考研高数习题集(上)

第二讲: 单元一: 定义求导

导数及应用

f(x)cosx 1

[ [f(x)cosx]'x 0 2]

x 0x

f(x)(cosx 1) f(x) f(0)

[lim 1 0 f'(0) 2]

x 0x

1. 设f(0) 1,f'(0) 2, 求: lim

2. 设f x 可导, f 0 1,f' 0 0, 求: lim

x 0

f(sinx) 1

lnf(x)

[lim

x 0

f(sinx) f(0)x 0sinx

1]

sinx 0lnf(x) lnf(0)x

3. 设lim

x a

f(x) bsinf(x) sinb. A, 求: lim

x ax ax a

sinf(x) sinbf(x) b

Acosb]

x af(x) bx a

[lim

4. 设f(x 1) af(x),f'(0) b(a,b 0), 求: f'(1). [f'(1) lim

x 0

f(x 1) f(1)a[f(x) f(0)]

lim ab] x 0xx

5. 设f(1 x) 3f(1 x) 8x(1 sinx), 并且f(x)可导, 求f'(1).

[f(1) 0,f'(1) 3f'(1) lim

x 0

8x(1 sinx)f(1 x) 3f(1 x)

lim 8,f'(1) 2] x 0xx

6. y y(x)满足: y ln[tan(x x)] x o( x), 求:y',y". [y' lim

1

2

y1

] ln(tanx),y"

x 0 xsinxcosx

7. 若y f(x)在x x0处有: y (1 x) 1 ( x)2, 则在x x0处有:dy ? [y' lim8. 求f' a ,其中f x 分别为:

(1)f(x) (x a) (x), 连续; [lim

y11

,dy dx]

x 0 x22

(x a) (x)

(a)]

x ax a

x a (x)x a

x a

(2)f(x) x a (x), 连续, (a) 0; [lim 0]

(x a)2 (x)

(3)f(x) (x a) (x), 有界. [lim 0]

x ax a

2

9. f(x) 2

a x

, 求: f'(a). [f (a) ln2,f (a) ln2,f'(a)不存在]

''

10. f(x),g(x)在( , )上满足: (1)f(x y) f(x)g(y) g(x)f(y) (2)f(0) 0,g(0) 1,f'(0) 1,g'(0) 0, 证明: f'(x) g(x). [f'(x) lim

x 0

f(x x) f(x)

f(x)g'(0) g(x)f'(0) g(x)]

x

11. 问f x 在x 0处是否连续?可导?

x

,x 01 ''

(1)f(x) 1 ex [f (0) 1,f (0) 0]

0,x 0

x 0''

(2)f(x) ,其中g x 有界 [f (0) f (0) f'(0) 0] x2g(x),x 0

2

x12xt 2,x 0

(3)f(x) [f'(0) lim limt2 0]

x 0t x2 0x 0

1

11 g(x)cos

g(x)cos,x 0 0] (4)f(x) , 且g(0) g'(0) 0. [f'(0) limx

x 0x 0x 0 (x sinx)f(x)

,x 0

12. 奇函数f(x)在x 0处可导, 问:F(x) 在x 0处是否连续? x

0x 0

可导? [f(0) 0,F'(0) lim

(x sinx)f(x)

2f'(0)]

x 0x2

1 2

xcos,x 0

13. 设 (x) 且f(x)在x 0处可导,令F(x) f[ (x)],求F'(0) x

0x 0

[ (0) 0, '(0) 0,F'(0) f'[ (0)] '(0) f'(0) 0 0] 14. 设函数 (x)在( , )上连续, 又f(x) cos (x),f'(x) sin (x), 证明: 对满 足 (x) n 的一切x, '(x) 1.

