2005
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TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
$?s
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à
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à
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￥ p?
T#1? b
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?l# M??
T
?l
!f
)(xfy = ? ￥
#×
=μ?l
0
x
0
xxx?=? )()(
00
xfxxfy+=?
)(
)()(
limlim
0
00
00
xf
x
xfxxf
x
y
xx
′
=
+
=
→?→?
0
0
0
0
)()(
lim)(
xx
xfxf
xf
x
=
′
→?
?
)(
0
xf
′
￥+il ML|
q b
?N?
í
)(
)(
0
xA
x
xf
+=
α
O )(
0
xfA
′
= b ? )( x?α
^ 0→?x
H￥í kl
b éB? V?1
xxxxfxf+?
′
=? )()()(
00
α
)()()()(
00
xxxfxfxf?+?
′
+= β
? xxx=? )()( αβ 1 0→?x
H￥ú¨í kl
b
?
?l￥
í
V[ ?Z ?31
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!·° Iù
T? www.tsinghuatutor.com - 1 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)(
)())((
lim)(
00
0
0
x
xfxxf
xf
x
+
=
′
→?
α
α
?l ?T
x
xfxxf
x
y
xx
+
=
→?→?
)()(
limlim
00
00
i 5?N ′1 )(xf )￥P?
:1 ?T
0
x )(
0
xf
′
x
xfxxf
x
y
xx
+
=
++
→?→?
)()(
limlim
00
00
i 5?N ′1 )(xf )￥·?
:1 b
0
x )(
0
xf
+
′
A ?? Ki ￥ 1Hq )(xf ) V?￥ sA1Hq
^
0
x )(xf )
￥P a·?
?i OM?
0
x
=
′
)(bf )(af
+
′
b
? á
ì
a )(xf > uW
V?
H
^·],[ ba )(xf
=
B?? V
?i O
),( ba
)(af
+
′
D )(bf
′
(i b
è
=+?+
∞→
)]
1
1ln(sin)
3
1ln([sinlim
xx
x
x
b
}3 ~
7
x
t
1
= 5
e K"
t
tt
t
)1ln(sin)31ln(sin
lim
0
+?+
→
2|])1ln(sin)31ln([sin
0
=
′
+?+=
=t
tt b
è ? kaf =
′
)( i 5
￡
!·° Iù
T? www.tsinghuatutor.com - 2 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
=
+∞→
)()
1
(lim af
h
afh
h
b
" k? b
#k b
$ b
%?i b 0
}3 ~
+∞→
)()
1
(lim af
h
afh
h
h
af
h
af
h
1
)()
1
(
lim
=
+∞→ t
aftaf
t
)()(
lim
0
+
=
→
.)()( kafaf?=
′
=
′
=
?Ka¨?
?
i ￥ 1Hq P·?
i OM?yN?ê
" b
è
!
≤?
>
=
0),1(
2
0,
1
arctan
)(
sin
xe
x
x
x
xf
x
π
)
)(xf ￥ V±? ? V± p )(xf
′
i)
??? b
}3 ~
n5 )(xf ) ?? b ??f
V??￥2
o? )
0=x )(xf
)￥ V±?1N I
n K 0=x
2
1
arctan
lim)0(
0
π
==
′
+
→
+
x
x
x
f
x
i
2
1
lim
2
)0(
sin
0
ππ
=
=
′
→
x
e
f
x
x
)0(
+
′
f
￡
!·° Iù
T? www.tsinghuatutor.com - 3 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
yN )(xf ) V±2
1 0=x
)(xf
)) V± b ),( +∞?∞
<
=
>
+
=
′
0,
2
cos
0,
2
0,
)1(2
1
arctan
)(
sin
2
3
xe
x
x
x
x
x
x
xf
x
π
π
)0(
2
)(lim
0
fxf
x
′
==
′
→
π
?
^ )(xf
′
0=x ) ?? b 2
1
)) ?? b )(xf ′
è
! 0)0( =f 5 )(xf 0=x ) V?￥ 1Hq1
b
"
)cos1(
1
lim
2
0
hf
h
h
→
b #
)1(
1
lim
0
h
h
ef
h
→
b
$
)(
1
lim
3
2
0
hf
h
h→
b %
)]()2([
1
lim
0
hfhf
h
h
→
b
}3 ~s??1 #
y1 #
￥ Ki ?N? K
h
e
e
ef
h
h
h
h
→
1
1
)1(
lim
0
h
e
e
fef
h
h
h
h
h
=
→→
1
lim
1
)0()1(
lim
00
￡
!·° Iù
T? www.tsinghuatutor.com - 4 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
i 7 1lim
1
lim
0
=
=
→
h
h
h
e
h
h
?i # K
h
h
h
e
fef
→
1
)0()1(
lim
0
i ?:
h
eh?=1)(α 5 )(hα t?
,￥Z
T
^ ?i￥ b V?)0(f
′
i i Oμ
)0(
1
)0()1(
lim
0
f
e
fef
h
h
h
′
=
→
b
"
?
^y1
H0→h
+
→ 0
2
h ???
?l b I
n $
K
=
→
)(
1
lim
3
2
0
hf
h
h
)(
1
lim
3
3
0
hf
h
h→
=
2
3
0
lim
h
h
h
→
0)(
1
lim
3
3
0
=
→
hf
h
h
i ?
￡
)(
1
lim
3
3
0
hf
h
h→
￥i y N $
g? bà ?ê [ %
KVr
T? ?
)0(f ? K
E5 I
n
))]0()(())0()2([(
1
lim
0
fhffhf
h
h
→
￥i ?
￡
h
hf
h
)2(
lim
0→
h
hf
h
)(
lim
0→
￥i ?
[ %
g? b
￡
!·° Iù
T? www.tsinghuatutor.com - 5 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
è
!
≤
>
=
0)(
0
cos1
)(
2
xxgx
x
x
xf ? )(xg
^μ?f
5 )(xf
)μ
% b 0=x
"
K?i b #
Ki ?? ?? b
$
???? V? b %
V? b
}3 ~
n5 I° )￥P· K b 0=x
0
2
lim
cos1
lim)(lim
2
000
==
=
+++
→→→ x
x
x
x
xf
xxx
y10)(lim)(lim
2
00
==
→→
xgxxf
xx
)(xg μ?
yN 0)0()(lim
0
==
→
fxf
x
# )(xf 0=x ) ?? b I °
)￥P·?
