广东专插本高等数学真题

广东专插本高等数学真题
广东专插本高等数学真题

2008年广东省普通高校本科插班生招生考试

《高等数学》试题

一、单项选择题(本大题共5小题,每小题3分,共15分。每小题给出的四个选项,只有

一项是符合题目要求的) 1、下列函数为奇函数的是

A. x x -2

B. x

x

e e -+ C. x

x

e e -- D. x x sin 2、极限()

x

x x 10

1lim -→+=

A. e

B. 1

-e C. 1 D.-1 3、函数在点0x 处连续是在该点处可导的

A.必要非充分条件

B. 充分非必要条件

C.充分必要条件

D. 既非充分也非必要条件 4、下列函数中,不是x x

e e 22--的原函数的是

A.

()

2

2

1x x

e e -+ B.

()

2

2

1x x

e e -- C.

()

x x

e e 222

1-+ D. ()

x x

e e 222

1-- 5、已知函数xy e z =,则dz =

A. ()dy dx e xy +

B. ydx +xdy

C. ()ydy xdx e xy +

D. ()xdy ydx e xy + 二、填空题(本大题共5小题,每小题3分,共15分) 6、极限x

x x e e x

-→-0lim

= 。

7、曲线y=xlnx 在点(1,0)处的切线方程是= 。 8、积分

()?-+22

cos sin π

πdx x x = 。

9、设y e v y e u x

x sin ,cos ==,则

x

v

y u ??+??= 。 10、微分方程

012

=+-x x dx dy 的通解是 。 三、计算题(本大题共8小题,每小题6分,共48分) 11、计算x

x x

x x sin tan lim 0

--→。

x e e x f x x 2)(--='-,

(4分)

2

22

)(2)(x x x

x e e e

e x

f -

--=-+=''>0,

于是)(x f '在),0(+∞内单调增加,从而)(x f '>)0(f '=0,

所以)(x f 在),0(+∞内单调增加,故)(x f >)0(f =0,即2x x e e -+>2

12

x +.

20、解:设?

--

=x

dt t f x x F 0

1)(2)(,则)(x F 在[0,1]上连续,

1)0(-=F ,因为0<f(x)<1,可证?

1

)(dx x f <1,于是?-=1

)(1)1(dt

t f F >0,

所以)(x F 在(0,1)内至少有一个零点.

又)(2)(x f x F -='>2﹣1>0,)(x F 在[0,1]上单调递增,

所以)(x F 在(0,1)内有唯一零点,即?=-x

dt t f x 01)(2在(0,1)内有唯一实根

(6分) (8分)

(10分)

(3分)

(6分) (9分)

(12分)

2009年广东省普通高校本科插班生招生考试

《高等数学》试题

一、单项选择题(本大题共5小题,每小题3分,共15分。每小题只有一个选项符合题目

要求) 1、设??

?≥-+=.

0,1,0,13)(x x x x x f π则=-+→x f x f x )

0()(lim 0 A. -1 B. 1 C. 3 D. ∞ 2、极限=??

?

?

?

+→x x x x x sin 22sin

lim 0

A. 0

B. 1

C. 2

D. ∞ 3、下列函数中,在点0=x 处连续但不可导的是 A. x y = B. 1=y C. x y ln = D. 1

1-=x y 4、积分?

=-'dx x f x )sin 21(cos

A. C x f +-)sin 21(2

B. C x f +-)sin 21(21

C. C x f +--)sin 21(2

D. C x f +--)sin 21(2

1

5、改变二次积分?

?

=1

2

),(x dy y x f dx I 的积分次序,则I =

A. ?

?

1

0),(y dx y x f dy B. ??1

01),(y

dx y x f dy C.

?

?

1

1),(y

dx y x f dy D. ??1

00

),(y

dx y x f dy

二、填空题(本大题共5小题,每小题3分,共15分)

6、若当0→x 时,2

2

2~11x ax --,则常数a = 。 7、曲线x

x y )

1ln(+=

的水平渐近线方程是 。 8、若曲线?????+=-=2

2

)

21(,

3t y t kt x 在t=0处的切线斜率为1,则常数k= 。

9、已知二元函数),(y x f z =的全微分,22

xydy dx y dz +=则y

x z

???2= 。

10、已知函数)(x f 满足==+=')(0)0(1)()(x f ,f ,x f x f 则且 。

=1121)21(102--+=-+

e y y e y .2

3-=e

(5分)

(2)??

--=10

1

2

2)1(dy y dy e V y

ππ

=

103102)1(3

2

y e y -+

π

π

=

.6

52

π

-

e 20、解:(1),,x

x f x x x x x f 42)(ln 424ln 442)(-

=''-=--+='Θ 当0<x <2时,<)(x f ''<0,所以)(x f 在(0,2)上的图形是凸的。 (2)Θ当0<x <2时,<)(x f ''<0, )(x f '∴在(0,2]上单调减少,由此知: 当0<x <2时,有,02ln 44)2()(φφ-=''f x f 故)(x f 在区间(0,2]上单调增加. 因此当0<x <2时,有

.04ln 442ln 8482ln 884)2()(ππ-=-=--+=f x f

(8分)

(2分) (10分)

(5分)

(8分)

(12分)

广东专插本高等数学2008-2010真题

2008年广东省普通高校本科插班生招生考试 《高等数学》试题 一、单项选择题(本大题共5小题,每小题3分,共15分。每小题给出的四个选项,只有 一项是符合题目要求的) 1、下列函数为奇函数的是 A. x x -2 B. x x e e -+ C. x x e e -- D. x x sin 2、极限() x x x 10 1lim -→+= A. e B. 1 -e C. 1 D.-1 3、函数在点0x 处连续是在该点处可导的 A.必要非充分条件 B. 充分非必要条件 C.充分必要条件 D. 既非充分也非必要条件 4、下列函数中,不是x x e e 22--的原函数的是 A. () 2 2 1x x e e -+ B. () 2 2 1x x e e -- C. () x x e e 222 1-+ D. () x x e e 222 1-- 5、已知函数xy e z =,则dz = A. ()dy dx e xy + B. ydx +xdy C. ()ydy xdx e xy + D. ()xdy ydx e xy + 二、填空题(本大题共5小题,每小题3分,共15分) 6、极限x x x e e x -→-0lim = 。 7、曲线y=xlnx 在点(1,0)处的切线方程是= 。 8、积分 ()?-+22 cos sin π πdx x x = 。 9、设y e v y e u x x sin ,cos ==,则 x v y u ??+??= 。 10、微分方程 012 =+-x x dx dy 的通解是 。 三、计算题(本大题共8小题,每小题6分,共48分) 11、计算x x x x x sin tan lim --→。

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广东省2011年专插本《高等数学》考试大纲(手录入版)

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