清华名师：微积分课程各章节知识点题解：第4章微分学基本定理及应用

分类：数学格式：pdf 日期：2005年11月22日

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x
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x

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^Kl′? b
y N V?f
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′
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ax
afxf
af
ax

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￡
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b
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|¤Kv Kl′ y 1],[ ba
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Cxf ≡)( (μ],[ bax∈? 0)( =
′
xf b?5 )(xf
￥KvKl′?à
μ=?-B
=?|¤?^
!],[ ba
P¤ )(max)(
],[
xff
bax∈
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1
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n
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T ( 1μ
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°
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5 ),( ba∈?ξ
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ab
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f
=
′
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)()(
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=
5

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:? ?￥HqO
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ξF '
ab
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f
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′
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r
T
))(()()( abfafbf?
′
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00
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′
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ξ
′
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′
xf 5 cxf =)( ?
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′
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),( ba
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9F O),( ba 0)( >
′
xf
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9F b7 ? 0)( ≤
′
xf
H 0)( <′ xf
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??h
b
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^

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Vw? í
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1
x
1
x )(xf
￥|′?
^
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μ
?? e?T¨b
],[ ba

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b

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′

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nxmxmxx
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=

b
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xf ?
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m
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0
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′
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è
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ùf

= V? O]1,0[
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7 )(max
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xfM
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Mf 2)( ≥
′
ξ b
}￡~
n5y )(xf
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ù1 ￥?
ùf
5o?￡
ü
( )1,0
0
∈ξ
P¤ Mf 2)(
0
≥
′
ξ b
¨ -BHSBOHF?′? ?
? )(xf
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= ???y]1,0[ ]1,0[
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i
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M
x )(max)(
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xfMxf
x
M
∈
== b

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?|¤
uW ),0(
M
x )1)((0
2 M
xfM?
′
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uW )1,(
M
x
M
xfM )(0
1
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M
x,0
1
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2 M
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xfxfM?
′
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′
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7
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210
ξξξ fff
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′

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0
∈ξ N
Hμ??
T )(2
0
ξfM
′
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0
∈+= ξξ
P¤ Mf 2)( ≥
′
ξ
T 1
h ?| ?ì μ °?′? ?￥ ?w
GQé?￡
ü ?|μ ù?
?B×1? ?￥ ?3D?¨b
è
! )(xf
μ=¨?
O]2,1[ 0)2()1( == ff
￡
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2
xfxxF?= )2,1(
0
∈?x
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0
=
′′
xF b

±
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′
￡ ZE

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5μ
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2
xfxxfxxF
′
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:? ?
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1
=
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′
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ZE

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xF )(xF
′′

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′′
xF )(xF
′

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f
b
]2,1[
I
n? #0)1( =
′
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′
xF '
?|￥2
b??
?
:? ? ?
)()2,1( xFx
′
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1
∈?ξ
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1
=
′
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′
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± ?
^eL
! ?? ? '
A
P¤)2,1(
0
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0
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! )13)(12)(1()(?++= xxxxxg
k??Z? 0)( =
′
xg
=μ+?
L?b)0,1(?
±
^ X?f
)(xg V r
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′
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1
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′
à
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3
1
,0(
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2
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2
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3
∈ξ b
? )(xg
′
1 Q[
T Kμ ?
L?y N )(xg
′
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=
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L?b
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! ￡
ü??
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24
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22
2
x
x
xx
x
<?+<
++
π
b
}￡~
ZE ? o3￡
ü??
T0>x
2
1
4
)1arctan(
22
1
2
<
+
<
++
x
x
xx
π

?i?
4
1arctan
π
= ?i ),0( ∞∈x
7
xxf arctan)( = 5μ
1)1(
)1()(
4
)1arctan(
+
=
+
x
fxf
x
x
π

¨ -BHSBOHF?′? ?5μ]1,1[ x+
)1,1(,
1
1
1)1(
4
)1arctan(
2
+∈
+
=
+
+
x
x
x
ξ
ξ
π

y
2
1
1
x+
1??hf
?
^
2
1
4
)1arctan(
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1
2
<
+
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++
x
x
xx
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xxx
xxxxf
++?
+++=
)22(
4
)1arctan()22()(
2
2
π

0)0( =f

0]
4
)1)[arctan(22(
)22(
4
)1arctan()22()(
>?++=
+?++=
′
π
π
xx
xxxxf

?
^?
H 'eP§??
T? ?b0>x 0>)(xf
7 0)0(,
24
)1arctan()( =+=?
π
x
xx
0)(,0
2
1
)1(1
1
)(
2
<?<?
++
=
′
x
x
x
'e·§??
T? ?b
è
! ]1,0[)( Cxf ∈
= V?O)1,0(
0)1()0( == ff ￡
üi )1,0(∈ξ
P
0)(')(2 =+ ξξξ ff b
}￡~/ ?^￡)()(
2
xfxxF =
]1,0[)( CxF ∈ O
= V?)1,0(
O ?¨
:? ?¤0)1()0( == FF )(xF
i 0)(')1,0( =∈ ξξ F
P '
0)(')(2
2
=+ ξξξξ ff
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# ￥S?
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f
￥ ?′ù5
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#×
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x )()(
0
xfxf ≤
))()(
0
xfxf ≥
5? )(
0
xf 1f
)(xf ￥B?v l
′b
v′al′d?1′
Pf
|′￥? ?1′?b
0
x
B ? V?f
) |¤′ ￥A1Hq
^
0
xx = 0)(
0
=
′
xf ?
?
^n:
? ?b
@ 0)(
0
=
′
xf ￥? ?￥1
0
x )(xf ￥?b
V+
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￡
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x
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′?￥ sHq
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! )(xf

#×
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0
x
0
x
)()()(
0
xfxfxf?=?
§?í ?M|5
0
x
0
xx = A
1 )(xf ￥ ′?bi O 0)( ≥? xf
H 1l′? 7
0
x
0)( ≤? xf
H 1v′?b
0
x
è p xxf?= 1)( ]1,1[?
=￥′?b
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≥?
<?
=
11
11
)(
xx
xx
xf ? 0)1( =f ¤
1<x 01)1(1 <?==? xfxf
1>x 01)1(1 <?==? xfxf

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? ? ?= ? sHq

!)(xf
??
#×
0
x ( ) ( )δδ +?
0000
,,xxxx U
= V ± ) ? ?5?
0
x )(xf
′

§M|
H 1
0
x
0
x )(xf ￥
′?iO

? ( )xxx,
0
δ?∈
H 0)( >
′
xf 7?
( )δ+∈
00
,xxx
H 0)( <
′
xf
0
xx = A1v ′?b

? ( )xxx,
0
δ?∈
H 0)( <
′
xf 7?
( )δ+∈
00
,xxx
H 0)( >
′
xf
0
xx = A1l ′?b
è
!Z? 03
3
=+? Axx )
A|′
H

Z?μB?
L?

