1、D201 0577-86593350 13958890700 Email 2015101218:30 6A107 : () , 2007.7. , , 2002. 5. 第一章 函数与极限 第一节 函数 ( function ) N Z Q R 区间区间: : . . ,.a bRab bxax , ),(ba bxax , ,ba oxab oxab ( bxax bxax , , ),ba ,(ba ),xaxa ),(bxxb oxa ox b 区间长度的定义区间长度的定义: : (). 3.3.邻域邻域: : . 0, a ( ,).U a ,a . ( ,).U ax axa xa
2、a a ,a ( ,)0.U axxa , aaxx y, .R)( Dx Dxxfy, )( ,Dx 1. , yx, , yx DxxfyyW),( ; range). ; , (2) (3): . , . (1) . sin sin xx yxy x )1 , 1( : D 2 1 1 x y : 2 1xy : 1,1,():0,1.Df D 2 (1) ( )2ln , ( )lnf xxxx (2) ( ), ( )f xxxx 22 (3) ( )sincos, ( )1f xxxx 定义定义: : ( , )( ), ( ). Cx y yf x xD yf x ox y ),
3、(yx x y W D 222 ayx ox y ),(yx x y W D ox y ),(yx x y W D ox y ),(yx x y W D 2 21,0 ,( ) 1,0 xx f x xx 12 xy 1 2 xy , , (piecewise function). 1. 1,1 10,2 )( xx xx xfy : )( 2 1 f . )( 1 t f f (x) , f (x) ),0D ),0W 2 1 2 1 2)(f2 )( 1 t f 10 t, 1 1 t 1t , 2 t x y O xy2 xy 1 1 10 sgn00 10 x yxx x . , (,
4、)D . 1 ,0 , 1 W xxx s g n 1 -1 x y O 1. ,Dx ,0M ,)(Mxf )(xf. M , f (x)D. ,y= sinx1y 1 01 , x y 1 1, x 1 1 x 1 1, x ,xR y = M 2121 ,xxIxx , )()( 21 xfxf )(xf I , )()( 21 xfxf )(xf I ; . 1 x 2 x x y O 2 yx , (-,0) . (0,+) 2 yx. (-,+) 3. ,Dx ,Dx f (x) ; f (x) . f (x)=x2+2R f (x)=x3R (-,0)U(0,+) (-,+) (
5、-,+) . 2:( ),( ) ; f xRf x ( )( )(), ( )( )()g xf xfx h xf xfx ( )( ) ( ) 2 g xh x f x 4. ,0,LDx ,DLx )(xf, xO2 y 2 L ( ). , y=sin x 2 y=tan x : . , Cxf)( (1) , , Dxxfy, )( Wxxfy ,)( 1 1 f f . 10yf (x) ; (2) 20 . , ),(,exy x , , . x y O )(xfy xy ),(abQ () () . 30 yf (x) 1. ; ; ; ; . (1) ( power func
6、tion ): yx :. a -3 y = x ; 0, x=0. , 13 yx 0 , 1 2 yx 1 2 yx 1 3 yx 0,; 1, 20 x. 30 x=0,y=1,(0,1). (2) exponential function (0,1) x yaaa (,) 0,+ 0a1 0a 0 x = 0 x 0 x y O 1 1 2. x y O4 12 321 P18 1 213 3 4 13 6 2 2 1 (1)sin 1 y x 2 2sin (2)ln(tane) xx y (1) 2 2 1 (1)sin 1 y x 2 yu sinuv 1 2 vw 2 1wx
7、1 22 2 ,sin ,1yu uv vwwx 2 2sin (2)ln(tane) xx y ln ,yu tanuv e w v 2 2sinwxx 2 ln ,tan ,e ,2sin w yu uv vwxx 2 2,0,1 ( ), ( ), 1,0,1 ( ). x xxex f xx xxxx fx ( ) ,( )1 ( ) ( ),( )1 x ex fx xx 0 1( )1,x , 0 x ( )21,xx ;20 x, 0 x 2 ( )1 1,xx ; 1 x ,1)(20 x , 0 x , 12)( xx ;2 x, 0 x, 11)( 2 xx ; 01 x