^?i b 0=x
0)(lim
)0()(
lim
00
==
→→
xxg
x
fxf
xx
x
fxf
x
)0()(
lim
0
+
→
0
2
lim
cos1
lim
2/3
2
00
==
=
++
→→
x
x
xx
x
xx
yN )0(
+
′
f D )0(
_
f
′
(i OM? b
?
^ )(xf ) V? O0=x 0)0( =
′
f
s?1 %
b
?f
B? V? %?￥f
??é
￡
!·° Iù
T? www.tsinghuatutor.com - 6 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
?é ? )(xf ) V?
H
0
x )(xf A ?i ) ?? b ?A? ?i?
0
x )(xf
) ??
H ??B? ) V? b
0
x
0
x
?é
!f
)(xf ?? O 0)0(' >f 5 i 0>δ
P¤ ? i￥
),0( δ∈x μ )0()( fxf > ?i￥ )0,( δ?∈x μ
)0()( fxf < b
￡ ? 0
0
)0()(
lim)0(
0
>
=
′
→
x
fxf
f
x
5? K ?? Vw
i 0>δ
P? )0,( δ?∈x ),0( δ∈x
H
0
0
)0()(
>
x
fxf
' )0()( fxf? D x ? ?]|yN ?i￥ ),0( δ∈x μ
)0()( fxf > ?i￥ )0,( δ?∈x μ )0()( fxf < b
? o?B?)￥?
?μ| ?
%?f
￥9h? bf
￥9h?
? uW
? ?é b
è
! )0(f
′
i 1)0( <f
?
e
x
xf
x
x
=
+
→
1
0
)
sin
)(cos1
1(lim
5 )0(f
′
=
b
"
b #
b $
0 1 2 b %
e b
<3 > s? $ b? e
x
xf
x
x
=
+
→
1
0
)
sin
)(cos1
1(lim
V[??? 0→x
H
μ
1)
sin
)(cos1
1ln(
1
lim
0
=
+?
→
x
xf
x
x
￡
!·° Iù
T? www.tsinghuatutor.com - 7 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
1
sin
)(cos11
lim
0
=
→
x
xf
x
x
y1 1)0( <f 5Aμ )0(0)(lim
0
fxf
x
==
→
?
^ 1
)(
lim
2
1
sin
)(cos11
lim
2
2
00
==
→→
x
xf
x
xf
x
xx
? V¨
ArE5
?y1 i
[ )0(f ′
2
)(
lim
)(
lim)]0([
00
2
=?=
′
→→
x
xf
x
xf
f
xx
¤? 2)0( =
′
f b
è
! ?
#×
= V ? O?)(xf 0=x 0≠x
H 0)( ≠xf X ?
2)0(,0)0( =
′
= ff p K
x
x
xf
sin
1
0
))(21(lim?
→
b
3
p K1 o p?
!E ?¨S KiD?
∞
1 2)0( =
′
f M ó" b
x
x
xf
sin
1
0
))(21(lim?
→
x
xf
xf
x
xf
sin
)(2
)(2
1
0
))(21(lim
→
=
?ˉ K? ?o? I
n K
x
x
x
xf
x
xf
xx
sin
)(2
lim
sin
)(2
lim
00
=
→→
? 2)0(,0)0( =
′
= ff i #
K V ?¨ K￥eE
p¤'μ
]
sin
lim[]
)0()(
lim[2
sin
)(2
lim
000
x
x
x
fxf
x
xf
xxx →→→
=
￡
!·° Iù
T? www.tsinghuatutor.com - 8 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
4)0(2?=
′
= f
?
^
4
sin
1
0
))(21(lim
→
=? exf
x
x
b
? ?¨?
?l p
t K
^B ?×15? ?
?
?l￥ K/?
T i?i ?¨ˉ
K? ?DX?×1 K￥2
b
±sà
QDMM
q
±sà
Q
??
￥?N?
í
á
ìXü?? V?f
)(xf )￥9
0
x
)()()(
000
xfxxfxf+=? V[V
U1
xxxxfxfxxf+
′
=+ )()()()(
000
α
? )( x?α 1
H￥í kl
b ?:0→?x xxx=? )()( αβ 5
)( x?β
^ ￥ú¨í kl
b?
^? V:1 x?
)()()()(
00
xxxfxfxf?+?
′
=? β
?l
!f
)(xfy = ? ￥
#×
=μ?l b ?i è
P¤f
9
0
x A
y? V[V1 )( xoxAy?+?=?
? AD í1
^x? )( xo? 0→?x
H￥ú¨ í kl
5?f
)(xfy = ? ) V± :1
0
x xAdy
x
=
0
)(
0
xdf= bf
￥±s
Yè:1 dxxfdxydydf )(
′
=
′
==
MM
q
?l
! 1 V?f
? K )(xfy =
￡
!·° Iù
T? www.tsinghuatutor.com - 9 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)(
)()(
limlim
00
00
xf
y
x
x
xfxxf
y
x
x
x
y
y
xx
′
=
+
=
→?→?
1 y ￥MM
q b x
ü6
?? ?l3 pf
)(PfQ = ? P1?ê
￥ Nì b3 p Nì ￥
MM
q1 )(Pf
Q
p
E
d
′
= T1Nì3 pQ?￥B?
|MM
q￥ '
′?l 1??
3 pN ì )(Pf
Q
p
E
d
′
= b
lmf
? l1
)(PPfPQR == b
?f
￥?
D±s
T
?
D±s
15
5
?T )(),( xgxf ? )?μ?
5 aμ a a
s
?1
,
H
?x x) (
μ?
O V± b
)()(])()([ xgxfxgxf
′
±
′
=
′
±
)()()()(])()([ xgxfxgxfxgxf
′
+
′
=
′
)(
)()()()(
]
)(
)(
[
2
xg
xgxfxgxf
xg
xf
′
′
=
′
E )(d)(d))()(( xgxfxgxf ±=±
E (x)d)((x)d)())()(( fxggxfxgxf +=
E (x)d))(( fcxcf = c1è
)(
)(d)((x)d)(
)(
)(
2
xg
xgxffxg
xg
xf
d
=?
