四川大学微积分1-2(2016)B卷

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(NEW)四川大学《690高等数学(微积分、级数)》历年考研真题汇编

6 （12分）一质量为m的物体，最初静止于x0处，在力F＝－k/x2 的作用下沿直线运动，试求物体在任意位置x处的速度．
7 （13分）质量为m的摩托车，在恒定的牵引力F的作用下工作，它所受的阻力与其速率的平方成正比，它能达到的最大速率是vm．试计算从静止加速到vm/2所需的时间以及所走过的路程．
3 求下列不定积分（共50分）：（1）（2）
（3）
（4）（5）（6）（7）（8）（9）（10）
4 用级数展开计算下列积分的近似值（计算前三项）（共20 分）：
（1）（2）
5 （5分）甲乙两船同时从一码头出发，甲船以30km/h的速度向北行驶，乙船以40km/h的速度向东行驶，求两船间距离增加的速度为多少？
2012年四川大学690高等数学（微积分、级数）考研真题
2013年四川大学690高等数学（微积分、级数）考研真题
2014年四川大学690高等数学（微积分、级数）考研真题
2015年四川大学690高等数学（微积分、级数）考研真题
2016年四川大学690高等数学（川大学690高等数学（微积分、级数）考研真题
1 请写出下列初等函数的级数展开式（共20分）：（1）ax （2）sin（x/2）（3）（4）ln（1＋x）（5）1/（1＋x）
2 求下列平面图形的面积（共30分）：（1）曲线y＝x3与y轴和直线y＝1所围成的图形；（2）曲线y＝x2与y＝2－x2所围成的图形．
目录
2012年四川大学690高等数学（微积分、级数）考研真题 2013年四川大学690高等数学（微积分、级数）考研真题 2014年四川大学690高等数学（微积分、级数）考研真题 2015年四川大学690高等数学（微积分、级数）考研真题 2016年四川大学690高等数学（微积分、级数）考研真题 2017年四川大学690高等数学（微积分、级数）考研真题

微积分试卷及标准答案6套

微积分试题 (A 卷)一. 填空题 (每空2分,共20分)1. 已知,)(lim 1A x f x =+→则对于0>∀ε，总存在δ>0，使得当时，恒有│ƒ(x )─A│< ε。

2. 已知2235lim2=-++∞→n bn an n ，则a = ，b = 。

3. 若当0x x →时，α与β 是等价无穷小量，则=-→ββα0limx x 。

4. 若f (x )在点x = a 处连续，则=→)(lim x f ax 。

5. )ln(arcsin )(x x f =的连续区间是。

6. 设函数y =ƒ(x )在x 0点可导，则=-+→hx f h x f h )()3(lim000______________。

7. 曲线y = x 2＋2x －5上点M 处的切线斜率为6，则点M 的坐标为。

8. ='⎰))((dx x f x d 。

9. 设总收益函数和总成本函数分别为2224Q Q R -=，52+=Q C ，则当利润最大时产量Q 是。

二. 单项选择题 (每小题2分,共18分)1. 若数列{x n }在a 的ε 邻域（a -ε,a +ε）内有无穷多个点，则（）。

(A) 数列{x n }必有极限，但不一定等于a (B) 数列{x n }极限存在，且一定等于a(C) 数列{x n }的极限不一定存在 (D) 数列{x n }的极限一定不存在 2. 设11)(-=x arctgx f 则1=x 为函数)(x f 的（）。

(A) 可去间断点 (B) 跳跃间断点 (C) 无穷型间断点(D) 连续点 3. =+-∞→13)11(lim x x x（）。

(A) 1 (B) ∞ (C)2e (D) 3e4. 对需求函数5p eQ -=，需求价格弹性5pE d -=。

当价格=p （）时，需求量减少的幅度小于价格提高的幅度。

(A) 3 (B) 5 (C) 6 (D) 105. 假设)(),(0)(lim ,0)(lim 0x g x f x g x f x x x x ''==→→；在点0x 的某邻域内(0x 可以除外)存在，又a 是常数，则下列结论正确的是（）。

四川大学《大学数学-微积分》期末考试试卷B(末尾含答案解析)

B．双叶双曲面 D．旋转抛物面
2. 设F (x y, y z, z x) 0，则 z ( ). x
A. F1 F3 F2 F3
B. F2 F1 F2 F3
C. F1 F3 F2 F3
D. F1 F3 F2 F3
3.函数z ln(x y)在抛物线y2 4x上点（1，2）处，沿着这抛物线在该点处偏向X轴正向的切线方向的方向导数为（）
A. 2 3
B. 3 3
C.1
D. 3
4.
1
dx
x
e
y
2
01
dy
（
）.
A. 1 (e1 1) 2
B. 1 (e 1) 2
C. 1 (e 1) 2
D. 1 (e1 1) 2
5.若 y1 和 y2 是二阶齐次线性方程 y p(x)y q(x)y 0 的两个特解,则
第 1 页共 6页
年级：装
5.级数
n1
(2
x
n
1)
n
的收敛区间是
______
.
三．计算题（每小题8分，共24分）
1.设函数 Q(x，y) 在 xoy 平面上具有一阶连续偏导数，曲线积分
L 2xydx Q(x, y)dy 与路径无关，并且对任意 t ，恒有
(1, t)
(t , 1)
(0,0) 2xydx Q(x, y)dy (0,0) 2xydx Q(x, y)dy
1.证明，变换
u v
x 2y x 3y
可把方程
6
2 x
z
2
2z xy
2z y 2
0 简化为 2 z 0 ． uv
第5页共6页
四川大学期末考试试卷

四川大学高数微积分I(上)考前复习用2018年期末真题试卷(含答案)

x

1
x4

2
x2

d 1
x
而
1 0
x4
2x 2x2
dx 1

1 2d x 2，
0
1
x4
2x 2x2
dx 1

1
2 x3
d
x

1，
故原无穷限广义积分也收敛.
三、解答题（每小题 10 分，共 20 分）
1.设两曲线为 l1 : y x2 ，l2 : x y 2 .
n1
n n1 n
(1)n1 1 xn
n1
n
f
(2017) (0)

a2017
2017!

2 2017
2017!
2 2016!
注前一问 6 分，后一问 2 分.
6.判断无穷限广义积分
0
x4
2x 2x2
d 1
x
的敛散性.
解 1

2x
f ( x2 y) (2xy x2 dy ) e x y (1 dy ) 1
dx
dx
解之得 dy dx
f
( x2 f (
y x
)
2
2xy e x y) x2 e
x
y

y
1
.
y) 2xy e x y f ( x2 y) x2 e x y
(2) 由(1)知， x0 为极值点，所以 f ( x0 ) 0. 将函数 f ( x) 在点 x x0 处展开，得
f (x)
f ( x0 )
f ( x0 )( x x0 )

【答案】2015-2016春四川大学微积分(I)-2

第 2页
解：令 F x2 y2 z2 1(x2 y2 1) 2 (x 2y 3z 6)
Fx Fy Fz F1 F2
2x 21x 2 0 2 y 21 y 22 0 2z 32 0 x2 y2 1 0
x 2y 3z 6 0
解：设辅助面1： z 1, ( x, y) Dxy:x 2 y 2 1 方向指向上侧，根据高斯公式，
( x2 y)dydz ( y2 z)dzdx ( x z2 )dxdy
[( x2 y)dydz ( y2 z)dzdx ( x z2 )dxdy]
1
1
(2
x
A 3, B 1 .
(2) 根据微分公式， (3 x2 y 6 xy2 y)dx (6 x2 y x3 x)dy d( x3 y 3 x2 y2 xy)
所以微分方程的通解： x3 y 3x2 y2 xy C .
(3) 曲线积分 (1,2) (3 x2 y 6 xy2 y)dx (6 x2 y x3 x)dy (0,0)
6．设平面闭曲线 L 是由 y 3x, y 0, x 1 所围成，则曲线积分 xyds ( 9 10 ).
L
2
7．微分方程 y y 2xy 满足 x 0 时 y 1 的特解是( y e x x2 ).

