英文版 微积分试卷答案 (1)

1、 (1) sin 2lim

x x x

→∞

= 0 . (2) d(arctan )x = 2

1d 1+x

x

(3)

2

1

d sin

x x

=

⎰

-cot +C

x x

(4).2()()x n e = 22n x e .

(5)0

x =⎰

26/3

2、

(6) The right proposition in the following propositions is ___A_____.

A. If lim ()x a

f x →exists and lim ()x a

g x →does not exist then lim (()())x a

f x

g x →+does not exist.

B. If lim ()x a

f x →,lim ()x a

g x →do bot

h not exist then lim (()())x a

f x

g x →+does not exist.

C. If lim ()x a

f x →exists and lim ()x a

g x →does not exist then lim ()()x a

f x

g x →does not exist.

D. If lim ()x a

f x →exists and lim ()x a

g x →does not exist then ()lim

()

x a

f x

g x →does not exist.

(7) The right proposition in the following propositions is __B______.

A. If lim ()()x a

f x f a →=then ()f a 'exists.

B. If lim ()()x a

f x f a →≠ then ()f a 'does not exist.

C. If ()f a 'does not exist then lim ()()x a

f x f a →≠.

D. If ()f a 'does not exist then the cure ()y f x =does not have tangent at (,())a f a .

(8) The right statement in the following statements is ___D_____.

A. sin lim

1x x x

→∞

= B. 1

lim (1)x x x e →∞

+=

C.

1

1d 1x x x

C

α

αα

+=

++⎰

D.

5

5

11d d 11b b a

a

x y

x

y

=

++⎰

⎰

(9) For continuous function ()f x , the erroneous expression in the following expressions

is ____D__. A.d (()d )()

d b

a f x x f

b b =⎰ B.

d (()d )()d b

a

f x x f a a =-⎰

C.

d (()d )0

d b

a

f x x x

=⎰ D.

d (()d )()()d b

a

f x x f b f a x

=-⎰

(10) The right proposition in the following propositions is __B______.

A. If ()f x is discontinuous on [,]a b then ()f x is unbounded on [,]a b .

B. If ()f x is unbounded on [,]a b then ()f x is discontinuous on [,]a b .

C. If ()f x is bounded on [,]a b then ()f x is continuous on [,]a b .

D. If ()f x has absolute extreme values on [,]a b then ()f x is continuous on [,]a b . 3、Evaluate 2

11lim (

)x

x e x

x

→--

2

1=lim (

)x

x e x

x

→--0

1=lim (

)2x

x e x

→-0

1=lim

=

2

2x

x e

→

（考点课本4.4节洛比达法则，每年都会有一道求极限的解答题，大多数都是用洛比达法则去求解，所以大家要注意4.4节的内容。注意洛比达法则的适用范围。）

4．Find 0d |x y =and (0)y ''if 2

x x

x y y t e

+=

+⎰

.

2

'()'

x x

x y y t e +=+⎰

（）

1'2()'2()1

x

x

y x y x e y x y x e +=⋅+⇒=⋅+-

(20(0)1)0x dy

y e dx dx

==⋅⋅+-=

''(2()1)'2()2'()x

x

y x y x e y x xy x e

=⋅+-=++

2

-(0)0-01

x x y y t e x y e =

+⇒=+=⎰

''02(0)20'(0)=3

y y y e =+⋅+（）

（考察微积分基本定理与微分，书上5.3节）

5、 Find 2

2

arctan d (1)

x x x x +⎰

=22

2

2

1)arctan d (1)

x x x

x

x x +-+⎰

（

2

2

arctan arctan =d d (1)

x x x x

x

x -

+⎰

⎰

-1

2

3

1

1=-arctan +d arctan +2

x x x x x x

-

⎰

2

2-1

2

2

1++1=-arctan +d arctan 1+2

x x

x x x x x x -⎰

（） -1

221

1=-arctan +d d arctan 1+2

x x x x x x x x --⎰

⎰（） -12

2

11=-arctan +In In 1+arctan 22

x x x x

x --

-1

2

1=-arctan +In

arctan +C 2

x x x -

（凑微分求不定积分，积分是微积分的重点及难点，大家一定要掌握透彻。）

6、 Given that 2

2()1

x

f x x =+.