￡
!·° Iù
T? www.tsinghuatutor.com - 10 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)()()(])()()([ xhxgxfxhxgxf
′
+
′
+
′
=
′
++
])()()([
′
xhxgxf
)()()()()()()()()( xhxgxfxhxgxf
′
+
′
+xhxgxf
′
=
è
! )(xyy = ?i? ),( +∞?∞∈x
@
)(
1
2
xox
x
y
y?+?
+
=? ?
4
)1(
π
π ey = 5
=)3(y
3
π
π e b
3 ?X??
T ?i ? ? 5μ),( +∞?∞∈x )(xf ),( +∞?∞
= V?
O
2
1 x
y
y
+
=
′
2
1 x
dx
y
dy
+
= sa¤?
Cxy lnarctanln += ?
x
Cey
arctan
=
?
4
)1(
π
π ey = ?
^ π=C b?
^
x
ey
arctan
π=
3
)3(
π
π ey = b
è
!f
( )xf ( )+∞∞?,
μ ?l u W [ ]2,0
() ( )4
2
= xxxf
? ?i￥ x ?
@ ( ) ( )2+= xkfxf ? k
1è
b
1 ( )xf [ )0,2?
￥Vr
T
2 ù 1′
H ) V? b k ()xf 0=x
3
? '02 <≤? x 220 <+≤ x
H
￡
!·° Iù
T? www.tsinghuatutor.com - 11 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
( ) ( )2+= xkfxf
()()[ ] ( )( )42422
2
++=?++= xxkxxxk
?5
!? ( ),00 =f
()
() ( ) ( )
,4
4
lim
0
0
lim0'
2
00
=
=
=
++
→→
+
x
xx
x
fxf
f
xx
()
() ( )
0
0
lim0'
0
=
→
x
fxf
f
x
()( )
k
x
xxkx
x
8
42
lim
0
=
++
=
→
7 ¤ b'() ()00
''
+?
= ff
2
1
=k
H ( )xf 0=x ) V? b
ú¨?
9
?Tf
)(xfy = ￥?f
)(xy
′
ˉμ?
])([
′′
xf 5?
])([
′′
xf 1 )(xfy = ￥=¨?
:1
2
2
),(,
dx
yd
xfy
′′′′
2
2
dx
fd
b
B?ü ￥?
?1)(
)1(
xf
n?
)(xf ￥ ¨?
:1n
n
n
nn
dx
yd
xfy ),(,
)()(
n
n
dx
fd
b
n¨?
?l1
x
xfxxf
xf
nn
x
n
+
=
→?
)()(
lim)(
)1()1(
0
)(
f
￥=¨
#=¨[
￥ò¨?
d?ú¨?
b
￡
!·° Iù
T? www.tsinghuatutor.com - 12 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
? ú¨?
￥?là pú¨?
o3??
￥'
TD
E5?¨é?9
b
è p 1è
￥ò¨?
b λ
λ
(
x
ey =
3 ,,,
2
L
xx
eyey
λλ
λλ =
′′
=
′
xnn
ey
λ
λ=
)(
b
è p ￥ò¨?
b xy sin=
3
+==
′
2
sincos
π
xxy
+=
+=
′′
2
2sin
2
cos
ππ
xxy
l l l l l
)
2
sin(
)(
π
nxy
n
+= b
?
?f
)(xf D )(xg eé ?ú¨?
Hè ¨?/
? ? L
!
)(xf )(xg (μ ¨?
?1 ??
DG
T n
)()()]()([
)()(
0
)(
xgxfCxgxf
kknk
n
n
k
n?
=
∑
=
!
)1()1(
!)!(
!
k
knnn
kkn
n
C
k
n
+
=
=
L
i?
b )()(
)0(
xfxf = )()(
)0(
xgxg =
è ￥ ¨?
^ xxy sin
2
=
3 ? ??
DG
T?i? ￥ ¨?
(1
,5μ
2
x 3≥n
￡
!·° Iù
T? www.tsinghuatutor.com - 13 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)99(2)100(2)100(
)(sin)(100)(sin xxxxy
′
+=
)98(2
)(sin)(
!2
99100
xx
′′
×
+
++
+=
2
99
sin200
2
100
sin
2
ππ
xxxx
+×+
2
98
sin99100
π
x
xxxxx sin9900cos200sin
2
= b
è p
22
1
ax
y
= ￥ ¨?
b n
3
)()(
22
)(
11
2
11
nn
n
axaxa
ax
y
+
=
=
+
=
++ 11
)(
!)1(
)(
!)1(
2
1
n
n
n
n
ax
n
ax
n
a
+
=
++ 11
)(
1
)(
1
2
!
)1(
nn
n
axax
a
n
è )32ln()( xxf?= ￥ ¨?
^
b
￡
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L 1101 è 62781785
2005
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!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
"
10
10
)32(
!103
x?
#
10
10
)32(
!93
x?
$
10
10
)32(
!103
x?
%
10
10
)32(
!93
x?
b
3 s?1 %
bo??i?
￥Q
Q a¨e￥2T+ ￥Z
a' V b
ˉf
p?E5D±sE
? ? ?T )(xu?= ? x)μ?
)();( ufyx
dx
du
=
′
=? ?
? ))(( xuu?= ) 9 μ?
)(uf
du
dy
′
= 5 ˉf
)]([ xfy?=
? x)9μ?
O
dx
du
du
dy
dx
dy
= )()(})]([{ xufxf
′
′
=
′
è p
x
ay
1
arctan
= D
)1ln(
2
++= xxy ￥?
b
3 )(
1
1
ln)(
2
2
1
arctan
1
arctan
+
=
′
x
x
aaa
xx
x
a
x
a
1
arctan
2
1
ln
+
=
￡
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T? www.tsinghuatutor.com - 15 - bvD ? S
L 1101 è 62781785
2005
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!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)2
12
1
1(
1
1
])1[ln(
22
2
x
xxx
xx
+
+
++
=
′
++
1
1
2
+
=
x
è pf
￥?
b )0(
sin
>= xxy
x
3 Z E
? ?f
?S
a·f
b
n5
H |
¤?f
xxy lnsinln = b
??f
p?E¤
x
x
xxy
y
sin
lncos
1
+=
′
V7 ]
sin
ln[cos
sin
x
x
xxxy
x
+=
′
b
??5 |
p ?￥ ZE? S |
p? E b"
a¨ ?
a ·f
?cμ ?y
TMe"{eZ a 7Z￥f
9
a¨ b
)(
)(
xv
xuy =
ZE
|
a·f
?1a ¨ˉf
p ?