二．计算题(每小题 9 分，共 45 分)
1．设
ez
2x
3y
z
1
0
确定的二元函数
f l
(0,0)
lim
0
f
(x,y )
f
(0, 0)
lim
0
f
( cos ,
cos )
lim
0
3 cos2 cos 3

2017春微积分I-2期末B卷试题

第1页，共2页四川大学半期考试试题（闭卷）（2016-2017学年第2学期）课程号：201138040课序号：课程名称：微积分（I ）-2任课教师：成绩：适用专业年级：学生人数：印题份数：学号：姓名：考生承诺我已认真阅读并知晓《四川大学考场规则》和《四川大学本科学生考试违纪作弊处分规定（修订）》，郑重承诺：1、已按要求将考试禁止携带的文具用品或与考试有关的物品放置在指定地点；2、不带手机进入考场；3、考试期间遵守以上两项规定，若有违规行为，同意按照有关条款接受处理。

考生签名：一、填空题（每小题4分，共20分）1.曲线220y x z ⎧=⎪⎨=⎪⎩绕y 轴旋转一周所成的曲面方程为__________.2.设(01)y z x x x =>≠，,则__________.dz =3.改变二次积分130()y y dy f x y d x ⎰⎰，的积分顺序为__________.4.函数2()f x y x y =，在点(11)，处方向导数的最大值为__________.5.曲线3z xy x y z =⎧⎪⎨++=⎪⎩上点(111)，，处的切线方程为__________. 二、解答题（每小题10分，共60分）1.设()()z z x y y x ==，由方程组()1z f y z x x y z =+⎧⎪⎨++=⎪⎩，确定，求. dz dy dx dx ，2.求由曲面222z x y =+及2262z x y =--所围成的立体的体积.3.求极限24301lim ln(4)rr xy dv r →Ω++⎰⎰⎰，其中2222. r x y z r Ω++≤：4.求过曲面2226x y z ++=上一点的切平面，且该切平面垂直于直线2. 2x y z x z --=⎧⎪⎨+=⎪⎩第2页，共2页。

川大高数半期考试试卷

x 0 y 0
x 0 y 0
(
( x0 , y0 ) f y ( x0 , y0 ) 0 . 若在(x0, y0)有 lim ) 4. 设 f x
z
0

0
则 z=f(x, y)在(x0, y0)是可微的，其中 ( x )2 ( y )2 .
第 1 页共 8 页
一、判断题：下列陈述，你认为正确者，请在前面括号里填入 Y ，否则填入 N.
阅卷人签名
(每小题 3 分 , 共 12 分 ) ( ) 1. 设连续函数 f(x)以 T 为周期. 若函数 f ( t )dt 仍以 T
a
T 0
x
为周期, 则 f ( x )dx 0 . ( ) 2. 设 f(x)是连续函数. 如果
a

f ( x )dx 收敛 lim f ( x ) 0.
x
f ( x )dx 收敛. a a a f ( x ) dx 收敛 a f ( x )dx 收敛. x f ( x ) dx 收敛 a a e f ( x )dx 收敛.
x 2 f ( x )dx 收敛
阅卷人签名
1. lim
n
n
k 1
1 nk
2. lim
( x 1) y x 1 | x 1 | | y |
y 0
第 4 页共 8 页
2u 3. 设 x+y+u+xy+u － 1=0. 求 . x y
2
ln(1 x 2 y 2 ) 4. I 2 2 1 x 2 y 2 dxdy . 2 0 x y R
r
H
六、应用题 :(共 10 分) 一个正圆柱形的水桶,内径为 2r, 内深为 H.里面装了一些水,

四川大学常微分方程 (张伟年著) 高等教育出版社课后答案

x2
dx,
两端积分得其通解为 √1 + y2 = C(1 + x2), 其中 C 为任意常数. 代入初值条件得 C = 2. 故所给初值问题的解为 y = 1 + 2x2.
(5) 令 u = ey, 原方程变为
du dx
=
x
+
x3,
容易求得其通解为
u
=
1 2
x2
+
1 4
x4
+ C,
从而原方程的通解为
解:
(1) 这里 M (x, y) = y, N (x, y) = y − x, 由于
E
=
∂M ∂y
−
∂N ∂x
=
2,
所以它不是恰当方程.
由于
−
E M
=
−
2 y
与 x 无关,
因此该方程有只依赖于 y 的积分因子
µ(x) = e−
2 y
dy
=
−
1 y2
.
因此方程
1 y
dx
+
y
− y2
x
dy
=
0
为恰当方程, 取 x0 = 0, y0 = 1, 可计算出
y = exp(
x
2 +
1
dx)(C
+
(x
+
1)
5 2
exp(−
x
2 +
1
dx)dx)
=
C(x + 1)2
+
2 3
(x
+
1)
7 2
,
即
y
=
C (x

四川大学2016独家真题

四川大学2016年博士研究生英语入学考试试题考生请注意：1.本试题共5大题，共11页，请考生注意检杏.考试时间为180分钟2. 1-70题答案请填写在机读卡相应处，否则不给分。

