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2019年江苏省普通高校“专转本”统一考试《高等数学》试卷

2019年江苏省普通高校“专转本”统一考试《高等数学》试卷
2019年江苏省普通高校“专转本”统一考试《高等数学》试卷

2019年江苏省普通高校“专转本”统一考试

一、选择题(本大题共

8小题,每小题

4

分,共32分)

l. 设当0→x 时,函数()

2()ln 1f x kx =+与()1cos g x x =-是等价无穷小,则常数k 的值为( ) A.

14 B.1

2

C.1

D.2 2. 0x =是函数()1

11

x

f x e =

+的( )

A. 跳跃间断点

B. 可去间断点

C. 无穷间断点

D. 振荡间断点 3. 设函数()f x 在0x =处连续,且()

0lim 1sin 2x f x x

→=,则()0f '=( )

A. 0

B.

1

2

C. 1

D. 2 4. 设()f x 是函数cos2x 的一个原函数,且()00f =,则()f x dx =?( )

A.1cos 24x C -

+ B.1

cos 22x C -+ C.cos2x C -+ D. cos2x C + 5. 设211ln 2ln 2

a dx x x +∞=

?,则积分下限a 的值为( ) A. 2 B. 4 C. 6 D. 8 6. 设()f x 为(),-∞+∞上的连续函数,则与

21

1f dx x ??

???

?

的值相等的定积分为( ) A.

()

22

1

f x dx x ?

B. ()

12

2

f x dx x ?

C. ()

112

2

f x dx x ?

D. ()

122

1

f x dx x ?

7.二次积分()011

,x

dx f x y dy --?

?

交换积分次序后得( )

A.()01

1

,y dy f x y dx --?? B.

()100,y

dy f x y dx -?

?

C.

()110

,y

dy f x y dx -?

? D.

()10

,y

dy f x y dx -?

?

8.设(

)1ln 1n

n u ?=-+

?

,1ln 1n v n ??

=+ ???,则( ) A.级数

1

n

n u

=∑与

1

n

n v

=∑都收敛 B. 级数

1

n

n u

=∑与

1

n

n v

=∑都发散

C. 级数

1

n

n u

=∑收敛,而级数

1

n

n v

=∑发散 D. 级数

1

n

n u

=∑发散,而级数

1

n

n v

=∑收敛

二、填空题{本大题共6小题,每小题4分,共24分)

9. 设函数()()1

12,1

,

1x x x f x a x -??-<=?≥??在点1x =处连续,则常数a = .

10. 曲线1t

t

x te y e ?=?=-?

在点()0,0处的切线方程为 . 11. 设()ln 1y x =+,若()

2018!n x y ==,则n = .

12.定积分

()1

4

1

cos

x x x dx -+?的值为 .

13.设()2,1,2a b →

?=-,3a b →→

?=,则向量a →

与向量b →

的夹角为 .

14.幂级数2

133n n

n x n

=+∑的收敛半径为 . 三、计算题(本大题共8小题,每小题8分,共64分)

15. 求极限()3

ln 1lim

1

x

x x t t dt

e →+-????-?.

16.求不定积分()2

x x

x e dx +?.

17.

计算定积分7

?.

18. 设()2

,z f x y x y =-,其中函数f 具有二阶连续偏导数,求22z

x

??.

19. 设(),z z x y =是由方程()2sin 1y x xy z +++=所确定的函数,求z x ??,z y

??.

20. 求通过()1,0,1M ,且与直线1111:123x y z L ---==

和21:2332x t

L y t z t

=+??

=+??=+?

都平行的平面方程.

21.求微分方程x

y y e '''-=的通解.

22. 计算二重积分??D

ydxdy ,其中D

是由曲线y =

与直线1y =及0x =所围成的平面闭区域.

四、证明题(本大题10分) 23.证明:当02x <<时,22x

x

e x

+<-.

五、综合题(本大题共2题,每小题10分,共20分)

24.已知函数()43f x ax bx =+在点3x =处取得极值27-,试求: (1)常数,a b 的值;

(2)曲线()y f x =的凹凸区间与拐点; (3)曲线()

1

y f x =的渐近线.

25.设()f x 为定义在[)0,+∞上的单调连续函数,曲线():C y f x =通过点()0,0及()1,1,过曲线C 上任一点(),M x y 分别作垂直于x 轴的直线x l 和垂直于y 轴的直线y l ,曲线C 与直线x l 及x 轴围成的平面图形的面积记为1S ,曲线C 与直线y l 及y 轴围成的平面图形的面积记为2S ,已知122S S =,试求: (1)曲线C 的方程;

(2)曲线C 与直线y x =围成的平面图形绕x 轴旋转一周所形成的旋转体的体积.

2016年江苏“专转本”高等数学试题及参考答案

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