8、. 2, 1 20 01 1 , , 2 , )( 2 1 2 2 xx x x x e x e xf x x 第二节 极限 “ 6n r 1 12sin 6 lr : 1 l .6 n ln 12sin 6 n ln r n :6 n ln 2 24sin 12 lr : 2 l :, 3 , 2 , 1 , 21n xxx (1) n x n x. . ;, 5 1 , 4 1 , 3 1 , 2 1 , 1 ;, 2 1 , 8 1 , 4 1 , 2 1 n ) 1( n n 1 2n ., 21 n xxx 1 x 2 x 3 x 4 x n x xnn ).(nfxn ;,)1(
9、, 1 , 1, 1 1 n ) 1( 1 n 111 2,1+,1+,1+,; 234 1 1+ n n n n 1 ) 1( 1 n xn “ . 1 ) 1( 1, n n n xn xn n a, , axn n lim xn a xn , xna, () n xa n 1 (1)lim; 3 n n (2) lim1 n n (3) lim2n n 1 (1)lim0 3 n n (2) lim1 n n (3) lim2n n n n qlim 1q 1q 11qq 0, 1, , 2 147(32) (2)lim; n n n (1)lim(1); n nn (1)lim(1)
10、n nn (1)(1) lim (1) n nnnn nn 1 0 2 147(32) (2)lim n n n 2 2 3 lim 2 n nn n 3 1/ lim 2 n n 3 2 xn xn xn A f (x) 1. 4f ( x )x0(x0) 2. (1) xx0 , f (x) 2: (1) xx0 , (2) x xx0 , xx0 , . ,)(Axf 0 lim( ) xx f xA 0 ( )()f xA xx 00 00 , 2 xx xx ; 0)()() 1 (AxfAxf 0 0 lim)2(xxf xx y=x+1 ( xR ) 2 1 -1 0 1 x y
11、 x 2 1 1 x y x y x1 y 2 x11 2 . 1, 1 ; 1,3 )( x xx xf 3 0 1 x y )(lim 1 xf x x x 3lim 1 )(lim 1 xf x |3)(|xf|33|x |1|3x0 ) 1(x 3 ) 13(lim) 1 ( 2 x x x x x 1 sinlim)2( 0 ) 13(lim) 1 ( 2 x x 5 x x x 1 sinlim)2( 0 0 |0 1 sin| x x 0 | 1 sin| x x | x )0(x 1.5 : Axf)0( 0 Axf xx )(lim 0 : . Axf xx )(lim 0
12、)(lim)(lim 00 xfxf xxxx xx0 , ,)(AxfA f (x) , 0 xx Axf)0( 0 Axf xx )(lim 0 xx0 , ,)(AxfA f (x) , 0 xx A 5. 0,1 0,1 0, 1 )( xx x xx xf 0 x )(xf : . )(lim 0 xf x ) 1(lim 0 x x 1 )(lim 0 xf x )1 (lim 0 x x 1 ,)00()00(ff 1)(lim 0 xf x 1 xy x y O 1 xy 1 x y O 1 6. 0,1 0,0 0,2 )( xx x xx xf 0 x, )(xf : )(
13、lim 0 xf x x x 2lim 0 0 )(lim 0 xf x ) 1(lim 0 x x 1 ,)00()00(ff )(lim 0 xf x xy2 1 xy (2) x , f (x) 1.6 x|x|, ,)(Axf A f (x) x , lim( ) x f xA ( )()f xA x lim ( ) x f xA lim( ) x f xA . Axf x )(lim Axfxf xx )(lim)(lim lim arctan x x 2 lim arctan x x 2 lim arctan. x x lim sin x x x sin11x lim sin x
14、x 1. , lim( )f xAlim( )g xB lim( )( )f xg x lim( )lim( )fxg x AB )()(limxgxf)(lim)(limxgxf AB lim( )lim( )kf xkf xkA )( )( lim xg xf )(lim )(lim xg xf )0(B B A 0 lim( ) n f x ( n , n A0 ) lim( ) n f x 0 lim( ) n f x lim( ) n f x ; 1 )2( lim) 1 ( 2 2 x xx x 45 43 lim)2( 2 2 1 xx xx x 1 )2( lim) 1 ( 2
15、 2 x xx x ) 1(lim )2(limlim 2 2 22 x xx x xx 14 02 0 45 43 lim)2( 2 2 1 xx xx x ) 1)(4( ) 1)(4( lim 1 xx xx x )4( )4( lim 1 x x x 3 5 ; 14 1 lim) 1 ( 2 x x x ; 12 13 lim)2( 2 2 xx x x 1 1 lim)3( 2 3 x x x 14 1 lim) 1 ( 2 x x x 2 2 1 4 11 lim x xx x 0 12 13 lim)2( 2 2 xx x x 2 2 11 2 1 3 lim xx x x 2
