复旦大学：数学分析（习题全角、教学教案）：ex4-3



5



? ? 15 ??Qf ?p?E5
? ¨?l￡ ü b (cos ) sinxx′ =? ￡ ?? 2 sin) 2 sin(2cos)cos( xx xxxx ?? +?=??+
 ?  ￥ ????sin x sin( ) ( 0) 22 xx x ?? ?→～
 V? 000 sin cos( ) cos 2 lim lim sin( ) lim sin 2 2 xxx x xx x x x x x x ?→ ?→ ?→ ? +? ? ? =? + ? =? ? ? b 2. ￡ ü
 ò xxx csccot)(csc ?= ′
ó xx 2 csc)(cot ?=′
? (arccos )x x ′=? ? 1 1 2
? 2 1 1 )cot(arc x x + ?=′
? (ch ) ? ′ = ? 1 2 1 1 x x
× (th ) (cth ) ?? ′ = ′ = ? 11 2 1 1 xx x 3
1
 xx x x x x x x csccot sin cos sin )'(sin sin 1 )'(csc 22 ' ?=?=?= ? ? ? ? ? ? = b
2
 x xx x x x x x 2 22 2 2 ' csc sin 1 tan sec tan )'(tan tan 1 )(cot ?=?=?=?= ? ? ? ? ? ? =′ b
3
 2 1 1 )'arcsin 2 ()(arccos x xx ? ?=?=′ π b
4
 2 1 1 )'arctan 2 ()cot(arc x xx + ?=?=′ π b
5
 1 22 11 1 1 (ch ) (ch )' sh ch 1 1 x yy yx ? ′=== = ? ? b
6
 1 22 11 11 (th ) = (th )' sech 1 th 1 x yyy ? ′== = 2 x? ?
 64 1 22 11 1 (cth ) (cth )' csch cth 1 1 x yyy ? ′==?=? = 2 x? ? b 3. p/ f ?￥?f ?
 ò xxxxf ?+= lnsin3)(
ó 3cos)( 2 ++= xxxxf
? fx x x x() ( )sin=+? 2 75
? )sec2tan3()( 2 xxxxf +=
? fx x x x x () esin cos=?+4 3
× fx xx x x () sin = + ?22 2 3
? fx x x () cos = + 1
ù fx xx x () sin ln = x? + 2 1
ú x xx xf ln cot )( 3 + =
? fx xx x x x x () sin cos sin cos = + ?
ü fx x x x ( ) (e log )arcsin=+ 3
Y xxxxxf sh)ln3(csc)( 2 ?=
T fx xx x x () sec csc = + ?
? x xx xf tanarc sin )( + =
3
1
 x x xxxxxf 2 11 cos3)'()'(ln)'sin3()(' ?+=?+= b
2
 b xxxxxxxxxxf 2sincos)'3()'()'(coscos')(' 2 +?=+++=
3
 )')(sin57(sin)'57()(' 22 xxxxxxxf ?++?+= xxxxx cos)57(sin)72( 2 ?+++= b
4
 )'sec2tan3()sec2tan3()'()(' 22 xxxxxxxf +++= )sectan2sec3()sec2tan3(2 22 xxxxxxx +++= b
5
 )' 3 ()'cos4()'(sinsin)'()(' x xxexexf xx +?+= 3 2 3 (sin cos ) 4sin 2 x ex x xx ? =++?b
6
 )')(2sin2()'2sin2()(' 3 2 3 2 ?? ?++?+= xxxxxxxf xx 65 3 5 3 2 )2sin2( 3 2 )2ln2cos21( ?? ?+??+= xxxxx xx b
7
 22 )cos( 1sin )cos( )'cos( )(' xx x xx xx xf + ? = + + ?= b
8
 2 )1( )'1)(ln2sin()1()'ln2sin( )(' + +??+? = x xxxxxxxx xf 2 2 )1(2 )ln2sin()1)(2cossin(2 + ??+?+ = xx xxxxxxxxx b
9
 33 2 (cot)'ln(cot)(ln) '( ) ln 'x xxx xx fx x +?+ = 22 3 2 (3 csc ) ln cot ln x xx x x x xx ?? = b
10
 )' cossin cos2 1()(' xxx x xf ? += 2 )cossin( )'cossin(cos2)cossin()'cos2( xxx xxxxxxxx ? ??? = 2 )cossin( )cossin(2 xxx xxx ? +? = b
11
 )')(arcsinlog(arcsin)'log()(' 33 xxexxexf xx +++= 2 1 1 ) 3ln ln (arcsin) 3ln 1 ( x x ex x e xx ? +++= b
12
 22 '( ) (csc 3ln )' sh (csc 3ln )( )'shf xxxxxxxx=? +? x 2 (csc 3ln ) (sh ) 'x xx x+? xxxxxxxxxx x xx ch)ln3(cscsh)2)(ln3(cscsh) 3 csc(cot 22 ?+?++?= )chsh2)(ln3(cscsh)3csccot( 2 xxxxxxxxxxx +?++?= b
13
 2 )csc( )'csc)(sec()csc()'sec( )(' xx xxxxxxxx xf ? ?+??+ = 2 )csc( )csccot1)(sec()csc)(sectan1( xx xxxxxxxx ? ++??+ = b 66
14
 x xxxxxx xf 2 arctan )')(arctansin(arctan)'sin( )(' +?+ = xx xxxxx 22 2 arctan)1( )sin(arctan)cos1)(1( + +?++ = b 4. pwL  (e )￥MLZ??ELZ?b yx= ln , )1 3 y1 ex ey ex 11 )(' == =
