嘉应数学分析课件7468437

1、 2 : f x0 ), f ().4.2 f x0 f U ( x0 ) .4.2 f x0 f U ( x0 ) . x0 H1 , G! 0 ,x0 |G , | | xf ( x)f ( x0 ) |1, |f ( x) | d |f ( x0 ) |1.4.2 f x0 f U ( x0 ) . x0 H1 , G! 0 ,x0 |G , | | xf ( x)f ( x0 ) |1, |f ( x) | d |f ( x0 ) |1. , H1 , H ! 0“,f (x) | . |4.3 f x0 , f ( x0 ) ! 0 ( f ( x0 )0 ) ,0r0) r, ,f

2、 ( x0 ) ( f ( x0 )rG)G ! 0, x ( x0G , x0f ( x) ! r( f ( x)r0),4.3 f x0 , f ( x0 ) ! 0 ( f ( x0 )0 ) ,0r0) r, ,f ( x0 ) ( f ( x0 )rG)G ! 0, x ( x0f ( x) ! r fG , x0( f ( x)r0), x0, GG , x0G)! 0,|f (x)x ( x0f ( x0 ) |H0f ( x0 )r, f ( x) ! r ! 0.f ( x0 ) . r 24.4 f ( x), g( x)x0, (1)f (x)g(x),(2)f (x)g

3、( x),f (x) g(x),f ( x)/ g( x),g( x0 ) z 0(3)(4)x0 . y = c y = x R , ,亩 xnP( x)aa xa01n. xnP( x)aaxa Q( x) 01nxmbb xb01m . xnP( x)aaxa Q( x) 01nxmbb xb01m . .4.5, u0g(u) f ( x) x0 f ( x0 ). g( f ( x) x0 .H !0 , g(u) u0 , G1u0 |G1 , ! 0 , | uH !0 , g(u) u0 , G1G1 , H ,! 0 , | uu0 | g(u)g(u0 ) | u0 |G,

4、| u! 0, | x| f ( x)x0 |f ( x0 ) |G1 ,H !0 , g(u) u0 , G1G1 , G1 ! 0 , ! 0 , | uu0| f ( x) x0G,! 0, | x| f ( x)x0 |f ( x0 ) |G1 ,g(u0 ) |H ,| uu0 | | g( f ( x)g( f ( x0) | g(u) g( f ( x) x0 .g(u) u0 , lim(2)f ( x) xo x0lim g( f (x)xo x0g(u0 )g( limf (x).xo x0(*) f( x0 )u04.5 f ( x) x0 . . limxo x0 f

5、( x)u0limf ( x)xo fu0 ,limf ( x)u0 limf ( x)xofu0,xo f(*). limsin(1x2).xo1 limsin(1x2).xo1 sin(1x2 ) g(u)x2 ) sinu, u(1 limsin(1x2 )xo1sin(lim(1x2 )xo10.sin x. limxo02 xsin x. limxo02 xu u 1, sin x)sin xxlim(2xo1limxo02211.xsin x. limxo02 xu u 1, sin x)sin xxlim(2xo1limxo02211.x limsin(11)x . xxofsi

6、n x. limxo02 xu u 1, sin x)sin xxlim(2xo1limxo02211.x limsin(11)x . x1 )xxof lim(x1e , sinu u exof1)xlimsin(1xofsine.x f a ,b. fa , b , . f ( x) D .x D,( f ( x) t f ( x0 ) ),x0 () x0 D , f ( x) df (x0 ) f ( x) D(),f ( x0 ) f ( x)D(). sgn x1,-1; y ysin x1,-1; . ,sin x (y(22 1, -1 ) sgn x1,-1; y ysin

7、x1,-1; . ,sin x (y22 1, -1 )(4.6 f ( x) f ( x) a ,b.a , b . f ( x)a , b , f ( x)a , b. f ( x)a , b , f ( x)a , b.4.6 , f ( x) f (x) a, b , . f ( x)a , b , f ( x)a , b.4.6 , f ( x) f (x) a, b , . f (x)1 , x (0 , 1)x (0, 1). . 4.7 f ( x) a ,b, f (a) zf (b) . P f (b) f (b)f (a) f (b) PP( f (a)f (a), x0

8、 (a , b) , P.f ( x0 ) yf ( x) y, .yf (a)yf ( x)xf (b)oaxx0b f ( x)a ,b ,f (a) f (b) x0 ,0 ,f ( x0 )0 f ( x)a ,b ,f (a) f (b) x0 ,0 , f ( x0 )0,4.7. , , 4.7 ., . : f ( x) a , b , M m , f ( a , b ) m , M .仈. x0, r ! 0, n xn0r. x0, r ! 0, n xn0r. n, limxo f f.xn x1 , xn r . xn f (x)1 0, x1 f (0)rf ( x1