16、 3 1 1 lim)3( 3 2 x x x 3 3 1 1 11 lim x xx x )( 0 1 1 lim 2 3 x x x 0 1 010 1 010 nn- n mm- x m , n m ).(lim 2 xxx x )(lim 2 xxx x xxx xxx x 2 22 lim xxx x x 2 lim 1 1 1 1 lim x x 2 1 0 )(lim 0 uxg xx )(),()(xguufyxgf 0 0 ),()(xUxgfy , 9 3 2 x x u u x3 lim 3 1 lim 3 x x = 6 1 6 6 6 1 g(x)=u 11 . :
17、1 ,xu , 1lim 1 u x 1 1 1 1 2 u u x x 1 u ) 1(lim 1 u u 2 2 1 ) 1)(1( lim 1 x xx x ) 1(lim 1 x x 2 0 sin lim1 x x x 1 0 lim(1) x x xe (1) (2) 00 sin( ) lim( )0lim1. ( ) xxxx x x x 00 1 ( ) lim( )0lim(1( ). x xxxx xxe 1 (lim(1) x x e x x 71828. 2e 0 tan3 lim; x x x ; cos1 lim 2 0 x x x 0 tan3 lim x x
18、x 0 sin 33 lim() 3cos3 x x xx 00 sin 31 3limlim 3cos3 xx x xx 3 2 0 cos1 lim x x x 2 2 0 2 sin2 lim x x x 2 2 0 ) 2 ( 2 sin lim 2 1 x x x 2 1 2 0 ) 2 2 sin lim( 2 1 x x x 1 0 lim(1); x x x lim(1) ; x x k x lim(1) x x k x lim(1) x k k x k x lim(1) x k k x k x k e 1 0 lim(1) x x x 1 1 0 lim(1) x x x 1
19、 e 1 1 0 lim(1) x x x 1 lim () ; 1 x x x x 1 lim () 1 x x x x 2 lim (1) 1 x x x 1 22 lim (1)(1) 11 x x xx 1 2 2 22 lim (1) lim (1) 11 x xx xx 2 e (1) , f (x) : 1/x ; )x , . )x 1. 0. x. . 0 (x) 0 xx . Axf xx )(lim 0 Axf)(,)(x 1.2 () , 2 1cos1 lim 2 0 x x x )( 2 1cos1 2 x x x )( 2 1 1cos 22 xxxx x0 2
20、2 1 1cosxxx 1: ; 2: ; (2) )(,),(),( 21 xfxfxf n )()()(lim 21 xfxfxf n )(lim)(lim)(lim 21 xfxfxf n 0 . 0 sin lim x x x 1 3. x x x 1 sinlim 0 3 x 1 sin, xx0, ; Mxfxf | )(|)( 0)()(xgxg |0)()(|xgxf| )(|xgM0 0)()(limxgxf 0 1 sinlim 0 x x x | )(|)(|xgxf (3) xxxxsin,0 2 0, x x x 2 0 lim , 0 2 0 lim x x x ,
21、 x x x sin lim 0 1 x0 x20 x0 x0 x20 sinx0 x0 (3) ,0lim , );(o ,lim , , ; 1.8 0, . ,0lim k ; x x x 3 lim 3 0 , 0 xx3 3x , );3( 3 xox 9 3 lim 2 3 x x x 3 1 lim 3 x x 6 1 93 2 xxx (4) , 1lim , x x x tan lim 0 1 ) cos 1sin (lim 0 xx x x x, tanx x , .tanxx x x x )1ln( lim 0 x x x 1 0 )1ln(lim )1 (limln 1
22、 0 x x x eln1 x, ln(1+x)x, .)1ln(xx , 2 2 1 x x0: 1 x a sin x cos1x1e x 1)1 ( x x , x x ,ln ax 4. : : : , lim : lim lim lim lim lim 5. x x x 5sin 2tan lim) 1 ( 0 x x x5 2 lim 0 5 2 . )0( sinsin lim x x x 1lim x x x 6() : : x, sinxx, sin ; xx x x 3 sin lim)2( 3 0 xx x x 3 lim 3 0 3 1 3 1 lim 2 0 x x