Z?μ=??]
L?

Z?μ???]
L?b
}3 ~
! 5 1?Q [
T K μ? ?
L ? b¨ ?
)
?f
￥9huWD′￥s?f ?b
Axxxf +?= 3)(
3
)(xf
)1)(1(333)(
2
+=?=
′
xxxxf

7 0)( =
′
xf 3¤? 1
1
=x
1
2
=x
iO
)1,(∞∈x
H 0)( >
′
xf )(xf ??9F
)1,1(?∈x
H 0)( <
′
xf )(xf ??h

),1( +∞∈x
H 0)( >
′
xf )(xf ??9Fb
￡
!·° 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
?
^ V? 1
1
=x 1 )(xf ￥v′? 7 1
2
=x 1 )(xf ￥
l′?b )(xf ￥vl′sY1

2)1( +=? Af 2)1(?= Af
Kv Kl′ (v ?
, l?
,
H )(xfy = ),1( +∞?
= í
,?
∞=
∞→
)(lim xf
x
' )(xf V )1,(∞
= V|¤ μ
′? ??f
￥
,?? ? V? )(xf ? )1,(∞
=μB?
,?b
0)2)(2()1()1( >?+= AAff
?N V¤ A?
@ 04
2
>?A ' 2>A

A ? ? )(xf ￥K vKl ′μB ?1
,
H'

H04)1()1(
2
=?= Aff
)(xf μ
?
,?N
H 2=A b

? )1(?f D )1(f |¤s|
H?i?
+∞=
+∞→
)(lim xf
x
?
^ )(xf μ ? ?
,? sYê ?
)1,1(),1,(∞ D ),1( +∞
=N
H 2<A b
? ? ?? ? sHq
?T 0)(
0
=
′
xf
0)(
0
<
′′
xf )0(> 5 )(
0
xf 1v′ l′
b
?éB?￥<l sHq1??j?B?a
ób
è p 543
3
2
)(
23
++?= xxxxf ￥′b
}3~
)2)(1(2462)(
2
=+?=
′
xxxxxf
￡
!·° 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
¤? b1
1
=x 2
2
=x
64)(?=
′′
xxf b 02)1( <?=
′′
f
[
3
20
)1( =f 1v′y 1 02)2( >=
′′
f
[
3
19
)2( =f 1l′b

>uWD 7uW
￥KvK′ù5

>uW
￥KvK′ù5
?)(xf
??
HA?
μKvK l′b7
μ W?? V??
H p
],[ ba ],[ ba
],[ ba )(xf
Kv Kl′
H ?
|?? V??#
?)￥f
′?Bé?1?7a??KvKl′b
],[ ba
è p 155
345
++?= xxxy ]2,1[?
￥ KvK′ b
}3~5p?
)3)(1(515205
2234
=+?=
′
xxxxxxy
7 0=
′
y 3¤ ? 0
1
=x 1
2
=x 3
3
=x O
?xxxy 306020
23
+?=
′′
3
x ]2,1[?
b
0)0( =
′′
y 010)1( <?=
′′
y
1)0( =y 2)1( =y
10)1(?=?y 7)2(?=y
yN )(xfy =
￥Kl ′1]2,1[? 10)1(?=?y K v ′
1 2)1( =y b
￡
!·° Iù
T? www.tsinghuatutor.com - 15 - bvD ? S
L 1101 è 62781785
ex 31 ≤≤ ￡
ü??
T è
!
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
1lnln)3(ln1
222
≤?≤? xx b
}￡~?B??
To?
7
xxxf
2
lnln2)(?= p )(xf
￥Kv Kl′b
5p?b
]3,1[ e
x
xx
xf ln
22
)(?=
′

7 0)( =
′
xf 3? ]3,1[
0
eex ∈=
112)()(
0
=?== efxf 0)1( =f

2
)3(ln)3ln(2)3( eeef?=
2
)3(ln1)3ln1)(3ln(?=?= e
[ 1max
]3,1[
=
∈ ex
f ?
a
ü 1￡￥
??
T? ?b
2
]3,1[
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∈ ex
f

f
7uW
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?/?ù5),( ba
7u W
=? ?
T ￥￡
ü è
#? 7u W
= ? ?￥f
￥ K v K
l ′
/? ù5 b
n5 7uW
= ? ?￥f
? AμK v′ K l
′byN I3 7uW
=f
￥K vKl′ù 511Au W
￥f ? ?¤ˉ
b
),( ba
),( ba
),( ba
B ? f ?/ V? ¨A1 ￥
m
I3 )(xf
= ￥
? D/ ?ù
5 8ZE
^
n5p
= ￥?D ? V?? ??
=￥
′?Q?éB? I°
),( ba
),( ba ),( ba
)(xf
=0uW
￥9h?[#
??),( ba
￡
!·° 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
§K Axf
ax
=
+
→
)(lim D Bxf
bx
=
→
)(lim Ka ? ?f
￥
KvKl′
?D/?b? ?ù5B?
^uW
=?|¤KvKl′b
è ￡
ü ? i )2,0(∈x ? ???
T
32ln4
2
+≥ xxxx
}￡~
7 32ln4)(
2
+= xxxxxf
I
n )(xf