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°La y x= ?DwL yx a = log MM
M?' ú
$ 3 !M?1
? ?),( 00 xx xy = ^ () log a yfx x= = ￥ML
 | q1 1
 ?[ 1 ln 1 )(' 0 0 == ax xf
# a x ln 1 0 = b? ? 0 00 ln ()log ln a x 0 f xx a x= ==
¤ ? 1ln 0 =x
' ex = 0
V7
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 ?[V? ￥ML1(, )11 1)1( +?= xny
 ? D xà??￥?US1 1 n n x n ? =
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 !"? yax bxc=++ 2
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S{(,)|(,) 2 = xy xyV o V[T??tL￥BHML }
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?^ !
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 ' b] ?? ),( yx cbxaxy ++< 2 0<a H ,
yN cbxaxy ++> 2 { }0)(|),( 2 1 >?++= ycbxaxayxS b V o V[T??tLBHML?O?? ??tL 
 ),( yx ),( yx ?[ { }0|),( 2 2 =?++= ycbxaxyxS b ?N¤? { }0)(|),()( 2 213 <?++== ycbxaxayxSSS C ∪ b 8. ò !  ) V?
 fx() xx= 0 gx() xx= 0 )? V?
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$ fx() gx() xx= 0 ))( 21 xgcxfc (+ xx= 0 3
1
: ))()( 21 xgcxfcxh (+=
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 5  )(xh xx= 0 21 /)]()([)( cxfcxhxg ?= xx= 0 )9 V?
V7á 3 ±b
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? 21 cc ?= H
 ))( 21 xgcxfc (+  ) ^ V?￥
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) ? fxgx()() xx= 0 )￥ V?f ?b 3 f ?  ) V ?
()fx c= 0x= () | |gx x=  0x= )? V?
 5 ? H ) V?
? fxgx()() 0c = 0x= 0c ≠ H 0x= )? V?b f ? () () | |fx gx x== 0x= )?? V?
? 2 ()()f xgx x=  ) V ?b f ?  0x= () () sgn| |fx gx x== 0x= )?? V?
 ()() sgn| |fxgx x=  0x= 68 )9? V?b 10
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1]BuW ￥ V?f ?
￡ ü )(xf ij nji ,,2,1, null= ∑ = ′′′= n k nnnn knkk n nnnn n n xfxfxf xfxfxf xfxfxf xfxfxf xfxfxf xfxfxf dx d 1 21 21 11211 21 22221 11211 )()()( )()()( )()()( )()()( )()()( )()()( null nullnullnull null nullnullnull null null nullnullnull null null b ￡ ? ?  T￥?l 11 12 1 21 22 2 12 () () () () () () () () () n n nn n f xfx fx f xfx fx d dx f xfx fx null null nullnull null null 12 12 () (1) () () () n n Nkk k kk k d f xf x f x dx =? ∑ null null 12 12 12 12 () 12 (1) [ () () () () () () () () ()] n nn n Nkk k k k nk k k nk kk k fxfx fxfxfx fx fxfx fx ′′=? + + ′+ ∑ null nullnull null null 11 12 1 21 22 2 12 '() '() '() () () () () () () n n nn n f xfx fx fx fx fx fx fx fx =+ null null nullnull null null 11 12 1 21 22 2 12 () () () '() '() '() () () () n n nn n fx fx fx fxfx fx fx fx fx + null null null nullnull null null 11 12 1 21 22 2 12 () () () () () () '() '() '() n n nn n f xfx fx f xfx fx f xf x fx + null null nullnull null null 11 12 1 12 1 12 () () () () () () () () () n n kk kn k nn n f xfx fx f xfx fx f xfx fx = ′′ ′= ∑ null nullnull null null nullnull null null b 69