlnx
lnx |sinx|dx =
integraldisplay pi
0
lnln(npi+z)
ln(npi+z) sinzdz greaterorequalslant
lnln(n+ 1)pi
ln(n+ 1)pi
integraldisplay pi
0
sinzdz = 2 lnln(n+ 1)piln(n+ 1)pi
a207
integraldisplay +∞
ee+pi
lnlnx
lnx dx>
integraldisplay +∞
ee+pi
lnlnx
x dx = lnx(lnlnx?1)
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
ee+pi
= +∞a167a75a100a133a220a7a79a123a167a26
∞summationdisplay
n=n0
lnln(n+ 1)pi
ln(n+ 1)pi a117
a209a167a117a180
+∞summationdisplay
n=n0
Ina117a209a167a108a13a6a200a169a94a135a194a241.
166
§2,a195a46a188a234a27a50a194a200a169
1,a101a15a200a169a180a196a194a241a186a88a74a194a241a167a166a217a138.
(1)
integraldisplay 1
2
0
cotxdx
(2)
integraldisplay 1
0
lnxdx
a41a181
(1) a207 lim
x→+0
cotx = ∞a167a75x = 0a143cotxa27a219a58
a113
integraldisplay 1
2
0+η
cotxdx = ln|sinx|
vextendsinglevextendsingle
vextendsinglevextendsingle
1
2
η
= ln
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
1
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle?ln|sinη|→ +∞(η → +0)a167a75a200a169
integraldisplay 1
2
0
cotxdxa117a209.
(2) a207 lim
x→+0
lnx = ∞a167a75x = 0a143lnxa27a219a58
a113
integraldisplay 1
0+η
lnxdx = x(lnx?1)
vextendsinglevextendsingle
vextendsinglevextendsingle
1
η
=?ηlnη?1 +η →?1(η → +0)a167a75a200a169
integraldisplay 1
0
lnxdxa194a241a117?1.
2,a63a216a101a15a200a169a27a194a241a53a181
(1)
integraldisplay 1
0
sinx
x32
dx
(2)
integraldisplay 1
0
dx
3radicalbigx2(1?x)
(3)
integraldisplay 1
0
lnx
1?x2 dx
(4)
integraldisplay pi
2
0
dx
sin2x·cos2x
(5)
integraldisplay 1
0
|lnx|pdx
(6)
integraldisplay pi
2
0
1?cosx
xm dx
(7)
integraldisplay 1
0
xa?1(1?x)b?1 dx
(8)
integraldisplay 1
0
xa?1(1?x)b?1 lnxdx
a41a181
(1) x = 0a143 sinx
x32
a27a219a58
a207 lim
x→+0
x12 · sinx
x32
= lim
x→+0
sinx
x = 1a167a75a226a133a220a7a79a123a167a26
integraldisplay 1
0
sinx
x32
dxa253a233a194a241.
(2) x = 0a167x = 1a254a143a26a200a188a234a27a219a58a167
integraldisplay 1
0
dx
3radicalbigx2(1?x) =
integraldisplay 1
2
0
dx
3radicalbigx2(1?x) +
integraldisplay 1
1
2
dx
3radicalbigx2(1?x)
a207 lim
x→+0
x23 · 13radicalbigx2(1?x) = lim
x→+0
1
3√1?x = 1a167a75a226a133a220a7a79a123a167a26
integraldisplay 1
2
0
dx
3radicalbigx2(1?x)a253a233a194a241a182
a113 lim
x→1?0
(1?x)13 dx3radicalbigx2(1?x) = lim
x→1?0
1
3√x2 = 1a167a75a226a133a220a7a79a123a167a26
integraldisplay 1
1
2
dx
3radicalbigx2(1?x)a253a233a194a241
a108a13
integraldisplay 1
0
dx
3radicalbigx2(1?x)a253a233a194a241.
167
(3) a207lim
x→1
lnx
1?x2 = limx→1
1
x
2x =? limx→1
1
2x2 =?
1
2 a167a75x = 1a216a180a219a58a167a117a180a100a200a169a144a107a152a135a219a580
a113 lim
x→+0
x12 · lnx1?x2 = lim
x→+0
x12 lnx = lim
x→+0
lnx
x?12
= lim
x→∞
1
x
12x?32
=?2 lim
x→+0
x12 = 0
a75a100a133a220a7a79a123a167a26
integraldisplay 1
0
lnx
1?x2 dxa194a241.
(4) x = 0a167x = pi2a254a143a26a200a188a234a27a219a58a167a75
integraldisplay pi
2
0
dx
sin2x·cos2x =
integraldisplay pi
4
0
dx
sin2x·cos2x +
integraldisplay pi
2
pi
4
dx
sin2x·cos2x
a207 lim
x→+0
x2· 1sin2x·cos2x = 1a167a133 1sin2x·cos2xgreaterorequalslant 0a167a75a226a133a220a7a79a123a167a26
integraldisplay pi
4
0
dx
sin2x·cos2xa117a209a150+∞
a113 1sin2x·cos2xgreaterorequalslant 0a167a75
integraldisplay pi
2
0
dx
sin2x·cos2xa117a209.
(5)
integraldisplay 1
0
|lnx|pdx =
integraldisplay 1
2
0
|lnx|pdx+
integraldisplay 1
1
2
|lnx|pdx
a233
integraldisplay 1
2
0
|lnx|pdxa167a8p> 0a158a1670a143a219a58
a207 lim
x→+0
x12|lnx|p = lim
x→+0
|lnx|p
x?12
= lim
x→+0
(?lnx)p
x?12
= (?1)p lim
x→+0
plnp?1x· 1x
12x?32
= (?1)p+12p lim
x→+0
lnp?1x
x?12
= 0(a132a214a254a254231a1441.(16))
a75a100a133a220a7a79a123a167a26
integraldisplay 1
2
0
|lnx|pdxa8p> 0a158a194a241
a8plessorequalslant 0a158a167
integraldisplay 1
2
0
|lnx|pdxa143a20a126a200a169a167a75
integraldisplay 1
2
0
|lnx|pdxa167a117a180pa143a63a219a138a158a167
integraldisplay 1
2
0
|lnx|pdxa254a194a241.
a233
integraldisplay 1
1
2
|lnx|pdxa167p< 0a158a1671a143a219a58
a207 lim
x→1?0
(1?x)?p|lnx|p = lim
x→1?0
|lnx|p
(1?x)p = limx→1?0
ln 1x
1?x
p
=
parenleftBigg
lim
x→1?0
1
x
parenrightBiggp
= 1
a75a226a133a220a7a79a123a167a26a8?p< 1a610 >p>?1a158a167
integraldisplay 1
1
2
|lnx|pdxa194a241a182
a8?pgreaterorequalslant 1a61plessorequalslant?1a158a167
integraldisplay 1
1
2
|lnx|pdxa117a209
a8pgreaterorequalslant 0a158a167
integraldisplay 1
1
2
|lnx|pdxa143a20a126a200a169a167a25a194a241
a117a180a8p>?1a158a167
integraldisplay 1
1
2
|lnx|pdxa194a241a182a8plessorequalslant?1a158a167
integraldisplay 1
1
2
|lnx|pdxa117a209
a110a220a127a167a8p>?1a158a167
integraldisplay 1
0
|lnx|pdxa194a241a182a8plessorequalslant?1a158a167
integraldisplay 1
0
|lnx|pdxa117a209.
(6) a207 lim
x→+0
1?cosx
xm =
0,mlessorequalslant 0
lim
x→+0
sinx
mxm?1 =
0,0 <m< 1
lim
x→+0
cosx
m(m?1)xm?2 =
0,1 <m< 2
1
2,m = 2∞,m> 2
a75 lim
x→+0
1?cosx
xm =
0,m< 2
1
2,m = 2∞,m> 2
a108a13a8mlessorequalslant 2a158a167
integraldisplay pi
2
0
1?cosx
xm dxa143a20a126a200a169a167a25a194a241
a8m> 2a158a167x = 0a143
integraldisplay pi
2
0
1?cosx
xm dxa27a219a58
a113 lim
x→+0
xm?2 1?cosxxm = lim
x→+0
1?cosx
x2 =
1
2
168
a75a80 <m?2 < 1a612 <m< 3a158a167a200a169a194a241a182a8m?2 greaterorequalslant 1a61mgreaterorequalslant 3a158a167a200a169a117a209
a108a13a8m< 3a158a167
integraldisplay pi
2
0
1?cosx
xm dxa194a241a182a8mgreaterorequalslant 3a158a167
integraldisplay pi
2
0
1?cosx
xm dxa117a209.
(7) a8agreaterorequalslant 1a133bgreaterorequalslant 1a158a167
integraldisplay 1
0
xa?1(1?x)b?1 dxa143a20a126a200a169a167a25a194a241
integraldisplay 1
0
xa?1(1?x)b?1 dx =
integraldisplay 1
2
0
xa?1(1?x)b?1 dx+
integraldisplay 1
1
2
xa?1(1?x)b?1 dx
a233a200a169
integraldisplay 1
2
0
xa?1(1?x)b?1 dx =
integraldisplay 1
2
0
(1?x)b?1
x1?a dx
a207 lim
x→+0
(1?x)b?1
x1?a =
0,a> 1
1,a = 1
∞,a< 1
a133 lim
x→+0
x1?a(1?x)
b?1
x1?a = limx→+0(1?x)
b?1 = 1
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a81?a< 1a61a> 0a158a200a169a194a241a182a81?agreaterorequalslant 1a61alessorequalslant 0a158a167a200a169a117a209a182
a233a200a169
integraldisplay 1
1
2
xa?1(1?x)b?1 dx =
integraldisplay 1
1
2
xa?1
(1?x)1?b dx
a207 lim
x→1?0
xa?1
(1?x)1?b =
0,b> 1
1,b = 1
∞,b< 1
a133 lim
x→1?0
(1?x)1?b x
a?1
(1?x)1?b = limx→1?0x
a?1 = 1
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a81?b< 1a61b> 0a158a200a169a194a241a182a81?bgreaterorequalslant 1a61blessorequalslant 0a158a167a200a169a117a209a182
a110a254a164a227a167a8a> 0a133b> 0a158a167
integraldisplay 1
0
xa?1(1?x)b?1 dxa194a241a167a217a123a156a47a200a169a254a117a209.
(8)
integraldisplay 1
0
xa?1(1?x)b?1 lnxdx =
integraldisplay 1
2
0
xa?1(1?x)b?1 lnxdx+
integraldisplay 1
1
2
xa?1(1?x)b?1 lnxdx
a233a200a169
integraldisplay 1
2
0
xa?1(1?x)b?1 lnxdx =
integraldisplay 1
2
0
(1?x)b?1 lnx
x1?a dx
a207 lim
x→+0
(1?x)b?1 lnx
x1?a =
braceleftbigg 0,a> 1
∞,alessorequalslant 1
a133a233?c> 0a167 lim
x→+0
x1?a+c(1?x)
b?1
x1?a |lnx| = limx→+0
lnx
x?c = limx→+0
1x
cx?c?1 = limx→+0
xc
c = 0
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a81?a+c< 1a61a>c> 0a158a194a241
a113 lim
x→+0
x1?a(1?x)
b?1
x1?a |lnx| =? limx→+0(1?x)
b?1 lnx = ∞
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a81?agreaterorequalslant 1a61alessorequalslant 0a158a117a209
a233a200a169
integraldisplay 1
1
2
xa?1(1?x)b?1 lnxdx =
integraldisplay 1
1
2
xa?1 lnx
(1?x)1?b dx
a207 lim
x→1?0
xa?1 lnx
(1?x)1?b =
0,b> 0
1,b = 0
∞,b< 0
a133 lim
x→1?0
(1?x)?b x
a?1
(1?x)1?b|lnx| = limx→1?0
lnx
1?x = limx→1?0
1
x = 1
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a8?b< 1a61b>?1a158a194a241a182a8?bgreaterorequalslant 1a61blessorequalslant?1a158a117a209
a110a254a164a227a167a26a8a> 0a133b>?1a158a167a200a169
integraldisplay 1
0
xa?1(1?x)b?1 lnxdxa194a241a167a217a123a156a47a200a169a254a117a209.
3,a121a178a195a46a188a234a50a194a200a169a27a133a220a7a79a123a57a217a52a129a47a170.
a121a178a181
(1) a133a220a7a79a123a181
(i) a100|f(x)|lessorequalslant C(x?a)p(C > 0),p< 1a167a127 lim
ε→+0
integraldisplay b
a+ε
|f(x)|dxlessorequalslant lim
ε→+0
integraldisplay b
a+ε
C
(x?a)p dx =
lim
ε→+0
C
1?p(1?a)
1?p
vextendsinglevextendsingle
vextendsinglevextendsingle
b
a+ε
= lim
ε→+0
bracketleftBigg
C
1?p(b?a)
1?p? C
1?p ε
1?p
bracketrightBigg
= C1?p(b?a)1?p
a61 lim
ε→+0
integraldisplay b
a+ε
|f(x)|dxa127a51a167a25
integraldisplay b
a
f(x)dxa253a233a194a241
(ii) a207a107
integraldisplay b
a+ε
|f(x)|dx greaterorequalslant
integraldisplay b
a+ε
C
(x?a)p dx =
bracketleftBigg
C
1?p(b?a)
1?p? C
1?p ε
1?p
bracketrightBigg
→ ∞(a8p > 1,C >
169
0a133ε→ 0a158)
a113a8p = 1a158a167
integraldisplay b
a
C
x?a dxa117a209a167a108a13
integraldisplay b
a
|f(x)|dxa117a209.
(2) a133a220a7a79a123a27a52a129a47a170a181
(i) a23lim
x→a
(x?a)p|f(x)| = k(0 <k<∞)
a75a233?k>ε> 0a167a127a51δ> 0a167a166a8a<x<a+δa158a167a1070 <k?ε< (x?a)p|f(x)|<k+ε
a61a107 k?ε(x?a)p <|f(x)|< k+ε(x?a)p
a117a180
integraldisplay b
a
dx
(x?a)pa134
integraldisplay b
a
|f(x)dxa211a158a194a241a189a117a209a163a56a40a143a133a220a7a79a123a164
a108a13a8p< 1a158a167
integraldisplay b
a
f(x)dxa253a233a194a241a182pgreaterorequalslant 1a158a167f(x)a107a189a210a167a75
integraldisplay b
a
f(x)dxa117a209
(ii) k = 0a158a167a18ε0 = 1a167a75?δ> 0a167a166a8a<x<a+δa158a167
|(x?a)pf(x)| = (x?a)p|f(x)|< 1a61|f(x)|< 1(x?a)pa167
a75a100a133a220a7a79a123a167a26p< 1a158a167
integraldisplay b
a
f(x)dxa253a233a194a241
(iii) k = ∞a158a167a18G = 1a167a75?δ> 0a167a166a8a<x<a+δa158a167a107|(x?a)pf(x)| = (x?a)p|f(x)|> 1
a61|f(x)|> 1(x?a)p
a75a100a133a220a7a79a123a167a26a8pgreaterorequalslant 1a158a167
integraldisplay b
a
|f(x)|dxa117a209a182a113f(x)a107a189a210a167a108a13
integraldisplay b
a
f(x)dxa117a209.
a110a254a167a26a1010 lessorequalslant k < +∞,p < 1a167a64a34
integraldisplay b
a
f(x)dxa253a233a194a241a182a1010 < k lessorequalslant +∞,p greaterorequalslant 1a167a64
a34
integraldisplay b
a
f(x)dxa117a209.
4,a63a216a101a15a200a169a27a194a241a53a181
(1)
integraldisplay +∞
0
dx
3radicalbig(x?1)2x(x?2)
(2)
integraldisplay +∞
0
ln(1 +x)
xα dx
(3)
integraldisplay +∞
0
dx
xp +xq
(4)
integraldisplay +∞
0
arctanx
xα dx
(5)
integraldisplay +∞
1
dx
xplnqx
(6)
integraldisplay +∞
∞
dx
|x?a1|p1|x?a2|p2 ···|x?an|pn
a41a181
(1) x = 0,1,2a254a143a26a200a188a234a27a219a58integraldisplay
+∞
0
dx
3radicalbig(x?1)2x(x?2) =
parenleftBiggintegraldisplay 1
2
0
+
integraldisplay 1
1
2
+
integraldisplay 3
2
1
+
integraldisplay 2
3
2
+
integraldisplay 3
2
+
integraldisplay +∞
3
parenrightBigg
dx
3radicalbig(x?1)2x(x?2)
a233a200a169
integraldisplay 1
2
0
dx
3radicalbig(x?1)2x(x?2)
a207 lim
x→+0
x23
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3radicalbig(x?1)2x(x?2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limx→+0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
(x?1)2(x?2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3
= 0
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a200a169
integraldisplay 1
2
0
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a233a200a169
integraldisplay 1
1
2
dx
3radicalbig(x?1)2x(x?2)
170
a207 lim
x→1?0
(1?x)56
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3radicalbig(x?1)2x(x?2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limx→1?0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1?x)16
[x(x?2)]13
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 0
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a200a169
integraldisplay 1
1
2
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a211a100a121a123a167a26a200a169
integraldisplay 3
2
1
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a233a200a169
integraldisplay 2
3
2
dx
3radicalbig(x?1)2x(x?2)
a207 lim
x→2?0
(2?x)23
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3radicalbig(x?1)2x(x?2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limx→2?0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
2?x
x(x?1)2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
3
= 0
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a200a169
integraldisplay 2
3
2
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a211a100a121a123a167a26
integraldisplay 3
2
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a207xgreaterorequalslant 3a158a167 13radicalbig(x?1)2x(x?2) lessorequalslant 1
(x?2)43
a133
integraldisplay +∞
3
dx
(x?2)43
a253a233a194a241
a75a100a39a22a7a79a123a167a26
integraldisplay +∞
3
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
a110a254a140a127a167
integraldisplay +∞
0
dx
3radicalbig(x?1)2x(x?2)a253a233a194a241
(2)
integraldisplay +∞
0
ln(1 +x)
xα dx =
integraldisplay 1
0
ln(1 +x)
xα dx+
integraldisplay +∞
1
ln(1 +x)
xα dx
a233
integraldisplay 1
0
ln(1 +x)
xα dx
a207 lim
x→+0
ln(1 +x)
xα =
0,αlessorequalslant 0
lim
x→+0
1
1+x
αxα?1 =
0,0 <α< 1
1,α = 1
∞,α> 1
a167a75a8α> 1a158a1670a143a219a58
a113 lim
x→+0
xα?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ln(1 +x)
xα
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limx→+0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ln(1 +x)
x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 1
a75a8α? 1 < 1a611 < α < 2a158a167
integraldisplay 1
0
ln(1 +x)
xα dxa253a233a194a241a182a8α greaterorequalslant 2a158a167
integraldisplay 1
0
ln(1 +x)
xα dxa117a209a182
a8αlessorequalslant 1a158a167
integraldisplay 1
0
ln(1 +x)
xα dxa143a20a126a200a169a167a25a55a194a241
a108a13a8α< 2a158a167
integraldisplay 1
0
ln(1 +x)
xα dxa253a233a194a241a182a8αgreaterorequalslant 2a158a167
integraldisplay 1
0
ln(1 +x)xα dxa117a209
a233
integraldisplay +∞
1
ln(1 +x)
xα dx
a18λ> 1a167a8α?λ> 0a158a167a207 lim
x→+∞
xλ ln(1 +x)xα = lim
x→+∞
1
1+x
(α?λ)xα?λ?1 = 0
a75a8α?λ> 0a61α>λ> 1a158a167a200a169
integraldisplay +∞
1
ln(1 +x)
xα dxa253a233a194a241a182
a113 lim
x→+∞
xα ln(1 +x)xα = +∞a167a75a8αlessorequalslant 1a158a167a200a169
integraldisplay +∞
1
ln(1 +x)
xα dxa117a209
a108a13a81 <α< 2a158a167
integraldisplay +∞
0
ln(1 +x)
xα dxa253a233a194a241a182a217a167a156a47a167
integraldisplay +∞
0
ln(1 +x)
xα dxa209a117a209.
(3)
integraldisplay +∞
0
dx
xp +xq =
integraldisplay 1
0
dx
xp +xq +
integraldisplay +∞
1
dx
xp +xq
a233a200a169
integraldisplay 1
0
dx
xp +xqa167a23min(p,q) = p
a101plessorequalslant 0a167a75
integraldisplay 1
0
dx
xp +xqa143a20a126a200a169a167a25
integraldisplay 1
0
dx
xp +xqa194a241
171
a101p> 0a167a100a117 lim
x→+0
xp 1xp +xq = lim
x→+0
1
1 +xq?p =
1
2,p = q1,pnegationslash= q
a25a200a169
integraldisplay 1
0
dx
xp +xqa61a8p< 1a61min(p,q) < 1a158a194a241
a233a200a169
integraldisplay +∞
1
dx
xp +xqa167a23max(p,q) = q
a100a117 lim
x→+∞
xq 1xp +xq = lim
x→+∞
1
x?(q?p) + 1 =
1
2,p = q1,pnegationslash= q
a25a200a169
integraldisplay +∞
1
dx
xp +xqa61a8q> 1a61max(p,q) > 1a158a194a241
a75a200a169
integraldisplay +∞
0
dx
xp +xqa8min(p,q) < 1a133max(p,q) > 1a158a194a241.
(4) a80 <αlessorequalslant 1a158a167 lim
x→+0
arctanx
xα = limx→+0
1
1+x2
αxα?1 = limx→+0
x1?α
α(1 +x2) = 0
a8αlessorequalslant 0a167 lim
x→+0
arctanx
xα = limx→+0x
αarctanx = 0
a75αlessorequalslant 1a158a1670a216a143a219a58
a113 lim
x→+∞
xα arctanxxα = lim
x→+∞
arctanx = pi2
a75αlessorequalslant 1a158a167a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
0
arctanx
xα dxa117a209
a8α> 1a158a167a207 lim
x→+0
arctanx
xα = limx→+0
1
1+x2
αxα?1 = limx→+0
1
αxα?1(1 +x2) = +∞a167a750a143a219a58
a75
integraldisplay +∞
0
arctanx
xα dx =
integraldisplay 1
0
arctanx
xα dx+
integraldisplay +∞
1
arctanx
xα dx
a233
integraldisplay 1
0
arctanx
xα dxa167α> 1a1670a143a219a58
a207 lim
x→+0
xα?1 arctanxxα = lim
x→+0
arctanx
x = 1
a75a100a133a220a7a79a123a27a52a129a47a170a167a26α?1 < 1a61α< 2a158a200a169a194a241a182a8αgreaterorequalslant 2a158a167a200a169a117a209
a233
integraldisplay +∞
1
arctanx
xα dx
a207
pi
4
xα lessorequalslant
arctanx
xα lessorequalslant
pi
2
xαa133
integraldisplay +∞
1
pi
2
xα dxa8α> 1a158a200a169a194a241a182
integraldisplay +∞
1
pi
4
xα dxa8αlessorequalslant 1a158a200a169a117a209
a75a100a39a22a7a79a123a167a26a8α> 1a158a167
integraldisplay +∞
1
arctanx
xα dxa194a241a182a8αlessorequalslant 1a158a167
integraldisplay +∞
1
arctanx
xα dxa117a209
a111a131a167a81 <α< 2a158a167
integraldisplay +∞
0
arctanx
xα dxa194a241a182a217a123a156a47a100a200a169a254a117a209.
(5)
integraldisplay +∞
1
dx
xplnqx =
integraldisplay 2
1
dx
xplnqx +
integraldisplay +∞
2
dx
xplnqx
a127a196
integraldisplay 2
1
dx
xplnqxa167a233a63a191a27p
a100a117 lim
x→1+0
bracketleftBigg
(x?1)q 1xplnqx
bracketrightBigg
= lim
x→1+0
1
xp ·
(x?1)q
lnqx = limx→1+0
(x?1)q
lnqx =
parenleftBigg
lim
x→1+0
x?1
lnx
parenrightBiggq
=
parenleftBigg
lim
x→1+0
1
1
x
parenrightBiggq
= 1
a75a200a169
integraldisplay 2
1
dx
xplnqxa61a8q< 1a133pa143a63a191a138a158a194a241
a127a196
integraldisplay +∞
2
dx
xplnqxa167a101p> 1a167a18α> 0a191a169a2a167a166p?α> 1a167a75a233a63a191qa167
a100a117 lim
x→+∞
parenleftBigg
xp?α 1xplnqx
parenrightBigg
= lim
x→+∞
1
xαlnqx = 0
a117a180a200a169
integraldisplay +∞
2
dx
xplnqxa8p> 1a133qa143a63a191a138a158a194a241a182
172
a101plessorequalslant 1,q< 1a167a100a117
integraldisplay +∞
2
dx
xplnqxgreaterorequalslant
integraldisplay +∞
2
dx
xlnqx =
1
1?q(lnx)
1?q
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
2
= +∞
a75a100a158a200a169
integraldisplay +∞
2
dx
xplnqxa117a209
a108a13a200a169
integraldisplay +∞
1
dx
xplnqxa8p> 1a133q< 1a158a194a241.
(6) a196a107a167a26a200a188a234a39a1171xa180
nsummationdisplay
i=1
pia63a195a161a2(a8x→±∞a158)
a217a103(a216a148a23a143inegationslash= ja158a167ai negationslash= aj)
a207 lim
x→ai
bracketleftBigg
|x?ai|pi 1|x?a
1|p1|x?a2|p2 ···|x?an|pn
bracketrightBigg
= ci,0 <ci < +∞(i = 1,2,···,n)
a25a200a169
integraldisplay +∞
∞
dx
|x?a1|p1|x?a2|p2 ···|x?an|pna61a8
nsummationdisplay
i=1
pi > 1a133pi < 1(i = 1,2,···,n)a158a194a241.
5,a23f(x)a8x→ +0a158a252a78a170a149a117+∞a167a193a121a178a181a101
integraldisplay 1
0
f(x)dxa194a241a167a55a76lim
x→0
xf(x)dx = 0.
a121a178a181 a100a75a23a1270a180f(x)a27a219a58a167a61
integraldisplay 1
0
f(x)dxa180a195a46a188a234a27a50a194a200a169a167a133a8xa191a169a130a670a158a167f(x) greaterorequalslant 0a167
a51[0,x]a254a252a78a126
a113
integraldisplay 1
0
f(x)dxa194a241a167a75a100a133a220a194a241a6a110a167a233?ε> 0,?δ> 0a167a80 < x2 <x<δa158a167
a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay x
x
2
f(x)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
integraldisplay x
x
2
f(x)dx< ε2
a100a49a152a200a169a165a138a189a110a167a26
integraldisplay x
x
2
f(x)dx = f(ξ)
parenleftBigg
x? x2
parenrightBigg
= x2f(ξ) > x2f(x)
parenleftBigg
x
2 <ξ<x
parenrightBigg
a117a180 x2f(x) <
integraldisplay x
x
2
f(x)dx< ε2 a610 lessorequalslantxf(x) <εa167a108a13lim
x→0
xf(x)dx = 0.
6,a63a216a101a15a200a169a27a253a233a194a241a218a94a135a194a241a53a181
(1)
integraldisplay +∞
0
xpsinx
1 +xq dx(qgreaterorequalslant 0)
(2)
integraldisplay +∞
0
esinxsin2x
xλ dx(λ> 0)
(3)
integraldisplay +∞
0
sinparenleftbigx+ 1xparenrightbig
xn dx
a41a181
(1)
integraldisplay +∞
0
xpsinx
1 +xq dx =
integraldisplay 1
0
xpsinx
1 +xq dx+
integraldisplay +∞
1
xpsinx
1 +xq dx
a233
integraldisplay 1
0
xpsinx
1 +xq dx
a207 lim
x→+0
x?(p+1)x
psinx
1 +xq = limx→+0
sinx
x
1 +xq = 1
a75a8?(p+ 1) < 1a61p>?2a158a167
integraldisplay 1
0
xpsinx
1 +xq dxa253a233a194a241a182a8plessorequalslant?2a158a167a200a169
integraldisplay 1
0
xpsinx
1 +xq dxa117a209
a233
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
xpsinx
1 +xq
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
xq sinx
1 +xq
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
xp
1 +xqa13 limx→+∞x
q?p x
p
1 +xq = limx→+∞
xq
1 +xq = 1
a75a8q?p> 1a61q>p+ 1a158
integraldisplay +∞
1
xp
1 +xq dxa194a241a167a117a180a100a39a22a7a79a123a167a26
integraldisplay +∞
1
xpsinx
1 +xq dxa253a233a194a241
a113 2x
q
1 +xq greaterorequalslant 1,
2xp
1 +xq greaterorequalslant
1
xq?p,
xp|sinx|
1 +xq greaterorequalslant
|sinx|
2xq?pa133a10056a1448.(2)a127a167
integraldisplay +∞
1
sinx
xq?p dxa8qlessorequalslantp+1a158a154a253a233
a194a241
173
a111a131a167a8p>?2,q>p+ 1a158a167
integraldisplay +∞
0
xpsinx
1 +xq dxa253a233a194a241
a127a196
integraldisplay +∞
1
xpsinx
1 +xq dx
a8q>pa158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 2a167 x
p
1 +xqa252a78a126a170a1170(x→ +∞)
a75a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
1
xpsinx
1 +xq dxa194a241
a8qlessorequalslantpa158a167a8q = pa158a167 x
p
1 +xq → 1(x→ +∞)a182a8q<pa158a167
xp
1 +xq → +∞(x→ +∞)
a75a233a191a169a140a27xa167a240a107 x
p
1 +xq greaterorequalslant
1
3
a117a180a233?A> 1a167a55?N ∈Z+a167a166a262Npi+ pi4 >Aa133a8xgreaterorequalslant 2Npi+ pi4a158a167a240a107 x
p
1 +xq greaterorequalslant
1
3
a108a13a233Aprime = 2Npi+ pi4,Aprimeprime = 2Npi+ pi2a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
xp
1 +xq sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
1
3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
√2
6
a75a100a133a220a6a110a167a26
integraldisplay +∞
1
xpsinx
1 +xq dxa117a209
a110a254a164a227a167a8q>p+1 >?1a158a167
integraldisplay +∞
0
xpsinx
1 +xq dxa253a233a194a241a182a8p+1 greaterorequalslantq>p>?2a158a167
integraldisplay +∞
0
xpsinx
1 +xq dxa94
a135a194a241.
(2) a207 lim
x→+0
esinxsin2x
xλ = limx→+0
sin2x
xλ =
0,λlessorequalslant 0
lim
x→+0
2cos2x
λxλ?1 =
0,0 <λ< 1
2,λ = 1
∞,λ> 1
=
0,λ< 1
2,λ = 1
∞,λ> 1
a75a8λ> 1a158a1670a143a219a58integraldisplay
+∞
0
esinxsin2x
xλ dx =
integraldisplay 1
0
esinxsin2x
xλ dx+
integraldisplay +∞
1
esinxsin2x
xλ dx
a233
integraldisplay 1
0
esinxsin2x
xλ dxa167a26a200a188a234a143a20a1670a143a219a58
a207 lim
x→+0
xλ?1e
sinxsin2x
xλ = limx→+0
esinxsin2x
x = 2
a75a100a133a220a7a79a123a27a52a129a47a170a167a26a8λ?1 < 1a61λ< 2a158a167
integraldisplay 1
0
esinxsin2x
xλ dxa253a233a194a241a182
a8λgreaterorequalslant 2a158a167
integraldisplay 1
0
esinxsin2x
xλ dxa117a209
a233
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
esinxsin2x
xλ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dx
a8λ> 1a158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
esinxsin2x
xλ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
e
xλa167a75
integraldisplay +∞
1
esinxsin2x
xλ dxa253a233a194a241
a117a180
integraldisplay +∞
0
esinxsin2x
xλ dxa81 <λ< 2a158a253a233a194a241a182
a8λlessorequalslant 1a158a167e
sinx|sin2x|
xλ >
e?1 sin2 2x
xλ = e
1
parenleftBigg
1?cos4x
2xλ
parenrightBigg
= 12e
parenleftBigg
1
xλ?
cos4x
xλ
parenrightBigg
a207
integraldisplay +∞
1
dx
xλa117a209,
integraldisplay +∞
1
cos4x
xλ dxa194a241a167a75
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
esinxsin2x
xλ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dxa117a209
a233
integraldisplay +∞
1
esinxsin2x
xλ dx
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
esinxsin2xdx
vextendsinglevextendsingle
vextendsinglevextendsingle= 2
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
esinxsinxdsinx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 4e
a113 1xλa252a78a126a170a1170a167a75a226a41a225a142a52a7a79a123a167a26a233λ> 0a107
integraldisplay +∞
1
esinxsin2x
xλ dxa194a241
a117a180
integraldisplay +∞
0
esinxsin2x
xλ dxa80 <λlessorequalslant 1a158a94a135a194a241
174
a108a13a81 <λ< 2a158
integraldisplay +∞
0
esinxsin2x
xλ dxa253a233a194a241a182a80 <λlessorequalslant 1a158
integraldisplay +∞
0
esinxsin2x
xλ dxa94a135a194a241.
(3)
integraldisplay +∞
0
sinparenleftbigx+ 1xparenrightbig
xn dx =
integraldisplay 1
0
sinparenleftbigx+ 1xparenrightbig
xn dx+
integraldisplay +∞
0
sinparenleftbigx+ 1xparenrightbig
xn dx = I1 +I2
a233I1a167a45x = 1t,dx =? dtt2 a167a75I1 =
integraldisplay 1
0
sinparenleftbigx+ 1xparenrightbig
xn dx =
integraldisplay +∞
1
sinparenleftbigx+ 1xparenrightbig
x2?n dx
a239a196I2
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinparenleftbigx+ 1xparenrightbig
xn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
xna167a75a8n> 1a158a167I2a253a233a194a241
a207
integraldisplay +∞
1
sinparenleftbigx+ 1xparenrightbig
xn dx =
integraldisplay +∞
1
sinparenleftbigx+ 1xparenrightbigparenleftbig1? 1x2parenrightbig
xnparenleftbig1? 1x2parenrightbig dx
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
1
sin
parenleftbigg
x+ 1x
parenrightbiggparenleftbigg
1? 1x2
parenrightbigg
dx
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglecos
parenleftBigg
x+ 1x
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
A
1
lessorequalslant 2
a133
bracketleftBigg
xn
parenleftBigg
1? 1x2
parenrightBiggbracketrightBiggprime
= nxn?1?(n?2)xn?3 = xn?3[nx2?(n?2)]
a75a8n∈ (0,1]a158a167xn
parenleftBigg
1? 1x2
parenrightBigg
a252a78a79a61 1xnparenleftbig1? 1
x2
parenrightbiga252a78a126a170a1170(x→ +∞)
a117a180a100a41a225a142a52a7a79a123a167a26a80 <nlessorequalslant 1a158a167I2a194a241
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinparenleftbigx+ 1xparenrightbig
xn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
sin2parenleftbigx+ 1xparenrightbig
xn =
1
2xn?
cos2parenleftbigx+ 1xparenrightbig
2xn
a80 <nlessorequalslant 1a158a167
integraldisplay +∞
1
dx
2xna117a209a167
integraldisplay +∞
1
cos2parenleftbigx+ 1xparenrightbig
2xn dxa194a241
a75a80 <nlessorequalslant 1a158a167
integraldisplay +∞
1
sin2parenleftbigx+ 1xparenrightbig
xn dxa117a209
a117a180a100a39a22a7a79a123a167a127
integraldisplay +∞
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinparenleftbigx+ 1xparenrightbig
xn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dxa80 <nlessorequalslant 1a158a117a209
a108a13a80 <nlessorequalslant 1a158a167
integraldisplay +∞
1
sinparenleftbigx+ 1xparenrightbig
xn dxa94a135a194a241
a8nlessorequalslant 0a158a167a233?A> 1a167a55?k ∈Z+a167a166a262kpi+ pi6 >Aa133a8xgreaterorequalslant 2kpi+ pi6a158a167
a240a107x?nsin
parenleftBigg
x+ 1x
parenrightBigg
greaterorequalslant sin
parenleftBigg
2kpi+ pi6
parenrightBigg
= 12
a117a180a167a233Aprime = 2kpi+ pi6,Aprimeprime = 2kpi+ pi3a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
sinparenleftbigx+ 1xparenrightbig
xn dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslant
integraldisplay Aprimeprime
Aprime
sin
parenleftBigg
2kpi+ pi6
parenrightBigg
dx = pi12
a75a100a133a220a194a241a6a110a167a26a8nlessorequalslant 0a158a167I2a117a209
a233I1a167a100I2a27a40a216a167a26a82?n > 1a61n < 1a158a253a233a194a241a182a81 greaterorequalslant 2?n > 0a611 lessorequalslant n < 2a158a94a135a194a241a182
a82?nlessorequalslant 0a61nlessorequalslant 2a158a117a209
a111a131a167
integraldisplay +∞
0
sinparenleftbigx+ 1xparenrightbig
xn dxa80 <n< 2a158a94a135a194a241.
7,a23f(x)a252a78a101a252a167 lim
x→+∞
f(x) = 0a167a88a74a19a234f prime(x)a51[0,+∞)a254a235a89a167a64a34a200a169
integraldisplay +∞
0
f prime(x)sin2xdxa194a241.
a121a178a181a207(sin2x)prime = sin2xa167a19a234f prime(x)a51[0,+∞)a254a235a89a167 lim
x→+∞
f(x)sin2x = 0
a75a100a169a220a200a169a250a170a167a26
integraldisplay +∞
0
f prime(x)sin2xdx = f(x)sin2x
vextendsinglevextendsingle
vextendsinglevextendsingle
+∞
0
integraldisplay +∞
0
f(x)sin2xdx =?
integraldisplay +∞
0
f(x)sin2xdx
a233a117
integraldisplay +∞
0
f(x)sin2xdxa167a100a174a127f(x)a252a78a101a252a167 lim
x→+∞
f(x) = 0a57
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
0
sin2xdx
vextendsinglevextendsingle
vextendsinglevextendsingle= 1
2|cos2A?1|lessorequalslant 1
a75a100a41a225a142a52a7a79a123a167a26
integraldisplay +∞
0
f(x)sin2xdxa194a241a167a108a13a200a169
integraldisplay +∞
0
f prime(x)sin2xdxa194a241.
8,a51a195a46a188a234a27a50a194a200a169(a200a169a129a143a107a129)a165a167a121a178a178a144a140a200a152a189a253a233a140a200a167a2a135a131a216a44.
a121a178a181a100a174a127f2(x)a140a200a167a75f
2(x)
2 a143a140a200
a207(|f(x)|?1)2 = f2(x)?2|f(x)|+ 1 greaterorequalslant 0a167a75|f(x)|lessorequalslant f
2(x) + 1
2
175
a117a180a100a39a22a7a79a123a167a26|f(x)|a140a200a61a178a144a140a200a189a253a233a140a200.
a135a131a216a44.
a126a181a10057a144a1261a167a26
integraldisplay 2
1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
(x?1)12
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle dxa194a241a61
integraldisplay 2
1
dx
(x?1)12
a253a233a194a241
a2
integraldisplay 2
1
dx
x?1a117a209a167a61
1
x?1 a51[1,2]a254a216a140a200.
9,a79a142a101a15a200a169a27a133a220a204a138a181
(1)
integraldisplay 3
0
dx
1?x
(2)
integraldisplay +∞
∞
sinxdx
a41a181
(1) P.V.
integraldisplay 3
0
dx
1?x = limη→0
bracketleftBiggintegraldisplay
1?η
0
dx
1?x +
integraldisplay 3
1+η
dx
1?x
bracketrightBigg
= lim
η→0
bracketleftBigg
ln(1?x)
vextendsinglevextendsingle
vextendsinglevextendsingle
1?η
0
ln(x?1)
vextendsinglevextendsingle
vextendsinglevextendsingle
3
1+η
bracketrightBigg
=?ln2
(2) P.V.
integraldisplay +∞
∞
sinxdx = lim
A→+∞
parenleftbiggintegraldisplay A
A
sinxdx
parenrightbigg
= lim
A→+∞
parenleftBigg
cosx
vextendsinglevextendsingle
vextendsinglevextendsingle
A
A
parenrightBigg
= lim
A→+∞
(cos(?A)?cosA) = 0
10,a121a178a50a194a200a169a57a133a220a204a138a131a109a27a39a88a181
(1) a101
integraldisplay +∞
∞
f(x)dxa194a241a167a217a138a143Aa167a75a133a220a204a138P.V.
integraldisplay +∞
∞
f(x)dxa127a51a167a133a31a117Aa167a2a135a131a216a44a182
(2) a101f(x) greaterorequalslant 0a167P.V.
integraldisplay +∞
∞
f(x)dxa127a51a167a217a138a143Aa167a75
integraldisplay +∞
∞
f(x)dxa194a241a167a133a194a241a117A.
a121a178a181
(1) a100
integraldisplay +∞
∞
f(x)dxa194a241a167a127
integraldisplay +∞
∞
f(x)dx =
integraldisplay 0
∞
f(x)dx+
integraldisplay +∞
0
f(x)dxa194a241a167
a75a107 lim
B→?∞
integraldisplay 0
B
f(x)dx+ lim
A→+∞
integraldisplay A
0
f(x)dxa127a51a167a65a79a18B =?Aa167a107 lim
A→+∞
integraldisplay A
A
f(x)dxa127a51a167a133a31
a117A
a249a76a178P.V.
integraldisplay +∞
∞
f(x)dxa127a51a167a133a31a117A
a2a135a131a216a44.
a126a88a181P.V.
integraldisplay +∞
∞
sinxdx = lim
A→+∞
integraldisplay A
A
sinxdx = 0a167a2
integraldisplay +∞
∞
f(x)dx =
integraldisplay +∞
∞
sinxdxa216a194a241.
(2) a94a135a121a123.
a101a216a44a167a75a100a117f(x) greaterorequalslant 0a167a26
integraldisplay a
∞
f(x)dxa218
integraldisplay +∞
a
f(x)dxa165a150a8a107a152a143+∞
a117a180
integraldisplay a
A
f(x)dxa218
integraldisplay A
a
f(x)dxa165a8A → +∞a158a150a8a107a152a170a117+∞a167a13a44a152a135a140a117a31a1170a167a108a13a167a130a27a218
a170a117+∞a167a249a134a174a127P.V.
integraldisplay +∞
∞
f(x)dxa127a51a103a241a167a75
integraldisplay +∞
∞
f(x)dxa194a241.
a113a100P.V.
integraldisplay +∞
∞
f(x)dx = Aa167a75a226a52a129a141a152a53a167a26
integraldisplay +∞
∞
f(x)dx = A.
176
a49a19a220a169 a188a234a145a63a234
a49a155a152a217 a188a234a145a63a234a33a152a63a234
§1,a188a234a145a63a234a27a152a151a194a241
1,a63a216a101a15a188a234a83a15a51a164a171a171a141a83a27a152a151a194a241a53a181
(1) fn(x) =
radicalbigg
x2 + 1n2,?∞<x<∞
(2) fn(x) = x2?x2n,0 lessorequalslantxlessorequalslant 1
(3) fn(x) = sin xn
(i)?l<x<l
(ii)?∞<x<∞
(4) fn(x) = xn(1?x)a167 0 lessorequalslantxlessorequalslant 1
(5) fn(x) = nx1 +nxa167 0 lessorequalslantxlessorequalslant 1
(6) fn(x) = xn ln xna167 0 <x< 1
a41a181
(1) a8?∞<x< +∞a158a167f(x) = lim
n→∞
fn(x) = lim
n→∞
radicalbigg
x2 + 1n2 = |x|
a75||fn?f|| = sup
x∈(?∞,+∞)
|fn(x)?f(x)| = sup
x∈(?∞,+∞)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
radicalbigg
x2 + 1n2?|x|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
n→ 0(n→∞)
a117a180a100a189a1942a167a26fn(x)a51(?∞,+∞)a83a152a151a194a241a117|x|.
(2) a8x = 1a158a167fn(1) = 0,f(x) = 0a182a80 lessorequalslantx< 1a158a167f(x) = lim
n→∞
fn(x) = 0a167a75f(x) = 0(0 lessorequalslantxlessorequalslant 1)
||fn?f|| = sup
x∈[0,1]
|fn(x)?f(x)| = sup
x∈[0,1]
|xn?x2n| = max
x∈[0,1]
|xn?x2n|
a45(xn?x2n)prime = nxn?1(1?2xn) = 0a167a75a26x = 0,x = n
radicalbigg
1
2
a113fn(0) = 0,fn
parenleftBigg
n
radicalbigg
1
2
parenrightBigg
= 14,fn(1) = 0a167a75||fn?f|| = 14 negationslash= 0a167a117a180a100a189a1942a167a26a100a188a234a83a15a51a164a171a171
a141a83a216a152a151a194a241.
(3) (i) a8?l<x<la158a167f(x) = lim
n→∞
fn(x) = 0
||fn?f|| = sup
x∈(?l,l)
|fn(x)?f(x)| = sup
x∈(?l,l)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglesin
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
l
n→ 0(n→∞)
a117a180a226a189a1942a167a26fn(x)a51(?l,l)a254a152a151a194a241a1170.
(ii) a8?∞<x< +∞a158a167f(x) = lim
n→∞
fn(x) = 0
a18ε0a1660 <ε0 < 1a167a216a216na245a140a167a144a135a18x = n2 pia167a210a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglef
parenleftBigg
n
2 pi
parenrightBigg
f
parenleftBigg
n
2 pi
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 1 >ε0
a75fn(x)a51(?∞,+∞)a254a216a152a151a194a241.
(4) a80 lessorequalslantx< 1a158a167f(x) = lim
n→∞
fn(x) = 0a182a8x = 1a158a167fn(1) = 0,f(1) = 0a167a75f(x) = lim
n→∞
fn(x) = 0
||fn?f|| = sup
x∈[0,1]
|fn(x)?f(x)| = sup
x∈[0,1]
|xn(1?x)| = max
x∈[0,1]
(xn?xn+1)
a45(xn?xn+1)prime = xn?1[n?(n+ 1)x] = 0a34a75a26x = 0,x = nn+ 1
a113fn(0) = fn(1) = 0,fn
parenleftBigg
n
n+ 1
parenrightBigg
=
parenleftBigg
n
n+ 1
parenrightBiggnparenleftBigg
1? nn+ 1
parenrightBigg
> 0
a75 max
x∈[0,1]
(xn?xn+1) =
parenleftBigg
n
n+ 1
parenrightBiggnparenleftBigg
1? nn+ 1
parenrightBigg
→ 0(n→∞)a61||fn?f||→ 0(n→∞)
a117a180a100a189a1942a167a26a100a188a234a83a15a51a164a171a171a141a83a152a151a194a241a1170.
177
(5) f(x) = lim
n→∞
fn(x) =
braceleftbigg 1,0 <xlessorequalslant 1
0,x = 0
a117a180f(x)a51[0,1]a254a216a235a89a167a13fn(x)a51[0,1]a254a235a89a167a75fn(x) = nx1 +nxa51[0,1]a254a216a152a151a194a241.
(6) a207 lim
t→+0
tlnt = 0a167a75f(x) = lim
n→∞
fn(x) = 0
|fn(x)?f(x)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
nln
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a233?ε> 0a167a207 lim
t→+0
tlnt = 0a167a75a127a51δ(ε) > 0a167a80 <t<δa158a167a107|tlnt?0|<ε
a18N =
bracketleftBigg
1
δ
bracketrightBigg
a167a8n>Na158a1671n<δ
a108a13a233a152a1310 <x< 1a167a1070 < xn<δa167a25|fn(x)?f(x)|lessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
n ln
x
n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε
a108a13a100a189a1941a167a26a100a188a234a51(0,1)a83a152a151a194a241a1170.
2,a63a216a101a15a63a234a27a152a151a194a241a53a181
(1)
∞summationdisplay
n=0
(1?x)xna167 0 lessorequalslantxlessorequalslant 1
(2)
∞summationdisplay
n=1
(?1)n?1x2
(1 +x2)n a167?∞<x< +∞
(3)
∞summationdisplay
n=1
sinnx
3√n4 +x4 a167?∞<x< +∞
(4)
∞summationdisplay
n=1
x
1 +n4x2 a167?∞<x< +∞
(5)
∞summationdisplay
n=1
sinnxsinx√
n+x a167 0 lessorequalslantxlessorequalslant 2pi
(6)
∞summationdisplay
n=1
(?1)n(1?e?nx)
n2 +x2 a167 0 lessorequalslantx< +∞
(7)
∞summationdisplay
n=1
2nsin 13nxa167 0 <x< +∞
a41a181
(1) a207a220a169a218Sn(x) =
nsummationdisplay
k=0
(1?x)xk = 1?xn+1a167a75S(x) = lim
n→∞
Sn(x) =
braceleftbigg 1,0 lessorequalslantx< 1
0,x = 1
a117a180S(x)a51[0,1]a254a216a235a89a167a13Sn(x)a51[0,1]a254a235a89a167a75
∞summationdisplay
n=0
(1?x)xna51[0,1]a254a216a152a151a194a241.
(2) a207a100a63a234a143a2a134a63a234a167a133 x
2
(1 +x2)n+1 lessorequalslant
x2
(1 +x2)n a167a75a123a170a27a253a233a138a216a172a135a76a167a27a196a145a27a253a233a138a167
a61|rn(x)|lessorequalslant x
2
(1 +x2)n =
x2
1 +nx2 +···+x2n <
1
n (?x∈ (?∞,+∞))
a108a13a233?ε> 0,?N =
bracketleftBigg
1
ε
bracketrightBigg
a167a8n>Na158a167a107|rn(x)|<εa167a75a100a63a234a51(?∞,+∞)a254a152a151a194a241.
(3) a8?∞<x< +∞a158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinnx
3√n4 +x4
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n43
a240a164a225a167a133a63a234
∞summationdisplay
n=1
1
n43
a194a241
a75a100a159a188a7a79a123a167a26a63a234
∞summationdisplay
n=1
sinnx
3√n4 +x4 a51(?∞,+∞)a254a152a151a194a241.
(4) a2070 lessorequalslant (1?n2|x|)2 = 1?2n2|x|+n4x2a167a752n2|x|lessorequalslant 1 +n4x2a61 2n
2|x|
1 +n4x2 lessorequalslant 1 (x∈ (?∞,+∞))
a108a13
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
1 +n4x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
2n2|x|
2n2(1 +n4x2) lessorequalslant
1
2n2
178
a113a63a234
∞summationdisplay
n=1
1
2n2 a194a241a167a75a226a159a188a7a79a123a167a26a63a234
∞summationdisplay
n=1
x
1 +n4x2 a51(?∞,+∞)a254a152a151a194a241.
(5) a8x = 0,2pia158a167
nsummationdisplay
k=1
sinkxsinx = 0
a8xnegationslash= 0,2pia158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
sinkxsinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= |sinx|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
sinkx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant|sinx|
1vextendsingle
vextendsinglesin x2vextendsinglevextendsingle= 2
vextendsinglevextendsinglecos x
2
vextendsinglevextendsinglelessorequalslant 2
a75a80 lessorequalslantxlessorequalslant 2pia158a167
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
sinkxsinx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant 2
a113 1√n+x a233x ∈ [0,2pi]a39a117na252a78a52a126a133a100 1√n+x lessorequalslant 1√n a26a8n → ∞a158a167 1√n+x a39a117xa51[0,2pi]a254
a152a151a47a170a1170(a100a189a1942)
a75a226a41a225a142a52a7a79a123a167a26a63a234
∞summationdisplay
n=1
sinnxsinx√
n+x a51[0,2pi]a254a152a151a194a241.
(6) a100a117a233?x∈ [0,+∞)a167a1070 lessorequalslant 1?e?nx < 1a167a75
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(?1)n(1?e?nx)
n2 +x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n2
a113a63a234
∞summationdisplay
n=1
1
n1 a194a241a167a75a226a159a188a7a79a123a167a26a63a234
∞summationdisplay
n=1
(?1)n(1?e?nx)
n2 +x2 a51[0,+∞)a254a152a151a194a241.
(7) a80un(x) = 2nsin 13nx
a80 <x< +∞a158a167a100a117|un(x)|lessorequalslant 1x
parenleftBigg
2
3
parenrightBiggn
a133
∞summationdisplay
n=1
1
x
parenleftBigg
2
3
parenrightBiggn
a194a241a167a25a6a63a234a253a233a194a241a167a108a13a194a241a167a2a167
a51(0,+∞)a83a191a216a152a151a194a241.
a88a101a216a44a167a23a167a152a151a194a241a167a75a233a63a137ε > 0a167a18ε = 1a167a55a127a51N = N(ε) ∈ Z+(a167a134xa195a39)a167a166
a8n>Na158a167a233a117(0,+∞)a83a27a152a131xa167a254a107|un+1(x) +un+2(x) +···+un+p(x)| <εa167a217a165pa143a63a191a20
a18a234
a56a18p = 1,n = Na167a75a233a152a131x∈ (0,+∞)a167a65a107|uN+1(x)|<ε = 1
a113a18x0 = 23N+1pi ∈ (0,+∞)a167a143a65a107|uN+1(x0)|< 1
a2a175a162a254a37a107uN+1(x0) = 2N+1 sin 13N+1x
0
= 2N+1 > 1a249a134|uN+1(x0)|< 1a103a241
a75a98a23a216a164a225a167a61a63a234
∞summationdisplay
n=1
2nsin 13nxa51(0,+∞)a254a194a241a2a154a152a151a194a241.
3,a121a178a152a151a194a241a189a1941a218a189a1942a27a31a100a53.
a121a178a181a189a1941=?a189a1942
a174a127a233a63a137a27ε> 0a167a127a51a144a157a54a117εa27a20a18a234N(ε)a167a166n>N(ε)a158a167a107|Sn(x)?S(x)|< ε2 a233a152a131x∈Xa209a164
a225
a117a180||Sn?S|| = sup
x∈X
|Sn(x)?S(x)|lessorequalslant ε2 <εa167a108a13 lim
n→∞
||Sn?S|| = 0.
a189a1942=?a189a1941
a174a127 lim
n→∞
||Sn?S|| = 0a167a61a233?ε> 0,?N(ε) ∈Z+a167a166a8n>Na158a167a233a152a131x∈Xa167a209a107vextendsingle
vextendsingle||Sn?S||?0vextendsinglevextendsingle= sup
x∈X
|Sn(x)?S(x)|<ε
a13|Sn(x)?S(x)|lessorequalslantvextendsinglevextendsingle||Sn?S||?0vextendsinglevextendsingle<εa233a152a131x∈Xa209a164a225.
(a17a28a97a113a47a140a121a178a188a234a145a63a234
∞summationdisplay
n=1
una189a1941a189a1942).
4,a193a121a63a234
∞summationdisplay
n=1
ln(1 +nx)
nxn a51a63a219a171a109[1 +α,∞),α> 0a143a152a151a194a241.
a121a178a181a207a8h> 0a158a167ln(1 +h) <ha167a75
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
ln(1 +nx)
nxn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
ln(1 +nx)
nxn <
nx
nxn =
1
xn?1 lessorequalslant
1
(1 +α)n?1 (1 +α lessorequalslant
x< +∞)
a113
∞summationdisplay
n=1
1
(1 +α)n?1 a194a241a167a75a226Ma7a79a123a167a26a6a63a234a51[1 +α,+∞)(α> 0)a254a152a151a194a241.
179
5,a101
∞summationdisplay
n=1
un(x)a27a152a132a145|un(x)| lessorequalslant cn(x)a167a191a133
∞summationdisplay
n=1
cn(x)a51Xa254a152a151a194a241a167a75
∞summationdisplay
n=1
un(x)a51Xa254a189a152a151a194a241a133a253a233
a194a241.
a121a178a181a207
∞summationdisplay
n=1
cn(x)a51Xa254a152a151a194a241
a75a100a152a151a194a241a27a133a220a191a135a94a135a167a26a233?ε > 0,?N = N(ε) ∈ Z+a167a166a8n > Na158a167a233a152a131x ∈ Xa218a63a191a27a20a18
a234pa167a107|cn+1(x) +cn+2(x) +···+cn+p(x)|<ε
a113
∞summationdisplay
n=1
un(x)a27a152a132a145|un(x)|lessorequalslantcn(x)
a75a233a254a227ε> 0a167a20a18a234N = N(ε)a167a166a8n>Na158a167a233a152a131x∈Xa218a254a227a20a18a234pa167a107
|un+1(x) +un+2(x) + ··· +un+p(x)| lessorequalslant vextendsinglevextendsingle|un+1(x)| + |un+2(x)| + ··· + |un+p(x)|vextendsinglevextendsingle lessorequalslant |cn+1(x) +cn+2(x) +
···+cn+p(x)|<ε
a100a152a151a194a241a27a133a220a191a135a94a135a167a26
∞summationdisplay
n=1
un(x)a51Xa254a152a151a194a241a133a253a233a194a241.