= ￥ ? μ| D ′ ù 5 b 5 p ? b
22ln44)(+=
′
xxxf

7 0)( =
′
xf 3? )2,0(1
0
∈=x éB? I°
? ?
§K￥f ?b
03)(lim
0
>=
+
→
xf
x

052ln8)(lim
2
>?=
→
xf
x

? 2
4
)(?=
′′
x
xf 02)(
0
>=
′′
xf yN
)(min0)1(
)2,0(
xff
x∈
== b
?i?" 0)( ≥xf 'e??
T? ?b
è p xxxf ln)( = ( )+∞,0
=￥KvKl′b
}3~
)ln2(
2
1
ln
2
1
)( x
x
x
xx
x
xf +=+=
′

7 0)( =
′
xf 3¤? ),0(
2
0
+∞∈=
ex
￡
!·° Iù
T? www.tsinghuatutor.com - 17 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
?
H),0(
2?
∈ ex 0)( <
′
xf )(xf ??h
?
H),(
2
+∞∈
ex 0)( >
′
xf )(xf ??9
?
^ 1
2
0
=ex )(xf ￥l′?
6??
0lnlim
0
=
+
→
xx
x
# +∞=
+∞→
xx
x
lnlim
V[ ? )(xf
=￥ Kl′1),0( +∞
12
)(

= eef íK
v′b
?
L=?¨ ù5 ?
L =ù5? μKv l
′ Of
?l×
= μ·B
?5??'1Kv l
′?b

wL￥?jD.?
? l ? w L
ê?
ò?￥ ML
/
Z 5
aNw L
^? j
￥bm8
V?f
Vm ?
A
"x￥9v ?wL )(xfy = ￥M L|
q
^ ??9￥ ' )(xf
′
??

6 7jwLML |
q??h l' )(xf
′
?? /? ?
^
?juW ￥×1ZE b7 ? )(xf
′′
i
H V[ ?¨
￥=¨?
?μ|
?wL
￥?j?b
)(xf
? ? u W I
? )0(0)( <>
′′
xf 5wL
)(xfy =
^? j
￥b
??wL
?j?s￥s???1wL￥.?b
? )(xfy = μ=¨ ?
5? ( ))(,
00
xfx 1.?￥A 1Hq
^
0)(
0
=
′′
xf b??.?9 V
C=¨?
?i￥?)b
??· ?f
)(xf ￥

#×
= μ@ ¨
￥?
H
0
x
￡
!·° Iù
T? www.tsinghuatutor.com - 18 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)(xf ￥,?ù5'1 )(xf
′
￥ ′?ù 5b yN,?￥
Y E V?v
′?￥
YZEb VB
,?/

?
§?M9h?5)(xf ′
0
x ( ))(,
00
xfx A1wL
￥.?b)(xfy =

? )(xf
′′

§ M| 5
0
x ( ))(,
00
xfx A1 wL
)(xfy = ￥.?b

? 0)( =
′′
xf 7 0)(
0
≠
′′′
xf
H( ))(,
00
xfx
A1wL ￥.?b)(xfy =
′?D.?÷B??￥2
^?/? ?b
? ? ? )(xfy =

5×
=μ ¨?
O
0
x n
0)()()(
0
)1(
00
===
′′
=
′
xfxfxf
n
L
? 50)(
0
)(
≠xf
n

? kn 2=
H 1
0
x )(xf ￥′?
O
H 1l′?0)(
0
)2(
>xf
k
0
x
0)(
0
)2(
<xf
k
H 1v′?
0
x ),2,1( L=k b

? 12?= kn
H( ))(,
00
xfx 1wL )(xfy = ￥
.?b
è

^?10=x
xeexf
xx
cos2)( ++=
￥′?$
}3~ xeexf
xx
sin2)(=
′
0)0( =
′
f
￡
!·° Iù
T? www.tsinghuatutor.com - 19 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
xeexf
xx
cos2)(?+=
′′
0)0( =
′′
f
xeexf
xx
sin2)( +?=
′′′
0)0( =
′′′
f
xeexf
xx
cos2)(
)4(
++=

04)0(
)4(
>=f
?? ? 10=x )(xf ￥l′?b
è
! 5
$b
5
)( xxf =
"
10=x )(xf ￥v′?b
#
10=x )(xf ￥l′?b
$
))0(,0( f 1wL )(xfy = ￥.?b %
"
#
$
( ? ?
?b
}3 ~ ?
?
^oμ $
??b
0)0()0()0()0(
)4(
==
′′′
=
′′
=
′
ffff
05)0(
)5(
≠=f
è
! )(xf

#×
=μ= ¨ ???
O0=x 0)0( =
′
f
1
cos1
)(
lim
0
=
′′
→
x
xf
x
5
#b
"
)0(f
^ )(xf ￥v′
#
)0(f
^ )(xf ￥l′
$
))0(,0( f
^wL )(xfy = ￥.? %
0=x ?
^
)(xf ￥′?
￡
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T? www.tsinghuatutor.com - 20 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
))0(,0( f 9?
^wL )(xfy = ￥.?b
}3~ ? 1
cos1
)(
lim
0
=
′′
→
x
xf
x
5 V? 0)0( =
′′
f ? ?@[
a
ü ))0(,0( f 1.?b
? K￥ ?? ?? ￥

# ×
= Aμ
0
x 0)( >
′′
xf ' ?
)(xf
′
?
#×
=1? ?9F ?y1 0)0( =f ?
^ ? ? )(xf
′

§M|'0=x )(xf ?M 9h?O ? YV 7M v
H x
0
x
)(xf ?? hM1?9 ?
^ V[ ? 0=x 1 )(xf ￥ l′?b 'o
μê[ #
??b
è ￡
ü??
T
).1,1(1)1(?>>+≥+ xxx λλ
λ

}￡~
! xxxf λ
λ
+= 1)1()(
ù5M1￡
ü,0)( ≥xf ??
]1)1[()1()(
11
+=?+=
′
λλ
λλλ xxxf
7 0)( =
′
xf ¤? )(xf ￥·B? 0=x b
?
H0<x 0)( <
′
xf
?
H0>x 0)( >
′
xf
[ 0)0( =f
^ )(xf ￥ l′
9
^Kl′b# 0)0()( =≥ fxf 'μ
xx λ
λ
+≥+ 1)1( b
è
! )(xf
μ=¨?