23、(1)1.9xx0 (x) , f (x)xx0(x), )(lim )( 0 xf x xx f (x) (2) , )( 1 xf ; , ,0)(xf )( 1 xf , . , : () 0 1. , , 0 x , .)( 0 xxf (1) (2) (3) : ; ; )()(lim 0 0 xfxf xx )(lim 0 xf xx 0 )(xxf 2. 2 1xy () xysinln 1cosxy 3. (1) (2) ; (3) , ),()(lim 0 0 xfxf xx , f (x) : , , f(x) . f (x) ; xytan 2 x y O 2 x 1x
24、: x y 1O (3) 0,1 0,0 0,1 )( xx x xx xfy x y O 1 1 , 1)00(f1)00(f 0 x ,)(lim 0 xf x , 21n xxxaxn n lim 2. 0 lim( ) xx f xA lim( ) x f xA Axf xx )(lim 0 Axf xx )(lim 0 Axf xx )(lim 0 )(lim)(lim 00 xfxf xxxx A 1. 2. ( )0 sin( ) lim1 ( ) x x x 1 ( ) ( )0 lim (1( ) x x xe ( ) ( ) 1 lim (1) ( ) x x e x 1.
25、 0 lim( )0 xx f x Axf xx )(lim 0 Axf)(,)(x 2. 0 0 lim( , ) xx yy f x yA 00 (,)f xy 1.2 124(1)(2) 781011 第二章 ( ) 微分学 第一节第一节 导数的概念导数的概念 s O 1. v )()( 0 tftf 0 tt lim 0 tt v )()( 0 tftf 0 tt 2 2 1 t gs 0 t )( 0 tf)(tf t () 播放播放 () () () () () () () () () () 2. N T 0 x M M x M N M T ( ) M N tan )()( 0 x
26、fxf 0 xx MT tanlim lim 0 xx k )()( 0 xfxf 0 xx x y )(xfy C O : . : N T 0 x M x x y )(xfy C O s O 0 t )( 0 tf)(tf t 1 . 0 lim xx 0 0) ()( xx xfxf x y x 0 lim , : ; 0 xx y ; )( 0 x f ; d d 0 xxx y 0 d )(d xxx xf , , . )( 0 x f x y x 0 lim (1) , . 0 x (2) (3) I , . : ; y ;)(x f ; d d x y . d )(d x xf :
27、, x, x, h )( 0 x f 0 )( xx xf x xf d )(d 0 I . 00 ()( )()( ) limlim xh f xxf xf xhf x y xh )(tfs 0 t )(:xfyC M )( 0 t f )( 0 x f s O 0 t )( 0 tf)(tf t N T 0 x M x x y )(xfy C O : ()( 1 ); x yf xxf step x ( ) ;2: )yf s xxf x ep x x t 0 0 ()( ) ( )( 3: )lim x y x x f xxf x y xfx st p x e 0 ()( ) ( )li
28、m h f xhf x fx h 0 lim xx 0 0) ()( xx xfxf : 1. (C ) . : y 2. : ax afxf )()( ax lim ax ax nn ax lim (lim ax 1n x 2 n xa 32 n xa) 1 n a x xfxxf )()( 0 lim x xy ( ) 1 )( xx )(x)( 2 1 x 2 1 2 1 x x2 1 x 1 )( 1 x 11 x 2 1 x ) 1 ( xx )( 4 3 x 4 7 4 3 x h xhx h sin)sin( lim 0 3. . : h xfhxf)()( 0 lim h 0
29、lim h ) 2 cos(2 h x ) 2 cos(lim 0 h x h xcos xxcos)(sin xxsin)(cos (hx) 4. . : h xfhxf)()( 0 lim hh xhx h ln)ln( lim 0 hh 1 lim 0 x x 1 )(ln 0 lim h h 1 x 1x 0 lim h eln )(tan 0 x f : : )0)( 0 x f x y O )(xfy C T 0 x M x y O1 1 1 1 5. , ? . : 3 2 3 1 x , 3 11 3 1 32 x ,1x,1y (1,1) , (1,1) 6. x = 0.
