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Unformatted text preview: 2007 -- 2008 8 0 8 ~ 10 0 0 0 120 0 A (2007 0 ) * (0 50 40 0 ) 1 1s x = lim x y 0 (1 + cos x ) ln(1 + x ) 3 sin x + x 2 cos 3 sin x + x 2 cos s s 1 1 1 3 sin x + x 2 cos x 2 cos x = lim x = lim 3 sin x + lim x =3 lim x y 0 (1 + cos x ) ln(1 + x ) x 0 x 0 x 0 2x 2x 2x 2 1 2s s y( x ) = e - x 2 + e d x (x>0 dy = , f ( x) lim = 0, lim x y 0 sin x x 0 ln(1 + f ( x) f ( x) ) sin x = lim sin x = lim f ( x ) = A x x 0 x ln a x 0 x 2 ln a a -1 3s b= f ( x ) = lim n x 2 n -1 + ax 2 + bx , s f (x) d x = 1 0 x 2n + 1 ,s a= , d d 1 x < -1 x -1 - b + a x = -1 2 f ( x ) = ax 2 + bx -1 < x < 1 , 1+ b + a x =1 2 1 x >1 x y ln(1 + f ( x) f ( x) ) sin x = A, ( A 0), 0 lim 2 = xy 0 x ax -1 ~* (s 6 s ) 4s 1 lim xy 0 f ( x) lim = 0, lim x y 0 sin x x 0 5. ln(1 + f ( x) f ( x) ) sin x = lim sin x = lim f ( x ) = A x x 0 x ln a x 0 x 2 ln a a -1 = , , y( x ) = (1 + x 2 )tan x dy x=0 6. 0 y( x ) = x 2 f (sin x ), 7s yy = = 1 0 (s x > 0, a > 0 ,s e - x ( x 2 - 2ax + 1) s ) f ( x ) = e x - x 2 + 2ax - 1, f y x ) = e x - 2 x + 2a , f ( x ) = e x - 2, ( f ( x ) = e x - 2 = 0, x = ln 2, (0, ln 2), ( x = ln 2), (ln 2, + ) 8s s f (0) = 1, f y 0) < 1, 0 ( x > 0 , f y( x ) < f ( x ), x > 0 , f ( x) e x .(s ) F ( x ) = ( f y x ) - f ( x ))e x , F (0) = f (0) - f (0) < 0, ( F y x ) = ( f ( x ) - f ( x ))e x < 0, f ( x ) - f ( x ) < 0, ( g ( x ) = f ( x )e - x , g (0) = 1, g ( x ) = ( f ( x ) - f ( x ))e - x < 0, g ( x ) = f ( x )e - x < g (0) = 1, * 1s (0 40 32 0 ) ( x) ay f ( x ) = ( x 2 - a 2 ) ( x ) . * B. s D. P * as . A. s , P * C. s s, s lim x a f ( x ) - f (a ) ( x 2 - a 2 ) ( x ) - 0 = lim = lim( x + a ) ( x ) = 2a (a ) x a x a x-a x-a 2 * (s 6 s ) 2s f ( x) h A . f ( 0) y f ( x ) y C (0, f (0)) y D f y 0) = 0, lim ( xy 0 f y( x ) =1 x . B. f (0) y f ( x ) y . . y = f ( x) , (0, f (0)) -* f (0) ,,P s f (0) = 0, f (0) = lim f ( x ) = lim xy 0 x 0 f y( x ) x = 00 x f y( x )- * lim xy 0 f y( x ) = 1, f y( x ) ~ x x y = ax 2 3s A 1 y y = ln x B e , s lim (1 + ( x - a )) 1 x 2e 1 sin( x - a ) dd C 0 1 sin( x - a ) D a 1 x -a x - a sin( x - a ) x lim (1 + ( x - a )) 1 2e = lim(1 + ( x - a )) x a =e y = ax 2 y = ln x ,s : 2ax = 1 1 2 1 1 1 1 ,x = , y = , x = e 2 , x2 = = e, a = x 2a 2 2a 2e 1 - cos x x 4s s f ( x ) = 2 x g( x) x>0 x 0 ,s g ( x ) - , s f ( x) y x = 0 0 0 A . ; - C - 6 * *; . B. s D 0 ,s . . 5s s lim xy a f ( x ) - f (a ) ( x - a) 1 3 = 1, 0 f ( x ) y x = a . * A . f (a ) y f ( x ) y 3 B. f (a ) y f ( x ) y * ... P (s 6 s ) C 0 s . D 0 . f (a ) = lim xy a f ( x ) - f (a ) f ( x ) - f (a ) 1 = lim = 1 2 3 x a ( x - a) ( x - a) ( x - a) 3 6s s f ( x ) = lim ny 1+ x ,0 1 + x 2n * B. f ( x )˪ D f ( x ) ع A . f ( x ) H * C f ( x )˪ . x = 1. x = -1. x = 0. 