6,a23f0(x)a51[0,a]a254a235a89a167a113fn(x) =
integraldisplay x
0
fn?1(t)dta167a121a178{fn(x)}a51[0,a]a254a152a151a194a241a117a34.
a121a178a181a207f0(x)a51[0,a]a254a235a89a167a75a217a107a46a167a61a127a51M > 0a167a107|f0(x)|lessorequalslantM
a113fn(x) =
integraldisplay x
0
fn?1(t)dta167a75
|f1(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay x
0
f0(t)dt
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay x
0
|f0(t)|dtlessorequalslant
integraldisplay x
0
M dt = MxlessorequalslantMa
|f2(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay x
0
f1(t)dt
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay x
0
|f1(t)|dtlessorequalslant
integraldisplay x
0
Mtdt = Mx
2
2 lessorequalslant
Ma2
2.......................................................
|fn(x) =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay x
0
fn?1(t)dt
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay x
0
|fn?1(t)|dtlessorequalslant
integraldisplay x
0
M t
n?1
(n?1)! dt = M
xn
n! lessorequalslantM
an
n!
a207 lim
n→∞
an
n! = 0a61a233?ε> 0,?N ∈Z
+a167a8n>Na158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
n!?0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
an
n! <ε
a117a180|fn(x)?0|<Mεa233a152a131x∈ [0,a]a254a164a225
a108a13a100a189a1941a167a26{fn(x)}a51[0,a]a254a152a151a194a241a117a34.
7,a121a178a63a234
∞summationdisplay
n=1
(?1)n?1 1n+x2 a39a117xa51(?∞,+∞)a254a143a152a151a194a241a167a2a233a63a219xa191a154a253a233a194a241a167a13a63a234
∞summationdisplay
n=1
x2
(1 +x2)n
a143a51x∈ (?∞,+∞)a254a253a233a194a241a167a2a191a216a152a151a194a241.
a121a178a181a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
(?1)k?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant 1a61
nsummationdisplay
k=1
(?1)k?1a51(?∞,+∞)a254a152a151a107a46
a113 1n+ 1 +x2 < 1n+x2 a167a75a188a234a15
braceleftBigg
1
n+x2
bracerightBigg
a233a117x∈ (?∞,+∞)a252a78a126
a113a233?ε> 0a167a18N =
bracketleftBigg
1
ε
bracketrightBigg
a167a75a8n>Na158a167a233a152a131x∈ (?∞,+∞)a167a209a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
1
n+x2?0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
n+x2 lessorequalslant
1
n<ε
a75
braceleftBigg
1
n+x2
bracerightBigg
a39a117x∈ (?∞,+∞)a152a151a194a241a1170a167a117a180a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
n=1
(?1)n?1 1n+x2 a51(?∞,+∞)a83
a152a151a194a241.
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle(?1)
n?1 1
n+x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
1
n+x2
a207 lim
n→∞
1
n+x2
1
n
= 1a133
∞summationdisplay
n=1
1
n a117a209a167a75a100a39a22a7a79a123a27a52a129a47a170a167a26
∞summationdisplay
n=1
1
n+x2 a117a209a167a117a180a233a63a219xa63a234a154a253a233
a194a241.
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x2
(1 +x2)n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
x2
(1 +x2)n
a233a27a189a27x∈ (?∞,+∞)a167a207 lim
n→∞
n
radicalBigg
x2
(1 +x2)n =
1
1 +x2,xnegationslash= 0
0,x = 0
a100a133a220a7a79a123a167a26
∞summationdisplay
n=1
x2
(1 +x2)n a51(?∞,+∞)a194a241a167a117a180a253a233a194a241.
180
a8xnegationslash= 0a158a167Sn(x) =
nsummationdisplay
k=1
x2
(1 +x2)k = 1?
1
(1 +x2)n a167S(x) = limn→∞Sn(x) = 1
a8x = 0a158a167Sn(0) = 0,S(0) = 0a167a75S(x) =
braceleftbigg 1,xnegationslash= 0
0,x = 0
a207Sn(x)a51(?∞,+∞)a254a235a89a167a13S(x)a51(?∞,+∞)a254a216a235a89a167a75
∞summationdisplay
n=1
x2
(1 +x2)n a51(?∞,+∞)a83a216a152a151a194a241.
8,a121a178a181
(1) a88a74
summationdisplay
|fn(x)|a51[a,b]a254a152a151a194a241a167a64a34
∞summationdisplay
1
fn(x)a51[a,b]a254a143a152a151a194a241a182
(2) a88a74
summationdisplay
fn(x)a51[a,b]a254a152a151a194a241a167a2
summationdisplay
|fn(x)|a153a55a152a151a194a241a167a177
∞summationdisplay
1
(?1)n(xn?xn+1),0 lessorequalslantxlessorequalslant 1a143a126
a53a96a178.
a121a178a181
(1) a100a133a220a79a75a57a75a23a167a26
a233?ε> 0,?N = N(ε) ∈Z+a167a166a8n>Na158a167a233a152a131x∈ [a,b]a218a63a191p∈Z=a167a107
|fn+1(x)|+|fn+2(x)|+···+|fn+p(x)|<ε
a108a13|fn+1(x) +fn+2(x) +···+fn+p(x)|lessorequalslant|fn+1(x)|+|fn+2(x)|+···+|fn+p(x)|<ε
a75a226a152a151a194a241a27a133a220a79a75a167a26
∞summationdisplay
1
fn(x)a51[a,b]a254a152a151a194a241.
(2) a126a181
∞summationdisplay
1
(?1)n(xn?xn+1)a51[0,1]a254a152a151a194a241
a207xn?xn+1 = 0(a8x = 0,1a158)a182a80 < x < 1a158a167xn?xn+1 = xn(1?x)a167a75xn?xn+1a51[0,1]a254a39
a117na252a78a126a8
a1001.(4)a167a26xn?xn+1 = xn(1?x)a51[0,1]a254a152a151a194a241a1170a167a75a100a41a225a142a52a7a79a123a167a26
∞summationdisplay
1
(?1)n(xn?xn+1) a51[0,1]a254a152a151a194a241
a2
∞summationdisplay
1
vextendsinglevextendsingle(?1)n(xn?xn+1)vextendsinglevextendsingle= ∞summationdisplay
1
(xn?xn+1)a51[0,1]a254a154a152a151a194a241(a1002.(1)a26).
9,a23a122a152a145?n(x)a209a180[a,b]a254a27a252a78a188a234a167a88a74
summationdisplay
n(x)a51[a,b]a27a224a58a143a253a233a194a241a167a64a34a249a63a234a51[a,b]a254a152
a151a194a241.
a121a178a181a207?n(x)a51[a,b]a254a252a78a167a25a107|?n(x)|lessorequalslant|?n(a)|+|?n(b)| (?x∈ [a,b])
a100a117
summationdisplay
|?n(a)|a218
summationdisplay
|?n(b)|a194a241a167a75
summationdisplay
(|?n(a)|+|?n(b)|)a194a241
a75a226Ma7a79a123a167a26a63a234
summationdisplay
n(x)a51[a,b]a254a152a151a194a241.
10,a101a15a188a234a15a180a196a152a151a194a241a186
(1) fn(x) = (sinx)n,0 lessorequalslantxlessorequalslantpi
(2) fn(x) = (sinx) 1n
(i) 0 lessorequalslantxlessorequalslantpi
(ii) δlessorequalslantxlessorequalslantpi?δ
(3) fn(x) = x
n
1 +xn
(i) 0 lessorequalslantxlessorequalslant 1?ε
(ii) 1?ε<x< 1 +ε (ε> 0)
(iii) 1 +εlessorequalslantx<∞
a41a181
(1) f(x) = lim
n→∞
fn(x) =
0,0 lessorequalslantxlessorequalslantpia133xnegationslash= pi2
1,x = pi2
a207f(x)a51[0,pi]a254a216a235a89a167a2fn(x)a51[0,pi]a254a235a89a167a75fn(x) = (sinx)na51[0,pi]a254a216a152a151a194a241.
181
(2) (i) f(x) = lim
n→∞
fn(x) =
braceleftbigg 0,x = 0,pi
1,0 <x<pi
a207f(x)a51[0,pi]a254a216a235a89a167a2fn(x)a51[0,pi]a254a235a89a167a75fn(x) = (sinx) 1na51[0,pi]a254a216a152a151a194a241.
(ii) a207f(x) = lim
n→∞
fn(x) = 1,|fn(x)?f(x)| = 1?(sinx) 1n lessorequalslant 1?(sinδ) 1n
a61||fn?f|| = sup
x∈[δ,pi?δ]
|fn(x)?f(x)| = 1?(sinδ) 1n → 0(n→∞)
a75a100a189a1942a167a26fn(x) = (sinx) 1na51[δ,pi?δ]a254a152a151a194a241a1171.
(3) (i) a80 lessorequalslantxlessorequalslant 1?εa158a167f(x) = lim
n→∞
fn(x) = 0a167a75|fn(x)?f(x)| = x
n
1 +xn lessorequalslantx
n lessorequalslant (1?ε)n
a117a180||fn?f|| = sup
x∈[0,1?ε]
|fn(x)?f(x)| = (1?ε)n → 0 (n→∞)
a75a100a189a1942a167a26fn(x) = x
n
1 +xn a51[0,1?ε]a254a152a151a194a241a1170.
(ii) f(x) = lim
n→∞
fn(x) =
0,1?ε<x< 1
1
2,x = 11,1 <x< 1 +ε
a207f(x)a51(1?ε,1+ε)a254a216a235a89a167a13fn(x)a51(1?ε,1+ε)a254a235a89a167a75fn(x) = x
n
1 +xna51(1?ε,1+ε)a254
a216a152a151a194a241.
(iii) a81 +εlessorequalslantx< +∞a158a167f(x) = lim
n→∞
fn(x) = 1a167a75|fn(x)?f(x)| = 11 +xn lessorequalslant 11 + (1 +ε)n
a117a180||fn?f|| = sup
x∈[1+ε,+∞)
|fn(x)?f(x)| = 11 + (1 +ε)n → 0 (n→∞)
a108a13a100a189a1942a167a26fn(x) = x
n
1 +xn a51[1 +ε,+∞)a254a152a151a194a241a1171.
11,a121a178
∞summationdisplay
1
ne?nxa51(0,+∞)a83a235a89.
a121a178a181a63a18x0 ∈ (0,+∞)a167a75a127a51α,β > 0a167a166α<x0 <βa167a51[α,β]a2540 <ne?nx lessorequalslantne?nα
a207α > 0a167a75eα > 1a167a117a180 lim
n→∞
(n+ 1)e?(n+1)α
ne?nα =
1
eα < 1a167a75a100a136a75a19a16a7a79a123a27a52a129a47a170a167a26a63
a234
∞summationdisplay
1
ne?nαa194a241a167a108a13a226Ma7a79a123a167a26
∞summationdisplay
1
ne?nxa51[α,β]a254a152a151a194a241.
a113ne?nxa51[α,β]a254a235a89a167a108a13
∞summationdisplay
1
ne?nxa51[α,β]a254a235a89
a207x0 ∈ [α,β]a167a75
∞summationdisplay
1
ne?nxa51x0a58a235a89
a100a117x0a180(0,+∞)a27a63a191a58a167a25
∞summationdisplay
1
ne?nxa51(0,+∞)a83a235a89.
12,a121a178a188a234f(x) =
∞summationdisplay
1
sinnx
n3 a51(?∞,+∞)a83a235a89a167a191a107a235a89a19a188a234.
a121a178a181a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sinnx
n3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n3 a133
∞summationdisplay
1
1
n3 a194a241a167a75a226Ma7a79a123a167a26f(x) =
∞summationdisplay
1
sinnx
n3 a51(?∞,+∞)a152a151a194a241
a113 sinnxn3 a51(?∞,+∞)a83a235a89a167a75f(x) =
∞summationdisplay
1
sinnx
n3 a51(?∞,+∞)a83a235a89
d
dx
parenleftBigg
sinnx
n3
parenrightBigg
= cosnxn2
a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosnx
n2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n2 a133
∞summationdisplay
1
1
n2 a194a241a167a75a226Ma7a79a123a167a26
∞summationdisplay
1
cosnx
n2 a51(?∞,+∞)a152a151a194a241
a117a180f prime(x) = ddx
parenleftBigg ∞summationdisplay
1
sinnx
n3
parenrightBigg
=
∞summationdisplay
1
cosnx
n2
a113 cosnxn2 a51(?∞,+∞)a83a235a89a167a75
∞summationdisplay
1
cosnx
n2 a51(?∞,+∞)a83a235a89
182
a61f prime(x)a51(?∞,+∞)a83a235a89a133f prime(x) =
∞summationdisplay
1
cosnx
n2,
13,a121a178a188a234ζ(x) =
∞summationdisplay
1
1
nx a51(1,+∞)a235a89a167a191a107a235a89a136a30a19a188a234.
a121a178a181a136a145a166a19a234a164a26a63a234a143?
∞summationdisplay
n=1
lnn
nx,a101a121a167a511 <alessorequalslantx< +∞a254a152a151a235a89(aa143a140a1171a27a63a219a234)
a8alessorequalslantx< +∞a158a167a1070 < lnnnx lessorequalslant lnnna
a100a117 lim
n→∞
lnn
na
1
n(a+1)/2
= 0a133
∞summationdisplay
n=1
1
n(a+1)/2 a194a241
a75a63a234
∞summationdisplay
n=1
lnn
na a194a241a167a117a180a100Ma7a79a123a167a26a63a234
∞summationdisplay
n=1
lnn
nx a51alessorequalslantx< +∞a254a152a151a194a241
a53a191a20a122a145 lnnnx a209a180xa27a235a89a188a234a167a75a63a234
∞summationdisplay
n=1
1
nx a51a lessorequalslant x < +∞a254a140a197a145a166a19a234a167a26ζ
prime(x) =?
∞summationdisplay
n=1
lnn
nx
a133ζprime(x)a51alessorequalslantx< +∞a254a235a89
a100a> 1a27a63a191a53a167a26ζprime(x) =?
∞summationdisplay
n=1
lnn
nx a233a152a1311 <x< +∞a164a225a133ζ
prime(x)a511 <x< +∞a254a235a89a167a8a44ζ(x)a141
a511 <x< +∞a254a235a89
a124a94a234a198a56a66a123a167a191a53a191a20a233a63a219a20a18a234ka167a63a234
∞summationdisplay
n=1
(lnn)k
na (a > 1)a209a194a241a167a149a236a254a227a167a140a121a181a233a63a219a20a18
a234ka167ζ(k)(x)a511 <x< +∞a254a209a127a51a133a235a89a167a133a140a100a6a63a234a197a145a166a19a234ka103a167a26
ζ(k)(x) = (?1)k
∞summationdisplay
n=1
(lnn)k
nx (1 <x< +∞).
14,a193a121a63a234
∞summationdisplay
1
sin(2npix)
2n a51a18a135a162a234a182a254a152a151a194a241a167a2a51a63a219a171a109a83a216a85a197a145a166a135a251.
a121a178a181a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(2npix)
2n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
2na233?x∈ (?∞< +∞)a27a164a225a133a63a234
∞summationdisplay
1
1
2na194a241a167a75a226Ma7a79a123a167a26
∞summationdisplay
1
sin(2npix)
2n
a51a18a135a162a234a182a254a152a151a194a241parenleftBigg
sin(2npix)
2n
parenrightBiggprime
= picos(2npix)
a101a121
∞summationdisplay
1
picos(2npix)a51a63a219a171a109a83a209a107a216a235a89a58
a63a18x∈ (?∞,+∞)a167a111a127a51k ∈Za167a166x = k+ya167a217a1650 lessorequalslanty< 1
a242a217a147a92a167a26
∞summationdisplay
1
cos(2npix) =
∞summationdisplay
1
cos(2npiy)a167a65a79a27a167a18y = 2?mha167a217a165m∈Z+,h = 0,1,2,···,2m?1
a8n>ma158a167cos(2npiy) = 1a167a100a158a63a234a152a132a145a216a170a1170a167a75
∞summationdisplay
1
cos(2npix) =
∞summationdisplay
1
cos(2npiy)a117a209a167a117a180
∞summationdisplay
1
picos(2npix)a117
a209
a113a51a63a219a171a109a83a209a127a51x = k+ 2?mh(h = 0,1,2,···,2m?1)a249a24a27a58a167ka143xa27a129a2a18a234a220a169
a75a63a234
∞summationdisplay
1
sin(2npix)
2n a51a63a219a171a109a83a216a85a197a145a166a135a251.
15,a107a121
1?r2
1?2rcosx+r2 = 1 + 2
∞summationdisplay
n=1
rncosnx
a8|r|< 1a158a164a225a167a108a13a121a178a181 integraldisplay
pi
pi
1?r2
1?2rcosx+r2 dx = 2pi (|r|< 1)
.
a121a178a181|rncosnx|lessorequalslant|r|na233?x∈ (?∞,+∞)a209a164a225
a207|r|< 1a167a75
∞summationdisplay
n=1
|r|na194a241a167a117a180a100Ma7a79a123a167a26
∞summationdisplay
n=1
rncosnxa51(?∞,+∞)a83a152a151a194a241
183
a108a13a23f(x) = 1 + 2
∞summationdisplay
n=1
rncosnx
a2071?2rcosx+r2 negationslash= 0a167a254a170a252a224a211a166a1771?2rcosx+r2a167a75a26
(1?2rcosx+r2)f(x) = (1?2rcosx+r2)
parenleftBigg
1 + 2
∞summationdisplay
n=1
rncosnx
parenrightBigg
=
bracketleftBigg
1?2rcosx+r2 + 2
∞summationdisplay
n=1
rncosnx?2
∞summationdisplay
n=1
rn+1(2cosnxcosx) + 2
∞summationdisplay
n=1
rn+2 cosnx
bracketrightBigg
=
bracketleftBigg
1?2rcosx+r2 + 2
∞summationdisplay
n=1
rncosnx?2
∞summationdisplay
n=1
rn+1 cos(n+ 1)x?2
∞summationdisplay
n=1
rn+1 cos(n?1)x+ 2
∞summationdisplay
n=1
rn+2 cosnx
bracketrightBigg
=
bracketleftBigg
1?r2 + 2
∞summationdisplay
n=1
rncosnx?2
parenleftBigg ∞summationdisplay
n=1
rn+1 cos(n+ 1)x+rcosx
parenrightBigg
2
parenleftBigg ∞summationdisplay
n=1
rn+1 cos(n?1)x?r2
parenrightBigg
+ 2
∞summationdisplay
n=1
rn+2 cosnx
bracketrightBigg
= 1?r2
a117a180f(x) = 1?r
2
1?2rcosx+r2 a61
1?r2
1?2rcosx+r2 = 1 + 2
∞summationdisplay
n=1
rncosnx
a100a117a254a170a109a224a63a234a51[?pi,pi]a254a152a151a194a241a167a133rncosnxa51[?pi,pi]a254a235a89a167a75a254a170a63a234a140a177a197a145a200a169a167a26integraldisplay
pi
pi
1?r2
1?2rcosx+r2 dx =
integraldisplay pi
pi
parenleftBigg
1 + 2
∞summationdisplay
n=1
rncosnx
parenrightBigg
dx = 2pi+ 2
∞summationdisplay
n=1
integraldisplay pi
pi
rncosnxdx = 2pi.
16,a94a107a129a67a88a189a110a121a178a41a90a189a110.
a121a178a181a207{Sn(x)}a51[a,b]a254a194a241a117S(x)a167a25a233?ε> 0,?x ∈ [a,b],?N(ε,x) ∈ Z+a167a166a26a8ngreaterorequalslantN(ε,x)a158a167a209
a65a107|Sn(x)?S(x)|<εa167a65a79a107|SN(ε,x)?S(x)|<ε
a100SN(ε,x)(x)?S(x)a51xa58a235a89a167a26a127a51xa58a27a109a25a141Oxa167a166a26|SN(ε,x)(y)?S(y)|,?y ∈Ox
a117a180braceleftbigOxvextendsinglevextendsinglex∈ [a,b]bracerightbiga8a164[a,b]a27a109a67a88(a233a224a58a,ba140a138a235a89a242a255)
a226a107a129a67a88a189a110a167a108a165a192a209a107a129a135a109a25a141Ox1,···,Oxma211a24a67a88[a,b]a133a247a118|SN(ε,xi)(y)?S(y)| < ε,?y ∈
Oxi,i = 1,2,···,m
a18N = max
ilessorequalslantilessorequalslantm
N(ε,xi)a167a75a8n > Na158a167a233?x ∈ [a,b]a167a100{Sn(x)}a252a78a53a218
muniontext
i=1
Oxi? [a,b]a167a55a127a51a44
a135Oxia167a166x∈Oxia167a133a107|Sn(x)?S(x)|lessorequalslant|SN(x)?S(x)|lessorequalslant|SN(ε,xi)(x)?S(x)|<ε
a61{Sn(x)}a51[a,b]a254a152a151a194a241a117S(x).
17,a101Sn(x)a51ca58a134a235a89(n = 1,2,3,···)a167a2{Sn(c)}a117a209a167a75a51a63a219a109a171a109(c?δ,c)a83(δ > 0)a167{Sn(x)}a55a216a152
a151a194a241.
a121a178a181a94a135a121a123.
a98a23a127a51δ0 > 0a167a166a26{Sn(x)}a51(c?δ0,c)a83a152a151a194a241
a100a152a151a194a241a27a133a220a6a110a167a26a233?ε> 0,?N(ε) ∈Z+a167a166a26a8n>N(ε)a158a167a233?x∈ (c?δ0,c)a218?p∈Z+a167a209a65
a107|Sn+p(x)?Sn(x)|< ε2 (?)a164a225
a207a122a152a135Sn(x)a51ca58a134a235a89a167a75Sn+p(x)?Sn(x)a143a51ca58a134a235a89
a117a180 lim
x→c?0
[Sn+p(x)?Sn(x)] = Sn+p(c)?Sn(c)
a51(?)a170a252a224a45x→c?0a167a26|Sn+p(c)?Sn(c)|lessorequalslant ε2 <ε
a100a234a15a27a133a220a194a241a6a110a167a26{Sn(c)}a194a241a167a134a174a127{Sn(c)}a117a209a103a241
a25a98a23a216a20a40a167a75a51a63a219a109a171a109(c?δ,c)a83(δ> 0)a167{Sn(x)}a55a216a152a151a194a241.
184
§2,a152a63a234
1,a166a101a15a136a152a63a234a27a194a241a171a109a181
(1)
∞summationdisplay
n=1
(2x)n
n!
(2)
∞summationdisplay
n=1
ln(n+ 1)
n+ 1 x
n+1
(3)
∞summationdisplay
n=1
bracketleftBiggparenleftBigg
n+ 1
n
parenrightBiggn
x
bracketrightBiggn
(4)
∞summationdisplay
n=1
xn2
2n
(5)
∞summationdisplay
n=1
[3 + (?1)n]n
n x
n
(6)
∞summationdisplay
n=1
3n + (?2)n
n (x+ 1)
n
a41a181
(1) an = 2
n
n!
a207R = lim
n→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= +∞a167a75a217a194a241a141a143(?∞,+∞).
(2)
∞summationdisplay
n=1
ln(n+ 1)
n+ 1 x
n+1 =
∞summationdisplay
n=2
lnn
n x
n a167an = lnn
n
a100a117 lim
y→+∞
(y+ 1)lny
yln(y+ 1) = limy→+∞
y+ 1
y limy→+∞
lny
ln(y+ 1) = 1a167a75R = limn→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle = 1a167a117a180a217a194a241a171a109
a143(?1,1)
a8x =?1a158a167a6a63a234a143
∞summationdisplay
n=2
(?1)n lnnn xn
a207
parenleftBigg
lnx
x
parenrightBiggprime
= 1?lnxx2 a133a8xgreaterorequalslant 3a158a167
parenleftBigg
lnx
x
parenrightBiggprime
< 0a167a75
braceleftBigg
lnn
n
bracerightBigg
a252a78a126a8
a113 lim
n→∞
lnn
n = 0a167a75a63a234
∞summationdisplay
n=2
(?1)n lnnn xn a143a52a217a90a91a63a234a167a117a180a63a234
∞summationdisplay
n=2
(?1)n lnnn xn a194a241
a8x = 1a158a167a6a63a234a143
∞summationdisplay
n=2
lnn
n x
n
a207 lim
n→∞
lnn
n
n = +∞a167a75a226a20a145a63a234a27a39a22a7a79a123a57a63a234
∞summationdisplay
n=1
1
na117a209a167a26a63a234
∞summationdisplay
n=2
lnn
n x
n a117a209
a75a100a63a234a27a194a241a141a143[?1,1).
(3) a207
∞summationdisplay
n=1
bracketleftBiggparenleftBigg
n+ 1
n
parenrightBiggn
x
bracketrightBiggn
=
∞summationdisplay
n=1
parenleftBigg
1 + 1n
parenrightBiggn2
xna167a75an =
parenleftBigg
1 + 1n
parenrightBiggn2
a113 lim
n→∞
nradicalbig|an| = ea167a75a217a194a241a140a187a143R = 1
ea167a194a241a171a109a143
parenleftBigg
1e,1e
parenrightBigg
.
a8x = ±1ea158a167a6a63a234a143
∞summationdisplay
n=1
(±1)n
parenleftBigg
1 + 1n
parenrightBiggn2 parenleftBigg
1
e
parenrightBiggn
a167a75un = (±1)n
parenleftBigg
1 + 1n
parenrightBiggn2 parenleftBigg
1
e
parenrightBiggn
a100a226a55a136a123a75a167a26 lim
n→∞
|un| = e?12 negationslash= 0
a75a63a234
∞summationdisplay
n=1
(±1)n
parenleftBigg
1 + 1n
parenrightBiggn2 parenleftBigg
1
e
parenrightBiggn
a117a209a167a117a180a6a63a234a27a194a241a141a143
parenleftBigg
1e,1e
parenrightBigg
.
185
(4) an = 12n
a100 lim
n→∞
n2radicalbig|an|xn2 = |x| lim
n→∞
n
radicalbigg
1
2 = |x|< 1a167a26a217a194a241a140a187a143R = 1a167a194a241a171a109a143(?1,1)
a8|x| = 1a61x = ±1a158a167a6a63a234a67a143
∞summationdisplay
n=1
(±1)n2
2n
a100a117a63a234
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(±1)n2
2n
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
1
2n a194a241a167a75a63a234
∞summationdisplay
n=1
(±1)n2
2n a253a233a194a241a75a194a241
a108a13a152a63a234
∞summationdisplay
n=1
xn2
2n a27a194a241a141a143[?1,1].
(5) an = [3 + (?1)
n]n
n
a207 lim
n→∞
n
radicalbigg
[3 + (?1)n]n
n = 4a167a75a63a234a194a241a140a187a143R =
1
4 a167a194a241a171a109a143
parenleftBigg
14,14
parenrightBigg
a8x = 14 a158a167a6a63a234a67a143
∞summationdisplay
n=1
[3 + (?1)n]n
n·4n =
∞summationdisplay
k=1
1
2k +
∞summationdisplay
k=1
1
(2k+ 1)22k+1
a233a63a234
∞summationdisplay
k=1
1
(2k+ 1)22k+1
a207 lim
k→∞
1
(2k+3)22k+3
1
(2k+ 1)22k+1
= 14 < 1 a167a75a226a136a75a19a16a7a79a123a167a26a63a234
∞summationdisplay
k=1
1
(2k+ 1)22k+1 a194a241
a113a63a234
∞summationdisplay
k=1
1
2k a117a209a167a75a63a234
∞summationdisplay
n=1
[3 + (?1)n]n
n·4n a117a209
a211a123a140a26a167a8x =?14 a158a167a63a234
∞summationdisplay
n=1
(?1)n[3 + (?1)
n]n
n·4n a117a209
a75a63a234
∞summationdisplay
n=1
[3 + (?1)n]n
n x
na27a194a241a141a143
parenleftBigg
14,14
parenrightBigg
.
(6) an = 3
n + (?2)n
n
a207 lim
n→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
3 a167a75a63a234a27a194a241a140a187a143R =
1
3 a167a194a241a171a109a143
parenleftBigg
43,?23
parenrightBigg
a8x =?43 a158a167a6a63a234a67a143
∞summationdisplay
n=1
(?1)n3
n + (?2)n
n
parenleftBigg
1
3
parenrightBiggn
=
∞summationdisplay
n=1
(?1)n
n +
∞summationdisplay
n=1
parenleftbig2
3
parenrightbign
n
a233a63a234
∞summationdisplay
n=1
parenleftbig2
3
parenrightbign
n
a207 lim
n→∞
parenleftbig2
3
parenrightbign+1/(n+ 1)
parenleftbig2
3
parenrightbign/n = 23 < 1 a167a75a226a136a75a19a16a7a79a123a167a26
∞summationdisplay
n=1
parenleftbig2
3
parenrightbign
n a194a241
a113a63a234
∞summationdisplay
n=1
(?1)n
n a194a241a167a75a8x =?
4
3 a158a167a6a63a234a194a241a182
a211a123a140a26a167a8x =?23 a158a167a6a63a234a117a209
a75a63a234
∞summationdisplay
n=1
3n + (?2)n
n (x+ 1)
na27a194a241a141a143
bracketleftBigg
43,?23
parenrightBigg
.
2,a166a63a234a27a194a241a140a187a181
(1)
∞summationdisplay
n=1
parenleftBigg
1 + 12 +···+ 1n
parenrightBigg
xn
(2)
summationdisplay(2n)!
(n!)2 x
n
a41a181
186
(1) an = 1 + 12 +···+ 1n
a2071 = n
radicalbigg
n· 1nlessorequalslant n
radicalbigg
1 + 12 +···+ 1nlessorequalslant n√n·1 → 1(n→∞)
a75 lim
n→∞
nradicalbig|an| = 1a167a117a180a217a194a241a140a187a143R = 1.
(2) an = (2n)!(n!)2
a207 lim
n→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= limn→∞
(n+ 1)2
2(n+ 1)(2n+ 1) =
1
4 a167a117a180a217a194a241a140a187a143R =
1
4,
3.
∞summationdisplay
n=1
3n + (?2)n
n (x+ 1)
n a23a152a63a234summationdisplayanxna27a194a241a140a187a143Ra167summationdisplaybnxna27a194a241a140a187a143Qa167a63a216a101a15a63a234a27a194a241
a140a187a181
(1)
summationdisplay
anx2n
(2)
summationdisplay
(an +bn)xn
(3)
summationdisplay
anbnxn
a41a181
(1) lim
n→∞
2nradicalbig|an| = lim
n→∞
parenleftBig
nradicalbig|an|
parenrightBig1
2 =
radicalbigg
1
R =
1√
Ra167a75a217a194a241a140a187a143R1 =
√R.
(2) a23An = an +bn
a75a107 nradicalbig|An| = nradicalbig|an +bn|lessorequalslant nradicalbig|an|+|bn|lessorequalslant nradicalbig2max(|an|,|bn|) = n√2· nradicalbigmax(|an,bn|) = n√2max( nradicalbig|an|,nradicalbig|bn|)
a207 lim
n→∞
n√2 = 1
a75 1R
2
= lim
n→∞
nradicalbig|An|lessorequalslant lim
n→∞
{ n√2max( nradicalbig|an|,nradicalbig|bn|)} = lim
n→∞
{max( nradicalbig|an|,nradicalbig|bn|)} = max
parenleftBig
lim
n→∞
nradicalbig|an|,lim
n→∞
nradicalbig|an|
parenrightBig
=
max
parenleftBigg
1
R,
1
Q
parenrightBigg
a108a13a167a26R2 greaterorequalslant 1
max
parenleftBigg
1
R,
1
Q
parenrightBigg= min(R,Q).
(3) a23Bn = anbn
a75a107 nradicalbig|Bn| = nradicalbig|anbn| = nradicalbig|an|· nradicalbig|bn|
a117a180 1R
3
= lim
n→∞
nradicalbig|Bn| = lim
n→∞
{ nradicalbig|an|· nradicalbig|bn|lessorequalslant lim
n→∞
nradicalbig|an|· lim
n→∞
nradicalbig|bn|} = 1
R ·
1
Q =
1
RQ
a108a13R3 greaterorequalslantRQ.
4,a23a233a191a169a140a27na167|an|lessorequalslant|bn|a167a64a34a63a234
summationdisplay
anxna27a194a241a140a187a216a2a117
summationdisplay
bnxna27a194a241a140a187.
a121a178a181a207a233a191a169a140a27na167|an|lessorequalslant|bn|a167a75 nradicalbig|an|lessorequalslant nradicalbig|bn|a167a117a180 lim
n→∞
nradicalbig|an|lessorequalslant lim
n→∞
nradicalbig|bn|
a23a63a234
summationdisplay
anxna27a194a241a140a187a143Ra167a63a234
summationdisplay
bnxna27a194a241a140a187a143Q
a75a80 < lim
n→∞
nradicalbig|an|lessorequalslant lim
n→∞
nradicalbig|bn|<∞a158a167a100R = 1
lim
n→∞
nradicalbig|an|,Q =
1
lim
n→∞
nradicalbig|bn| a167a26RgreaterorequalslantQa182
a80 = lim
n→∞
nradicalbig|an|lessorequalslant lim
n→∞
nradicalbig|bn|a167a75R = ∞,Qlessorequalslant∞a167a117a180RgreaterorequalslantQa182
a8 lim
n→∞
nradicalbig|an|lessorequalslant lim
n→∞
nradicalbig|bn| = ∞a158a167a75Rgreaterorequalslant 0,Q = 0a167a117a180RgreaterorequalslantQ
a110a254a127a167a63a234
summationdisplay
anxna27a194a241a140a187a216a2a117
summationdisplay
bnxna27a194a241a140a187.
5,a121a178a152a63a234a27a53a1591a218a53a1592.
a121a178a181a53a1591.
a23xa143(x0?R,x0 +R)a83a63a152a58a167a111a140a177a192a180 <r<Ra167a166a26|x?x0|lessorequalslantr
a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=0
an(x?x0)na51[x0?r,x0 +r]a254a152a151a194a241
a113an(x?x0)n(n = 0,1,2,···)a51[x0?r,x0+r]a235a89a167a75a100a188a234a145a63a234a27a218a27a235a89a53a127S(x)a51[x0?r,x0+r]a235
a89a167a8a44a51xa249a152a58a235a89
187
a13xa143(x0?R,x0 +R)a254a63a152a58a167a75S(x)a51(x0?R,x0 +R)a235a89
a113a101
∞summationdisplay
n=0
an(x?x0)na51x0 + Ra194a241a167a75a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=0
an(x?x0)na51[a,x0 + R](a18a ∈ (x0?
R,x0 +R))a254a152a151a194a241
a100a117an(x?x0)n(n = 0,1,2,···)a51[a,x0 +R]a235a89a167a75a100a188a234a145a63a234a27a218a27a235a89a53a189a110a167a26
S(x)a51[a,x0 +R]a235a89a167a8a44a143a51x0 +Ra235a89a167a117a180S(x)a51(x0?R,x0 +R]a254a235a89
a211a110a101
∞summationdisplay
n=0
an(x?x0)na51x0?Ra194a241a167a75S(x)a51[x0?R,x0 +R)a254a235a89.
a53a1592.
(1) a23xa143(x0?R,x0+R)a83a63a152a58a167a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=0
an(x?x0)na51[x0,x]a254a152a151a194a241(a101x<x0a167
a75a18[x,x0 ]a61a140)
a113an(x?x0)n(n = 0,1,2,···)a51[x0,x]a235a89
a75a100a188a234a145a63a234a197a145a166a200a169a189a110a167a26integraldisplay
x
x0
S(x)dx =
integraldisplay x
x0
parenleftBigg ∞summationdisplay
n=0
an(x?x0)n
parenrightBigg
dx =
∞summationdisplay
n=0
integraldisplay x
x0
[an(x?x0)n]dx =
∞summationdisplay
n=0
an
n+ 1 (x?x0)
n+1
(2) a100a495a144a83a753(2)a127a167a101{xn}a194a241a167a133 lim
n→∞
xn = 0a167a75a233a63a219{yn}a167a107 lim
n→∞
(xn·yn) = lim
n→∞
xn· lim
n→∞
yn
a75 lim
n→∞
nradicalbig|nan| = lim
n→∞
nradicalbig|an|
a249a96a178a181
∞summationdisplay
n=1
nan(x?x0)n?1a134
∞summationdisplay
n=1
an(x?x0)na107a131a211a27a194a241a140a187R
a23xa180(x0?R,x0 +R)a83a63a152a58a167a111a140a192a18a152a580 <r<Ra167a166a26|x?x0|lessorequalslantr
a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=0
an(x?x0)na51[x0?r,x0 +r]a254a152a151a194a241a167a207a13a194a241
a113
∞summationdisplay
n=1
nan(x?x0)n?1a27a194a241a140a187a143Ra167a75a100a67a19a16a49a19a189a110a167a26
∞summationdisplay
n=1
nan(x?x0)n?1a51[x0?r,x0+r]a254
a152a151a194a241
a113nan(x?x0)n?1(n = 1,2,···)a51[x0?r,x0 +r]a235a89a167a75a100a188a234a145a63a234a197a145a135a169a189a110a167a26
a51[x0?r,x0 +r]a8a44a143a210a51xa58a167a107 ddx S(x) = ddx
parenleftBigg ∞summationdisplay
n=0
an(x?x0)n
parenrightBigg
=
∞summationdisplay
n=1
nan(x?x0)n?1
a50a100xa51(x0?R,x0 +R)a27a63a191a53a167a26a51(x0?R,x0 +R)a254a170a143a164a225
(3) a23
∞summationdisplay
n=0
an
n+ 1 (x?x0)
n+1a194a241a140a187a143Rprime
a100(1)a167a26a8
∞summationdisplay
n=0
an(x?x0)na51(x0?R,x0 +R)a194a241(a194a241a20S(x))a158a167a107
∞summationdisplay
n=0
an
n+ 1 (x?x0)
n+1a51(x0?R,x0 +R)a254a194a241
parenleftbigg
a194a241a20
integraldisplay x
x0
S(x)dx
parenrightbigg
a167a64a34RlessorequalslantRprime
a44a152a144a161a167a100(2)a167a8
∞summationdisplay
n=0
an
n+ 1 (x?x0)
n+1a51(x0?Rprime,x0 +Rprime)a254a194a241
parenleftbigg
a194a241a20
integraldisplay x
x0
S(x)dx
parenrightbigg
a158a167a107
∞summationdisplay
n=0
an(x?x0)na51(x0?Rprime,x0 +Rprime)a194a241(a194a241a20S(x))a167a64a34Rprime lessorequalslantR
a117a180R = Rprime
6,a23
∞summationdisplay
0
ana194a241a117Aa167
∞summationdisplay
0
bna194a241a117Ba167a88a74a167a130a27a133a220a166a200
∞summationdisplay
0
cn =
∞summationdisplay
0
(a0bn +a1bn?1 +···+anb0)
a194a241a167a75a152a189a194a241a117AB.
a121a178a181a138A(x) =
∞summationdisplay
0
anxn,B(x) =
∞summationdisplay
0
bnxn,C(x) =
∞summationdisplay
0
cnxn
a8x = 1a158a167A = A(1) =
∞summationdisplay
0
an,B = B(1) =
∞summationdisplay
0
bn,C = C(1) =
∞summationdisplay
0
cn
188
a61a152a63a234
∞summationdisplay
0
anxn,
∞summationdisplay
0
bnxn,
∞summationdisplay
0
cnxna51x = 1a194a241
a100Abela49a152a189a110a167a26a254a227a27a152a63a234a51|x|< 1a83a253a233a194a241
a100a133a220a189a110a167a26a63a234
∞summationdisplay
0
cnxna194a241a117
parenleftBigg ∞summationdisplay
0
anxn
parenrightBiggparenleftBigg ∞summationdisplay
0
bnxn
parenrightBigg
a61C(x) = A(x)B(x)
a207
∞summationdisplay
0
anxn,
∞summationdisplay
0
bnxn,
∞summationdisplay
0
cnxna51x = 1a194a241
a100a152a63a234a97a113a53a1591a167a75A(x),B(x),C(x)a51x = 1a134a235a89
C(1) = lim
x→1?0
C(x) = lim
x→1?0
A(x)B(x) = A(1)B(1)
a75C = ABa167a117a180
∞summationdisplay
0
cn = AB.
7,a23f(x) =
∞summationdisplay
0
anxna8|x|<ra158a194a241a167a64a34a8
∞summationdisplay
0
an
n+ 1 r
n+1a194a241a158a164a225
integraldisplay r
0
f(x)dx =
∞summationdisplay
0
an
n+ 1 r
n+1
a216a216
∞summationdisplay
0
anxna8x = ra158a180a196a194a241.
a121a178a181a207f(x) =
∞summationdisplay
0
anxna8|x|<ra158a194a241a167a75a217a194a241a140a187a143Ra167a133rlessorequalslantRa167a108a13f(x)a51(?r,r)a83a194a241.
a75a226a53a1592a167a8x∈ (?r,r)a158a167a107
integraldisplay θ
0
f(x)dx =
integraldisplay θ
0
bracketleftBigg ∞summationdisplay
0
anxn
bracketrightBigg
dx =
∞summationdisplay
0
an
n+ 1 θ
n+1,θ ∈ (0,r)
a61
integraldisplay θ
0
f(x)dx =
∞summationdisplay
0
an
n+ 1 θ
n+1 θ ∈ (0,r)
a207
∞summationdisplay
0
an
n+ 1 r
n+1a194a241a167a75
∞summationdisplay
0
an
n+ 1 θ
n+1a51θ = ra194a241a167a117a180a217a218S(θ)a51ra58a134a235a89
S(r) =
∞summationdisplay
0
an
n+ 1 r
n+1 = lim
θ→r?0
S(θ) = lim
θ→r?0
integraldisplay θ
0
f(x)dx =
integraldisplay r
0
f(x)dx
a108a13a216a216
∞summationdisplay
0
anxna8x = ra158a180a196a194a241a167a254a107
integraldisplay r
0
f(x)dx =
∞summationdisplay
0
an
n+ 1 r
n+1
8,a124a94a254a75a121a178
integraldisplay 1
0
ln(1?t)
t dt =?
∞summationdisplay
n=1
1
n2,
a121a178a181a207ln(1 +x) =
∞summationdisplay
n=1
(?1)n+1x
n
n (?1 <x< 1)a167a75
ln(1?x)
x =?
∞summationdisplay
n=1
xn?1
n (?1 <x< 1a133xnegationslash= 0)
a61f(x) =
ln(1?x)
x,x∈ (?1,0)
uniontext(0,1)
1,x = 0
a167f(x) =?
∞summationdisplay
n=0
xn
n+ 1 (?1 <x< 1)
a207
∞summationdisplay
n=0
1n+1
n+ 1 =?
∞summationdisplay
n=0
1
(n+ 1)2 a194a241a167a75a100a254a75a40a216a167a26
integraldisplay 1
0
ln(1?x)
x dx =?
∞summationdisplay
n=1
1
(n+ 1)2 =?
∞summationdisplay
n=1
1
n2
9,a166f(x) =
∞summationdisplay
n=1
sin(2n ·x)
n! a27a240a142a78a21a63a234a167a96a178a167a27a240a142a78a21a63a234a191a216a76a171a249a135a188a234.
a121a178a181a207
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(2n ·x)
n!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
n! (x∈ (?∞,+∞))a167a133
∞summationdisplay
n=0
1
n!a194a241a167a75a100Ma7a79a123a167a26
∞summationdisplay
n=1
sin(2n ·x)
n! a51(?∞,+∞)a83
a152a151a194a241a167a108a13a194a241
f(0) = 0,
∞summationdisplay
n=1
bracketleftBigg
sin(2n ·x)
n!
bracketrightBiggprime
=
∞summationdisplay
n=1
2ncos(2n ·x)
n!
a113
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
2ncos(2n ·x)
n!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
2n
n! (x∈ (?∞,+∞))a167a133
∞summationdisplay
n=0
2n
n! a194a241a167a75a100Ma7a79a123a167a26
∞summationdisplay
n=1
2ncos(2n ·x)
n! a51(?∞,+∞)a83
a152a151a194a241
189
a1132
ncos(2n ·x)
n! (n = 0,1,2,···) a51(?∞,+∞)a83a235a89a167a75a100a197a145a166a19a189a110a167a26a51(?∞,+∞)a254
f prime(x) = ddx
bracketleftBigg ∞summationdisplay
n=1
sin(2n ·x)
n!
bracketrightBigg
=
∞summationdisplay
n=1
2ncos(2n ·x)
n!
a117a180f prime(0) =
∞summationdisplay
n=1
2n
n! = e
2?1
a88a100a101a22a167a94a234a198a56a66a123a167a26
f(m)(x) =
∞summationdisplay
n=1
2mnsinparenleftbigmpi2 + 2npiparenrightbig
n!,
f(m)(0) =
0,m = 2k
∞summationdisplay
n=1
(?1)k2
(2k+1)n
n! = (?1)
k(e22k+1?1),m = 2k+ 1
a75f(x)a27a240a142a78a21a63a234a143
∞summationdisplay
k=0
(?1)k(e22k+1?1) x
2k+1
(2k+ 1)! a217a194a241a140a187a143R = limk→∞
(e22k+1?1)/(2k+ 1)!
(e22k+3?1)/(2k+ 3)!
a2070 lessorequalslant (e
22k+1?1)/(2k+ 1)!
(e22k+3?1)/(2k+ 3)! = (2k+ 2)(2k+ 3)
e22k+1?1
e22k+3?1 lessorequalslant (2k+ 2)(2k+ 3)
e22k+1
e22k+3 =
(2k+ 2)(2k+ 3)
e6·22k a167
a133 lim
x→∞
(2x+ 2)(2x+ 3)
e6·22x = 0a167a75 limk→∞
(2k+ 2)(2k+ 3)
e6·22k = 0
a117a180R = 0a167a61a217a240a142a78a21a63a234a61a51x = 0a194a241
a2a100a99a161a140a127a217a51(?∞,+∞)a83a254a194a241a167a75a167a27a240a142a78a21a63a234a191a216a76a171a100a188a234.
10,a121a178a181
(1)
∞summationdisplay
n=0
x4n
(4n)! a247a118y
(IV) = ya182
(2)
∞summationdisplay
n=0
xn
(n!)2 a247a118xy
primeprime +yprime?y = 0.
a121a178a181
(1) an = n
radicalBigg
1
(4n)! a167R = limn→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= +∞
a75a127a233a63a152xa167a152a63a234a209a194a241a167a61a217a194a241a141a143(?∞,+∞)
a51a194a241a141a83a197a145a135a169a131a167a26
yprime =
∞summationdisplay
n=1
x4n?1
(4n?1)!,y
primeprime =
∞summationdisplay
n=1
x4n?2
(4n?2)!,y
primeprimeprime =
∞summationdisplay
n=1
x4n?3
(4n?3)!,y
(4) =
∞summationdisplay
n=1
x4n?4
(4n?4)! =
∞summationdisplay
n=0
x4n
(4n)! = y
a61y(IV) = y.
(2) an = 1(n!)2 a167a75R = lim
x→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= +∞
a75a127a233a63a152xa167a152a63a234a209a194a241a167a61a217a194a241a141a143(?∞,+∞)
a51a194a241a141a83a197a145a135a169a131a167a26yprime =
∞summationdisplay
n=1
xn?1
(n?1)!n! =
∞summationdisplay
n=0
xn
n!(n+ 1)!,y
primeprime =
∞summationdisplay
n=1
xn?1
(n?1)!(n+ 1)!
a117a180xyprimeprime +yprime =
∞summationdisplay
n=1
xn
(n?1)!(n+ 1)! +
∞summationdisplay
n=0
xn
n!(n+ 1)! = 1 +
∞summationdisplay
n=1
bracketleftBigg
1
(n?1)!(n+ 1)! +
1
n!(n+ 1)!
bracketrightBigg
xn =
1 +
∞summationdisplay
n=1
n+ 1
n!(n+ 1)! x
n = 1 +
∞summationdisplay
n=1
xn
(n!)2 =
∞summationdisplay
n=0
xn
(n!)2 = y
a61xyprimeprime +yprime?y = 0.
11,a208a109a181
(1) f(x) = 1a?x (anegationslash= 0)a164a143xa27a152a63a234a167a191a40a189a194a241a137a140a182
(2) f(x) = lnxa143(x?2)a27a152a63a234.
a41a181
190
(1) a207f(x) = 1a?x = 1a
parenleftBigg
1
1? xa
parenrightBigg
a167a133
∞summationdisplay
n=0
parenleftBigg
x
a
parenrightBiggn
= 11? x
a
a167a100a158
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x
a
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle< 1
a75f(x) = 1a
bracketleftBigg ∞summationdisplay
n=0
parenleftBigg
x
a
parenrightBiggnbracketrightBigg
=
∞summationdisplay
n=0
xn
an+1 a167|x|<|a|
(2) f(x) = lnx = ln[2 + (x?2)] = ln2 + ln
parenleftBigg
1 + x?22
parenrightBigg
a207ln
parenleftBigg
1 + x?22
parenrightBigg
=
∞summationdisplay
n=1
(?1)n+1
n
parenleftBigg
x?2
2
parenrightBiggn
=
∞summationdisplay
n=1
(?1)n+1
2n ·n (x?2)
n a1670 <xlessorequalslant 4
a75f(x) = ln2 +
∞summationdisplay
n=1
(?1)n+1
2n ·n (x?2)
n a167a194a241a141a143(0,4].
12,a124a94a174a127a208a109a170a208a109a101a15a188a234a143a152a63a234a167a191a40a189a194a241a137a140a181
(1) e
x?e?x
2
(2) sin2x = 1?cos2x2
a41a181
(1) a207ex =
∞summationdisplay
n=0
xn
n! (?∞<x< +∞),e
x =
∞summationdisplay
n=0
(?1)nxn
n! (?∞<x< +∞)
a75f(x) = e
x?e?x
2 =
1
2
bracketleftBigg ∞summationdisplay
n=0
xn
n!?
∞summationdisplay
n=0
(?1)nxn
n!
bracketrightBigg
a8n = 2ka158a167f(x) = 0a182a8n = 2k+ 1a158a167f(x) =
∞summationdisplay
k=0
x2k+1
(2k+ 1)!
a110a254a140a127a167f(x) =
∞summationdisplay
n=0
x2n+1
(2n+ 1)! a167a194a241a141a143(?∞,+∞).
(2) a207cos2x =
∞summationdisplay
n=0
(?1)n(2x)2n
(2n)! (?∞<x< +∞)
a75sin2x = 1?cos2x2 =
∞summationdisplay
n=1
(?1)n+122n?1
(2n)! x
2na167a194a241a141a143(?∞,+∞).
13,a208a109 ddx
parenleftBigg
ex?1
x
parenrightBigg
a143xa27a152a63a234a167a191a237a2091 =
∞summationdisplay
n=1
n
(n+ 1)!,
a41a181a207ex =
∞summationdisplay
n=0
xn
n! (?∞<x< +∞)a167a75
ex?1
x =
∞summationdisplay
n=0
xn
(n+ 1)! (xnegationslash= 0)
a45f(x) =
ex?1
x,xnegationslash= 01,x = 0 a167a75
∞summationdisplay
n=0
xn
(n+ 1)! a143f(x)a27a152a63a234a167a217a194a241a137a140a143(?∞,+∞)
a100a152a63a234a27a197a145a166a19a189a110a167a26
∞summationdisplay
n=0
xn
(n+ 1)! a51(?∞,+∞)a83a197a145a166a19
d
dx f(x) =
∞summationdisplay
n=1
n
(n+ 1)! x
n?1,xnegationslash= 0
1
2,x = 0
a117a180 ddx
parenleftBigg
ex?1
x
parenrightBigg
=
∞summationdisplay
n=0
n
(n+ 1)! x
n?1
a207 ddx
parenleftBigg
ex?1
x
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x=1
= (x?1)e
x + 1
x2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x=1
= 1a167a75
∞summationdisplay
n=0
n
(n+ 1)! x
n?1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x=1
=
∞summationdisplay
n=0
n
(n+ 1)! = 1
14,a166a101a15a188a234a27a152a63a234a208a109a170a167a191a237a209a194a241a140a187a181
(1)
integraldisplay x
0
sint
t dt
191
(2)
integraldisplay x
0
cost2 dt
a41a181
(1) a207sint =
∞summationdisplay
n=0
(?1)nt2n+1
(2n+ 1)! a167a75
sint
t =
∞summationdisplay
n=0
(?1)nt2n
(2n+ 1)! (tnegationslash= 0)
a45f(t) =
sint
t,tnegationslash= 01,t = 0 a167a75
∞summationdisplay
n=0
(?1)nt2n
(2n+ 1)! a143f(t)a27a152a63a234a167a194a241a141a143(?∞,+∞)
a100a152a63a234a197a145a200a169a189a110a167a26
∞summationdisplay
n=0
(?1)nt2n
(2n+ 1)! a51(?∞,+∞)a83a197a145a200a169
integraldisplay x
0
f(t)dt =
∞summationdisplay
n=0
(?1)nt2n+1
(2n+ 1)(2n+ 1)!,xnegationslash= 0
0,x = 0
a75
integraldisplay x
0
sint
t dt =
∞summationdisplay
n=0
(?1)nt2n+1
(2n+ 1)(2n+ 1)! a167a217a194a241a140a187a143R = +∞.
(2) a207cost2 =
∞summationdisplay
n=0
(?1)n(t2)2n
(2n)! =
∞summationdisplay
n=0
(?1)nt4n
(2n)! a167a217a194a241a141a143(?∞,+∞)a167a194a241a140a187a143R = ∞
a100a152a63a234a27a197a145a200a169a189a110a167a26
∞summationdisplay
n=0
(?1)nt4n
(2n)! a51(?∞,+∞)a83a197a145a200a169
integraldisplay x
0
cost2 dt =
∞summationdisplay
n=0
(?1)nx4n+1
(4n+ 1)(2n)! a167a217a194a241a140a187a143R = ∞.
15,a166a101a15a63a234a27a218a181
(1)
∞summationdisplay
n=1
xn
n!
(2)
∞summationdisplay
n=1
(?1)n+1 x
n+1
n(n+ 1)
(3)
∞summationdisplay
n=1
n2xn?1
(4)
∞summationdisplay
n=0
(2n+ 1)x2n
n!
a41a181
(1) a207ex =
∞summationdisplay
n=0
xn
n! (?∞<x< +∞)a167a75
∞summationdisplay
n=1
xn
n! =
∞summationdisplay
n=0
xn
n!?1 = e
x?1(?∞<x<∞)
(2)
∞summationdisplay
n=1
(?1)n+1 x
n+1
n(n+ 1) =
∞summationdisplay
n=1
(?1)n+1
n x
n+1 +
∞summationdisplay
n=1
(?1)n+1
n+ 1 x
n+1
a207ln(1 +x) =
∞summationdisplay
n=1
(?1)n+1
n x
n (?1 <xlessorequalslant 1)
a75
∞summationdisplay
n=1
(?1)n+1 x
n+1
n(n+ 1) = x
∞summationdisplay
n=1
(?1)n+1
n x
n +
∞summationdisplay
n=2
(?1)n+1
n x
n = xln(1 +x) +
∞summationdisplay
n=1
(?1)n+1
n x
n?x =
(1 +x)ln(1 +x)?x(?1 <xlessorequalslant 1)
x =?1a158a167f(x) =
∞summationdisplay
n=1
1
n(n+ 1) = limN→∞
Nsummationdisplay
n=1
parenleftBigg
1
n?