O],[ ba )()( bfaf =
￡
ü
￡
!·° 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
à
iB? ),( ba∈ξ
P
ξ
ξ
ξ
′
=
′′
b
f
f
)(2
)(
}￡~ ? 3PMMF? ?
P),(
0
bax ∈? 0)(
0
=
′
xf 4 3
1￡￥?
Tcμ
))(()(2 ξξξ?
′′
=
′
bff
?¨M[/￡ùf
￥ZE|
)()()(
2
xfxbxF
′
=
0)()(
0
== xFbF Q? 3PMMF? ?
),(
0
bx∈?ξ
P 0)( =
′
ξF
)()()()(2)(
2
xfxbxfxbxF
′′
+
′
=
′

7 ξ=x 5μ
)()(2 ξξ fb
′
)()(
2
ξξ fb
′′
=
? ),(
0
bx∈ξ b≠ξ yN
)(2))(( ξξξ fbf
′
=?
′′

'μ
ξ
ξ
ξ
′
=
′′
b
f
f
)(2
)( ),( ba∈ξ b
è
! )(xf 1
=¨ V?f
O],[ ba
0)(),()( ≠
′′
= xfbfaf 5/

5?p￥
^
% b

" (μ),( bax∈? 0)()( ≠? bfxf b
# )(xf

=??M?j?b),( ba

$à
iB? ),,( ba∈ξ
P¤ 0)( =
′
ξf b
%à
B ?
),,( ba∈ξ
P¤ 0)( =ξf b
￡
!·° 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
}3~
n5 "
D $
^￥y1 ?μ ),(
0
bax ∈
P¤
0)()(
0
=? bfxf ' )(xf ?? ?
μM]f
′5 ?
3PMMF? ? V¤? )(xf ′ μ
?
,? é7 )(xf
′′

= μB?
,?
?D
),( ba
0)( ≠
′′
xf
±b
?y 1 0)(),()( ≠
′′
= xfbfaf ??
,?? ?? ?
)(xf
′′

=?M|yN #
??b?ê
% b),( ba
è
!?iè
0>k f

k
e
x
xxf +?= ln)(
?l×
=￥
,??
1
$ b
"
b #
b $
b %
b
}3~
ex
xf
11
)(?=
′
μ·B? ex =
0

? O
H
∞=+∞>=?∞=
+
)(,0)(,)0( fkeff
),0( ex∈ ↑?>
′
)(,0)( xfxf )(xf μB ?
,? ),0(
1
ex ∈
),( ∞+∈ ex
H ↓?<
′
)(,0)( xfxf )(xf μ B
?
,? yNs?1
$ b),(
2
∞+∈ ex
è
! )(xf ),( ∞+?∞
= V ?O 2)(lim =
′
∞→
xf
x

X?K?
T
[])()1(limlim xfxf
cx
cx
x
x
x
+=
+
∞→∞→

￡
!·° 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
pè
cb
}3~
x
x
cx
cx
+
∞→
lim
c
cx
cx
c
cx
x
e
cx
c
2
2
22
1lim =
+=
∞→

? -BHSBOHF?′? ?

[ ] 21)(lim)()1(lim =?
′
=?+
∞→∞→
ξfxfxf
xx

? )1,( +∈ xxξ ? X??
T 5μ 2
2
=
c
e 3
2
2ln
=c
è
! ￡
üba <<0

ba
ab
a
b
+
>
)(2
ln
}￡~1￡
a
b
a
b
a
b
+
>
1
)1(2
ln
)1(2ln)1(?>+
a
b
a
b
a
b
b
7 )1(2ln)1()(,ttttf
a
b
t+==
5μ 0)1(,1 => ft
￡
!·° Iù
T? www.tsinghuatutor.com - 24 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)1(2ln
1
)(+
+
=
′
tt
t
t
tf ,0)1( =
′
f
0)
1
1(
1
)( >?=
′′
tt
tf
0)(,0)( >?>
′
tftf b
víL
?g?[/? ?víL

?iè
C
P¤ Cxf
x
=
+∞→
)(lim
Cxf
x
=
∞→
)(lim 5 Cy = 1 )(xfy = ￥
￡
üvíLb

? i
P
0
x ∞=
+
→
)(lim
0
xf
xx
∞=
→
)(lim
0
xf
xx
5
1
0
xx = )(xfy = ￥<°víLb
è?wL
1
sin
=
x
x
y ￥víLy1 0
1
sin
lim =
∞→
x
x
x

[ 0=y 1wL￥
￡
üvíLb?y1 ∞=
→
1
sin
lim
1
x
x
x

[
1wL￥#°víLb1=x

|vé L?i°L baxy +=
P¤
0))((lim =
±∞→
baxxf
x
5 baxy += 1
)(xfy = ￥|vé Lb
L¨ ?￥ZE I3
^ ?i K
a
x
xf
x
=
±∞→
)(
lim DK
￡
!·° Iù
T? www.tsinghuatutor.com - 25 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
baxxf
x
=?
±∞→
))((lim b
?oμ a
x
xf
x
=
±∞→
)(
lim ￥i?@[￡|véL￥iQ è
xxy sin+= a
x
xf
x
==
∞→
1
)(
lim i?
=?
±∞→
))((lim axxf
x
x
x
sinlim
∞→
?ib

b?i y7
àμ|véLb
è pf
1
)1)(13(
)(
2
+
=
x
ex
xf
x
￥víLb
}3~

∞=
+
=

→→
1
)1)(13(
lim)(lim
2
11
x
ex
xf
x
xx

+∞=
+
=
++
→→
1
)1)(13(
lim)(lim
2
11
x
ex
xf
x
xx

#μ<°víL 1=x

+∞=
+
=
∞→?∞→
1
)1)(13(
lim)(lim
2
x
ex
xf
x
xx

+∞=
+
=
+∞→+∞→
1
)1)(13(
lim)(lim
2
x
ex
xf
x
xx

[í
￡
üvíLb

￡
!·° Iù
T? www.tsinghuatutor.com - 26 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
+∞=
+
=
+∞→+∞→
)1(
)1)(13(
lim
)(
lim
2
xx
ex
x
xf
x
xx

