数学分析试题集锦（一）：04

? 1c± D±s x1± àQ#9 ? 1
p?tLy = x2A(1;1)??B(?2;4)?￥MLZ??ELZ?
 2
?S = vt? 12gt2
p
1
t = 1;t = 1+￠t-W￥ü ( ?
!￠t = 1;0:1;0:01


2
t = 1￥ § H ?
 3
 k??wLy = lnx't?￥MLü??/ °L

1
y = x?1

2
y = 2x?3
 4
 !f(x) = 8 < : x2;x ? 3 ax+b;x < 3; k??a;b￥′
 Pf(x)x = 3) V?
 5
p/ wL·??P￥MLZ??ELZ?

1
y = x24 ;P(2;1)

2
y = cosx;P(0;1)
 6
p/ f ?￥?f ?.
1
f(x) = jxj3

2
f(x) = 8 < : x+1; x ? 0; 1; x < 0; 1 7
 !f ?f(x) = 8 < : xm sin 1x; x 6= 0 0; x = 0
m1?? ?

 kù

1
m???′ H
f(x)x = 0 ??

2
m???′ H
f(x)x = 0 V?

3
m???′ H
f0(x)x = 0 ??
 8
 !g(0) = g0(0) = 0
f(x) = 8< : g(x)sin 1x; x 6= 0; 0; x = 0: pf0(0)
 9
￡ ü
?f0(x0)i
5 lim ￠x!0 f(x0 +￠x)?f(x0 ?￠x) 2￠x = f 0(x0) 10
 !f(x) ^?l(?1;+1) ￥f ?
O?ix1;x2 2 (?1;+1)
 μ f(x1 +x2) = f(x1)f(x2): ?f0(0) = 1
￡ ü?ix 2 (?1;+1)
μf0(x) = f(x)
 11
 !f(x) ^}f ?
Of0(0)i
￡ ü
f0(0) = 0
 12
 !f(x) ^f ?
Of0(x0) = 3
pf0(?x0). 13
¨?l￡ ü
 V?￥}f ?￥?f ? ^f ?
 V?￥f ? ￥?f ? ^}f ?
 14
p/ f ?￥?f ?

1
y = x2 sinx

2
y = xcosx+3x2

3
y = xtanx?7x+6
2
4
y = ex sinx?7cosx+5x2

5
y = 4px+ 1x ?2x3

6
y = 3x+5px+ 7x3

7
y = 1+x21?x2

8
y = 11+x+x2

9
y = x(1?x)(2?x)

10
y = 11+px ? 11?px

11
y = 1+ px 1?px

12
y = 13px + 3px

13
y = x3 lnx? 1nxn

14
y = cosxx4 ln 1x

15
y = (x+ 1x)lnx

16
y = xcosx?lnxx+1

17
y = 1x+cosx

18
y = xsinx+cosxxsinx?cosx

19
y = xex?1sinx

20
y = xsinxlnx
 15
p/ ˉ?f ?￥?f ?

1
y = (x3 ?4)3
3
2
y = x(a2 +x2)pa2 ?x2

3
y = xpa2?x2

4
y = 3 q 1+x3 1?x3

5
y = ln(lnx)

6
y = 12 lnflfla+xa?xflfl

7
y = ln(x+pa2 +x2)

8
y = lntan x2

9
y = cos(cospx)

10
y = cos3 x?cos3x

11
y = 1p2…e?3x2

12
y = arcsin(sinxcosx)

13
y = arctan 2x1?x2

14
y = e?x2+2x

15
y = ln q (x+2)(x+3) x+1

16
y = e2x sin3x+ x22

17
y = e?kx sin!x1+x (k;!1è ?

18
y = xpa2 ?x2 + xpa2?x2

19
y = sinn xcosnx

20
y = ln p1+x?p1?x p1+x+p1?x
 4 16
¨ ?p?Ep/ f ?￥?f ?

1
y = x q 1?x 1+x

2
y = x21?x q 1+x 1+x+x2

3
y = (x+p1+x2)n

4
y = xx;(x > 0)

5
y = xlnx;(x > 0)

6
y = (1+x)1x;(x > 0)

7
y = xtanx;(x > 0)

8
y = asinx;(a > 0)
 17
 !f(x) ^x Vp?￥f ?
pdydx

1
y = f(x2)

2
y = f(ex)ef(x)

3
y = f(f(f(x)))
 18
 !’(x)??(x) ^x Vp?￥f ?
pdydx

1
y = p’2(x)+?2(x)

2
y = arctan ’(x)?(x)(?(x) 6= 0)

3
y = ’(x)p?(x)(?(x) > 0;’(x) 6= 0)

4
y = log’(x) ?(x) (?(x) > 0;’(x) > 0;’(x) 6= 1)
 19
p/ f ?￥?f ?
 5
1
y = eax(cosbx+sinbx)

2
y = xarctanx? 12 ln(1+x2)

3
y = arctan p1?x2?1 x +arctan 2x 1?x2

4
y = arctan(tan2 x)

5
y = (ab)x(bx)a(xa)b(a;b > 0)

6
y = x2pa2 ?x2 + a22 arcsin xa(a > 0)

7
y = x2pa2 +x2 + a22 ln flfl flx+ pa2 +x2flfl fl(a > 0)

8
y = ln(arccos 1px)

9
y = xaa +axa +aax(a > 0)