1
n+ 1
parenrightBigg
= 1
(3)
∞summationdisplay
n=1
n2xn?1 =
∞summationdisplay
n=0
(n+ 1)2xn,an = (n+ 1)2
a75 lim
n→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
an
an+1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= 1a167a117a180a217a194a241a140a187a143R = 1
a8|x| = 1a158a167a100a117(n+ 1)2 → +∞a167a75a63a234a117a209a167a117a180a63a234a27a194a241a141a143(?1,1)
192
a8x∈ (?1,1)a158a167a45f(x) =
∞summationdisplay
n=1
n2xn?1,|x|< 1
a100a53a1592a167a26
∞summationdisplay
n=1
n2xn?1a51(?1,1)a140a197a145a200a169a167
integraldisplay x
0
f(x)dx =
∞summationdisplay
n=1
nxna167a133a217a194a241a140a187a216a67a167a69a1431.
a113a100a53a1592a167a26
∞summationdisplay
n=1
nxna51(?1,1)a254a140a197a145a200a169
integraldisplay x
0
parenleftbiggintegraldisplay x
0
f(x)dx
parenrightbigg
dx =
∞summationdisplay
n=1
integraldisplay x
0
nxndx =
∞summationdisplay
n=1
n
n+ 1 x
n+1 = x2
∞summationdisplay
n=0
xn +
∞summationdisplay
n=1
parenleftBigg
x
n
n
parenrightBigg
+x =
x2
1?x + ln(1?x) +x,|x|< 1
a75
integraldisplay x
0
f(x)dx =
parenleftBigg
x2
1?x + ln(1?x) +x
parenrightBiggprime
= x(1?x)2
a117a180f(x) =
parenleftBigg
x
(1?x)2
parenrightBiggprime
= 1 +x(1?x)3,|x|< 1
(4)
∞summationdisplay
n=0
(2n+ 1)x2n
n! = 1 +
∞summationdisplay
n=1
(2n+ 1)x2n
n! = 1 +
∞summationdisplay
n=1
2x2
(n?1)! x
2(n?1) +
∞summationdisplay
n=1
x2n
n!
a207ex2 = 1 +
∞summationdisplay
n=1
x2n
n! =
∞summationdisplay
n=1
x2(n?1)
(n?1)!
a75
∞summationdisplay
n=0
(2n+ 1)x2n
n! = (2x
2 + 1)ex2,(?∞<x< +∞)
193
§3,a37a67a189a110
1,a51a52a171a109[?1,1]a254a94a203a12a100a34a245a145a170B4(x)a37a67a188a234f(x) = x+|x|2 a167a138a209a188a234y = x+|x|2 a218y = B4(x)a27a227
a47.
a41a181a45x =?1 + 2ya167a75a80 lessorequalslantylessorequalslant 1a158a167?1 lessorequalslantxlessorequalslant 1a167a100a158y = x+ 12,1?y = 1?x2,f(x) = f(?1 + 2y)
a75f(x)a51[?1.1]a254a94a203a12a100a34a245a145a170a143Bn(x) =
nsummationdisplay
k=0
f
parenleftBigg
1 + 2· kn
parenrightBigg
Ckn(x+ 1)
k(1?x)n?k
2n
a75B4(x) =
4summationdisplay
k=0
f
parenleftBigg
1 + k2
parenrightBigg
Ck4 (x+ 1)
k(1?x)4?k
24
a113f(x)a8?1 lessorequalslantxlessorequalslant 0a158a167f(x) = 0a167
a75B4(x) = f
parenleftBigg
1
2
parenrightBigg
C34 (x+ 1)
3(1?x)
24 +f(1)C
4
4
(x+ 1)k
24 =
1
8(1?x)(x+ 1)
3 + 1
16(1 +x)
4.
a45
a54
a0
a0a0
-1 10
1
x
y
2,a23f(x)a180[a,b]a254a27a235a89a188a234a167a121a178a127a51a107a110a88a234a27a245a145a170P(x)a167a166a26 max
x∈[a,b]
|f(x)?P(x)|<ε.a217a165εa180a253a107a137
a189a27a63a191a20a234.
a121a178a181a207f(x)a180[a,b]a254a27a235a89a188a234
a75a100a37a67a189a110a167a26a233a63a191a137a189a27ε> 0a167a189a127a51a245a145a170Q(x)a167a166a26||f(x)?Q(x)|| = max
x∈[a,b]
|f(x)?Q(x)|< ε2
a217a165Q(x) = a0 +a1x+···+anxn(a0,a1,···,ana254a143a162a234)
a23C = max(|a|,|b|)a167a100a162a234a27a200a151a53a167a26a55a127a51a107a110a234bia167a166a26|bi?ai|< ε4(n+ 1)2Ci(i = 0,1,···,n)
a191a23P(x) = b0 +b1x+···+bnxn
a75|P(x)?Q(x)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
i=0
(bi?ai)xi
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
i=1
|bi?ai||x|i <
nsummationdisplay
i=0
ε
4(n+ 1)2Ci C
i < ε
2
a117a180||P(x)?Q(x)|| = max
x∈[a,b]
|P(x)?Q(x)|< ε2
a108a13||f(x)?P(x)||lessorequalslant||f(x)?Q(x)||+||Q(x)?P(x)||<ε
a61a127a51a107a110a88a234a27a245a145a170P(x)a167a166a26 max
x∈[a,b]
|f(x)?P(x)|<ε
194
a49a155a19a217 a76a112a68a63a234a218a76a112a68a67a134
§1,a76a112a68a63a234
1,a121a178a181
(1) 1,cosx,cos2x,···,cosnx,···
(2) sinx,sin2x,sin3x,···,sinnx,···
a180[0,pi]a254a27a20a2a88a182a21,cosx,sinx,cos2x,sin2x,···,cosnx,sinnx,···a216a180[0,pi]a254a27a20a2a88.
a121a178a181
(1) a207
integraldisplay pi
0
1·coskxdx = 0 (k = 1,2,···),
integraldisplay pi
0
coskx·coslxdx =
0,k negationslash= l,k,l = 1,2,···
pi
2,k = l = 1,2,···integraldisplay pi
0
12 dx = pi
a751,cosx,cos2x,···,cosnx,···a180[0,pi]a254a27a20a2a88
(2) a207
integraldisplay pi
0
sinkxsinlxdx =
0,k negationslash= l,k,l = 1,2,···
pi
2,k = l = 1,2,···
a75sinx,sin2x,sin3x,···,sinnx,···a180[0,pi]a254a27a20a2a88
a113
integraldisplay pi
0
1·sinxdx = 2 negationslash= 0a167a751,cosx,sinx,cos2x,sin2x,···,cosnx,sinnx,···a216a180[0,pi]a254a27a20a2a88.
2,a121a178a181sinx,sin3x,···,sin(2n+ 1)x,···a180
bracketleftBigg
0,pi2
bracketrightBigg
a254a27a20a2a88a167a21a209a167a27a73a79a20a2a88
parenleftBigg
a61a216a61a20a2a167a13a133a122a135a188a234a27a178a144a51
bracketleftBigg
0,pi2
bracketrightBigg
a254a27a200a169a1431
parenrightBigg
a167a191a19a209sin pix2l,sin 3pix2l,···,sin (2n+ 1)pix2l,···a180[0,l]a254
a27a20a2a88.
a121a178a181a207
integraldisplay pi
2
0
[sin(2k+ 1)xsin(2l+ 1)x]dx =
0,k negationslash= l,k,l = 1,2,···
pi
4,k = l = 1,2,···
a75sinx,sin3x,···,sin(2n+ 1)x,···a180
bracketleftBigg
0,pi2
bracketrightBigg
a254a27a20a2a88
a113a100
integraldisplay pi
2
0
bracketleftBigg
sin(2k+ 1)x
a
bracketrightBigg2
dx = pi4a2 = 1(k = 1,2,···)a167a26a =
√pi
2
a75a51
bracketleftBigg
0,pi2
bracketrightBigg
a254a167a27a73a79a20a2a88a1432sinx√pi,2sin3x√pi,···,2sin(2n+ 1)x√pi,···
a113
integraldisplay l
0
sin (2k+ 1)pix2l sin sin(2m+ 1)pix2l dx =
0,k negationslash= m,k,m = 1,2,···
l
2 negationslash= 0,k = m = 1,2,···
a75sin pix2l,sin 3pix2l,···,sin (2n+ 1)pix2l,···a180[0,l]a254a27a20a2a88.
3,a23f(t)a180a177a207a143Ta27a144a197a167a167a51
bracketleftBigg
T2,T2
bracketrightBigg
a254a27a188a234a76a171a170a143
f(t) =
E,a80 lessorequalslantt< T2 a158
0,a8? T2 lessorequalslantt< 0a158
a242a249a135a144a197a208a109a164a76a112a68a63a234.
a41a181a207ω = T2,f(t) =
E,a80 lessorequalslantt< T2 a158
0,a8? T2 lessorequalslantt< 0a158
195
a75ak = 2T
integraldisplay T
2
T2
f(t)coskωtdt = 2T
integraldisplay T
2
0
Ecos 2kpiT tdt = 0
a0 = 2T
integraldisplay T
2
T2
dt = 2T
integraldisplay T
2
0
Edt = E
bk = 2T
integraldisplay T
2
T2
f(t)sinkωtdt = T2
integraldisplay T
2
0
Esin 2kpiT tdt =
0,ka143a243
2E
kpi,ka143a219
a75f(x) ～ E2 + 2Epi
∞summationdisplay
k=1
1
2k?1 sin
(4k?2)pix
T =
E,0 <x< T2
0,?T2 <x< 0
E
2,x = 0,±
T
2
4,a23f(t)a180a177a207a143Ta27a140a197a18a54a197a167a167a51
bracketleftBigg
T2,T2
parenrightBigg
a254a27a188a234a76a171a170a143
f(t) =
Umsinωt,a80 lessorequalslantt< T2a158
0,a8? T2 lessorequalslantt< 0a158
a114a249a140a197a18a54a197a208a109a164a76a112a68a63a234.
a41a181a207ωprime = 2piT,f(t) =
Umsinωt,a80 lessorequalslantt< T2a158
0,a8? T2 lessorequalslantt< 0a158
a75a0 = 2T
integraldisplay T
2
T2
f(t)dt = 2UmωT
parenleftBigg
1?cos Tω2
parenrightBigg
ak = 2T
integraldisplay T
2
T2
f(t)coskωprimetdt = UmωT + 2kpi
parenleftBigg
1?cos Tω+ 2kpi2
parenrightBigg
+ UmωT?2kpi
parenleftBigg
1?cos Tω?2kpi2
parenrightBigg
bk = 2T
integraldisplay T
2
T2
f(t)sinkωprimetdt = UmωT?2kpi sin Tω?2kpi2? UmωT + 2kpi sin Tω+ 2kpi2
a75f(t) ～ 2UmωT
parenleftBigg
1?cos Tω2
parenrightBigg
+
∞summationdisplay
k=1
parenleftBigg
akcos 2kpiT t+bksin 2kpiT t
parenrightBigg
=
Umsinωt,0 lessorequalslantt< T2
0,?T2 <t< 0
Um
2 sin
Tω
2,t = ±
T
2
5,a23f(t)a1772pia143a177a207a167a51[?pi,pi)a83
f(t) =
braceleftbigg t,a8?pilessorequalslantt< 0a158
0,a80 lessorequalslantt<pia158
a114f(t)a208a109a164a76a112a68a63a234.
a41a181a207a0 = 1pi
integraldisplay pi
pi
f(t)dt =?pi2
ak = 1pi
integraldisplay pi
pi
f(t)cosktdt = 1k2pi[1?(?1)k]
bk = 1pi
integraldisplay pi
pi
f(t)sinktdt = (?1)
k+1
k
a75f(t) ～?pi4 +
∞summationdisplay
k=1
(?1)k+1
k sinkt+
1
pi
∞summationdisplay
k=1
1?(?1)k
k2 coskt =
pi4 +
∞summationdisplay
k=1
(?1)k+1
k sinkt+
2
pi
∞summationdisplay
k=0
1
(2k+ 1)2 cos(2k+ 1)t =
t,?pi<t< 0
0,0 lessorequalslantt<pi
pi2,t = ±pi
6,a23f(t)a180a177a207a1432pia33a112a143ha27a231a184a47a197a167a167a51[0,2pi)a254a27a188a234a76a171a170a143f(t) = h2pi ta167a242a249a135a231a184a47a197a208a109a164a76
a112a68a63a234.
196
a41a181a207a0 = 1pi
integraldisplay 2pi
0
f(t)dt = h
ak = 1pi
integraldisplay 2pi
0
f(t)cosktdt = 0
bk = 1pi
integraldisplay 2pi
0
f(t)sinktdt =? hkpi
a75f(t) ～ h2 +
∞summationdisplay
k=1
(akcoskt+bksinkt) = h2? hpi
∞summationdisplay
k=1
sinkt
k =
h
2pi t,0 <t< 2pih
2,t = 0,2pi
7,a242a176a221a143τa33a112a143ha33a177a207a143Ta27a221a47a197a208a109a164a123a117a63a234.
a41a181a51a152a135a177a207
bracketleftBigg
T2,T2
bracketrightBigg
a83a221a47a197a188a234a76a136a170a143f(t) =
0,?T2 lessorequalslantt<?τ2
h,?τ2 lessorequalslanttlessorequalslant τ2
0,τ2 <tlessorequalslant T2
a75a0 = 2T
integraldisplay T
2
T2
f(t)dt = 2hT τ
ak = 2T
integraldisplay T
2
T2
f(t)cos 2kpiT tdt = 2hkpi sin kτT pi
bk = 2T
integraldisplay T
2
T2
f(t)sin 2kpiT tdt = 0
a117a180f(t) ～ hT τ +
∞summationdisplay
k=1
2h
kpi sin
kpi
T τ cos
2kpi
T t
8,a21a209a88a22712-5a164a171a27a177a207a143Ta27a110a14a197a51
bracketleftBigg
0,T2
parenrightBigg
a83a27a188a234a76a171a170a167a191a242a167a208a109a164a20a117a63a234.
a45
a54
a64
a64
a64
a64
a64a64a0a0
a0
a0
a0a0a64a64
a64
a64
a64a64
0 T4 T2
T4
T2
y
x
a41a181a88a227a164a171a27a177a207a143Ta27a110a14a197a51
bracketleftBigg
0,T2
parenrightBigg
a27a188a234a76a136a170a143f(t) =
4E
T t,0 lessorequalslantt<
T
4
4E
T
parenleftBigg
T
2?t
parenrightBigg
,T4 lessorequalslantt< T2
a107a114f(t)a242a255a164
bracketleftBigg
T2,T2
bracketrightBigg
a254a27a188a234a167a50a226a75a191a167a132a55a76a114a167a242a255a164a219a188a234a167a117a180a0 = ak = 0
bk = 4T
integraldisplay T
2
0
f(t)sin 2kpiT tdt = 8Ek2pi2 sin k2 pi =
0,ka143a243
(?1)k?12 ·8E
k2pi2,ka143a219
a75f(t) ～ 8Epi2
∞summationdisplay
n=1
(?1)n+1
(2n?1)2 sin
2(2n?1)pi
T t
9,a51a171a109(0,2pi)a165a208a109f(x) = pi?x2 a164a76a112a68a63a234.
a41a181a207a0 = 1pi
integraldisplay 2pi
0
pi?x
2 dx = 0
ak = 1pi
integraldisplay 2pi
0
pi?x
2 coskxdx = 0
bk = 1pi
integraldisplay 2pi
0
pi?x
2 sinkxdx =
1
k
197
a75f(x) ～
∞summationdisplay
k=1
1
k sinkx =
pi?x
2 (0 <x< 2pi)
10,a51a171a109(?pi,pi)a165a208a109f(x) = pi2?x2a164a76a112a68a63a234.
a41a181a207a51(?pi,pi)a254a167f(x) = pi2?x2a143a243a188a234a167a75bk = 0
a113a0 = 2pi
integraldisplay pi
0
(pi2?x2)dx = 43 pi2
ak = 2pi
integraldisplay pi
0
(pi2?x2)coskxdx = (?1)k+1 4k2
a75f(x) ～ 23 pi2 + 4
∞summationdisplay
k=1
(?1)k+1
k2 coskx = pi
2?x2 (?pi<x<pi)
11,a242f(x) = sgn(cosx)a208a109a164a76a112a68a63a234.
a41a181a207f(x+ 2pi) = sgn[cos(x+ 2pi)] = sgn(cosx) = f(x)a167a75f(x)a180a1772pia143a177a207a27a177a207a188a234
a100f(?x) = f(x)a167a75f(x)a143a243a188a234a167a117a180bk = 0
a113a0 = 2pi
integraldisplay pi
0
sgn(cosx)dx = 2pi
bracketleftBiggintegraldisplay pi
2
0
dx+
integraldisplay pi
pi
2
(?1)dx
bracketrightBigg
= 0
ak =
integraldisplay pi
0
sgn(cosx)coskxdx = 4kpi sin kpi2 =
0,k = 2n
(?1)n 4(2n+ 1)pi,k = 2n+ 1 (n = 0,1,2,···)
a75f(x)a51(?∞,+∞)a254a140a208a143f(x) ～ 4pi
∞summationdisplay
n=0
(?1)n
(2n+ 1) cos(2n+ 1)x = sgn(cosx)
12,a65a8a88a219a114a171a109
parenleftBigg
0,pi2
parenrightBigg
a83a27a140a200a188a234f(x)a242a255a0a167a166a167a208a109a164a27a76a112a68a63a234a27a47a71a88a101a181
f(x) ～
∞summationdisplay
n=1
ancos(2n?1)x (?pi<x<pi)
a41a181a207a208a109a170a165a195a20a117a145a167a75f(x)a242a255a0a65a143a243a188a234
a23f(x)a242a255a20
parenleftBigg
pi
2,pi
parenrightBigg
a83a27a220a169a143?(x)
a207a208a109a170a165a243a234a145a27a88a234a2n = 0a61a2n = 2pi
bracketleftBiggintegraldisplay pi
2
0
f(x)cos2nxdx+
integraldisplay pi
pi
2
(x)cos2nxdx
bracketrightBigg
= 0
a75
integraldisplay pi
2
0
f(x)cos2nxdx+
integraldisplay pi
pi
2
(x)cos2nxdx = 0
a51a134a224a99a152a200a169a165a138a67a254a147a134a167a45x = pi?t
a75?
integraldisplay pi
2
pi
f(pi?t)cos2n(pi?t)dt+
integraldisplay pi
pi
2
(x)cos2nxdx = 0a61
integraldisplay pi
pi
2
[f(pi?x) +?(x)]cos2nxdx = 0
a135a166a254a170a164a225a167a75a55a76a8x∈
parenleftBigg
pi
2,pi
parenrightBigg
a158a34a107f(pi?x) +?(x) = 0a61?(x) =?f(pi?x)
a117a180a210a166a209a10a242a255a0a27a188a234a51
parenleftBigg
pi
2,pi
parenrightBigg
a83a27a76a136a170a143?f(pi?x)
a113a242a255a0a27a188a234a143a243a188a234a167a75a167a51
parenleftBigg
pi2,0
parenrightBigg
a27a76a136a170a143f(?x)a167a51
parenleftBigg
pi,?pi2
parenrightBigg
a27a76a136a170a143?f(pi+x)
a216a148a23a242a255a0a27a188a234a143ψ(x)a167a75ψ(x) =
f(pi+x),?pi<x<?pi2
f(?x),?pi2 <x< 0
f(x),0 <x< pi2
f(pi?x),pi2 <x<pi
13,a211a254a152a75a167a2a208a109a27a76a112a68a63a234a47a71a143a181
f(x) ～
∞summationdisplay
n=1
bnsin(2n?1)x (?pi<x<pi)
198
a41a181a207a208a109a170a165a195a123a117a145a167a75f(x)a242a255a0a65a143a219a188a234
a23f(x)a242a255a20
parenleftBigg
pi
2,pi
parenrightBigg
a83a27a220a169a143?(x)
a207a208a109a170a165a243a234a145a27a88a234b2n = 0a61b2n = 2pi
bracketleftBiggintegraldisplay pi
2
0
f(x)sin2nxdx+
integraldisplay pi
pi
2
(x)sin2nxdx
bracketrightBigg
= 0
a75
integraldisplay pi
2
0
f(x)sin2nxdx+
integraldisplay pi
pi
2
(x)sin2nxdx = 0
a51a134a224a99a152a200a169a165a138a67a254a147a134a167a45x = pi?t
a75?
integraldisplay pi
2
pi
f(pi?t)sin2n(pi?t)dt+
integraldisplay pi
pi
2
(x)sin2nxdx = 0a61
integraldisplay pi
pi
2
[?f(pi?x) +?(x)]sin2nxdx = 0
a135a166a254a170a164a225a167a75a55a76a8x∈
parenleftBigg
pi
2,pi
parenrightBigg
a158a34a107?f(pi?x) +?(x) = 0a61?(x) = f(pi?x)
a117a180a210a166a209a10a242a255a0a27a188a234a51
parenleftBigg
pi
2,pi
parenrightBigg
a83a27a76a136a170a143f(pi?x)
a113a242a255a0a27a188a234a143a219a188a234a167a75a167a51
parenleftBigg
pi2,0
parenrightBigg
a27a76a136a170a143?f(?x)a167a51
parenleftBigg
pi,?pi2
parenrightBigg
a27a76a136a170a143?f(pi+x)
a216a148a23a242a255a0a27a188a234a143ψ(x)a167a75ψ(x) =
f(pi+x),?pi<x<?pi2
f(?x),?pi2 <x< 0
f(x),0 <x< pi2
f(pi?x),pi2 <x<pi
14,a23f(x)a140a200a33a253a233a140a200a167a121a178a181
(1) a88a74a188a234f(x)a51[?pi,pi]a254a247a118f(x+pi) = f(x)a167a64a34a2m?1 = b2m?1 = 0
(2) a88a74a188a234f(x)a51[?pi,pi]a254a247a118f(x+pi) =?f(x)a167a64a34a2m = b2m = 0
a121a178a181
(1) a207f(x)a140a200a33a253a233a140a200a133a188a234f(x)a51[?pi,pi]a254a247a118f(x+pi) = f(x)
a75f(x)a51[?pi,pi]a254a140a200a33a253a233a140a200a133a177pia143a177a207
a117a180ak = 1pi
integraldisplay pi
pi
f(x)coskxdx = 1pi
bracketleftbiggintegraldisplay 0
pi
f(x)coskxdx+
integraldisplay pi
0
f(x)coskxdx
bracketrightbigg
a233a109a224a49a19a170a138a67a254a147a134a181t = x?pia167a75a217a67a1431pi
integraldisplay pi
0
f(x)coskxdx = 1pi
integraldisplay 0
pi
f(t)cosk(t+pi)dt
a117a180ak = 1pi
integraldisplay 0
pi
[1 + (?1)k]f(x)coskxdx
a108a13a167a26a2m?1 = 0(m = 1,2,···)
a211a110a167a26b2m?1 = 0(m = 1,2,···)
(2) a207f(x)a140a200a33a253a233a140a200a133a188a234f(x)a51[?pi,pi]a254a247a118f(x+pi) =?f(x)a167a75f(x+ 2pi) = f(x)
a117a180f(x)a51[?pi,pi]a254a140a200a33a253a233a140a200a133a1772pia143a177a207
a117a180ak = 1pi
integraldisplay pi
pi
f(x)coskxdx = 1pi
bracketleftbiggintegraldisplay 0
pi
f(x)coskxdx+
integraldisplay pi
0
f(x)coskxdx
bracketrightbigg
a233a109a224a49a19a170a138a67a254a147a134a181t = x?pia167a75a217a67a1431pi
integraldisplay pi
0
f(x)coskxdx =?1pi
integraldisplay 0
pi
f(t)cosk(t+pi)dt
a117a180ak = 1pi
integraldisplay 0
pi
[1 + (?1)k+1]f(x)coskxdx
a108a13a167a26a2m = 0(m = 1,2,···)
a211a110a167a26b2m = 0(m = 1,2,···)
15,a177a207a1432pia27a140a200a218a253a233a140a200a188a234f(x)a27a76a112a68a88a234a143an,bna167a79a142a181
(1) a188a234f(x+k) (ka143a126a234)a27a76a112a68a88a234an,bna182
(2) F(x) = 1pi
integraldisplay pi
pi
f(t)f(x?t)dta27a76a112a68a88a234An,Bna167a23a107a39a27a200a169a94a83a140a2a134.
199
a41a181
(1) a100a174a127a167a26a0 = 1pi
integraldisplay pi
pi
f(x)dx,an = 1pi
integraldisplay pi
pi
f(x)cosnxdx,bn = 1pi
integraldisplay pi
pi
f(x)sinnxdx
a75a138a147a134x+k = ya133f(x)a180a1772pia143a177a207a27a188a234a167a107
a0 = 1pi
integraldisplay pi
pi
f(x+k)dx = 1pi
integraldisplay pi+k
pi+k
f(y)dy = a0
an = 1pi
integraldisplay pi
pi
f(x+k)cosnxdx = 1pi
integraldisplay pi+k
pi+k
f(y)cosn(y?k)dy = ancosnk+bnsinnk
a61an = ancosnk+bnsinnk (n = 0,1,2,···)
a211a110a167a140a166a26bn = bncosnk?ansinnk
(2) a207f(x)a180a177a207a1432pia27a140a200a218a253a233a140a200a188a234
a75F(x+ 2pi) = 1pi
integraldisplay pi
pi
f(t)f(x+ 2pi?t)dt = F(x)a167a117a180F(x)a69a180a1772pia143a177a207a27a188a234
a113a0 = 1pi
integraldisplay pi
pi
f(x)dx,an = 1pi
integraldisplay pi
pi
f(x)cosnxdx,bn = 1pi
integraldisplay pi
pi
f(x)sinnxdx
a75A0 = 1pi
integraldisplay pi
pi
F(x)dx = 1pi2
integraldisplay pi
pi
f(t)dt
integraldisplay pi
pi
f(x?t)dx
a233
integraldisplay pi
pi
f(x?t)dxa138a147a134x?t = ya133f(x)a180a1772pia143a177a207a27a188a234a167a107
A0 = 1pi2
integraldisplay pi
pi
f(t)dt
integraldisplay pi?t
pi?t
f(y)dy = 1pi2
bracketleftbiggintegraldisplay pi
pi
f(t)dt
bracketrightbigg2
= a20
a211a110a167a140a166a26An = a2n?b2n
Bn = 2anbn
16,a88a74?(?x) = ψ(x)a167a175?(x)a134ψ(x)a27a76a112a68a88a234a131a109a107a159a111a39a88a186
a41a181a188a234?(x)a134ψ(x)a27a76a112a68a88a234a169a143an = 1pi
integraldisplay pi
pi
(x)cosnxdx,bn = 1pi
integraldisplay pi
pi
(x)sinnxdx
αn = 1pi
integraldisplay pi
pi
ψ(x)cosnxdx,βn = 1pi
integraldisplay pi
pi
ψ(x)sinnxdx
a233an = 1pi
integraldisplay pi
pi
(x)cosnxdxa109a224a138a67a254a147a134y =?xa167a191a242?(?x) = ψ(x)a147a92a167a26
an = 1pi
integraldisplay?pi
pi
(?y)cosn(?y)d(?y) = 1pi
integraldisplay pi
pi
ψ(x)cosnxdx = αn (n = 0,1,2,···)
a211a110a167a26bn =?βn (n = 1,2,···)
17,a88a74?(?x) =?ψ(x)a167a175?(x)a134ψ(x)a27a76a112a68a88a234a131a109a107a159a111a39a88a186
a41a181a188a234?(x)a134ψ(x)a27a76a112a68a88a234a169a143an = 1pi
integraldisplay pi
pi
(x)cosnxdx,bn = 1pi
integraldisplay pi
pi
(x)sinnxdx
αn = 1pi
integraldisplay pi
pi
ψ(x)cosnxdx,βn = 1pi
integraldisplay pi
pi
ψ(x)sinnxdx
a233an = 1pi
integraldisplay pi
pi
(x)cosnxdxa109a224a138a67a254a147a134y =?xa167a191a242?(?x) =?ψ(x)a147a92a167a26
an = 1pi
integraldisplay?pi
pi
(?y)cosn(?y)d(?y) =?1pi
integraldisplay pi
pi
ψ(x)cosnxdx =?αn (n = 0,1,2,···)
a211a110a167a26bn = βn (n = 1,2,···)
18,a23f(t)a51(?pi,pi)a254a169a227a235a89a167a8t = 0a235a89a133a107a252a253a19a234a167a121a178a8p→∞a158integraldisplay
pi
pi
f(t) cos
t
2?cospt
2sin t2 dt→
1
2
integraldisplay pi
0
[f(t)?f(?t)]cot t2 dt
a121a178a181
integraldisplay pi
pi
f(t) cos
t
2?cospt
2sin t2 dt =
integraldisplay 0
pi
f(t) cos
t
2?cospt
2sin t2 dt+
integraldisplay pi
0
f(t) cos
t
2?cospt
2sin t2 dt
a51a109a224a99a152a200a169a165a45t =?xa167a75
integraldisplay 0
pi
f(t) cos
t
2?cospt
2sin t2 dt =?
integraldisplay pi
0
f(?t) cos
t
2?cospt
2sin t2 dt
a147a163a6a170a167a26
integraldisplay pi
pi
f(t) cos
t
2?cospt
2sin t2 dt =?
integraldisplay pi
0
f(?t) cos
t
2?cospt
2sin t2 dt +
integraldisplay pi
0
f(t) cos
t
2?cospt
2sin t2 dt =
1
2
integraldisplay pi
0
[f(t)?f(?t)]cot t2 dt? 12
integraldisplay pi
0
f(t)?f(?t)
2sin t2 cosptdt
200
a101a121 lim
p→∞
integraldisplay pi
0
f(t)?f(?t)
2sin t2 cosptdt = 0
a207
integraldisplay pi
0
f(t)?f(?t)
2sin t2 cosptdt =
bracketleftbiggintegraldisplay δ
0
+
integraldisplay pi
δ
bracketrightbiggf(t)?f(?t)
2sin t2 cosptdt(a217a1650 <δ<pi)
a233a117
integraldisplay pi
δ
f(t)?f(?t)
2sin t2 cosptdt
a207f(t)a51(?pi,pi)a254a169a227a235a89a167 12sin t
2
a51(δ,pi)a254a235a89a167a75f(t)?f(?t)2sin t
2
a51(δ,pi)a254a169a227a235a89a207a13a140a200
a75a100a105a249a218a110a167a26 lim
p→∞
integraldisplay pi
δ
f(t)?f(?t)
2sin t2 cosptdt = 0
a233a117
integraldisplay δ
0
f(t)?f(?t)
2sin t2 cosptdt
lim
p→∞
integraldisplay δ
0
f(t)?f(?t)
2sin t2 cosptdt = limp→∞
integraldisplay δ
0
[f(t)?f(?t)]
parenleftBigg
1
2sin t2?
1
t
parenrightBigg
cosptdt+ lim
p→∞
integraldisplay δ
0
f(t)?f(?t)
t cosptdt
a207lim
t→0
parenleftBigg
1
2sin t2?
1
t
parenrightBigg
= 0a167a214a191a189a194a167t = 0a158a167a188a234 12sin t
2
1t a27a138a1430a167a75 12sin t
2
1t a180[0,δ]a254a27a235a89
a188a234
a113f(t)a143(?pi,pi)a254a27a169a227a235a89a188a234a167a75[f(t)?f(?t)]
parenleftBigg
1
sin t2?
1
t
parenrightBigg
a51[0,δ]a254a169a227a235a89a167a207a13a140a200a167a75a100a105a249
a218a110a167a26 lim
p→∞
integraldisplay δ
0
f(t)?f(?t)
2sin t2 cosptdt = 0
a207f prime(+0),f prime(?0)a127a51a167a75 lim
t→+0
f(t)?f(?t)
t = f
prime(+0) +f prime(?0)a127a51
a214a191a189a194a167t = 0a158a167a188a234f(t)?f(?t)t a138a143f prime(+0) +f prime(?0)a167a75f(t)?f(?t)t a180[0,δ]a254a27a169a227a188a234a167a207a13
a140a200a167a117a180a100a105a249a218a110a167a26 lim
p→∞
integraldisplay δ
0
f(t)?f(?t)
t cosptdt = 0
a110a254a140a26a167a8p→∞a158a167
integraldisplay pi
0
f(t)?f(?t)
2sin t2 cosptdt→
1
2
integraldisplay pi
0
[f(t)?f(?t)]cot t2 dt
19,a23Tn(x) = 12 +
nsummationdisplay
v=1
cosvx,T0(x) = 12,σn(x) = T0(x) +···+Tn(x)n+ 1
a121a178
(1) σn(x) = 12(n+ 1)
parenleftBigg
sin n+12 x
sin x2
parenrightBigg2
(2)
integraldisplay pi
pi
σn(x)dx = pi
a121a178a181
(1) a2072sin x2
parenleftBigg
1
2 +
nsummationdisplay
v=1
cosvx
parenrightBigg
= sin 2n+ 12 xa167a75Tn(x) = sin
2n+1
2 x
2sin x2
a117a180σn(x) = T0(x) +···+Tn(x)n+ 1 =
1
2 +
nsummationdisplay
k=1
Tk(x)
n+ 1 =
1
n+ 1
parenleftBigg
1
2 +
nsummationdisplay
k=1
sin 2n+12 x
2sin x2
parenrightBigg
= 12(n+ 1)sin2 x
2
parenleftBigg
sin2 x2 +
nsummationdisplay
k=1
sin x2sin 2k+ 12 x
parenrightBigg
= 12(n+ 1)sin2 x
2
bracketleftBigg
1
2?
1
2 cos(n+ 1)x
bracketrightBigg
=
1
2(n+ 1)
parenleftBigg
sin 2n+12 x
2sin x2
parenrightBigg2
(2)
integraldisplay pi
pi
σn(x)dx =
integraldisplay pi
pi
1
2 +
nsummationdisplay
k=1
Tk(x)
n+ 1 dx =
1
n+ 1
integraldisplay pi
pi
bracketleftBigg
1
2 +
nsummationdisplay
k=1
parenleftBigg
1
2 +
nsummationdisplay
v=1
cosvx
parenrightBiggbracketrightBigg
dx =
1
n+ 1
bracketleftBigg
pi+
nsummationdisplay
k=1
parenleftBigg
pi+
nsummationdisplay
v=1
integraldisplay pi
pi
cosvxdx
parenrightBiggbracketrightBigg
= pi.
201
20,a23?(x)a51[a,b]a254a143a252a78a79a92a188a234a167a121a178
(1) a88a74a = 0,b< 0a167a1071pi
integraldisplay b
a
(z) sinpzz dz →?12?(?0) (p→∞)
(2) a88a74a< 0,b> 0a167a1071pi
integraldisplay b
a
(z) sinpzz dz →?(+0) +?(?0)2 (p→∞)
a121a178a181
(1) a207?(x)a51[a,b]a254a143a252a78a79a92a188a234a167a75?(?t)a51[?b,?a]a254a143a252a78a126a8a188a234
a8a = 0,b< 0a158a167?(?t)a51[0,?b](?b> 0)a254a143a252a78a79a92a188a234
a233
integraldisplay b
a
(z) sinpzz dza138a67a254a147a134z =?ta167a75
integraldisplay b
a
(z) sinpzz =?
integraldisplay?b
0
(?t) sinptt dt
a75a100a41a225a142a52a218a110a167a26 lim
p→∞
integraldisplay?b
0
(?t) sinptt dt = pi2?(?0)a61 lim
p→∞
integraldisplay b
a
(z) sinpzz dz =?pi2?(?0)
a117a1801pi
integraldisplay b
a
(z) sinpzz dz →?12?(?0) (p→∞)
(2) a207a< 0,b> 0a167?(x)a51[a,b]a254a143a252a78a79a92a188a234a167
integraldisplay b
a
(z) sinpzz dz =
integraldisplay 0
a
(z) sinpzz dz+
integraldisplay b
0
(z) sinpzz dz
a226(1)a167a26 lim
p→∞
integraldisplay a
0
(z) sinpzz dz =?pi2?(?0)a167a75 lim
p→∞
integraldisplay 0
a
(z) sinpzz dz = pi2?(?0)
a113a100a41a225a142a52a218a110a167a26 lim
p→∞
integraldisplay b
0
(z) sinpzz dt = pi2?(+0)
a75 lim
p→∞
integraldisplay b
a
(z) sinpzz dz = pi2[?(+0) +?(?0)]
a117a1801pi
integraldisplay b
a
(z) sinpzz dz →?(?0) +?(+0)2 (p→∞)
202
§2,a76a112a68a67a134
1,a23f(x)a51(?∞,+∞)a83a253a233a140a200a167a121a178hatwidef(ω)a51(?∞,+∞)a83a235a89.
a121a178a181a233?ω ∈ (?∞,+∞)a167a111a107Aprime,Aprimeprimea167a166a26ω ∈ [Aprime,Aprimeprime ]
a100a117
vextendsinglevextendsingle
vextendsinglehatwidef(ω)
vextendsinglevextendsingle
vextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
∞
f(x)e?iωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay +∞
∞
|f(x)|dx
a0a246a194a241a133a216a185a235a254ωa167a249a76a178a200a169hatwidef(ω) =
integraldisplay +∞
∞
f(x)e?iωxdxa51[Aprime,Aprimeprime ]a254a152a151a194a241
a226a152a151a194a241a200a169a27a235a89a53a167a26hatwidef(ω)a51[Aprime,Aprimeprime ]a254a235a89a167a108a13a51a58ωa63a235a89
a100ωa27a63a191a53a167a26hatwidef(ω)a51(?∞,+∞)a83a235a89.
2,a23f(x)a51(?∞,+∞)a83a253a233a140a200a167a121a178 lim
ω→∞
hatwidef(ω) = 0.
a121a178a181a100f(x)a51(?∞,+∞)a83a253a233a140a200a167a26a233a117a63a137a27ε > 0a167a127a51A > 0a167a166a107
integraldisplay +∞
A
|f(x)|dx < ε3
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
A
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay +∞
A
|f(x)|dx< ε3
a23f(x)a51[0,A]a83a195a215a58a167a75a51[0,A]a165a2a92a169a580 = t0 <t1 <···<tm = Aa167a191a23f(x)a51[tk?1,tk]a254a27a101a40
a46a143mka167a117a180integraldisplay
A
0
f(x)sinωxdx =
msummationdisplay
k=1
integraldisplay tk
tk?1
f(x)sinωxdx =
msummationdisplay
k=1
integraldisplay tk
tk?1
[f(x)?mk]sinωxdx+
msummationdisplay
k=1
mk
integraldisplay tk
tk?1
sinωxdx
a108a13
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
0
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
msummationdisplay
k=1
ωk?tk +
msummationdisplay
k=1
|mk||cosntk?1?cosntk|n lessorequalslant
msummationdisplay
k=1
ωk?tk + 2ω
msummationdisplay
k=1
|mk|
a217a165ωka143f(x)a51a171a109[tk?1,tk]a254a27a8a204a167?tk = tk?tk?1
a100a117f(x)a51[0,A]a254a140a200a167a25a140a18a44a152a169a123a167a166a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
msummationdisplay
k=1
ωk?tk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<
ε
3
a233a117a249a24a27a189a27a169a123a167
msummationdisplay
k=1
|mk|a143a152a189a138a167a207a13a127a51δ> 0a167a166a8ω>δa158a167a240a1072ω
msummationdisplay
k=1
|mk|< ε3
a117a180a233a254a227a164a192a18a27δa167a8ω>δa158vextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay +∞
0
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
0
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsingle+
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
A
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslantεa61 lim
ω→∞
integraldisplay +∞
0
f(x)sinωxdx = 0
a217a103a167a23f(x)a51a171a109[0,A]a165a107a215a58a167a143a123a66a229a132a167a216a148a23a144a107a152a135a215a58a133a1430
a117a180a233a63a137a27ε> 0a167a127a51η> 0a167a166a107
integraldisplay η
0
|f(x)|dx< ε3
a113f(x)a51[η,A]a254a195a97a58a167a25a65a94a254a227a40a74a140a26a127a51δa167a166a8ω>δa158a167a240a107
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
η
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsingle< ε
3
a117a180a8ω>δa158a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
0
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay η
0
|f(x)|dx+
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay A
η
f(x)sinωxdx
vextendsinglevextendsingle
vextendsinglevextendsingle+
integraldisplay +∞
A
|f(x)|dx<ε
a61 lim
ω→∞
integraldisplay +∞
0
f(x)sinωxdx = 0
a211a123a167a26a8f(x)a51(?∞,+∞)a83a253a233a140a200a158a167a254a107 lim
ω→∞
integraldisplay +∞
∞
f(x)sinωxdx = 0
a211a123a140a121a26a8f(x)a51(?∞,+∞)a83a253a233a140a200a158a167 lim
ω→∞
integraldisplay +∞
∞
f(x)cosωxdx = 0
a117a180 lim
ω→∞
hatwidef(x) = 0.
3,a166a101a15a188a234a27a76a112a68a67a134a181
(1) f(x) =
Esinω0x,|x|< piω
0
0,|x|greaterorequalslant piω
0
(2) f(x) =
0,?∞<xlessorequalslant?pi2
2h
τ x+h,?
τ
2 <x< 0
2hτ x+h,0 lessorequalslantx< τ2
0,τ2 lessorequalslantx< +∞
203
a41a181
(1) hatwidef(ω) =
integraldisplay ∞
∞
f(x)e?iωxdx =
integraldisplay pi
ω0
piω0
Esinω0xe?iωxdx = E
integraldisplay pi
ω0
piω0
sinω0x(cosωx?isinωx)dx =
2Ei
integraldisplay pi
ω0
0
sinω0xsinωxdx = iE
integraldisplay pi
ω0
0
[cos(ω0+ω)x?cos(ω?ω0)x]dx = iE
sin(ω0 +ω)xω
0 +ω
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
pi
ω0
0
sin(ω?ω0)xω?ω
0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
pi
ω0
0
=
2Eω0i
ω2?ω20 sin
ω
ω0 pi(ω negationslash= ±ω0)
a207hatwidef(ω)a143(?∞,+∞)a83a27a235a89a188a234a167a75hatwidef(?ω0) = lim
ω→?ω0
hatwidef(ω) = ±iEpi
ω0,
(2) hatwidef(ω) =
integraldisplay ∞
∞
f(x)e?iωxdx =
integraldisplay 0
τ2
parenleftBigg
2h
τ x+h
parenrightBigg
e?iωxdx+
integraldisplay τ
2
0
parenleftBigg
2hτ x+h
parenrightBigg
e?iωxdx =
2h
τ
bracketleftBiggintegraldisplay
0
τ2
xe?iωxdx?
integraldisplay τ
2
0
xe?iωxdx
bracketrightBigg
+ 2hω sin ωτ2 = 4hτω2? 4hτω2 cos ωτ2 (ω negationslash= 0)
a207hatwidef(ω)a143(?∞,+∞)a83a27a235a89a188a234a167a75hatwidef(0) = lim
ω→0
hatwidef(ω) = hτ
2,
204
a49a111a159 a245a67a254a135a200a169a198
a49a152a220a169 a245a3a188a234a27a52a129a216
a49a155a110a217 a245a3a188a234a27a52a129a134a235a89
§1,a178a161a58a56
1,a121a178(xn,yn) → (x0,y0)a27a191a135a94a135a180a181xn →x0,yn →y0(n→∞)
a121a178a181?
a207 lim
n→∞
Mn = M0a167a75a233?ε> 0,?N ∈Z+a167a8n>Na158a167a107r(Mn,M0) <ε
a61radicalbig(xn?x0)2 + (yn?y0)2 <ε
a117a180a152a189a107|xn?x0|lessorequalslantr(Mn,M0) <ε,|yn?y0|lessorequalslantr(Mn,M0) <εa61xn →x0,yn →y0(n→∞)
a207(|xn?x0|+|yn?y0|)2 greaterorequalslant|xn?x0|2 +|yn?y0|2a610 lessorequalslantradicalbig|xn?x0|2 +|yn?y0|2 lessorequalslant|xn?x0|+|yn?y0|
a113xn →x0,yn →y0(n→∞)a167a75radicalbig|xn?x0|2 +|yn?y0|2 → 0(n→∞)a61(xn,yn) → (x0,y0)(n→∞)
2,a121a178a181a101a178a161a254a27a58a15{Mn}a194a241a167a75a167a144a107a152a135a52a129.
a121a178a181a23 lim
n→∞
Mn = M0a167a98a23a113a107 lim
n→∞
Mn = M0 prime
a100a189a194a167a233?ε> 0,?N ∈Z+a167a8n>Na158a167a107r(Mn,M0) < ε2,r(Mn,M0 prime) < ε2
a100a110a14a216a31a170a167a107r(M0,M0 prime) lessorequalslantr(Mn,M0) +r(Mn,M0 prime) <ε
a113M0,M0 primea143a27a189a27a252a58a167a100εa27a63a191a53a167a26r(M0,M0 prime) = 0a61M0 = M0 prime.
3,a121a178a181a101Mn →M0(n→∞)a167a64a111a167a27a63a219a152a135a102a15Mnk →M0.
a121a178a181a207Mn →M0(n→∞)a167a75a233?ε> 0,?N ∈Z+a167a8n>Na158a167a107r(Mn,M0) <ε
a56a18K = Na167a75a233a152a131k>Ka167a107nk >nK = nN greaterorequalslantNa167a103a44a107r(Mnk,M0) <εa61Mnk →M0(k →∞).
4,a166a101a15a58a56Ea27a83a58a167a9a58a167a62a46a58a181
(1) Ea100a247a118y<x2a27a58a164a124a164a182
(2) Ea100a247a1181 lessorequalslantx2 + y
2
4 < 4a27a58a164a124a164a182
(3) Ea100a247a1180 <x2 +y2 < 1a27a58a164a124a164a182
(4) Ea100a164a107a249a24a27a58(x,y)a164a124a164a167a217a165xa218ya209a180a107a110a234.
a41a181
(1) a133a247a118y<x2a27a58(x,y)a180Ea27a83a58a182a133a247a118y>x2a27a58(x,y)a180Ea27a9a58a182a133a247a118y = x2a27a58(x,y)a180Ea27
a62a46a58.
(2) a133a247a1181 <x2 + y
2
4 < 4a27a58(x,y)a180Ea27a83a58a182a133a247a118x
2 + y
2
4 < 1a189x
2 + y
2
4 > 4a27a58(x,y)a180Ea27a9a58a182
a133a247a118x2 + y
2
4 = 1a189x
2 + y
2
4 = 4a27a58(x,y)a180Ea27a62a46a58.
(3) a133a247a1180 < x2 + y2 < 1a27a58(x,y)a180Ea27a83a58a182a133a247a118x2 + y2 > 1a27a58(x,y)a180Ea27a9a58a182a6a58θa57a247
a118x2 +y2 = 1a27a58(x,y)a180Ea27a62a46a58.
(4) a100a107a110a234a57a195a110a234a27a200a151a53a167a26a178a161a254a164a107a58(x,y)a209a180Ea27a62a46a58.
5,a121a178a181a101M0a180a178a161a58a56Ea27a224a58a167a75a51Ea165a127a51a58a15Mn →M0(n→∞).
a121a178a181a174a127M0a180a178a161a58a56Ea27a224a58a167a18δn = 1na167a51O(M0,δ1)a165a189a127a51Ea27a58M1 negationslash= M0a182a51O(M0,δ2)a165a189a127
a51Ea27a58M2,M2 negationslash= Mi(inegationslash= 0,1)
a88a100a63a49a101a22a167a26a20a58a15{Mn}(Mn negationslash= Mi)(i = 0,1,···,n?1)a133r(M0,Mn) < 1n
a117a180a8n→∞a158a167r(M0,Mn) → 0a61Mn →M0(n→∞).
6,a121a178a178a161a58a15a27a194a241a6a110.
a121a178a181?
a23Mn →M0(n→∞)a167a75a233?ε> 0,?N ∈Z+a167a8n,m>Na158a167a107r(Mn,M0) < ε2,r(Mm,M0) < ε2
205
a100a229a108a27a110a14a216a31a170a167a26r(Mm,Mn) lessorequalslantr(Mn,M0) +r(Mm,M0) <ε
a23a58a15{Mn}a247a118a233?ε> 0,?N ∈Z+a167a8n,m>Na158a167a107r(Mn,Mm) <ε
a242{Mn}a169a79a221a75a20a252a138a139a73a182a254a167a26a234a15{xn},{yn}
a207|xm?xn|<r(Mm,Mn) <ε,|ym?yn|<r(Mm,Mn) <ε
a100R1a254a27a133a220a194a241a6a110a167a26{xn},{yn}a209a194a241
a23xn →x0,yn →y0(n→∞)a167a75 lim
n→∞
Mn = M0(M0(x0,y0))a61{Mn}a194a241.
7,a94a178a161a254a27a107a129a67a88a189a110a121a178a159a16a100a65a46a100a189a110.
a121a178a181
(1) a101{Mn(xn,yn)}a180a107a46a107a129a58a56a167a189a110a164a225a182
(2) a101{Mn(xn,yn)}a180a107a46a195a161a58a56a167a2265a167a144a73a121E = {Mn(xn,yn)vextendsinglevextendsinglen = 1,2,···}a165a150a8a107a152a135a224a58.
a135a121.a23Ea118a107a224a58.
a100a117a lessorequalslant xn lessorequalslant b,c lessorequalslant yn lessorequalslant d(n = 1,2,···)a167a13a221a47a141R = {(x,y)vextendsinglevextendsinglea lessorequalslant x lessorequalslant b,c lessorequalslant d}a180a107a46a52a171a141
a133E?R
a233?M(x,y) ∈Ra167a209a216a180Ea27a224a58a167a207a13a127a51δMa167a166a26O(M,δM)a150a245a107Ea165a107a129a135a58a167
{O(M,δM)vextendsinglevextendsingleM ∈R}a67a88R
a226a107a129a67a88a189a110a167a127a51a107a129a135a109a56O(M1,δM1),···,O(Mk,δMk)a211a24a67a88Ra167a217a165a122a135O(Mi,δMi)(i =
1,2,···,k)a165a150a245a107a107a129a135Ea165a27a58
a117a180
kuniontext
i=1
O(Mi,δMi)a150a245a185Ea165a107a129a135a58
a2a100a117
kuniontext
i=1
O(Mi,δMi)?R?Ea167a117a180a103a241.
206
§2,a245a3a188a234a27a52a129a218a235a89a53
1,a40a189a191a177a209a101a15a188a234a131a189a194a141a181
(1) u = √x?√1?y
(2) u = √x?y+ 1
(3) u = ln(?x?y)
(4) uradicalbigsin(x2 +y2)
(5) uradicalbigR2?x2?y2?z2 +radicalbigx2 +y2 +z2?r2
a41a181
(1) a189a194a141a143xgreaterorequalslant 0a133ylessorequalslant 1
(2) a189a194a141a143a247a118a216a31a170ylessorequalslantx+ 1a27a58a56
(3) a189a194a141a143a140a178a161x+y< 0
(4) a189a194a141a143a247a118a216a31a1702kpilessorequalslantx2 +y2 lessorequalslant (2k+ 1)pi(k = 0,1,2,···)a27a58a56
(5) a189a194a141a143a247a118a216a31a170r2 lessorequalslantx2 +y2 +z2 lessorequalslantR2a27a58a56
2,a166a101a15a52a129a181
(1) limx→0
y→0
x2 +y2
|x|+|y|
(2) limx→0
y→0
x2 +y2radicalbig
x2 +y2 + 1?1
(3) limx→0
y→0
1 +x2 +y2
x2 +y2
(4) limx→0
y→0
sin(x3 +y3)
x2 +y2
(5) limx→+∞
y→+∞
(x2 +y2)e?(x+y)
(6) limx→1
y→0
ln(x+ey)radicalbig
x2 +y2
a41a181
(1) a2070 lessorequalslant x
2 +y2
|x|+|y|lessorequalslant
(|x|+|y|)2
|x|+|y| = |x|+|y|a133limx→0y→0(|x|+|y|) = 0a167a75limx→0y→0
x2 +y2
|x|+|y| = 0
(2) a207 lim
t→+0
t√
t+ 1?1 = limt→+0(
√t+ 1 + 1) = 2a167a75lim
x→0
y→0
x2 +y2radicalbig
x2 +y2 + 1?1 = 2
(3) a207 lim
t→+0
1 +t
t = +∞a167a75limx→0y→0
1 +x2 +y2
x2 +y2 = +∞
(4) a2070 lessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
sin(x3 +y3)
x2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
|x3 +y3|
x2 +y2 lessorequalslant
|x|3 +|y|3
x2 +y2 =
|x|3
x2 +y2 +
|y|3
x2 +y2 lessorequalslant|x|+|y|a133limx→0y→0(|x|+|y|) = 0
a75limx→0
y→0
sin(x3 +y3)
x2 +y2 = 0
(5) a207 lim
t→+∞
t
et = 0,limt→+∞
t2
et = 0
a75 limx→+∞
y→+∞
(x2 +y2)e?(x+y) = limx→+∞
y→+∞
bracketleftBigg
(x+y)2
e?(x+y)?2
x
ex ·
y
ey
bracketrightBigg
= 0
(6) limx→1
y→0
ln(x+ey)radicalbig
x2 +y2 = ln2
207
3,a193a121a101limy→a
x→b
f(x,y) = Aa127a51a167a13a8xa18a63a219a134aa25a67a131a138a158a167a52a129lim
y→b
f(x,y) =?(x)a127a51a167a75a19a103a52a129a127a51a167
a133a31a117Aa181
lim
x→a
lim
y→b
f(x,y) = limy→a
x→b
f(x,y) = A
.
a121a178a181a207a19a173a52a129a127a51a167a75a233?ε > 0,?δ > 0a167a8|x?a| < δ,|y?b| < δa133(x?a)2 + (y?b)2 negationslash= 0a158a167a240
a107|f(x,y)?A|<ε
a121a510 <|x?a|<δa165a27a189xa167a13a51a254a170a165a45y →ba167a61a26|?(x)?A|lessorequalslantεa167a249a210a121a178a10lim
x→a
(x) = A
a117a180lim
x→a
lim
y→b
f(x,y) = lim
x→a
(x) = A = limy→a
x→b
f(x,y)
4,(1) a193a222a209a252a135a19a103a52a129a216a131a31a27a126a102a182
(2) a193a222a209a144a107a152a135a19a103a52a129a127a51a27a126a102a182
(3) a193a222a209a19a173a52a129a127a51a167a2a19a103a52a129a216a28a127a51a27a126a102.
a41a181
(1) a126a181f(x,y) =
x?y
x+y,x+y negationslash= 0
0,x+y = 0
a51a58(0,0)a27a19a103a52a129
lim
x→0
lim
y→0
f(x,y) = lim
x→0
x
x = 1,limy→0 limx→0f(x,y) = limy→0
y
y =?1
a75lim
x→0
lim
y→0
f(x,y) negationslash= lim
y→0
lim
x→0
f(x,y).
(2) a126a181f(x,y) = xsin
1
x +y
x+y a51a58(0,0)a27a19a103a52a129
lim
y→0
lim
x→0
f(x,y) = lim
y→0
y
y = 1
a2lim
x→0
lim
y→0
f(x,y) = lim
x→0
xsin 1x
x = limx→0sin
1
x a216a127a51.
(3) a126a181f(x,y) =
xsin 1y,y negationslash= 0
0,y = 0
a51a58(0,0)a27a19a103a52a129a218a19a173a52a129
a2070 lessorequalslant|f(x,y)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglexsin
1
y
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant|x|a167a75limx→0y→0 f(x,y) = 0a61a217a19a173a52a129a127a51
lim
y→0
lim
x→0
f(x,y) = 0a167a13a8y → 0a158a167xsin 1y a52a129a216a127a51a167a61lim
x→0
lim
y→0
f(x,y)a216a127a51.
5,a63a216a101a15a188a234a51a58(0,0)a27a19a103a52a129a218a19a173a52a129a181
(1) f(x,y) = x
2y2
x2y2 + (x?y)2
(2) f(x,y) = (x+y)·sin 1x ·sin 1y
a41a181
(1) lim
x→0
lim
y→0
f(x,y) = 0,lim
y→0
lim
x→0
f(x,y) = 0
a101a85y = kx→ 0a27a144a149a18a52a129a167a75a107 lim
y=kx
x→0
f(x,y) = lim
x→0
x2k2
x2k2 + (1?k)2
a65a79a27a167a169a79a18k negationslash= 1a57k = 1a167a66a26a20a216a211a27a52a1290a571a167a207a100limx→0
y→0
f(x,y)a216a127a51.