[? +∞→x
H
àμ|víLb
a
xx
ex
x
xf
x
xx
=?=
+
=
∞→?∞→
3
)1(
)1)(13(
lim
)(
lim
2
])([lim kxxfb
x
=
∞→

]3
1
)(13(
[lim
2
x
ex
x
x?
+
=
∞→
)1
x+
1
)1(3)1)(13(
lim
2
+?+
=
∞→
x
xxex
x
x
3
1
13)13(
lim
2
=
+
=
∞→ x
xex
x
x
[∞→x
Hμ|víL
)1(3 +?= xy
?víLù5?V
??HKsYé? I3b
8 è5
è Z? xe
x
=?
2
1 ),0( ∞+
=
L?￥
?
1
b
"
#
$
%

}3~
7 xexf
x
=
2
1)( 0)0( =f ?
012)(
2
=?=
′
x
exf p¤?
2
2ln
0
=x
04)(
2
<?=
′′
x
exf #
^·B?
0
x
￡
!·° Iù
T? www.tsinghuatutor.com - 27 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
OμKv′1
0)2ln1(
2
1
2
2ln
1)
2
2ln
(
2ln
>?==
ef b
?y1 ?K￥???∞=
+∞→
)(lim xf
x
i ),
2
1
(
1
∞+∈x
P¤ 0)(
1
<xf yNi
),
2
2ln
(
12
xx ∈

P¤ 0)(
2
=xf b
? )
2
2ln
,0(∈x
H 0)( >
′
xf ??9F)(xf
0)0()( => fxf í
,?b
? ),
2
2ln
( ∞+∈x
H 0)( <
′
xf ??h
KμB?
,?b)(xf
?
^
^
2
x )(xf ),
2
2ln
( ∞+
=￥·B
,?b
8

s?
^
^
2
x )(xf ),0( ∞+
=￥·B
L?b
[
????|¨?
)(xf
′
￥ ???7'5?ó )(xf
′
￥ ? ?
?HqKíE9
2Tb?
^¨
ArE5
H￥ènpb
è
! )(xf =¨ V? B M),0[ ∞+ ),0( ∞+∈x μ
0)( ≠
′′
xf ￡
ü ),0( ∞+
=wL )(xfy =
B?
))(,(
00
xfx )￥MLD?wL"M??í??b
<￡>
ZE
!MLZ?1
))(()()(
000
xxxfxfxy?
′
+=
i|￡ùf

)()()( xyxfxF?=
))(()()(
000
xxxfxfxf?
′
=
￡
!·° Iù
T? www.tsinghuatutor.com - 28 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
5o?￡
ü )(xF
),0( ∞+
="M?
))(,(
00
xfx ?í
,?b
￡
ü
=?M|b)(xF ),0( ∞+
),()()(
0
xfxfxF
′
′
=
′
0)(
0
=
′
xF
0)()( ≠
′′
=
′′
xfxF ?^L
! 0)( >
′′
xF
0)(
0
>
′′
xF ?
^ 0)(
0
=xF 1 ￥l′? y 1
H
)(xF
0
0 xx<<
0)()(0)( >↓<
′
xFxFxF
0
xx>
H 0)()(0)( >↑>
′
xFxFxF
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0
=xF 1 )(xF ),0( ∞+
=￥Kl′
[ )(0)(
0
xxxF ≠> b￡8b

ZE Q￡Eb
L
! )(xfy = ))(,(
00
xfx )￥MLD?wL"M??

μ B???
01
xx ≠ ?^
!
¨ -BHSBOHF
01
xx > ],[
10
xx
?′? ? ),(
10
xx∈?ξ
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01
01
)()(
)(
xx
xfxf
f
=
′
ξ
?i? ))(,()),(,(
0011
xfxxfx ?ML
yN?μ
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)()(
)(
0
01
01
xfk
xx
xfxf
f
′
==
=
′
M
ξ
? 3PMMF? ?
Ai ),(
01
ξξ x∈?
P¤
0)(
1
=
′′
ξf ?D 0)( ≠
′′
xf
±b￡8b
￡
!·° Iù
T? www.tsinghuatutor.com - 29 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
è
! )(xf
^ )1,1(?
=￥ ??f
O
a
x
xf
x
=
+
→
)(
lim
0

5 )(xf )￥?
1
"b0=x
"
b #
a a? b $
0b %
?ib
}3~
a
x
xf
x
xf
x
xf
xxx
==
=
+
→→→
)(
lim
)(
lim
)(
lim
000
#
)(xf )￥?
1 ab0=x
è ￡
üZ? 0122
2
=?++ xx
x
àμ
??]
L?
}￡~ L
! 0122
2
=?++ xx
x
μ? ???[
?]￥
L
?b
! 5122)(
2
++= xxxf
x
)(xf ??? ?[
?]
￥
,?b
?
^ à
μB?
,?b4)2(ln2)("
2
+=
x
xf
7 v?
,#e Z?àμ
??]￥
L
?b
4)2(ln2)("
2
+=
x
xf
è
! )(xf
@
[] xxfxf sin2)()(
2
=
′
+
′′

5
bs?$
"
? 0)0( ≠
′′
f 5 0=x A1 )(xfy = ￥′?b
#
? 0)0( =
′
f 5 0=x A1 )(xfy = ￥′?b
$
? 0)0( =
′
f 5 ))0(,0( f A1 wL )(xfy = ￥.?b
%
? 0)0( ≠
′′
f 5 ))0(,0( f A1 wL )(xfy = ￥.?b
￡
!·° Iù
T? www.tsinghuatutor.com - 30 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
}3~? 0)0( =
′
f ?
[] xxfxf sin2)()(
2
=
′
+
′′
5 0)0( =
′′
f
[ ] )()(2cos2)( xfxfxxf
′′
′
=
′′′

02)0( >=
′′′
f
yN? ))0(,0( f A1 )(xfy = ￥.?b #
??ê $
b
? 0)0( ≠
′′
f 5 0)0( ≠
′
f
?
^ "
D %
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±sD'? ?#?¨ ??sDef