3.翻译和作文题答在答题纸上，答在试题上不给分。

书写要求字迹消楚、工整。

I. Reading Comprehension (30%; one mark each)Directions: Read the following six passages. Answer the questions below each passage by choosing [A], [B], [C], or [D]. Write your answers on the Answer Sheet.Passage OneWhen a person begins a mediatedor immediate encounter, he already stands in some kind of social relationship to the others concerned, and expects to stand in a given relationship to them after the particular encounter ends. This, of course, is one of the ways in which social contacts are geared into the wider society. Much of the activity occurring during an encounter can be understood as an effort on everyone’s part to get through the occasion and all the unanticipated and unintentional events that can cast participants in an undesirable light, without disrupting the relationships of the participants. And if relationships are in the process of change，the object will be to bring the encounter to a satisfactory close without altering the expected course of development. The perspective nicely accounts, for example, for the little ceremonies of greeting and farewell which occur when people begin a conversational encounter or depart from one. Greetings provide a way of showing that a relationship is still what it was at the termination of the previous co-participation, and, typically, that this relationship involves sufficient suppression of hostility for the participants temporarily to drop their guards and talk. Farewells sum up the effect of the encounter upon the relationship and show what the participants may expect of one another when they next meet. The enthusiasm of greetings compensates for the weakening of the relationship caused by the absence just terminated, while the enthusiasm of farewells compensates the relationship for the harm that is about to be done to it by separation.It seems to be a characteristic obligation of many social relationships that each of the members guarantees to support a given face for the other members in given situations. To prevent disruption of these relationships, it is therefore necessarily for each member to avoid destroying the others’ face. At the same time, it is often the person’s social relationship with others that leads him to participate in certain encounters with them, where incidentally he will be dependent upon them for supporting his face. Furthermore, in many relationships, the members come to share a face, so that in the presence of third parties an improper act on the part of one member becomes a source of acute embarrassment to the other members. A social relationship, then, can be seen as a way in which the person is more than ordinarily forced to trust his self-image and face to the tact and good conduct of others.1 .The last word of the first sentence, nam ely “ends' is most likely B .[A] a noun, meaning “purposes” or “objectives”[B] a verb, meaning “comes to a finish”[C] a postpositional adjective, me aning “finishing”[D] an adjective, meaning “purposeful”2. According to the author, if any unexpected difficulties occur in a social contact,B .[A] the relationships between the participants break up[B] those who participate will be in an unintentional event[C] all participants would try to maintain their relationships|D] the participants will certainly get through an activity3. Which of the following is NOT an idea of the author? C[A] The participants hope their relationship would be the same as they met last.[B] Greetings are just as important as farewells in a social encounter.[C] Before every greeting there is always sufficient hostility to suppress.[D] If their relationship changes, the participants want it to change as they hoped.4. The last sentence of the second paragraph means that . A[A] one’s self-image is dependent on how others behave[B] face and self-image are two different kinds of relationships[C] social relationship is something that is forced on all participants[D] to get along well with others is a process of giving each other face5. The best title for this passage may well be . A[A] Face and Social Relationship [B] How to Conduct Socially[C] Greetings and Farewells [D] Conversational SociologyPassage Two（Tips:出现人名字的地方用笔圈出来，数字用笔圈出来）The poet William Blake wrote in the early nineteenth century: “Great things are done when men and mountains meet.”Great things indeed were done on Mount Everest in May of 1996. Also poignant things, foolish things, deadly things：Hundreds of climbers from eleven different expeditions were on the mountain ---- thirty-one near the summit ---- when a freakish and fierce-some storm blew in. Eight climbers perished, the highest one-day death toll since the first expedition tried to reach the top of the world’s tallest peak in 1921.Adventurers have always sought challenges: deeper jungles, wider oceans, newer worlds. But mountains have been special. Perhaps it’s their size, the ir power, their resistance to conquest. InPatrick Mey ers’s play K2, a marooned climber on the Himalaya peak that gives the play its name delivers this li ne: “Mountains are metaphors.” And so they are. Climbers search not just for summits but also for themselves. They reach up to reach in.That helps explain why Everest has been enveloped by “Mountain Madness，”the name of a Seattle company that offers guided tours of the peak for about $65,000 (plus airfare to Nepal). New technology and equipment have also helped: lighter gear, warmer clothing, better radios and telephones. And the adventure can be shared, practically in real time, with Internet browsers around the world.But the community of high-mountain explorers now is gripped by soul-searching and second-guessing. Everest, after all, is not a theme peak. Some of the dead were experienced guides who lost their lives trying to save less agile amateurs. Said Mark Bryant, editor of Outside Magazines:“Some of us have been asking: Is it right that an average climber can order an ascent of Everest out of a catalog?” An Australian mountaineer, Tim McCartney-Snape, told the Associated Press: “Some things should remain sacred, and Everest is one of them. Even the strongest and toughest have found it can be extremely difficult just existing at that altitude, without other people depending on you.On Everest, dependency can lead to heroism and to tragedy. One frostbitten amateur, Seaborne Weathers of Dallas, was plucked from a rocky ledge at 22,000 feet by a Nepalese army helicopter — an act of incredible bravery. And Rob Hall, a guide who had climbed Everest several times, stayed on its slope with a dying customer. After learning they were hopelessly trapped, Hall managed to place a satellite telephone call to his pregnant wife, Jan, in New Zealand. “Hey, look,” he told her, “don’t worry about me.” At that moment, Hall remembered Harold, the character in K2 who muses: “Understanding has no meaning. Holding on, just holding on, that has meaning. Like Harold, he knew the mountain was still a mountain. Still a goal. Still a dream. And he couldn’t hold on. Rob Hall died before rescuers could reach him.6. Which of the following statements best describes the author’s point of view?[A] Amateurs should not be encouraged to climb Mount Everest.[B] Guided tours of Mount Everest have become a source of enormous profits.[C] In the past, Mount Everest has made heroes out of ordinary men and women.[D] Mount Everest should remain a metaphor to be talked about.7. The sentence “They reach up to reach in”may be best paraphrased by “”[A] To know the inherent meaning of a mountain, one needs to climb up to the top[B] In conquering the mountain, one finds a proof of oneself[C] Mountaineers must climb upward in order to climb inside it[D] One needs to climb up to the top to see why a mountain is metaphorical8. All of the following statements are true EXCEPT .[A] Mountain climbing can have a live report simultaneously[B] Mountaineers can be located instantly during their climbing[C] New technology significantly reduces risks and dangers in mountain climbing {D} It is more dangerous to have someone depending on you during mountain climbing9. In the last paragraph,Harold’s saying “Understanding has no meaning” means.[A] determination is more vital than thinking[B] persistence and action require reasoning[C] one should avoid misunderstanding the situation[D] it is meaningless to think in mountaineering10. Most likely, the author of this passage is .. [A] a historian[B] a mountaineer[C] a tourist guide [D] a reporterPassage 3The Hertz Corporation, the U.S. Air Force, Hyatt Hotels Corporation, the City of Dallas, and the Neiman-Marcus Group, Inc., have one thing in common----all have purchased the services of Feedback Plus. Feedback Plus is an agency that dispatches profe ssional shoppers who pose as customers. These “mystery” shoppers visit the client’s business, purchase products or services and report back to the client on the quality of service they receive. The City of Dallas hired Feedback Plus to see how car-pound employees treat citizens picking up their cars. The Air Force is using professional shoppers to assess customer service at their on-base supply stores. Banks, hospitals, and public utilities are also hiring mystery shoppers. Vickie Henry, chief executive of Feedback Plus, notes that many similar firms compete for client’s business, and service really differentiates one firm from another. Although Henry has a database of 8,800 people who serve as professional shoppers, she sometimes assumes the role of mystery shopper herself. During a recent visit to an upscale women's apparel store, she observed the type of customer service most companies attempt to avoid. None of the many salespeople on the sales floor said hello when she entered the store. When she removed a skirt from a clothing rack, none of the salespeople approached her. Finally, several minutes after entering the store, Henry approached a salesperson and asked to use the dressing room. Needless to say, service at this firm did not receive high marks from Feedback Plus.As organizations experience increased competition for clients, patients, and customers, awareness of the importance of public contact increases. They are giving new attention to the old adage “First impressions are lasting impressions.” Re search indicates that initial impressions do indeed tend to linger. Therefore, a positive first impression can set the stage for a long-term relationship.We are indebted to Susan Bixler, president of Professional Image, Inc., and author of Professional Presence, for giving us a better understanding of what it means to possess professional presence. Professional presence is a dynamic blend of poise, self-confidence, control and style that empowers us to be able to command respect in any situation. Once acquired, it permits us to be perceived as self-assured and thoroughly competent. We project a confidence that others can quickly perceive the first time they meet us.Bixler points out that, in most cases, the credentials we present during a job interview or when we are being considered for a promotion are not very different from those of other persons being considered, [t is oar professional presence that permits us to rise above the crowd. Debra Benton, a career consultant, says, “Any boss with a choice of two people with equal qualifications will choose the one with style as well as substance.” Learning to create a professional presence is one of the most valuable skills we can acquire.The development of professional presence begins with a full appreciation ofthe power of first impressions. The tendency to form impressions quickly at the time of an initial meeting illustrates what social psychologists call a primacy effect in the way people perceive one another. The general principle is that first impressions establish the mental framework within which a person is viewed, and later evidence is either ignored or reinterpreted to coincide with this framework.11. For all of the following walks of life EXCEPT the the professional presence has beendiscussed in this passage.[A] economic [B] military[C] academic [D] medical12. The underlined word “apparel” in the first paragraph means ““[A] apparatus [B] cosmetics[C] clothing [D] specialty13. Which of the following statements can be inferred from the passage?[A] No first impressions would ever change in the later contacts.[B] How one composes oneself determines how one is evaluated by others.[C] Social psychology is a science that stipulates the principles for social behaviors.[D] Opportunities in jobs or promotions are for those who differ from their competitors.14. The underlined word : “poise” in the third paragraph means:[A] proper comportment [B]desirable position[C] careful pause [D] positive assurance15. Which of the following is likely to be the title of this article?