(2) a2070 lessorequalslant|f(x,y)|lessorequalslant|x+y|lessorequalslant|x|+|y|a167a75limx→0
y→0
f(x,y) = 0a61a217a19a173a52a129a127a51
a113lim
y→0
(x + y) · sin 1x · sin 1y a216a127a51
parenleftBigg
a8xnegationslash= 1kpi
parenrightBigg
(k = ±1,±2,···)a167lim
x→0
(x + y) · sin 1x · sin 1y a216a127
a51
parenleftBigg
a8y negationslash= 1kpi
parenrightBigg
(k = ±1,±2,···)
a61lim
x→0
lim
y→0
f(x,y)a57lim
y→0
lim
x→0
f(x,y)a209a216a127a51.
208
6,a63a216a101a15a188a234a27a235a89a137a140a181
(1) u = 1radicalbigx2 +y2
(2) u = ln(1?x2?y2)
(3) u = 1sinxsiny
(4) u = ln 1(x?1)a + (y?b)2 + (z?c)2
a41a181
(1) a188a234u = 1radicalbigx2 +y2 a51a58(0,0)a195a189a194a167a25a6a58(0,0)a143a100a188a234a27a216a235a89a58a167a216a100a58a9a254a235a89a182
(2) a252a160a11a83a27a58a167a61a247a118x2 +y2 < 1a27a136a58a143a188a234u = ln(1?x2?y2)a27a235a89a58a182
(3) a235a89a137a140a143xnegationslash= mpi,y negationslash= npi(m,n = 0,±1,±2,···).
(4) a216a58(a.b.c)a9a254a235a89.
7,a121a178a188a234
f(x,y) =
2xy
x2 +y2,x
2 +y2 negationslash= 0
0,x2 +y2 = 0
a169a79a233a117a122a152a67a254xa218ya180a235a89a27a167a2a154a39a117a19a67a254a27a235a89a188a234.
a121a178a181a107a27a189y = anegationslash= 0a167a75a26xa27a188a234g(x) = f(x,a) = 2axx2 +a2 (?∞<x< +∞)
a167a180a63a63a107a189a194a27a107a110a188a234
a113a8y = 0a158a167f(x,0) ≡ 0a167a167a119a44a180a235a89a27
a117a180a8a67a234ya27a189a158a167a188a234f(x,y)a233a117a67a234xa180a235a89a27
a211a110a140a121a167a8a67a234xa27a189a158a167a188a234f(x,y)a233a117a67a234ya180a235a89a27
a138a143a19a3a188a234a167f(x,y)a143a51a216a58(0,0)a9a27a136a58a254a235a89a167a2a51a58(0,0)a216a235a89
a8a196a58P(x,y)a247a19a130y = mxa170a117a6a58a158a167a107 limy=mx
x→0
f(x,y) = lim
x→0
2mx2
(1 +m2)x2 =
2m
1 +m2
a18a216a211a27ma167a75a52a129a138a216a211a167a96a178a217a19a173a52a129a216a127a51a167a117a180limx→0
y→0
f(x,y) negationslash= f(0,0)
a75a217a39a117a19a67a254a27a188a234a51(0,0)a58a216a235a89a167a108a13a217a154a39a117a19a67a254a27a235a89a188a234.
8,a121a178a188a234
f(x,y) =
x2y
x4 +y2,x
2 +y2 negationslash= 0
0,x2 +y2 = 0
a51(0,0)a58a247a122a152a94a19a130x = tcosθ,y = tsinθ(0 lessorequalslantt+∞)a235a89a167a2a167a51(0,0)a58a216a235a89.
a121a178a181a8sinθ = 0a158a167cosθ = 1a189?1a167a117a180a8tnegationslash= 0a158a167f(tcosθ,tsinθ) = 0a167a13f(0,0) = 0
a75a107lim
t→0
f(tcosθ,tsinθ) = f(0,0)
a8sinθ negationslash= 0a167a107lim
t→0
f(tcosθ,tsinθ) = 0a167a25a107lim
t→0
f(tcosθ,tsinθ) = f(0,0)
a217a103a167a23a196a58P(x,y)a247a14a212a130y = x2a170a117a6a58a167a26 lim
y=x2
x→0
f(x,y) = 12 negationslash= f(0,0)a167a75a188a234f(x,y)a51a58(0,0)a216a235
a89.
9,a101f(x,y)a51a44a152a171a141Ga83a233a67a254xa143a235a89a167a233a67a254ya247a118a111a202a70a91a94a135a167a61a233a63a219
(x,yprime) ∈G,(x,yprimeprime) ∈G
a107|f(x,yprime)?f(x,yprimeprime)|lessorequalslantL|yprime?yprimeprime|
a217a165La143a126a234a167a75a100a188a234a51Ga83a235a89.
a121a178a181a207f(x,y)a51a171a141Ga83a233a67a254xa143a235a89a167a75a233Ga83a63a152a58(x0,y0)a167a233?ε< 0,?δ1 > 0a167a8|x?x0|<δ1a158a167
a107|f(x,y0)?f(x0,y0)|< ε2
a113a207f(x,y)a51Ga83a233ya247a118a111a202a70a91a94a135a167a75a233a63a219(x,y) ∈G,(x,y0) ∈Ga167a107|f(x,y)?f(x,y0)|lessorequalslantL|y?y0|
209
a45L|y?y0|< ε2 a167a75|y?y0|< ε2L
a18δ = min
parenleftBigg
δ1,ε2L
parenrightBigg
a167a8|x?x0|<δ,|y?y0|<δa158a167a189a107
|f(x,y)?f(x0,y0)|lessorequalslant|f(x,y)?f(x,y0)|+|f(x,y0)?f(x0,y0)|<ε
a61a100a188a234a51Ga83a235a89.
210
a49a155a111a217 a160a19a234a218a28a135a169
§1,a160a19a234a218a28a135a169a27a86a103
1,a166a101a15a188a234a27a160a19a234a181
(1) z = x2 ln(x2 +y2)
(2) u = exy
(3) z = xy+ xy
(4) u = arctan yx
(5) u = x2 +y2 +z2 + 2xy+ 2yz + 2zx
(6) u = eθ cos(θ+?)
a41a181
(1) zx = 2x
bracketleftBigg
ln(x2 +y2) + x
2
x2 +y2
bracketrightBigg
,zy = 2x
2y
x2 +y2.
(2) ux = yexy,uy = xexy.
(3) zx = y+ 1y,zy = x(y
2?1)
y2,
(4) ux =? yx2 +y2,uy = xx2 +y2,
(5) ux = 2(x+y+z),uy = 2(x+y+z),uz = 2(x+y+z).
(6) u? = eθ[cos(θ+?)?sin(θ+?)],uθ =?eθ[sin(θ+?) + cos(θ+?)].
2,a23f(x,y) = x2y2?2ya167a166fx(x,y),fy(x,y),fx(2,3),fy(0,0),fy(x,y)
vextendsinglevextendsingle
vextendsinglex=yy=x,
a41a181fx(x,y) = 2xy2,fy(x,y) = 2x2y?2,fx(2,3) = 36,fy(0,0) =?2,fy(x,y)
vextendsinglevextendsingle
vextendsinglex=yy=x = 2xy2?2
3,a23z = ln(√x+√y)a167a121a178x?z?x +y?z?y = 12,
a121a178a181a207z = ln(√x+√y)a167a75?z?x = 12√x(√x+√y),?z?y = 12√y(√x+√y)
a117a180x?z?x +y?z?y = 12,
4,a166a101a15a188a234a51a137a189a58(x0,y0)a27a28a135a169a181
(1) u = x4 +y4?4x2y2,(0,0),(1,1)
(2) u = xradicalbigx2 +y2,(1,0),(0,1)
(3) u = xsin(x+y),(0,0),
parenleftBigg
pi
4,
pi
4
parenrightBigg
(4) u = ln(x+y2),(0,1),(1,1)
a41a181
(1) a207du = 4x(x2?2y2)dx+ 4y(y2?2x2)dya167a75
a51(0,0)a58du = 0a182a51(1,1)a58du =?4dx?4dy.
(2) a207du = y
2
(x2 +y2)32
dx? xy
(x2 +y2)32
dya167a75
a51(1,0)a58du = 0a182a51(0,1)a58du = dx.
(3) a207du = [sin(x+y) +xcos(x+y)]dx+xcos(x+y)dya167a75
a51(0,0)a58du = 0a182a51
parenleftBigg
pi
4,
pi
4
parenrightBigg
a58du = dx.
211
(4) a207du = dxx+y2 + 2yx+y2 dya167a75
a51(0,1)a58du = dx+ 2dya182a51(1,1)a58du = dx2 + dy.
5,a166a101a15a188a234a27a28a135a169a181
(1) u = sin(x2 +y2)
(2) u = xm ·yn
(3) u = exy
(4) u = xy
(5) u =radicalbigx2 +y2 +z2
(6) u = ln(x2 +y2 +z2)
a41a181
(1) du = 2cos(x2 +y2)(xdx+ydy)
(2) du = xm?1yn?1(mydx+nxdy)
(3) du = exy(ydx+xdy)
(4) du = xy?1(ydx+xlnxdy)
(5) du = xdx+ydy+zdzradicalbigx2 +y2 +z2
(6) du = 2(xdx+ydy+zdz)x2 +y2 +z2
6,a121a178a181f(x,y) =radicalbig|xy|a51(0,0)a235a89a167fx(0,0),fy(0,0)a127a51a167a2a51(0,0)a58a216a140a135.
a121a178a181a100limx→0
y→0
radicalbig|xy| = 0a167a26lim
x→0
y→0
f(x,y) = 0 = f(0,0)a167a75f(x,y)a51(0,0)a58a235a89
a207fx(0,0) = lim
x→0
f(?x,0)?f(0,0)
x = 0,fy(0,0) = lim?y→0
f(0,?y)?f(0,0)
y = 0
a75fx(0,0),fy(0,0)a127a51
a2f(x,y) =radicalbig|xy|a51(0,0)a58a216a140a135.a101a140a135a167a75a107?f = fx(0,0)?x+fy(0,0)?y+o(ρ)a61?f = o(ρ)
a127a196a58P(x,y)a247y = xa170a1170a158a167a107?fρ =
radicalbig|?x?y|
radicalbig?x2 +?y2 = 1√2 negationslash→ 0(ρ → 0)a103a241a167a117a180a98a23a216a164a225a167
a75f(x,y)a51(0,0)a58a216a140a135.
7,a121a178a181f(x,y) =
xyradicalbig
x2 +y2,x
2 +y2 negationslash= 0
0,x2 +y2 = 0
a51(0,0)a58a27a25a141a165a235a89a167fx(x,y),fy(x,y)a107a46a167a2a51(0,0)a58
a216a140a135.
a121a178a181a100a117 xyradicalbigx2 +y2 a180a19a3a208a31a188a234a167a51a217a189a194a141a83a55a235a89a167a75f(x,y)a51x2 +y2 negationslash= 0a235a89
a1130 <
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
xyradicalbig
x2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle =
|xy|radicalbig
x2 +y2 lessorequalslant
radicalbigx2 +y2
2,f(0,0) = 0a167a75limx→0y→0 f(x,y) = f(0,0)a167a117a180f(x,y)a51(0,0)a58
a235a89a167a108a13f(x,y)a51(0,0)a58a27a63a219a25a141a83a235a89
a207fx(0,0) = lim
x→0
f(?x,0)?f(0,0)
x = 0,fy(0,0) = lim?y→0
f(0,?y)?f(0,0)
y = 0
a8x2 +y2 negationslash= 0a158a167fx(x,y) = y
3
(x2 +y2)32
,|fx(x,y)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
y3
(x2 +y2)32
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant 1a167a75fx(x,y)a107a46
a211a110a140a26fy(x,y)a107a46
a2f(x,y)a51(0,0)a58a216a140a135.a101a140a135a167a75a107?f = fx(0,0)?x+fy(0,0)?y+o(ρ)a61?f = o(ρ)
a127a196a58P(x,y)a247y = xa170a1170a158a167a107?fρ =
x?y√
x2+?y2radicalbig
x2 +?y2 =
1
2 negationslash→ 0(ρ → 0)a103a241a167a117a180a98a23a216a164a225a167
a75f(x,y)a51(0,0)a58a216a140a135.
212
8,a23f(x,y) =
(x2 +y2)sin 1x2 +y2,x2 +y2 negationslash= 0
0,x2 +y2 = 0
a121a178fx(x,y),fy(x,y)a127a51a2a216a235a89a167a51(0,0)a58a27a63a219a25a141a165a195a46a167a2a51(0,0)a58a140a135.
a121a178a181a8x2 +y2 negationslash= 0a158a167fx(x,y) = 2xsin 1x2 +y2? 2xx2 +y2 cos 1x2 +y2
fx(0,0) = lim
x→0
f(?x,0)?f(0,0)
x = 0a167a75fx(0,0)a127a51
a127a9a51a58
parenleftBigg
1√
2npi,0
parenrightBigg
a27a160a19a234
fx
parenleftBigg
1√
2npi,0
parenrightBigg
=?2√2npi →?∞(n→∞)
a249a96a178fx(x,y)a51(0,0)a58a27a63a219a25a141a83a195a46a167a75a217a51(0,0)a58a216a235a89a167a117a180fx(x,y)a216a235a89
a211a110a140a26fy(x,y)a127a51a2a216a235a89a133fy(0,0) = 0a167a51(0,0)a58a27a63a219a25a141a165a195a46
a113?f?fx(0,0)?x?fy(0,0)?yρ =radicalbig?x2 +?y2 sin 1?x2 +?y2 → 0(ρ→ 0)
a75f(x,y)a51(0,0)a58a140a135.
9,a166a101a15a188a234a27a112a30a160a19a234a181
(1) u = xsin(x+y) +ycos(x+y)a167a164a107a19a30a160a19a234
(2) u = 12 ln(x2 +y2)a167 a164a107a19a30a160a19a234
(3) u = xln(xy)a167?
3u
x2?y
(4) u = ln(ax+by+cz)a167?
4u
x4,
4u
x2?y2
(5) u = (x?x0)p ·(y?y0)qa167?
p+qu
xp?yq
(6) u = x·y·zex+y+za167?
p+q+ru
xp ·?yq ·?zr
a41a181
(1) a207ux = (1?y)sin(x+y) +xcos(x+y),uy =?ysin(x+y) + (x+ 1)cos(x+y)
a75ux2 = (2?y)cos(x+y)?xsin(x+y),uxy = uyx = (1?y)cos(x+y)?(x+ 1)sin(x+y),
uy2 =?ycos(x+y)?(x+ 2)sin(x+y)
(2) a207ux = xx2 +y2,uy = yx2 +y2
a75ux2 = y
2?x2
(x2 +y2)2,uxy = uyx =?
2xy
(x2 +y2)2,uy2 =
x2?y2
(x2 +y2)2
(3) a207?u?y = xy a167a75?
2u
x?y =
1
y a167a117a180
3u
x2?y = 0
(4)?u?x = aax+by+cz,?u?y = bax+by+cz
2u
x2 =?
a2
(ax+by+cz)2,
2u
y2 =?
b2
(ax+by+cz)2
3u
x3 =
2a3
(ax+by+cz)3,
3u
x?y2 =
2ab2
(ax+by+cz)3
4u
x4 =?
6a4
(ax+by+cz)4,
4u
x2?y2 =?
6a2b2
(ax+by+cz)4
(5) a207?
qu
yq = q!(x?x0)
pa167a75?
p+qu
xp?yq = p!q!(p,qa254a143a103a44a234)
(6)?
p+q+ru
xp ·?yq ·?zr =
p
xp(xe
x)?
q
yq(ye
y)?
r
zr(ze
z) = ex+y+z(x+p)(y+q)(z +r)
10,a23
213
(1) u = x2?2xy?3y2
(2) u = xy2
(3) u = arccos
radicalBigg
x
y
a8a121a164a225a31a170?
2u
x?y =
2u
y?x
a121a178a181
(1) a207ux = 2x?2y,uy =?2x?6ya167a75?
2u
x?y =?2,
2u
y?x =?2a167a117a180
2u
x?y =
2u
y?x
(2) a207ux = y2xy2?1,uy = 2yxy2 lnxa167a75?
2u
x?y = 2yx
y2?1(1 +y2 lnx),?
2u
y?x = 2yx
y2?1(1 +y2 lnx)
a117a180?
2u
x?y =
2u
y?x
(3) a80 <xlessorequalslantya158a167u = arccos
radicalBigg
x
y = arccos
√x
√y
a75ux =? 12radicalbigx(y?x),uy =
√x
2y√y?x
a117a180?
2u
x?y =
1
4√x(y?x)32
,?
2u
y?x =
1
4√x(y?x)32
a108a13?
2u
x?y =
2u
y?x
a211a110a140a121a167a8ylessorequalslantx< 0a158a167?
2u
x?y =
2u
y?xa143a164a225
a110a254a167a26?
2u
x?y =
2u
y?x.
214
§2,a166a69a220a188a234a160a19a234a27a243a170a123a75
1,a166a101a15a188a234a27a160a19a234a181
(1) u = f(x,y)a167a217a165x = rcosθ,y = rsinθa167a166?u?r,?
2u
r2 a182
(2) u = f(x,y)a167a217a165x = aξ,y = bηa167a166?u?ξ,?
2u
ξ2,
2u
ξ?η,
u
η,
2u
η2
(3) u = f(x2 +y2 +z2)a167a166?u?x,?
2u
x2,
2
x?y,
u
y,
u
z
(4) u = f
parenleftBigg
x,xy
parenrightBigg
a167a166?u?x,?
2u
x2,
u
y,
a41a181
(1)?u?r = fx cosθ+fy sinθ
2u
r2 = fx2 cos
2θ+fxy sin2θ+f
y2 sin
2θ
(2)?u?ξ = afx,?
2u
ξ2 = a
2f
x2,
2u
ξ?η = abfxy,
u
η = bfy,
2u
η2 = b
2f
y2
(3)?u?x = 2xfprime(x2+y2+z2),?
2u
x2 = 2f
prime(x2+y2+z2)+4x2fprimeprime(x2+y2+z2),?
2
x?y = 4xyf
primeprime(x2+y2+z2),
u
y = 2yf
prime(x2 +y2 +z2),?u
z = 2zf
prime(x2 +y2 +z2)
(4)?u?x = f1 + 1y f2,?
2u
x2 = f11 +
2
y f12 +
1
y2 f22,
u
y =?
x
y2 f2
2,a23Φ = Φ(x,y,z),x = u+v,y = u?v,z = uva167a166Φu,Φv.
a41a181Φu = Φx + Φy +vΦz,Φv = Φx?Φy +uΦz
3,a166a101a15a188a234a27a28a135a169(a23a217a140a135)a181
(1) u = f(x+y)
(2) u = f(x+y,x?y)
(3) u = f(ax2 +by2 +cz2)
a41a181
(1) du = fprime(x+y)(dx+ dy)
(2) du = (f1 +f2)dx+ (f1?f2)dy
(3) du = 2fprime(ax2 +by2 +cz2)(axdx+bydy+czdz)
4,a8a121a101a15a136a170a181
(1) a23z =?(x2 +y2)a167a75y?z?x?x?z?y = 0a182
(2) a23u = y?(x2?y2)a167a75y?u?x +x?u?y = xuy a182
(3) a23u = x?(x+y) +yψ(x+y)a167a75?
2u
x2?2
2u
x?y +
2u
y2 = 0.
a121a178a181
(1) a207?z?x = 2x?prime(x2 +y2),?z?y = 2y?prime(x2 +y2)
a75y?z?x?x?z?y = 0.
(2) a207?u?x = 2xy?prime(x2?y2),?u?y =?(x2?y2)?2y2?prime(x2?y2)
a75y?u?x +x?u?y = x?(x2?y2) = xuy,
215
(3) a207?u?x =?(x+y) +x?prime(x+y) +yψprime(x+y),?u?y = x?prime(x+y) +ψ(x+y) +yψprime(x+y)
a75?
2u
x2 = 2?
prime(x+y)+x?primeprime(x+y)+yψprimeprime(x+y),?
2u
x?y =?
prime(x+y)+ψprime(x+y)+x?primeprime(x+y)+yψprimeprime(x+y)
2u
y2 = 2ψ
prime(x+y) +x?primeprime(x+y) +yψprimeprime(x+y)
a117a180?
2u
x2?2
2u
x?y +
2u
y2 = 0.
5,a166?u =?
2u
x2 +
2u
y2 +
2u
z2,u = f(x+y+z,x
2 +y2 +z2).
a41a181a207?u?x = f1 + 2xf2a167a75?
2u
x2 = f11 + 4xf12 + 4x
2f22 + 2f2
a226a233a161a53a167a26?
2u
y2 = f11 + 4yf12 + 4y
2f22 + 2f2,?
2u
z2 = f11 + 4zf12 + 4z
2f22 + 2f2
a117a180?u =?
2u
x2 +
2u
y2 +
2u
z2 = 3f11 + 4(x+y+z)f12 + 4(x
2 +y2 +z2)f22 + 6f2.
6,a101u = f(r),r =radicalbigx2 +y2a167a217a165f(r)a19a103a140a135a167a193a121a178
2u
x2 +
2u
y2 =
d2u
dr2 +
1
r
du
dr
a121a178a181a207?u?x = xradicalbigx2 +y2 fprime(r)a167a75?
2u
x2 =
y2
(x2 +y2)32
fprime(r) + x
2
x2 +y2 f
primeprime(r)
a226a233a161a53a167a26?
2u
y2 =
x2
(x2 +y2)32
fprime(r) + y
2
x2 +y2 f
primeprime(r)
a117a180?
2u
x2 +
2u
y2 =
d2u
dr2 +
1
r
du
dr
7,a101u,va143x,ya27a188a234a167x = rcosθ,y = rsinθa167a193a100
u
x =
v
y,
u
y =?
v
x
a121a178a31a170?u?r = 1r?v?θ,?v?r =?1r?u?θ,
a121a178a181a207u,va143x,ya27a188a234a167x = rcosθ,y = rsinθ
a75?u?r = cosθ?u?x + sinθ?u?y,?v?r = cosθ?v?x + sinθ?v?y,
u
θ =?rsinθ
u
x +rcosθ
u
y,
v
θ =?rsinθ
v
x +rcosθ
v
y
a113?u?x =?v?y,?u?y =v?xa167a75?u?r = 1r?v?θ,?v?r =?1r?u?θ,
8,a23f(tx,ty) = tnf(x,y)a167a75a107
x?f?x +y?f?y = nf
a228a107a249a24a53a159a27a188a234a167a161a143na103a224a103a188a234.a124a94a249a40a74a167a233z =radicalbigx2 +y2a167a166x?z?x +y?z?y,
a121a178a181a207f(tx,ty) = tnf(x,y)a167a75a252a224a233ta166a160a19a167a26f1(tx,ty)x+f2(tx,ty)y = ntn?1f(x,y)
a45t = 1a167a75f1(x,y)x+f2(x,y)y = nf(x,y)a61x?f?x +y?f?y = nf
a207z(x,y) =radicalbigx2 +y2a167a75z(tx,ty) = tradicalbigx2 +y2 (tgreaterorequalslant 0)
a117a180x?z?x +y?z?y = z =radicalbigx2 +y2.
9,a23?a134ψa180a63a191a27a19a30a140a19a188a234a167a121a178a181
z = x?
parenleftBigg
y
x
parenrightBigg
+ψ
parenleftBigg
y
x
parenrightBigg
216
a247a118x2?
2z
x2 + 2xy
2z
x?y +y
2?
2z
y2 = 0
a121a178a181a207?z?x =?
parenleftBigg
y
x
parenrightBigg
yx?prime
parenleftBigg
y
x
parenrightBigg
yx2 ψprime
parenleftBigg
y
x
parenrightBigg
,?z?y =?prime
parenleftBigg
y
x
parenrightBigg
+ 1x ψprime
parenleftBigg
y
x
parenrightBigg
a75?
2z
x2 =
y2
x3?
primeprime + 2y
x3 ψ
prime + y
2
x4 ψ
primeprime,?
2z
x?y =?
y
x2?
primeprime? 1
x2 ψ
prime? y
x3 ψ
primeprime,?
2z
y2 =
1
x?
primeprime + 1
x2 ψ
primeprime
a117a180x2?
2z
x2 + 2xy
2z
x?y +y
2?
2z
y2 = 0
10,a23u =?(x+at) +ψ(x?at)a167a217a165?,ψa180a63a191a27a19a103a140a135a188a234a167a166a121
2u
t2 = a
2?
2u
x2,
a121a178a181a207u =?(x+at) +ψ(x?at)a167?,ψa180a63a191a27a19a103a140a135a188a234
a75?u?t = a(?prime?ψprime),?u?x =?prime +ψprimea167a117a180?
2u
t2 = a
2(?primeprime +ψprimeprime),?
2u
x2 =?
primeprime +ψprimeprime
a108a13?
2u
t2 = a
2?
2u
x2.
217
§3,a100a144a167(a124)a164a40a189a27a188a234a27a166a19a123
1,a166a100a101a15a144a167a164a40a189a27a188a234z = f(x,y)a27a152a30a218a19a30a27a160a19a234a181
(1) x+y+z = ez
(2) xyz = x+y+z
a41a181
(1) a252a62a39a117xa166a19a167a261 +zx = zxeza167a75zx = 1ez?1 a167a117a180zx2 = e
z
(1?ez)3
a211a123a140a26a167zy = 1ez?1,zy2 = e
z
(1?ez)3,zxy = zyx =
ez
(1?ez)3
(2) a252a62a39a117xa166a19a167a26yz +xyzx = 1 +zx (?)a167a75zx = yz?11?xy
a242(?)a170a252a62a39a117xa166a19a167a262yzx +xyzx2 = zx2a167a75zx2 = 2yzx1?xy = 2y(yz?1)(xy?1)2
a211a123a140a26a167zy = xz?11?xy,zy2 = 2x(xz?1)(xy?1)2,zxy = zyx = 2z(xy?1)2
2,a166a100a101a15a144a167a164a40a189a27a188a234a27a28a135a169a189a160a19a234a181
(1) f(x+y,y+z,z +x) = 0a167a166?z?x,?z?y a182
(2) z = f(xz,z?y)a167a166dza182
(3) F(x?y,y?z,z?x) = 0a167a166?z?x,?z?y a182
(4) F(x,x+y,x+y+z) = 0a167a166?z?x,?z?y,?
2z
x2,
a41a181
(1) a252a62a39a117xa166a19a167a133z = z(x,y)a167a26f1 +f2zx +f3(zx + 1) = 0a167a75zx =?f1 +f3f
2 +f3
a211a123a140a26a167zy =?f1 +f2f
2 +f3
(2) a252a224a135a169a167a26dz = (xdz +zdx)f1 + (dz? dy)f2a167a75dz = zf1 dx?f2 dy1?xf
1?f2
(3) a252a62a39a117xa166a19a167a133z = z(x,y)a167a26F1?F2zx +F3(zx?1) = 0a167a75zx = F1?F3F
2?F3
a211a123a140a26a167zy = F2?F1F
2?F3
(4) a252a62a39a117xa166a19a167a133z = z(x,y)a167a26F1 +F2 +F3(1 +zx) = 0 (?)a167a75zx =?F1 +F2 +F3F
3
a51(?)a170a252a62a50a39a117xa166a19a167a26
F11 +F12 +F13(1 +zx) +F21 +F22 +F23(1 +zx) +zx2F3 + (1 +zx)[F13 +F23 +F33(1 +zx)] = 0
a75zx2 =? 1F3
3
[F23 (F11 + 2F12 +F22)?2F3(F1 +F2)(F13 +F23) +F33(F1 +F2)2]
a211a123a140a26a167zy =?F2 +F3F
3
3,a23a100a144a167z = x+y·?(z)a40a189a188a234z = z(x,y)a167a231?y?prime(z) negationslash= 0a167a121a178
z
y =?(z)·
z
x
a121a178a181a144a167a252a224a135a169a167a133z = z(x,y)a167a26dz = dx+?(z)dy+y?prime(z)dz
a1131?y?prime(z) negationslash= 0a167a75dz = dx+?(z)dy1?y?prime(z)
a117a180?z?y =?(z)1?y?prime(z),?z?x = 11?y?prime(z)a167a108a13?z?y =?(z)·?z?x
218
4,a121a178a100a144a167ax+by +cz = Φ(x2 +y2 +z2)a164a189a194a27a188a234z = z(x,y)a247a118a144a167(cy?bz)?z?x + (az?cx)?z?y =
bx?aya167a217a165Φ(u)a180ua27a140a135a188a234a167a,b,ca143a126a234.
a121a178a181a144a167a252a224a135a169a167a133z = z(x,y)a167Φ(u)a180ua27a140a135a188a234
a75a26adx+bdy+cdz = 2(xdx+ydy+zdz)Φprime
a117a180?z?x = 2xΦ
prime?a
c?2zΦprime,
z
y =
2yΦprime?b
c?2zΦprime
a108a13(cy?bz)?z?x + (az?cx)?z?y = bx?ay
5,a23?a143a63a191a27a140a135a188a234a167a121a178a100a144a167?(cx?az,cy?bz) = 0a164a189a194a27a188a234z = z(x,y)a247a118a?z?x +b?z?y = c.
a121a178a181a233a144a167a252a224a169a79a39a117x,ya166a19a167a133z = z(x,y)a167a26
c?1?a?1zx?b?2zx = 0,?a?1zy +c?2?b?2zy = 0
a117a180?z?x = c?1a?
1 +b?2
,?z?y = c?2a?
1 +b?2
a108a13a?z?x +b?z?y = c.
6,a121a178a100a144a167F(x+zy?1,y+zx?1) = 0a164a40a189a27a188a234z = z(x,y)a247a118x?z?x +y?z?y = z?xy.
a121a178a181a233a144a167a252a224a169a79a39a117x,ya166a19a167a133z = z(x,y)a167a26
F1
parenleftBigg
1 + zxy
parenrightBigg
+F2
parenleftBigg
zx
x?
z
x2
parenrightBigg
= 0,F1
parenleftBigg
zy
y?
z
y2
parenrightBigg
+F2
parenleftBigg
1 + zyx
parenrightBigg
= 0
a117a180?z?x = yzF2?x
2yF1
x(xF1 +yF2),
z
y =
xzF1?xy2F2
y(xF1 +yF2)
a108a13x?z?x +y?z?y = z?xy.
7,a166a101a15a144a167a124a164a40a189a27a188a234a27a19a234a189a160a19a234a189a28a135a169a181
(1)
braceleftbigg x+y+z = 0,
x·y·z = 1,a166
dy
dx,
dz
dx,
d2y
dx2 a182
(2)
x+y = u+v,
x
y =
sinu
sinv,
a166du,dva182
(3)
braceleftbigg xu+yv = 0,
yu+xv = 1,a166
u
x,
u
y,
v
x,
v
y,
2u
x?y a182
(4)
x = cosθcos?,
y = cosθsin?,
z = sinθ,
a166?z?x,?z?y a182
(5)
braceleftbigg u = f(u,x,v+y),
v = g(u?x,u2 ·y),a166
u
x,
v
x,
a41a181
(1) a233xa166a19a167a26
1 + dydx + dzdx = 0
yz +xz dydx+xy dzdx = 0
(?)
a233a225a166a41a167a26 dydx = y(z?x)x(y?z),dzdx = z(x?y)x(y?z)
(?)a170a50a233xa166a19a167a26
d2y
dx2 +
d2z
dx2 = 0
z dydx +y dzdx +z dydx +x dydx · dzdx +xz d
2y
dx2 +y
dz
dx +x
dy
dx ·
dz
dx+xy
d2z
dx2 = 0
a233a225a167a26 d
2y
dx2 =
2z dydx + 2y dzdx + 2x dydx · dzdx
x(y?z)
a242 dydx,dzdxa147a92a167a26 d
2y
dx2 =
yz[(x?y)2 + (x?z)2 + (y?z)2]
x2(z?y)3
219
(2) a242a6a170a85a21a143
braceleftbigg u+v = x+y
ysinu = xsinv a135a169a167a26
braceleftbigg du+ dv = dx+ dy
sinudy+ycosudu = sinvdx+xcosvdv
a75du = 1xcosv+ycosu[(sinv+xcosv)dx?(sinu?xcosv)dy]
dv = 1xcosv+ycosu[?(sinv?ycosu)dx+ (sinu+ycosu)dy]
(3) a135a169a167a26
braceleftbigg xdu+ydv =?udx?vdy
ydu+xdv =?vdx?udy
a117a180du = 1x2?y2[(yv?xu)dx+ (yu?xv)dy],dv = 1x2?y2[(yu?xv)dx+ (yv?xu)dy]
a75?u?x = yv?xux2?y2,?u?y = yu?xvx2?y2,?v?x = yu?xvx2?y2,?u?x = yv?xux2?y2
a117a180?
2u
x?y =
(yux?v?xvx)(x2?y2)?2x(yu?xv)
(x2?y2)2
a242?u?x,?v?xa147a92a167a26?
2u
x?y =
2(x2v+y2v?2xyu)
(x2?y2)2
(4) a100x,ya233xa166a160a19a234a167a26
1 =?sinθ·cosθ?x?cosθ·sinx
0 =?sinθ·sinθ?x + cosθ·cosx
a75?θ?x =? cos?sinθ,x =? sin?cosθ a167a117a180?z?x =?z?θ ·?θ?x =?cotθcos? =?xz
a211a110a140a26a167?z?y =?yz
(5) a233xa166a160a19a167a26
u
x = f1
u
x+f2 +f3
v
x
v
x = g1
parenleftBigg
u
x?1
parenrightBigg
+ 2vyg2?v?x
a75?u?x = f2(1?2vyg2)?g1f3(f
1?1)(2vyg2?1)?g1f3
,?v?x = g1(f1 +f2?1)(f
1?1)(2vyg2?1)?g1f3
.
8,a144a167x = u+v,y = u2 +v2,z = u3 +v3a189a194za143x,ya27a188a234a167a166?z?x,?z?y,
a41a181a207x2?y = 2uva167a75z = (u+v)(u2?uv+v2) = x2 (3y?x2)
a117a180?z?x = 32 (y?x2),?z?y = 32 x.
9,a23x = rcosθ,y = rsinθa167a67a134a144a167
dx
dt = y+kx(x
2 +y2)
dy
dt =?x+ky(x
2 +y2)
a143a52a139a73a144a167.
a41a181a100a144a167a127a167x,ya180ta27a188a234a167a108a52a139a73a67a134a127r,θa143a180ta27a188a234a167x = rcosθ,y = rsinθ
a252a224a233ta166a19a167a26
dx
dt = cosθ
dr
dt?rsinθ
dθ
dtdy
dt = sinθ
dr
dt +rcosθ
dθ
dt
a242x,y,dxdt,dydt a147a92a6a144a167a124a167a26
cosθ drdt?rsinθ dθdt = rsinθ+krcosθ·r2
sinθ drdt +rcosθ dθdt =?rcosθ+krsinθ·r2
a117a180 drdt = kr3,dθdt =?1.
10,a23x = eu cosθ,y = eu sinθ a167a67a134a144a167?
2z
x2 +
2z
y2 = 0.
a41a181a207x = eu cosθ,y = eu sinθa167a75u = ln(x2 +y2),θ = arctan yx
220
a75?u?x = xx2 +y2,?u?y = yx2 +y2 ;?θ?x =? yx2 +y2,?θ?y = xx2 +y2 a167a117a180?u?x =?θ?y,?u?y =θ?x
a113?z?x =?z?u ·?u?x +?z?θ ·?θ?x,?z?y =?z?u ·?u?y +?z?θ ·?θ?y
a75?
2z
x2 =
2z
u2
parenleftBigg
u
x
parenrightBigg2
+ 2?
2z
θ?u ·
u
x ·
θ
x +
2z
θ2
parenleftBigg
θ
x
parenrightBigg2
+?z?u ·?
2u
x2 +
z
θ ·
2θ
x2
2z
y2 =
2z
u2
parenleftBigg
u
y
parenrightBigg2
+ 2?
2z
θ?u ·
u
y ·
θ
y +
2z
θ2
parenleftBigg
θ
y
parenrightBigg2
+?z?u ·?
2u
y2 +
z
θ ·
2θ
y2
a113?
2u
x2 =
x
parenleftBigg
θ
y
parenrightBigg
=y
parenleftBigg
θ
x
parenrightBigg
=y
parenleftBigg
u?y
parenrightBigg
=
2u
y2
a211a123a140a26a167?
2θ
x2 =?
2θ
y2
a75?
2θ
x2 +
2θ
y2 =
2u
x2 +
2u
y2 = 0
a113
parenleftBigg
u
x
parenrightBigg2
+
parenleftBigg
u
y
parenrightBigg2
=
parenleftBigg
θ
x
parenrightBigg2
+
parenleftBigg
θ
y
parenrightBigg2
,?u?x ·?θ?x =u?y ·?θ?y
a75?
2z
x2 +
2z
y2 = e
2u
parenleftBigg
2z
u2 +
2z
θ2
parenrightBigg
= 0
a61?
2z
u2 +
2z
θ2 = 0.
11,a23x = rcosθ,y = rsinθa167a75f(x,y) = Φ(r,θ)a167a94Φa39a117r,θa27a160a19a234a53a76a171?
2f
x2 +
2f
y2,
a41a181a242f(x,y) = Φ(r,θ)a39a117r,θa166a160a19a167a26
f
x ·
x
r +
f
y ·
y
r =
Φ
r a61
f
x cosθ+
f
y sinθ =
Φ
r
f
x ·
x
θ +
f
y ·
y
θ =
Φ
θ a61?
f
x rsinθ+
f
y rcosθ =
Φ
θ
a75?
2Φ
r2 =
2f
x2 cos
2θ+?
2f
x?y sin2θ+
2f
y2 sin
2θ
2Φ
θ2 = r
2
parenleftBigg
2f
x2 sin
2θ
2f
x?y sin2θ+
2f
y2 cos
2θ
parenrightBigg
f?x rcosθf?y rsinθ
a117a180?
2Φ
r2 +
1
r2
2Φ
θ2 =
2f
x2 +
2f
y2?
1
r
Φ
r
a108a13?
2f
x2 +
2f
y2 =
2Φ
r2 +
1
r2
2Φ
θ2 +
1
r
Φ
r
12,a23x = eξ,y = eηa167a67a134a144a167ax2?
2z
x2 + 2bxy
2z
x?y +cy
2?
2z
y2 = 0(a,b,ca143a126a234).
a41a181a207x = eξ,y = eηa167a75ξ = lnx,η = lnya167a117a180 dξdx = 1x,dηdy = 1y
a75?z?x = 1x?z?ξ,?z?y = 1y?z?η
a117a180?
2z
x2 =
1
x2
parenleftBigg
2z
ξ2?
z
ξ
parenrightBigg
,?
2z
y2 =
1
y2
parenleftBigg
2z
η2?
z
η
parenrightBigg
,?
2z
x?y =
1
xy
2z
ξ?η
a147a92a6a144a167a167a122a123a18a110a167a26a
parenleftBigg
2z
ξ2?
z
ξ
parenrightBigg
+ 2b?
2z
ξ?η +c
parenleftBigg
2z
η2?
z
η
parenrightBigg
= 0.
13,a23ξ = x,η = x2 +y2a167a67a134a144a167y?z?x?x?z?y = 0,
a41a181a100a144a167a127za180x,ya27a188a234a167a13ξ,ηa113a180x,ya27a188a234a167a108a13za140a119a164a180a207a76a165a109a67a254ξ,ηa39a117x,ya27a69a220a188a234
a117a180?z?x =?z?ξ + 2x?z?η,?z?y = 2y?z?η
a207a13y?z?x?x?z?y = y?z?ξ
221
a207y negationslash≡ 0a167a75a100y?z?x?x?z?y = 0a167a26?z?ξ = 0.
14,a23ξ = x,η = y?x,ζ = z?xa167a67a134a144a167?u?x +?u?y +?u?z = 0.
a41a181a100a144a167a127ua180x,y,za27a188a234a167a13ξ,η,ζa113a180x,y,za27a188a234a167a108a13ua140a119a164a180a207a76a165a109a67a254ξ,η,ζa39a117x,y.za27a69
a220a188a234
a117a180?u?x =?u?ξu?ηu?ζ,?u?y =?u?η,?u?z =?u?ζ
a75a100?u?x +?u?y +?u?z = 0a167a26?z?ξ = 0
15,a23a130a53a67a134ξ = x + λ1y,η = x + λ2ya167a121a51a135a114a144a167A?
2u
x2 + 2B
2u
x?y + C
2u
y2 = 0(A,B,Ca143a126a234a167
a133AC?B2 < 0)a67a134a143?
2u
ξ?η = 0a167a121a178λ1,λ2a143a144a167Cλ
2 + 2Bλ+A = 0a27a252a135a131a201a162a138.
a121a178a181a100a144a167a127ua180x,ya27a188a234a167a207a13a140a177a114ua192a143a177ξ,ηa143a165a109a67a254a27a39a117x,ya27a69a220a188a234a167a117a180
u
x =
u
ξ +
u
η,
u
y = λ1
u
ξ +λ2
u
η
2u
x2 =
2u
ξ2 + 2
2u
ξ?η +
2u
η2,
2u
y2 = λ
2
1
2u
ξ2 + 2λ1λ2
2u
ξ?η +λ
2
2
2u
η2,
2u
x?y = λ1
2u
ξ2 + (λ1 +λ2)
2u
ξ?η +λ2
2u
η2
a207a13A?
2u
x2 + 2B
2u
x?y +C
2u
y2 =
2u
ξ2 (A+ 2Bλ1 +Cλ
2
1) + 2
2u
ξ?η [A+B(λ1 +λ2) +Cλ1λ2] +
2u
η2(A+
2Bλ2 +Cλ22) = 0
a135a166A?
2u
x2 + 2B
2u
x?y +C
2u
y2 = 0a67a134a143
2u
ξ?η = 0a167a55a76
A+ 2Bλ1 +Cλ21 = 0
A+ 2Bλ2 +Cλ22 = 0
A+B(λ1 +λ2) +Cλ1λ2 negationslash= 0
a100a99a252a135a144a167a167a26λ1,λ2a180a144a167Cλ2 + 2Bλ+A = 0a27a138
a13a100a49a110a135a144a167a167a26λ1 negationslash= λ2a167a75λ1,λ2a180Cλ2 + 2Bλ+A = 0a27a252a135a131a201a162a138
a113a207λ1 +λ2 =?2BC,λ1λ1 = AC a167a75A+B(λ1 +λ2) +Cλ1λ2 = 2C (AC?B2) negationslash= 0
a117a180a144a167A?
2u
x2 + 2B
2u
x?y + C
2u
y2 = 0a51a130a53a67a134ξ = x + λ1y,η = x + λ2ya101a40a162a67a134a143
2u
ξ?η = 0a167
a133λ1,λ2a143a144a167Cλ2 + 2Bλ+A = 0a27a252a135a131a201a162a138.
16,a121a178a46a202a46a100a144a167?w ≡?
2w
x2 +
2w
y2 = 0a51a67a122x =?(u,v),y = ψ(u,v)
parenleftBigg
a167a130a247a118u =?ψ?v,v =ψ?u
parenrightBigg
a101
a47a71a2a177a216a67.
a121a178a181a108a144a167a127ωa180x,ya27a188a234a167x,ya180u,va27a188a234a167a75wa180a177x,ya143a165a109a67a254a27u,va27a188a234
a117a180?w?u =?w?x ·u +?w?y ·?ψ?u,?w?v =?w?x ·v +?w?y ·?ψ?v
2w
u2 =
2w
x2
parenleftBigg
u
parenrightBigg2
+ 2?
2w
x?y ·
u ·
ψ
u +
2w
y2
parenleftBigg
ψ
u
parenrightBigg2
+?w?x ·?
2?
u2 +
w
y ·
2ψ
u2
2w
v2 =
2w
x2
parenleftBigg
v
parenrightBigg2
+ 2?
2w
x?y ·
v ·
ψ
v +
2w
y2
parenleftBigg
ψ
v
parenrightBigg2
+?w?x ·?
2?
v2 +
w
y ·
2ψ
v2
a53a191a154a242a122a94a135u =?ψ?v,v =ψ?u a167a75?
2?
u2 =
2ψ
u?v,
2?
v2 =?
2ψ
v?u,
2ψ
u2 =?
2?
u?v,
2ψ
v2 =
2?
v?u
a242?
2w
u2,
2w
v2 a131a92a167a191a242a254a227a136a170a147a92a167a26
2w
u2 +
2w
v2 =
parenleftBigg
2w
x2 +
2w
y2
parenrightBiggbracketleftBiggparenleftBigg
u
parenrightBigg2
+
parenleftBigg
v
parenrightBigg2bracketrightBigg
a207?
2w
x2 +
2w
y2 = 0a167a75
2w
u2 +
2w
v2 = 0
a249a76a178a46a202a46a100a144a167?w ≡?
2w
x2 +
2w
y2 = 0a51a67a122x =?(u,v),y = ψ(u,v)a101a47a71a2a177a216a67.
17,a23ξ = x?at,η = x+ata167a67a134a144a167?
2u
t2 = a
2?
2u
x2,
222
a41a181a100a144a167a127ua180t,xa27a188a234a167ξ,ηa143a180t,xa27a188a234a167a25a140a242ua192a143a177ξ,ηa143a165a109a67a254a27a39a117t,xa27a188a234
a75?u?t =?a?u?ξ +a?u?η,?u?x =?u?ξ +?u?η
2u
t2 = a
2?
2u
ξ2?2a
2?
2u
ξ?η +a
2?
2u
η2,
2u
x2 =
2u
ξ2 +
2u
ξ?η +
2u
η2
a117a180a100?
2u
t2 = a
2?
2u
x2 a167a264a
2?
2u
ξ?η = 0
a113anegationslash≡ 0a167a75?
2u
ξ?η = 0.
18,a138a103a67a234a218a207a67a234a27a67a134a167a18u,va143a35a27a103a67a234a167w = w(u,v)a143a35a27a207a67a234a181
(1) a23u = x2 +y2,v = 1x + 1y,w = lnz?(x+y)a167a67a134a144a167
y?z?x?x?z?y = (y?x)·z
(2) a23u = x+y,v = yx,w = zxa167a67a134a144a167
2z
x2?2
2z
x?y +
2z
y2 = 0
(3) a23x = u,y = u1 +uv,z = u1 +u·w a167a67a134a144a167
x2?z?x +y2?z?y = z2
(4) a23u = xy,v = x,w = xz?ya167a67a134a144a167
y?
2z
y2 + 2
z
y =
2
x
a41a181
(1) a100a174a127a167a26du = 2xdx+ 2ydy,dv =? 1x2 dx? 1y2 dy,dw = 1z dz? dx? dy
a44a152a144a161a167dw =?w?u du+?w?v dv
a751z dz? dx? dy =?w?u (2xdx+ 2ydy) +?w?v
parenleftBigg
1x2 dx? 1y2 dy
parenrightBigg
a18a110a167a26dz =
parenleftBigg
2xz?w?u? zx2 ·?w?v +z
parenrightBigg
dx+
parenleftBigg
2yz?w?u? zy2 ·?w?v +z
parenrightBigg
dy
a242a254a170a164a40a189a27?z?x,?z?y a147a92a6a144a167a167a26z
parenleftBigg
x
y2?
y
x2
parenrightBigg
w
v = 0
a113z
parenleftBigg
x
y2?
y
x2
parenrightBigg
negationslash≡ 0a167a75?w?v = 0.
(2) a100a174a127a167a26du = dx+ dy,dv =? yx2 dx+ 1x dy,dw =? zx2 dx+ 1x dz
a44a152a144a161a167dw =?w?u du+?w?v dv
a75? zx2 dx+ 1x dz =?w?u (dx+ dy) +?w?v
parenleftBigg
yx2 dx+ 1x dy
parenrightBigg
a18a110a167a26dz =
parenleftBigg
x?w?u? yx ·?w?v + zx
parenrightBigg
dx+
parenleftBigg
x?w?u +?w?v
parenrightBigg
dy
a75?z?x = x?w?u? yx ·?w?v + zx,?z?y = x?w?u +?w?v
223
a45R =?z?xz?y = w?(1 +v)?w?v
a75?
2z
x2? 2
2z
x?y +
2z
y2 =
x
parenleftBigg
z
x?
z
y
parenrightBigg
y
parenleftBigg
z
x?
z
y
parenrightBigg
=?R?xR?y =?R?u
parenleftBigg
u
x?
u
y
parenrightBigg
+
R
v
parenleftBigg
v
x?
v
y
parenrightBigg
=v
bracketleftBigg
w?(1 +v)?w?v
bracketrightBiggparenleftBigg
yx2? 1x
parenrightBigg
= (1 +v)
2
x
2w
v2 = 0
a207(1 +v)
2
x negationslash≡ 0a167a75a6a144a167a67a143
2w
v2 = 0.
(3) a207x = u,y = u1 +uv,z = u1 +u·w a167a75u = x,v = 1y? 1x,w = 1z? 1x
a117a180du = dx,dv = 1x2 dx? 1y2 dy,dw = 1x2 dx? 1z2 dz
a44a152a144a161a167dw =?w?u du+?w?v dv
a75 1x2 dx? 1z2 dz =?w?u dx+?w?v
parenleftBigg
1
x2 dx?
1
y2 dy
parenrightBigg
a18a110a167a26dz = z2
parenleftBigg
1
x2?
w
u?
1
x2 ·
w
v
parenrightBigg
dx+ z
2
y2 ·
w
v dy
a242a254a170a164a40a189a27?z?x,?z?y a147a92a6a144a167a167a26x2z2?w?u = 0
a113xz negationslash≡ 0a167a75?w?u = 0.
(4) a207u = xy,v = x,w = xz?ya167a75?w?y = x?z?y?1,?w?y =?w?u ·?u?y +?w?v ·?v?y,?u?y =? xy2,?v?y = 0
a117a180?w?y =? xy2?w?u a167a75?z?y = 1x? 1y2?w?u
a117a180y?
2z
y2 + 2
z
y =
2
x+
x
y3
2w
u2 =
2
x
a113xy3 negationslash≡ 0a167a75a6a144a167a67a134a143?
2w
u2 = 0.
224
§4,a152a109a173a130a27a131a130a134a123a178a161
1,a166a101a15a173a130a51a164a171a58a63a27a131a130a134a123a178a161a181
(1) x = asin2t,y = bsint·cost,z = ccos2ta167a51t = pi4 a27a58a63a182
(2) x2 +y2 +z2 = 6,x+y+z = 0a167a51a58(1,?2,1).
a41a181
(1) x0 = a2,y0 = b2,z0 = c2,xprime(t0) = a,yprime(t0) = 0,zprime(t0) =?c
a75a173a130a51t = pi4 a27a58a63a27a131a130a144a167a143
x? a2
a =
z? c2
c
y = b2
a61
x
a +
z
c = 1
y = b2
a123a178a161a144a167a143a
parenleftBigg
x? a2
parenrightBigg
c
parenleftBigg
z? c2
parenrightBigg
= 0a61ax?cz = 12 (a2?c2).
(2) a207D(F1,F2)D(y,z)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingle2y 2z1 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
=?6,D(F1,F2)D(z,x)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingle2z 2x1 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
= 0,
D(F1,F2)
D(x,y)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingle2x 2y1 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,?2,1)
= 6
a75a173a130a51a58(1,?2,1)a27a131a130a144a167a143
braceleftbigg x+z?2 = 0
y =?2
a123a178a161a144a167a143x?z = 0.
2,a51a173a130x = t,y = t2,z = t3a254a166a209a152a58a167a166a100a58a27a131a130a178a49a117a178a161x+ 2y+z = 4.
a41a181a23a164a166a58a143(t0,t20,t30)a167a75xprime(t0) = 1,yprime(t0) = 2t0,zprime(t0) = 3t20
a117a180a173a130a27a131a130a144a149a165a254a143v = {1,2t0,3t20}
a113a178a161a123a165a254n = {1,2,1}a167a75a226a75a191a167a65a107v·n = 1 + 4t0 + 3t20 = 0a167a117a180t0 =?1,t0 =?13
a75a164a166a58a143(?1,1,?1),
parenleftBigg
13,19,? 127
parenrightBigg
.
3,a121a178a173a130x = aet cost,y = aet sint,z = aeta134a73a161x2 +y2 = z2a27a49a130a131a2a164a211a152a14.
a121a178a181a242x,y,za147a92x2 +y2 = z2a167a26a2e2t cos2t+a2e2t sin2t = a2e2t = z2a167a75a173a130a65a51a173a161a254
a11a73x2 +y2 = z2a27a186a58a51a6a58a167a76a11a73a254a63a152a58P(x0,y0,z0)a27a49a130a143a76a6a58
a75a49a130a27a144a149a165a254a143v1 = {x0,y0,z0}
a113a173a130a51a58Pa27a131a149a254a143v2 = {aet0(cost0?sint0),aet0(sint0 + cost0),aet0} = {x0?y0,x0 +y0,z0}
x20 +y20 = z20
a75cos(hatwiderv1,v2) = v1 ·v2|v
1||v2|
= 2√6 =
√6
3 a167a249a134a173a130a254a58(x,y,z)a27a160a152a118a107a39a88
a207a13a173a130a134a73a161a27a49a130a131a2a164a211a152a14.
4,a166a101a15a136a173a130a51a164a171a58a27a131a130a27a144a149a123a117a181
(1) x = t2,y = t3,z = t4a167a51t = 1a27a58a254a182
(2) xyz = 1,y2 = xa167a51a58(1,1,1).
a41a181
(1) a207xprime(t0) = 2,yprime(t0) = 3,zprime(t0) = 4a167a75a131a149a254a143{2,3,4}
a117a180a144a149a123a117a143a181cosα = ± 229 √29,cosβ = ± 329 √29,cosγ = ± 429 √29.
(2) a207D(F1,F2)D(y,z)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingle xz xy?2y 0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
= 2,D(F1,F2)D(z,x)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
=
vextendsinglevextendsingle
vextendsinglevextendsinglexy yz0 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
= 1,
D(F1,F2)
D(x,y)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
=
vextendsinglevextendsingle
vextendsinglevextendsingleyz xz1?2y
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1,1)
=?3a167a75a131a149a254a143{2,1,?3}
a117a180a144a149a123a117a143a181cosα = ±
√14
7,cosβ = ±
√14
14,cosγ = ±
3
14
√14.
225
§5,a173a161a27a131a178a161a134a123a130
1,a166a101a15a173a161a51a164a171a58a27a131a178a161a57a123a130a144a167a181
(1) x = asin?cosθ,y = asin?sinθ,z = acos?a167a51(θ0,?0)a182
(2) exz +eyz = 4a167a51a58(ln2,ln2,1)a182
(3) z = 2x2 + 4y2a167a51a58(2,1,12)a182
(4) ax2 +by2 +cz2 +d = 0a167a51a58(x0,y0,z0).
a41a181
(1) a207D(y,z)D(θ,?)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(θ0,?0)
=
vextendsinglevextendsingle
vextendsinglevextendsingleasin?cosθ acos?sinθ0?asin?
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(θ0,?0)
=?asin2?0 cosθ0,
D(z,x)
D(θ,?)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(θ0,?0)
=?a2 sin2?0 sinθ0,D(x,y)D(θ,?)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(θ0,?0)
=?a2 sin?0 cos?0
a75a131a178a161a144a167a143sin?0 cosθ0x+ sin?0 sinθ0y+ cos?0z = a
a123a130a144a167a143x?asin?0 cosθ0sin?
0 cosθ0
= y?asin?0 sinθ0sin?
0 sinθ0
= z?acos?0cos?
0
.
(2) a207a51(ln2,ln2,1)a58fx = 2,fy = 2,fz =?ln16
a75a131a178a161a144a167a143x+y?2ln2·z = 0a182a123a130a144a167a143x?ln21 = y?ln21 = z?1?2ln2,
(3) a207zx(2,1) = 8,zy(2,1) = 8
a75a131a178a161a144a167a1438x+ 8y?z = 12a182a123a130a144a167a143x?28 = y?18 = z?12?1,
(4) a207a51(x0,y0,z0)a58fx = 2ax0,fy = 2by0,fz = 2cz0
a75a131a178a161a144a167a143ax0x+by0y+cz0z +d = 0a182a123a130a144a167a143x?x0ax
0
= y?y0by
0
= z?z0cz
0
.