ü à
TD
ArE5
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SD 9
D ?
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? dL? ù5 ? á ￥
9
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Tf
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af
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μí [
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f
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Tf
? / í í
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×1￥
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m
b
6? f
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ü à
T 5BZMPS

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ó ?
G b è ? ? ′ ù5 [# ¨?p K ￥
A r E 5?
^ [
ü à
T1 ?
G 7¤?
tμr￥ZEb
)(xf
)(xp
n
? )(xfy = ) ￥ V?? D V±?à
Qb ?í￥
0x 0
x )(xf V
[V
U1
)())(()()(
0000
xxxxxfxfxf?+?
′
+= α

? )(
0
xx?α 1
H￥íkl
b
0
xx →
?
T V?C9
￥í
′ V|)(xf
))(()()(
000
xxxfxfxf?
′
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?
?￥μ )(
0
xx?α ?
@ú1p5 V
!X
^? D -
0
x x
￡
!·° Iù
T? www.tsinghuatutor.com - 31 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
Wi ξ
P¤
2
0000
))(())(()()( xxfaxxxfxfxf?
′′
+?
′
+= ξ
? 1 è

Y
L
a
2
1
=a b? ?
^
ü à
T ￥'
±Xb 9
^±s ?′?
?￥éB?w<b
¨
ü à
Tn
? ?
ü à
T
!f
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= μ ¨?

),( ba 1+n
2
0
0
000
)(
!2
)(
))(()()( xx
xf
xxxfxfxf?
′′
+?
′
+=
)()(
!
)(
0
0
)(
xRxx
n
xf
n
n
n
+?++L
?
1
0
)1(
)(
)!1(
)(
)(
+
+
+
=
n
n
n
xx
n
f
xR
ξ

ξ
^o? -W￥
?
b
0
x x )(xR
n
?1 ¨
ün
à?[ μ ?ì μ°?
T￥?[
b

H￥
ü à
T?S
X ?

T'0
0
=x
2
!2
)0(
)0()0()( x
f
xffxf
′′
+
′
+=
1
)1()(
)!1(
)(
!
)0(
+
+
+
+++
n
n
n
n
x
n
f
x
n
f ξ
L
b
μ
e
v ? [?
T ￥
ü à
T 1 N
H
o1pf
)(xf u W
= μ ¨?

1
),( ba
n
￡
!·° Iù
T? www.tsinghuatutor.com - 32 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
2
0
0
000
)(
!2
)(
))(()()( xx
xf
xxxfxfxf?
′′
+?
′
+=
nn
n
xxoxx
n
xf
)()(
!
)(
00
0
)(
+?++L

?
' [?7y μ ?ì μ°?
T?[￥
ü à
Tè¨ ?￡
ü??
Ts
Bt ?
ù5b

μ
e
v?
T?[￥
ü à
Tè¨ ?pK I° ?
′ù5b

μ ?ì μ°?[￥
ü à
T V[
j1 ?ì μ°±s?′? ?￥w<7
?ì μ°±s?′? ?
Lé
^ ¨
ü à
Tb

ó?￥f
é?
ü àZ 7
HB ?μ
?ZE'¨
ü à
T?lp
ò¨ ?
￥ZE ?-1° ¤E 7 ? ¨?f
ü à
T￥ 2
1G ?¨
f
￥}

Dˉ
é?
ü àZ 7￥ZE d?1W¤ ZEb 5
"? ?
T
àμ+Y·
ü¨°¤ZE5 V ?¨W¤ZEb
è |
2
)( xxf = 1
0
=x )Z 71B¨
ü à
T μ ?ì μ
°?[
e
v?[
b
3 2)(,2)1(,1)1( =
′′
=
′
= xfff
)￥ Z 7
T1 μ ?ì μ°?[

1
0
=x
22
)1()1(21?+?+= xxx
μ
e
v?[?
T￥Z 7
T1
B¨
))1(()1(21
2
+?+= xoxx b
è p xxxf?= sin)( 0
0
=x )￥ ?¨
ü à
T μ
e
v?[
b
3o? ?¨ ￥Z 7
Té?
'μxsin
￡
!·° Iù
T? www.tsinghuatutor.com - 33 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)(
6
1
)(
!3
1
sin
3333
xoxxxoxxx +?=?+?=?
è p )￥?¨
ü à
T μ
e
v?[?
T
bxtan 0=x
3
1sec)(tan,0tan
0
2
00
==
′
=
=== xxx
xxx
0tansecsec2)(tan
00
==
′′
== xx
xxxx
2sec2tansec4)(tan
0
422
0
=+?=
′′′
== xx
xxxx
yN¤??¨Z 7
T
)(
!3
2
tan
33
xoxxx ++= b
è p ) μ ?ì μ°?[?
T￥ ¨
ü à
Tb
x
e 1?=x
0
n
3 ¨W¤E | 1
x
e
111
1
+?+
==
xxx
e
e
ee |
j1
'
7
1+x
e
X
e 1+= xX
?¨'Z 7
T'μ
))1(
!
1
)1(
!2
1
)1(1(
1
2
LL ++++++++=
nx
x
n
xx
e
e
n
x
en
x
e
x
ee
)1(
!
1
)1(
!2
1
)1(
11
2
+
+++
+++= L
1
)1(
)!1(
+
+
+
++
n
x
en
e
ξ
L
? ξ D -Wx 1
0
=x ),( +∞?∞∈x b
￡
!·° Iù
T? www.tsinghuatutor.com - 34 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
è
! 0
)(6sin
lim
3
0
=
+
→
x
xxfx
x
p
2
0
)(6
lim
x
xf
x
+
→
b
3 )()6(
!3
1
66sin
33
xoxxx +?=
yN?X?Hq
+
→
3
0
)(6sin
lim
x
xxfx
x
3
33
0
)()(366
lim
x
xxfxoxx
x
++?
=
→

0036
)(6
lim
2
0
=+?
+
=
→
x
xf
x

Ka¤? 36
)(6
lim
2
0
=
+
→
x
xf
x
b
+
→
3
0
)(6sin
lim
x
xxfx
x
3
0
)(666sin
lim
x
xxfxxx
x
++?
=
→