E5#eE
T
xxx
exy
lnsinsin
==
è
!
1
lnarctan
2
2
+
=
x
x
x
e
e
ey
5
1
1
2
1
+
=
=
e
e
dx
dy
x
b
3 |f
Vr
T?1
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T? www.tsinghuatutor.com - 16 - bvD ? S
L 1101 è 62781785
2005
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!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)1ln(
2
1
arctan
2
++?=
xx
exey 5
1
1
1
2
2
2
+
+?
+
=
′
x
x
x
x
e
e
e
e
y
1
1
1
2
2
2
1
+
+?
+
=
=
e
e
e
e
dx
dy
x
1
1
2
+
=
e
e
b
ˉf
p?9
H 31? ??WM
üf
s3?B1X??
￥f
¨ˉ
f
p?E5 Ka1ü? ?￥?WM
¨1M
￥f
9} b ?
1g?
ˉf
￥s3
p?E5a V[?? ??WM
:| o1T???μ
s3Ba p?BQ °?K?
1M
1? b
ˉf
￥±sE5 B¨±s?
T?M?
b
! )(ufy = V±?u 11M
H )(ufy = ￥ ± s
uufdy d)(
′
= b
? u1?WM
u
^M
￥ V±f
x )(xu?=
H5 )]([ xfy?= ￥±s
uufxxufxxfy d)(d)()(d})]([{d
′
=
′′
=
′
= Vní
^1M
^?WM
f
u )(ufy = ￥±s?
T?M b
?f
p?ED±sE
? )(xyy = ?B? ?f
Z ? 0),( =yxF ?? 5 V
j1
0))(,( ≡xyxF °¤ ?¨ˉf
p?E5é? p?
3 ' V b
x
y′
/
è
a
ü p?f
￥=¨?
￥ZE b
è
! p4
22
=++ yxyx y
′′
b
3
!Xü
??￥f
4
22
=++ yxyx )(xyy = } ?Z?5¤?
T 4)()(
22
=++ xyxxyx
￡
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T? www.tsinghuatutor.com - 17 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
Z?
H1? p?¤ x
022 =
′
+
′
++ yyyxyx
3¤
yx
yx
y
2
2
+
+
=
′
b
H1? Q p?¤ x
02)(22
2
=
′′
+
′
+
′′
+
′
+
′
+ yyyyxyy
3
yx
yy
y
2
)(222
2
+
′
+
′
+
=
′′
| y
′
￥Vr
T} ?i? ?¤
33
22
)2(
24
)2(
)(6
yxyx
yxyx
y
+
=
+
++
=
′′
b
è
! )(xyy = ?Z?
1ln =+ yxy
?? p wL )(xyy =
)￥ELZ? b 1=x
3 ?X?Z?
7 5 μ1=x 1ln =+ yy A ?μ 1=y b ?X?Z?
HsY
1? x p?
0
1
=
′
++
′
xx
y
y
yyx
7 1,1 == yx ?N V¤? 012
1
=+
′
=xx
y
yN
2
1
1
=
′
=xx
y
[ )(xyy =
)EL|
q 2=k bE LZ?1
)1(21?=? xy
12?= xy b
Qf
D?
Z?￥ p?E5
￡
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T? www.tsinghuatutor.com - 18 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
? ? Qf
p?E5
? )(yx?=
uW
=?? a V? O 0)( ≠
′
y? 5
Qf
)(xfy = ?￥ uW
=9 V? O
)(
1
)(
y
xf
′
=
′
b
? ? ú￥Qf
àμ?Me ?f
￥M )(xfy =
:| b
è
! 1??f
1 Qf
O)(xf )(xg
3
1
)1(,2)1(?=
′
= ff
1)1( =
′′
f
p )2(g
′′
p
x
fxf
x
2
)1()1(
lim
0
→
b
<3 >
h?i )(xg 1 )(xf ￥Qf
￥Hq?Xü?M
M
:|
1 ?¨Qf
?
T?| )(xg ^1 )(yg ? )(xfy = b
?Qf
?
T V¤? 1)()( =
′′
ygxf
H1? x Q p? 0)()()()( =
′′′′
+
′′′
x
yygxfygxf
0)()]([)()(
2
=
′′′
+
′′′
ygxfygxf
7 ?μ1=x 2=y b?i?
3
)1(
1
)2(?=
′
=
′
f
g yN¤?
33)2( =
′′
g b
6
3
)1(
2
1
2
)1()1(
lim
0
=
′
=
→
f
x
fxf
x
b?
^¨ ?
?
?l b
￡
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L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
? ? ?
Z? ??￥f
￥ p?E5
? Tttyytxx ∈== ),(),( ?
V? O 0)( ≠
′
t? 5??
Z?
=
=
),(
),(
tyy
txx
Tt∈
??￥f
9 V? O
)(
)(
tx
ty
x
y
dx
dy
t
t
′
′
=
′
′
= b
?+Y?i
)(
)(
2
2
tx
ty
dx
yd
′′
′′
≠ ? ?
T1
)(
1
)]([
)()()()(
)(
22
2
tx
tx
txtytxty
x
y
dt
d
dx
yd
t
t
′
′
′′′
′′′
=
′
′
=
3
)]([
)()()()(
tx
txtytxty
′
′′′
′′′
=
è p?L
=
=
),cos1(
),sin(
tay
ttax
2
π
=t )￥ MLZ? b
3 ??
t
t
ta
ta
x
y
dx
dy
t
t
cos1
sin
)cos1(
sin
=
=
′
′
= )2( πkt ≠
[?L
2
π
=t )￥ ML|
q1
￡
!·° Iù
T? www.tsinghuatutor.com - 20 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
1
cos1
sin
22
=
=
==
ππ
tt
t
t
dx
dy
?L
??
2
π
=t ￥?
