《2013厦门大学616数学分析真题》
【免费下载】厦门大学研究生入学考试历年真题及答案
厦门大学研究生入学考试历年真题及答案点击蓝色字体查看原文温馨提示：内容较多，(Ctrl+H)搜索所需内容2015年厦门大学701语言文学基础全套资料本专业课考试科目的全套资料主要包括：1．历年考研真题及详解（即考研拥有的本专业课考试科目的全部考研真题，部分可能由于官方未公布而无法取得，但是我们在不断更新，如有会第一时间补发给学员）；2．本科生、研究生课堂笔记；3．本科生、研究生课堂作业；4．教师授课讲义及课件；5．期中期末考试试题；6．指定教材（参考教材）配套资料（课后习题答案、模拟试题等）。
2015年厦门大学802文学理论与文学评论写作全套资料本专业课考试科目的全套资料主要包括：1．历年考研真题及详解（即考研拥有的本专业课考试科目的全部考研真题，部分可能由于官方未公布而无法取得，但是我们在不断更新，如有会第一时间补发给学员）；2．本科生、研究生课堂笔记；3．本科生、研究生课堂作业；4．教师授课讲义及课件；5．期中期末考试试题；6．指定教材（参考教材）配套资料（课后习题答案、模拟试题等）。
2015年厦门大学861语言学全套资料本专业课考试科目的全套资料主要包括：1．历年考研真题及详解（即考研拥有的本专业课考试科目的全部考研真题，部分可能由于官方未公布而无法取得，但是我们在不断更新，如有会第一时间补发给学员）；2．本科生、研究生课堂笔记；3．本科生、研究生课堂作业；4．教师授课讲义及课件；5．期中期末考试试题；6．指定教材（参考教材）配套资料（课后习题答案、模拟试题等）。
2015年厦门大学313历史学基础全套资料本专业课考试科目的全套资料主要包括：1．历年考研真题及详解（即考研拥有的本专业课考试科目的全部考研真题，部分可能由于官方未公布而无法取得，但是我们在不断更新，如有会第一时间补发给学员）；2．本科生、研究生课堂笔记；3．本科生、研究生课堂作业；4．教师授课讲义及课件；5．期中期末考试试题；6．指定教材（参考教材）配套资料（课后习题答案、模拟试题等）。
厦门大学2006年数学分析
厦门大学2006年招收攻读硕士学位研究生入学考试试题(一)数学分析部分1.判断题(1)在闭区间上定义的连续函数一定一致连续.(2)设()f x 为可微函数,则''()(())f x dx f x dx =⎰⎰.(3)一个绝对收敛的级数改变其求和顺序后仍然收敛,且收敛值不变.(4)因为有理数集是可数集,所以我们可以将非负有理数按大小排列成一个数列:12.n r r r <<<<(5)有限闭区间上的一个具有连续导数的有界函数,其导数也有界.2.我们将所有有理数排成一个数列1{}n n r ∞=,试讨论函数1sgn()()2n n n x r f x ∞=-=∑的连续性. 3.设函数()f x 在(,)-∞+∞上连续,(,)α∈-∞+∞,证明:(,)x ∀∈-∞+∞,都有01lim [()()]()().xh f t h f t dt f x f h αα→+-=-⎰ 4.设012(,,)n x a a a = 是n 元实函数 12,1()(,,,)n n ij i j i j f x f x x x a x x ===∑ 在单位球2121{(,,,):1}n n n i i x x x x R x ==∈≤∑ 内的极值点.则存在R λ∈使得00Ax x λ=,其中111212122212n n n n nn a a a a a a A a a a ⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭. 5.设()f t 为连续函数,,,a b c 为常数.证明 2221211()(1)()x y z f ax by cz dxdydz u f ku du π-++≤++=-⎰⎰⎰⎰,其中k = 6.设ϕ为可微函数,,,a b c 为常数.证明由方程222()ax by cz x y z ϕ++=++确定的函数(,)z z x y =满足方程()().z z cy bz az cx bx ay x y∂∂-+-=-∂∂ (二)实变函数部分1.证明有理数集是0测度集.2.设[,]k E a b ⊂的测度(),1,2,.k m E b a k =-= 证明1()[,]k k m E a b ∞=⋂=. 3.设(1,2,)k f k = 为[0,1]上的一列可测函数.若()0,.1()k k f x a e f x →+证明k f 以测度收敛于0.(三)常微分部分1.求解方程(sin sin )cos 0.x e x y dx ydy ++=2. 求解方程21.2dy x y dx x y --=+- 3. .求解方程'''320.y y y x -+-=。
《2013厦门大学616数学分析真题》
《2013厦门大学616数学分析真题》入学考试试题科目代码：616科目名称：数学分析招生专业：数学科学学院各专业一、（16分）设数列{x n }单调递增、非负，并且lim n →∞x n =a ,证明lim n →∞(a 21+a 22+...+a n n )1n =a二、（15分）设函数f 在[a,b]上为单调函数，且f (a )>a,f (b )<b 。
求证存在x 0∈[a,b ]使得f (x 0)=x 0。
三、（15分）设f ∈C [a,b ],且∫10f (x )dx =∫10xf (x )dx =0,∫10x 2f (x )dx =1，求证存在x 0∈[0,1]，满足 f (x 0) ≥12四、（15分）函数u=u(x,y)在整个平面上有二阶连续的偏导数。
求证∆u =∂2u ∂2x +∂2u ∂2y =0的充分必要条件是 C ∂u ∂−→n ds =0，其中−→n 为光滑封闭曲线C 的单位外法向量。
五、（15分）设函数f 在区间[a,b]可导，f (a +b 2)=0，且 f ′(x ) ≤M ,证明 ∫b a f (x )dx ≤M 2(a −b )21六、（20分）设{f n }为闭区间[a,b]上的一个函数列，并且满足(1)对任何z ∈[a,b ],f n (z )是一个有界数列;(2)∀ε>0,∃δ>0，使得当 x −y <σ时，对一切自然数，有‘ f n (x )−f n (y ) <ε七、（20分）设f 在区间[a,b]上非负，连续且lim x →∞f (x )=0(1)证明f 在区间[a,b]上取到最大值;(2)f 在区间[a,b]上能否取到最小值？（回答问题并说明理由）八、（20分）设f 在[0,∞)可微且有界，证明存在x n ⊂[0,+∞)使得x n →∞并且f (x n )→0九、（15分）计算二重积分∫∫∑zdxdy ，其中∑是三角形{(x,y,z ):x,y,z ≥0,x +y +z =1}。
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厦门大学真题2013年_真题-无答案
厦门大学真题2013年(总分100,考试时间90分钟)Part Ⅰ V ocabulary and StructureDirections: There are 30 sentences in this part. For each sentence there are four choices marked A, B, C and D. Choose the ONE answer that **pletes the sentence or substitute the underlined expression.1. ______ one time, Manchester was the home of the most productive cotton mills in the world.A. OnB. ByC. AtD. Of2. If **e to Tokyo, I can put you ______ in an apartment near my company.A. acrossB. downC. outD. up3. It seems oil ______ from this pipe for some time. We'll have to take the machine apart to put it right.A. had leakedB. is leakingC. leakedD. has been leaking4. He will agree to do what you require ______ him.A. ofB. fromC. toD. for5. Though he was born and brought up in America, he can speak ______ Chinese.A. fluidB. smoothC. fluentD. flowing6. We look forward to ______ to the opening