3
0
2
0
66sin
lim
)(6
lim
x
xx
x
xf
xx
+
+
=
→→

3
3
0
2
0
)6(
lim
6
1)(6
lim
x
x
x
xf
xx →→
+
=
￡
!·° Iù
T? www.tsinghuatutor.com - 35 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
Ka¤? 36
)(6
lim
2
0
=
+
→
x
xf
x
b
?/
TE
^p￥
3
0
3
0
)(6
lim
)(6sin
lim
x
xxfx
x
xxfx
xx
+
=
+
→→

0
)(6
lim
2
0
=
+
=
→
x
xf
x
p ey?? B??|a ￥íkl
9D?y 0ê?
dE 9D s? g1
pb

ArE5

A rE5
^p K￥×1 ZE
^ ? O?′ ? ? üf
1M1 ?
17?￥ZEb
? ?
ArE5
?T

0)(lim)(lim == xgxf ∞

K??í )(),( xgxf
′′
?iO 0)( ≠
′
xg

)(
)(
lim
xg
xf
′
′
i1í kv 5
)(
)(
lim
xg
xf
i1ík v
O??
)(
)(
lim
xg
xf
′
′
b
?
[
K?￥t_1
?H∞→
0
xx t_b

h A ? i ¨
ArE5
^B?
kV???
1′?μ K
5e K ?'μ s? 7? ?
1′ K?i
ù? V
i ??
?X?
HqpNK5
ArE5
k
>??
6ù
ZE3 %ù5b
è
pK
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!·° Iù
T? www.tsinghuatutor.com - 36 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
.1
3
cos21
lim
3
0
+
→
x
x
x
x

3 e
T

3
3
cos2
ln
0
1
lim
x
e
x
x
x
+
→
2
0
3
cos2
ln
lim
x
x
x
+
=
→

( )
2
0
3lncos2ln
lim
x
x
x
+
=
→
()
x
x
x
x
2
sin
cos2
1
lim
0

+
=
→

.6
1sin
cos2
1
lim
2
1
0
=?
+
=
→
x
x
x
x

3E e
T
2
0
3
3
cos2
ln
0
3
cos2
ln
lim
1
lim
x
x
x
e
x
x
x
x
+
=
=
→
+
→

.
6
1
3
1cos
lim
3
1cos
1ln
lim
2
0
2
0
=
=
+
=
→→
x
x
x
x
xx
?|
5?1
￡
!·° Iù
T? www.tsinghuatutor.com - 37 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
+

→
1
3
cos2
)1)(cos1(2
1
lim
sin
0
x
x
x
x
ex
5? 5|s
T íkl9D é 7E
°¤¨
A rE5?á ￥ˉp? 9
b s
??Mb

è pK
)1(
2sin
2
1
cos1
lim
2
2
2
0

→
x
x
ex
xxx
b
3
5 I
n?Níkl}Da I
n
ArE5
)1(
2sin
2
1
cos1
lim
2
2
2
0

→
x
x
ex
xxx
b
3
1
3
sin
lim
cossin
lim
)cos(sinsin
lim
2
0
3
0
4
0
==
=
=
→→
→
x
xx
x
xxx
x
xxxx
xx
x

è pK
ax
ax
ee
axx
→
ln
lncos
lim b
3
ax
axax
ax
ax
ee
ax
x
ee
axx
=
→→→
ln
ln
limcoslim
ln
lncos
lim
￡
!·° Iù
T? www.tsinghuatutor.com - 38 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
ax
x
ax
ee
e
ax
a
=
→
1
limcos
ax
ee
ea
ax
ax
x
ax
=
→
→
limlimcos
aeae
aa
coscos ==
b

e?￥ ??
T
?a
0
0 ∞?∞ ? ?
0
0
∞
1 ?
0
∞ ?

A?YV
a?￥?M?1
0
0

∞
∞
?a?
P¨
ArE5é?
kb
è pK
→
xx
x
x
ln
1
1
lim
1
b
3
ZE
xx
xxx
xx
x
xx
ln)1(
1ln
lim
ln
1
1
lim
11
+?
=
→→

2
1
11
1
lim
1
ln
ln
lim
2
11
=
+
=
+
=
→→
x
x
x
x
x
x
x
xx
b

ZE
)11ln()1(
1ln
lim
ln
1
1
lim
11
+?
+?
=
→→
xx
xxx
xx
x
xx

￡
!·° Iù
T? www.tsinghuatutor.com - 39 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
2
1
)1(
1ln
lim
+?
=
→
x
xxx
x
2
1
)1(2
1
lim
)1(2
ln
lim
11
=
=
=
→→
x
x
x
x
xx
b
8 è5
è
! =¨ V?O)(xf 0
)(
≠+
x
xf
x X?
,)
)(
1(lim
3
1
0x
e
x
xf
x
x
=++
→

p )0(),0(),0( fff
′′′
b
s? 4)0(,0)0()0( =
′′
=
′
= fff
<3>?5¤
3
)
)(
1ln(
1
0
1
0
lim)
)(
1(lim ee
x
xf
x
x
xf
x
x
x
x
x
==++
++
→→
é7¤
3
)
)(
1ln(
lim
0
=
++
→ x
x
xf
x
x
0)
)(
(lim0)
)(
1ln(lim
00
=+?=++?
→→ x
xf
x
x
xf
x
xx
yN
￡
!·° Iù
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L 1101 è 62781785
2005
￡
!·°
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L 1101 è 62781785
x
x
xf
x
x
)
)(
1ln(
lim
0
++
→
3
)(
lim
0
=
+
=
→ x
x
xf
x
x
' 3
)(
lim
2
2
0
=
+
→
x
xfx
x

3
)(
lim1
)(
lim
2
0
2
2
0
=+=
+
→→
x
xf
x
xfx
xx

[ 2
)(
lim
2
0
=
→
x
xf
x
O 2
2
)(
lim
0
=
′
→
x
xf
x

? ???D?
?l¤?
4)0('',0)0(',0)0( === fff b
0)0(),(2)(
22
=+= fxoxxf#
4)0('',0)0(' == ff b
è
! )(xf 0=x