^
aa,1
2
π
#
p￥ MLZ?1 axay
=? 1
2
π
' 0
2
2 =
+? ayx
π
b
è
! tbytax sin,cos == p
x
y
′′
b
3 t
a
b
ta
tb
x
y
y
t
t
x
cot
sin
cos
=
=
′
′
=
′
)cos(
)cot(
)(
′
′
=
′
′′
=
′′
ta
t
a
b
x
y
y
t
tx
x
ta
b
ta
t
a
b
32
2
sin
1
sin
sin
1
=
=
è
!
3
1
)(
+
=
x
x
xf p
( )
)(1
10
f
3 1ZL9
5 Sí) ? ?/ )(xf
3
4
1
3
43
3
1
)(
+
=
+
+
=
+
=
xx
x
x
x
xf
￡
!·° Iù
T? www.tsinghuatutor.com - 21 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
11
11
)10(
)3(
!9)1(4
)(
+
=
x
xf
911
)10(
2
!9
2
!94
)5( =
=?f
è
! 0>δ )(xf ],[ δδ?
μ?l 1)0( =f
O
@
0
1
)(sin)1ln(
lim
2
0
=
+?
→
x
x
e
xfxx
1 ￡
ü K
1
sin)(sin
lim
2
0
→
x
x
e
xxfx
i i pN K′ b
2 ￡
üf
)(xf ) V±i p0=x )0(f
′
b
<3 >
ZE
n5
x
xf
e
xxfx
x
x
x
1)(
lim
1
sin)(sin
lim
00
2
=
→→
b
?X?
1
)(sin)1ln(
lim
2
0
+?
→
x
x
e
xfxx
2
0
)(sin)1ln(
lim
x
xxfxxx
x
++?
=
→
+
+?
=
→
2
0
)1ln(
lim
x
xx
x
2
0
)(sin
lim
x
xxfx
x
→
x
x
x
xf
x
x
x
x
xxx
sin
)(
lim
sin
lim
2
1
1
1
lim
000
+
+
=
→→→
0
sin
11)(
lim
22
1
lim
00
=
+?
+
=
→→
x
x
x
xf
x
xx
￡
!·° Iù
T? www.tsinghuatutor.com - 22 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
1
sin
1
lim
1)(
lim
22
1
lim
000
x
x
x
x
xf
x
xxx
+
+
=
→→→
xx
xx
x
xf
xx
sin
sin
lim
1)(
lim
2
1
00
+
+?=
→→
2
3
00
6
1
lim
1)(
lim
2
1
x
x
x
xf
xx →→
+
+?=
00
)0()(
lim
2
1
0
=+
+?=
→
x
fxf
x
¨ K
E5 Vw K
x
fxf
x
)0()(
lim
0
→
i O
2
1)0()(
lim
0
=
→
x
fxf
x
b
pSE
1
)(sin)1ln(
lim0
2
0
+?
=
→
x
x
e
xfxx
2
0
)(
lim
x
xfxx
x
+?
=
→
0
1)(
lim
0
=
=
→
x
xf
x
yN 0
1
sin)(sin
lim
2
0
=
→
x
x
e
xxfx
b
￡
!·° Iù
T? www.tsinghuatutor.com - 23 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
ZE ?¨
ü à
T5μ
1
)(sin)1ln(
lim
2
0
+?
→
x
x
e
xfxx
2
22
0
)(sin)(
2
1
lim
x
xfxxoxx
x
++
=
→
??
?l
K
x
xf
x
1)(
lim
0
→
2
1)0()(
lim
0
=
=
→
x
fxf
x
i
?
^ )(xf ) V±i O 0=x
)0(f
′
2
1)0()(
lim
0
=
=
→
x
fxf
x
b
è
! )(xyy =
^=¨è±sZ?
x
eqyypy
3
=+
′
+
′′
@
SHq 0)0()0( =
′
= yy
￥+35?
Hf
0→x
)(
)1ln(
2
xy
x+
￥ K
b
"
?i b #
?? b
#
$
?? b %
?? b
<3 >
)(
lim
)(
)1ln(
lim
2
0
2
0
xy
x
xy
x
xx →→
=
+
2
1
2
)(
2
lim
)(
2
lim
00
==
′′
=
′
=
→→
xyxy
x
xx
b
ê $
b
￡
!·° Iù
T? www.tsinghuatutor.com - 24 - bvD ? S
L 1101 è 62781785
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
$?s
?B [ ±s
? c ?
à
Q a?éD9
?
à
Q
?
?lDà
Q
^Bíf
±sD￥?
= ?
￥ ?Dà
Q ?V K￥? ? a
M
i O?ü?
￥?l ATB?S K
T b
??
à
Q'
& V[¤?B"
×1?é 7?t?é
^ù?f
?
￥×1G Dy b
9
Z
?T
? y
￥?
9
? b D ?1g?z'?f
￥?
T ?
￥
15
E5ˉf
￥ p?E5 [#Qf
a ?f
??
Z? ??￥f
￥ p?
T#1? b
?
?l# M??
T
?l
!f
)(xfy = ? ￥
#×
=μ?l
0
x
0
xxx?=? )()(
00
xfxxfy+=?
)(
)()(
limlim
0
00
00
xf
x
xfxxf
x
y
xx
′
=
+
=
→?→?
0
0
0
0
)()(
lim)(
xx
xfxf
xf
x
=
′
→?
?
)(
0
xf
′
￥+il ML|
q b
?N?
í
)(
)(
0
xA
x
xf
+=
α
O )(
0
xfA
′
= b ? )( x?α
^ 0→?x
H￥í kl
b éB? V?1
xxxxfxf+?
′
=? )()()(
00
α
)()()()(
00
xxxfxfxf?+?
′
+= β
? xxx=? )()( αβ 1 0→?x
H￥ú¨í kl
b
?
?l￥
í
V[ ?Z ?31
￡
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T? www.tsinghuatutor.com - 1 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)(
)())((
lim)(
00
0
0
x
xfxxf
xf
x
+
=
′
→?
α
α
?l ?T
x
xfxxf
x
y
xx
+
=
→?→?
)()(
limlim
00
00
i 5?N ′1 )(xf )￥P?
:1 ?T
0
x )(
0
xf
′
x
xfxxf
x
y
xx
+
=
++
→?→?
)()(
limlim
00
00
i 5?N ′1 )(xf )￥·?
:1 b
0
x )(
0
xf
+
′
A ?? Ki ￥ 1Hq )(xf ) V?￥ sA1Hq
^
0
x )(xf )
￥P a·?