ceremony.A. inviteB. be invitedC. having been invitedD. being invited7. If people feel hopeless, they don't bother to ______ the skills they need to succeed.A. adoptB. acquireC. accumulateD. assemble8. Professor Wang, ______ for his informative lectures, was warmly received by his students.A. knowingB. knownC. to be knownD. having known9. She just had no faith in me. It was William ______ she still had her faith.A. thatB. whoC. whomD. in whom10. The conference ______ a full week by the time it ends.A. must have lastedB. will have lastedC. would lastD. has lasted11. "Bob certainly has a low opinion of Sue. " "It can't be any worse than ______ of him. "A. herB. hersC. sheD. she does12. The woman has not yet ______ the loss of her son.A. got upB. got byC. got overD. got round13. Eighty percent of mothers cradle their ______ in their left arms, holding them against the left side of their bodies.A. infantsB. hosesC. handkerchiefsD. fences14. The explorers came forward with gifts of ducks and flour-cakes and ______ troughs of water for the horses to drink.A. held inB. held withC. held underD. held up15. He likes to take a hand in everything, even those that hardly concern him.A. offer help toB. get mixed up inC. have a part inD. make a fuss over16. A ______ examination for the post of department manager will be held next Tuesday.A. classifyingB. comparingC. contrastingD. competitive17. Mother was so weak after her operation that the doctors wondered if they would be able to ______ her through.A. pullB. cureC. pushD. save18. Go and see what your mother is ______ now.A. forB. atC. aboutD. busy19. With three young children to take care of, Cathy is kept on the run every minute of the day.A. walkingB. at full speedC. busyD. on foot20. Since his retirement, Peter Smith, who was ______ a teacher, has written four novels.A. latelyB. usuallyC. formerlyD. already21. We must ______ on our reputation to expand the business.A. improveB. buildC. developD. weigh22. ______ it or not, his discovery has created a stir in scientific circles.A. BelieveB. To believeC. BelievingD. Believed23. ______ does business with that fellow is bound to lose money.A. WhoeverB. WhoC. No matter whoD. However24. Ann never dreams of ______ for her to be sent abroad very soon.A. there being a chanceB. there to be a chanceC. there be a chanceD. being a chance25. Frequently single-parent children ______ some of the functions that the absent adult in the house would have served.A. take offB. take afterC. take inD. take on26. Whenever a **pany ______ a small one, the product almost always gets worse.A. gets on withB. cuts downC. takes overD. puts up with27. Samuel was obliged to compromise on lesser questions.A. **pelledB. was delightedC. was preparedD. was only too ready28. Children tend to ______ while playing, even if they make a promise before.A. lose all count of timeB. keep all count of timeC. be aware of the passage of timeD. waste time29. A survey was carried out on the death rate of new-horn babies in that region, ______ were surprising.A. as resultsB. which resultsC. the results of itD. the results of which30. Our manager is so ______ in his thinking, he never listens to new ideas.A. stiffB. rigidC. tenseD. tightPart Ⅱ Reading ComprehensionSection ADirections: There are 3 passages in this part. Each passage is followed by some questions or unfinished statements. For each of them there are four choices marked A, B, C and D.Statuses are marvelous human inventions that enable us to get along with one another and to determine where we "fit" in society. As we go about our everyday lives, we mentally attempt to place people in terms of their statuses. For example, we must judge whether the person in the library is a reader or a librarian, whether the telephone caller is a friend or a salesman, whether