30、: h fhf)0()0( h h 0h,1 0h,1 h fhf h )0()0( lim 0 , h fhf h )0()0( lim 0 1lim 0 h 1 h fhf h )0()0( lim 0 ) 1(lim 0 h 1 2. )( 0 x f )( 0 xf 3. . () () )(bf , : a , b , . 1. : x , , 0 x x . : x . : xy x = 0 , . x y O , )( 0 xf () () )0( x )0( x )( 0 xf 0 x , xxf)( x = 0 2 . , , x y O xy ).(, 0, 0,sin )
31、(xf xx xx xf ( )cosfxx ( )1,fx (0)f (0)f (0)1. f cos ,0 ( ). 1,0 xx fx x x0 , x=0 , 0 sin0 lim h h h 0 0 lim h h h 1 , a, b, , : x=3 , (3)f 3 3 lim 3 x axa x a (3)f 2 3 9 lim 3 x x x 6 6,9ab , , (30)9(30)3ffab 1. : 3. : 4. , ; 5. : 6. , . ; . ) (C ) ( x ) (sinx ) (cosx axf)( 0 2. axfxf )()( 00 ) (ln
32、x ;0 ; 1 x ;cos x;sin x x 1 ; ; 3, 4, 5, 6 , 7 1. : )(x f , )( 0 x f ; : 0 )( xx xf)( 0 x f : ? )()( 00 xfxf 函数的微分导数的应用 第三节 2. 1. , 0 0 ( ) : (): (1) )( )( lim) 3 xg xf ax ( ) . )( )( lim )( )( lim xg xf xg xf axax ,)()()2xgxf 1. 0 0 () f (x), g (x) 1. 1 ax : , ax 3. )( )( lim xg xf , ,x 2. 3 0 0 1.
33、 : 0 0 2 3 : ! 26 6 lim 1x x x 1 6 6 lim 1 x 33 2 x 123 2 xx lim 1 x 26 6 lim 1 x x x 2. : x lim 0 0 2 2 1 lim x x x 1 2 1 1 x 2 1 x (2) )( )( lim) 3 xg xf ax () )( )( lim xg xf ax 2. . )( )( lim xg xf ax () ,)()()2xgxf f (x), g (x) 0 lim x : = 0 x x x e 2 lim x x e 2 lim . e lim 2 x x x 0 lnsin lim
34、,. lnsin x mx m nN nx 1. 0 cos lim cos x mnnx nmmx 0 cossin lim cossin x mmxnx nnxmx 0 sin lim sin x mnx nmx mx mxm sin cos nx nxn sin cos m n n m 5. : x lim xsin1 xcos1 () x xcos 1 x xsin 1 : 0 0 0 0 0 1 0 6. ).0(lnlim 0 nxx n x 0 : n xx x ln lim 0 0 lim x 0)(lim 0 n x n x x 1 1 n xn . )tan(seclim
35、2 xx x : ) cos sin cos 1 (lim 2 x x xx x x xcos sin1 lim 2 2 lim x 7. 0 0 0 0 0 1 0 xcos xsin 0 0 8. xx x lnlim 0 x x x1 ln lim 0 2 01 1 lim x x x .lim 0 x x x 0 0 : x x x 0 lim xx x ln 0 elim 0 e1 0 0 0 0 0 1 0 )(lim 0 x x 0 2. (a, b) 1. a, b , )0)( x f () . a, b , (1) x y o )(xfy a b A B 0)( x f x
36、 y o )(xfy 0)( x f a b B A (a, b) , f ( x ) 9. . : 12186)( 2 xxxf)2)(1(6xx ,0)( x f 2, 1xx x )(x f )(xf ) 1,( 2 00 1)2,1 (),2( 21 , 1,();,2 .2,1 1 2 xO y 12 2 22 ln(1) 11 xx xx xx x0 22 ( )1ln(1)1f xxxxx ( )fx ( )fx 0,( )(0)xf xf 0 x 22 1ln(1)1xxxx ( )0,fx 22 1ln(1)1xxxx 2 ln(1)xx 2 (ln(1)xxx 2 1 x