?i OM?
0
x
=
′
)(bf )(af
+
′
b
? á
ì
a )(xf > uW
V?
H
^·],[ ba )(xf
=
B?? V
?i O
),( ba
)(af
+
′
D )(bf
′
(i b
è
=+?+
∞→
)]
1
1ln(sin)
3
1ln([sinlim
xx
x
x
b
}3 ~
7
x
t
1
= 5
e K"
t
tt
t
)1ln(sin)31ln(sin
lim
0
+?+
→
2|])1ln(sin)31ln([sin
0
=
′
+?+=
=t
tt b
è ? kaf =
′
)( i 5
￡
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T? www.tsinghuatutor.com - 2 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
=
+∞→
)()
1
(lim af
h
afh
h
b
" k? b
#k b
$ b
%?i b 0
}3 ~
+∞→
)()
1
(lim af
h
afh
h
h
af
h
af
h
1
)()
1
(
lim
=
+∞→ t
aftaf
t
)()(
lim
0
+
=
→
.)()( kafaf?=
′
=
′
=
?Ka¨?
?
i ￥ 1Hq P·?
i OM?yN?ê
" b
è
!
≤?
>
=
0),1(
2
0,
1
arctan
)(
sin
xe
x
x
x
xf
x
π
)
)(xf ￥ V±? ? V± p )(xf
′
i)
??? b
}3 ~
n5 )(xf ) ?? b ??f
V??￥2
o? )
0=x )(xf
)￥ V±?1N I
n K 0=x
2
1
arctan
lim)0(
0
π
==
′
+
→
+
x
x
x
f
x
i
2
1
lim
2
)0(
sin
0
ππ
=
=
′
→
x
e
f
x
x
)0(
+
′
f
￡
!·° Iù
T? www.tsinghuatutor.com - 3 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
yN )(xf ) V±2
1 0=x
)(xf
)) V± b ),( +∞?∞
<
=
>
+
=
′
0,
2
cos
0,
2
0,
)1(2
1
arctan
)(
sin
2
3
xe
x
x
x
x
x
x
xf
x
π
π
)0(
2
)(lim
0
fxf
x
′
==
′
→
π
?
^ )(xf
′
0=x ) ?? b 2
1
)) ?? b )(xf ′
è
! 0)0( =f 5 )(xf 0=x ) V?￥ 1Hq1
b
"
)cos1(
1
lim
2
0
hf
h
h
→
b #
)1(
1
lim
0
h
h
ef
h
→
b
$
)(
1
lim
3
2
0
hf
h
h→
b %
)]()2([
1
lim
0
hfhf
h
h
→
b
}3 ~s??1 #
y1 #
￥ Ki ?N? K
h
e
e
ef
h
h
h
h
→
1
1
)1(
lim
0
h
e
e
fef
h
h
h
h
h
=
→→
1
lim
1
)0()1(
lim
00
￡
!·° Iù
T? www.tsinghuatutor.com - 4 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
i 7 1lim
1
lim
0
=
=
→
h
h
h
e
h
h
?i # K
h
h
h
e
fef
→
1
)0()1(
lim
0
i ?:
h
eh?=1)(α 5 )(hα t?
,￥Z
T
^ ?i￥ b V?)0(f
′
i i Oμ
)0(
1
)0()1(
lim
0
f
e
fef
h
h
h
′
=
→
b
"
?
^y1
H0→h
+
→ 0
2
h ???
?l b I
n $
K
=
→
)(
1
lim
3
2
0
hf
h
h
)(
1
lim
3
3
0
hf
h
h→
=
2
3
0
lim
h
h
h
→
0)(
1
lim
3
3
0
=
→
hf
h
h
i ?
￡
)(
1
lim
3
3
0
hf
h
h→
￥i y N $
g? bà ?ê [ %
KVr
T? ?
)0(f ? K
E5 I
n
))]0()(())0()2([(
1
lim
0
fhffhf
h
h
→
￥i ?
￡
h
hf
h
)2(
lim
0→
h
hf
h
)(
lim
0→
￥i ?
[ %
g? b
￡
!·° Iù
T? www.tsinghuatutor.com - 5 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
è
!
≤
>
=
0)(
0
cos1
)(
2
xxgx
x
x
xf ? )(xg
^μ?f
5 )(xf
)μ
% b 0=x
"
K?i b #
Ki ?? ?? b
$
???? V? b %
V? b
}3 ~
n5 I° )￥P· K b 0=x
0
2
lim
cos1
lim)(lim
2
000
==
=
+++
→→→ x
x
x
x
xf
xxx
y10)(lim)(lim
2
00
==
→→
xgxxf
xx
)(xg μ?
yN 0)0()(lim
0
==
→
fxf
x
# )(xf 0=x ) ?? b I °
)￥P·?
^?i b 0=x
0)(lim
)0()(
lim
00
==
→→
xxg
x
fxf
xx
x
fxf
x
)0()(
lim
0
+
→
0
2
lim
cos1
lim
2/3
2
00
==
=
++
→→
x
x
xx
x
xx
yN )0(
+
′
f D )0(
_
f
′
(i OM? b
?
^ )(xf ) V? O0=x 0)0( =
′
f
s?1 %
b
?f
B? V? %?￥f
??é
￡
!·° Iù
T? www.tsinghuatutor.com - 6 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
?é ? )(xf ) V?
H
0
x )(xf A ?i ) ?? b ?A? ?i?
0
x )(xf
) ??
H ??B? ) V? b
0
x
0
x
?é
!f
)(xf ?? O 0)0(' >f 5 i 0>δ
P¤ ? i￥
),0( δ∈x μ )0()( fxf > ?i￥ )0,( δ?∈x μ
)0()( fxf < b
￡ ? 0
0
)0()(
lim)0(
0
>
=
′
→
x
fxf
f
x
5? K ?? Vw
i 0>δ
P? )0,( δ?∈x ),0( δ∈x
H
0
0
)0()(
>
x
fxf
' )0()( fxf? D x ? ?]|yN ?i￥ ),0( δ∈x μ
)0()( fxf > ?i￥ )0,( δ?∈x μ )0()( fxf < b
? o?B?)￥?
?μ| ?
%?f
￥9h? bf
￥9h?