[A] The Power of First Impression[B] The Primacy Effect in Marketing[C] Social Psychology in Business[D] The Importance of FeedbackPassage FourYou can’t drive if you’re blind, or blind drunk, but an alarming number of Americans find themselves, at least occasionally, driving in a blind rage. “It’s a major social issue，”says Dr. Ricardo Martinez, administrator of the National Highway Traffic Safe ty Administration. “A 3,000-pound car in the hands of rude, hostile person is a lethal weapon.”A report on “road rage” to be released this week by the American Automobiles Association concluded that “motorists ... are increasingly being shot, stabbed, beaten, and run over for i nane reasons.” And inanity is not confined to young louts in “Baywatch” T-shirt: young men are by far the most common perpetrators, but middle-aged men and women can be equally big jerks. The most common manifestation of road rage was aggressive tailgating, followed by headlight flashing, “obscene gestures”, blocking other vehi cles, and verbal abuse. Drivers have been assaulted with weapons ranging from partially eaten burritos to canes (“a favorite with the elderly and disabled”）to golf clubs ---- and other vehicles, including buses, bulldozers, forklifts, and military tanks. “In terms of fatal crashes, drunks are a muchbigger menace,”says David Willis, president of the AAA Foundation of Traffic Safety. “But the average motorist doesn’t encounter a drunk very often, while in a place like Washington, D.C., at least once a week yo u’ll have an encounter with some crazy guy on the road.”Naturally, the phenomenon has given rise to its own therapeutic movement, whose leading practitioner is a Whittier, California, psychologist named Arnold Nerenburg. Nerenburg, who calls himself “America’s Road Rage Therapist”, has identified four stimuli that provoked road rage. The most common is feeling endange red by someone else’s driving --- for example, when another driver cuts you off or follows too closely. Others are resentment at being forced to slow down, righteous indignation at someone who breaks traffic rules or steals your parking space and ----perhaps the most dangerous, because it opens the door to an escalating exchange of hostilities ----anger at another driver who takes his own road rage out on you.The fact that most drivers are mutual strangers contributes to the volatility of highway confrontations. “There’s a deep psychological urge to release aggre ssion against an anonymous other,” Nerenburg says.Road-rage therapy tends toward the common-sensical---- “Take a deep breath and just let it go,” Nerenburg recommends. But it might help to consider that you might not be all that anonymous to the other driver. One of his patients realized the depth of his problem after he yelled an obscenity at the woman in the next car ----who turned out to be his boss s wife.16 .Which of the following statements is true according to the passage?[A] More and more Americans are using their cars to express their anger.[B] Old people and women are milder in temperament during driving.[C] Common sense might be the basis for overcoming road rage.[D] If people know each other, road rage would not happen.17. According to American Automobile Association, thepeople are more likely than all the others to be road-angered.[A] young [B] middle-aged[C]old [D] handicapped18. Which of the following is a justifiable cause for road rage, according to Dr. Nerenburg?[A] Another driver fails to observe a traffic law.[B] The parking space is occupied by another car.[C] The lane is taken by a slowly-moving car.[D] Another driver flashes the head-light.19. The underlined word “lethal” in the first paragraph means:[A] powerful[B] illegal[C] dangerous[D] deadly20. What might be the deep problem that one of Nerenburg’s patients had realized?[A] He faces a lawsuit of sex harassment.[B] He is in danger of being fired.[C] He will be fined by traffic police.[D] He falls ill and has to see a doctor.Passage FiveMost people, asked if they can think without -speech, would probably answer, “Yes, but it is not easy for me to do so. Still I know it can be done.” Language is but a garment! But what if language is not so much a garment as a prepared road or groove? It is, indeed, in the highest degree likely that language is an instrument originally put to uses lower than the conceptual plane and that thought arises as a refined interpretation of its content. The product grows, in other words, with the instrument and the thought may be no more conceivable, in its genesis and daily practice, without speech than is mathematical reasoning practicable without the lever of an appropriate mathematically symbolism. No one believes that even the most difficult mathematical proposition is inherently dependent on an arbitrary set of symbols, but it is impossible to suppose that the human mind is capable of arriving at or holding such a proposition without the symbolism.The writer, for one, is strongly of the opinion that the feeling entertained by so many that they can think, or even reason, without language is an illusion. The illusion seems to be due to a number of factors. The simplest of these is the failure to distinguish between imagery and thought. As a matter of fact, no sooner do we try to put an image into conscious relation with another than we find ourselves slipping into a silent flow of words. Thought may be a natural domain apart from the artificial one of speech, but speech would seem to the only road we know of that leads to it.21. In the first line, the underlined phrase can best be replaced by[A] most of the people [B] more people[C] more than one people [D] many people22. In line 3, the word ‘groove” is probably closest in meaning to[A] later growth [B] designated slot[C] particular path [D] ready-made viaduct23. Which of the following statements would the author of the passage agree?[A] Thought came into being earlier than language[B] It is language that makes conceptual thought possible[C] Thought is no different from mathematics because it depends on symbolism[D] Both thought and language are the interpretation of propositions.24. Which of the following statements is true according to the author?[A] The imagery is the precondition of thought[B] Thought has to be bore in words.[C] Imagery is an explanation of words.[D] Once we use words, we make mistakes.25. The idea of this passage is held by[A]some people including the writer.[B] the writer himself alone[C] most people but the writer[D] one of the people other than the writerPassage SixStrikes and strikebreaking, lockouts and boycotts, all pit one side against the other in labor disputes. Ultimately, the negative effects of such actions ---- including resentment, fear, and distrust ---- linger for months or years after a dispute has been resolved.Increasingly, more productive techniques such as mediation and arbitration are being used to settle disagreements between labor and management. Either one may come into play before a labor contract expires or after some other strategy, such as a strike, has proven ineffective.Mediation is the use of a neutral third party to assist management and the union during their negotiations. This third party (the mediator) listens to both sides, trying to find common ground for agreement. The mediator also tries to encourage communication between the two sides to promote compromise, and generally keep the negotiation moving. Initially, the mediator may meet privately with each side. Eventually, however, the goal is to get the two sides to settle their differences at the bargaining table.Unlike mediation, the arbitration step involves a formal hearing. Just as it may be the final step in a grievance procedure, it may also be used in contract negotiations when the two sides cannot agree on one or more issues. At this point, the arbitrator hears the formal positions of both parties on outstanding, unresolved issues. The arbitrator then analyzes these positions and makes a decision on the possible resolution of the issues. If both sides have agreed in advance that the arbitration will be binding, that means they must accept the arbitrator’s decision.If mediation and arbitration are unsuccessful, then according to the Taft-Hartley Act, the president of the United States can obtain a temporary injunction to prevent or to stop a strike if it would endanger national health or security.26. According to the author, with the solution of a labor crisis[A] the tension between labor and management can continue[B] labor and management do not trust each other any more[C] the negative effect of actions like strike would be resolved[D] a long time is needed to bring the dispute to an end27. To resolve a labor dispute, one should[A] resort to mediation or arbitration from the start[B] make sure that the labor contract expires[C] take other steps before going to mediation or arbitration[D] strike first and then accept mediation or arbitration28. Which of the following is true of mediation?[A] The mediator makes final decisions after meeting with both sides.[B] To avoid bias, the mediator cannot meet either side in advance.[C] The primary task is to help both sides bargain with each other.[D] The mediator can be a representative from either the labor or the management.29. The pre-condition for an arbitration to be authoritative is that[A] both sides agree in advance to abide by the decision made[B] a formal hearing must be conducted just like in a court trial[C] the decision is focused on the most outstanding and unresolved issues[D] there is no bargaining allowed by arbitration，unlike mediation30. In the last sentence, the underlined word “injunction” most likely refers to[A] a formal declaration [B] a subpoena [C] a lawsuit protocol [D] an official orderVI Vocabulary (10%; 0-5 mark each)31. Dr. Norman Bethune came from Canada to help the Chinese people in their war against Japanese aggression.[A] in the way [B] all the way [C] along the way [D] by the way32. Throughout history, he who knows the art of war uses force only as the last[A] resort [B] rescue [C] refrain [D] recant33. After a careful investigation and evaluation，the city hall decided to the old house.[A] dismantle [B] destroy [C] demolish [D] delineate34. The monument was so small that it appears to be more for than for memorial.[A] significance [B] indifference [C] oblivion [D] memory35. Millions of years ago the Vesuvian volcano destroyed Pompeii, but today it is[A] dormant [B] Pacifying [C] ignited [D] pictured36. Because humans have to talk about the limitless world by means of limited language sounds, there is a great between the world and language.[A] apathy [B]anomaly [C] asymmetry [D] agnosticism37. If an organism is quite successful in getting used with the environment that is new to it, we say that the organism is very[A] agile [B] adjustable [C] adoptive [D] adaptive38. When we do Planning, we Should take a11 relevant factors into consideration in order to as much as possible the difficult cases or even failures.[A] forestall [B] forerun [C] foretell [D] forecast39. is the practice of putting yourself in a position of another person in order to understand his/her feelings.[A] affectionate [B]empathy[C] pathology [D] affiliation40. In schools and universities, some courses are compulsory, which one has to take, while the others are( ) , either freely of as required.[A] optimal [B] opaque [C] optical [D] optional41. When one applies for a job position, one needs to ( ) one's resume, describing one's educational as well as working experiences.[A] submit [B] subject [C] submerge [D] subside42. As industry grew，so ( ) did the need for more and more skilled industrial workers.[A] much[B] as to[C] too[D] as for43. The rescue team worked hard to search for the missing mountaineers,( ) the heavy and icy snowstorm.[A] in spite [B] despite [C] although [D] disregard44. “Breaking Bad” is the most thrilling TV drama series ( ) I have watched in several years.[A] as [B] what [C] which [D] that45. The revised feasibility report handed in by the draft team is good enough, ( )a few spelling errors on some pages.[A] except for [B] except that [C] excepting [D] except46. It is mandatory that the engineering project ( ) accomplished by the end of this year.[A] is [B] has been [C] be[D] will be47. ( ) the advice from the councilor, we would not have finished the task so smoothly.[A] out of [B] what with [C] but for [D] instead of48. The traffic accident has claimed 5 lives, the cause of which is still ( ) investigation.[A] under [B] beyond [C] for [D] beneath49. Even though the bell for dismissing class has rung，the teacher is still talking ( )[A] over [B] forward [C] off [D] away50. Since the negotiation with the management has come to a deadlock, the worker's union decide to take ( ) the street.[A] to [B] over [C] down [D] away withIII Cloze (10%; 0.5 mark each)In the last decade, giving birth at home has become an increasingly popular option for some couples. Assisted by a physician or a nurse-midwife，many women have successfully given birth at home (51) A to healthy babies. In fact, some studies indicate that ―一for (52)<C>uncomplicated pregnancies --- home delivery is as safe as hospital delivery. (53) [B] Advocates of home birth argue that the atmosphere in a hospital --- with all its forbidding machines，rules, regulations, and general lack of “homeyness”---- is stressful. (54) [B] Therefore ，giving birth in a hospital detracts from what should be a joyous, natural human experience. Supporters of home birth further argue that hospitals are (55) [C] meant to deal with illness and that the delivery of a baby should not be viewed as an illness.On the other side of the argument, critics of home birth argue that if emergency medical (56) [A] procedures are necessary, giving birth at home may be (57) [B] downright dangerous. Furthermore, hospital practices in labor and delivery have changed (58) [D] radically in the last decade, particularly with the increased popularity of the Lamaze method. Thus hospitals are not the strange, forbidding environments they once were. (59) [A] Most hospitals, for example, allow fathers to be present (60) [A] at the entire labor and delivery, and many allow the father to be present (61) [A] in the operating room during the cesarean deliveries. Many hospitals have (62) [D] altogether created birth centers, homelike rooms with comfortable beds and armchairs, that allow labor and delivery to (63) [C] occur in a。