2,a51a173a161z = xya254a166a152a58a167a166a249a58a27a123a130a82a134a117a178a161x+ 3y+z + 9 = 0a167a191a21a209a100a123a130a144a167.
a41a181a76a173a161a254a63a152a58M0(x0,y0,z0)a27n1 = {y0,x0,?1}a167a123a130a27a131a149a254a143n2 = {1,3,1}
a135a166a123a130a82a134a117a254a227a178a161a167a75n1 bardbl n2a61?y1 =?x3 = 11
a117a180a164a166a58a143(?3,?1,3)a167a75a123a130a144a167a143x+ 31 = y+ 13 = z?31,
3,a121a178a173a161√x+√y+√z = √a,(a> 0)a254a63a219a152a58a27a131a178a161a51a136a139a73a182a254a27a31a229a131a218a31a117a.
a121a178a181a51a173a161a254a63a18a152a58P0(x0,y0,z0)
a75a173a161a51a84a58a27a131a178a161a144a167a143 12√x
0
(x?x0) + 12√y
0
(y?y0) + 12√z
0
(z?z0) = 0
a61√y0z0(x?x0) +√x0z0(y?y0) +√x0y0(z?z0) = 0
a117a180a131a178a161a51a139a73a182a254a27a31a229a169a143√ax0,√ay0,√az0a167a217a218a143√a(√x0 +√y0 +√z0) = a.
4,a166a252a173a161x2 +y2 = a2,bz = xya27a2a14.
a41a181a23a252a173a161a63a152a2a58M0(x0,y0,z0)
a100a252a173a161a51M0a58a27a123a149a254a143n1 = {2x0,2y0,0},n2 = {y0,x0,?b}
a117a180a2a14?a247a118cos? = n1 ·n2|n
1||n2|
= 2bz0|a|√a2 +b2,
226
§6,a144a149a19a234a218a70a221
1,a166u = x2?xy+y2a51(1,1)a63a247a144a149l = (cosα,sinα)a27a144a149a19a234.a191a63a152a218a166a181
(1) a51a61a135a144a149a254a217a19a234a107a129a140a138a182
(2) a51a61a135a144a149a254a217a19a234a107a129a2a138a182
(3) a51a61a135a144a149a254a217a19a234a1430a182
(4) a166ua27a70a221.
a41a181a207ux = 2x?y,uy =?x+ 2ya167a75ux(1,1) = 1,uy(1,1) = 1
a113?u?l = ux(1,1)cosα+uy(1,1)sinαa167a75?u?l = cosα+ sinα = √2sin
parenleftBigg
α+ pi4
parenrightBigg
a117a180
(1) a8α = pi4 a158a167a51a144a149l =
parenleftBigg√
2
2,
√2
2
parenrightBigg
a254a217a19a234a107a129a140a138√2a182
(2) a8α =?34 pia158a167a51a144a149l =
parenleftBigg
√2
2,?
√2
2
parenrightBigg
a254a217a19a234a107a129a2a138?√2a182
(3) a8α =?pi4,34 pia158a167a51a144a149l =
parenleftBigg√
2
2,?
√2
2
parenrightBigg
a189l =
parenleftBigg
√2
2,
√2
2
parenrightBigg
a254a217a19a234a1430a182
(4) gradu = ux(1,1)i +uy(1,1)j = i + j.
2,a166u = xyza51a58M(1,1,1)a167a247l = (2,?1,3)a27a144a149a19a234a57a70a221.
a41a181a207ux = yz,uy = xz,uz = xya167a75a51(1,1,1)a58ux = uy = uz = 1
a113a149a254la27a144a149a123a117cosα = 2√14,sinβ =? 1√14,cosγ = 3√14
a75?u?l = ux(1,1,1)cosα+uy(1,1,1)cosβ +uz(1,1,1)cosγ = 27 √14 a182gradu = i + j + k.
3,a166a234a254a188a234u = x2 + 2y2 + 3z2 +xy+ 3x?2y?6za51O(0,0,0)a57A(1,1,1)a27a70a221a57a217a140a2.
a41a181a207ux = 2x+y+ 3,uy = 4y+x?2,uz = 6z?6
a75a51O(0,0,0)a58a181ux = 3,uy =?2,uz =?6a167a117a180gradu = 3i?2j + 6k,|gradu| = 7
a51A(1,1,1)a58a181ux = 6,uy = 3,uz = 0a167a117a180gradu = 6i + 3j,|gradu| = 3√5.
4,a121a178a181
(1) grad(αu+βv) = αgradu+βgradva167a217a165α,βa209a180a126a234a182
(2) grad(uv) = ugradv+vgradua182
(3) gradF(u) = F prime(u)gradu
a121a178a181a177a19a3a188a234a143a126a53a121.a45u = u(x,y),v = v(x,y)
(1) a207?(αu+βv)?x = α?u?x +β?v?x,?(αu+βv)?y = α?u?y +β?v?y
a75grad(αu+βv) =
parenleftBigg
(αu+βv)
x,
(αu+βv)
y
parenrightBigg
= α
parenleftBigg
u
x,
v
y
parenrightBigg
+β
parenleftBigg
u
x,
v
y
parenrightBigg
= αgradu+βgradv.
(2) a207?(uv)?x = v?u?x +u?v?x,?(uv)?y = v?u?y +u?v?y
a75grad(uv) =
parenleftBigg
(uv)
x,
(uv)
y
parenrightBigg
= v
parenleftBigg
u
x,
u
y
parenrightBigg
+u
parenleftBigg
v
x,
v
y
parenrightBigg
= ugradv+vgradu.
(3) gradF(u) =
parenleftBigg
F
x,
F
y
parenrightBigg
=
parenleftBigg
F prime(u)?u?x,F prime(u)?u?y
parenrightBigg
= F prime(u)
parenleftBigg
u
x,
u
y
parenrightBigg
= F prime(u)gradu
a245a3a188a234a140a149a19a3a188a234a121a131.
5,a121a178grad1r =? rr3 a167r =radicalbigx2 +y2 +z2,r = xi +yj +zk.
a121a178a181a207?r?x = xr,?r?y = yr,?r?z = zr
a75grad1r = ddr
parenleftBigg
1
r
parenrightBigg
gradr =? 1r2
parenleftBigg
r
x i +
r
y j +
r
z k
parenrightBigg
=? 1r2 · 1r(xi +yj +zk) =? rr3,
227
6,a23a234a254a188a234u = ln 1r,r =radicalbig(x?a)2 + (y?b)2 + (z?c)2a167a51a152a109a165a61a10a58a254a164a225|gradu| = 1a186
a41a181a207?r?x = x?ar,?r?y = y?br,?r?z = z?cr
a75gradu = ddr
parenleftBigg
ln 1r
parenrightBigg
gradr =?1r
parenleftBigg
x?a
r i +
y?b
r j +
z?c
r k
parenrightBigg
=? 1r2 [(x?a)i + (y?b)j + (z?c)k]
a117a180|gradu| = 1r = 1a167a75r = 1a61(x?a)2 + (y?b)2 + (z?c)2 = 1
a249a76a178a51a177(a,b,c)a143a165a37a167a140a187a1431a27a165a161a254a164a225|gradu| = 1.
228
§7,a18a86a250a170
1,a21a209a58(1,?2)a78a67a188a234f(x,y) = 2x2?xy?y2?6x?3y+ 5a27a18a86a250a170.
a41a181a207fx = 4x?y?6,fy =?x?2y?3,fx2 = 4,fxy =?1,fy2 =?2a167a164a107a110a30a160a19a254a1430
a75a51a58(1,?2)a167f = 5,fx = 0,fy = 0,fx2 = 4,fxy =?1,fy2 =?2
a117a180f(x,y) = 5 + 2(x?1)2?(x?1)(y+ 2)?(y+ 2)2.
2,a85xa57ya27a166a152a208a109a188a234f(x,y) = ex ln(1 +y)a20a110a145a143a142.
a41a181a207fx = ex ln(1 +y),fy = e
x
1 +y,fx2 = e
x ln(1 +y),f
y2 =?
ex
(1 +y)2,fxy =
ex
1 +y
fx3 = ex ln(1 +y),fy3 = 2e
x
(1 +y)3,fxy2 =?
ex
(1 +y)2,fyx2 =
ex
1 +y
a75a51a58(0,0)a63a167f = 0,fx = fx2 = fx3 = 0,fy = 1,fxy = 1,fy2 =?1,fxy2 =?1,fyx2 = 1,fy3 = 2
a117a180f(x,y) = y+xy? 12 y2 + 12 x2y? 12 xy2 + 13 y3a34
229
a49a155a202a217 a52a138a218a94a135a52a138
§1,a52a138a218a129a2a19a166a123
1,a166a101a15a188a234a27a52a138a181
(1) z = x2?(y?1)2
(2) z = (x?y+ 1)2
(3) z = 3axy?x3?y3 (a> 0)
(4) z = sinx+ cosy+ cos(x?y)
parenleftbigg x
0 lessorequalslant y lessorequalslant pi2
parenrightbigg
(5) z = xy·
radicalbigg
1? x
2
a2?
y2
b2 (a,b> 0)
a41a181
(1) a45zx = 2x = 0,zy =?2(y?1) = 0
a75a26x = 0,y = 1a167a117a180a58(0,1)a143a140a85a52a138a58
a113zx2 = 2,zxy = 0,zy2 =?2a167a75A = 2,B = 0,C =?2a167a117a180H =?4 < 0a167a108a13a100a188a234a195a52a138.
(2) a45zx = 2(x?y+ 1) = 0,zy =?2(x?y+ 1) = 0
a75a8a58a169a217a51x?y+ 1 = 0a254a158a167a188a234a140a85a107a52a138
a113A = 2,B =?2,C = 2a167a75H = 0a167a25a73a63a152a218a7a228
a207a233a134a130x?y+ 1 = 0a254a27a58a254a107z = 0a167a133z greaterorequalslant 0a240a164a225
a75a188a234za51a134a130x?y+ 1 = 0a254a136a58a18a26a52a2a138z = 0.
(3) a45zx = 3ay?3x2 = 0,zy = 3ax?3y2 = 0
a75a26
braceleftbigg x
1 = 0
y1 = 0
braceleftbigg x
2 = a
y2 = a a61a51a58(0,0),(a,a)a63a140a85a107a52a138
a113zx2 =?6x,zxy = 3a,zy2 =?6y
a75a51a58(0,0)a167A = 0,B = 3a,C = 0a167a117a180H =?9a2 < 0a167a100a158a188a234a195a52a138a182
a51a58(a,a)a167A =?6a< 0,B = 3a,C =?6aa167a117a180H = 27a2 > 0a167a100a158a188a234a107a52a140a138z = a3.
(4) a45zx = cosx?sin(x?y) = 0,zy =?siny+ sin(x?y) = 0
a75a26cosx = sinya167a117a180y = pi2?xa167a25cosx?sin(x?y) = cos?sin
parenleftBigg
2x? pi2
parenrightBigg
= 2cos x2 cos 32 x = 0
a2070 lessorequalslantxlessorequalslant pi2a167a75cos x2 negationslash= 0,cos 32 x = 0a167a117a180x = pi3,y = pi6 a167a61a51a58
parenleftBigg
pi
3,
pi
6
parenrightBigg
a63a140a85a107a52a138
a113zx2 =?sinx?cos(x?y),zxy = cos(x?y),zy2 =?cosy?cos(x?y)
a75A =?√3 < 0,B =
√3
2,C =?
√3a167a117a180H = 9
4 > 0a167a61a51a58
parenleftBigg
pi
3,
pi
6
parenrightBigg
a63a188a234a107a52a140a138z = 32 √3.
(5) a45zx = y
radicalbigg
1? x
2
a2?
y2
b2?
x2y
a2
radicalbigg
1? x
2
a2?
y2
b2
= 0,zy = x
radicalbigg
1? x
2
a2?
y2
b2?
xy2
b2
radicalbigg
1? x
2
a2?
y2
b2
= 0
a75a26
braceleftbigg x
1 = 0
y1 = 0
x2 = a√3
y2 = b√3
x3 =? a√3
y3 =? b√3
x4 = a√3
y4 =? b√3
x5 =? a√3
y5 = b√3
a117a180a51a58P1(0,0),P2
parenleftBigg
a√
3,
b√
3
parenrightBigg
,P3
parenleftBigg
a√3,? b√3
parenrightBigg
,P4
parenleftBigg
a√
3,?
b√
3
parenrightBigg
,P5
parenleftBigg
a√3,b√3
parenrightBigg
a63a140a85a18a52
a138
a113zx2 =?3a
2b2xy+ 2b2x3y+ 3a2xy3
a4b2
parenleftBig
1? x2a2? y2b2
parenrightBig3
2
,zxy = a
4b4?3a2b4x2 + 2b4x4?3a4b2y2 + 3a2b2x2y2 + 2a4y4
a4b4
parenleftBig
1? x2a2? y2b2
parenrightBig3
2
zy2 = 3b
2x3y?3a2b2xy+ 2a2xy3
a2b4
parenleftBig
1? x2a2? y2b2
parenrightBig3
2
230
a51a58P2
parenleftBigg
a√
3,
b√
3
parenrightBigg
,P3
parenleftBigg
a√3,? b√3
parenrightBigg
a63a167A =?4
√3b
3a < 0,B =?
2
3
√3,C =?4√3a
3b
a100a158H = 4 > 0a167a117a180a188a234a107a52a140a138z =
√3
9 aba182
a51a58P4
parenleftBigg
a√
3,?
b√
3
parenrightBigg
,P5
parenleftBigg
a√3,b√3
parenrightBigg
a63a167A = 4
√3b
3a > 0,B =?
2
3
√3,C = 4√3a
3b
a100a158H = 4 > 0a167a117a180a188a234a107a52a2a138z =?
√3
9 aba182
a51a58P1(0,0)a63a167A = 0,B = 1,C = 0a167a100a158H =?1 < 0a167a117a180a188a234a195a52a138.
2,a121a178a188a234z = (1 +ey)cosx?yeya107a195a161a245a135a52a140a138a167a2a195a52a2a138.
a121a178a181a45zx =?(1 +ey)sinx = 0 zy = ey cosx?ey?yey = 0
a2071 +ey negationslash= 0a167a75sinx = 0a167a117a180x = kpi (k ∈Z)
a113ey negationslash= 0a167a75cosx?1?y = 0a61a107y = cosx?1
a8ka143a243a234a158a167y = 0a182a8ka143a219a234a158a167y =?2a167a75a140a85a27a52a138a58a143
braceleftbigg x
1 = 2kpi
y1 = 0
braceleftbigg x
2 = (2k+ 1)pi
y2 =?2 (k =
0,±1,±2,···)
a113zx2 =?(1 +ey)cosx,zxy =?ey sinx,zy2 = ey cosx?2ey?yey
a75a51a58(2kpi,0)a167A =?2 < 0,B = 0,C =?1a167a100a158H = 2 > 0a167a75a100a158a188a234a107a52a140a138z = 2
a51a58((2k+ 1)pi,?2)a167A = 1 + 1e2,B = 0,C =? 1e2 a167a100a158H =? 1e2
parenleftBigg
1 + 1e2
parenrightBigg
< 0a167a75a100a158a188a234a195a52a138
a110a254a140a127a167a188a234z = (1 +ey)cosx?yeya107a195a161a245a135a52a140a138a167a2a195a52a2a138.
3,a51a174a127a177a127a1432pa27a152a131a110a14a47a165a166a209a161a200a129a140a27a110a14a47.
a41a181a23a110a14a47a27a62a127a169a79a143x,y,za167a75C = x+y+z = 2pa167D = x+y+z2 = pa167a117a180z = 2p?x?y
a75S =radicalbigD(D?x)(D?y)(D?z) =radicalbigp(p?x)(p?y)(x+y?p)
a127a196a188a234u = S2 = p(p?x)(p?y)(x+y?p),0 <x,y<p
Sa27a52a138a254a143ua27a52a138a133a8ua51a58(x,y)a18a26a27a52a138a216a1430a158a167Sa143a51a58(x,y)a18a26a52a138
a45ux = p(p?y)(2p?2x?y) = 0,uy = p(p?x)(2p?x?2y) = 0
a207pnegationslash= 0,0 <x,y<pa167a75a41a26x = y = 23 pa167a117a180z = 23 p
a75a8x = y = z = 23 pa158a167ua107a52a138a61Sa107a52a138
a108a13a8x = y = z = 23 pa158a167a161a200a129a140a133a138a143S =
√3
9 p
2.
4,a173a161z = 12 x2?4xy+ 9y2 + 3x?14y+ 12 a51a219a63a107a129a112a58a189a129a36a58a186
a41a181a100
braceleftbigg z
x = x?4y+ 3 = 0
zy =?4x+ 18y?14 = 0 a167a41a26
braceleftbigg x = 1
y = 1 a61a51a58(1,1)a140a85a107a52a138
a113zx2 = 1,zxy =?4,zy2 = 18a167a75A = 1 > 0,B =?4,C = 18a167a117a180H = 2 > 0
a75a100a158a188a234a107a52a2a138z =?5a167a108a13a173a161a107a129a36a58(1,1,?5)
a113a8x2 +y2 → +∞a158a167z → +∞a167a25a173a161a195a129a112a58.
5,a174a127y = ax2 +bx+ca167a121a255a26a152a124a234a226(xi,yi),i = 1,2,···,na167a124a94a129a2a19a166a123a167a166a88a234a,b,ca164a247a118a27a110a3
a152a103a144a167a124.
a41a181a100a174a127a167a26ε =
nsummationdisplay
i=1
(yi?ax2i?bxi?c)2a167a143a166a111a160a11ε(a,b,c)a136a20a129a2a167a100a52a138a27a55a135a94a135a167a107
ε
a =?2
nsummationdisplay
i=1
(yi?ax2i?bxi?c)x2i = 0,?ε?b =?2
nsummationdisplay
i=1
(yi?ax2i?bxi?c)xi = 0,?ε?c =?2
nsummationdisplay
i=1
(yi?ax2i?bxi?c) = 0
a61a,b,ca247a118a101a15a110a3a152a103a144a167a124a181
parenleftBigg nsummationdisplay
i=1
x4i
parenrightBigg
a+
parenleftBigg nsummationdisplay
i=1
x3i
parenrightBigg
b+
parenleftBigg nsummationdisplay
i=1
x2i
parenrightBigg
c =
nsummationdisplay
i=1
x2iyi
parenleftBigg nsummationdisplay
i=1
x3i
parenrightBigg
a+
parenleftBigg nsummationdisplay
i=1
x2i
parenrightBigg
b+
parenleftBigg nsummationdisplay
i=1
xi
parenrightBigg
c =
nsummationdisplay
i=1
xiyi
parenleftBigg nsummationdisplay
i=1
x2i
parenrightBigg
a+
parenleftBigg nsummationdisplay
i=1
xi
parenrightBigg
b+nc =
nsummationdisplay
i=1
yi
6,a173a130y = x2a51[0,1]a254a135a94a159a111a24a27a134a130η = ax+ba53a147a79a167a226a85a166a167a27a178a144a216a11a27a200a169
J(a,b) =
integraldisplay 1
0
(y?η)2 dxa143a52a2a27a191a194a101a143a129a90a67a113a186
231
a41a181J(a,b) =
integraldisplay 1
0
(y?η)2 dx =
integraldisplay 1
0
(x2?ax?b)2 dx = 15 + a
2
3 +b
2? a
2?
2
3 b+ab
a143a10a192a74a,ba166a178a144a216a11a27a200a169J(a,b)a136a20a52a2a167a100a52a138a27a55a135a94a135a167a107
a45?J?a =?12 + 23 a+b = 0,?J?b =?23 +a+ 2b = 0
a75a = 1,b =?16
a117a180a173a130y = x2a94a134a130η = x? 16 a53a147a79a167a140a136a20a129a90a67a113a27a135a166.
232
§2,a94a135a52a138
1,a166a101a15a188a234a51a164a137a94a135a101a52a138a181
(1) f = x+y,a101x2 +y2 = 1a182
(2) f = x?2y+ 2z,a101x2 +y2 +z2 = 1a182
(3) f = xyz,a1011x + 1y + 1z = 1a (x> 0,y> 0,z> 0,a> 0)a182
(4) f = 1x + 1y,a101x+y = 2a182
(5) f = xyz,a101x2 +y2 +z2 = 1,x+y+z = 0.
a41a181
(1) a138a188a234L = x+y+λ(x2 +y2?1)
a41a144a167a124
Lx = 1 + 2λx = 0
Ly = 1 + 2λy = 0
Lλ = x2 +y2?1 = 0
a167a26
x1 =
√2
2
y1 =
√2
2
λ1 =?
√2
2
x2 =?
√2
2
y2 =?
√2
2
λ2 =
√2
2a113L
x2 = 2λ,Lxy = 0,Ly2 = 2λ
a75d2L
parenleftBigg√
2
2,
√2
2
parenrightBigg
=?√2(dx2 + dy2) < 0a167a117a180a188a234a51
parenleftBigg√
2
2,
√2
2
parenrightBigg
a63a18a26a52a140a138√2a182
a211a110a140a26a167a188a234a51
parenleftBigg
√2
2,?
√2
2
parenrightBigg
a63a18a26a52a2a138?√2.
(2) a138a188a234L = x?2y+ 2z +λ(x2 +y2 +z2?1)
a41a144a167a124
Lx = 1 + 2λx = 0
Ly =?2 + 2λy = 0
Lz = 2 + 2λz = 0
Lλ = x2 +y2 +z2?1 = 0
a167a26
x1 = 13
y1 =?23
Z1 = 23
λ1 =?32
x2 =?13
y2 = 23
z2 =?23
λ2 = 32
a113Lx2 = Ly2 = Lz2 = 2λ,Lxy == Lxz = Lyz = 0
a75d2L(x2,y2,z2) = 3(dx2 + dy2 + dz2) > 0a167a117a180a188a234a51
parenleftBigg
13,23,?23
parenrightBigg
a63a18a26a52a2a138?3a182
a211a110a140a26a167a188a234a51
parenleftBigg
1
3,?
2
3,
2
3
parenrightBigg
a63a18a26a52a140a1383.
(3) a138a188a234L = xyz +λ
parenleftBigg
1
x +
1
y +
1
z?
1
a
parenrightBigg
a41a144a167a124
Lx = yz? λx2 = 0
Ly = xz? λy2 = 0
Lz = xy? λz2 = 0
Lλ = 1x + 1y + 1z? 1a = 0
a167a26x = y = z = 3a,λ = 81a4
a113Lx2(3a,3a,3a) = Ly2(3a,3a,3a) = Lz2(3a,3a,3a) = 6a,
Lxy(3a,3a,3a) = Lxz(3a,3a,3a) = Lyz(3a,3a,3a) = 3a
a75d2L(3a,3a,3a) = 3a[(dx+ dy+ dz)2 + dx2 + dy2 + dz2] > 0a167a117a180a188a234a51(3a,3a,3a)a63a18a26a52a2
a13827a3.
(4) a138a188a234L = 1x + 1y+λ(x+y?2)
233
a41a144a167a124
Lx =? 1x2 +λ = 0
Ly =? 1y2 +λ = 0
Lλ = x+y?2 = 0
a167a26x = y = λ = 1
a113Lx2(1,1) = Ly2(1,1) = 2,Lxy(1,1) = 0
a75d2L(1,1) = 2(dx2 + dy2) > 0a167a117a180a188a234a51(1,1)a63a18a26a52a2a1382.
(5) a138a188a234L = xyz +u(x2 +y2 +z2?1) +v(x+y+z)
a41a144a167a124
Lx = yz + 2ux+v = 0
Ly = xz + 2uy+v = 0
Lz = xy+ 2uz +v = 0
Lu = x2 +y2 +z2?1 = 0
Lv = x+y+z = 0
a167a26
x1 =
√6
6
y1 =
√6
6
z1 =?
√6
3
u1 =
√6
12
v1 = 16
x2 =?
√6
6
y2 =?
√6
6
z2 =
√6
3
u2 =?
√6
12
v2 = 16
x3 =?
√6
3
y3 =
√6
6
z3 =
√6
6
u3 =
√6
12
v3 = 16
x4 =
√6
3
y4 =?
√6
6
z4 =?
√6
6
u4 =?
√6
12
v4 = 16
x5 =
√6
6
y5 =?
√6
3
z5 =
√6
6
u5 =
√6
12
v5 = 16
x6 =?
√6
6
y6 =
√6
3
z6 =?
√6
6
u6 =?
√6
12
v6 = 16
a113d2L = 2u(dx2 + dy2 + dz2) + 2(zdxdy+ydxdz +xdydz)
a75a51a58(x1,y1,z1)a63a167d2L =
√6
6 (dx
2 + dy2 + dz2?4dxdy+ 2dxdz + 2dydz)
a100x2 +y2 +z2 = 1a167a262xdx+ 2ydy+ 2zdz = 0a167a75a51a58(x1,y1,z1)a63a167a107dx+ dy = 2dz
a113a100x+y+z = 0a167a26dx+ dy+ dz = 0a167a75dx =?dy,dz = 0a167a117a180d2L(x1,y1,z1) = √6 dx2 > 0a167
a75a188a234a51
parenleftBigg√
6
6,
√6
6,?
√6
3
parenrightBigg
a63a18a26a52a2a138?
√6
18
a211a110a140a26a167a188a234a51(x3,y3,z3),(x5,y5,z5)a63a18a26a52a2a138?
√6
18
a188a234a51(x2,y2,z2),(x4,y4,z4),(x6,y6,z6)a63a18a26a52a140a138
√6
18,
2,a166f = xmynzpa51a94a135x+y+z = a,a> 0,m> 0,n> 0,p> 0,x> 0,y> 0,z> 0a131a101a27a129a140a138.
a41a181a207x> 0,y> 0,z> 0a167a75f = xmynzp a129a140a158a167lnf = mlnx+nlny+plnza143a129a140a167a135a131a189a44a167a25a144a73
a166lnfa27a52a140a58a167a167a143a180fa27a52a140a58
a45L = mlnx+nlny+plnz +λ(x+y+z?a)
a75a41a144a167
Lx = mx +λ = 0
Ly = ny +λ = 0
Lz = pz +λ = 0
Lλ = x+y+z?a = 0
a167a26
x = mam+n+p
y = nam+n+p
z = pam+n+p
λ =?m+n+pa
a75
parenleftBigg
ma
m+n+p,
na
m+n+p,
pa
m+n+p
parenrightBigg
a143a140a85a52a138a58
a113Lx2 =?mx2,Lxy = Lyz = Lxz = 0,Ly2 =? ny2,Lz2 =? pz2,d2L =
parenleftBigg
mx2 dx2? ny2 dy2? pz2 dz2
parenrightBigg
< 0
a25a51
parenleftBigg
ma
m+n+p,
na
m+n+p,
pa
m+n+p
parenrightBigg
a63lnfa107a52a140a138a167a61fa107a52a140a138 m
mnnpp
(m+n+p)m+n+p a
m+n+p
a113f = xmnnzpa8(x,y,z)a170a117a62a46
braceleftbigg x+y = a
z = 0
braceleftbigg x+z = a
y = 0
braceleftbigg y+z = a
x = 0 a158a167f → 0a167a25fa27a141a152a52a140a58
a143a180a167a27a129a140a58.
3,a166a253a11x2 + 3y2 = 12a27a83a26a31a27a110a14a47a167a166a217a46a62a178a49a117a253a11a27a127a182a167a13a161a200a129a140.
a41a181a100a117a75a165a110a14a47a83a26a117a253a11 x
2
(2√3)2 +
y2
4 = 1a180a31a27a110a14a47a167a133a46a62a178a49a117a127a182
234
a25a217a46a62a164a233a186a58a55a180a225a182a254a253a11a27a186a58(0,±2)
a23a110a14a47a27a44a152a135a186a58a139a73a143(x,y) (x,y> 0)a167a75a217a83a26a31a27a110a14a47a46a62a127a1432xa167a112a143y+ 2
a31a27a110a14a47a110a186a58a139a73a143A(0,?2),B(x,y),C(?x,y)a167a100a253a11a27a233a161a53a167a26A(0,2),B(x,?y),C(?x,?y)a143a180
a217a186a58
a75S = x(y+ 2)a167a58(x,y)a51a253a11x2 + 3y2 = 12a254
a113a207a100a175a75a180a166S = x(y+ 2)a51a129a155a94a135x2 + 3y2 = 12a254a27a129a140a138(x,y> 0)
a138a188a234L = x(y+ 2) +λ(x2 + 3y2?12)
a75a41a144a167
Lx = y+ 2 + 2λx = 0
Ly = x+ 6λy = 0
Lλ = x2 + 3y2?12 = 0
a167a26
x = 3
y = 1
λ =?12
a117a180a217a186a58a139a73a143A(0,2),B(3,?1),C(?3,?1)a189A(0,?2),B(3,1),C(?3,1)
a207a100a175a75a143a162a83a175a75a167a129a140a138a55a127a51a167a75a51(0,2),(3,?1),(?3,?1)a189(0,?2),(3,1),(?3,1)a63a217a161a200a129a140a167a217
a138a1439.
4,a193a166a14a212a130y2 = 4xa254a27a58a167a166a167a134a134a130x?y+ 4 = 0a131a229a129a67.
a41a181a23a164a166a58a139a73a143(x,y)a167a75a167a20a134a130a27a229a108a143d = 1√2 |x?y+ 4|a167a217a165y2 = 4x
a134a130x?y+ 4 = 0a242a178a161a169a143a134a33a109a252a220a169a167a134a161x?y+ 4 < 0a167a109a161x?y+ 4 > 0
a13a14a212a130y2 = 4xa51a109a161a220a169a167a207a13a58(x,y)a20a167a27a229a108a143d = 1√2 (x?y+ 4)
a45L = 1√2 (x?y+ 4) +λ(y2?4x)
a75a41a144a167a124
Lx = 1√2?4λ = 0
Ly =? 1√2 + 2λy = 0
Lλ = y2?4x = 0
a167a26
x = 1
y = 2
λ = 14√2
a113Lx2 = 0,Ly2 = 12√2,Lxy = 0,d2L(1,2) = Lx2 dx2 + 2Lxy dxdy+Ly2 dy2 = dy
2
2√2 > 0
a25(1,2)a143a52a2a58a167a61a58(1,2)a20a134a130a27a229a108a129a67.
5,a14a212a161z = x2 +y2a26a178a161x+y+z = 1a31a164a152a253a11a167a166a6a58a20a249a253a11a27a129a127a134a129a225a229a108.
a41a181a226a75a191a167a166a229a108d =radicalbigx2 +y2 +z2a51a129a155a94a135z = x2 +y2,x+y+z = 1a27a129a138
a61a166u = x2 +y2 +z2a51a129a155a94a135a101a27a129a138
a138L = x2 +y2 +z2 +λ(z?x2?y2) +γ(x+y+z?1)
a75a41a144a167a124
Lx = 2x?2λx+γ = 0
Ly = 2y?2λy+γ = 0
Lz = 2z +λ+γ = 0
Lλ = z?x2?y2 = 0
Lγ = x+y+z?1 = 0
a167a26
x1 =?1 +
√3
2
y1 =?1 +
√3
2
z1 = 2?√3
λ1 =?5
√3 + 9
3
γ1 =?7 + 113 √3
,
x2 =?1?
√3
2
y2 =?1?
√3
2
z2 = 2 +√3
λ2 = 5
√3 + 9
3
γ2 =?7? 113 √3
a117a180d(x1,y1,z1) =
radicalbig
9?5√3,d(x2,y2,z2) =
radicalbig
9 + 5√3
a226a175a75a27a162a83a191a194a167a129a127a33a129a225a229a108a127a51
a75a129a127a229a108a143a6a58a20a58
parenleftBigg
1 +
√3
2,?
1 +√3
2,2 +
√3parenrightBigga27a229a108a167a143radicalbig9 + 5√3a182
a129a225a229a108a143a6a58a20a58
parenleftBigg
1 +√3
2,
1 +√3
2,2?
√3parenrightBigga27a229a108a167a143radicalbig9?5√3.
6,a166a152a109a152a58(a,b,c)a20a178a161Ax+By+Cz +D = 0a27a129a225a229a108.
a41a181a23(x,y,z)a143a178a161Ax+By+Cz+D = 0a254a63a152a58a167a75a167a134(a,b,c)a58a27a229a108a143d =radicalbig(x?a)2 + (y?b)2 + (z?c)2a167
a217a165(x,y,z)a247a118Ax+By+Cz +D = 0
a207d> 0a167a25da129a140d2a129a140
a85a75a23a167a65a166d2 = (x?a)2 + (y?b)2 + (z?c)2a51a94a135Ax+By+Cz +D = 0a101a27a52a138
a45L = (x?a)2 + (y?b)2 + (z?c)2 +λ(Ax+By+Cz +D)
235
a75a41a144a167a124
Lx = 2(x?a) +λA = 0
Ly = 2(y?b) +λB = 0
Lz = 2(z?c) +λC = 0
Lλ = Ax+By+Cz?D = 0
a167a26
x = a? 12 λA
y = b? 12 λB
z = c? 12 λC
λ = 2(Aa+Bb+Cc+D)A2 +B2 +C2
a117a180d = |Aa+Bb+Cc+D|√A2 +B2 +C2
a113a8x,y,za165a107a63a152a170a117∞a158a167d→∞a167a207a100a51braceleftbig(x,y,z)vextendsinglevextendsingle(x?a)2 + (y?b)2 + (z?c)2 <dbracerightbiga83a55a18a129a2a138
a75a58(a,b,c)a20a178a161Ax+By+Cz +D = 0a27a129a225a229a108a143d = |Aa+Bb+Cc+D|√A2 +B2 +C2,
236
a49a155a56a217 a219a188a234a127a51a189a110a33a188a234a131a39
§1,a219a188a234a127a51a189a110
1,a101a51a219a188a234a127a51a189a110a165a94a135a85a143a181
(1) a51a171a141D,(x0?alessorequalslantxlessorequalslantx0 + 1,y0?blessorequalslantylessorequalslanty0 +b)a254a235a89a182
(2) F(x0,y0) = 0a182
(3) a8xa27a189a158a167a188a234F(x,y)a180ya27a252a78a188a234a182a75a140a26a20a159a111a24a27a40a216a167a193a121a178a131.
a121a178a181a40a216a57a121a178a181
(1) a51a58(x0,y0)a27a44a152a25a141a83a167a100a144a167F(x,y) = 0a85a141a152a40a189y = f(x)a180xa27a252a78a188a234a133y0 = f(x0).
a100a94a135(3)a127a167a8xa27a189a158a167F(x,y)a180ya27a238a130a252a78a188a234.a216a148a23a167a180ya27a238a130a252a79a188a234
a27a189x0a167a188a234F(x0,y)a180ya27a238a130a79a188a234a167a133F(x0,y0) = 0a167a207a100a107F(x0,y0 +b) > 0,F(x0,y0?b) < 0
a100a94a135(1)a127a167F(x,y)a51a171a141Da254a235a89a167a207a13a127a51η> 0a167a166a8|x?x0|<ηa158a167a189a107
F(x,y0 +b) > 0,F(x,y0?b) < 0
a64a34a233?x∈O(x0,η)a167a100a188a234F(x,y)a51[y0?b,y0 +b]a27a235a89a53a57F(x,y0 +b) > 0,F(x,y0?b) < 0
a226a34a58a127a51a189a110a167a55a127a51y ∈ (y0?b,y0 +b)a167a166F(x,y) = 0
a100a117F(x,y)a51[y0?b,y0 +b]a238a130a252a78a167a108a13a8y>ya158a167F(x,y) > 0a182a8y<ya158a167F(x,y) < 0
a25a254a227ya180a141a152a27
a249a76a178a233?x ∈ O(x0,η)a167a207a76a254a227a144a123a167a107a141a152a27ya134xa233a65a167a133a247a118F(x,y) = 0a167a117a180a40a189a10a189a194
a51O(x0,η)a254a27a252a138a188a234y = f(x)a247a118F(x,f(x)) = 0a167a65a79a107F(x0,y0) = 0a61y0 = f(x0).
(2) f(x)a180a235a89a188a234.
x1 ∈O(x0,a)a167a101a121y = f(x)a51x1a58a235a89.
a233?ε> 0(ε<b)a167a23y1 = f(x1)a167a117a180F(x1,y1) = 0
a113a100a94a135(3)a167F(x,y)a180ya27a238a130a252a79a188a234
a207a100F(x1,y1 +ε) > 0,F(x1,y1?ε) < 0
a75a100Fa27a235a89a53a167a127a127a51a25a141O(x1,δ)?O(x0,a)a167a166a26a8x∈O(x1,δ)a158a167a240a107
F(x,y1 +ε) > 0,F(x,y1?ε) < 0
a117a180a226a34a58a127a51a189a110a167a26a55a107y ∈O(y1,ε)a167a166F(x,y) = 0a61y = f(x)
a61a144a135|x?x1|<δa167a210a107|f(x)?f(x1)| = |y?y1|<εa61y = f(x)a51x1a58a235a89
a100x1 ∈O(x0,a)a27a63a191a53a167a26f(x)a143a235a89a188a234.
2,a188a234F(x,y) ≡y2?x2(1?x2) = 0a51a61a10a58a67a11a140a141a152a47a251a189a252a138a235a89a167a133a107a235a89a19a234a27a188a234y = y(x).
a41a181a19a3a188a234F(x,y) = y2?x2(1?x2)a51a18a135a19a145a152a109a235a89a167a167a27a160a19a234Fx = 4x3?2x,Fy = 2ya143a235a89
a100y2?x2(1?x2) = 0a167a101y = 0a167a75x2(1?x2) = 0a167a41a26x = 0,x = ±1
a113y2 greaterorequalslant 0,x2 greaterorequalslant 0a167a251?x2 greaterorequalslant 0a61?1 lessorequalslantxlessorequalslant 1
a8y negationslash= 0a158a167Fy negationslash= 0
a100a219a188a234a127a51a189a1101a167a51a63a219a247a118{(x,y)vextendsinglevextendsingle|x|< 1,xnegationslash= 0,y2?x2(1?x2) = 0}a67a11a140a141a152a47a251a189a252a138a235a89a133a107
a235a89a19a234a27a188a234y = y(x).
3,a121a178a107a141a152a140a19a27a188a234y = y(x)a247a118a144a167 siny+ sinhy = xa167a191a166a209a19a234yprime(x).
a121a178a181a19a3a188a234F(x,y) = siny+ sinhy?xa51a18a135a19a145a152a109a235a89a167Fx =?1,Fy = cosy+ coshya143a235a89
a113coshy = e
y +e?y
2 greaterorequalslant 1 a133a31a210a144a51y = 0a158a164a225a167a13a100a158cosy = 1a167a51a152a132a156a185a101|cosy|lessorequalslant 1
a75a233a152a131a58(x,y)a167a240a107Fy = cosy+ coshy> 0a167a117a180Fy negationslash= 0
a100a219a188a234a127a51a189a1101a167a51a63a219a247a118a254a227a144a167a27a58(x,y)a167a107a141a152a140a19a27a188a234a247a118a144a167siny+ sinhy = x
a217a19a188a234a143yprime =?FxF
y
= 1cosy+ coshy.
4,a23Da180a58P0,(x0,y0,z0,u0,v0)a27a25a141a167a101
(1) F(x0,y0,z0,u0,v0) = 0,G(x0,y0,z0,u0,v0) = 0a182
(2) F,Ga39a117a152a131a67a254a27a160a19a234a51Da165a235a89a182
(3) J =
vextendsinglevextendsingle
vextendsinglevextendsingleFu FvG
u Gv
vextendsinglevextendsingle
vextendsinglevextendsinglea51P0a58a216a143a34a182
a75a51(x0,y0,z0)a27a25a141Ra83a127a51a141a152a27a152a233a188a234
u = f(x,y,z);v = g(x,y,z)
a247a118a181
237
(1) u0 = f(x0,y0,z0),v0 = g(x0,y0,z0)
(2) F(x,y,z,f,g) ≡ 0,G(x,y,z,f,g) ≡ 0
(3) u = f(x,y,z),v = g(x,y,z)a51a58(x0,y0,z0)a27a25a141Ra83a107a233a152a131a67a254a27a160a19a234a167a133
f
x =?
1
J
D(F,G)
D(x,v),
f
y =?
1
J
D(F,G)
D(y,v),
f
z =?
1
J
D(F,G)
D(z,v)
g
x =?
1
J
D(F,G)
D(u,x),
g
y =?
1
J
D(F,G)
D(u,y),
g
z =?
1
J
D(F,G)
D(u,z)
a121a178a181a100a94a135(3)a127a167Fu,Fva165a150a8a107a152a135a51P0a58a216a31a1170
a216a148a23Fv(P0) negationslash= 0a167a75a100a219a188a234a127a51a189a1102a167a127a51P0a58a27a44a135a25a141a83a140a177a114va108F(x,y,z,u,v) = 0a165a41a209a53.
a23v =?(x,y,z,u)a133v0 =?(x0,y0,z0,u0)a51(x0,y0,z0,u0)a27a44a135a25a141a83a180a141a152a27a167a228a107a39a117x,y,z,ua27a235a89
a160a19a234
a114v =?(x,y,z,u)a147a92G(x,y,z,u,v)a165a26G(x,y,z,u,?(x,y,z,u)) = ψ(x,y,z,u)
a25ψu = Gu +Gv ·vu = Gu +Gv
parenleftBigg
FuF
v
parenrightBigg
=? JF
v
a100a98a23Fv(P0) negationslash= 0a133a51P0a58J negationslash= 0a167a25ψu(x0,y0,z0,u0) negationslash= 0
a75a100a189a1102a167a26a51(x0,y0,z0,u0)a27a44a25a141a83a140a108a144a167G = G(x,y,z,u,?) ≡ψ(x,y,z,u) = 0a165a41a209ua53.
a23u = f(x,y,z)a167a167a51(x0,y0,z0)a27a44a25a141a83a107a235a89a160a19a234a167a133u0 = f(x0,y0,z0)
a114u = f(x,y,z)a147a92?(x,y,z,u)a165a26v =?(x,y,z,f(x,y,z)) = g(x,y,z)
a75a107g(x0,y0,z0) =?(x0,y0,z0,u0) = v0
a25u = f(x,y,z),v = g(x,y,z)a61a143a164a166
a233a144a167a124
braceleftbigg F(x,y,z,u,v) = 0
G(x,y,z,u,v) = 0 a252a224a39a117xa166a19a167a26
F
x +
F
u ·
f
x +
F
v ·
g
x = 0?G
x +
G
u ·
f
x +
G
v ·
g
x = 0
a41a131a167a26?f?x =?1J D(F,G)D(x,v),?g?x =?1J D(F,G)D(x,u)
a211a110a140a26?f?y =?1J D(F,G)D(y,v),?f?z =?1J D(F,G)D(z,v),?g?y =?1J D(F,G)D(u,y),?g?z =?1J D(F,G)D(u,z)
5,a23?i(x)(i = 1,2,···,n)a180xa27a235a89a140a19a188a234a167a133
Gi(x1,···,xn) = Fi(?1(x1),···,?n(xn))
a121a178?(G1,G2,···,Gn)?(x
1,x2,···,xn)
=?(?(x1),···,?n(xn))
nproducttext
i=1
i prime(xi)
a217a165?(x1,x2,···,xn) = D(F1,F2,···,Fn)D(x
1,x2,···,xn)
nproducttext
i=1
i prime(xi) =?1 prime(x1)?2 prime(x2)···?n prime(xn).
a121a178a181a207?i(x)(i = 1,2,···,n)a180xa27a235a89a140a19a188a234a167a133Gi(x1,···,xn) = Fi(?1(x1),···,?n(xn))
a75?Gi?x
j
=?Fi
j
·j?x
j
=?Fi
j
i prime(xj)(i,j = 1,2,···,n)
a117a180?(G1,G2,···,Gn)?(x
1,x2,···,xn)
=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
G1
x1
G1
x2 ···
G1
xn
G2
x1
G2
x2 ···
G2
xn
..........................
Gn
x1
Gn
x2 ···
Gn
xn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
F1
1(x1)?1
prime(x1)?F1
2(x2)?2
prime(x2) ···?F1
n(xn)?n
prime(xn)
F2
1(x1)?1
prime(x1)?F2
2(x2)?2
prime(x2) ···?F2
n(xn)?n
prime(xn)
.............................................................
Fn
1(x1)?1
prime(x1)?Fn
2(x2)?2
prime(x2) ···?Fn
n(xn)?n
prime(xn)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
=
1 prime(x1)?2 prime(x2)···?n prime(xn)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
F1
1(x1)
F1
2(x2) ···
F1
n(xn)
F2
1(x1)
F2
2(x2) ···
F2
n(xn)
......................................
Fn
1(x1)
Fn
2(x2) ···
Fn
n(xn)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
=
nproducttext
i=1
i prime(xi) D(F1,F2,···,Fn)D(?
1(x1),?2(x2),···,?n(xn))
=
(?(x1),···,?n(xn))
nproducttext
i=1
i prime(xi).
6,a23F(x,y,z)a107a19a30a235a89a160a19a234a167a191a100F(x,y,z) = 0a140a40a189z = f(x,y).a63a216z = f(x,y)a27a52a138a27a55a135a218a191a169a94
238
a135.a50a166a100
x2 +y2 +z2?2x+ 2y?4z?10 = 0
a164a40a189a27z = f(x,y)a27a52a138.
a121a178a181a207a188a234z = f(x,y)a18a26a52a138a27a55a135a94a135a143
braceleftbigg z
x = fx(x,y) = 0
zy = fy(x,y) = 0
a113zx =?FxF
z
,zy =?FyF
z
a167a75F(x,y,z)a18a26a52a138a27a55a135a94a135a143
braceleftbigg F
x = 0
Fy = 0
a113a219a188a234a18a52a138a27a27a191a169a94a135a134a119a188a234a97a211a167a144a180a166a19a30a160a19a234a158a135a94a219a188a234a27a112a30a160a19a234a166a123
a45
braceleftbigg F
x = 2x?2 = 0
Fy = 2y+ 2 = 0 a41a26
braceleftbigg x = 1
y =?1 a167a233a65a27za138a143z1 =?2,z2 = 6
a113?z?x = x?12?z,?z?y = 1 +y2?z,?
2z
x2 =
(x?1)2 + (2?z)2
(2?z)3,
2z
y2 =
(1 +y)2 + (2?z)2
(2?z)3,
2z
x?y =
(x?1)(1 +y)
(2?z)3
a117a180a51a58(1,?1,?2),?
2z
x2 =
1
4,
2z
y2 =
1
4,
2z
x?y = 0a167a100
1
4 ·
1
4?0 =
1
16 > 0a57
1
4 > 0a167a75z1 =?2a143a52a2a138a182
a51a58(1,?1,6),?
2z
x2 =?
1
4,
2z
y2 =?
1
4,
2z
x?y = 0a167a100
parenleftBigg
14
parenrightBiggparenleftBigg
14
parenrightBigg
0 = 116 > 0a57?14 < 0a167a75z2 = 6a143a52
a140a138
239
§2,a188a234a49a15a170a27a53a159a33a188a234a131a39
1,a121a178
x = rcosθcos?
y = rcosθsin?
z = rsinθ
a188a234a213a225
a121a178a181a207D(x,y,z)D(r,θ,?) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
cosθcosrsinθcosrcosθsin?
cosθsinrsinθsin? rcosθcos?
sinθ rcosθ 0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle=?r2 cosθ
a75a51r negationslash= 0a133θ negationslash= kpi+ pi2a27a171a141Da83D(x,y,z)D(r,θ,?) negationslash= 0
a117a180a226a189a1105a167a26a6a188a234a124a51a171a141Da83a188a234a213a225.
2,a121a178
y1 = x1 +x2 +···+xn
y2 = x21 +x22 +···+x2n
y3 = x1x2 +x1x3 +···+xn?1xn
a188a234a131a39a167a191a21a209a217a188a234a39a88a170.
a121a178a181a207a127a51a188a234?(t1,t2) = 12(t21?t2)a167a166a26y3 =?(y1,y2) = 12(y21?y2)a51a18a135na145a152a109(x1,x2,···,xn)a83
a164a143a240a31a170
a75a188a234a124a51a18a135na145a152a109a165a188a234a131a39a167a217a188a234a39a88a170a143y3 = 12(y21?y2).
3,a101a15a188a234a180a196a131a39a186
(1) x?yx?z,y?zy?x,z?xz?y
(2) x1?x?y?z,y1?x?y?z,z1?x?y?z
a41a181
(1) a207f1 ·f2 ·f3 =?1a167a75a188a234a131a39.
(2) a45f1(x,y,z) = x1?x?y?z,f2(x,y,z) = y1?x?y?z,f3(x,y,z) = z1?x?y?z
a75Jocobia221a10a143
f1
x
f1
y
f1
z
f2
x
f2
y
f2
z
f3
x
f3
y
f3
z
=
1?y?z
(1?x?y?z)2
x
(1?x?y?z)2
x
(1?x?y?z)2
y
(1?x?y?z)2
1?x?z
(1?x?y?z)2
y
(1?x?y?z)2
z
(1?x?y?z)2
z
(1?x?y?z)2
1?x?y
(1?x?y?z)2
a113a100a221a10a27a157a1433a167a75a226a189a1106a167a26a188a234a124a188a234a213a225.
240
a49a110a220a169 a185a235a67a254a27a200a169a218a50a194a200a169
a49a155a212a217 a185a235a67a254a27a200a169
1,a23F(y) =
integraldisplay y2
y
e?x2y dxa167a79a142F prime(y).
a41a181a207a189a1104a94a135a247a118a167a65a94a189a1104a167a107
F prime(y) =
integraldisplay y2
y
(?x2)e?x2y dx+ 2ye?y5?e?y3 = 52 ye?y5? 32e?y3? 12y F(y).
2,a23F(y) =
integraldisplay y
0
(x+y)f(x)dxa167a217a165f(x)a143a140a135a188a234a167a166F primeprime(y).
a41a181a207f(x)a143a140a135a188a234a167a75f(x)a235a89a167a117a180(x+y)f(x)a235a89a167a75a189a1104a94a135a247a118
a117a180F prime(y) =
integraldisplay y
0
f(x)dx+ 2yf(y)a167F primeprime(y) = 3f(y) + 2yfprime(y).
3,a101F(y) =
integraldisplay 1
0
ln
radicalbig
x2 +y2 dxa167a134a26a79a142a200a169a167a166a209F(y)a167a50a166a209F prime(0)a167a191a117a8a65a94a189a1104a79a142F prime(0)a27a20a40
a53.
a41a181a8y negationslash= 0a158a167a107F(y) =
integraldisplay 1
0
ln
radicalbig
x2 +y2 dx = xln
radicalbig
x2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
integraldisplay 1
0
x2
x2 +y2 dx = ln
radicalbig
1 +y2? 1 +
y
integraldisplay 1
0
dxy
1 +
parenleftBigg
x
y
parenrightBigg2 dx = ln
radicalbig
1 +y2?1 +yarctan 1y,
a207F(0) =
integraldisplay 1
0
lnxdx =?1
a75F+ prime(0) = lim
y→+0
F(y)?F(0)
y =
pi
2,F?
prime(0) = lim
y→?0
F(y)?F(0)
y =?
pi
2
a117a180F prime(0)a216a127a51
a44a152a144a161a167a8x> 0a158a167ylnradicalbigx2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
y=0
= yx2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
y=0
= 0a167a75
integraldisplay 1
0
parenleftBigg
yln
radicalbig
x2 +y2
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
y=0
dx = 0
a113F+ prime(0) = pi2 negationslash= 0 =
integraldisplay 1
0
parenleftBigg
yln
radicalbig
x2 +y2
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
y=0
dx,F? prime(0) =?pi2 negationslash= 0 =
integraldisplay 1
0
parenleftBigg
yln
radicalbig
x2 +y2
parenrightBiggvextendsinglevextendsingle
vextendsinglevextendsingle
y=0
dx
a75a8y = 0a158a167a216a85a51a200a169a210a101a166a19a234a167a61a166a166a134a33a109a19a234a143a216a49.
4,a166a188a234
E(k) =
integraldisplay pi
2
0
radicalBig
1?k2 sin2?d?a218F(k) =
integraldisplay pi
2
0
d?radicalbig
1?k2 sin2?
(0 <k< 1)
a27a19a234a133a121a178E(k)a247a118a144a167a181
Eprimeprime(k) + 1k Eprime(k) + E(k)1?k2 = 0
.
a41a181Eprime(k) =?
integraldisplay pi
2
0
ksin2?radicalbig
1?k2 sin2?
d? = 1k
integraldisplay pi
2
0
bracketleftBiggradicalBig
1?k2 sin2 1radicalbig
1?k2 sin2?
bracketrightBigg
d? = 1k[E(k)?F(k)]
F prime(k) =
integraldisplay pi
2
0
ksin2?
(1?k2 sin2?)32
d? =?1k
integraldisplay pi
2
0
bracketleftBigg
1radicalbig
1?k2 sin2?
1
(1?k2 sin2?)32
bracketrightBigg
d? =?1kF(k)+1k
integraldisplay pi
2
0
1
(1?k2 sin2?)32
d?
a207 dd?
parenleftBigg
k2 sin?cos?radicalbig
1?k2 sin2?
parenrightBigg
= k
2(cos2sin2?)(1?k2 sin2?) +k4 sin2?cos2?
(1?k2 sin2?)32
= k
2?1 + (1?k2 sin2?)2
(1?k2 sin2?)32
=
k2?1
(1?k2 sin2?)32
+
radicalbig
1?k2 sin2?
a75 1
(1?k2 sin2?)32
= 11?k2
radicalbig
1?k2 sin2 11?k2 dd?
parenleftBigg
k2 sin?cos?radicalbig
1?k2 sin2?
parenrightBigg
a117a180
integraldisplay pi
2
0
d?
(1?k2 sin2?)32
= 11?k2
integraldisplay pi
2
0
radicalBig
1?k2 sin2?d? = 11?k2 E(k)
241
a75F prime(k) = 1k(1?k2) E(k)? 1k F(k)
a117a180Eprimeprime(k) = (E
prime(k)?F prime(k))k?(E(k)?F(k))
k2 =?
E(k)
k2(1?k2)?
F(k)
k2
a108a13Eprimeprime(k) + 1k Eprime(k) + E(k)1?k2 =? E(k)k2(1?k2) + F(k)k2 + E(k)?F(k)k2 + E(k)1?k2 = 0.
5,a239a196a188a234
F(y) =
integraldisplay 1
0
yf(x)
x2 +y2 dx,(ygreaterorequalslant 0)
a27a235a89a53a167a217a165f(x)a180[0,1]a254a235a89a133a143a20a27a188a234.
a41a181a23d>c> 0a167a18y ∈ [c,d]a167a75a26a200a188a234 yf(x)x2 +y2 a51[0,1]×[c,d]a254a235a89
a100a189a1101a167a26F(y) =
integraldisplay 1
0
yf(x)
x2 +y2 dxa51[c,d]a254a235a89a167a100c,da27a63a191a53a167a26F(y)a51y> 0a235a89
a113f(x)a180[0,1]a254a235a89a133a143a20a27a188a234a167a75f(x)a51[0,1]a254a55a107a129a2a138m> 0
a100a117F(y) greaterorequalslantm
integraldisplay 1
0
y
x2 +y2 dx = marctan
1
ya57 limy→+0arctan
1
y =
pi
2 a167a75 limy→+0F(y) greaterorequalslant
mpi
2 > 0
a113F(0) = 0a167a75F(y)a8y = 0a158a216a235a89.