[f'(x) lim

cos (x x) cos (x)

sin (x) '(x)]

x 0 x

1

xarctan,x 0 x15. 考察函数f(x) 在x 0处的连续性,可导性,以及f'(x)的连续性. 0x 0

[f'(0) limarctan

x 0

x1 1

,f'(x) arctan (x 0),limf'(x) f'(0)] 2x 0x2x1 x2

1x

tf(t)dt

16. 若f(x)有连续的导数,且f(0) 0,设F(x) x2 0

c

F(x)连续,并问此时F'(x)是否连续?

x 0x 0

,确定常数c,使

xf(x) 2 tf(t)dtf'(0)f'(0)0

,F'(x) ,limF'(x) [limF(0) 0 c,F'(0) ]

x 0x 03x33

单元二: 公式与法则 1. 设y f(

2

x

dy3x 2

),且f'(x) arctanx2,求:

dx3x 2

. [ f'(

x 0

3x 212

)

3x 2(3x 2)2

x 0

3 ] 4

2. f(x)在x 1处具有连续导数, 且f'(1) 2,

求lim

d

f.

x 0 dx

x 0

[limf f(x)

1]

3. f可导,f(0) f(1) 2,f'(0) f'(1) 1,F(x) e [dF e4. 求y

':(1)y ln

f(x)

f(lnx),求:dF(x)x 1

(f'(x)f(lnx)

f'(lnx)

)dx,dFx

2 edx] x 1

xe [y' 1

e2x 1

] 2

x2(x x 1)

1ex1

[y' 2 cotex]]

(2)y xx 12

(3)y x

(4)y

x 1 x 2

[y' x

(x 1)(x 2)

[(2x 3)lnx

(x 1)(x 2)

]]

x

[y' 11 sinx ]] 2x4(x cosx)

x2 a2122

,y' 2 2x] 5. 求y

':(1)y

[y 2aa (2)y

b

a

e(x y)dy [y

2

b x

a x

etdt,y' e(b x) e(a x)]

222

1

,x 0 x 1 (3)y ln(1 x) [y' ]

1,x 0 x 1

(4)y f(x (x x)),求y' [y' f'(x (x x))[1 (1 2x) '(x x)]]

222

ln(x e)x 0

,,(a 0),求a.使f'(0)存在. 6. f(x) x

ax 0

[f(0 ) f(0 ) 1,f (0)

'

1'

,f (0) lna,a e

7. 选定参数A,C, 使立方抛物线:y A(x a)(x b)(x C),(a x b)与曲线

k1(x a)x ak k2k1b k2a

,,C y 光滑连接起来. [A 1] 2

k(x b)x b(a b)k k 212

[f (a) k1,f (a) A(a b)(a C);f (b) A(b a)(b C),f (b) k2] 8. f(x) lim

n

'

'

'

'

x2e

n x 1

ax b

, 问a,b为何值时,f(x)可导, 并求f'(x)

en(x 1) 1

x 1 ax b

1 2x 1

] [f(x) (1 a b),x 1,a 2,b 1,f'(x) ,

2xx 1 22

xx 1

9. (1)y xsinx,求y

2

2

10

1281028 10!(10)

( x ),y(0) 45 28] x 0. [y 28!2 8!

x3(5)5(5)

(2)f(x) ,求f(0). [f(x) x ,f(0) 5! 120]

1 x sinx

x 0121

(3)f(x) x,, 求f"(0); [f(x) 1 x ,f"(0) ]

x 03!3 1 ex 1

,x 011 (n)

xn ,f(n)(0) (4)f(x) x, 求:f(0). [f(x) ]

(n 1)!n 1 1x 0

10. (1)y cos3x, 求

2

dy1(n)

[(n 1)y (1 cos6x),y n

dx2

n

6ncos(6x

2

n

)]

23

(2)f(x) (1 x)(1 x)(1 x),求f'''(x)x 1

[f (1 x)(1 x)(1 x x),f"' 6(1 x)(1 x x) ,f"'(1) 36]

322

x5(5)

(3)f(x) , 求: f(x).

x 1

[f(x) x x x x 1

4

3

2

15!

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