the unfamiliar person on our property is a thief or a meter reader, and so on.The statuses we assume often vary with the people we encounter, and change throughout life. Most of us can, at very high speed, assume the statuses that various situations require. Much of social interaction consists of identifying and selecting among appropriate statuses and allowing other people to assume their statuses in relation to us. This means that we fit our actions to those of other people based on a constant mental process of appraisal and interpretation. Although some of us find the task more difficult than others, most of us perform it rather effortlessly.A status has **pared to ready-made clothes. Within certain limits, the buyer can choose style and fabric. But an American is not free to choose the costume of a Chinese peasant or that of a Hindu prince. We must choose from among the clothing presented by our society. Furthermore, our choice is limited to a size that will fit, as well as by our pocketbook. Having made a choicewithin these limits we can have certain alterations made, hut apart from minor adjustments, we tend to be limited to what the stores have on their racks. Statuses **e ready made, and the range of choice among them is limited.1. In the first paragraph, the writer tells us that statuses can help us ______.A. determine whether a person is fit for a certain jobB. behave appropriately in relation to other peopleC. protect ourselves in unfamiliar situationsD. make friends with other people2. According to the writer, people often assume different statuses ______.A. in order to identify themselves with othersB. in order to better identify othersC. as their mental processes changeD. as the situation changes3. The word "appraisal" (Sentence 4, Paragraph 2) most probably means ______.A. involvementB. appreciationC. assessmentD. presentation4. In the last sentence of the second paragraph, the pronoun "it" refers to "______".A. fitting our actions to those of other people appropriatelyB. identification of other people's statusesC. selecting one's own statusesD. constant mental process5. By saying that "an American is not free to choose the costume of a Chinese peasant or that of a Hindu prince" (Sentence 3, Paragraph 3), the writer means ______.A. different people have different styles of clothesB. ready-made clothes may need alterationsC. **e ready made just like clothesD. our choice of statuses is limitedHumanity uses a little less than half the water available worldwide. Yet occurrences of shortages and droughts are causing famine and distress in some areas, and industrial and agricultural by-products are polluting water supplies. Since the world's population is expected to double in the next 50 years, many experts think we are on the edge of a widespread water crisis.But that doesn't have to be the outcome. Water shortages do not have to trouble the world—if we start valuing water more than we have in the past. Just as we began to appreciate petroleum more after the 1970s oil crises, today we must start looking at water from a fresh economic perspective. We can no longer afford to consider water a virtually free resource of which we can use as much as we like in any way we want.Instead, for all used except the domestic demand of the poor, governments should price water to reflect its actual value. This means charging a fee for the water itself as well as for the supply costs.Governments should also protect this resource by providing water in more economically and environmentally sound ways. For example, often the cheapest way to provide irrigation water in the dry tropics is through small-scale projects, such as gathering rainfall in depressions and pumping it to nearby cropland.No matter what steps governments take to provide water more