四川大学高数微积分I(下)考前复习用2016年期末真题试卷(含答案)

四川大学期末考试试卷A（2015‐2016年第二学期）科目：微积分II 课程号：考试时间：120分钟注：请将答案写在答题纸对应的方框内，否则记0分。

一、填空（每小题3分，共18分）1. 22011xy xy y x -+→→lim=41. 2. 设0=-，则--),,(x z z y y x F x z ∂∂= 0232313≠---'''''',F F F F F F . 3. 若0d ，则d 022=+⎰⎰xx y t t t t e sin )(cos x yd d = 22y ex x cos )(sin cos - . 4. 函数y x 在),(01取极y xy x y x f +-+-=222),( 小值. 5. 21)'(的通解是 ''y y +=))sin(ln(21C x C y +-= .6. 若区域D 由0=x ，0=y ，21=+y x ，1=+y x 围成，且，y x d 12，y x d d 12，+∈Z n ，依从小到大的顺序给321I I I ,,排序为 ⎰⎰++=D n y x y x I d d 121)][ln(⎰⎰+=DI 2+n y ]x [d I 3⎰⎰=Dx [sin(++n y )]231I I I << .二、计算题（每小题8分，共48分）1. 求x x 的通解.e y y x sin ''432+=-解：齐次问题的特征方程为1， 1012-==⇒=-λλλ,则齐次问题的通解为。

x x e C e C y -+=21特解可分解为，x e y y 23=-''x x y y sin ''4=-的特解之和。

x e y y 23=-''的特解为，x e 2x x y y sin ''4=-的特解为)cos sin (x x x +-2，则所求方程的通解为。

四川大学期末考试试卷A卷

四川大学期末考试试卷（A 卷）（2014—2015年第二学期）科目：微积分（I ）-2 课程号：考试时间：120分钟注：请将答案写在答题纸规定的方框内，否则记0分。

一、填空题(每小题3分，共18分)1．函数22ln(2)z x y =++在x =2，y =1时的全微分为2．已知曲线23,,x t y t z t ===上的点M 处的切线平行于平面24x y z ++=，则M 的坐标是3．二重积分()22222sin 34x y a x x y d σ+≤-++⎰⎰的值等于4．设L 为连接(1,0), (0,1)两点的线段，曲线积分()L x y ds +⎰的值等于5. 设∑为平面1x y z ++=在第一卦限的部分，曲面积分2(1)dS x y ∑++⎰⎰的值等于 6．微分方程ln dy y x y dx x=的通解是二、计算题 (每小题8分，共48分)1．设 5431z xz yz -+=，求2(0,0)z x y ∂∂∂． 2．设(2,sin )z f x y y x =-，其中f 具有连续二阶偏导数，求2 , 24x y z x yπ==∂∂∂． 3．计算2z dxdydz Ω⎰⎰⎰，其中Ω是两个球2222xy z R ++≤，2222 (0)x y z Rz R ++≤>所围成的闭区域．4．利用格林公式计算积分232()(2)Lx xy dx y xy dy -+-⎰Ñ，其中L 顶点为(0,0), (2,0), (2,2)和(0,2)的正方形区域的正向边界．5．计算222()()()SI y x dydz z y dzdx x z dxdy =-+-+-⎰⎰,其中S 为抛物面222z x y =--位于0z ≥内的部分的上侧．6．求微分方程tan sec dy y x x dx-=满足初始条件00x y ==的特解．三、应用题 (每小题10分，共20分)1．抛物面22z x y =+被平面1x y z ++=截成一椭圆，求原点到此椭圆的最长和最短距离．2．设函数()x ϕ连续，且满足00()()()x x x x e t t dt x t dt ϕϕϕ=+-⎰⎰，求()x ϕ．四、分析证明题 (每小题7分，共14分)1．设222222),0(,)0,0x y x y f x y x y ++≠=⎨⎪+=⎩，讨论(,)f x y 在(0,0)处的可微性．2．设()[,],()0f x C a b f x ∈>，证明2()()()bb a a dx f x dx b a f x ≥-⎰⎰．。