0 0 0 f ( x) = 1 + x 1 0 f ( x) y x < -1 x = -1 -1 < x < 1 , x =1 x >1 7s ( f y( x ) + ( f ( x )) 2 = x ,s f y 0) = 0, 0 H * A . f ( 0) y f ( x ) y C (0, f (0)) y s . B. f (0) y f ( x ) y . D (0, f (0)) * . y = f ( x) f (0) = 0,^ f ( x ) + ( f ( x )) 2 = x, f (0) = 0, f ( x ) = x - ( f ( x )) 2 f ^ ( x ) = 1 - 2( f ( x )) f ( x ), f (0) = 1, * ft ( x ) = x - ( f ( x )) 2 , x < 0, f ( x ) < 0; x > 0, f ( x )??? 0; 8s s g( x) - e - x f ( x) = x 0 x 0 ,s g ( x* ) x -- ( , s g (0) = 1, g y 0) = -1, x=0 f y x ) y (- , + ) ( A.s . C 0 . s 4 * B. -- * D 0 --, * -- (s 6 s ) ( f ( x ) - f (0) g( x) - e - x 0 gx ) + e - x 0 g ( x ) - e - x g ( 0) - 1 f (0) = lim = lim = lim = lim = x 0 x 0 x 0 x 0 ( x - 0) x2 2x 2 2 x 0, f ( x ) = xg ( x ) + xe - x - g ( x ) + e - x x2 0 0 lim f ( x ) = lim xg ( x ) + xe - x - g ( x ) + e - x xg ( x ) + xe - x g( x) - e - x = lim - lim x 0 x 0 x 0 x 0 x2 x2 x2 g ( x ) + e - x g (0) - 1 = lim - x 0 x 2 -x ( * g x ) - e g (0) - 1 g (0) - 1 = lim - = = f (0) x 0 1 2 2 P ... (0 28 0 ) 1s (8 0 ) 0 x > 0 ,s ( x 2 - 1) ln x 0 ( x - 1) 2 0 F ( x ) = ( x 2 - 1) ln x - ( x - 1) 2 , F (1) = 0, 1 F ( x ) = ( x 2 - 1) + 2 x ln x - 2( x - 1) = (2 - ( x + x (0, 1), F ( x ) <, F ( x ) F (1) = 0, ( x 2 -) ln x 0 1 (1, + ), F ( x ) = F ( x) = - 1 )) + 2 x ln x , F (1) = 0 x ( x -1) 2 1 1 - 1 2 2 ln x + - = + 1 2 ln x, F (1) 2, - = 2 x x2 F (1) = 0, 1 1 ( + 1) < 0, (1, + ), F y x ) > 2, F ( x ) > 0, F ( x ) ( x x2 ( x 2 - 1) ln x ( x - 1) 2 2s (10 0 )0 f (x) d [0,1] ... P (0,1) P f (0) = 0, f (1) = 1, a d b, s (0,1) f ( 0) = 0 < P ... * Ȫ 5 , a < 1 = f (1), a+b , c a b + =a+b f ( ) f ( ) (0, 1), 0 f (c ) = a , a+b * ...P (s 6 s ) f (c ) - f (0) = f ( )c, 0: (0 < < c ), f (1) - f (c ) = f ( )(1 - c ), (c < < 1), a c c (a + b ) = = , f ( ) f (c ) a 1 1-c 1-c (1 - c )(a + b) = = = , s f ( ) 1 - f (c ) a b 1- a+b 3s (10 0 )0 f (x) d [-1, 1] 0 0 ; f (-1) = 0, f (1) = 1, f y( 0 ) = 0 0 (-1, 1) 0 f ( ) = 3 0 : * (( f ( x ) = f (0) + f 0) x + 0 = f (-1) = f (0) + 1 1 f (0) x 2 + f ( ) x 3 , 2! 3! (0, x ) 1 1 ( f (0) - f (1 ), 1 -1, 0) ( 2! 3! 1 1 ( 1 = f (1) = f (0) + f (0) + f ( 2 ), 2 (0, 1) 2! 3! A H f (1 ) + f ( 2 ) = 6, * ( 0 f y ( x ) y [-1 1]s ,s f y ( x ) y [1 ,2 ]A H M HA m,d m Ȫ 0 1 ( ) ( f* (1+ f ( 2 )) 2 ,s M, 1 ,2 ] (-1, 1), [ f ( ) = * f 1 ) + f ( 2 ) ( = 3. 2 6 * (s 6 s ) ...
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This note was uploaded on 06/24/2008 for the course MATH In USTB taught by Professor Fan during the Fall '07 term at UVA.

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