37、x 2 ln(1)xxx 2 1 1xx 2 (1) 1 x x 2 1 x x 0, +)f (x) 0, 2 2 1 () 1 xx x . , (1) , ; (2) , . . , 0)( 0 x f)(xf 0 x )(xf 0 x 0 x0)( 0 xf（）. . : 52 ,xx 641 ,xxx 3 x 2) 1) . 31292)( 23 xxxxf , , , 1 2 xO y 12 ox y a b )(xfy 1 x 2 x 3 x 4 x 5 x 6 x () , 0)( x f , 0)( x f ),( 0 0 xUx ,)( 0 xxfx0 , (1) “ (2
38、) “ .)( 0 xxf 00 (,)xxx, 00 (,)xxx , , 0)( x f .)( 0 xxf 00 (,)xxx , 00 (,)xxx , , 0)( x f (3) x0, ,)(x f .)( 0 xxf x y o 0 x x y o 0 x );() 1 (x f (2)( )0fx ;,)() 3(x f .)4( x y o x y o x y o 0 x 0 x x y o . 23 ( )(1) (1)f xxx 223 ) 1() 1( 3) 1)(1(2)(xxxxxf ) 15() 1)(1( 2 xxx 0)( x f : 1, 5 1 1x :
39、)(x f ( )f x ( )fx (1,) 1 ( ,1) 5 1 5 1 1 ( 1, ) 5 1(, 1) x 13456 ( ) 53125 f(1)0f () , ; . : (1) )( 0 x f 0 0) ()( lim 0 xx xfxf xx 0 )( lim 0 xx xf xx ,0)( 0 x f ,0,0 0 xx 00 xxx;0)( x f 00 xxx,0)( x f 0 x 0 x 0 x .)( 0 xxf (2) . 12. . : 1) ,) 1(6)( 22 xxxf) 15)(1(6)( 22 xxxf 2) ,0)( x f 1,0, 1 32
40、1 xxx 3) ,06)0( f ; ,0) 1 () 1( ff. 1x y 1O (3) f (x (1) (2) maxM, )(af)(bf a, b. 21 2 33 ( )(1) 2, 2f xxx 24 2 33 12 2 33 2 (1) ( ) 3 (1) xx fx xx ( )0,fx 24 2 33 (1)0 xx 1 2 x 01xx 1 () 2 f 3 1 ( )4 2 f xx (0)f( 1)f ( 2)f 33 ( )243f xx 21 2 33 ( )(1) 2, 2f xxx 1,1, 33 43, 3 4 12 2 (P66): 7 1. )( )
41、( lim xg xf , )( )( xg xf )( )( xg xf ? . , x xx x x 1 2 0 cossin3 lim 2 1 xx x )1ln( 0 )03( 2 1 2 3 : 3) 2cos1x : 2 0 3 cos1 lim x x x 3 0 lim x x 3. xsin x 1coslim 0 x xxxsin 2 2 2 1 03 lim x x x xcos1 2 2 1 x 6 1 6 1 xx xxx x 2 0 sin )sin(cos lim , 1 x t 2 0 11221 lim t tt t 4. : t t 2 lim 0 2 1
42、)21 ( t 2 1 )1 ( t 2 )1 ()21 ( lim 2 3 2 3 2 1 0 tt t 4 1 , axxaxf3sin 3 1 sin)( 3 2 x , : )(xf 0)( 3 2 f 2a )(xf 2)(axf3)( 3 2 f 5. )(3cos)cos( 3 2 3 2 a ,3sin3sin2xx , . 01 2 1 a 1 ,0)(xf Nnxxnxf n ,)1 ()( ).(limnM n : )(xf ,0)( x f ) 1(1 )1 ( 1 xnxn n 6. n xn)1 ( 1 )1 ( n xnxn ,)(xfx )(nM 1 ) 1 (