10
y = 16 ln (x+1)2x2?x+1 + 1p3 arctan 2x?1p3
 x2±sàQ#9 ? 1
p/ f ?·??￥±s

1
y = anxn +an?1xn?1 +￠￠￠+a1x+a0
pdy(0);dy(1)

2
y = secx+tanx
pdy(0);dy(…4)?dy(…)

3
y = 1a arctan xa
pdy(0);dy(a)

4
y = 1x + 1x2
pdy(0:1);dy(0:01)
 2
p/ f ?￥±s

1
y = x1?x2

2
y = xlnx?x
6
3
y = px+lnx? 1px

4
y = arcsinp1?x2

5
y = esinx2

6
y = lnflfltan(x2 + …4)flfl
 3
 !u;v ^x￥ V±f ?
pdy

1
y = arctan uv

2
y = lnpu2 +v2

3
y = lnsin(u+v)

4
y = 1pu2+v2
 4
p/ f ?￥±sdy

1
y = sin2 t;t = ln(3x+1)

2
y = ln(3t+1);t = sin2 x

3
y = e3u;u = 12 lnt;t = x3 ?2x+5

4
y = arctanu;u = (lnt)2;t = 1+x2 ?cotx
 x3?f ?D? ?Z?±sE 1.p/ ?f ?￥? ?dydx

1
x2a2 + y2b2 = 1(a;b1è ?)

2
y2 = 2px(p1è ?)

3
x2 +xy +y2 = a2(a1è ?)
7
4
x3 +y3 ?xy = 0

5
y = x+ 12 siny

6
x23 +y23 = a23(a1è ?)

7
y = cos(x+y)

8
y = x+arctany

9
y = 1?ln(x+y)+ey

10
arctan yx = lnpx2 +y2
 2
p/ ? ?Z?￥? ?

1
 8 < : x = t1+t y = 1?t1+t

2
 8 < : x = sin2 t y = cos2 t

3
 8 < : x = e2t cos2 t y = e2t sin2 t

4
 8< : x = a(lntan t2 +cost) y = asint
 3
pf ?y = y(x)·??￥? ?

1
y = cosx+ 12 siny;(…2;0)

2
yex +lny = 1;(0;1)

3
 8 < : x = t?sint; y = 1?cost; t = …2;…)
8
4
 8 < : x = 1?t2; y = t?t3; t = p2 2 ; p3 3)
 4
B????
 '110m
  ?￥????14m.
1
9? ￡ H
p ￡￥8V ￡ ?úh￥M? q

2
p8V?? ????R￥M? q
 5
 !x = acos3 t;y = asin3 t

1
py0(t)

2
￡ üwL￥ML$USà ??￥é1B?è ?
 6
￡ ü
wL 8 < : x = a(cost+tsint) y = a(sint?tcost) ?B?￥EL?e?￥  ? ???a. x4ú¨± Dú¨±s 1.p/ f ?·??￥ú¨? ?

1
f(x) = 3x3 +4x2 ?5x?9
pf00(1);f000(1);f(4)(1)

2
f(x) = xp1+x2;pf00(0);f00(1);f00(?1)
 2
p/ f ?￥ú¨? ?

1
y = xlnx
py00

2
y = e?x2
py000

3
y = x2e2x;py(n)

4
y = arcsinxp1?x2
py(n)
9
5
y = x5 cosx
py(50)

6
y = x3ex?e?x2
py(30)
 3
p/ f ?￥n¨? ?

1
y = ax

2
y = lnx
 4
p/ f ?￥n¨? ?

1
y = 1x(1?2x)

2
y = sin2 x

3
y = 1x2?2x?8

4
y = exx

5
y = ln x+21?x

6
y = 2x lnx
 5
 !f(x)￥ò¨? ?i
py00#y000

1
y = f(x2)

2
y = f(1x)

3
y = f(e?x)

4
y = f(lnx)

5
y = f(f(x))
 6
?f(x) = 8< : e? 1x2;x 6= 0 0;x = 0
￡ üf(n)(0) = 0
 10 7
p/ f ?￥=¨±s

1
y = 1px

2
y = xarctanx

3
y = f(u) = eu;u = ’(x) = x2
 8
p/ f ?￥?¨±s

1
 !u(x) = lnx;v(x) = ex;pd3(uv);d3(uv)

2
 !u(x) = ex2;v(x) = cos2x
pd3(uv);d(uv)
 9
p/ ? ?Z?￥=¨? ?

1
 8 < : x = 2t?t2 y = 3t?t3

2
 8 < : x = acost y = asint

3
 8 < : x = a(t?sint) y = a(1?cost)

4
 8< : x = et cost y = et sint

5
 8< : x = acos3 t y = asin3 t

6
 8 < : x = f0(t) y = tf0(t)?f(t)
 10
p/ ?f ?￥=¨? ?d2ydx2
 11
1
ex+y ?xy = 0

2
x3 +y3 ?3axy = 0

3
y2 +2lny ?x4 = 0
 11
 !f ?y = f(x)?x=¨ V?
Of0(x) 6= 0
?f(x)iQf ?x = f?1(y)
 kp(f?1)00(y)
 12
 !y = c1 sinx+c2 cosx
￡ üy ?@Z?y00 +y = 0
 13
 !y = arctanx

1
￡ üy ?@Z?(1+x2)y00 +2xy0 = 0

2
py(n)(0)
 14
 !y = y(x)iQf ?
O ?@Z? d2y dx2 +( dy dx) 3 = 0 ￡ ü
Qf ?x = x(y) ?@d2xdy2 = 1
iO?NpB?y = y(x)
 12