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重庆市普通高校专升本统一选拔考试《高等数学》试题

重庆市普通高校专升本统一选拔考试《高等数学》试题
重庆市普通高校专升本统一选拔考试《高等数学》试题

2006年重庆市普通高校专升本统一选拔考试《高等数学》试题 一、填空题(本大题共5小题,每小题4分,满分20分)

1、=-→x

x x 1

)31(lim _________________。 2、n n n x n ∑∞

=13

的收敛半径为

____________________。

3、dx x x ?-22

2

sin π

π=_________________。

4、0145=-'-''y y y 的通解为____________________。 5

?

???????????---534

1112332122131的秩为

_________________。

二、单项选择题(本大题共5小题,

每小题4分,满分20分)

6、函数

x x y 33

-=的减区间为( )。

A 、]1,(--∞

B 、]1,1[-

C 、),1[+∞

D 、

),(+∞-∞

7、函数)(x f y =的切线斜率为2x

,通过

)2,2(则曲线方程为( )。

A 、

3

4

12

+=x y B 、

12

12

+=

x y C 、

32

12

+=x y D 、1412+=x y 8、设

3

21

n

u n =

,n n

n v 53

=,则( )。

A 、∑∞

=1

n n u 收敛,∑∞

=1

n n

v 发散 B 、∑∞

=1

n n u 发散,∑∞

=1

n n v 收敛

C 、∑∞

=1n n u 发散,∑∞

=1n n v 发散 D 、∑∞

=1

n n

u

收敛,∑∞

=1

n n v 收敛

9、函数

b ax ax x f +-=6)(2在区间]2,1[-上

的最大值为3,最小值为29-,且0>a 则( )。 A 、

15

311

,1532=

-=b a B 、

15

311

,1532-

==b a C 、15

179

,1532-==b a D 、

15

179

,1532=-

=b a

10、

n 元齐次线性方程组0=Ax 的系数矩阵A 的秩为r ,则0=AX 有非零解的充要条件是( )

A 、n r <;

B 、n r =;

C 、n r ≥

D 、n r >

三、计算与应用题(本大题共10个小题,11-20每题8分,满分80分)

11、求极限2

cos 1lim 0-+--→x x x e e x

12、设,arctan 22)1ln(2

x x x

x y +-+=求y '

13、设函数11222

34++--=x x x x y ,求函

数的凹凸区间与拐点 14、求定积分dx e

x ?+4

01

2

15、设二元函数xy y z x

sin +=,求全微

分dz

16、求二重积分dxdy x

y D

??

22

,其中区域D 是

由直线2,==x x y 和曲线x

y 1

=围成

17、解微分方程

152=-'-''y y y ,

70='=x y ,30

==x y

18、曲线x

y =

的一条切线过点)0,1(-,

求该切线与x 轴及x

y =所围成平面

图形的面积

19、求线性方程组???

??=+++=+++=+++1

321

24322

53432143214321x x x x x x x x x x x x 的

通解。

20、若n 阶方阵

A

B

满足

E B A AB =++(E 为n 阶单位矩阵)。

证明

(1)E B +为可逆矩阵 (2))(2

1

)

(1

E A E B +=+-

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