43、 n n n ) 1 1 ()( n fnM )(limnM n 1 e 1 ) 1 1 1 (lim n nn 1 1 n x 函数的微分(之一) 第三节 : , ? x , A , , 2 xA 0 x x xx 0 2 0 xA xx 0 2 )( x x 0 x 0 x x 0 x x, 0 x , 0 xx , : 0 x ( A x ) )(xfy xA .dxAy )( xoxA , (1) dy x xAyd (2) Axf (x)x0 y dy () 1 ox Ax 1 (5),().xydy y=x3x0 x (4)A0dyy 33 00 ()yxxx 223 00 33(
44、)() .xxxxx )1()2( x0dy. .3d 2 0 xxy ,x.3 2 0 xxdyy (0).x (3)()ydyox x : : “ , )()( 00 xfxxfy ) )( (limlim 00 x xo A x y xx A )( xoxA , 0 x , xxfy)(d 0 : 0 x , xxfy)(d 0 “ )(lim 0 0 xf x y x )( 0 xf x y )0lim( 0 x xxxfy)( 0 )()( 0 xoxxf xxfy)(d 0 , : .dxx )( d d xf x y (d x x). dxxfdy) ( 1. 01. 0, 1x
45、x 2 xy yd 01. 0 1 x x x2x 01. 0 1 x x 02. 0 . y=x3x=2 . xxyd)(d 3 1 . 0 2 2 1 . 0 23d x x x xxxy1 . 023 2 3 2 xx x=0.1x=2 2 d x y xx d3 2 )d(xx 2 2 d3 x xxxd12 2 . 1 ox y 0 M 0 x 0 xx y x dy M P N tany dyyxNP y dy dxxfdy) ( 1( )d C ln; x aa dx ; ln dx xa 1 ;xdx 2()d x 3() x d a4(log) a dx 0; cos;x d
46、xsin;x dx ; x e dx; dx x 7()d8()d 5()d6()d x eln x sin xcosx 11(sec )dx 2 ; 1 dx x 2 ; 1 dx x 2 ; 1 dx x 2 . 1 dx x sectan;xx dx csccot;xx dx12(csc )dx 13(arcsin )dx 14(arccos )dx 15(arctan )dx 16(arccot )dx 2 sec;xdx 2 csc;xdx910(tan )dx(cot )dx ud u(x) , v(x) , (C ) , xxgufd)()()(u f 3. vudd vuuvd
47、d : u , . duufdy) ( ., 1),1 (ln 2 dyxxy )1ln()1ln(2xdxdy )1ln(2x dx x x 1 )1ln(2 4. : )(cosd 31x exdy )cos(d 31 xe x dxex x ) 3()(cos 31 xxxe x d)sincos3( 31 xdxe x sin 31 )1 ( 1 1 xd x 5. : , 0)d(cos()sin( dyxxy xxyyxdcosdsin)sin(yx0)d(d yx xyd d )sin(cosyxxy xyxsin)sin( 6. : xxd) d() 1 ( tt dcos)
48、d()2( 2 2 1 x t sin 1 : . C C . )(C : )()( 0 xoxxfy x, y xxf)( 0 xxfxfxxf)()()( 000 xxx 0 )()()( 000 xxxfxfxf : (1) (2) (3) )()( 00 xfxxf 1. . : ,sin)(xxf ooo 130cos30sin 2 1 2 3 180 7. )130sin( 00 )sin()( 00 xxxxf,cossin 00 xxx ,30 0 o x o x1 xxfxfxxf)()()( 000 x n 1 1 ) x( x x x x1 : ,1)( n xxf (4
49、) ,x ,0 0 x , xffxf)0()0()( (3) ),)()()( 000 xxxfxfxf x , 1)0(f)0(f 0 1 1 |)1 ( 1 xn n x n 1 (4) . : 3 10001 3 ) 1000 1 1 (1000 10 ) 1000 1 3 1 1(00033.10 8. )|(| 1 11xx n x n 3 1000 1 110 1. 2. : uufufd)()(d ( u ) 3. : 1,2 1. () :, 2. 3. xx ed)d(arctane x2 e1 1 xd x x 2 e1 e x x sind tand . 4 x 3 sec xxd2sin) (d. 5 Cx2cos 2 1
