专插本模拟试题4

2011专插本高等数学模拟试题（四）

一．选择题(本大题共5小题，每小题3分，共15分)

1． 函数()f x 的定义域为[]0,1，则函数1

1

()()5

5

f x f x ++-的定义域是（ ）.

()

A 14,55??-???? ()

B 16,55??

????

()

C 14,55??

????

()D []0,1 2．设0

()()x F x f t dt =

?

，其中2,01

()1,12

x x f x x ?≤≤=?≤≤?，则下面结论中正确的是（ ）.

()

A 31,01()3, 12x x F x x x ?≤≤?=??≤≤? ()

B 311

,01

()33

, 12x x F x x x ?-≤≤?=??≤≤?

()

C 3

1,01()3

1,12x x F x x x ?≤≤?=??-≤≤?

()D 3

1,013

()2,12

3x x F x x x ?≤≤??=??-≤≤??

3．曲线(1)(2),(02)y x x x x =--≤≤与x 轴所围图形的面积可表示为 （ ）.

()A 2

(1)(2)x x x dx ---? ()

B 1 2 0

1

(1)(2)(1)(2)x x x dx x x x dx ---

--?

?

()C 1

2

0 1

(1)(2)(1)(2)x x x dx x x x dx ---+--?? ()

D 2 0

(1)(2)x x x dx --?

4.广义积分3

(1)

x dx x +∞+?

为（ ）.

()A 1- ()B 0 1()2

C - 1()2

D

5.积分(2)x a

f u du '?

为（ ）.

()A 2[()()]f x f a - ()B (2)(2)f x f a - ()C 2[(2)(2)]f x f a - ()

D 1[(2)(2)]2

f x f a -

二．填空题：(本大题共5小题，每小题3分，共15分)

6. 若 3sin 41

,0()0

ax x e x f x x

a x -?+-≠?

=??=?

在 0x =连续，则 a = .

7. 曲线2

3

1x t y t

?=+?=?在2t =处的切线方程为 .

8、2

2

(1cos )x x dx -+?＝ .

9、当a = 时，抛物线2y x =与三直线,1,0x a x a y ==+=所围成的图形面积最小。 10、 微分方程 450y y y '''-+=的通解为 .

三．计算题：(本大题共8小题，每小题6分，共48分)

11．计算1

2

3lim (

)

6

x x x x -→∞

++。

12．设[cos(ln )sin(ln )]y x x x =+，求dy dx

。

13．由方程2

2ln

arctan

y

x x y +=所确定的y 是x 的函数，求dx

dy 。

14．计算不定积分2

2

1sin cos dx x x ?.

15．计算定积分 1 0

x

x dx

e e

-+?

。

16．设函数)sin()cos(y x xy z ++=，求偏导数x

z ??和

y

x z ???2

.

17．计算D

I dxdy =

??，其中{}(,)

1D x y x y =+≤.

18、计算微分方程

2

2

(1)(1)

dy x y dx

y x +=

+满足初始条件 (0)1y =的特解.

四．综合题： (共2小题，第一题12分，第二小题10分，共22分)

19、设函数22

()()f x x f x dx =-

?

，求()f x 在区间[0,2]上的最大值与最小值.

20、设1sin ,0()0,0

x x f x x

x α

?≠?

=??=?

， （α为实数）

试问α在什么范围时, （1）()f x 在点0x =连续； （2）()f x 在点0x =可导.

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