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江南大学高等数学第3阶段

江南大学高等数学第3阶段
江南大学高等数学第3阶段

一.选择题(每题4分)

1. 设22(,)x f y x y x y

-=-, 则(,)f x y = ( D ). (a) 2(1)1y x x +- (b) 2(1)1y x x -+ (c) 2(1)1x x x

+- (d) 2(1)1x y y +- 2. 设函数 (,)z f x y = 在点 00(,)x y 的某邻域内有定义, 且存在一阶偏导数, 则

00x x y y z y ==?=?( B ) (a) 00000(,)(,)lim y f x x y y f x y y ?→+?+?-? (b) 00000(,)(,)lim y f x y y f x y y

?→+?-? (c) 000()()

lim y f y y f y y ?→+?-? (d) 0000(,)(,)

lim y f x x f x y y ?→+?-?

3. 若D 是平面区域22{19}x y ≤+≤, 则D

dxdy ??=( B )

(a) 7π (b) 8π (c) 9π (d) 10π

4. 下面各微分方程中为一阶线性方程的是 ( B )

(a) 32xy y '+= (b) 2cos y xy x '+= (c) 2yy x '= (d) 21y xy '-=

5. 微分方程 ()0x y y x y '++-= 的通解是 ( D ). (a) 221arctan ln()2y x y C x ++= (b) 22

arctan ln()y x y C x -+= (c) 22arctan ln()y x y C x ++= (d) 2

21arctan ln()2y x y C x -+=

二.填空题(每题4分)

6. 设

z =则 13

x y z x ==?=?

____327. 设 2cot()z y xy =-, 则z y ?=?___22(2)csc ()y x y xy ---_____

8. 设sin y

x z e x y =+, 则2z x y ???=_____3cos y x x y e y x +-+_____

9. 设 2ln(32)x y z y x e =-+, 则 dz =_2223

(2)()3232x y x xe dx e dy y x y x -+++--__________.

10. 交换二次积分次序 ln 10(,)e x I dx f x y dy =??=10(,)y

e

e dy

f x y dx ??_____________.

11. 微分方程 4

43d u u v dv += 的自变量为_v_____, 未知函数为____u____, 方程的阶数为____4___

12. 微分方程 10dy dx xy -= 的通解是

___y =C 为任意常数_____

三. 解答题 (满分52分)

13. 设

(,)z z x y = 是由方程 2cos()0z e x y x z -+-= 所确定的隐函数, 求 dz

解答: 2

2

2y sin()sin()sin()2y sin()则有=+sin()sin()

z z z z z z dz dx dy x y z x x z x e x z z x y e x z x x z x dz dx dy e x z e x z ??=

+???+-=?+-?=?+-+-+-+-

14. 求函数

(3),(0,0)z xy x y x y =-->>的极值。

解答: x xx xy yy 2z =(3)0

(1)z =x (3)0

(2)联立上面(1),(2)两等式,且考虑x>0,y>0,得驻点:(1,1)

A=z =-2y ,=z 322,z 2在驻点(1,1)处,AC-B 30,且A=-2<0,此处取得极大值z=1

y y x y xy x y xy B x y C x

---=---==--==-=> 15. 计算

2D

xy dxdy ??, 其中D 是由曲线 21,,3xy y x y === 围成的平面区域。 解答:

222

123

1

313

222

001

00

dy x x dy x

9

D D D

y

xy dx xy dx xy dx

y dx y dx y dx

=++

=++

=-

??????

????

16. 计算

22

x y

D

e dxdy

+

??, 其中D是由22

25

x y

≤+≤确定。

解答:

2

2

52

2,原式=d()

d e e

π

ρ

ρθπ

ρρπ

<=<=<=<=

=-

?

17. 求微分方程

2

dy y

dx y x

=

-

的通解。

解答:

2

2

3

dy(y-x)=ydx

y()

1

3

为任意常数

dy xdy ydx d xy

y

xy C

y C

=+=

=+

=

18. 求微分方程cos

dy y

x

dx x

+=的通解。

解答:

对于齐次方程+=

dy y

dx x

0,得到齐次方程解:

,为任意常数

C

y C

x

=

2

用常数变易法,把换成u(x),则有y=

u

-,代入原一阶线性微分方程,得

x

=x cos x,积分得到u=xsinx+cos x+C,C为任意常数,则通解

xsinx+cos x+C

y=(C为任意常数)

u

C

x

u

y

x

u

x

'

'=

'

19. 求微分方程(sin)tan0

y x dx xdy

-+=满足初始条件()1

6

y

π

=的解。

解答:

tan sin cot cos 对于奇次方程cot 0,通解为:ln ||ln |sin |1

,为任意常数 (1)sin dy x y x dx

dy y x x dx

dy y x dx

y x C C y C x

+=+=+==-+= 222利用常数变易法,将C 代替为u(x),则有:

u y=csc ,求导:sin y =u cscx-ucscxcotx,(2)

将(1)、(2)代入原微分方程,得到(sin x )u =cosxsinx ,积分得到u=+C ,则微分方程通解:2

(sin x )3y=(+C )csc x ,由于y ()=1,得到=,则微分方程特解:268

(sin x )3sin x 3y=(+)csc x=+csc x 2828

u x x

C π='''

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江南大学大学英语一阶段

年江南大学-大学英语一阶段

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答案: 1-10: A D B A C A A A A D 11-20: C B D A D B C A A A 21-30: B B B A C B B D B A 31-40: B A D D C D B A B B 41-50: D B B A D D C B B D Part I V ocabulary and Structure Directions: There are a number of incomplete sentences in this part. For each sentence there are four choices marked A), B), C) and D). Choose the ONE answer that best completes the sentence. Then mark the corresponding letter on the Answer Sheet with a single line through the centre. 1.The ship _____ South America. A. is bound for B. bound for C. bound to D. is bound to 2.The lifeboat went out to rescue a ship in _____. A. disgrace B. disappointment C. despair D. distress 3.I must _____ my lawyer before I make my final decisions. A. contrast B. contact C. contract D. compare 4.The two cars were broken into pieces in the _____. A. collision B. crash C. collapse D. confusion 5.You’d better _____ the warm weather by going for a walk this afternoon. A. take use of B. take the advantage of C. take advantage of D. take the use of 6.Jean _____ own a car but I don’t believe her. A. claims B. claims to C. claims that D. claims of 7.We tried to _____ him _____climbing the mountain without a guide. A. discourage from

江南大学大学英语三第2阶段答案

请联系网站删除资料收集于网络,如有侵权b 20171203第二阶段大学英语(三)2017-10-27 08:09:38 交卷时间:一、完形填空1. For using words. things without actually here are various ways to say yoevent, a you at sports to __1__ a seat next to example, if you want __2_your you do this, or jacket across the seat. When u put a sweater mahas sweater or jacket seat is not __3__. Your _ tells people that the __given message is for a while. The same rked the place as __4__ you __6which telling people a beach. You are 5__ you spread a blanket on that 分界线) boundary (界限, you __ are claiming. You are laying out a blanket only the It includes not a marks section of the sand as __7__. it. __8__ a also small area area, but bea middle of the to the park and occupy happen __9__ you to go yourselto entire bench that you want the showing nch, you are __10__ aryou bench, side of the you while. __11__ if slide over to the a f for otwith the bench you that are willing to __12__ letting e people know hers. own. their __13__ areas, of (human All beings need territories 领土), __14__

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