9、 ) ,n rx(0, x ) , xnr . x x01000 r n ).(: xn 0 ,f ) f ( x) x, y, 0 d x.y, ynxnx)( yn1yn 2 xyxn2xn1)( y 0,f ( y). f ( x) f a ,bf (a ,b) a ,b . : x0 a ,b , f ( x0 )x0 . f a ,bf (a ,b) a ,b . : x0 a ,b , f ( x0 )x0 .a df (a) ,f (b) d b.f (b), .aaf (a) bf ( x)x,F ( x)a)( f (b)f (a), f (b)b.F(a)F(b) ( f

10、(a)b)0. f ( x) a, b , F ( x) a,b x0 (a ,b), F(x0)0f ( x0 )x0 . f (x) a ,b ,4.8 y.1 ( x) f (a) , f (b) f (b) , f (a)f f (x) a ,b ,4.8 y. x1 ( x) f (a) , f (b) f (b) , f (a)f , f (a) , f (b) f (x) 1( y) .f1 ( y ) f (a ), f (b) xff1( y) ,0f (b), f (a)y0 y0x0( )y ax0b .f (b) y2y0y1 Gminy2f (a)y0,y0y1xHx

11、HxxaOb0 00 H ,a- Hx0Hx0H ) , y2b, f (x0H ) ,y1 ! 0,y1Gf ( x0min y2y0 , y0GfH (dy2 ) ( y1 d) y0y1( y1)y0f y1( y)f1( y 2 ),Hff1( y)H .0f1( y)0 1( y) f (a), xf. f ( x)sin x S S 2 2arcsin x 1 ,1 y y. arccos x , yarctan x , yarccot x ,. f ( x)sin x S S 2 2arcsin x 1 ,1 y y. arccos x , yarctan x , yarccot

12、 x ,. yxn (n) 0 f) 10 , f) , .xn y . 2. f ( x)I x2 I, x1, H ! 0,G! 0 , G ,| x1 x2 |f ( x2 ) |H ,|f ( x1 ) f ( x) I.俆仈.x1, f) . f ( x)俆仈.x1, f) . f ( x) x1 , x2 1 , f) ,| x2x1 |d| x- x|,|xx|1221xx12 H ! 0, GH , x1 |H ,| x1x2 | G| d | x2 |x1x2x 1 f) 1 y (0, 1).x1 y (0, 1).x 俆 f (x) I (G ) H0 ! 0, x1,

13、x2 I, | x1|f (x1 )傼 y. Gx2 |G, f ( x2 ) | t H0 .1,x (0, 1) x1 ),2G ,H1 G(Gy G2G , xx, | x- x|12121 ! 1.11 G1 (0 ,1)xx21x1xx2O1yx f ( x) I f ( x) ,I :(1) 俆, H ! 0, f ( x) I ,G H , x , GG ( x0 ,H ). f ( x) I. . 1G yH , (0, 1), 0xGmin2H , x0, H , x . 200x2 yx 1 f) H(2)GG ,G x0 . f (x) x0 IH ! 0 ,G , G

14、(H , x0 ),| x|f (x)x0 |f ( x0 ) |H .G (H , x0 ) x0 ( H ,I. f (x) x0),.4.9f a,b a,b, f: , .4.9f a,b a,b, f: , . 1 c 1 , 2 c , c 2 . f ( x) 1 , 2 f ( x) 1, 2. fG1 ! 0 , G2 H ! 0 , 1 ,2! 0 , f ( x2 ) |H ,f ( x2 ) |H .|G1 , |x1 , x2 1 , | x1x1 , x2 2 , | x1f (x1 )f (x1 )x2x2|G 2 ,| G| x11minG1 , G 2 ! 0

15、, x1, x2 1 2, G ,x, x2 2 .x2 |x1 , x2 1f ( x2 ) |H .|f ( x1 )2.2 2 .x1 1,c |G , | x2G ,c 1 2 , | x1c | |f ( x1 )f ( x2 ) | d |f ( x1 )f (c) |f ( x2 )f (c) |d HH2H . f ( x) 1 2. c. 1 21 d x d 21 ,f ( x)0 ,x d 32, 1 , 2 (2 , 3 1, 3 , .f ( x) a, f) limf ( x) A.xo ff ( x) a, f) f ( x) a, f) limf ( x) A.xo ff ( x) a, f) H ! 0, limf (x)A , X f (x1