6,a65a94a233a235a234a166a19a123a79a142a200a169a181
(1)
integraldisplay pi
2
0
ln(a2?sin2x)dx (a> 1)(a216a55a189a126a234a167a101a79a142a158a209a121a195a46a156a185a167a18a52a129a79a142)a182
(2)
integraldisplay pi
0
ln(1?2acosx+a2)dx (|a|< 1)
a41a181
(1) a23I(a) =
integraldisplay pi
2
0
ln(a2?sin2x)dx
a207a26a200a188a234ln(a2?sin2x)a51
bracketleftBigg
0,pi2
bracketrightBigg
×[1,+∞]a254a216a235a89
a75a216a85a94a189a1102a167a143a10a85a94a189a110a167a160a2a137a140
bracketleftBigg
0,pi2
bracketrightBigg
×[b,c](b> 1,c→ +∞)
a249a158f(x,a) = ln(a2?sin2x)a57fa = 2aa2?sin2xa209a51a52a221a47
bracketleftBigg
0,pi2
bracketrightBigg
×[b,c]a254a235a89
a100a189a1102a167a107Iprime(a) =
integraldisplay pi
2
0
2a
a2?sin2x dx =
2√
a2?1
bracketleftBigg
arctan a+ 1√a2?1 + arctan a?1√a2?1
bracketrightBigg
= pi√a2?1
a233aa200a169a167a26I(a) = piln|a+√a2?1|+C
a207a∈ [b,c]a167a100b,ca27a63a191a53a167a75I(a) = piln|a+√a2?1|+C
(2) a23I(a) =
integraldisplay pi
0
ln(1?2acosx+a2)dx
a8|a|< 1a158a167a100a1171?2acosx+a2 greaterorequalslant 1?2|a|+a2 = (1?|a|)2 > 0
a75f(x,a) = ln(1?2acosx+a2)a57fa =?2cosx+ 2a1?2acosx+a2 a209a51a52a221a47[0,pi]×[?b,b]a254a235a89(|a|lessorequalslantb< 1)
a100a189a1102a167a107Iprime(a) =
integraldisplay pi
0
2cosx+ 2a
1?2acosx+a2 dx =
pi
a?
1?a2
a
integraldisplay pi
0
dx
(1 +a2)?2acosx =
pi
a?
1?a2
a(1 +a2)
integraldisplay pi
0
dx
1 +
parenleftBig
2aa2+1
parenrightBig
cosx
a138a147a134t = tan x2
a75
integraldisplay pi
0
dx
1 +
parenleftBig
2aa2+1
parenrightBig
cosx
= 2
integraldisplay +∞
0
1 +a2
(1?a)2 + (1 +a)2t2 dt =
1 +a2
1?a2 pi
a117a180Iprime(a) = 0a167a108a13I(a) = C
a113I(0) = 0a167a75C = 0a167a117a180I(a) = 0
a207a∈ [?bb]a167a100ba27a63a191a53a167a26a8|a|< 1a158a167I(a) = 0.
7,a65a94a200a169a210a101a166a200a169a144a123a79a142a200a169a181
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xb?xa
lnx dx (a> 0,b> 0)
242
(a101a209a121a195a46a156a185a134a99a161a211a24a63a110).
a41a181a216a148a23a<b
a207
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xb?xa
lnx dx =
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
dx
integraldisplay b
a
xy dy =
integraldisplay b
a
dy
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xy dx.
a249a112a167a8x = 0a158a167sin
parenleftBigg
ln 1x
parenrightBigg
xya110a41a1430a167a108a13sin
parenleftBigg
ln 1x
parenrightBigg
xya510 lessorequalslantxlessorequalslant 1,alessorequalslantylessorequalslantba254a235a89
a75a140a65a94a200a169a210a101a27a200a169a123a2a134a200a169a103a83
a138a147a134x = e?ta167a140a26
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xy dx =
integraldisplay +∞
0
e?(y+1)t sintdt = 11 + (1 +y)2
a117a180
integraldisplay 1
0
sin
parenleftBigg
ln 1x
parenrightBigg
xb?xa
lnx dx =
integraldisplay b
a
dy
1 + (1 +y)2 = arctan(1+b)?arctan(1+a) = arctan
b?a
1 + (1 +b)(1 +a)
8,a121a178
integraldisplay 1
0
dx
integraldisplay 1
0
x2?y2
(x2 +y2)2 dy negationslash=
integraldisplay 1
0
dy
integraldisplay 1
0
x2?y2
(x2 +y2)2 dx.
a121a178a181a207
integraldisplay 1
0
dx
integraldisplay 1
0
x2?y2
(x2 +y2)2 dy =
integraldisplay 1
0
dx
1 +x2 =
pi
4 a167
integraldisplay 1
0
dy
integraldisplay 1
0
x2?y2
(x2 +y2)2 dx =?
integraldisplay 1
0
dy
1 +y2 =?
pi
4
a75
integraldisplay 1
0
dx
integraldisplay 1
0
x2?y2
(x2 +y2)2 dy negationslash=
integraldisplay 1
0
dy
integraldisplay 1
0
x2?y2
(x2 +y2)2 dx.
9,a23a188a234f(x,y)a51D = [a,A;b,B]a107a46a167a216a22Da83a107a129a94a235a89a173a130y =?i(x)a167fa51Da235a89a167a121a178a181
F(x) =
integraldisplay B
b
f(x,y)dy
a51[a,A]a235a89.
a121a178a181a216a148a23a144a107a152a94a235a89a173a130y =?1(x)a167f(x,y)a51a249a94a173a130a254a109a228
a207f(x,y)a107a46a167a80M = sup
[a,A;b,B]
|f(x,y)|
a63a18x0 ∈ [α,β]? [a,A],y0 =?1(x0) ∈ [b,B]
a101a121F(x) =
integraldisplay B
b
f(x,y)dya51x0a58a235a89a167a61a121?ε> 0,?δ> 0a167a8|x?x0|<δa158a167a107
|F(x)?F(x0)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay B
b
f(x,y)dy?
integraldisplay B
b
f(x0,y)dy
vextendsinglevextendsingle
vextendsinglevextendsingle<ε
a100a117y =?1(x)a51x0a58a235a89a167a75a233?ε1 > 0,?δ1 > 0a167a8|x?x0|< 2δ1a158a167a107|y?y0| = |?1(x)1(x0)|<ε1
a117a180a51x0?δ1 lessorequalslantxlessorequalslantx0 +δ1,blessorequalslantylessorequalslantBa27a145a141a83a166f(x,y)a109a228a27a58a144a185a117a177(x0,y0)a143a165a37a27a221a47a141x0?δ1 lessorequalslant
xlessorequalslantx0 +δ1,y0?ε1 <y <y0 +ε1a51a249a145a141a27a254a33a101a252a253(a101y0?ε1a84a208a31a117ba189y0 +ε1a84a208a31a117Ba167a75a144a107
a254a253a189a101a253)a167a52a141a165f(x,y)a143a235a89
a207a13a51a249a252a135(a189a152a135)a52a141a165f(x,y)a143a152a151a235a89a167a65a79a233?ε2 > 0,?δ2 > 0a167a8|x?x0|<δ2a158a167a107
|f(x,y)?f(x0,y)|<ε2
a121a18δ = min(δ1,δ2)a167a8|xprime?x0|<δa158a167a107
|F(xprime)?F(x)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay B
b
f(xprime,y)dy?
integraldisplay B
b
f(x0,y)dy
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay y0?ε1
b
|f(xprime,y)?f(x0,y)|dy+
integraldisplay y0+ε1
y0?ε1
|f(xprime,y)|dy+
integraldisplay y0+ε1
y0?ε1
|f(x0,y)|dy+
integraldisplay B
y0+ε1
|f(xprime,y)?f(x0,y)|dylessorequalslantε2(B?b) + 4ε1M
a101a18ε1 = ε8M,ε1 = ε2(B?b)
a75a8|xprime?x0|<δa158a167a107|F(xprime)?F(x0)|<ε
a8x0 ∈ [a,α]a189x0 ∈ [β,A]a158a167F(x)a51x0a235a89a167a25F(x)a51[a,A]a235a89
a101f(x,y)a107a109a228a27a235a89a173a130a107a65a94a167a75a144a73a114a166f(x,y)a140a85a164a143a109a228a27a58a94a150a245a65a135a2a221a47a133a109a210a49a10
a217a123a216a121a131a211
a100a117f(x,y)a107a46a133a150a245a107a65a94a109a228a130a167a75F(x) =
integraldisplay B
b
f(x,y)dya127a51a133a51[a,A]a235a89.
243
a49a155a108a217 a185a235a67a254a27a50a194a200a169
1,a121a178a181a101a51[a,∞;c,d]a83a164a225|f(x,y)|lessorequalslantF(x,y)a167a191a133a39a117y ∈ [c,d]a200a169
integraldisplay +∞
a
F(x,y)dxa152a151a194a241a167a75
integraldisplay +∞
a
f(x,y)dxa39
a117y ∈ [c,d]a189a152a151a194a241a167a133a253a233a194a241.
a121a178a181a207a200a169
integraldisplay +∞
a
F(x,y)dxa39a117y ∈ [c,d]a152a151a194a241a167a75a100a185a235a67a254a27a50a194a200a169a27a133a220a152a151a194a241a6a110a167a26
a233?ε> 0a167a127a51a134ya195a39a27A0(ε) >aa167a8A,Aprime greaterorequalslantA0a158a167a233a152a131y ∈ [c,d]a167a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprime
A
F(x,y)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε
a13
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprime
A
f(x,y)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprime
A
|f(x,y)|dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprime
A
F(x,y)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<εa233a152a131y ∈ [c,d]a209a164a225
a100a195a161a129a185a235a67a254a50a194a200a169a27a133a220a152a151a194a241a6a110a167
integraldisplay +∞
a
f(x,y)dxa39a117y ∈ [c,d]a152a151a194a241a167
integraldisplay +∞
a
|f(x,y)|dxa39
a117y ∈ [c,d]a152a151a194a241
a75
integraldisplay +∞
a
f(x,y)dxa39a117y ∈ [c,d]a152a151a194a241a133a253a233a194a241.
2,a121a178a101a15a200a169a51a164a137a189a27a171a109a83a152a151a194a241a181
(1)
integraldisplay +∞
0
cosxy
x2 +y2 dx (ygreaterorequalslanta> 0)
(2)
integraldisplay +∞
0
cosxy
x2 + 1 dx (?∞<y< +∞)
(3)
integraldisplay 1
0
lnxydx
parenleftBigg
1
blessorequalslantylessorequalslantb,b> 1
parenrightBigg
a121a178a181
(1) a207ygreaterorequalslanta> 0a167a75
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosxy
x2 +y2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
x2 +a2 a13
integraldisplay +∞
0
dx
x2 +a2 =
pi
2aa194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
0
cosxy
x2 +y2 dxa39a117ya51[a,+∞)(a> 0)a83a152a151a194a241.
(2) a207y ∈ (?∞,+∞),
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
cosxy
x2 + 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
1
x2 + 1 a13
integraldisplay +∞
0
dx
x2 + 1 =
pi
2 a194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
0
cosxy
x2 + 1 dxa39a117ya51(?∞,+∞)a83a152a151a194a241.
(3) x = 0a180a219a58a167a81blessorequalslantylessorequalslantb,b> 1,0 <xlessorequalslant 1a158a167|lnxy|lessorequalslant|lnx|+|lny|lessorequalslant?lnx+ lnb = ln bx
a207 lim
x→+0
x14 ln bx = lim
x→+0
ln bx
x?14
= 0a167a75a100a195a46a188a234a50a194a200a169a7a79a123a27a52a129a47a170a167a26
integraldisplay 1
0
ln bx dxa194a241
a108a13a100a159a188a7a79a123a167a26
integraldisplay 1
0
lnxydxa39a117ya51[ 1b,b](b> 1)a254a152a151a194a241.
3,a23f(x,y)a51[a,+∞;c,d]a235a89a167a233[c,d)a254a122a152a135ya167
integraldisplay +∞
a
f(x,y)dxa194a241a167a2a200a169a51y = da117a209.
a121a178a249a200a169a51[c,d]a154a152a151a194a241.
a121a178a181a100
integraldisplay +∞
a
f(x,d)dxa117a209a167a26?ε0 > 0,?A0 >a,?Aprime,Aprimeprime greaterorequalslantA0a167a166
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
f(x,d)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslantε0
a249a76a178a233y = d∈ [c,d]a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
f(x,y)dx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglegreaterorequalslantε0a167a96a178
integraldisplay +∞
a
f(x,y)dxa51[c,d]a154a152a151a194a241.
4,a63a216a101a15a200a169a51a141a189a171a109a27a152a151a194a241a53a181
(1)
integraldisplay +∞
1
xαe?x dx (alessorequalslantαlessorequalslantb;a,ba143a63a191a162a234)
(2)
integraldisplay +∞
0
√αe?αx2 dx (0 <α< +∞)
244
(3)
integraldisplay +∞
∞
e?(x?α)2 dx
(i) a<α<b
(ii)?∞<α< +∞
(4)
integraldisplay 1
0
xp?1 ln2xdx
(i) pgreaterorequalslantp0 > 0
(ii) p> 0
(5)
integraldisplay +∞
0
e?αx sinxdx (α> 0)
a41a181
(1) a207α∈ [a,b],x∈ (1,+∞)a167a750 <|xαe?x|lessorequalslantxbe?x
a113 lim
x→+∞
x2 ·xbe?x = 0a167a75a226a195a161a129a50a194a200a169a27a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
xbe?x dxa152a151a194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
1
xαe?x dxa39a117α∈ [a,b](a,ba143a63a191a162a234)a152a151a194a241.
(2)
integraldisplay +∞
0
√αe?αx2 dx = √pi
2 a194a241a167a2a167a51(0,+∞)a39a117αa154a152a151a194a241
a233?A> 0a167a207 lim
α→+0
integraldisplay +∞
A
√αe?αx2 dx = lim
α→+0
integraldisplay +∞
√αA e
t2 dt =
integraldisplay +∞
0
e?t2 dt =
√pi
2
a25a233a1170 < ε0 <
√pi
2 a167a55a127a51α0 > 0a167a166a26
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay +∞
A
√α
0e?α0x
2 dx
vextendsinglevextendsingle
vextendsinglevextendsingle =
integraldisplay +∞
A
√α
0e?α0x
2 dx > ε
0a167
a61
integraldisplay +∞
0
√αe?αx2 dxa39a117αa51(0,+∞)a254a216a152a151a194a241.
(3) a233a63a191a27a189a27α∈ (?∞,+∞)a167a200a169
integraldisplay +∞
∞
e?(x?α)2 dxa209a194a241a167a133
integraldisplay +∞
∞
e?(x?α)2 dx = √pi
(i) |x|a191a169a140a158a167a233a152a131a<α<ba167a1070 <e?(x?α)2 < 2e?x
2
4
a207
integraldisplay +∞
∞
e?x
2
4 dx = 2
integraldisplay +∞
0
e?x
2
4 dxa194a241
a75a100a159a188a7a79a123a167a26
integraldisplay +∞
∞
e?(x?α)2 dxa233a<α<ba152a151a194a241.
(ii) a233?A> 0a167a107 lim
α→+∞
integraldisplay +∞
A
e?(x?α)2 dx = lim
α→+∞
integraldisplay +∞
A?α
e?t2 dt = √pi
a75a8αa191a169a140a158a167
integraldisplay +∞
A
e?(x?α)2 dx>
√pi
2
a100a100a167a26
integraldisplay +∞
0
e?(x?α)2 dxa51?∞<α< +∞a254a154a152a151a194a241
a108a13
integraldisplay +∞
∞
e?(x?α)2 dxa51?∞<α< +∞a254a154a152a151a194a241.
(4) (i) |xp?1 ln2x| = xp?1 ln2xlessorequalslantxp0?1 ln2x (pgreaterorequalslantp0 > 0,0 lessorequalslantxlessorequalslant 1)
a200a169
integraldisplay 1
0
xp?1 ln2xdx =
integraldisplay +∞
0
e?p0zz2 dz
lim
z→+∞
z2 ·e?p0zz2 = lim
z→+∞
z4
ep0z = 0 (p0 > 0)
a75a100a133a220a7a79a123a27a52a129a47a170
integraldisplay +∞
0
e?p0zz2 dza194a241a167a117a180
integraldisplay 1
0
xp0?1 ln2xdxa194a241
a108a13a100a159a188a7a79a123a167a26
integraldisplay 1
0
xp?1 ln2xdxa39a117pa51pgreaterorequalslantp0 > 0a254a152a151a194a241.
(ii) a207a8x∈
parenleftBigg
0,1e
parenrightBigg
,ln2xgreaterorequalslant 1
a25a107
integraldisplay 1
0
xp?1 ln2xdx>
integraldisplay 1
e
0
xp?1 ln2xdx>
integraldisplay 1
e
0
xp?1 dx = 1p
parenleftBigg
1
e
parenrightBiggp
→ +∞ (p→ +0)
a117a180
integraldisplay 1
0
xp?1 ln2xdxa51p> 0a158a216a152a151a194a241.
245
(5) a94a135a121a123.
a98a23
integraldisplay +∞
0
e?αx sinxdxa39a117α > 0a152a151a194a241a167a75a233?ε > 0,?M > 0a167a8Aprimeprime > Aprime > Ma158a167a233a152
a131α> 0a164a225
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
e?αx sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε
a108a13a233a117α∈ (0,1)a189a164a225
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
e?αx sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle<ε
a51a216a31a170a252a62a45α→ 0a167a75a107
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay Aprimeprime
Aprime
sinxdx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslantεa167a108a13
integraldisplay +∞
0
sinxdxa194a241
a13
integraldisplay A
0
sinxdx = 1?cosAa167a8A→ +∞a158a167cosAa27a52a129a216a127a51a167a117a180
integraldisplay +∞
0
sinxdxa117a209a167a75a103a241a167a25
a98a23a216a164a225
a108a13
integraldisplay +∞
0
e?αx sinxdxa39a117α> 0a216a152a151a194a241.
5,a121a178a181
(1)
integraldisplay +∞
0
α
x2 +α2 dxa51a216a185α = 0a27a63a219a171a109a254a180a235a89a188a234a182
(2) F(p) =
integraldisplay pi
0
sinx
xp(pi?x)2?p dxa51(0,2)a83a235a89.
a121a178a181
(1) a23F(α) =
integraldisplay +∞
0
α
x2 +α2 dx.
a233a63a219α0 negationslash= 0a167a216a148a23α0 > 0a167a56a18δ> 0a167a166a26α0?δ> 0a167a101a121F(α)a51[α0?δ,α0 +δ]a83a152a151a194a241
a175a162a254a167a8α∈ [α0?δ,α0 +δ]a158a167 αx2 +α2 lessorequalslant α0 +δx2 + (α
0?δ)2
a207a200a169
integraldisplay +∞
0
α0 +δ
(α0?δ)2 +x2 dxa194a241a167a75a100a159a188a7a79a123a167a26F(α)a51[α0?δ,α0 +δ]a254a39a117αa152a151a194a241
a117a180a100a235a89a189a110a167a26F(α)a51a84a171a109a254a180αa27a235a89a188a234a167a65a79a51α0a58a235a89
a100a117α0 negationslash= 0a27a63a191a53a167a26
integraldisplay +∞
0
α
x2 +α2 dxa233a63a219αnegationslash= 0a235a89a167a100a100a140a127F(α)a51a63a219a216a185α = 0a27a171a109a254
a209a235a89
a2a100 lim
α→+0
integraldisplay +∞
0
α
α2 +x2 dx =
pi
2,limα→?0
integraldisplay +∞
0
α
α2 +x2 dx =?
pi
2
a26F(α)a51α = 0a63a216a235a89a167a75
integraldisplay +∞
0
α
x2 +α2 dxa51a216a185α = 0a27a63a219a171a109a254a180a235a89a188a234.
(2) a63a18p∈ (0,2)a167a75a127a510 <p1,p2 < 2a167a1660 <p1 lessorequalslantplessorequalslantp2 < 2
a2070a218pia254a140a85a180a219a58a167a242a200a169a169a143a110a227integraldisplay
pi
0
sinx
xp(pi?x)2?p dx =
integraldisplay 1
0
sinx
xp(pi?x)2?p dx+
integraldisplay pi?1
1
sinx
xp(pi?x)2?p dx+
integraldisplay pi
pi?1
sinx
xp(pi?x)2?p dx
a233a117
integraldisplay 1
0
sinx
xp(pi?x)2?p dx
a207 sinxxp(pi?x)2?p lessorequalslant sinxxp
2(pi?x)2?p2
(0 lessorequalslantxlessorequalslant 1,0 <p1 lessorequalslantplessorequalslantp2 < 2)
a133 lim
x→+0
xp2?1 sinxxp
2(pi?x)2?p2
= 1pi2?p
2
a207p2 < 2a167a75p2?1 < 1a167a117a180a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay 1
0
sinx
xp2(pi?x)2?p2 dxa194a241
a108a13a100a159a188a7a79a123a167a26
integraldisplay 1
0
sinx
xp(pi?x)2?p dxa39a117p∈ [p1,p2 ]a152a151a194a241
a113a26a200a188a234 sinxxp(pi?x)2?pa51(0,1]×[p1,p2 ]a254a235a89a167a75a100a235a89a53a189a110a167a26
integraldisplay 1
0
sinx
xp(pi?x)2?p dxa51[p1,p2 ]a235
a89integraldisplay
pi?1
1
sinx
xp(pi?x)2?p dxa180a185a235a67a254a27a126a194a200a169
246
a207a26a200a188a234 sinxxp(pi?x)2?pa51[1,pi?1]×[p1,p2 ]a235a89a167a75a100a235a89a53a189a110a167a26
integraldisplay pi?1
1
sinx
xp(pi?x)2?p dxa51[p1,p2 ]a235
a89
a233a117
integraldisplay pi
pi?1
sinx
xp(pi?x)2?p dx
a207 sinxxp(pi?x)2?p lessorequalslant sin(pi?x)xp
1(pi?x)2?p1
(pi?1 lessorequalslantxlessorequalslantpi,0 <p1 lessorequalslantplessorequalslantp2 < 2)
a133 lim
x→pi?0
(pi?x)1?p1 sin(pi?x)xp
1(pi?x)2?p1
= 1pip
1
a207p1 > 0a167a751?p1 < 1a167a117a180a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay pi
pi?1
sin(pi?x)
xp?1(pi?x)2?p1 dxa194a241
a108a13a100a159a188a7a79a123a167a26
integraldisplay pi
pi?1
sinx
xp(pi?x)2?p dxa39a117p∈ [p1,p2 ]a152a151a194a241
a113a26a200a188a234 sinxxp(pi?x)2?pa51[pi?1,pi)×[p1,p2 ]a254a235a89a167a75a100a235a89a53a189a110a167a26
integraldisplay pi
pi?1
sinx
xp(pi?x)2?p dxa51[p1,p2 ]a235
a89
a110a220a177a254a167a26F(p)a51[p1,p2 ]a235a89a167a108a13a51a217a254a63a152a58pa235a89
a113a100p∈ (0,2)a27a63a191a53a167a26F(p) =
integraldisplay pi
0
sinx
xp(pi?x)2?p dxa51(0,2)a83a235a89..
6,a23f(t)a8t> 0a158a235a89.a88a74
integraldisplay +∞
0
tλf(t)dta8λ = a,λ = ba158a209a194a241a167a64a34
integraldisplay +∞
0
tλf(t)dta39a117λa51[a,b]a254a152a151a194
a241.
a121a178a181a207f(t)a8t> 0a158a235a89a167a75a26a200a188a234tλf(t)a27a219a58a144a140a85a1800
a117a180
integraldisplay +∞
0
tλf(t)dt =
integraldisplay 1
0
tλf(t)dt+
integraldisplay +∞
1
tλf(t)dt
a233a117
integraldisplay 1
0
tλf(t)dt =
integraldisplay 1
0
tλ?ataf(t)dt
a207
integraldisplay +∞
0
taf(t)dta194a241a167a75
integraldisplay 1
0
taf(t)dta194a241a167a108a13a39a117λ ∈ (?∞,+∞)a152a151a194a241a167a13tλ?aa233a117λ greaterorequalslant aa252a78a126
a133|tλ?a|lessorequalslant 1 (0 lessorequalslanttlessorequalslant 1,λgreaterorequalslanta)
a75a100Abela7a79a123a167a26
integraldisplay 1
0
tλf(t)dta39a117λgreaterorequalslantaa152a151a194a241
a233a117
integraldisplay +∞
1
tλf(t)dt =
integraldisplay +∞
1
tλ?btbf(t)dt
a207
integraldisplay +∞
0
tbf(t)dta194a241a167a75
integraldisplay +∞
1
tbf(t)dta194a241a167a108a13a39a117λ ∈ (?∞,+∞)a152a151a194a241a167a13tλ?ba233a117λ lessorequalslant ba252a78a126
a133|tλ?b|lessorequalslant 1 (1 lessorequalslantt< +∞,λlessorequalslantb)
a75a100Abela7a79a123a167a26
integraldisplay +∞
1
tλf(t)dta39a117λlessorequalslantba152a151a194a241
a117a180
integraldisplay +∞
0
tλf(t)dta39a117λa51[a,b]a254a152a151a194a241.
7,a108a31a170e
ax?e?bx
x =
integraldisplay b
a
e?xy dya209a117a167a79a142a200a169
integraldisplay +∞
0
e?ax?e?bx
x dx(b>a> 0)
a41a181a207e
ax?e?bx
x =
integraldisplay b
a
e?xy dya167a75
integraldisplay +∞
0
e?ax?e?bx
x dx =
integraldisplay +∞
0
dx
integraldisplay b
a
e?xy dy
a188a234e?xya51[0,+∞)×[a,b]a254a235a89
a113a233y ∈ [a,b](a> 0)a167a75|e?xy|lessorequalslante?axa133
integraldisplay +∞
0
e?ax dx = 1aa194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
0
e?xy dxa39a117ya51[a,b]a152a151a194a241
a100a200a169a2a134a94a83a189a110a167a26
integraldisplay +∞
0
e?ax?e?bx
x dx =
integraldisplay +∞
0
dx
integraldisplay b
a
e?xy dy =
integraldisplay b
a
dy
integraldisplay +∞
0
e?xy dx =
integraldisplay b
a
dy
y =
ln ba (b>a> 0)
247
8,a193a121a178Γ(s)a27a19a234a127a51a167a166a209Γprime(s)a27a200a169a76a136a170a167a96a178a237a19a76a167a180a220a110a27.
a121a178a181Γ(s) =
integraldisplay +∞
0
xs?1e?x dx
a207xs?1e?xa57s (xs?1e?x) = xs?1e?x lnxa510 <x< +∞,s> 0a254a235a89
integraldisplay +∞
0
xs?1e?x dx =
integraldisplay 1
0
xs?1e?x dx+
integraldisplay +∞
1
xs?1e?x dx
a233a117a63a191a27s> 0a167a111a140a180 <s0 lessorequalslantslessorequalslantS0
xs?1e?x lessorequalslantxs0?1e?x(0 lessorequalslantxlessorequalslant 1)
a207a101s0 < 1a1670a143a219a58a167a100 lim
x→+0
x1?s0xs0?1e?x = 1a57a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay 1
0
xs0?1e?xxdxa194a241a182
a101s0 greaterorequalslant 1a167a75
integraldisplay 1
0
xs0?1e?x dxa143a126a194a200a169a167a25a194a241
a111a131
integraldisplay 1
0
xs0?1e?x dxa194a241a167a108a13a100a159a188a7a79a123a167a26
integraldisplay 1
0
xs?1e?x dxa39a117sa51sgreaterorequalslants0a254a152a151a194a241
a113xs?1e?x lessorequalslantxS0?1e?x(1 lessorequalslantx< +∞)
a207 lim
x→+∞
x2xS0?1e?x = 0a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
xS0?1e?x dxa194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay +∞
1
xs?1e?x dxa39a117sa51slessorequalslantS0a254a152a151a194a241
a108a13
integraldisplay +∞
0
xs?1e?x dxa51[s0,S0 ]a254a152a151a194a241a167a25a194a241.
integraldisplay +∞
0
xs?1e?x lnxdx =
integraldisplay 1
0
xs?1e?x lnxdx+
integraldisplay +∞
1
xs?1e?x lnxdx
a233a254a161a270 <s0 lessorequalslantslessorequalslantS0a167|xs?1e?x lnx|lessorequalslantxs0?1|lnx| (0 <xlessorequalslant 1)
a207 lim
x→+0
x1?s02 xs0?1 lnx = lim
x→+0
lnx
x?s02
= 0a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay 1
0
xs0?1e?x|lnx|dx =?
integraldisplay 1
0
xs0?1e?x lnxdxa194a241
a117a180a100a159a188a7a79a123a167a26
integraldisplay 1
0
xs?1e?x lnxdxa51sgreaterorequalslants0a254a152a151a194a241
a113xs?1e?x lnx = xse?x lnxx <xS0e?x (1 lessorequalslantx< +∞)
a207 lim
x→+∞
x2xS0e?x = lim
x→+∞
xS0+2
ex = 0a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay +∞
1
xS0e?x dxa194a241a167a117a180a100a159a188
a7a79a123a167a26
integraldisplay +∞
1
xs?1e?x lnxdxa51slessorequalslantS0a254a152a151a194a241
a108a13
integraldisplay +∞
0
xs?1e?x lnxdxa51[s0,S0 ]a254a152a151a194a241
a75a100a200a169a210a101a166a19a189a110a167a26Γ(s)a51[s0,S0 ]a254a140a19a167a8a44a51sa140a19a167a133Γprime(s) =
integraldisplay +∞
0
xs?1e?x lnxdx
a50a100s> 0a27a63a191a53a167a26Γ(s)a51s> 0a140a19a133Γprime(s) =
integraldisplay +∞
0
xs?1e?x lnxdx.
9,(1) a108
integraldisplay +∞
0
e?y2 dy =
√pi
2 a237a209L(c) =
integraldisplay +∞
0
e?y2?c
2
y2 dy =
√pi
2 e
2ca182
(2) a124a94a200a169a210a101a166a19a27a123a75a218a209 dLdc =?2La53a166a26a211a152a40a74a167a191a237a209
integraldisplay +∞
0
e?ay2? by2 dy (a > 0,b > 0)a131
a138.
a121a178a181
(1) L(c) =
integraldisplay +∞
0
e?y2?c
2
y2 dy =
integraldisplay +∞
0
e?
parenleftBig
y?cy
parenrightBig2
2c dy = e?2c
integraldisplay +∞
0
e?
parenleftBig
y?cy
parenrightBig2
dy =
e?2c
integraldisplay +∞
0
e?
parenleftBig
y?cy
parenrightBig2
d
parenleftBigg
y? cy
parenrightBigg
+e?2c
integraldisplay +∞
0
e?
parenleftBig
y?cy
parenrightBig2
dcy
a51a99a152a200a169a165a45u = y? cy a167a51a0a152a200a169a165a45v = cy
a75L(c) = e?2c
integraldisplay +∞
∞
e?u2 du?e?2c
integraldisplay +∞
0
e?(v?cv)2 dv = √pie?2c?L(c)
248
a117a180L(c) =
integraldisplay +∞
0
e?y2?c
2
y2 dy =
√pi
2 e
2c.
(2) L(c) =
integraldisplay +∞
0
e?y2?c
2
y2 dy,
dL
dc = 2
integraldisplay +∞
0
e?y2?c
2
y2
parenleftBigg
cy2
parenrightBigg
dy
a45v = cy a167a75 dLdc =?2
integraldisplay +∞
0
e?v2?c
2
v2 dv =?2L(c)
a117a180lnL =?2c+ lnc0a61ln Lc
0
=?2ca189a61L = c0e?2c
a113L(0) =
integraldisplay +∞
0
e?y2 dy =
√pi
2 a167a75c0 =
√pi
2 a167a117a180L(c) =
√pi
2 e
2c
a75a45u = √aya167a107
integraldisplay +∞
0
e?ay2? by2 dy = 1√a
integraldisplay +∞
0
e?u2?(
√ab)2
u2 du =
1√
a ·
√pi
2 e
2√ab = 1
2
radicalBigg
pi
a e
2√ab (a> 0,b> 0).
249
a49a111a220a169 a245a67a254a200a169a198
a49a155a202a217 a200a169(a19a173a33a110a173a200a169a167a49a152a97
a173a130a33a173a161a200a169)a27a189a194a218a53a159
§2,a200a169a27a53a159
1,a121a178a165a138a189a110a181a101f(M),g(M)a51?a254a235a89a167g(M)a51?a216a67a210a167a75
integraldisplay
f(M)g(M)d? = f(P)
integraldisplay
g(M)d?
a217a165P ∈?.
a121a178a181a23?a180a107a46a52a171a141a133a107a221a254
a207f(M),g(M)a51?a254a235a89a167g(M)a51?a216a67a210
a75f(M),g(M)a51?a254a140a200a167a133a140a23g(M) greaterorequalslant 0a167M = max
M∈?
{f(M)},m = min
M∈?
{f(M)}
a100a53a1594a167a26m
integraldisplay
g(M)d? lessorequalslant
integraldisplay
f(M)g(M)d? lessorequalslantM
integraldisplay
g(M)d?
a101
integraldisplay
g(M)d? = 0a167a100a117g(M) greaterorequalslant 0a133a235a89a167a75a55a107g(M) ≡ 0,M ∈?a167a108a13
integraldisplay
f(M)g(M)d? = 0a61a135a121a216
a31a170a164a225a182
a101
integraldisplay
g(M)d? > 0a167a75mlessorequalslant
integraldisplay
f(M)g(M)d?
integraldisplay
g(M)d?
lessorequalslantM
a100a235a89a188a234a27a48a138a189a110a167a26a55a127a51P ∈?a167a166
integraldisplay
f(M)g(M)d?
integraldisplay
g(M)d?
= f(P)
a61
integraldisplay
f(M)g(M)d? = f(P)
integraldisplay
g(M)d?
a211a110a167a8g(M) lessorequalslant 0a158a167a189a107
integraldisplay
f(M)g(M)d? = f(P)
integraldisplay
g(M)d?.
2,a121a178a181a101f(M)a51?a254a235a89a167f(M) greaterorequalslant 0a167a2f(M) negationslash≡ 0a167a75
integraldisplay
f(M)d? > 0
.
a121a178a181a207f(M) greaterorequalslant 0a167f(M) negationslash≡ 0a167a75a150a8a127a51a152a58M0 ∈?a167a166a26f(M0) > 0
a113f(M)a51?a254a235a89a167a8a44a51M0a235a89a167a75a55a127a51δ> 0a167a8M ∈O(M0,δ)a158a167a107f(M) > 0
a117a180
integraldisplay
f(M)d? =
integraldisplay
\O(M0,δ)
f(M)d? +
integraldisplay
O(M0,δ)
f(M)d? greaterorequalslant
integraldisplay
O(M0,δ)
f(M)d? > 0
3,a121a178a181a101f(M)a51?a254a235a89a167a51?a27a63a219a220a169a171a141?primea254
integraldisplay
prime
f(M)d? = 0
a75f(M) ≡ 0
a100a100a121a178a181a101f(M),g(M)a51?a254a235a89a167a51?a27a63a219a220a169a171a141?primea254a164a225a181
integraldisplay
prime
f(M)d? =
integraldisplay
prime
g(M)d?
a75a51?a254a164a225a181f(M) ≡g(M).
a121a178a181a94a135a121a123.a101a127a51a58Mprime ∈?a167a166f(Mprime) negationslash= 0a167a216a148a23f(Mprime) > 0
a100a117f(M)a51?a254a235a89a167a75a127a51Mprimea27a25a141?prime = O(Mprime,δ)(δ> 0)a167a166a26f(M) > f(M
prime)
2 > 0,?M ∈?
prime
a117a180a107
integraldisplay
prime
f(M)d? greaterorequalslant f(M
prime)
2 ||?
prime||> 0a134a75a23
integraldisplay
prime
f(M)d? = 0a103a241
250
a75a98a23a216a164a225a167a61a107f(M) ≡ 0
a45F(M) = f(M)?g(M)a167a75a51?a27a63a219a220a169a171a141?primea254
integraldisplay
prime
F(M)d? =
integraldisplay
prime
f(M)d
integraldisplay
prime
g(M)d? = 0
a108a13a100a254a161a164a121a40a216a167a107F(M) ≡ 0a167a61f(M)?g(M) ≡ 0a189a61f(M) ≡g(M).
4,a101|f(M)|a51?a254a140a200a167a64a34f(M)a51?a254a180a196a140a200a186a127a9a188a234f(x,y) =?1a167a8xa218ya165a150a8a107a152a135a180a195a110a234
a158a182f(x,y) = 1a167a8xa218ya209a180a107a110a234a158a167a51[0,1;0,1]a254a27a200a169.
a41a181a153a55.
a175a162a254a167f(x,y)a51[0,1;0,1]a254a27a254a218a33a101a218a169a79a143Sprime =
summationdisplay
ik
Mikik = 1,S =
summationdisplay
ik
mikik =?1
a217a165Mik = max
[0,1;0,1]
f(x,y) = 1,mik = min
[0,1;0,1]
f(x,y) =?1
a108a13f(x,y)a51[0,1;0,1]a254a216a140a200
a44a13|f(x,y)|≡ 1a51[0,1;0,1]a254a140a200.
251
a49a19a155a217 a173a200a169a27a79a142a57a65a94
§1,a19a173a200a169a27a79a142
1,a122a19a173a200a169
I =
integraldisplayintegraldisplay
D
f(x,y)dσ
a143a19a103a200a169(a169a79a15a209a233a252a135a67a254a107a0a103a83a216a211a27a19a103a200a169)a167a217a165a200a169a141Da169a79a143a181
(1) Da180a100xa182a134x2 +y2 = r2(y> 0)a164a140a164a27a171a141a182
(2) Da180a100y = 0,y = x2(x> 0)a57x+y = 2a164a140a164a27a171a141a182
(3) Da180a100y = x,x = 2a57y = 1x (x> 0)a164a140a164a27a171a141a182
(4) Da180a11a1301 lessorequalslantx2 +y2 lessorequalslant 4
a41a181
(1) I =
integraldisplay r
r
dx
integraldisplay √r2?x2
0
f(x,y)dy =
integraldisplay r
0
dy
integraldisplay √r2?y2
√
r2?y2
f(x,y)dx
(2) I =
integraldisplay 1
0
dy
integraldisplay 2?y
3√y
f(x,y)dx =
integraldisplay 1
0
dx
integraldisplay x3
0
f(x,y)dy+
integraldisplay 2
1
dx
integraldisplay 2?x
0
f(x,y)dy
(3) I =
integraldisplay 1
1
2
dy
integraldisplay 2
1
y
f(x,y)dx+
integraldisplay 2
1
dy
integraldisplay 2
y
f(x,y)dx =
integraldisplay 2
1
dx
integraldisplay x
1
x
f(x,y)dy
(4) I =
integraldisplay?1
2
dx
integraldisplay √4?x2
√
4?x2
f(x,y)dy+
integraldisplay 1
1
dx
bracketleftBiggintegraldisplay
√
1?x2
√
4?x2
f(x,y)dy+
integraldisplay √4?x2
√
1?x2
f(x,y)dy
bracketrightBigg
+
integraldisplay 2
1
dx
integraldisplay √4?x2
√
4?x2
f(x,y)dy =
integraldisplay?1
2
dy
integraldisplay √4?y2
√
4?y2
f(x,y)dx+
integraldisplay 1
1
dy
bracketleftBiggintegraldisplay
√
1?y2
√
4?y2
f(x,y)dx+
integraldisplay √4?y2
√
1?y2
f(x,y)dx
bracketrightBigg
+
integraldisplay 2
1
dy
integraldisplay √4?y2
√
4?y2
f(x,y)dx
2,a23f(x,y)a51a171a141Da254a235a89a167a217a165Da180a100y = x,y = aa57x = b(b>a)a164a140a164a27a167a121a178
integraldisplay b
a
dx
integraldisplay x
a
f(x,y)dy =
integraldisplay b
a
dy
integraldisplay b
y
f(x,y)dx
a121a178a181a100f(x,y)a51Da254a235a89a167a25a140a200
a45f(x,y) =
braceleftbigg f(x,y),(x,y) ∈D
0,(x,y) ∈ [a,b;a,b]\D a216y = xa9a235a89a167a25a55a140a200
a75
integraldisplayintegraldisplay
D
f(x,y)dxdy =
integraldisplay
[a,b;a,b]
f(x,y)dxdy =
integraldisplay b
a
dx
integraldisplay b
a
f(x,y)dy =
integraldisplay b
a
dx
integraldisplay x
a
f(x,y)dy
integraldisplayintegraldisplay
D
f(x,y)dxdy =
integraldisplay
[a,b;a,b]
f(x,y)dxdy =
integraldisplay b
a
dy
integraldisplay b
a
f(x,y)dx =
integraldisplay b
a
dy
integraldisplay b
y
f(x,y)dx.
3,a51a101a15a200a169a165a85a67a197a103a200a169a27a103a83a181
(1)
integraldisplay 2a
0
dx
integraldisplay √2ax
√
2ax?x2
f(x,y)dya182
(2)
integraldisplay 2pi
0
dx
integraldisplay sinx
0
f(x,y)dya182
(3)
integraldisplay 1
0
dy
integraldisplay 2y
0
f(x,y)dx+
integraldisplay 3
1
dy
integraldisplay 3?y
0
f(x,y)dxa182
(4)
integraldisplay 1
0
dx
integraldisplay x2
0
f(x,y)dy+
integraldisplay 2
1
dx
integraldisplay √1?(x?1)2
0
f(x,y)dy.
a41a181
(1)
integraldisplay 2a
0
dx
integraldisplay √2ax
√
2ax?x2
f(x,y)dy =
integraldisplay a
0
dy
bracketleftBiggintegraldisplay
a?
√
a2?y2
y2
2a
f(x,y)dx+
integraldisplay 2a
a+
√
a2?y2
f(x,y)dx
bracketrightBigg
+
integraldisplay 2a
a
dy
integraldisplay 2a
y2
2a
f(x,y)dx.
252
(2)
integraldisplay 2pi
0
dx
integraldisplay sinx
0
f(x,y)dy =
integraldisplay pi
0
dx
integraldisplay sinx
0
f(x,y)dy+
integraldisplay 2pi
pi
dx
integraldisplay sinx
0
f(x,y)dy =
integraldisplay pi
0
dx
integraldisplay sinx
0
f(x,y)dy?
integraldisplay 2pi
pi
integraldisplay 0
sinx
f(x,y)dy =
integraldisplay 1
0
dy
integraldisplay pi?arcsiny
arcsiny
f(x,y)dx?
integraldisplay 0
1
dy
integraldisplay 2pi+arcsiny
pi?arcsiny
f(x,y)dx.
(3)
integraldisplay 1
0
dy
integraldisplay 2y
0
f(x,y)dx+
integraldisplay 3
1
dy
integraldisplay 3?y
0
f(x,y)dx =
integraldisplay 2
0
dx
integraldisplay 3?x
x
2
f(x,y)dy.
(4)
integraldisplay 1
0
dx
integraldisplay x2
0
f(x,y)dy+
integraldisplay 2
1
dx
integraldisplay √1?(x?1)2
0
f(x,y)dy =
integraldisplay 1
0
dy
integraldisplay 1+√1?y2
√y f(x,y)dx.
4,a79a142a101a15a19a173a200a169a181
(1)
integraldisplayintegraldisplay
[a,b;c,d]
xyex2+y2 dxdya182
(2)
integraldisplayintegraldisplay
xy2 dxdy,?a180a100a14a212a130y2 = 2pxa218a134a130x = ρ2 (ρ> 0)a164a46a27a171a141a182
(3)
integraldisplayintegraldisplay
dxdy√
2a?x (a > 0),?a180a100a11a37a51a58(a,a)a140a187a143aa133a134a139a73a182a131a131a27a11a177a27a22a225a152a227a108a218a139a73a182a164a140
a164a27a171a141a182
(4)
integraldisplayintegraldisplay
(x2 +y2)dxdy,?a180a177y = x,y = x+a,y = aa218y = 3a (a> 0)a143a62a27a171a141.
a41a181
(1)
integraldisplayintegraldisplay
[a,b;c,d]
xyex2+y2 dxdy =
integraldisplay b
a
xex2 dx
integraldisplay d
c
yey2 dy = 14 (eb2?ea2)(ed2?ec2).
(2)
integraldisplayintegraldisplay
xy2 dxdy =
integraldisplay ρ
2
0
xdx
integraldisplay √2px
√2px
y2 dy = pρ
3
21
√pρ.
(3)
integraldisplayintegraldisplay
dxdy√
2a?x =
integraldisplay a
0
dx√
2a?x
integraldisplay a?√2ax?x2
0
dy =
parenleftBigg
2√2? 83
parenrightBigg
a√a.
(4)
integraldisplayintegraldisplay
(x2 +y2)dxdy =
integraldisplay 3a
a
dy
integraldisplay y
y?a
(x2 +y2)dx = 14a4.
5,a121a178
J =
integraldisplay b
a
dx
integraldisplay x
a
f(y)dy =
integraldisplay b
a
f(y)(b?y)dy =
integraldisplay b
a
f(x)(b?x)dx
a121a178a181a242
integraldisplay b
a
dx
integraldisplay x
a
f(y)dya197a145a200a169a167a26
integraldisplayintegraldisplay
f(y)dxdya167a217a165?a180x = b,x = y,y = aa164a140a164a27a171a141
a233a100a200a169a140a122a143a107a233xa0a233ya27a200a169a167a75a26integraldisplay
b
a
dx
integraldisplay x
a
f(y)dy =
integraldisplay b
a
dy
integraldisplay b
y
f(y)dx =
integraldisplay b
a
f(y)(b?y)dy =
integraldisplay b
a
f(x)(b?x)dx.
6,a23a178a161a254a171a141Da51xa182a218ya182a254a27a221a75a127a221a143lx,ly,Da27a161a200a143|D|a167(α,β)a143Da83a63a152a58a167a121a178a181
(1)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplayintegraldisplay
D
(x?α)(y?β)dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslantlxly|D|a182
(2)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplayintegraldisplay
D
(x?α)(y?β)dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
l2xl2y
4,
a121a178a181
(1) a100a117(x?α)(y?β)a51Da254a235a89a167a25a100a200a169a165a138a189a110a167a127a51(ξ,η) ∈Da167a166a26vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplayintegraldisplay
D
(x?α)(y?β)dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle(ξ?α)(η?β)
integraldisplayintegraldisplay
D
dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslantlxly|D|
253
(2) a23lx = b?a,ly = d?ca167a75vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplayintegraldisplay
D
(x?α)(y?β)dxdy
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplayintegraldisplay
D
|x?α||y?β|dxdylessorequalslant
integraldisplayintegraldisplay
[a,b;c,d]
|x?α||y?β|dxdy =
integraldisplay b
a
|x?α|dx
integraldisplay d
c
|y?β|dy =
parenleftbiggintegraldisplay α
a
(α?x)dx+
integraldisplay b
α
(x?α)dx
parenrightbiggparenleftbiggintegraldisplay β
c
(β?y)dy+
integraldisplay d
β
(y?β)dy
parenrightbigg
=
(b?α)2 + (α?a)2
2 ·
(d?β)2 + (β?c)2
2 lessorequalslant
(b?a)2
2 ·
(d?c)2
2 =
l2xl2y
4
7,a94a52a139a73a79a142
integraldisplayintegraldisplay
f(x,y)dxdya158a167a200a169a129a88a219a27a152(a21a209a101a15a171a141a254a27a252a171a197a103a200a169)a186
(1)?,a140a11x2 +y2 lessorequalslanta2,ygreaterorequalslant 0a182
(2)?,a140a130a2 lessorequalslantx2 +y2 lessorequalslantb2,xgreaterorequalslant 0a182
(3)?,a11x2 +y2 lessorequalslantay (a> 0)a182
(4)?,a20a144a47a1810 lessorequalslantxlessorequalslanta,0 lessorequalslantylessorequalslanta.
a41a181
(1)
integraldisplayintegraldisplay
f(x,y)dxdy =
integraldisplay pi
0
dθ
integraldisplay |a|
0
f(rcosθ,rsinθ)rdr =
integraldisplay |a|
0
rdr
integraldisplay pi
0
f(rcosθ,rsinθ)dθ.
(2)
integraldisplayintegraldisplay
f(x,y)dxdy =
integraldisplay pi
2
pi2
dθ
integraldisplay |b|
|a|
f(rcosθ,rsinθ)rdr =
integraldisplay |b|
|a|
rdr
integraldisplay pi
2
pi2
f(rcosθ,rsinθ)dθ.
(3)
integraldisplayintegraldisplay
f(x,y)dxdy =
integraldisplay pi
0
dθ
integraldisplay asinθ
0
f(rcosθ,rsinθ)rdr =
integraldisplay a
0
rdr
integraldisplay pi?arcsin r
a
arcsin ra
f(rcosθ,rsinθ)dθ.
(4)
integraldisplayintegraldisplay
f(x,y)dxdy =
integraldisplay pi
4
0
dθ
integraldisplay a
cosθ
0
f(rcosθ,rsinθ)rdr+
integraldisplay pi
2
pi
4
dθ
integraldisplay a
sinθ
0
f(rcosθ,rsinθ)rdr
=
integraldisplay a
0
rdr
integraldisplay pi
2
0
f(rcosθ,rsinθ)dθ+
integraldisplay √2 a
a
rdr
integraldisplay arcsin a
r
arccos ar
f(rcosθ,rsinθ)dθ.
8,a51a101a15a200a169a165a218a63a35a67a254u,va167a67a134a101a15a200a169.
(1)
integraldisplay b
a
dx
integraldisplay βx
αx
f(x,y)dy (0 <a<b,0 <α<β)a167a101
u = x,
v = yx ;
(2)
integraldisplay 2
0
dx
integraldisplay 2?x
1?x
f(x,y)dya167a101u = x+y,v = x?ya182
(3)
integraldisplayintegraldisplay
f(x,y)dxdya167a217a165?a180a100a173a130√x+√y = √aa134a139a73a182a164a46a27a171a141.a101
braceleftbigg x = ucos4v
y = usin4v
a41a181
(1) a207
braceleftbigg x = u
y = uv a167a75|J| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
D(x,y)
D(u,v)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle= u> 0
a117a180
integraldisplay b
a
dx
integraldisplay βx
αx
f(x,y)dy =
integraldisplay b
a
udu
integraldisplay β
α
f(u,uv)dv
(2) a207
x = u+v2
y = u?v2
a167a75|J| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
D(x,y)
D(u,v)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1
2
a117a180
integraldisplay 2
0
dx
integraldisplay 2?x
1?x
f(x,y)dy = 12
integraldisplay 2
1
du
integraldisplay 4?u
u
f
parenleftBigg
u+v
2,
u?v
2
parenrightBigg
dv
(3) a207
braceleftbigg x = ucos4v
y = usin4v a167a75|J| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
D(x,y)
D(u,v)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
usin3 2v
2
a117a180
integraldisplayintegraldisplay
f(x,y)dxdy = 12
integraldisplay a
0
udu
integraldisplay pi
2
0
sin3 2vf(ucos4v,usin4v)dv = 12
integraldisplay pi
2
0
sin3 2vdv
integraldisplay a
0
uf(ucos4v,usin4v)du.
254
9,a65a94a52a139a73a79a142a101a15a19a173a200a169a181
(1)
integraldisplayintegraldisplay
x2+y2lessorequalslantR2
e?(x2+y2) dxdya182
(2)
integraldisplayintegraldisplay
pi2lessorequalslantx2+y2lessorequalslant4pi2
sin
radicalbig
x2 +y2 dxdya182
(3)
integraldisplayintegraldisplay
(x+y)dxdy,(?a180a11x2 +y2 lessorequalslantx+ya27a83a220).
a41a181
(1)
integraldisplayintegraldisplay
x2+y2lessorequalslantR2
e?(x2+y2) dxdy =
integraldisplay 2pi
0
dθ
integraldisplay R
0
re?r2 dr = pi(1?e?R2).
(2)
integraldisplayintegraldisplay
pi2lessorequalslantx2+y2lessorequalslant4pi2
sin
radicalbig
x2 +y2 dxdy =
integraldisplay 2pi
0
dθ
integraldisplay 2pi
pi
rsinrdr =?6pi2
(3) a138a67a134x = 12 +rcosθ,y = 12 +rsinθa167a75|J| = r
a117a180
integraldisplayintegraldisplay
(x+y)dxdy =
integraldisplay 2pi
0
dθ
integraldisplay 1√
2
0
[r+r2(sinθ+ cosθ)]dr = pi2,
10,a166a100a73a161z = hRradicalbigx2 +y2a33a178a161z = 0a57a11a206a161x2 +y2 = R2a164a140a27a225a78a78a200.
a41a181a73a161z = hR radicalbigx2 +y2a33a178a161z = 0a57a11a206a161x2 + y2 = R2a164a140a27a225a78a51XOYa178a161a254a27a19a75a141a180a11
a141? = {(x,y)vextendsinglevextendsinglex2 +y2 lessorequalslantR2}a167a51a49a152a150a129a220a169a80a143?1
a75a124a94a233a161a53a167a26a164a166a225a78a78a200a143
V =
integraldisplayintegraldisplay
zdxdy = 4
integraldisplayintegraldisplay
1
zdxdy = 4hR
integraldisplayintegraldisplay
1
radicalbig
x2 +y2 dxdy = 4hR
integraldisplay pi
2
0
dθ
integraldisplay R
0
r2 dr = 23 piR2h.
11,a166a165a161x2 +y2 +z2 = a2a134a11a206a161x2 +y2 = ax (a> 0)a250a1a220a169a27a78a200.
a41a181a100a233a161a53a167a26V = 2
integraldisplayintegraldisplay
radicalbig
a2?x2?y2 dxdy = 2
integraldisplay pi
2
pi2
dθ
integraldisplay acosθ
0
r
radicalbig
a2?r2 dr = 23 a3
parenleftBigg
pi? 43
parenrightBigg
.
12,a166a100a14a212a130y2 = mx,y2 = nx (0 <m<n)a218a134a130y = αx,y = βx (0 <α<β)a164a140a164a171a141a27a161a200.
a41a181a138a67a134a181u = y
2
x,v =
y
xa167a75|J| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
D(x,y)
D(u,v)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle=
1vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingleD(u,v)D(x,y)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
= 1y2
x3
= uv4
a117a180a164a166a161a200a143D =
integraldisplayintegraldisplay
dxdy =
integraldisplay β
α
dv
v4
integraldisplay n
m
udu = 16 (n2?m2)
parenleftBigg
1
α3?
1
β3
parenrightBigg
.
13,a166a173a130
parenleftBigg
x2
a2 +
y2
b2
parenrightBigg2
= xyc2 a164a140a27a161a200.
a41a181a100a173a130a144a511a333a150a129a133a39a117a6a58a233a161a167a25a144a73a79a142a227a47a51a49a152a150a129a165a27a161a200a167a502a21a61a140
a45x = arcosθ,y = brsinθa167a75|J| = |ab|r,r =
radicalbig|ab|
|c|
√sinθcosθ
a117a180D =
integraldisplayintegraldisplay
D
dxdy = 2
integraldisplay pi
2
0
dθ
integraldisplay √|ab|
|c|
√sinθcosθ
0
|ab|rdr = a
2b2
2c2,
14,a166a152a212a78a27a78a200a167a100a212a78a27a46a161a143a181a178a161z = 0a167a14a212a1612z = x
2
a +
y2
b a167a177a57a177a165a161x
2 +y2 +(z?c)2 = c2a134
a249a135a14a212a161a27a2a130a143a79a130a27a20a206a161(a,b,c> 0).
a41a181a242z = x
2
2a +
y
2b2a147a92a165a144a167a167a26x
2 +y2 +
parenleftBigg
x2
2a +
y2
2b?c
parenrightBigg2
= c2
255
a45x = √a rcosθ,y = √b rsinθa167a75r = 2radicalbigc?(acos2θ+bsin2θ),|J| = √ab r
a117a180V =
integraldisplayintegraldisplay
D
1
2
parenleftBigg
x2
a +
y2
b
parenrightBigg
dxdy = 4
integraldisplay pi
2
0
dθ
integraldisplay 2√c?(acos2 θ+bsin2 θ)
0
√ab
2 r
3 dr = 4√abpi
parenleftBigg
3
8 a
2 + 3
8 b
2 + 1
4 ab?ac?bc+c
2
parenrightBigg
.