? uW
? ?é b
è
! )0(f
′
i 1)0( <f
?
e
x
xf
x
x
=
+
→
1
0
)
sin
)(cos1
1(lim
5 )0(f
′
=
b
"
b #
b $
0 1 2 b %
e b
<3 > s? $ b? e
x
xf
x
x
=
+
→
1
0
)
sin
)(cos1
1(lim
V[??? 0→x
H
μ
1)
sin
)(cos1
1ln(
1
lim
0
=
+?
→
x
xf
x
x
￡
!·° Iù
T? www.tsinghuatutor.com - 7 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
1
sin
)(cos11
lim
0
=
→
x
xf
x
x
y1 1)0( <f 5Aμ )0(0)(lim
0
fxf
x
==
→
?
^ 1
)(
lim
2
1
sin
)(cos11
lim
2
2
00
==
→→
x
xf
x
xf
x
xx
? V¨
ArE5
?y1 i
[ )0(f ′
2
)(
lim
)(
lim)]0([
00
2
=?=
′
→→
x
xf
x
xf
f
xx
¤? 2)0( =
′
f b
è
! ?
#×
= V ? O?)(xf 0=x 0≠x
H 0)( ≠xf X ?
2)0(,0)0( =
′
= ff p K
x
x
xf
sin
1
0
))(21(lim?
→
b
3
p K1 o p?
!E ?¨S KiD?
∞
1 2)0( =
′
f M ó" b
x
x
xf
sin
1
0
))(21(lim?
→
x
xf
xf
x
xf
sin
)(2
)(2
1
0
))(21(lim
→
=
?ˉ K? ?o? I
n K
x
x
x
xf
x
xf
xx
sin
)(2
lim
sin
)(2
lim
00
=
→→
? 2)0(,0)0( =
′
= ff i #
K V ?¨ K￥eE
p¤'μ
]
sin
lim[]
)0()(
lim[2
sin
)(2
lim
000
x
x
x
fxf
x
xf
xxx →→→
=
￡
!·° Iù
T? www.tsinghuatutor.com - 8 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
4)0(2?=
′
= f
?
^
4
sin
1
0
))(21(lim
→
=? exf
x
x
b
? ?¨?
?l p
t K
^B ?×15? ?
?
?l￥ K/?
T i?i ?¨ˉ
K? ?DX?×1 K￥2
b
±sà
QDMM
q
±sà
Q
??
￥?N?
í
á
ìXü?? V?f
)(xf )￥9
0
x
)()()(
000
xfxxfxf+=? V[V
U1
xxxxfxfxxf+
′
=+ )()()()(
000
α
? )( x?α 1
H￥í kl
b ?:0→?x xxx=? )()( αβ 5
)( x?β
^ ￥ú¨í kl
b?
^? V:1 x?
)()()()(
00
xxxfxfxf?+?
′
=? β
?l
!f
)(xfy = ? ￥
#×
=μ?l b ?i è
P¤f
9
0
x A
y? V[V1 )( xoxAy?+?=?
? AD í1
^x? )( xo? 0→?x
H￥ú¨ í kl
5?f
)(xfy = ? ) V± :1
0
x xAdy
x
=
0
)(
0
xdf= bf
￥±s
Yè:1 dxxfdxydydf )(
′
=
′
==
MM
q
?l
! 1 V?f
? K )(xfy =
￡
!·° Iù
T? www.tsinghuatutor.com - 9 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)(
)()(
limlim
00
00
xf
y
x
x
xfxxf
y
x
x
x
y
y
xx
′
=
+
=
→?→?
1 y ￥MM
q b x
ü6
?? ?l3 pf
)(PfQ = ? P1?ê
￥ Nì b3 p Nì ￥
MM
q1 )(Pf
Q
p
E
d
′
= T1Nì3 pQ?￥B?
|MM
q￥ '
′?l 1??
3 pN ì )(Pf
Q
p
E
d
′
= b
lmf
? l1
)(PPfPQR == b
?f
￥?
D±s
T
?
D±s
15
5
?T )(),( xgxf ? )?μ?
5 aμ a a
s
?1
,
H
?x x) (
μ?
O V± b
)()(])()([ xgxfxgxf
′
±
′
=
′
±
)()()()(])()([ xgxfxgxfxgxf
′
+
′
=
′
)(
)()()()(
]
)(
)(
[
2
xg
xgxfxgxf
xg
xf
′
′
=
′
E )(d)(d))()(( xgxfxgxf ±=±
E (x)d)((x)d)())()(( fxggxfxgxf +=
E (x)d))(( fcxcf = c1è
)(
)(d)((x)d)(
)(
)(
2
xg
xgxffxg
xg
xf
d
=?
￡
!·° Iù
T? www.tsinghuatutor.com - 10 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)()()(])()()([ xhxgxfxhxgxf
′
+
′
+
′
=
′
++
])()()([
′
xhxgxf
)()()()()()()()()( xhxgxfxhxgxf
′
+
′
+xhxgxf
′
=
è
! )(xyy = ?i? ),( +∞?∞∈x
@
)(
1
2
xox
x
y
y?+?
+
=? ?
4
)1(
π
π ey = 5
=)3(y
3
π
π e b
3 ?X??
T ?i ? ? 5μ),( +∞?∞∈x )(xf ),( +∞?∞
= V?
O
2
1 x
y
y
+
=
′
2
1 x
dx
y
dy
+
= sa¤?
Cxy lnarctanln += ?
x
Cey
arctan
=
?
4
)1(
π
π ey = ?
^ π=C b?
^
x
ey
arctan
π=
3
)3(
π
π ey = b
è
!f
( )xf ( )+∞∞?,
μ ?l u W [ ]2,0
() ( )4
2
= xxxf
? ?i￥ x ?
@ ( ) ( )2+= xkfxf ? k
1è
b
1 ( )xf [ )0,2?
￥Vr
T
2 ù 1′
H ) V? b k ()xf 0=x
3
? '02 <≤? x 220 <+≤ x
H
￡
!·° Iù
T? www.tsinghuatutor.com - 11 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
( ) ( )2+= xkfxf
()()[ ] ( )( )42422
2
++=?++= xxkxxxk
?5
!? ( ),00 =f
()
() ( ) ( )
,4
4
lim
0
0
lim0'
2
00
=
=
=
++
→→
+
x
xx
x
fxf
f
xx
()
() ( )
0
0
lim0'
0
=
→
x
fxf
f
x
()( )
k
x
xxkx
x
8
42
lim
0
=
++
=
→
7 ¤ b'() ()00
''
+?