四川大学2015-2016学年第二学期课程考试试卷答案（A卷）课程名称

四川大学2015-2016学年第二学期课程考试试卷答案（A卷）课程名称四川大学2015-2016学年第二学期课程考试试卷答案(A 卷)课程名称：运筹学考试时间：120分钟年级：xxx 级专业： xxx题目部分，（卷面共有47题，0分，各大题标有题量和总分）一、判断（7小题，共0分）1、具有中间型效用曲线的决策者，对收入的增长和对金钱的损失都不敏感。

答案：对2、具有中间型效用曲线的决策者，对收入的增长以及对损失的金额都不敏感；（）答案：错3、应用最小机会损失准则决策时，如果用一般的损益矩阵来代替机会损失矩阵，则Savage 准则将建立在maxmin 条件，而不是minmax 条件上；（）答案：对4、在不确定型决策中Laplace 准则较之Savage 准则具有较小的保守性；（）答案：对5、不管决策问题如何变化，一个人的效用曲线总是不变的。

答案：错6、决策树比决策矩阵更适宜于描绘序列决策过程。

（）答案：对7、不管决策问题怎么变化，一个人的效用曲线总是不变的；（）答案：错二、问答（7小题，共0分）(a)若乐观系数矩阵中的数字是利润，请用非确定型决策的各种决策准则分别确定出相应的最优方案。

(b)若表中的数字为成本，问对应于上述各决策准则所选择的方案有何变化？答案：（a ）悲观主义准则：3S 乐观主义准则：3S Laplace 准则：3S Savage 准则：4S 折衷主义准则: 3S .(b)悲观主义准则: 2S 乐观主义准则: 1S Laplace 准则: 1S Savage 准则: 1S 折衷主义准则: 1S 或2S .2、某公司有50000元多余资金，如用于某项开发事业估计成功率为96%，成功时一年可获利12%，但一旦失败，有丧失全部资金的危险。

如把资金存放到银行中，则可稳得年利6%。

为获取更多情报，该公司求助于咨询服务，咨询费用为500元，但咨询意见只是提供试用决策树法分析：(a)该公司是否值得求助于咨询服务； (b)该公司多余资金应如何合理使用？答案：多余资金用于开发事业成功时可获利6000元，如存入银行可获利3000元。

高等数学微积分习题册上册答案

三、根据函数极限的定义证明下列极限.
（1）
lim
x→∞
1 x2
= 0；
证明：对任意ε>0，解不等式
|
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− 0 |=
1 x2
<ε
→|
x |>
1 ε
四川大学数学学院高等数学教研室编
3
学院
姓名
学号
日期
取X
= 1 ，当| x |> ε
X,
|
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−
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，所以
lim
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1.3 函数的极限
证明：对任意ε>0，解不等式 | x2 − 4 + 4 |=| x + 2 |< ε x+ 2
取δ = ε ，当 0 <| x + 2 |< δ , | x2 − 4 + 4 |< ε ，所以 lim x2 − 4 = −4 。
x+2
x→−2 x + 2
二、证明 lim(4x −1) = 11，并求正数δ ，使得当| x − 3 |< δ 时，就有| (4x −1) −11|< 0.001. x→3
学院
姓名
学号
一、根据数列极限的定义证明下列极限：
日期
1.2 数列的极限
（1）
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四川大学期末考试试题(闭卷)2018-2019春微积分