#×
= V?O
2)0(,1)0( =
′
= ff
pK
∞→
n
f
n
n
n
n
1
1
1
sinlim b
3 I
nK
()()xfx
x
x
x
→
1
1
0
sin
1
lim
))(1(
sin
sin
0
2
sin
1lim
xfx
xx
xx
x
x
x
xx?
→
+=
￡
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T? www.tsinghuatutor.com - 41 - bvD ? S
L 1101 è 62781785
2005
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!·°
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ê?7<v0 1006 bvD ? S
L 1101 è 62781785
??K? ?o3pK
))(1(
6
1
lim
))(1(
sin
lim
2
3
0
2
0
xfx
x
xfx
xx
xx
=
→→
12
1
)0(6
1
)(1
lim
6
1
0
=
′
=
=
→ fxf
x
x

?i[
????|
^¨
K?lb
?ˉK? ?¤?
()()
12
1
1
1
0
sin
1
lim ex
x
xfx
x
=
→
yN
12
1
1
1
1
sinlim e
n
n
n
f
n
n
=
∞→
b
? h +Y?i?=? ?|a￥ K9
?
¨
Ar E5 y1 ?
1′
￥KíEp¤ '5
àμ ??￥Hq
b)(xf ′
'/
TE
^p￥
12
1
)0(6
1
)(
1
lim
6
1
)(1
lim
6
1
00
=
′
=
′
=
→→
fxfxf
x
xx
[
????|¨?
)(xf
′
￥ ???7'5?ó )(xf
′
￥ ???H
qKíE9
2Tb?
^¨
ArE5
H￥ènpb
è
! )(xf =¨ V? B M),0[ ∞+ ),0( ∞+∈x μ
0)( ≠
′′
xf ￡
ü ),0( ∞+
=wL )(xfy =
B?
))(,(
00
xfx )￥MLD?wL"M??í??b
<￡>
ZE 5BZMPS
TZEb|
ZE ?￥
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L 1101 è 62781785
2005
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)()()( xyxfxF?=
))(()()(
000
xxxfxfxf?
′
=
)Z 71 5BZMPS
T
0
xx=
,))((
2
1
)(
2
0
xxfxF?
′′
= ξ ξ -Wb
0
xxD
y1 0)( ≠
′′
xf ??
,?? ? )(xf
′′
?M| 5 g
?M|b￡8b
)(xF
è
! ￡
ü b4
U
!,eab >>
ab
ba >
xaaxxf lnln)(?=
x
x
xf
ln
)( = b
<￡>/ ex
x
x
xf >= ?
ln
)( 5
2
ln1
)('
x
x
xf
= ^¤ 0)(' <xf
[ )(xf
^?ì?h ￥ μ,eab >>
)()( afbf < '
a
a
b
b lnln
< ??¤
ba
ababba lnln,lnln << '
?
f
￥?é¤ b
ab
ba >

è X?f
μ=¨ ???
)(xf 0)1()0( == ff b￡
ü
?
H]1,0[∈x
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T? www.tsinghuatutor.com - 43 - bvD ? S
L 1101 è 62781785
2005
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!·°
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ê?7<v0 1006 bvD ? S
L 1101 è 62781785
})({max
2
1
)(
]1,0[
xfxf
x
′′
≤
′
∈
b
￡
ü |]1,0[∈?a )(xf ax = )Z 71
() () ( ) ()( )
2
2
1
)( axfaxafafxf?
′′
+?
′
+= ξ

ξ -Wxa,
sY
7 ¤?1=x 0=x
() () ( ) ()( )
2
1
1
2
1
1)(10 afaafaff?
′′
+?
′
+== ξ
? )1,(
1
a∈ξ
() () ( ) ()()
2
2
2
1
)(00 afaafaff ξ
′′
+?
′
+==
? ),0(
2
a∈ξ

TMhiM[? ?
()( ) ()( )
2
2
2
1
1
2
1
)( afafaf ξξ
′′

′′
=
′

?
H0=a 1=a ()
1
2
1
)( ξfaf
′′
=
′

()
2
2
1
)( ξfaf
′′
=
′
e??
T￥?|? ?b
?
H?????
T¤?)1,0(∈a
()() ()
2
2
2
1
1
2
1
)( afafaf ξξ
′′
+?
′′
≤
′

7 ( ) ( ){ }
21
,max)( ξξξ fff
′′′′
=
′′

￡
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T? www.tsinghuatutor.com - 44 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
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ê?7<v0 1006 bvD ? S
L 1101 è 62781785
? )1,0(∈ξ byN
()() ()ξξ faafaf
′′
≤+?
′′
≤
′
2
1
1
2
1
)(
22
})({max
2
1
]1,0[
xf
x
′′
≤
∈
b
?
^
?iB?a ]1,0[
[ })({max
2
1
)(
]1,0[
xfxf
x
′′
≤
′
∈
b
è
! )(,0 xf>δ uW ),( δδ?
=μ?l
??
),( δδ?∈x
H
μ
2
)( xxf ≤

5 A
^0=x )(xf ￥<$>b
"
W?b #
??7? V?￥?b
$
V?￥?
O 0)0( =
′
f b
%
V?￥?
? 0)0( ≠
′
f b
}3~? ),( δδ?∈x
Hy1
2
)( xxf ≤
7 0=x ¤?
0)0( =f b
6?μ x
x
xf
≤≤
)(
0

[
?C/5¤ 0
)(
lim
0
=
→
x
xf
x

O )0(0)(
lim
0
fxf
x
==
→
b
￡
!·° Iù
T? www.tsinghuatutor.com - 45 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
6??μ,
)(
x
x
xf
x ≤≤? Q?C/5¤
)0(0
0
)0()(
lim
)(
lim
00
f
x
fxf
x
xf
xx
′
==
=
→→

[s?1
$ b

￡
!·° Iù
T? www.tsinghuatutor.com - 46 - bvD ? S
L 1101 è 62781785

课件简介

课件名称：	清华名师：微积分课程各章节知识点题解
课件分类：	数学
课件类型：	电子教案
文件大小：	1.83MB
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