15,a166a62a127a143aa27a20a144a47a0a134a27a159a254a167a23a0a134a254a122a152a58a27a151a221a134a84a58a229a20a144a47a44a152a186a58a27a229a108a164a20a39a167a133a51a20a144a47
a27a165a58a63a151a221a143ρ0.
a41a181a23a44a152a186a58a143a6a58(0,0)a167a75ρ = kradicalbigx2 +y2a133a8x = y = a2 a158a167ρ = ρ0a167a117a180k =
√2 ρ
0
a
a75a151a221a188a234a143ρ(x,y) =
√2
a ρ0
radicalbigx2 +y2
a117a180a124a94a497a75(4)a167a26
m =
integraldisplayintegraldisplay
[0,a;0,a]
√2 ρ
0
a
radicalbig
x2 +y2 ddy =
integraldisplay pi
4
0
dθ
integraldisplay a
cosθ
0
√2 ρ
0
a r
2 dr+
integraldisplay pi
2
pi
4
dθ
integraldisplay a
sinθ
0
√2 ρ
0
a r
2 dr
= ρ0a
2
3 [2 +
√2 ln(1 +√2)].
256
§2,a110a173a200a169a27a79a142
1,a79a142a101a15a110a173a200a169a181
(1)
integraldisplayintegraldisplayintegraldisplay
V
xy2z3 dxdydz,Va181a100a173a161z = xy,y = x,z = 0,x = 1a164a140a164a182
(2)
integraldisplayintegraldisplayintegraldisplay
V
xyzdxdydz,Va181a100a173a161x2 +y2 +z2 = 1,xgreaterorequalslant 0,ygreaterorequalslant 0,z greaterorequalslant 0a140a164.
a41a182
(1)
integraldisplayintegraldisplayintegraldisplay
V
xy2z3 dxdydz =
integraldisplay 1
0
xdx
integraldisplay x
0
y2 dy
integraldisplay xy
0
z3 dz = 1364,
(2)
integraldisplayintegraldisplayintegraldisplay
V
xyzdxdydz =
integraldisplay 1
0
xdx
integraldisplay √1?x2
0
ydy
integraldisplay √1?x2?y2
0
zdz = 148,
2,a141a171a101a15a110a173a200a169a27a171a141Va27a47a71a191a85a67a200a169a103a83a181
(1)
integraldisplay 1
0
dx
integraldisplay 1?x
0
dy
integraldisplay x+y
0
f(x,y,z)dza182
(2)
integraldisplay 1
0
dx
integraldisplay x
0
dy
integraldisplay xy
0
f(x,y,z)dza182
(3)
integraldisplay 2
1
dx
integraldisplay 1
0
dy
integraldisplay 0
1?x?y
f(x,y,z)dza182
(4)
integraldisplay 1
1
dx
integraldisplay √1?x2
√
1?x2
dy
integraldisplay 1
√
x2+y2
f(x,y,z)dza182
(5)
integraldisplay 1
0
dx
integraldisplay 1
0
dy
integraldisplay x2+y2
0
f(x,y,z)dz.
a41a182
(1)
integraldisplay 1
0
dx
integraldisplay 1?x
0
dy
integraldisplay x+y
0
f(x,y,z)dz =
integraldisplay 1
0
dy
integraldisplay 1?y
0
dx
integraldisplay x+y
0
f(x,y,z)dz
=
integraldisplay 1
0
dx
integraldisplay x
0
dz
integraldisplay 1?x
0
f(x,y,z)dy+
integraldisplay 1
0
dx
integraldisplay 1
x
dz
integraldisplay 1?x
z?x
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay z
0
dx
integraldisplay 1?x
z?x
f(x,y,z)dy+
integraldisplay 1
0
dz
integraldisplay 1
z
dx
integraldisplay 1?x
0
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay z
0
dy
integraldisplay 1?y
z?y
f(x,y,z)dx+
integraldisplay 1
0
dz
integraldisplay 1
z
dy
integraldisplay 1?y
0
f(x,y,z)dx
=
integraldisplay 1
0
dy
integraldisplay y
0
dz
integraldisplay 1?y
0
f(x,y,z)dx+
integraldisplay 1
0
dy
integraldisplay 1
y
dz
integraldisplay 1?y
z?y
f(x,y,z)dx
(2)
integraldisplay 1
0
dx
integraldisplay x
0
dy
integraldisplay xy
0
f(x,y,z)dz =
integraldisplay 1
0
dy
integraldisplay 1
y
dx
integraldisplay xy
0
f(x,y,z)dz
=
integraldisplay 1
0
dx
integraldisplay x2
0
dz
integraldisplay x
z
x
f(x,y,z)dy =
integraldisplay 1
0
dz
integraldisplay 1
√z dx
integraldisplay x
z
x
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay 1
√z dy
integraldisplay 1
y
f(x,y,z)dx+
integraldisplay 1
0
dz
integraldisplay √z
z
dy
integraldisplay 1
z
y
f(x,y,z)dx
=
integraldisplay 1
0
dy
integraldisplay y2
0
dz
integraldisplay 1
y
f(x,y,z)dx+
integraldisplay 1
0
dy
integraldisplay y
y2
dz
integraldisplay 1
z
y
f(x,y,z)dx
(3)
integraldisplay 2
1
dx
integraldisplay 1
0
dy
integraldisplay 0
1?x?y
f(x,y,z)dz =
integraldisplay 1
0
dy
integraldisplay 2
1
dx
integraldisplay 0
1?x?y
f(x,y,z)dz
=
integraldisplay 1
0
dy
integraldisplay 0
y
dz
integraldisplay 2
1
f(x,y,z)dx+
integraldisplay 1
0
dy
integraldisplay?y
1?y
dz
integraldisplay 2
1?y?z
f(x,y,z)dx
=
integraldisplay?1
2
dz
integraldisplay 1
1?z
dy
integraldisplay 2
1?y?z
f(x,y,z)dx+
integraldisplay 0
1
dz
integraldisplay?z
0
dy
integraldisplay 2
1?y?z
f(x,y,z)dx+
integraldisplay 0
1
dz
integraldisplay 1
z
dy
integraldisplay 2
1
f(x,y,z)dx
257
=
integraldisplay?1
2
dz
integraldisplay 2
z
dx
integraldisplay 1
1?x?z
f(x,y,z)dy+
integraldisplay 0
1
dz
integraldisplay 1?z
1
dx
integraldisplay 1
1?x?z
f(x,y,z)dy+
integraldisplay 0
1
dz
integraldisplay 2
1?z
dx
integraldisplay 1
0
f(x,y,z)dy
=
integraldisplay 2
1
dx
integraldisplay 2
1?x
dz
integraldisplay 1
0
f(x,y,z)dy+
integraldisplay 2
1
dx
integraldisplay 1?x
x
dz
integraldisplay 1
1?x?z
f(x,y,z)dy
(4)
integraldisplay 1
1
dx
integraldisplay √1?x2
√
1?x2
dy
integraldisplay 1
√
x2+y2
f(x,y,z)dz =
integraldisplay 1
1
dy
integraldisplay √1?y2
√
1?y2
dx
integraldisplay 1
√
x2+y2
f(x,y,z)dz
=
integraldisplay 1
1
dx
integraldisplay 1
|x|
dz
integraldisplay √z2?x2
√
z2?x2
f(x,y,z)dy =
integraldisplay 1
0
dz
integraldisplay z
z
dx
integraldisplay √z2?x2
√
z2?x2
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay z
z
dy
integraldisplay √z2?y2
√
z2?y2
f(x,y,z)dx =
integraldisplay 1
1
dy
integraldisplay 1
|y|
dz
integraldisplay √z2?y2
√
z2?y2
f(x,y,z)dx
(5)
integraldisplay 1
0
dx
integraldisplay 1
0
dy
integraldisplay x2+y2
0
f(x,y,z)dz =
integraldisplay 1
0
dy
integraldisplay 1
0
dx
integraldisplay x2+y2
0
f(x,y,z)dz
=
integraldisplay 1
0
dx
integraldisplay x2
0
dz
integraldisplay 1
0
f(x,y,z)dy+
integraldisplay 1
0
dx
integraldisplay x2+1
x2
dz
integraldisplay 1
√
z?x2
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay √z
0
dx
integraldisplay 1
√
z?x2
f(x,y,z)dy+
integraldisplay 1
0
dz
integraldisplay 1
√z dx
integraldisplay 1
0
f(x,y,z)dy+
integraldisplay 2
1
dz
integraldisplay 1
√z?1 dx
integraldisplay 1
√
z?x2
f(x,y,z)dy
=
integraldisplay 1
0
dz
integraldisplay √z
0
dy
integraldisplay 1
√
z?y2
f(x,y,z)dx+
integraldisplay 1
0
dz
integraldisplay 1
√z dy
integraldisplay 1
0
f(x,y,z)dx+
integraldisplay 2
1
dz
integraldisplay 1
√z?1 dy
integraldisplay 1
√
z?y2
f(x,y,z)dx
=
integraldisplay 1
0
dy
integraldisplay y2
0
dz
integraldisplay 1
0
f(x,y,z)dx+
integraldisplay 1
0
dy
integraldisplay y2+1
y2
dz
integraldisplay 1
√
z?y2
f(x,y,z)dx
3,a79a142a101a15a110a173a200a169a181
(1)
integraldisplayintegraldisplayintegraldisplay
V
zdxdydza167a217a165a200a169a171a141Va180a100a165a161x2 +y2 +z2 = 4a134a14a212a161z = 13 (x2 +y2)a164a140a164a27a225a78a182
(2)
integraldisplayintegraldisplayintegraldisplay
V
(x2 +y2 +z2)dVa167a217a165Va180x2 +y2 +z2 lessorequalslant 1a182
(3)
integraldisplayintegraldisplayintegraldisplay
V
z2 dxdydz,Va100a252a135a165x2 +y2 +z2 lessorequalslantR2,x2 +y2 +z2 lessorequalslant 2Rza27a250a1a220a169a164a124a164a182
(4)
integraldisplayintegraldisplayintegraldisplay
V
radicalBigg
1? x
2
a2?
y2
b2?
z2
c2 dxdydz,Va143a253a165
x2
a2 +
y2
b2 +
z2
c2 lessorequalslant 1.
a41a182
(1) a124a94a206a161a139a73a167a26
integraldisplayintegraldisplayintegraldisplay
V
zdxdydz =
integraldisplay 2pi
0
dθ
integraldisplay √3
0
dr
integraldisplay √4?r2
r2
3
rzdz = 134 pi
(2) a124a94a165a161a139a73a167a26
integraldisplayintegraldisplayintegraldisplay
V
(x2 +y2 +z2)dV =
integraldisplay 2pi
0
dθ
integraldisplay pi
0
sin?d?
integraldisplay 1
0
ρ4 dρ = 45 pi
(3) a124a94a165a161a139a73a167a26integraldisplayintegraldisplayintegraldisplay
V
z2 dxdydz =
integraldisplay 2pi
0
dθ
integraldisplay pi
3
0
cos2?sin?d?
integraldisplay R
0
ρ4 dρ+
integraldisplay 2pi
0
dθ
integraldisplay pi
2
pi
3
cos2?sin?d?
integraldisplay 2Rcos?
0
ρ4 dρ
= 59480 piR5
(4) a100a50a194a165a161a139a73a167a26
integraldisplayintegraldisplayintegraldisplay
V
radicalBigg
1? x
2
a2?
y2
b2?
z2
c2 dxdydz = abc
integraldisplay 2pi
0
dθ
integraldisplay pi
0
sin?d?
integraldisplay 1
0
ρ2
radicalbig
1?ρ2 dρ = pi
2
4 abc.
4,a124a94a165a161a139a73a189a206a161a139a73a79a142a101a15a173a161a164a46a78a200a181
(1) x2 +y2 +z2 = 4R2a27a83a220a26x2 +y2 = 2Rxa164a121a209a27a220a169a182
(2) (x2 +y2 +z2)3 = 3xyz.
258
a41a182
(1) a124a94a206a161a139a73x = rcosθ,y = rsinθ,z = za133|J| = ra167a51a100a67a134a101a167a173a161a144a167a67a143a181
r2 +z2 = 4R2,r = 2rcosθ
a75V =
integraldisplayintegraldisplayintegraldisplay
V
dxdydz = 2
integraldisplay pi
2
pi2
dθ
integraldisplay 2Rcosθ
0
rdr
integraldisplay √4R2?r2
0
dz = 163 R3
parenleftBigg
pi? 43
parenrightBigg
(2) a100a75a127a225a78a51a49a152a33a49a110a33a49a56a57a49a108a37a129a83a167a233a117a249a10a37a129a169a79a107
x,y,z greaterorequalslant 0;x,y lessorequalslant 0,z greaterorequalslant 0;x,z lessorequalslant 0,y greaterorequalslant 0;xgreaterorequalslant 0,y,z lessorequalslant 0 a207a6a170a134a224a57a109a224a8x,y,za165a63a252a135a211a158a67a210
a158a31a170a69a164a225a167a25a225a78a51a249a111a135a37a129a83a27a136a220a169a167a152a233a152a233a47a233a161a117a139a73a182a131a152.
a100a165a161a139a73x = ρsin?cosθ,y = ρsin?sinθ,z = ρcos?,|J| = ρ2 sin?
a173a161a144a167a67a143a181ρ6 = 3ρ3 sin2?cos?sinθcosθa61ρ3 = 3sin2?cos?sinθcosθa167
a133a51a49a152a37a129a83a167ρgreaterorequalslant 0,0 lessorequalslantθlessorequalslant pi2,0 lessorequalslant?lessorequalslant pi2
a117a180V = 4
integraldisplay pi
2
0
dθ
integraldisplay pi
2
0
sin?d?
integraldisplay 3√3sin2?cos?sinθcosθ
0
ρ2 dρ = 12,
5,a124a94a183a8a27a139a73a67a134a79a142a101a15a173a161a164a140a78a200a181
(1)
parenleftBigg
x2
a2 +
y2
b2 +
z2
c2
parenrightBigg2
= x
2
a2 +
y2
b2
(2)
parenleftBigg
x
a +
y
b
parenrightBigg2
+
parenleftBigg
z
c
parenrightBigg2
= 1,(x> 0,y> 0,z> 0,a,b,c> 0)
(3) z = x2 +y2,z = 2(x2 +y2),xy = a2,xy = 2a2,x = 2y,2x = ya167(a217a165x,y> 0)
a41a182
(1) a100a50a194a165a161a139a73:x = aρsin?cosθ,y = bρsin?sinθ,z = cρcos?a167a217a165ρgreaterorequalslant 0,0 lessorequalslantθ lessorequalslant 2pi,0 lessorequalslant?lessorequalslantpia167
a249a158|J| = abcρ2 sin?
a173a161a144a167a67a143a181ρ = sin?
a75V = abc
integraldisplay 2pi
0
dθ
integraldisplay pi
0
sin?d?
integraldisplay sin?
0
ρ2 dρ = pi
2
4 abc
(2) a138a67a134a181x = arcos2θcos?,y = brsin2θcos?,z = crsin?a167a217a1650 lessorequalslantr lessorequalslant 1,0 lessorequalslantθ lessorequalslant pi2,0 lessorequalslant?lessorequalslant pi2 a167
a249a158|J| = 2abcr2 cosθsinθcos?
a75V =
integraldisplay pi
2
0
dθ
integraldisplay pi
2
0
d?
integraldisplay 1
0
(2abcr2 cosθsinθcos?)dr = abc3
(3) a45z = u(x2 +y2),xy = v,x = wya167a75x = √wv,y =
radicalbigg
v
w,z = u
parenleftBigg
wv+ vw
parenrightBigg
a100a158|J| = v2 + v2w2 a167a1331 lessorequalslantulessorequalslant 2,a2 lessorequalslantvlessorequalslant 2a2,12 lessorequalslantwlessorequalslant 2
a117a180V =
integraldisplay 2
1
du
integraldisplay 2a2
a2
vdv
integraldisplay 2
1
2
parenleftBigg
1
2 +
1
2w2
parenrightBigg
dw = 94 a4
6,a166a228a107a252a160a78a2000 lessorequalslantxlessorequalslant 1,0 lessorequalslantylessorequalslant 1,0 lessorequalslantz lessorequalslant 1a27a212a78a27a159a254a167a101a212a78a51a58M(x,y,z)a27a151a221a143μ = x+y+z.
a41a182m =
integraldisplayintegraldisplayintegraldisplay
V
(x+y+z)dxdydz =
integraldisplay 1
0
dx
integraldisplay 1
0
dy
integraldisplay 1
0
(x+y+z)dz = 32,
259
§3,a200a169a51a212a110a254a27a65a94
1,a166a101a15a173a130a164a46a0a134a27a159a37a139a73a181
(1) ay = x2,x+y = 2a (a> 0)
(2) r = a(1 + cos?) (0 lessorequalslant?lessorequalslantpi)
a41a181
(1) a151a221ρa143a126a234a167a75xG =
integraltextintegraltext
xd?
integraltextintegraltext
d?,yG =
integraltextintegraltext
yd?
integraltextintegraltext
d?
a100
integraldisplayintegraldisplay
d? =
integraldisplay a
2a
dx
integraldisplay 2a?x
x2
a
dy = 92 a2
integraldisplayintegraldisplay
xd? =
integraldisplay a
2a
xdx
integraldisplay 2a?x
x2
a
dy =?94 a3
integraldisplayintegraldisplay
yd? =
integraldisplay a
2a
dx
integraldisplay 2a?x
x2
a
ydy = 365 a3
a75xG =?a2,yG = 85 a
(2) a151a221ρa143a126a234a167a75xG =
integraltextintegraltext
rcos?d?
integraltextintegraltext
d?,yG =
integraltextintegraltext
rsin?d?
integraltextintegraltext
d?
a100
integraldisplayintegraldisplay
d? =
integraldisplay pi
0
d?
integraldisplay a(1+cos?)
0
rdr = 34 a2pi
integraldisplayintegraldisplay
rcos?d? =
integraldisplay pi
0
d?
integraldisplay a(1+cos?)
0
r2 cos?dr = 58 a3pi
integraldisplayintegraldisplay
rsin?d? =
integraldisplay pi
0
d?
integraldisplay a(1+cos?)
0
r2 sin?dr = 43 a3
a75xG = 56 a,yG = 16a9pi,
2,a166a100a101a15a173a161a164a46a27a212a78a27a159a37a181
(1) x
2
a2 +
y2
b2 +
z2
c2 = 1,xgreaterorequalslant 0,ygreaterorequalslant 0,z greaterorequalslant 0
(2) z = x2 +y2,x+y = a,x = 0,y = 0,z = 0
a41a181
(1) a151a221ρa143a126a234a167a75xG =
integraltextintegraltextintegraltext
V
xdxdydz
integraltextintegraltextintegraltext
V
dxdydz,yG =
integraltextintegraltextintegraltext
V
ydxdydz
integraltextintegraltextintegraltext
V
dxdydz,zG =
Dintegraltextintegraltextintegraltext
V
zdxdydz
integraltextintegraltextintegraltext
V
dxdydz
a45x = aρsin?cosθ,y = bρsin?sinθ,z = cρcos?a167a217a1650 lessorequalslant ρ lessorequalslant 1,0 lessorequalslant θ lessorequalslant pi2,0 lessorequalslant? lessorequalslant pi2 a167a100
a158|J| = abcρ2 sin?
a75
integraldisplayintegraldisplayintegraldisplay
V
dxdydz = abc
integraldisplay pi
2
0
dθ
integraldisplay pi
2
0
sin?d?
integraldisplay 1
0
ρ2 dρ = pi6 abc
integraldisplayintegraldisplayintegraldisplay
V
xdxdydz = a2bc
integraldisplay pi
2
0
cosθdθ
integraldisplay pi
2
0
sin2?d?
integraldisplay 1
0
ρ3 dρ = pi16 a2bc
a117a180xG = 38 aa167a100a233a161a53a167a26yG = 38 b,zG = 38 c
(2) a151a221ρa143a126a234a167a75xG =
integraltextintegraltextintegraltext
V
xdxdydz
integraltextintegraltextintegraltext
V
dxdydz,yG =
integraltextintegraltextintegraltext
V
ydxdydz
integraltextintegraltextintegraltext
V
dxdydz,zG =
integraltextintegraltextintegraltext
V
zdxdydz
integraltextintegraltextintegraltext
V
dxdydz
a100
integraldisplayintegraldisplayintegraldisplay
V
dxdydz =
integraldisplay a
0
dx
integraldisplay a?x
0
dy
integraldisplay x2+y2
0
dz = a
4
6
260
integraldisplayintegraldisplayintegraldisplay
V
xdxdydz =
integraldisplay a
0
xdx
integraldisplay a?x
0
dy
integraldisplay x2+y2
0
dz = a
5
15
integraldisplayintegraldisplayintegraldisplay
V
ydxdydz = a
5
15,
integraldisplayintegraldisplayintegraldisplay
V
zdxdydz = 7180 a6
a75xG = 25 a,yG = 25 a,zG = 730 a2.
3,a166a254a33a169a217a117a252a135a11r = 2sinθa57r = 4sinθa131a109a27a171a141a254a27a159a254a27a159a37.
a41a181a100a233a161a53a167a26x = 0
a113y = 13pi
integraldisplay pi
0
dθ
integraldisplay 4sinθ
2sinθ
r2 sinθdr = 73 a167a75a164a166a47a37a143
parenleftBigg
0,73
parenrightBigg
.
4,a51a44a152a41a23a76a167a165a167a135a51a140a11a47a27a134a62a254a86a254a152a135a62a134a134a187a31a127a27a221a47a167a166a18a135a178a161a227a47a27a159a37a225a51a11a37
a254a167a193a166a221a47a27a44a152a62a127.
a41a181a23a151a221ρa143a126a234a167a221a47a27a44a152a62a127a143la167a11a37a51a139a73a6a58(0,0)a167a18a11a160a117xa182a254a144a167a18a221a47a160a117xa182a101a144
a117a180x =
ρ
integraldisplay R
R
xdx
integraldisplay √R2?x2
l
dy
ρparenleftbig12 piR2 + 2Rlparenrightbig = 0
y =
ρ
integraldisplay R
R
dx
integraldisplay √R2?x2
l
ydy
ρparenleftbig12 piR2 + 2Rlparenrightbig ==
2
piR+ 4l
parenleftBigg
2
3 R
2?l2
parenrightBigg
a45y = 0a167a75a26l =
√6
3 R.
5,a166a254a33a169a217a51a100y = x2a134y = 1a164a140a164a27a178a161a227a47a254a27a159a254a39a117a134a130y =?1a27a61a196a46a254.
a41a181Iy=1 =
integraldisplayintegraldisplay
(y+ 1)2 d? =
integraldisplay 1
1
dx
integraldisplay 1
x2
(y+ 1)2 dy = 368105,
6,a166a100a101a15a173a161a164a46a254a33a78a233a117a164a171a182a27a61a196a46a254a181
(1) z = x2 +y2,x+y = ±1,x?y = ±1,z = 0a39a117za182a182
(2) a127a144a78a39a117a167a27a152a99.
a41a181
(1) a173a161a164a46a254a33a212a78a233a117OZa182a27a61a196a46a254a80a143IOZ
a75IOZ =
integraldisplayintegraldisplayintegraldisplay
V
(x2 +y2)dxdydz
=
integraldisplay 1
0
dx
integraldisplay 1?x
x?1
dy
integraldisplay x2+y2
0
(x2 +y2)dz +
integraldisplay 0
1
dx
integraldisplay x+1
(1+x)
dy+
integraldisplay x2+y2
0
(x2 +y2)dz = 1445
(2) a23a127a144a780 lessorequalslantz lessorequalslantc,0 lessorequalslantylessorequalslantb,0 lessorequalslantxlessorequalslanta
a39a117za182a27a61a196a46a254a143IOZ =
integraldisplay a
0
dx
integraldisplay b
0
dy
integraldisplay c
0
(x2 +y2)dz = abc3 (a2 +b2).
7,a166a254a33a0a161x2 +y2 lessorequalslantR2,z = 0a233a117za182a254a152a58(0,0,c) (c> 0)a63a252a160a159a254a27a218a229.
a41a181a218a229a51OX,OYa182a254a27a19a75a1430a167a61Fx = Fy = 0a167a23ρ = ρ0
a75Fz = k
integraldisplayintegraldisplay
ρ0 cd3 d? = kρ0
integraldisplay 2pi
0
dθ
integraldisplay R
0
cr
(r2 +c2)32
dr = 2kρ0pic
bracketleftBigg
1
c?
1√
R2 +c2
bracketrightBigg
.
8,a166a254a33a206a78x2 +y2 lessorequalslanta2,0 lessorequalslantz lessorequalslantha233a117p(0,0,c) (c>h)a58a63a27a252a160a159a254a27a218a229.
a41a181a23ρ = ρ0a167a100a233a161a53a167a26a218a229a51OX,OYa182a254a27a19a75a1430a167a61Fx = Fy = 0
a124a94a206a161a139a73a167a26a218a229a51OZa182a254a27a19a75a143a181
Fz = kρ0
integraldisplayintegraldisplay
dxdy
integraldisplay h
0
z?c
(x2 +y2 + (z?c)2)32
dz = kρ0
integraldisplay 2pi
0
dθ
integraldisplay a
0
rdr
integraldisplay h
0
z?c
[r2 + (z?c)2]32
dz
= 2pikρ0(
radicalbig
a2 +c2?
radicalbig
a2 + (c?h)2?h).
261
§4,a50a194a173a200a169
1,a79a142a101a15a50a194a173a200a169a131a138a181
(1)
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq
(2)
integraldisplayintegraldisplay
x2+y2lessorequalslant1
dxdyradicalbig
1?x2?y2
(3)
integraldisplay +∞
∞
integraldisplay +∞
∞
e?(x2+y2) dxdy.a191a100a100a121a178a86a199a200a169
1√
pi
integraldisplay +∞
∞
e?x2 dx = 1
a41a181
(1) a100a117a26a200a188a234a154a75a167a25I =
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq =
integraldisplay +∞
1
dx
xp
integraldisplay +∞
1
x
dy
yq
a8qlessorequalslant 1a158a167a100xgreaterorequalslant 1a167a1270 < 1xlessorequalslant 1a167a75a26a200a169
integraldisplay +∞
1
x
dy
yq a117a209a133a107
integraldisplay +∞
1
x
dy
yq = +∞a167
a117a180I =
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq = +∞
a8q> 1a158a167
integraldisplay +∞
1
x
dy
yq =
xq?1
q?1
a100a158a167a8p>qa158a167I =
integraldisplay +∞
1
dx
xp
integraldisplay +∞
1
x
dy
yq =
1
q?1
integraldisplay +∞
1
xq?p?1 dx = 1(q?1)(p?q)
a8plessorequalslantqa158a167(p+ 1)?qlessorequalslant 1a167a75a200a169 1q?1
integraldisplay +∞
1
xq?p?1 dx = +∞a167a108a13a26I =
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq = +∞
a110a254a140a127a167a8p>q> 1a158a167I =
integraldisplayintegraldisplay
xygreaterorequalslant1
xgreaterorequalslant1
dxdy
xpyq =
1
(q?1)(p?q) a167a217a123a156a185I = +∞.
(2)
integraldisplayintegraldisplay
x2+y2lessorequalslant1
dxdyradicalbig
1?x2?y2 = limε→1
integraldisplay 2pi
0
dθ
integraldisplay ε
0
r√
1?r2 dr = 2pi.
(3) a138a67a134x = rcosθ,y = rsinθ (0 lessorequalslantθlessorequalslant 2pi,r> 0),|J| = r
a75
integraldisplay +∞
∞
integraldisplay +∞
∞
e?(x2+y2) dxdy =
integraldisplay 2pi
0
dθ
integraldisplay +∞
0
re?r2 dr = pi.
a100
integraldisplay +∞
∞
integraldisplay +∞
∞
e?(x2+y2) dxdy =
integraldisplay +∞
∞
e?x2 dx
integraldisplay +∞
∞
e?y2 dya133
integraldisplay +∞
∞
e?x2 dx =
integraldisplay +∞
∞
e?y2 dya143a44a152
a138
a75
integraldisplay +∞
∞
e?x2 dx = √pia61 1√piintegraltext+∞?∞ e?x2 dx = 1.
2,a63a216a101a161a50a194a173a200a169a27a194a241a53a181
(1)
integraldisplay +∞
∞
integraldisplay +∞
∞
dxdy
(1 +|x|p)(1 +|y|q)
(2)
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
(x,y)
(1 +x2 +y2)p dxdy,0 <mlessorequalslant|?(x,y)|lessorequalslantM
(3)
integraldisplay a
0
integraldisplay a
0
(x,y)
|x?y|p dxdy,0 <mlessorequalslant|?(x,y)|lessorequalslantM
262
(4)
integraldisplayintegraldisplay
x2+y2lessorequalslant1
(x,y)
(x2 +xy+y2)p dxdy,0 <mlessorequalslant|?(x,y)|lessorequalslantM
a41a181
(1) a207a26a200a188a234a143a20a133a39a117OX,OYa182a233a161a167a75
integraldisplay +∞
∞
integraldisplay +∞
∞
dxdy
(1 +|x|p)(1 +|y|q) = 4
integraldisplay +∞
0
integraldisplay +∞
0
dxdy
(1 +xp)(1 +yq)
a113 lim
x→+∞
xp 11 +xp = 1a167a75a100a195a161a129a50a194a200a169a133a220a7a79a123a27a52a129a47a170a167a26
a8p> 1a158a167a200a169
integraldisplay +∞
0
dx
1 +xp a194a241a182a8plessorequalslant 1a158a167a200a169
integraldisplay +∞
0
dx
1 +xp a117a209
a211a110a140a26a167a8q> 1a158a167a200a169
integraldisplay +∞
0
dy
1 +yq a194a241a182a8qlessorequalslant 1a158a167a200a169
integraldisplay +∞
0
dy
1 +yq a117a209
a110a254a140a127a167a8p> 1a133q> 1a158a167a200a169
integraldisplay +∞
∞
integraldisplay +∞
∞
dxdy
(1 +|x|p)(1 +|y|q) a194a241a167a217a123a156a185a254a117a209.
(2) a207 m(1 +x2 +y2)p lessorequalslant?(x,y)(1 +x2 +y2)p lessorequalslant M(1 +x2 +y2)p
a75a100a50a194a173a200a169a27a39a22a7a79a123a57a50a194a173a200a169a53a159a167a26
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
(x,y)
(1 +x2 +y2)p dxdya134
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
dxdy
(1 +x2 +y2)p
a211a241a209
a100a26a200a188a234a27a233a161a53a57a154a75a53a167a26
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
dxdy
(1 +x2 +y2)p = 2
integraldisplay 1
0
dy
integraldisplay +∞
0
dx
(1 +x2 +y2)p
a100a1170 lessorequalslantylessorequalslant 1a167a75
a101pgreaterorequalslant 0a167a75
integraldisplay +∞
0
dx
(2 +x2)p lessorequalslant
integraldisplay +∞
0
dx
(1 +x2 +y2)p lessorequalslant
integraldisplay +∞
0
dx
(1 +x2)p
a101p< 0a167a75
integraldisplay +∞
0
dx
(2 +x2)p greaterorequalslant
integraldisplay +∞
0
dx
(1 +x2 +y2)p greaterorequalslant
integraldisplay +∞
0
dx
(1 +x2)P
a233a117α> 0a167a100a117 lim
x→+∞
x2p 1(α2 +x2)p = 1a167a75a200a169
integraldisplay +∞
0
dx
(α2 +x2)p a8p>
1
2 a158a194a241a182a8plessorequalslant
1
2 a158a117a209
a117a180
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
dxdy
(1 +x2 +y2)p a8p>
1
2 a158a194a241a182a8plessorequalslant
1
2 a158a117a209
a108a13
integraldisplayintegraldisplay
0lessorequalslantylessorequalslant1
(x,y)
(1 +x2 +y2)p dxdya8p>
1
2 a158a194a241a182a8plessorequalslant
1
2 a158a117a209.
(3) a2070 < m|x?y|p lessorequalslant?(x,y)|x?y|p lessorequalslant M|x?y|p
a75a100a50a194a173a200a169a27a39a22a7a79a123a57a50a194a173a200a169a53a159a167a26
integraldisplay a
0
integraldisplay a
0
(x,y)
|x?y|p dxdya134
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p a211a241a209
a100a26a200a188a234a27a233a161a53a57a154a75a53a167a26
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p dxdy = 2
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p
a8p< 1a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p =
a2?p
(2?p)(1?p)
a75
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p = 2
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p =
2a2?p
(2?p)(1?p)
a8pgreaterorequalslant 1a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p = limε→+0
integraldisplay a
ε
dx
integraldisplay x?ε
0
dy
(x?y)p
a8p = 1a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p = limε→+0
integraldisplay a
ε
dx
integraldisplay x?ε
0
dy
x?y = limε→+0(alna?a+ε?alnε) = +∞
a75
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p a8p = 1a158a117a209a182
a8p = 2a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p = limε→+0
integraldisplay a
ε
dx
integraldisplay x?ε
0
dy
(x?y)2 = limε→+0
parenleftBigg
a+εlnε
ε?1?lna
parenrightBigg
=?∞
a75
integraldisplay a
0
integraldisplay a
0
dxdy
|x?y|p a8p = 2a158a117a209a182
263
a8p> 1a133pnegationslash= 2a158a167
integraldisplay a
0
dx
integraldisplay x
0
dy
(x?y)p = limε→+0
integraldisplay a
ε
dx
integraldisplay x?ε
0
dy
(x?y)p
= lim
ε→+0
bracketleftBigg
1
(p?1)εp?1
parenleftBigg
a? p?1p?2 ε
parenrightBigg
+ 1(p?1)(p?2)ap?2
bracketrightBigg
= +∞
a75
integraldisplay a
0
integraldisplay a
0
dxdy)
|x?y|p a8p> 1a133pnegationslash= 2a158a117a209
a110a254a140a127
integraldisplay a
0
integraldisplay a
0
(x,y)
|x?y|p dxdya8p< 1 a158a194a241a182a8pgreaterorequalslant 1a158a117a209.
(4) (0,0)a180a219a58a167a100a117x2 + +xy+y2 > 0(a8(x,y) negationslash= (0,0))a167a75
m
(x2 +xy+y2)p lessorequalslant
(x,y)
(x2 +xy+y2)p lessorequalslant
M
(x2 +xy+y2)p
a100a50a194a173a200a169a27a39a22a7a79a123a57a50a194a173a200a169a53a159a167a26
integraldisplay a
0
integraldisplay a
0
(x,y)
(x2 +xy+y2)p dxdya134
integraldisplay a
0
integraldisplay a
0
dxdy
(x2 +xy+y2)p
a211a241a209integraldisplayintegraldisplay
x2+y2lessorequalslant1
dxdy
(x2 +xy+y2)p = limε→+0
integraldisplay 1
ε
dr
r2p?1
integraldisplay 2pi
0
dθ
(1 + sinθcosθ)p = N limε→+0
integraldisplay 1
ε
dr
r2p?1
=
N lim
ε→+0
(?lnε) = +∞,p = 1
N lim
ε→+0
1?ε2?2p
2?2p =
N
2?2p,p< 1
∞,p> 1
parenleftBigg
a217a165N =
integraldisplay 2pi
0
dθ
(1 + sinθcosθ)pa143a126a194a200a169a167a143a126a254
parenrightBigg
a111a131a167
integraldisplayintegraldisplay
x2+y2lessorequalslant1
dxdy
(x2 +xy+y2)p a8p< 1a158a194a241a182a8pgreaterorequalslant 1a158a117a209
a108a13
integraldisplayintegraldisplay
x2+y2lessorequalslant1
(x,y)
(x2 +xy+y2)p dxdya8p< 1 a158a194a241a182a8pgreaterorequalslant 1a158a117a209.
3,a121a178a23Da180a100a51a49a152a150a129a27a14a212a130y = x2a167a11a177x2 +y2 = 1a57xa182a164a140a164a27a171a141a167a75
integraldisplayintegraldisplay
D
dxdy
x2 +y2 a127a51.
a121a178a181(0,0)a180a219a58integraldisplayintegraldisplay
D
dxdy
x2 +y2 =
integraldisplay θ0
0
dθ
integraldisplay 1
sinθ
cos2 θ
dr
r =
integraldisplay θ0
0
ln cos
2θ
sinθ dθ,0a180a219a58
parenleftBigg
a217a165θ0a247a118 sinθ0cos2θ
0
= 1a61sinθ0 =
√5?1
2
parenrightBigg
a207 lim
θ→+0
θ12 ln cos
2θ
sinθ = 0a167a75a100a133a220a7a79a123a27a52a129a47a170a167a26
integraldisplay θ0
0
ln cos
2θ
sinθ dθ a194a241
a108a13a6a200a169
integraldisplayintegraldisplay
D
dxdy
x2 +y2 a127a51.
4,a166a254a33a20a11a73a78a39a117a51a167a27a186a58a63a27a159a254a143ma27a159a58a27a218a229.
a41a181a218a229a51OX,OYa182a254a27a19a75a1430a167a61Fx = Fy = 0a167
Fz =
integraldisplayintegraldisplayintegraldisplay
V
mz
gr3 dV =
m
g
integraldisplay 2pi
0
dθ
integraldisplay R
0
dρ
integraldisplay h
R ρ
0
ρz
(ρ2 +z2)32
dz = 2mRpigl (l?g).
264
a49a19a155a152a217 a173a130a200a169a218a173a161a200a169a27a79a142
§1,a49a152a97a173a130a200a169a27a79a142
1,a79a142
integraldisplay
l
(x+y)dsa167la180a177O(0,0),A(1,0),B(0,1)a143a186a58a27a110a14a47.
a41a181I =
integraldisplay
l
(x+y)ds =
braceleftbiggintegraldisplay
OA
+
integraldisplay
AB
+
integraldisplay
BO
bracerightbigg
(x+y)ds
a51a134a130a227OAa254a167y = 0,ds = dxa167a75
integraldisplay
OA
(x+y)ds =
integraldisplay 1
0
xdx = 12 a182
a51a134a130a227ABa254a167y = 1?x,ds = √2 dxa167a75
integraldisplay
AB
(x+y)ds =
integraldisplay 1
0
√2dx = √2 a182
a51a134a130a227BOa254a167x = 0,ds = dya167a75
integraldisplay
BO
(x+y)ds =
integraldisplay 1
0
ydy = 12
a117a180I = 1 +√2,
2,a79a142
integraldisplay
l
(x2 +y2)dsa167la180a177a6a58a143a165a37a167a140a187a143Ra27a134a140a11a177.
a41a181a207l,x = Rcosθ,y = Rsinθ,pi2 lessorequalslantθlessorequalslant 32 pia167a75ds =radicalbigx2θ +y2θ dθ = Rdθ
a117a180
integraldisplay
l
(x2 +y2)ds = piR3.
3,a79a142
integraldisplay
l
(x2 +y2 +z2)dsa167la180a11a218a94a130a181x = acost,y = asint,z = bt (0 lessorequalslanttlessorequalslant 2pi).
a41a181a207ds =radicalbigxprime2(t) +yprime2(t) +zprime2(t) dt = √a2 +b2 dt
a75I =
integraldisplay
l
(x2 +y2 +z2)ds = 23 pi(3a2 + 4pi2b2)
radicalbig
a2 +b2.
4,a79a142
integraldisplay
l
x2 dsa167la180a165a161x2 +y2 +z2 = a2a134a178a161x+y+z = 0a131a2a27a11a177.
a41a181a100a233a161a53a167a26
integraldisplay
l
x2 ds =
integraldisplay
l
y2 ds =
integraldisplay
l
x2 dsa167a75
integraldisplay
l
x2 ds = 13
integraldisplay
l
(x2 +y2 +z2)ds = a
2
3
integraldisplay
l
ds = 23 pia3.
5,a79a142
integraldisplay
l
z2
x2 +y2 dsa167la180a218a130a181x = acost,y = asint,z = at,(0 lessorequalslanttlessorequalslant 2pi).
a41a181a207ds =radicalbigxprime2(t) +yprime2(t) +zprime2(t) dt = √2 dta167a75I =
integraldisplay
l
z2
x2 +y2 ds =
8√2
3 pi
3a.
6,a23a152a55a225a106la27a144a167a143a181
x = et cost,y = et sint,z = et,(0 lessorequalslanttlessorequalslantt0)
a167a51a122a152a58a27a151a221a134a84a58a27a165a187a178a144a164a135a39a167a133a51a58(1,0,1)a63a1431a167a166a167a27a159a254.
a41a181a207ρ = kx2 +y2 +z2 a133a51a58(1,0,1)a63ρ = 1a167a75k = 2a167a117a180ρ = 2x2 +y2 +z2 = e?2t
a113ds =radicalbigxprime2(t) +yprime2(t) +zprime2(t) dt = √3et dta167a75m =
integraldisplay
l
ρds = √3(1?e?t0).
7,a166a253a11x = acost,y = bsinta177a46a27a159a254(0 lessorequalslanttlessorequalslant 2pi)a167a101a173a130a51a58M(x,y)a27a130a53a151a221a143ρ = |y|.
a41a181M =
integraldisplay
l
|y|dsa167a217a165la143a253a11x = acost,y = bsint(0 lessorequalslanttlessorequalslant 2pi)
(1) a23a>ba167a75ds =radicalbigxprime2(t) +yprime2(t) dt = aradicalbig1?ε21 cos2t dta167a217a165ε1 =
√a2?b2
a
a117a180M =
integraldisplay
l
|y|ds =
integraldisplay pi
0
absint
radicalbig
1?ε1 cos2t dt+
integraldisplay 2pi
pi
a(?bsint)
radicalBig
1?ε21 cos2t dt =
2ab
radicalBig
1?ε21 + 2abε
1
arcsinε1 = 2b2 + 2abε
1
arcsinε1
(2) a23a<ba167a75ds =radicalbigxprime2(t) +yprime2(t) dt = aradicalbig1 +ε22 cos2t dta167a217a165ε2 =
√b2?a2
a
a117a180M =
integraldisplay
l
|y|ds =
integraldisplay pi
0
absint
radicalBig
1 +ε22 cos2t dt+
integraldisplay 2pi
pi
a(?bsint)
radicalBig
1 +ε22 cos2t dt =
2ab
radicalBig
1 +ε22 + 2abε
2
ln(ε2 +
radicalBig
1 +ε22) = 2b2 + 2abε
2
ln(ε2 +
radicalBig
1 +ε22)
265
(3) a101a = ba167a75ds =radicalbigxprime2(t) +yprime2(t) dt = adta167a117a180M =
integraldisplay
l
|y|ds =
integraldisplay pi
0
a2 sintdt+
integraldisplay 2pi
pi
a(?asint)dt =
4a2
a108a13M =
2b2 + 2abε
1
arcsinε1,a>b
4a2,a = b
2b2 + 2abε
2
ln(ε2 +radicalbig1 +ε22),a<b
266
§2,a49a152a97a173a161a200a169a27a79a142
1,a79a142a101a15a173a161a161a200a181
(1) z = axya157a185a51a11a206x2 +y2 = a2a83a27a220a169a182
(2) a73a161x2 +y2 = 13 z2a134a178a161x+y+z = 2a(a> 0)a164a46a220a169a27a76a161a182
(3) a206a161x2 +y2 = a2a26a19a178a161x+z = 0,x?z = 0(x> 0,y> 0)a164a31a220a169.
a41a181
(1) a100zx = ay,zy = axa167a26radicalbig1 +z2x +z2y =radicalbig1 +a2x2 +a2y2
a100a233a161a53a167a191a124a94a206a161a139a73a167a26
S = 4
integraldisplayintegraldisplay
σxy
radicalbig
1 +a2x2 +a2y2 dxdy = 4
integraldisplay pi
2
0
dθ
integraldisplay a
0
radicalbig
1 +a2r2rdr = 23a2pi
bracketleftBig
(1 +a4)32?1
bracketrightBig
.
(2) a173a161a27a2a130a51xoya178a161a254a27a19a75a1433x2 + 3y2 = (2a?x?y)2a61x2 +y2?xy+ 2a(x+y) = 2a2
a45x = 1√2 (xprime?yprime),y = 1√2(xprime +yprime)a167a75a144a167a67a143(x
prime + 2√2a)2
(2√3a)2 +
yprime2
(2a)2 = 1
a100a100a140a132a167a173a161a164a46a27a212a78a51xoya178a161a254a27a19a75a141a143a1772aa143a225a140a182a1672√3aa143a127a140a182a27a253a11
a212a78a27a76a161a200a100a31a161a218a31a209a27a73a161a252a220a169a124a164
a233a117z = 2a?x?y,z =radicalbig3x2 + 3y2a169a79a107radicalbig1 +z2x +z2y = √3,radicalbig1 +z2x +z2y = 2
a117a180a212a78a27a76a161a200a143S =
integraldisplayintegraldisplay
√3dxdy+integraldisplayintegraldisplay
2dxdy = (√3 + 2)pi·2a·2√3pi = 4pi(3 + 2√3)a2.
(3) a100yx =?xy,yz = 0a167a26radicalbig1 +y2x +y2z =
radicaltpradicalvertex
radicalvertexradicalbt
1 +
parenleftBigg
x
y
parenrightBigg2
= |a|√a2?x2
a75S =
integraldisplayintegraldisplay
σxz
|a|√
a2?x2 dxdz =
integraldisplay |a|
0
dx
integraldisplay x
x
|a|√
a2?x2 dz = 2a
2.
2,a79a142a49a152a97a173a161a200a169a181
(1)
integraldisplayintegraldisplay
S
(x+y+z)dS,S,a134a140a165a161x2 +y2 +z2 = a2,ylessorequalslant 0a182
(2)
integraldisplayintegraldisplay
S
xdS,S,a218a94a161x = ucosv,y = usinv,z = cva254a27a152a220a1690 lessorequalslantulessorequalslanta,0 lessorequalslantvlessorequalslant 2pia182
(3)
integraldisplayintegraldisplay
S
dS,S,a165a161x2 +y2 +z2 = 2cz(c> 0)a89a51a73a161x2 +y2 = z2a83a27a220a169a182
(4)
integraldisplayintegraldisplay
S
(x2 +y2)dS,S,a78a200radicalbigx2 +y2 lessorequalslantz lessorequalslant 1a27a62a46a182
(5)
integraldisplayintegraldisplay
S
dS
r2,Sa143a11a206a161x
2 +y2 = R2a48a117z = 0a218z = Ha131a109a27a220a169a167a217a165ra143a173a161a254a27a58a20a6a58a27a229a108.
a41a181
(1) a242x2 +y2 +z2 = a2a221a75a20xoza178a161a167a100a158a107y =?√a2?x2?z2
a75yx = x√a2?x2?z2,yz = z√a2?x2?z2a167a117a180radicalbig1 +y2x +y2z = a√a2?x2?z2
a117a180
integraldisplayintegraldisplay
S
(x+y+z)dS =
integraldisplay a
a
dx
integraldisplay √a2?x2
√
a2?x2
a√
a2?x2?z2 (x?
radicalbig
a2?x2?z2 +z)dz =?pia3
(2) E = x2u +y2u +z2u = 1,F = xuxv +yuyv +zuzv = 0,G = x2v +y2v +z2v = u2 +c2
a75
integraldisplayintegraldisplay
S
xdS =
integraldisplayintegraldisplay
Σ
ucosv
radicalbig
u2 +c2 dudv =
integraldisplay a
0
u
radicalbig
u2 +c2 du
integraldisplay 2pi
0
cosvdv = 0
267
(3) a207x2 +y2 +z2 = 2cza167a75x2 +y2 + (z?c)2 = c2,z = c+radicalbigc2?x2?y2
a117a180zx =? xradicalbigc2?x2?y2,zy =? yradicalbigc2?x2?y2 a167a75radicalbig1 +z2x +z2y = cradicalbigc2?x2?y2
a117a180
integraldisplayintegraldisplay
S
dS =
integraldisplayintegraldisplay
σ
cradicalbig
c2?x2?y2 dσ =
integraldisplay 2pi
0
dθ
integraldisplay c
0
cr√
c2?r2 dr = 2pic
2.
(4) a169a143a252a220a169a181
a49a152a220a169a181z = 1,radicalbig1 +z2x +z2y = 1a182a49a19a220a169a181z =radicalbigx2 +y2,radicalbig1 +z2x +z2y = √2
a75
integraldisplayintegraldisplay
S
(x2 +y2)dS =
integraldisplay 2pi
0
dθ
integraldisplay 1
0
r3 dr+
integraldisplay 2pi
0
dθ
integraldisplay 1
0
√2r3 dr = pi
2 (1 +
√2).
(5) x = Rcosθ,y = Rsinθ,z = z (0 lessorequalslantθ2pi,0 lessorequalslantz lessorequalslantH)
a75E = x2θ +y2θ +z2θ = R2,F = xθxz +yθyz +zθzz = 0,G = x2z +y2z +z2z = 1
a117a180√EG?F2 = Ra167a108a13
integraldisplayintegraldisplay
S
dS
r2 =
integraldisplay 2pi
0
dθ
integraldisplay H
0
R
R2 +z2 dz = 2piarctan
H
R,
3,a166a14a212a161a138z = 12 (x2 +y2),0 lessorequalslantz lessorequalslant 1a27a159a254.a100a138a27a151a221a143ρ = z.
a41a181a207z = 12 (x2 +y2)a167a75zx = x,zy = ya167a117a180radicalbig1 +z2x +z2y =radicalbig1 +x2 +y2
a75a159a254M =
integraldisplayintegraldisplay
S
ρdS = 12
integraldisplayintegraldisplay
x2+y2lessorequalslant2
(x2+y2)
radicalbig
1 +x2 +y2 dxdy = 12
integraldisplay 2pi
0
dθ
integraldisplay √2
0
r3
radicalbig
1 +r2 dr = 2(1 + 6
√3)
15 pi.
268
§3,a49a19a97a173a130a200a169
1,a79a142a101a15a49a19a97a173a130a200a169a181
(1)
integraldisplay
l
(x2?2xy)dx+ (y2?2xy)dy,la143y = x2a108(1,1)a20(?1,1)a182
(2)
contintegraldisplay
l
(x2 +y2)dx+ (x2?y2)dy,la143a177A(1,0),B(2,0)C(2,1),D(1,1)a143a186a58a27a20a144a47a167a20a149a182
(3)
integraldisplay
l
(2a?y)dx+ dy,la143a94a211a130x = a(t?sint),y = a(1?cost)a167a108(0,0)a20(2pi,0)a182
(4)
integraldisplay
l
ydx?xdy+ (x2 +y2)dz,la143a173a130x = et,y = e?t,z = ata108(1,1,0)a20(e,e?1,a)
a41a181
(1)
integraldisplay
l
(x2?2xy)dx+ (y2?2xy)dy =
integraldisplay?1
1
[x2?2x3 + 2x(x4?2x3)]dx = 1415,
(2) I =
contintegraldisplay
l
(x2 +y2)dx+ (x2?y2)dy =
parenleftbiggintegraldisplay
AB
+
integraldisplay
BC
+
integraldisplay
CD
+
integraldisplay
DA
parenrightbigg
(x2 +y2)dx+ (x2?y2)dy
a247ABa167y = 0a167a25
integraldisplay
AB
(x2?y2)dy = 0
a211a24a167a107
integraldisplay
BC
(x2 +y2)dx =
integraldisplay
CD
(x2?y2)dy =
integraldisplay
DA
(x2 +y2)dx = 0
a75I =
integraldisplay 2
1
x2 dx+
integraldisplay 1
0
(4?y2)dy+
integraldisplay 1
2
(x2 + 1)dx =
integraldisplay 0
1
(1?y2)dy = 2.
(3)
integraldisplay
l
(2a?y)dx+ dy =
integraldisplay 2pi
0
[(a+acost)·a(1?cost) +asint]dt = a2pi.
(4)
integraldisplay
l
ydx?xdy+ (x2 +y2)dz =
integraldisplay 1
0
[e?t ·et?et(?e?t) + (e2t +e?2t)a]dt = 2 + a2 (e2?e?2).
2,a166a200a169
J =
integraldisplay (1,1,1)
(0,0,0)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
1 1 1
x y z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle· dr
a217a165dra143a165a187a144a149a167a200a169a180a187a169a79a143a181
(1) a247a134a130a182
(2) a247a173a130r = isin?+ j(1?cos?) + k 2?pi,
parenleftBigg
0 lessorequalslant?lessorequalslant pi2
parenrightBigg
.
a41a181
(1) a134a130a144a167a143a181x = y = z
a75J =
integraldisplay (1,1,1)
(0,0,0)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
1 1 1
x y z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle· dr =
integraldisplay (1,1,1)
(0,0,0)
(z?y)dx+ (x?z)dy+ (y?x)dz =
integraldisplay 1
0
(x?x)dx+
integraldisplay 1
0
(y?y)dy+
integraldisplay 1
0
(z?z)dz = 0.
(2) J =
integraldisplay (1,1,1)
(0,0,0)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
1 1 1
x y z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle·dr =
integraldisplay pi
2
0
braceleftBiggbracketleftBigg
2?
pi?(1?cos?)
bracketrightBigg
cos?+
parenleftBigg
sin 2?pi
parenrightBigg
sin?+ [(1?cos?)?sin?]· 2pi
bracerightBigg
d? =
1? pi2? 8pi,
3,a23a49a119a52a173a130La51a49a119a173a161Sa254a167Sa27a144a167a143z = f(x,y)a167a173a130La51XYa161a254a27a221a75a173a130a143la167a188a234P(x,y,z)a51La254
a235a89a167a121a178 contintegraldisplay
L
P(x,y,z)dx =
contintegraldisplay
l
P[x,y,f(x,y)]dx
a121a178a181a216a148a23Sa143a173a161a27a254a253a167z = f(x,y),(x,y) ∈D
a75a173a161a27a62a46La51XYa178a161a254a27a221a75a65a180a95a158a2a144a149a27a173a130l,x =?(t),y = ψ(t),a<b,alessorequalslanttlessorequalslantb
a152a109a173a130La27a144a167a145a131a140a76a143L,x =?(t),y = ψ(t),z = ω(t) = f[?(t),ψ(t)],alessorequalslanttlessorequalslantb
a117a180
contintegraldisplay
L
P(x,y,z)dx =
integraldisplay b
a
P (?(t),ψ(t),f[?(t),ψ(t)])?prime(t)dt =
contintegraldisplay
l
P[x,y,f(x,y)]dx.
269
4,a121a178a181a233a117a173a130a200a169a27a15a79a170a143
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
l
P dx+Qdy
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslantLM,(a170a165La143a200a169a173a130a227a127a221)
M = max
(x,y)∈l
radicalbig
P2 +Q2
a124a94a249a135a216a31a170a15a79a181
IR =
contintegraldisplay
x2+y2=R2
ydx?xdy
(x2 +xy+y2)2
a191a121a178 lim
R→∞
IR = 0.
a121a178a181
(1)
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
l
P dx+Qdy
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
l
[P cosα+Qsinα]dS
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
l
|(P,Q)·(cosα,sinα)| dS lessorequalslant
integraldisplay
l
|(P,Q)||(cosα,sinα)|dS =
integraldisplay
l
radicalbig
P2 +Q2 dS =
radicalbig
P2(ξ,η) +Q2(ξ,η)
integraldisplay
l
dS lessorequalslantML.
(2) a207P = y(x2 +xy+y2)2,Q =?x(x2 +xy+y2)2 a167a75radicalbigP2 +Q2 =
radicalbigx2 +y2
(x2 +xy+y2)2 =
R
(R2 +xy)2
a117a180M = max
(x,y)∈l
radicalbigP2 +Q2 = R
(R2 +xy)2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
x = ±
√2
2 R
y =?
√2
2
= 4R3
a750 lessorequalslant|IR| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
contintegraldisplay
x2+y2=R2
ydx?xdy
(x2 +xy+y2)2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslantLM =
8pi
R2
a113 lim
R→∞
8pi
R2 = 0a167a75 limR→∞IR = 0.