四川大学期末考试试题（闭卷）（2018——2019 学年第 2 学期） A 卷课程号：201138040 课序号：01~50 课程名称：微积分（I）-2 任课教师：成绩：第 1 页，共 2 页第 2 页，共 2 页⎨2018-2019 春微积分（I ）-2 A 卷参考答案一．填空题(每小题 4 分，共 5 分)1．曲面 z = 2x 3 - y 2 在点(1,1,1)处的切平面方程是 ( 6x - 2 y - z- 3 = 0 ).解: dz = 6x 2dx - 2 ydy , 6x - 2 y - z - 3 = 02. 设= lnarctan y, 则 x=( 3 ).x = 2, y =1 5解 :∂z = 1 2x - 1 - y =x + y , ∂z =3∂x 2 x 2+y2y 2x21 +x2x 2 + y 2∂x x = 2, y =1 53 设 f ( x , y ) =+ ⎰⎰ x 2+ y 2≤1f (x , y )dxdy ,则⎰⎰x 2+ y2≤1f ( x ,y )dxdy =2π ().3(π-1)解:f ( x , y )dxdy = k = k )dxdy = 2π k π ⎰⎰⎰⎰ 3 +x 2 + y 2 ≤1x 2 + y 2 ≤14. f (x ) 是周期为2π的周期函数， f (x ) = ⎧cos x - 2, -π≤ x < 0, ⎩sin x + 2, 0 < x < π设 s (x ) 是其傅里叶级数的和函数, 则 s (π) = ( -1).2解: s (π) = 1 [ lim f ( x ) + lim f ( x )] = - 12 x →-π+x →π-2 5. 二阶微分方程 y y ''+ y '2= 1 的通解是(y 2 = x 2 + c x + c ).∂z ∂xy 2 = 2 ( x + c )dx = x 2 + ⎰ yy '=x + c ,12二．计算题(每小题 7 分，共 28 分)6.设e z + x - 2xy + z - 1 = 0 确定的函数 z = z ( x , y ) ，∂z ∂ 2z求(1) ∂y, (2)∂y ∂x. ( 0,0) 解：(1)方程两边同时对 y 求偏导：(1) (e z+ 1)∂z- 2x = 0,∂y…………………………………………..(2 分)∂z=2x, .............................. (1 分)∂y e z + 1(2) 方程两边再同时对 x 求偏导：∂z ∂z ∂ 2z (2) e z⋅ + (e z + 1) - 2 = 0 …………………………..(2 分)∂x ∂y ∂x ∂y把 x =0,y =0 代入原方程可得 z = 0 ，( 0,0) = 2xe z + 1 = 0 , …(1 分)∂ 2 z=2=再代入方程(2), 得 ∂x ∂y e z +11 ................ (1 分)7. 空间闭曲面∑ 由 x 2 + y 2 = 1, z = 0和z = 4 + y 围成, 求它的表面积.解: S = π+⎰x 2 + y 2 =1(4 + y )ds +⎰⎰x 2 + y 2 ≤11dxdy…………………………..(4 分)= π+ 8π+2π=（9 + 2）π ……………………………..…………..(3 分)8. 计算曲线积分 I = ⎰L(e x sin y - 2 y )dx + (e x cos y - 3)dy ,其中 L 为由点 A:(2,0)到点 B;(1,2)再到原点 O (0,0)的折线段.解： ∂P =∂y ∂Q = ∂x ∂ (e x sin y - 2 y ) = e x cos y - 2 ,∂y∂(e x cos y - 3) = e x cos y ,∂x ∴ ∂P ≠ ∂y∂Q ∂x , 曲线积分与路径有关 .................... (1 分)∂z ∂y ( 0,D⎰⎰ =⎰⎰ ( ∂Q - ∂P )dxdy = 2⎰⎰ dxdy = 4,………………………… (3 分)ABOA ∂x ∂y D⎰OA= 20 ⋅ dx + (e x- 3) ⋅ 0 ⋅ dx = 0 , ............................ (1 分) 0因此∴ I =⎰AMOA- ⎰OA= 4- 0 = 4. ………………………… (1 分)⎰⎨⎩xy9. 设可导函数φ(x ) 满足φ(x ) c os x + 2 xφ(t )sin t d t = x +1, 求φ(x ) .解：设 y =φ(x ) , 两边对 x 求导,得y 'cos x + ysin x = 1其通解为 y = c cos x +sin x .………………………… (2 分)………………………… (4 分)因为 x = 0时, y = 1, 得c = 1. 所以 y = cos x + sin x .…………(1 分)三. 解答题(每小题 9 分，共 27 分)⎧ x k y10. 设二元函数 f ( x , y ) = ⎪ 4 x 2 + y 2, ( x , y ) ≠ (0, 0) . ⎪ 0,( x , y ) = (0, 0)(1) 当k 为何值时 f ( x , y ) 在点(0,0)处连续;(2) 当k 为何值时 f ( x , y ) 在点(0,0)处可微. 解：(1)令 x = ρcos θ, y = ρsin θ,lim f ( x , y ) = lim ρk -1 cos θsin θx →0 y →0ρ→0 1+ 3cos 2 θ 因为∀θ, cos θsin θ 有界,1+ 3cos 2 θ所以当k > 1时 f ( x , y ) 在点(0,0)处连续 ............. (3 分)(2)根据偏导数定义f (0, 0) = lim h →0f (0, 0) = lim k →0f (h , 0) - f (0, 0) = 0 hf (0, k ) - f (0,0)= 0 k…………………………… (2 分)讨论极限ρ→0 ρ= lim ρk -2 cos θsinθρ→0 1+3cos2 θ所以当k > 2 时f ( x, y) 在点(0,0)处可微............ (2 分)1+1+∞n =0⎩ 1 21 211. 设 f (x )=1x1+x 2∞+ arctan x , (1)把 f (x ) 展开成 x 的幂级数; (2)求 f (2019) (0) .n n解:= ∑(-1)x ,n =0x ∈(-1,1)…………………… (1 分)∴ x = 1+ x 2x ∑ n =0 (-1)nx 2n = ∑ n =0 (-1)nx 2n+1, x ∈(-1,1)…………………… (2 分)1 1 1 ∞ n 2narctan x = ⎰0 2 dx = ⎰0 ( ∑(-1) xn =0)dx= ∑(-1)nx 2n +1 , x ∈[-1,1]…………………… (3 分)n =02n +1f (x ) = x+ arctan x = ∞2n + 2 (-1)n x 2n +11+ x 2f (2019) (0) = 2019!a2019∑ 2n+1= -2019! 2020= -2020 ⋅ 2018!2019…………………… (1 分)…………………… (2 分)⎧ y ' - 3 y ' + 2 y = - xe x12 求初值问题⎨y (0) =y '(0) = 1的特解.解：对应特征方程为r 2 - 3r + 2 = 0 ，其特征根为r = 1, r = 212所以齐次方程的通解为 y = C e x+ C e 2 x ................................. (3 分)因为λ= 1 是单根, 设非齐次方程特解为 y * = x (ax +b )e x……… (2 分)代入原方程，化简得a =1, b = 1 . 所以 y * = ( 122x 2 + x )e x ， .... (2 分)从而原方程的通解为 y = C e x+ C e 2 x+ ( 1 x 2 + x )e x…………………… (1 分)∞ ∞由y(0) =y '(0) = 1 , 得c1=-1, c2= 2所以初值问题的特解为y = ( 12x 2 +x + 2)e x -e 2 x ........................... (1 分)四. 证明题(7 分)13.证明级数∑n=1(-1)n-11条件收敛于ln 2 .n∞n n ⎰ 1 2∞ n -1 1 ∞ 1证明: ∑(-1)n =1 =∑ 是调和级数, 发散; n =1 …………………… (1 分)又由莱布尼茨判别法,交错级数∑(-1) n =1 n -1 1 收敛, n…………………… (1 分)则∑(-1)n =1 n -1 1 条件收敛. n …………………… (1 分)考虑幂级数 s (x ) = ∑(-1)n =1n -1 1x n n ,收敛半径为 1, ....................................... (1 分)在绝对收敛区间为(-1,1)内,s '(x ) = ∑(-1)n =1 n -11 (x nn ) ' = ∑(-1) n =1 n -1x n -1 = 1 , 1+ x …………………… (3 分) s (x ) =x 1 dx =ln(1 + x ) 0 1+ x因为 x=1 时, 级数收敛,则 s (1) = lim ln(1+ x ) = ln 2. x →1-…………………… (1 分)五．应用题(每小题 9 分，共 18 分)14. 求曲面 x 2 + y 2 - 4x - 4 y + 7 = 0 和平面 x + z = 8 交线上的点到 y 轴的最长距离和最短距离.解：点( x , y , z ) 到 y 轴的距离为. …………………… (2 分)令F = x 2 + z 2 + λ( x 2 + y 2 - 4x - 4 y - 7) + λ( x + z - 8) , (2 分) ⎧F = 2x + λ(2x - 4) + λ = 0⎪ x 1 2 ⎪F y = λ1 (2 y - 4) = 0 F = 2z + λ = 0 …………………… (2 分)⎨ z 2 ⎪F = x 2 + y 2 - 4x - 4 y - z - 7 = 0 ∞ ∞ ∞ ∞ ∞⎪ λ1 ⎪F = x + z - 8 = 0⎩ λ2解得(x , y , z ) = (1, 2, 7)或(3, 2, 5) ....................................................................(1分) 将这两个点分别代入目标函数，可得最大值5 和最小值 .……… (2 分) 34 2∑121 2 2 15. 设空间曲面∑： z 2 = x 2 + y 2 (1 ≤ z ≤ 2部分) ，方向指向外侧，计算曲面积分 I = ⎰⎰ cos( y + z )dydz + y 2dzdx + ( x + z 2 )dxdy .解：作辅助面∑1： z = 1, ( x , y ) ∈ D : x 2 + y 2 ≤ 1 ,方向指向下侧;辅助面∑2： z = 2, ( x , y ) ∈ D : x 2 + y 2 ≤ 4 ,方向指向上侧 (2 分) ∑+∑1+∑2 围成空间闭区域Ω ,方向指向外侧, 根据高斯公式, 有⎰⎰∑+∑ +∑ = ⎰⎰⎰Ω (2 y + 2z )dxdydz = 2⎰⎰⎰Ωzdxdydz = ⎰ ⎰⎰ ⎰2 2 15 2 zdz dxdy = 2π 1 D z 1 z ⋅ z dz = π, 2…………………… (4 分) ⎰⎰∑⎰⎰∑ = -⎰⎰x 2 + y 2 ≤ 1 ( x + 1)dxdy = -π,= ⎰⎰x 2 + y 2 ≤ 4 ( x + 4)dxdy =16π,…………………… (1 分) …………………… (1 分) ∴ I = ⎰⎰1 1 2∑+∑1 +∑2-⎰⎰∑-⎰⎰∑2=15π+π- 16π=-15π.2 2.……… (1 分)。