= ff
2
1
=k
H ( )xf 0=x ) V? b
ú¨?
9
?Tf
)(xfy = ￥?f
)(xy
′
ˉμ?
])([
′′
xf 5?
])([
′′
xf 1 )(xfy = ￥=¨?
:1
2
2
),(,
dx
yd
xfy
′′′′
2
2
dx
fd
b
B?ü ￥?
?1)(
)1(
xf
n?
)(xf ￥ ¨?
:1n
n
n
nn
dx
yd
xfy ),(,
)()(
n
n
dx
fd
b
n¨?
?l1
x
xfxxf
xf
nn
x
n
+
=
→?
)()(
lim)(
)1()1(
0
)(
f
￥=¨
#=¨[
￥ò¨?
d?ú¨?
b
￡
!·° Iù
T? www.tsinghuatutor.com - 12 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
? ú¨?
￥?là pú¨?
o3??
￥'
TD
E5?¨é?9
b
è p 1è
￥ò¨?
b λ
λ
(
x
ey =
3 ,,,
2
L
xx
eyey
λλ
λλ =
′′
=
′
xnn
ey
λ
λ=
)(
b
è p ￥ò¨?
b xy sin=
3
+==
′
2
sincos
π
xxy
+=
+=
′′
2
2sin
2
cos
ππ
xxy
l l l l l
)
2
sin(
)(
π
nxy
n
+= b
?
?f
)(xf D )(xg eé ?ú¨?
Hè ¨?/
? ? L
!
)(xf )(xg (μ ¨?
?1 ??
DG
T n
)()()]()([
)()(
0
)(
xgxfCxgxf
kknk
n
n
k
n?
=
∑
=
!
)1()1(
!)!(
!
k
knnn
kkn
n
C
k
n
+
=
=
L
i?
b )()(
)0(
xfxf = )()(
)0(
xgxg =
è ￥ ¨?
^ xxy sin
2
=
3 ? ??
DG
T?i? ￥ ¨?
(1
,5μ
2
x 3≥n
￡
!·° Iù
T? www.tsinghuatutor.com - 13 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)99(2)100(2)100(
)(sin)(100)(sin xxxxy
′
+=
)98(2
)(sin)(
!2
99100
xx
′′
×
+
++
+=
2
99
sin200
2
100
sin
2
ππ
xxxx
+×+
2
98
sin99100
π
x
xxxxx sin9900cos200sin
2
= b
è p
22
1
ax
y
= ￥ ¨?
b n
3
)()(
22
)(
11
2
11
nn
n
axaxa
ax
y
+
=
=
+
=
++ 11
)(
!)1(
)(
!)1(
2
1
n
n
n
n
ax
n
ax
n
a
+
=
++ 11
)(
1
)(
1
2
!
)1(
nn
n
axax
a
n
è )32ln()( xxf?= ￥ ¨?
^
b
￡
!·° Iù
T? www.tsinghuatutor.com - 14 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
"
10
10
)32(
!103
x?
#
10
10
)32(
!93
x?
$
10
10
)32(
!103
x?
%
10
10
)32(
!93
x?
b
3 s?1 %
bo??i?
￥Q
Q a¨e￥2T+ ￥Z
a' V b
ˉf
p?E5D±sE
? ? ?T )(xu?= ? x)μ?
)();( ufyx
dx
du
=
′
=? ?
? ))(( xuu?= ) 9 μ?
)(uf
du
dy
′
= 5 ˉf
)]([ xfy?=
? x)9μ?
O
dx
du
du
dy
dx
dy
= )()(})]([{ xufxf
′
′
=
′
è p
x
ay
1
arctan
= D
)1ln(
2
++= xxy ￥?
b
3 )(
1
1
ln)(
2
2
1
arctan
1
arctan
+
=
′
x
x
aaa
xx
x
a
x
a
1
arctan
2
1
ln
+
=
￡
!·° Iù
T? www.tsinghuatutor.com - 15 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)2
12
1
1(
1
1
])1[ln(
22
2
x
xxx
xx
+
+
++
=
′
++
1
1
2
+
=
x
è pf
￥?
b )0(
sin
>= xxy
x
3 Z E
? ?f
?S
a·f
b
n5
H |
¤?f
xxy lnsinln = b
??f
p?E¤
x
x
xxy
y
sin
lncos
1
+=
′
V7 ]
sin
ln[cos
sin
x
x
xxxy
x
+=
′
b
??5 |
p ?￥ ZE? S |
p? E b"
a¨ ?
a ·f
?cμ ?y
TMe"{eZ a 7Z￥f
9
a¨ b
)(
)(
xv
xuy =
ZE
|
a·f
?1a ¨ˉf
p ?
E5#eE
T
xxx
exy
lnsinsin
==
è
!
1
lnarctan
2
2
+
=
x
x
x
e
e
ey
5
1
1
2
1
+
=
=
e
e
dx
dy
x
b
3 |f
Vr
T?1
￡
!·° Iù
T? www.tsinghuatutor.com - 16 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)1ln(
2
1
arctan
2
++?=
xx
exey 5
1
1
1
2
2
2
+
+?
+
=
′
x
x
x
x
e
e
e
e
y
1
1
1
2
2
2
1
+
+?
+
=
=
e
e
e
e
dx
dy
x
1
1
2
+
=
e
e
b
ˉf
p?9
H 31? ??WM
üf
s3?B1X??
￥f
¨ˉ
f
p?E5 Ka1ü? ?￥?WM
¨1M
￥f
9} b ?
1g?
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L 1101 è 62781785
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L 1101 è 62781785
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L 1101 è 62781785
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L 1101 è 62781785
2005
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!·°
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L 1101 è 62781785
2005
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!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
11
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￡
!·° Iù
T? www.tsinghuatutor.com - 22 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
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ê?7<v0 1006 bvD ? S
L 1101 è 62781785
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￡
!·° Iù
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L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
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￡
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T? www.tsinghuatutor.com - 24 - bvD ? S
L 1101 è 62781785