5,a23a178a161a171a141Da100a152a94a235a89a52a173a130La164a140a164a167a171a141Da27a161a200a23a143Sa167a237a19a94a173a130a200a169a79a142a161a200Sa27a250a170a181
S = 12
contintegraldisplay
L
xdy?ydx
a121a178a181
(1) a196a107a127a196a227a47D = PQRSa167a217a165QR,PS bardblYa182a167PQ,y = y0(x);SR,y = y1(x)a133a51[a,b]a254a235a89
a249a158L,PSRQP
a242Da27a161a200a119a138a252a173a62a70a47abPSa218abQPa27a161a200a131a11(a217a165a,ba169a143SP,RQa134Xa182a27a2a58)
a117a180a107S =
integraldisplay b
a
[y1(x)?y0(x)]dx
a44a152a144a161a167a226IIa46a173a130a79a142a250a170a167a107
integraldisplay
slurabovePQydx =
integraldisplay b
a
y0(x)dx,
integraldisplay
sluraboveSRydx =
integraldisplay b
a
y1(x)dx
a191a53a191a20
integraldisplay
PS
ydx =
integraldisplay
PQ
ydx = 0
a75?
integraldisplay
L
ydx =
integraldisplay
PSRQP
ydx =
parenleftbiggintegraldisplay
PS
+
integraldisplay
sluraboveSR +
integraldisplay
RQ
+
integraldisplay
sluraboveQR
parenrightbigg
ydx =
integraldisplay b
a
y1(x)dx?
integraldisplay b
a
y0(x)dx = S
a61S =?
integraldisplay
L
ydx.
(2) a233a117a171a141D = PQRSa167a217a165PQ,RS bardblXa182a167a211a110a167a107
integraldisplay
L
xdy = S.
(3) a233a117a141a69a44a27a171a141a156a47a140a122a143a254a252a171a156a47a167a211a24a79a142a195a2a172a161a200a167a44a0a131a92a167a53a191a173a69a180a130a131a112a45a158a167
a211a24a140a26a254a252a171a40a74.
a110a254a164a227a167a107S = 12
contintegraldisplay
L
xdy?ydx.
6,a79a142a101a15a173a130a164a140a171a141a27a161a200a181
(1) a253a11a181x = acost,y = bsint,(0 lessorequalslanttlessorequalslant 2pi)a182
(2) a40a47a130a181x = acos3t,y = asin3t,(0 lessorequalslanttlessorequalslant 2pi).
270
a41a181
(1) S = 12
contintegraldisplay
L
xdy?ydx = 12
integraldisplay 2pi
0
abdt = piab.
(2) S = 12
contintegraldisplay
L
xdy?ydx = 32 a2
integraldisplay 2pi
0
sin2tcos2tdt = 38 pia2.
271
§4,a49a19a97a173a161a200a169
1,a79a142
integraldisplayintegraldisplay
S
(x+y)dydz + (y+z)dzdx+ (z +x)dxdy
Sa180a177a6a58a143a165a37a27a20a144a78(a122a62a127a221a1432)a27a62a46a167a141a149a9a253.
a41a181I =
integraldisplayintegraldisplay
S
(x+y)dydz+(y+z)dzdx+(z+x)dxdy =
integraldisplayintegraldisplay
S
(x+y)dydz+
integraldisplayintegraldisplay
S
(y+z)dzdx+
integraldisplayintegraldisplay
S
(z+x)dxdy
a79a142I =
integraldisplayintegraldisplay
S
(x+y)dydz
a207a20a144a78a56a135a161a165a107a111a135a161a82a134a117YOZa178a161a167a75a100a111a135a161a27a161a200a1430
a117a180
integraldisplayintegraldisplay
S
(x+y)dydz =
integraldisplayintegraldisplay
1lessorequalslantylessorequalslant1
1lessorequalslantzlessorequalslant1
(1 +y)dydz?
integraldisplayintegraldisplay
1lessorequalslantylessorequalslant1
1lessorequalslantzlessorequalslant1
(?1 +y)dydz = 2
integraldisplayintegraldisplay
1lessorequalslantylessorequalslant1
1lessorequalslantzlessorequalslant1
dydz = 8
a211a110a140a26
integraldisplayintegraldisplay
S
(y+z)dzdx = 8,
integraldisplayintegraldisplay
S
(z +x)dxdy = 8
a117a180I =
integraldisplayintegraldisplay
S
(x+y)dydz + (y+z)dzdx+ (z +x)dxdy = 24.
2,a79a142
integraldisplayintegraldisplay
S
f(x)dydz +g(y)dxdz +h(z)dxdy
a170a165f,g,ha143a235a89a188a234a167Sa143a178a49a56a161a78(0 lessorequalslantxlessorequalslanta,0 lessorequalslantylessorequalslantb,0 lessorequalslantz lessorequalslantc)a27a62a46a167a141a149a9a253.
a41a181a23S1,x = a;S2,x = 0;S3,y = b;S4,y = 0;S5,z = c;S6,z = 0
a75I =
integraldisplayintegraldisplay
S
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S1
+
integraldisplayintegraldisplay
S2
+
integraldisplayintegraldisplay
S3
+
integraldisplayintegraldisplay
S4
+
integraldisplayintegraldisplay
S5
+
integraldisplayintegraldisplay
S6
f(x)dydz +g(y)dxdz +h(z)dxdy
a207
integraldisplayintegraldisplay
S1
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S1
f(x)dydz = f(a)bc
integraldisplayintegraldisplay
S2
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S2
f(x)dydz =?f(0)bc
integraldisplayintegraldisplay
S3
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S3
g(y))dzdx = g(b)ac
integraldisplayintegraldisplay
S4
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S4
g(y)dzdx =?g(0)ac
integraldisplayintegraldisplay
S5
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S5
h(z)dxdy = h(c)ab
integraldisplayintegraldisplay
S6
f(x)dydz +g(y)dxdz +h(z)dxdy =
integraldisplayintegraldisplay
S6
h(z)dxdy =?h(0)ab
a75I =
integraldisplayintegraldisplay
S
f(x)dydz +g(y)dxdz +h(z)dxdy = abc
bracketleftBigg
f(a)?f(0)
a +
g(b)?g(0)
b +
h(c)?h(0)
c
bracketrightBigg
.
3,a79a142
integraldisplayintegraldisplay
S
yzdzdx
S,x
2
a2 +
y2
b2 +
z2
c2 = 1a27a254a140a76a161a27a254a253.
a41a181a242a253a165a161a76a143a235a234(?,θ)a47a170a181x = asin?cosθ,y = bsin?cosθ,z = ccos?
parenleftBigg
0 lessorequalslant?lessorequalslant pi2,0 lessorequalslantθlessorequalslant 2pi
parenrightBigg
I =
integraldisplayintegraldisplay
S
yzdzdx = ±
integraldisplayintegraldisplay
bcsin?cos?sinθ·Bd?dθa167a217a165?a143?θa178a161a254a27a171a1410 lessorequalslant?lessorequalslant pi2,0 lessorequalslantθlessorequalslant 2pi
a133B = z?xθ?x?zθ = acsin2?sinθ
a207a200a169a247a254a253a167a65a18a20a210a167a61a26I = abc2
integraldisplay pi
2
0
sin3?cos?d?
integraldisplay 2pi
0
sin2θdθ = pi4 abc2
272
4,a79a142
integraldisplayintegraldisplay
S
zdxdy+xdydz +ydxdz
Sa143a206a161x2 +y2 = 1a26a178a161z = 0a57z = 3a164a31a220a169a27a9a253.
a41a181a100a117a206a161x2 +y2 = 1a51XOYa178a161a254a27a221a75a143a152a11a177a167a25a217a161a200a1430a167a108a13
integraldisplayintegraldisplay
S
zdxdy = 0
a113
integraldisplayintegraldisplay
S
xdydz =
integraldisplayintegraldisplay
Sa99
+
integraldisplayintegraldisplay
Sa0
xdydz =integraldisplayintegraldisplay
Syz
radicalbig
1?y2 dydz?
integraldisplayintegraldisplay
Syz
(?
radicalbig
1?y2)dydz =
2
integraldisplay 2
0
dz
integraldisplay 1
1
radicalbig
1?y2 dy = 3pi
integraldisplayintegraldisplay
S
ydxdz =
integraldisplayintegraldisplay
Sa109
+
integraldisplayintegraldisplay
Sa134
ydxdz = 2integraldisplay 3
0
dz
integraldisplay 1
1
radicalbig
1?x2 dx = 3pi
a75
integraldisplayintegraldisplay
S
zdxdy+xdydz +ydxdz = 6pi.
5,a79a142
integraldisplayintegraldisplay
S
x3 dydz +y3 dzdx+z3 dxdy
Sa143a165a161x2 +y2 +z2 = a2a27a9a253.
a41a181a226a211a134a233a161a167a144a73a79a142
integraldisplayintegraldisplay
S
x3 dydza167a133
integraldisplayintegraldisplay
S
x3 dydz =
integraldisplayintegraldisplay
S1
x3 dydz +
integraldisplayintegraldisplay
S2
x3 dydz
a217a165S1a57S2a169a79a76a171a101a140a165a161a57a254a140a165a161a167a61S2,x =radicalbiga2?y2?z2a65a18a254a253a182S1,x =?radicalbiga2?y2?z2a65
a18a101a253
a75
integraldisplayintegraldisplay
S
x3 dydz =
integraldisplayintegraldisplay
S2
x3 dydz +
integraldisplayintegraldisplay
S1
x3 dydz = 2
integraldisplayintegraldisplay
y2+z2lessorequalslanta2
(a2?y2?z2)32 dydz =
2
integraldisplay 2pi
0
dθ
integraldisplay a
0
r(a2?r2)32 dr = 45 pia5
a117a180
integraldisplayintegraldisplay
S
x3 dydz +y3 dzdx+z3 dxdy = 125 pia5.
273
a49a19a155a19a217 a136a171a200a169a109a27a233a88a218a124a216a208a218
§1,a136a171a200a169a109a27a233a88
1,a124a94a130a21a250a170a79a142a173a130a200a169a181
(1)
contintegraldisplay
l
xy2 dx?x2ydy,la181a11a177x2 +y2 = a2a182
(2)
contintegraldisplay
l
(x+y)dx?(x?y)dy,la181a253a11a177x
2
a2 +
y2
b2 = 1a182
(3)
contintegraldisplay
l
(x+y)2 dx?(x2 +y2)dy,la181a186a58a143A(1,1),B(3,2),C(2,5)a27a110a14a47a27a62a46a182
(4)
integraldisplay
sluraboveAMO(e
x siny?my)dx+ (ex cosy?m)dya167
a217a165 sluraboveAMOa143a100a58A(a,0)a150a58O(0,0)a178a76a254a140a11a177x2 +y2 = axa27a23a180a182
(5)
contintegraldisplay
l
ex[(1?cosy)dx?(y?siny)dy],la181a171a1410 <x<pi,0 <y< sinxa27a62a46.
a41a181
(1) a100a130a21a250a170a167a100a158P = xy2,Q =?x2y
a75
contintegraldisplay
l
xy2 dx?x2ydy =
integraldisplayintegraldisplay
D
(?2xy?2xy)dxdy =?4
integraldisplay a
a
xdx
integraldisplay √a2?x2
√
a2?x2
ydy = 0
(2) P = x+y,Q =?(x?y)a167a75
contintegraldisplay
l
(x+y)dx?(x?y)dy =?2
integraldisplayintegraldisplay
D
dxdy =?2piab
(3) AB,BC,CAa27a144a167a169a79a143a181AB,x?2y+ 1 = 0;BC,3x+y?11 = 0;CA,4x?y?3 = 0
P = (x+y)2,Q =?(x2 +y2)
a75I =
contintegraldisplay
l
(x+y)2 dx?(x2 +y2)dy =?2
integraldisplayintegraldisplay
D
(2x+y)dxdy
=?2
bracketleftBiggintegraldisplay
2
1
dx
integraldisplay 4x?3
x+1
2
(2x+y)dy+
integraldisplay 3
2
dx
integraldisplay 11?3x
x+1
2
(2x+y)dy
bracketrightBigg
=?4632
(4) a51Oxa182a254a235a26a58O(0,0)a134A(a,0)a167a249a24a66a8a164a181a52a27a140a11a47AMOAa167a133a51a130a227OAa254integraldisplay
OA
(ex siny?my)dx+ (ex cosy?m)dy = 0
a75
integraldisplay
sluraboveAMOA(e
x siny?my)dx+ (ex cosy?m)dy =
integraldisplay
sluraboveAMO(e
x siny?my)dx+ (ex cosy?m)dy
a124a94a130a21a250a170a167a26
integraldisplay
sluraboveAMOA(e
x siny?my)dx+ (ex cosy?m)dy = m
integraldisplayintegraldisplay
D
dxdy = pim8 a2
a117a180
integraldisplay
sluraboveAMO(e
x siny?my)dx+ (ex cosy?m)dy = pim
2
8 a
2
(5) P = ex(1?cosy),Q = (y?siny)(?ex)
a75
contintegraldisplay
l
ex[(1?cosy)dx?(y?siny)dy] =?
integraldisplayintegraldisplay
D
yex dxdy =?
integraldisplay pi
0
ex dx
integraldisplay sinx
0
ydy =?15 (epi?1).
2,a124a94a130a21a250a170a79a142a101a15a173a130a164a140a161a200a181
(1) a40a47a130a181x = acos3t,y = bsin3ta182
(2) a14a212a130a181(x+y)2 = ax(a> 0)a218xa182
a41a181
(1) a100a130a21a250a170a167a161a200Da143D =
integraldisplayintegraldisplay
D
dxdy = 12
contintegraldisplay
l
xdy?ydx
a113x = acos3t,y = bsin3t (0 lessorequalslanttlessorequalslant 2pi)a167a75D = 12
contintegraldisplay
l
xdy?ydx = 38 ab
integraldisplay 2pi
0
sin2 2tdt = 38 piab
274
(2) a138a147a134y = txa167a75a6a144a167a122a143x2(1 +t)2 = ax (a> 0,x> 0)
a117a180a26a173a130a235a234a144a167x = a(1 +t)2,y = at(1 +t)2 (0 lessorequalslantt< +∞)
a167a134Oxa182a27a2a58a143(a,0)a134(0,0)
a51Oxa182a254a108(0,0)a58a20(a,0)a58a27a152a227a254a107xdy?ydx = 0a182a51a14a212a130a254a107xdy?ydx = a
2
(1 +t)4 dt
a117a180a161a200D = 12
contintegraldisplay
l
xdy?ydx = a
2
2
integraldisplay +∞
0
dt
(1 +t)4 =
a2
6,
3,a121a178a101Ca143a178a161a254a181a52a173a130a167la143a63a191a144a149a75
contintegraldisplay
c
cos(l,n)ds = 0
a170a165na143Ca27a9a123a130a144a149.
a121a178a181a216a148a23Ca27a144a149a143a95a158a2a144a149
a207(l,n) = (l,x)?(n,x)a167a75cos(l,n) = cos(l,x)cos(n,x) + sin(l,x)sin(n,x)
a113sin(n,x) =?cos(t,x),cos(n,x) = sin(t,x)a133cos(t,x) = dxds,sin(t,x) = dyds
a75cos(l,n)ds = cos(l,x)dy?sin(l,x)dx
a117a180
contintegraldisplay
C
cos(l,n)ds =
contintegraldisplay
C
[?sin(l,x)dx+ cos(l,x)dy]
a100P =?sin(l,x),Q = cos(l,x)a167a26?P?y = 0 =?Q?x
a117a180
contintegraldisplay
C
cos(l,n)ds =
integraldisplayintegraldisplay
D
parenleftBigg
Q
x?
P
y
parenrightBigg
dxdy = 0
4,a23u(x,y),v(x,y)a180a228a107a19a30a235a89a160a19a234a27a188a234a167a191a23
u≡?
2u
x2 +
2u
y2
a121a178a181
(1)
integraldisplayintegraldisplay
σ
udxdy =
integraldisplay
l
u
n ds
(2)
integraldisplayintegraldisplay
σ
v?udxdy =?
integraldisplayintegraldisplay
σ
parenleftBigg
u
x
v
x +
u
y
v
y
parenrightBigg
dxdy+
contintegraldisplay
l
v?u?n ds
(3)
integraldisplayintegraldisplay
σ
(u?v?v?u)dxdy =?
integraldisplay
l
parenleftBigg
v?u?n?u?v?n
parenrightBigg
ds
a217a165σa143a52a173a130la164a140a27a178a161a171a141a167?u?n,?v?na143a247la9a123a130a144a149a19a234.
a121a178a181
(1)
integraldisplay
l
u
n ds =
integraldisplay
l
parenleftBigg
u
x cos(n,x) +
u
y sin(n,x)
parenrightBigg
ds =
integraldisplay
l
u
x sin(t,x)ds?
integraldisplay
l
u
y cos(t,x)ds
=
integraldisplay
l
u
x dy?
integraldisplay
l
u
y dx =
integraldisplayintegraldisplay
σ
bracketleftBigg
2u
x2 +
2u
y2
bracketrightBigg
dxdy =
integraldisplayintegraldisplay
σ
udxdy
(2) a207
contintegraldisplay
l
v?u?n ds =
contintegraldisplay
l
v
parenleftBigg
u
x cos(n,x) +
u
y sin(n,x)
parenrightBigg
ds =
contintegraldisplay
l
bracketleftBigg
v?u?x sin(t,x)?v?u?y cos(t,x)
bracketrightBigg
ds
=
contintegraldisplay
l
v?u?x dy?v?u?y dx =
integraldisplayintegraldisplay
σ
bracketleftBigg
x
parenleftBigg
v?u?x
parenrightBigg
y
parenleftBigg
v?u?y
parenrightBiggbracketrightBigg
dxdy
=
integraldisplayintegraldisplay
σ
parenleftBigg
u
x
v
x +
u
y
v
y
parenrightBigg
dxdy+
integraldisplayintegraldisplay
σ
v?udxdy
a75
integraldisplayintegraldisplay
σ
v?udxdy =?
integraldisplayintegraldisplay
σ
parenleftBigg
u
x
v
x +
u
y
v
y
parenrightBigg
dxdy+
contintegraldisplay
l
v?u?n ds
275
(3) a100(2)a167a26
integraldisplayintegraldisplay
σ
u?vdxdy =?
integraldisplayintegraldisplay
σ
parenleftBigg
v
x
u
x +
v
y
u
y
parenrightBigg
dxdy+
contintegraldisplay
l
u?v?n ds
a75
integraldisplayintegraldisplay
σ
(u?v?v?u)dxdy =?
integraldisplay
l
parenleftBigg
v?u?n?u?v?n
parenrightBigg
ds
5,a166a177a101a200a169a131a138
I =
contintegraldisplay
l
[xcos(n,x) +ycos(n,y)]ds
la181a157a140a107a46a171a141a27a123a252a181a52a173a130a167na143a167a27a9a123a130a144a149.
a41a181I =
contintegraldisplay
l
[xcos(n,x) +ycos(n,y)]ds =
contintegraldisplay
l
[xsin(t,x)?ycos(t,x)]ds
=
contintegraldisplay
l
xdy?ydx =
integraldisplayintegraldisplay
σ
parenleftBigg
dx
dx+
dy
dy
parenrightBigg
dxdy = 2
integraldisplayintegraldisplay
σ
dxdy = 2S
6,a121a178a181 contintegraldisplay
l
cos(r,n)
r ds = 0
a217a165la180a152a252a235a207a171a141σa27a62a46a13ra180la254a27a152a58a20σa9a44a152a189a58a27a229a108.a101ra76a171la254a152a58a20σa83a44a152a189a58a27a229a108a167
a64a34a249a200a169a131a138a31a1172pi.
a121a178a181a23ra143a58A(x,y)a20la254a27a58M(ξ,η)a27a149a254a167n,ra134Oxa182a27a89a14a169a79a143α,β
a75(r,n) = α?βa167a117a180cos(r,n) = cosαcosβ + sinαsinβ = ξ?xr cosα+ η?yr sinα
a75
contintegraldisplay
l
cos(r,n)
r ds =
contintegraldisplay
l
parenleftBigg
η?y
r2 sinα+
ξ?x
r2 cosα
parenrightBigg
ds =
contintegraldisplay
l
ξ?x
r2 dη?
η?y
r2 dξ
a207P =?η?yr2,Q = ξ?xr2 a167a75?P?η =?(ξ?x)
2 + (η?y)2
r4 =
Q
ξ
a207a13P,Qa27a160a19a234a216a22a58A(a100a63r = 0)a9a167a51a28a178a161a254a180a235a89a27a167a133?Q?ξ =?P?η
a117a180a124a94a130a21a250a170a167a127a8a58Aa51a173a130la131a9a158a167
contintegraldisplay
l
cos(r,n)
r ds = 0
a8a58a51a173a130 l a131a83a158a167P,Q,?P?η,?Q?ξ a254a51(x,y)a216a235a89a167a75a216a85a134a26a166a94a130a21a250a170a167a143a100a51 l a164a157a140a27a171
a141σa83a167a177Aa143a11a37a167Ra143a140a187a138a152a11a167a177a217a11a177a138a143a173a130 lprimea167a191a166a217a157a140a27a171a141σprime? σa167a50a242σa42a140a143σprimeprimea167
a166σ?σprimeprime
a207P,Q,?P?η,?Q?ξ a254a51a216(x,y)a9a27a18a135a178a161a254a235a89a133?P?η =?Q?ξ
a75a51a69a235a207a171a141σprimeprime\(x,y)a165a235a89a133?P?η =?Q?ξ
a249a158
contintegraldisplay
l
ξ?x
r2 dη?
η?y
r2 dξ =
contintegraldisplay
lprime
ξ?x
r2 dη?
η?y
r2 dξ
a13
contintegraldisplay
lprime
ξ?x
r2 dη?
η?y
r2 dξ =
integraldisplay 2pi
0
dθ = 2pi
a75
contintegraldisplay
l
cos(r,n)
r ds = 2pi
(a8a58Aa51la254a158a167
contintegraldisplay
l
cos(r,n)
r ds = pi)
7,a124a94a112a100a250a170a67a134a177a101a200a169a181
(1)
integraldisplayintegraldisplay
S
xydxdy+xzdxdz +yzdydz
(2)
integraldisplayintegraldisplay
S
parenleftBigg
u
x cosα+
u
y cosβ +
u
z cosγ
parenrightBigg
dS
a217a165cosα,cosβ,cosγa180a173a161a27a9a123a130a144a149a123a117.
a41a181
276
(1) a207P = yz,Q = xz,R = xya167a75Px = Qy = Rz = 0
a117a180a100a112a100a250a170a167a26
integraldisplayintegraldisplay
S
xydxdy+xzdxdz +yzdydz = 0
(2) a207P = ux,Q = uy,R = uza167a75a100a112a100a250a170a167a26integraldisplayintegraldisplay
S
parenleftBigg
u
x cosα+
u
y cosβ +
u
z cosγ
parenrightBigg
dS =
integraldisplayintegraldisplayintegraldisplay
V
parenleftBigg
2u
x2 +
2u
y2 +
2u
z2
parenrightBigg
dxdydz
8,a124a94a112a100a250a170a79a142a173a161a200a169a181
(1)
integraldisplayintegraldisplay
S
x2 dydz +y2 dxdz +z2 dxdy,Sa181a225a144a780 lessorequalslantx,y,z lessorequalslantaa27a9a76a161a182
(2)
integraldisplayintegraldisplay
S
x3 dydz +y3 dxdz +z3 dxdy,Sa181a252a160a165a9a76a161a182
(3)
integraldisplayintegraldisplay
S
(x2 cosα+y2 cosβ +z2 cosγ)dS,Sa181x2 +y2 = z2,0 lessorequalslantz lessorequalslanth;cosα,cosβ,cosγa143a100a173a161a9a123a130a144
a149a123a117.
a41a181
(1) a207P = x2,Q = y2,R = z2a167
a75a100a112a100a250a170a167a26
integraldisplayintegraldisplay
S
x2 dydz +y2 dxdz +z2 dxdy = 2
integraldisplay a
0
dx
integraldisplay a
0
dy
integraldisplay a
0
(x+y+z)dz = 3a4
(2) a207P = x3,Q = y3,R = z3
a75a100a112a100a250a170a167a26
integraldisplayintegraldisplay
S
x3 dydz +y3 dxdz +z3 dxdy = 3
integraldisplayintegraldisplayintegraldisplay
V
(x2 +y2 +z2)dxdydz
= 3
integraldisplay 2pi
0
dθ
integraldisplay pi
0
sin?d?
integraldisplay 1
0
r4 dr = 125 pi
(3) a100a112a100a250a170a167a26integraldisplayintegraldisplay
S
(x2 cosα+y2 cosβ +z2 cosγ)dS = 2
integraldisplayintegraldisplayintegraldisplay
V
(x+y+z)dxdydz
= 2
integraldisplay 2pi
0
d?
integraldisplay h
0
rdr
integraldisplay h
r
[r(cos?+ sin?) +z]dz = pih
4
2,
9,a121a178a181a101
u≡?
2u
x2 +
2u
y2 +
2u
z2
Sa180Va27a62a46a173a161a167a75a164a225a101a161a250a170a181
(1)
integraldisplayintegraldisplayintegraldisplay
V
udxdydz =
integraldisplayintegraldisplay
S
u
n dS
(2)
integraldisplayintegraldisplay
S
u?u?n dS =
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBiggparenleftBigg
u
x
parenrightBigg2
+
parenleftBigg
u
y
parenrightBigg2
+
parenleftBigg
u
z
parenrightBigg2bracketrightBigg
dxdydz +
integraldisplayintegraldisplayintegraldisplay
V
u?udxdydz
a170a165ua51V +Sa254a107a235a89a19a30a19a234a167?u?na143a247a173a161Sa9a123a130a144a149a27a19a234.
a121a178a181
(1)
integraldisplayintegraldisplay
S
u
n dS =
integraldisplayintegraldisplay
S
bracketleftBigg
u
x cos(n,x) +
u
y cos(n,y) +
u
z cos(n,z)
bracketrightBigg
dS =
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBigg
2u
x2 +
2u
y2 +
2u
z2
bracketrightBigg
dxdydz =
integraldisplayintegraldisplayintegraldisplay
V
udxdydz
(2)
integraldisplayintegraldisplay
S
u?u?n dS =
integraldisplayintegraldisplay
S
bracketleftBigg
u?u?x cos(n,x) +u?u?y cos(n,y) +u?u?z cos(n,z)
bracketrightBigg
dS
=
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBigg
x
parenleftBigg
u?u?x
parenrightBigg
+y
parenleftBigg
u?u?y
parenrightBigg
+z
parenleftBigg
u?u?z
parenrightBiggbracketrightBigg
dxdydz
277
=
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBiggparenleftBigg
u
x
parenrightBigg2
+
parenleftBigg
u
y
parenrightBigg2
+
parenleftBigg
u
z
parenrightBigg2bracketrightBigg
dxdydz +
integraldisplayintegraldisplayintegraldisplay
V
u
bracketleftBigg
2u
x2 +
2u
y2 +
2u
z2
bracketrightBigg
dxdydz
=
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBiggparenleftBigg
u
x
parenrightBigg2
+
parenleftBigg
u
y
parenrightBigg2
+
parenleftBigg
u
z
parenrightBigg2bracketrightBigg
dxdydz +
integraldisplayintegraldisplayintegraldisplay
V
u?udxdydz
10,a121a178a100a173a161Sa164a157a140a27a78a200a31a117
V = 13
integraldisplayintegraldisplay
S
(xcosα+ycosβ +zcosγ)dS
a170a165cosα,cosβ,cosγa143a173a161Sa27a9a123a130a27a144a149a123a117.
a121a178a181a100a112a100a250a170a167a26
V =
integraldisplayintegraldisplayintegraldisplay
V
dxdydz = 13
integraldisplayintegraldisplay
S
(xdydz +ydxdz +zdxdy) = 13
integraldisplayintegraldisplay
S
(xcosα+ycosβ +zcosγ)dS.
11,a124a94a100a247a142a105a250a170a79a142a173a130a200a169a181
(1)
contintegraldisplay
l
ydx+zdy+xdz,la181a11a177
braceleftbigg x2 +y2 +z2 = a2
x+y+z = 0 a108xa182a20a149a119a22a11a177a180a95a158a2a144a149a27a182
(2)
contintegraldisplay
l
(z?y)dx+ (x?z)dy+ (y?x)dz,la180a108(a,0,0)a178(0,a,0)a218(0,0,a)a163a20(a,0,0)a27a110a14a47.
a41a181
(1) a114a178a161x+y+z = 0a254la164a157a140a27a171a141a80a143σa167a75σa27a123a130a144a149a143(1,1,1)
a75a217a144a149a123a117a143cosα = cosβ = cosγ = 1√3
a117a180
contintegraldisplay
l
ydx+zdy+xdz =
integraldisplayintegraldisplay
S
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
cosα cosβ cosγ
x
y
z
y z x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
=?
integraldisplayintegraldisplay
S
(√3dS =?√3 pia2
(2) a114la164a157a140a27a171a141a80a143σa167a75σa27a123a130a144a149a143(1,1,1)a167a75a217a144a149a123a117a143cosα = cosβ = cosγ = 1√3
a113P = z?y,Q = x?z,R = y?xa167a75?R?yQ?z = 2,?P?zR?x = 2,?Q?xP?y = 2
a117a180
contintegraldisplay
l
(z?y)dx+ (x?z)dy+ (y?x)dz = 2√3
integraldisplayintegraldisplay
S
dS = 3a2
278
§2,a173a130a200a169a218a180a187a27a195a39a53
1,a23a51a44a52a221a47a171a141Da83?P?y =?Q?x a167a193a121
U(x,y) =
integraldisplay x
x0
P(x,y)dx+
integraldisplay y
y0
Q(x0,y)dy+C
a143P dx+Qdya27a6a188a234a167a217a165C = U(x0,y0).
a121a178a181a207U(x,y) =
integraldisplay x
x0
P(x,y)dx+
integraldisplay y
y0
Q(x0,y)dy+C
a75?U?x = P(x,y),?U?y =
integraldisplay x
x0
Py(x,y)dx+Q(x0,y) =
integraldisplay x
x0
Qx(x,y)dx+Q(x0,y) = Q(x,y)
a117a180dU = P(x,y)dx+Q(x,y)dy
a113U(x0,y0) = Ca167a75U(x,y) =
integraldisplay x
x0
P(x,y)dx +
integraldisplay y
y0
Q(x0,y)dy + Ca143P dx + Qdya27a6a188a234a167a217a165C =
U(x0,y0)
2,a79a142a101a15a28a135a169a170a27a130a200a169a181
(1)
integraldisplay (1,1)
(0,0)
(x?y)(dx? dy)
(2)
integraldisplay (a,b)
(0,0)
f(x+y)(dx+ dy)a167a170a165f(u)a180a235a89a188a234a182
(3)
integraldisplay (1,2)
(2,1)
ydx?xdy
x2 a167a247a216a218Oya182a131a2a27a229a187a182
(4)
integraldisplay (0,1,1)
(1,2,3)
yzdx+xzdy+xydz
(5)
integraldisplay (6,8)
(1,0)
xdx+ydyradicalbig
x2 +y2 a167a247a216a207a76a6a58a27a229a187a182
(6)
integraldisplay (1,2)
(2,1)
(x)dx+ψ(y)dya167a217a165?,ψa143a235a89a188a234.
a41a181
(1) a207(x?y)(dx? dy) = d(x?y)
2
2 a167a75
integraldisplay (1,1)
(0,0)
(x?y)(dx? dy) = (x?y)
2
2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,1)
(0,0)
= 0
(2) a207P +Q = f(x+y)a167a75?P?y = fprime(x+y) =?Q?xa167a117a180a108(0,0)a20(a,b)a200a169a134a180a187a195a39
a18(0,0) → (a,0) → (a,b)a167
a75
integraldisplay (a,b)
(0,0)
f(x+y)(dx+ dy) =
integraldisplay a
0
f(x+ 0)dx+
integraldisplay b
0
f(a+y)dy =
integraldisplay a+b
0
f(u)du
(3) a8xnegationslash= 0a158a167ydx?xdyx2 = d
parenleftBigg
yx
parenrightBigg
a167a75
integraldisplay (1,2)
(2,1)
ydx?xdy
x2 =?
y
x
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,2)
(2,1)
=?32
(4) a207yzdx+xzdy+xydz = dxyza167a75
integraldisplay (0,1,1)
(1,2,3)
yzdx+xzdy+xydz = 0
(5) a8(x,y) negationslash= (0,0)a158a167xdx+ydyradicalbigx2 +y2 = dradicalbigx2 +y2a167a75
integraldisplay (6,8)
(1,0)
xdx+ydyradicalbig
x2 +y2 = 9
(6) a100a117?,ψa180a235a89a188a234a167a25a107?(x)dx+ψ(y)dy = d(F(x) +G(x))
a217a165F(x) =
integraldisplay x
2
(u)du,G(y) =
integraldisplay y
1
ψ(v)dv
a117a180a107
integraldisplay (1,2)
(2,1)
(x)dx+ψ(y)dy = (F(x)+G(y))
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(1,2)
(2,1)
=
integraldisplay 1
2
(u)du+
integraldisplay 2
1
ψ(v)dv =
integraldisplay 2
1
[ψ(x)(x)]dx
3,a166a6a188a234ua181
279
(1) (x2 + 2xy?y2)dx+ (x2?2xy?y2)dy
(2) (2xcosy?y2 sinx)dx+ (2ycosx?x2 siny)dy
(3) az dx+ bz dy+?by?axz2 dz
(4) (x2?2yz)dx+ (y2?2xz)dy+ (z2?2xy)dz
(5) ex[ey(x?y+ 2) +y]dx+ex[ey(x?y) + 1]dy
a41a181
(1) a207?P?y = 2x?2y,?Q?x = 2x?2ya167a75?P?y =?Q?x
a117a180u =
integraldisplay x
0
(x2 + 2xy?y2)dx+
integraldisplay y
0
(?y2)dy+C = x
3
3 +x
2y?xy2? y
3
3 +C
(2) a207(2xcosy?y2 sinx)dx + (2ycosx?x2 siny)dy = cosydx2 + x2 dcosy + y2 dcosx + cosxdy2 =
d(x2 cosy+y2 cosx)
a75u = x2 cosy+y2 cosx+C
(3) a207az dx+ bz dy+?by?axz2 dz = azdx?xdzz2 +bzdy?ydzz2 = dax+byz
a75u = 1z (ax+by) +C
(4) a207(x2?2yz)dx+ (y2?2xz)dy+ (z2?2xy)dz = 13 (dx3 + dy3 + dz3)?2(yzdx+xzdy+xydz) =
d
parenleftBigg
x3 +y3 +z3
3?2xyz
parenrightBigg
a75u = 13 (x3 +y3 +z3)?2xyz +C
(5) a207ex[ey(x?y + 2) + y]dx + ex[ey(x?y) + 1]dy = (x?y)ex+y(dx + dy) + 2ex+y dx + d(yex) =
d((x?y)ex+y) +ex+y d(x+y) + d(yex) = d((x?y+ 1)ex+y) + d(yex)
a75u = (x?y+ 1)ex+y +yex +C
4,a8a121a181
P dx+Qdy = 12 xdy?ydxAx2 + 2Bxy+Cy2
(A,B,Ca143a126a234a167a133AC?B2 > 0)a183a220a94a135a181
P
y =
Q
x
a166a181a39a117a219a58(0,0)a27a204a130a126a234.
a121a178a181a207P =? y2(Ax2 + 2Bxy+Cy2),Q = x2(Ax2 + 2Bxy+Cy2)
a75?P?y = Cy
2?Ax2
2(Ax2 + 2Bxy+Cy2)2 =
Q
x
a117a180ω =
contintegraldisplay
x2+y2=1
P dx+Qdy =
integraldisplay pi
2
pi2
dt
2(Acos2t+ 2Bsintcost+Csin2t) =
1√
AC?B2 arctan
Cparenleftbigtant+ BCparenrightbig√
AC?B2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
pi
2
pi2
=
pi√
AC?B2,C > 0
pi√AC?B2,C < 0
5,a121a178a181 integraldisplay
xdx+ydy
x2 +y2
a39a117a219a58(0,0)a27a204a130a126a234a1430a167a108a13xdx+ydyx2 +y2 a27a200a169a134a180a187a195a39.
a121a178a181a207P = xx2 +y2,Q = yx2 +y2 a167a75?P?y =? 2xy(x2 +y2)2 =?Q?x
a117a180ω =
contintegraldisplay
x2+y2=1
P dx+Qdy = 0
280
(1) a101a52a180la216a157a140(0,0)a58a167a140a242a219a58(0,0)a134a171a141Da27a62a46a94a152a94a173a130Ca235a26a229a53a167a117a180a69a235a207a171a141a67a164a10
a252a235a207a171a141
a113?P?y =?Q?x a167a75a100a31a100a94a135a167a26
contintegraldisplay
l
xdx+ydy
x2 +y2 = 0
(2) a101a52a180la157a140a219a58(0,0)a167a207a247a130a55a219a58a27a63a152a52a180a27a200a169a31a117a204a130a126a234a167a75
contintegraldisplay
l
xdx+ydy
x2 +y2 = 0
a111a131xdx+ydyx2 +y2 a27a200a169a134a180a187a195a39.
281
§3,a124a216a208a218
1,a23H(t) = eta +e?tba167a217a165a,ba143a126a149a254a167ta143a235a234a167
(1) a166 dHdt
(2) a121a178 d
2H
dt2 = H
a41a181a207H(t) = eta +e?tba167a,ba143a126a149a254a167a75
(1) dHdt = eta?e?tb
(2) d
2H
dt2 =
d
dt
parenleftBigg
dH
dt
parenrightBigg
= eta +e?tb = H
2,a121a178a181 ddt[A·(B×C)] = dAdt ·(B×C) + A·
parenleftBigg
dB
dt ×C
parenrightBigg
+ A·
parenleftBigg
B× dCdt
parenrightBigg
a121a178a181a23A = Axi +Ayj +Azk,B = Bxi +Byj +Bzk,C = Cxi +Cyj +Czk
a75A·(B×C) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
Ax Ay Az
Bx By Bz
Cx Cy Cz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsinglea167a233a31a170a252a224a166a19a167a109a224a94a233a49a15a170a166a19a123a75a167a26
d
dt[A·(B×C)] =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
Axt Ayt Azt
Bx By Bz
Cx Cy Cz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle+
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
Ax Ay Az
Bxt Byt Bzt
Cx Cy Cz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle+
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
Ax Ay Az
Bx By Bz
Cxt Cyt Czt
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
= dAdt ·(B×C) + A·
parenleftBigg
dB
dt ×C
parenrightBigg
+ A·
parenleftBigg
B× dCdt
parenrightBigg
3,a23a = 3i + 20j?15ka167a233a101a15a234a254a124φa169a79a166a209gradφa57div(φa).
(1) φ = (x2 +y2 +z2)?12
(2) φ = x2 +y2 +z2
(3) φ = ln(x2 +y2 +z2)
a41a181
(1) gradφ = φxi +φyj +φzk =?x
(x2 +xy+y2)32
i +?y
(x2 +xy+y2)32
j +?z
(x2 +xy+y2)32
k
div(φa) = φdiva + gradφ·a = gradφ·a =?3x?20y+ 15z
(x2 +xy+y2)32
(2) gradφ = 2xi + 2yj + 2zk,div(φa) = 6x+ 40y?30z
(3) gradφ = 2xx2 +y2 +z2 i + 2yx2 +y2 +z2 j + 2zx2 +y2 +z2 k,div(φa) = 6x+ 40y?30zx2 +y2 +z2
4,a23U(x,y,z) = xyz
(1) a166U(x,y,z)a51a58P1(0,0,0),P2(1,1,1)a57P3(2,1,1)a63a247b = 2i + 3j?4ka27a144a149a19a234a182
(2) a51a254a227a110a58a63a167U(M)a27a129a140a144a149a19a234a143a219a138a186
(3) a51a254a227a110a58a63a167a166divgradU(M)a57rotgradU(M).
a41a181
(1) a207ba27a144a149a123a117a143cosα = 2√29,cosβ = 3√29,cosγ =? 4√29
a75?U?b = yzcosα+xzcosβ +xycosγ = 1√29 (2yz + 3xz?4xy)
a117a180a51P1(0,0,0)a58?U?b = 0a182a51P2(1,1,1)a58?U?b =
√29
29 a182a51P3(2,1,1)a58
U
b = 0
(2) a207?U?b = gradU ·b0 = |gradU|cos(gradU,b0)a167a217a165b0a180ba144a149a27a252a160a149a254
a75U(M)a27a129a140a144a149a19a234a143|gradU| =radicalbigy2z2 +x2z2 +x2y2
a117a180a51P1(0,0,0)a58|gradU| = 0a182a51P2(1,1,1)a58|gradU| = √3a182a51P3(2,1,1)a58|gradU| = 3
282
(3) a207gradU = yzi +xzj +xyk
a75divgradU =?(yz)?x +?(xz)?y +?(xy)?z = 0
rotgradU =
parenleftBigg
(xy)
y?
(xz)
z
parenrightBigg
i +
parenleftBigg
(yz)
z?
(xy)
x
parenrightBigg
j +
parenleftBigg
(xz)
x?
(yz)
y
parenrightBigg
k = 0
a117a180a51a254a227a110a58a63a167divgradU(M) = 0,rotgradU(M) = 0.
5,a166a149a254a = x2i +y2j +z2ka66a76a165x2 +y2 +z2 = 1,x> 0,y> 0,z> 0a27a54a254.
a41a181Φ =
integraldisplayintegraldisplay
S
x2 dydz +y2 dxdz +z2 dxdya167
integraldisplayintegraldisplay
S
z2 dxdy =
integraldisplay pi
2
0
dθ
integraldisplay 1
0
(1?r2)rdr = pi8
a97a113a47a167a169a79a149XOZ,YOZa178a161a221a75a167a140a26
integraldisplayintegraldisplay
S
y2 dxdz =
integraldisplayintegraldisplay
S
x2 dydz = pi8 a167a117a180Φ = 38 pi.
6,a166a = yzi +xzj +xyka207a76Sa27a54a254a167a23
(1) Sa143a11a206a78x2 +y2 lessorequalslanta2,0 lessorequalslantz lessorequalslantha27a253a161a182
(2) Sa143(1)a165a11a206a78a27a254a46a161a182
(3) Sa143(1)a165a11a206a78a27a76a161.
a41a181
(3)
integraldisplayintegraldisplay
S
an dS =
integraldisplayintegraldisplayintegraldisplay
V
divadV =
integraldisplayintegraldisplayintegraldisplay
V
bracketleftBigg
(yz)
x +
(xz)
y +
(xy)
z
bracketrightBigg
dV = 0
a117a180a149a254aa66a76a11a206a78a76a161a27a54a254a1430
(2)a207a51a11a206a78a27a254a33a101a46a161an = xya167a75
integraldisplayintegraldisplay
Sa254
an dS =
integraldisplayintegraldisplay
x2+y2lessorequalslanta2
xydxdy =
integraldisplay 2pi
0
dθ
integraldisplay a
0
r3 sinθcosθdr = 0
a211a110
integraldisplayintegraldisplay
Sa101
an dS = 0a167a117a180a149a254aa66a76a11a206a78a254a46a161a27a54a254a1430
(1)a207
integraldisplayintegraldisplay
S
an dS =
integraldisplayintegraldisplay
Sa253
an dS +
integraldisplayintegraldisplay
Sa254
an dS +
integraldisplayintegraldisplay
Sa101
an dSa167a75
integraldisplayintegraldisplay
Sa253
an dS = 0.
7,a166a = grad
parenleftBigg
arctan yx
parenrightBigg
a247a173a130la27a130a54a254a181
(1) la143(x?2)2 + (y?2)2 = 1,z = 0a182
(2) la143x2 +y2 = 4,z = 1.
a41a181
(1) a100a174a127a167a107a = grad
parenleftBigg
arctan yx
parenrightBigg
=? yx2 +y2 i + xx2 +y2 j
a75rota =
parenleftBig
x
x2+y2
parenrightBig
x?
parenleftBig
yx2+y2
parenrightBig
y
k = 0(a216x = y = 0a61Oza182a254a27a58)
a207l,(x?2)2 + (y?2)2 = 1,z = 0a180a216a140a55za182a27a173a130a167a25a140a117la254a220a152a173a161Sa167a166Sa134Oza182a216a131a2
a75a226a100a247a142a105a250a170a167a107a130a54a254
contintegraldisplay
l
adl =
integraldisplayintegraldisplay
S
n·rotadS = 0
(2) a207l,x2 + y2 = 4,z = 1a167a100a158la20a208a140a55Oza182a94a61a152a177a167a18a126a234c > 0a191a169a2(c < 2)a167a166la160a117a178
a161z = ca27a254a144a167a51a178a161z = ca254a140a55Oza182a18a152a11a177lr,x2 + y2 = r2,z = ca167ra191a169a2a167a166ra2a1172a167
a177la134lra143a62a46a220a254a152a173a161Sa167a166Sa134Oza182a216a131a2
a100a100a247a142a105a250a170a167a26
contintegraldisplay
l
a· dr +
contintegraldisplay
lr
a· dr =
integraldisplayintegraldisplay
S
n·rotadS = 0a167a217a165?lra76a171a247a94a158a2a144a149
a117a180a130a54a254
contintegraldisplay
l
a· dr =
contintegraldisplay
lr
a· dr
a113a18lra27a235a234a144a167x = rcosθ,y = rsinθ,z = ca167a26
contintegraldisplay
lr
a· dr =
integraldisplay 2pi
0
dθ = 2pi
a108a13
contintegraldisplay
l
a· dr = 2pi.
283
8,a166a149a254a =?yi +xj +ck(ca143a126a234)a27a130a54a254a181
(1) a247a11a177x2 +y2 = 1,z = 0a182
(2) a247a11a177(x?2)2 +y2 = 1,z = 0.
a41a181
(1) a207l,x2 +y2 = 1,z = 0a167a75l = costi + sintj + 0k(0 lessorequalslanttlessorequalslant 2pi)
a117a180a·dl = dta167a108a13a164a166a130a54a254a143
contintegraldisplay
l
a· dl =
integraldisplay 2pi
0
dt = 2pi
(2) a207l,(x?2)2 +y2 = 1,z = 0a167a75l = (2 + cost)i + sintj + 0k(0 lessorequalslanttlessorequalslant 2pi)
a117a180a·dl = (2cost+ 1)dta167a108a13a164a166a130a54a254a143
contintegraldisplay
l
a· dl =
integraldisplay 2pi
0
(2cost+ 1)dt = 2pi
9,a121a178a181
(1) rot(uA) = urotA + gradu×Aa182
(2) div(φa) = φdiva + gradφ·a
(3) graddiva?rotrota =?a
a121a178a181
(1) a207rotx(uA) = u
parenleftBigg
Az
y?
Ay
z
parenrightBigg
+
parenleftBigg
Az?u?y?Ay?u?z
parenrightBigg
= urotxA + (gradu×A)x
a211a123a140a26a167roty(uA) = urotyA + (gradu×A)y,rotz(uA) = urotzA + (gradu×A)z
a117a180rot(uA) = urotA + gradu×A
(2) a207?(φax)?x = φ?ax?x +ax?φ?x,?(φay)?y = φ?ay?y +ay?φ?y,?(φaz)?z = φ?az?z +az?φ?z
a75div(φa) = φdiva + gradφ·a
(3) a207graddiva = grad
parenleftBigg
ax
x +
ay
y +
az
z
parenrightBigg
=
parenleftBigg
2ax
x2 +
2ay
x?y +
2az
x?z
parenrightBigg
i +
parenleftBigg
2ax
x?y +
2ay
y2 +
2az
y?z
parenrightBigg
j +
parenleftBigg
2ax
x?z +
2ay
y?z +
2az
z2
parenrightBigg
k
rotrota = rot
bracketleftBiggparenleftBigg
az
y?
ay
z
parenrightBigg
i +
parenleftBigg
ax
z?
az
x
parenrightBigg
j +
parenleftBigg
ay
x?
ax
y
parenrightBigg
k
bracketrightBigg
=
parenleftBigg
2ay
x?y +
2az
x?z?
2ax
y2?
2ax
z2
parenrightBigg
i+
parenleftBigg
2ax
x?y +
2az
y?z?
2ay
x2?
2ay
z2
parenrightBigg
j+
parenleftBigg
2ax
x?z +
2ay
y?z?
2az
x2?
2az
y2
parenrightBigg
k
a75graddiva?rotrota =?axi +?ayj +?azk =?a
10,a166rot(w×r)a167wa143a126a165a254a167ra143a165a187a149a254.
a41a181a23w = (w1,w2,w3),r = (x,y,z)
a117a180w×r =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
w1 w2 w3
x y z
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= (w2z?w3y)i + (w3x?w1z)j + (w1y?w2x)k
rot(w×r) = 2w1i + 2w2j + 2w2k = 2w
11,a121a178a = yz(2x+y+z)i +xz(x+ 2y+z)j +xy(x+y+ 2z)ka143a2a197a124a167a191a166a217a179a188a234.
a121a178a181a233a152a109a63a152a58(x,y,z)a167a107
rota =
braceleftBigg
y [xy(x+y+ 2z)]?
z [xz(x+ 2y+z)]
bracerightBigg
i
+
braceleftBigg
z [yz(2x+y+z)]?
x [xy(x+y+ 2z)]
bracerightBigg
j
+
braceleftBigg
x [xz(x+ 2y+z)]?
y [yz(2x+y+z)]
bracerightBigg
k
= 0
a75aa143a2a197a124
a100a117a179φa247a118dφ = a·dl = ax dx+ay dy+az dz = d[(xyz(x+y+z)]
a75a217a179a188a234a143u(x,y,z) = xyz(x+y+z) +Ca167a217a165Ca143a63a191a126a234.
284
12,a166a149a254a = ra247a218a130r = acosti +asintj +btk(0 lessorequalslanttlessorequalslant 2pi)a27a152a227a164a138a27a245.
a41a181a207a = xi +yj +zk,l,x = acost,y = asint,z = bt(0 lessorequalslant 2pi)
a75a164a166a27a245a143W =
integraldisplay
l
xdx+ydy+zdz = 2b2pi2
13,a23φa143a140a135a188a234a167a79a142a181gradφ(r),div(φ(r)r)a57rot(φ(r)r).
a41a181gradφ(r) = φprime(r)gradr = φprime(r)· rr
div(φ(r)r) = φ(r)divr + r·gradφ(r) = 3φ(r) +rφprime(r)
rot(φ(r)r) = φ(r)rotr + gradφ(r)×r = 0
14,a166a247a118a94a135div(φ(r)r) = 0a27a188a234φ(r).
a41a181a100a254a75a167a26div(φ(r)r) = 3φ(r) +rφprime(r)
a135a166div(φ(r)r) = 0a167a144a1353φ(r) +rφprime(r) = 0a61a135φ
prime(r)
φ(r) =?
3
r
a75a26φ(r) = cr3 (ca143a126a234)
15,a166a177a101a136a149a254a27a209a221a57a94a221(a,ba143a126a149a254)a181
(1) (a·r)b
(2) a×r
(3) φ(r)(a×r)
(4) r×(a×r)
a41a181
(1) div[(a·r)b] = (a·r)divb + grad(a·r)·b = grad(a·r)·b = a·b
rot[(a·r)b] = (a·r)rotb + grad(a·r)×b = grad(a·r)×b = a×b
(2) a207(a×r)x = ayz?azy,(a×r)y = azx?axz,(a×r)z = axy?ayx
a75div(a×r) =x (ayz?azy) +y (azx?axz) +z(axy?ayx) = 0
rot(a×r) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
i j k
x
y
z
ayz?azy azx?axz axy?ayx
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
= 2a
(3) div[φ(r)(a × r)] = φ(r)div(a × r) + grad(φ(r))(a × r) = φ(r)(r·rota? a·rotr) + (rφprime(r) · (a × r) =
φprime(r)rr ·(a×r) = φ
prime(r)
r [r·(a×r)] = 0rot[φ(r)(a×r)] = φ(r)rot(a×r) + grad(φ(r))×(a×r) =
φ(r)[?(axi +ayj +azk) + 3a] +
parenleftBigg
r
r φ
prime(r)
parenrightBigg
×(a×r) = 2φ(r)a + φ
prime(r)
r [r
2a?(a·r)r]
(4) a207r×(a×r) = |r|2a?(a·r)r
a75div[r×(a×r)] = |r|2diva+grad|r|2·a?[(r·a)divr+r·grad(a·r)] = (ax+ay+az)?4(xax+yay+zaz)
rot[r×(a×r)] = |r|2rotr + grad|r|2 ×a?(r·a)rotr?grad(r·a)×r = 1r (r×a)?a×r.
16,a121a178a177a101a31a170a181
(1) grad(a·b) = a×rotb + b×rota + (b·?)a + (a·?)b
(2) rot(a×b) = (b·?)a?(a·?)b + (divb)a?(diva)b
(3) c·grad(a·b) = a·(c·?) + b·(c·?)a
(4) (c·?)(a×b) = a×(c·?)b?b×(c·?)a
a121a178a181
(1) grad(a·b) = grad(axbx +ayby +azbz) =parenleftBigg
ax?bx?x +bx?ax?x +ay?by?x +by?ay?x +az?bz?x +bz?az?x
parenrightBigg
i +
parenleftBigg
ax?bx?y +bx?ax?y +ay?by?y +by?ay?y +az?bz?y +bz?az?y
parenrightBigg
j +
285
parenleftBigg
ax?bx?z +bx?ax?z +ay?by?z +by?ay?z +az?bz?z +bz?az?z
parenrightBigg
k
= a×rotb + b×rota + (b·?)a + (a·?)b
(2) rot(a×b) =
parenleftBigg
bxx +byy +bzz
parenrightBigg
a?
parenleftBigg
axx +ayy +azz
parenrightBigg
b + (divb)a?(diva)b
= (b·?)a?(a·?)b + (divb)a?(diva)b
(3) c·grad(a·b) =
cx
parenleftBigg
ax?bx?x +bx?ax?x +ay?by?x +by?ay?x +az?bz?x +bz?az?x
parenrightBigg
+
cy
parenleftBigg
ax?bx?y +bx?ax?y +ay?by?y +by?ay?y +az?bz?y +bz?az?y
parenrightBigg
+
cz
parenleftBigg
ax?bx?z +bx?ax?z +ay?by?z +by?ay?z +az?bz?z +bz?az?z
parenrightBigg
= a·(c·?) + b·(c·?)a
(4) (c·?)(a×b) =
parenleftBigg
cxx +cyy +czz
parenrightBigg
[(aybz?azby)i + (azbx?axbz)j + (axby?aybx)k]
=
parenleftBigg
bzcx?ay?x +aycx?bz?x?bycx?az?x?azcx?by?x +bzcy?ay?y +aycy?bz?y?bycy?az?y?azcy?by?y +
bzcz?ay?z +aycz?bz?z?bycz?az?z?azcz?by?z
parenrightBigg
i +
parenleftBigg
bxcx?az?x +azcx?bx?x?bzcx?ax?x?axcx?bz?x +bxcy?az?y +azcy?bx?y?bzcy?ax?y?axcy?bz?y +
bxcz?az?z +azcz?bx?z?bzcz?ax?z?axcz?bz?z
parenrightBigg
j +
parenleftBigg
bycx?ax?x +axcx?by?x?bxcx?ay?x?aycx?bx?x +bycy?ax?y +axcy?by?y?bxcy?ay?y?aycy?bx?y +
bycz?ax?z +axcz?by?z?bxcz?ay?z?aycz?bx?z
parenrightBigg
k
= a×(c·?)b?b×(c·?)a
17,a193a121divgradsin2ra140a76a171a164F(r)a27a47a170a167a191a21a209F(x).
a121a178a181divgradsin2r =?sin2r =?
2 sin2r
x2 +
2 sin2r
y2 +
2 sin2r
z2 =
x
parenleftBigg
sin2r· xr
parenrightBigg
+y
parenleftBigg
sin2r· yr
parenrightBigg
+z
parenleftBigg
sin2r· zr
parenrightBigg
= 2cos2r+ 2sin2r· 1r = 2r (sin2r+rcos2r)
a61F(r) = 2r (sin2r+rcos2r)
a75F(x) = divgradsin2xr = d
2 sin2x
dx2 = 2cos2x
18,a121a178a181a8|a|2 =a126a234a158a167a107(a·?)a =?a×rota.
a121a178a181a207|a|2 =a126a234a167a75grad|a|2 = 0
a10016a75(1)a167a26grad(a·a) = 2[a×rota + (a·?)a] = 0
a75(a·?)a =?a×rota.