四川大学期末考试试题(闭卷)2016-2017春微积分

（四川大学期末考试试题（闭卷）（2016——2017 学年第 2 学期）A 卷印题份数：学号：课程号：201138040 课序号：课程名称：微积分（I ）-2 任课教师：成绩：适用专业年级：学生人数：姓名：⎨ ⎩3. 求微分方程 y ' + 2 y ' + y = 6 x e- x的通解．4. 设 f ( x , y ) 的二阶偏导都连续，f (0,0) = 0 ， f x'(0,0) = f y '(0,0) = 1 ， f x 'x (0,0) = 2 ，函数 z = z ( x , y ) 由 z =f ( x + y , yz ) 确定，求 z 'x (0,0)、 z 'y (0,0)、 z 'y 'x (0,0)．三、综合题 (每小题 9 分，共 18 分)⎧ x 2 y 21．讨论函数 f ( x , y ) = ⎪ (2 x 2 + 7 y 2 )3/ 2, ( x , y ) ≠ (0,0) 在点(0,0) 处的下列性质：⎪0 , ( x , y ) = (0,0) （1）偏导数的存在性；（2）函数的连续性；（3）函数的可微性．2．设 f '( x ) 连续， f (1) = 2017 ，当 x ≠ 0 时 f ( x ) > 0 ，曲线积分 ⎰ Ly d x - x d yf ( x ) + 2017 y 2y d x - x d y在不含原点的单连通区域上与路径无关，求：（1） f ( x ) 的表达式；（2）⎰ Lf ( x ) + 2017 y 2，其中 L 为 x2+ 2017 y 2 = 1 ， L 的方向规定为逆时针方向．四、应用题 (每小题 9 分，共 18 分)1. 设曲面∑ 是由 yoz 平面上的曲线 z = y 2绕 z 轴旋转产生的，曲面∑ 与平面 x +zy + z= 12围成的立体记为Ω ，求：（1）曲面∑ 的方程；（2）曲面∑ 与平面 x +y + = 1 的交线在 xoy2平面上的投影的曲线方程；（3）计算Ω 的体积．(提示：利用 x + 1 = ρ cos θ 、y + 1 = ρ sin θ )zy 2 2. 在椭圆抛物面= x 2 +与平面 z = 20 围成的空间区域中内置一个长方体，假设该长方204体的一个面位于 z = 20 上，长方体的其它面都与某个坐标平面平行，求长方体的体积的最大值．五、证明题 (7 分)设区域 D 为 x2+y 2:(1， I = ⎰⎰sin( x 2 + Dy 2 )5/ 2 d x d y ，求证：d t ；（2）I <2π2/ 7 ；（3）I >π34· ˝ 解：直接代入曲线方程，´ y dx ´+ x d y = ˜¸L LL/ 4 ．参考解答及其评分标准一、填空题：（每题3分，共21分）1、曲面 z = 2x y 上点(1, 0, 2)处的切平面方程为.解：z x (1, 0) = 0，z y (1, 0) = 0，故切平面方程为z − 2 = 0.2、 20x 3 + 17y 3limx →0 y →0x 2 + y =.解：limx →0 y →020x 3 + 17y 3x 2 + y 2= lim ρ (20 cos 3 θ + 17 sin 3ρ→0+θ) = 0.3、设 Ω为x 2 + y 2 + z 2� 1，则 (x 2+ y 2 + z 2)dxdydz =.Ω2π π 1 2 21 4解：原式= ´0 dθ ´0 dϕ ´0 r · r sin ϕdr = 2π · 2 · 5 = 5 π. 4、设L 是y = x 2 − 1上从(0, −1)到(2, 3)的有向曲线，则´ ydx + xdy = .2− 1 + x · 2x )dx = 6. 5、设区域D 是由y = x 2与y = x 围成的，则 xydxdy = .D1 x 1 1 3 51解：˜ xydxy = ´0 dx ´x 2 xydy = ´0 2 (x − x )dx = 24 . 6、设曲线L 的方程为x 2 + y 2 = 1，则 (x 2 + 7y 2)ds = .L解：由曲线L 的对称性 ¸ x 2ds = ¸ y 2ds ，2L 20 D5∵∴ ¸ (x 2 + 7y 2)ds = 4¸ (x 2 + y 2)ds = 4¸ ds = 8π.7、微分方程 xy ， + y = x 2满足y (3) = 4的特解为 .解： (xy )， = x 2，xy = 1 x 3 + C ，由y (3) = 4可得： 312 = 9 + C ，于是C = 3，∴ y = 1 x 2 + 3.3xL L L6✓ ˜˜Σx 2+y 2�120 0 2、设曲面}为z = ✓1 − x 2 − y 2，方向规定为上侧，求˜ x 2dydz +}˝ ´ ˜ 1 2 ˝π/2 1 2 ΣΣ二、计算题：（每题9分，共36分）1、设曲面Σ为z =x 2 + y 2（x2+ y 2� 1），求˜(20xy + 17y 2)dS .解：由曲面Σ的对称性 xydS = 0， ............ 1分Σ˜ x 2dS = ˜y 2dS ， ....................... 1分 ∴ (20xy + 17y 2)dS Σ 17 2 = 2˜(x + y 2 )dS ..................................................... 2分 =17 √2 ˜ (x 2 + y 2)dxdy .................................... 2分= 17 √2 ´ 2π dθ ´ 1 ρ3dρ ..............................................2分=17 √2π .................................................................... 1分4y 2dzdx + 5z 3dxdy.艺解：补充平面z = 0，使得它们与曲面围成封闭的立体Ω，曲面方向都指向外侧； ............................................................ 1分显然在补充的平面上，曲面积分为零， ..................................... 1分于是由高斯公式可得˜ x 2dydz +y 2dzdx +5z 3dxdy = ˝(2x +2y +15z 2)dxdydz ; ........... 1分艺Ω由立体Ω的对称性，˝ xdxdy dz = ˝y dxdy dz = 0; ......................... 2分ΩΩ2 1 2用截面法计算三重积分z Ωdxdydz = 0 dz z D zdxdy .......................... 1分= ´0 z · π(1 − z 2) = π .................................................................... 2分15因此，˜ x 2dy dz +y 2dz dx +5z 3dxdy = 15˝ z 2dxdy dz = 2π .............. 1分艺Ω另解：前面解答步骤习题，用球面坐标计算三重积分 z 2dxdydz Ω2ππ/21 222= ´0 dθ ´0 dϕ ´0 r cos ϕ· r sin ϕdr ............................................... 1分= 2π ´0 cos ϕ sin ϕdϕ ´0 r dr = π ................................................. 2分 15 因此，˜ x 2dy dz +y 2dz dx +5z 3dxdy = 15˝ z 2dxdy dz = 2π .............. 1分艺Ω2Σ22 473、求微分方程y ，， + 2y ， + y = 6xe −x 的通解。

四川大学锦江学院《高等数学》2016-2017第一学期期末试卷

四川大学锦江学院《高等数学》2016-2017第一学期期末试卷考试类型：（闭卷）考试考试时间： 120 分钟学号姓名年级专业一、填空题（本大题共5小题，每小题3分，共15分） 1．二元函数2ln(21)z y x =-+的定义域为。

2. 设向量(2,1,2)a =，(4,1,10)b =-，c b a λ=-，且a c ⊥，则λ= 。

3．经过(4,0,2)-和(5,1,7)且平行于x 轴的平面方程为。

4．设yz u x =，则du = 。

5．级数11(1)npn n ∞=-∑，当p 满足条件时级数条件收敛。

二、单项选择题（本大题共5小题，每小题3分，共15分）1．微分方程2()'xy x y y +=的通解是（）A ．2x y Ce =B ．22x y Ce =C ．22y y e Cx =D ．2y e Cxy = 2．求极限(,)(0,0)limx y →= （）A ．14 B ．12-C ．14-D ．123．直线:327x y zL ==-和平面:32780x y z π-+-=的位置关系是（）A ．直线L 平行于平面πB ．直线L 在平面π上C ．直线L 垂直于平面πD ．直线L 与平面π斜交4．D 是闭区域2222{(,)|}x y a x y b ≤+≤，则Dσ= （）A ．33()2b a π-B ．332()3b a π-C ．334()3b a π-D ．333()2b a π-5．下列级数收敛的是（）A ．11(1)(4)n n n ∞=++∑ B ．2111n n n ∞=++∑ C ．1121n n ∞=-∑ D．1n ∞=三、计算题（本大题共7小题，每小题7分，共49分） 1. 求微分方程'x y y e +=满足初始条件0x =，2y =的特解。

2. 计算二重积分22Dx ydxdy x y++⎰⎰，其中22{(,)1,1}D x y x y x y =+≤+≥。