试题2011gj(1st)

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武汉理工大学考试试卷（A卷）

2010 ~ 2011 学年1 学期高等数学(双语) 课程时间120分钟

I. Choose the best answer from the given choices. (3'515'

?=)

1.Let

sin

,0

()

2,0

x

x

f x x

x

?

≠

?

=?

?=

?

，then we can conclude that ( ).

A．)

(

lim

x

f

x→

does not exist. B．)

(

lim

x

f

x→

exists，but)

(x

f is not continuous at0

x= C．)

(x

f is continuous but not differentiable at 0

x=.

D．)

(x

f is differentiable at0

x=

2.When 1

1

,02-

+

→ax

x is the equivalent infinitesimal of 1cos3x

-, then ( ) .

A. 3

a=B. 1/3

a=C. 1/9

a=D. 9

a=

3.The increasing interval of 2

()x

f x xe

=is （）

A.

1

(,)

2

-+∞B. )1,1

(-C.

1

(,)

2

-∞D. )1

,

(-

-∞

4.Suppose that ()

f x

'is continuous，then()

2

f x dx

'=

?( ) A.()

2

f x C

+； B. ()

2f x C

+； C. ()

1

2

2

f x

C

+； D. ()

2

xf x C

+

5.Suppose that the improper integral

()

2

d

1k

x

x

+∞

-

?converges, then （）

A.1

>

k B.k≥1 C.k≤1 D. 1

<

k

II. Fill the blanks (3'515'

?=)

6.

x

x x

x2

1

lim?

?

?

?

?+

∞

→

=_______________

7.x

x

x

y cos

2

1

sin+

=, then =

=

4

π

x

dy________________

8.The normal line of curve

2

1

4

y

x

=

+

at point )5/1,1(is

9.

Let an accumulation function 30

()x G x t =?

, then ()1G '=

10.

1

1

sin )x x dx -?

=

III 、Find the following limits and derivatives (7'428'?=)

11. x

e e

x

x x sin 2lim 0-→-

12. 2lim arctan x

x x π→+∞??

???

13. Suppose that )(x y is defined by sin()xy x y =+, find (0)y '.

14. Let )(x y be defined by the parametric equation ?

??-=+=t t y t x arctan )1ln(2，find 22,dy d y

dx dx .. IV . Find the following integrals and area. (7'428'?=)

15. ?+-+x x x x d 231

2

16. dx x x ?

-12

12

2

1 17. ?

∞+-0

2d x e x x

18. .4 and 2by bounded

region the of area the Find 2

-==x y x y R

V. (4') Let ()x f be continuous on [0,1] and differentiable on (0,1). If ()00=f

and ()11=f ，then prove that there exists ∈ξ(0,1)，such that ()()='?ξξf e f 1e -.

bent to form a square, and the other is bent to form an equilateral triangle. Where should

the cut be made if (a) the sum of the two areas is to be a minimum;(b)a maximum?(Allow

the possibility of no cut.).

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卷

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订

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答

题

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不

要

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写

考

生

信

息

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武汉理工大学教务处

试题标准答案及评分标准用纸

课程名称：高等数学（双语）（ A 卷）

I 、1.B; 2.D; 3.A; 4.C ; 5.A.

II. 1. 2e , 2. dx ???

?

??+π8242, 3. 012310125=--y x ,

4. 5. 2π III. Find the limits of the following functions. ('412'7=?)

1. 000lim

lim lim 12sin 22

x x x x x x

x x x e e e e e e x x ---→→→--+===(7') 2. )]arctan 2

ln(lim ex p[)arctan 2(lim x x x x x x π

π

+∞

→+∞

→==]1)

arctan 2

ln(lim

exp[x

x x π

+∞

→（2’）

=2

2exp[lim ](1)arctan x x x x

→+∞-+（5’） =π2

-e ( 7’) 3. cos()()1xy y xy y ''+=+ （3’） c o s ()1

1c o s ()

y x y y x x y -'=

- （5’）

00x y =?= （6’） (0)y '1=- （7’）

4. 222

12,1t

t dt dx t

t dt dy +=+=----------2’

t dx dy 2

1

=--------------4’ t t dx dt dx

y d 41212

2

2+==----------7’ IV . 1.

?+-+x x x x d 2312?---=x x x d )1223( (4’)C x x +---=1ln 22ln 3 (7’)

2. Let t x sin =, I=dt t )1(csc 2

4

2

-?π

π(4’)＝24

)

cot (π

π

t t --＝4

1π

-

(7’)

3．?

∞+-0

2dx e x x ?

∞+--=0

2x de x (2’) x d xe e x x x ?

∞+-+∞-+-=0

22 (4’)

?

∞+--=0

2x x d e 2220

=+-=?

∞+-+∞-x d e xe x x

(7’)

4. ???-==4

22x y x y )4,8(),2,2(-? (2’) ?-???

? ??-+=4

2224dy y y A (5’) .18=(7’) V . let ()

()e

f x F x =(2’)

since F(x) satisfies the condition of Lagrange Theorem on [ 0,1]，then

(1)(0)()1F F F ξ'-=?

That is, ()

()='?ξξf e

f 1e - (4’)

VI. let x

be the length of one piece, then []2

2

1000,100434x x S x -???

=+∈? ?

????

(3’)

'270013

011

S x -=?=

(6’)

When 0625x S ==,

when 100x S ==

3

10000

x S =

=

(8’)

So , min S S =

=??

()m a x 6250S S == (10’)