efficiently, they must change their institutional and legal approaches to water use. Rather than spread control among hundreds or even thousands of local, regional, and national agencies that watch various aspects of water use, countries should set up central authorities to coordinate water policy.6. What is the real cause of the potential water crisis?A. Only half of the world's water can be used.B. The world population is increasing faster and faster.C. Half of the world's water resources have been seriously polluted.D. Humanity has not placed sufficient value on water resources.7. As indicated in the passage, the water problem ______.A. is already serious in certain parts of the worldB. has been exaggerated by some experts in the fieldC. poses a challenge to the technology of building reservoirsD. it underestimated by government organizations at different levels8. According to the author, the water price should ______.A. be reduced to the minimumB. stimulate domestic demandC. correspond to its real valueD. take into account the occurrences of droughts9. The author says that in some hot and dry areas it is advisable to ______.A. build big lakes to store waterB. construct big pumping stationsC. build small and cheap irrigation systemsD. channel water from nearby rivers to cropland10. In order to raise the efficiency of the water supply, measures should be taken to ______.A. guarantee full protection of the environmentB. centralize the management of water resourcesC. increase the sense of responsibility of agencies at all levelsD. encourage local and regional protection of water resourcesEvery night she listened to her father going around the house, locking the doors and windows. She listened, the back door closed; she could hear the fastener of the kitchen window's click, and the restless pad of his feet going back to try the front door. It wasn't only the outside doors he locked; he locked the empty kitchen too. He was looking something out, but obviously it was something capable of entering into his first defenses. He raised his second line all the way up to bed.In fourteen years, she thought unhappily, the house will be his; he had paid twentyfive pounds down and the rest he was paying month by month as rent. "Of course," he was in the habit of saying, "I've improved the property." "Yes," he repeated, "I've improved the property," looking around for a nail to drive in, a weed to uproot. It was more than a sense of property; it was a sense of honesty. Some people who bought their homes through the society let them go to rack and ruin and then cleared out.She stood with her ear against the wall, a small, dark, angry, immature figure. There was no more to be heard from the other room; but in her inner ear she still heard the footsteps of aproperty owner, the tap-tap of a hammer, the scrape of a spade, the whistle of radiator steam, a key turning, a bolt pushed home, the little busy sounds of men building barriers. She stood planning.11. Which of the following is TRUE of the father in the passage?A. He thought a lot about his daughter's future.B. He saved a lot of money for his daughter.C. He thought that he was secure.D. He avoided his neighbors on purpose.12. From the passage we can see that the father is ______.A. kind to his daughter and neighborsB. cruel to his daughter and neighborsC. systematical in his actionsD. careful about his appearance13. According to the passage, the daughter's attitude toward his father is of ______.A. slight dislikeB. great disapprovalC. strong loveD. grateful acceptance14. Which of the following feelings is conveyed in the passage?A. Tenseness.B. Peace.C. Nervousness.D. Happiness.15. All of the following are true EXCEPT that ______.A. the father built his defenses carefullyB. some property owners let their homes go worseC. the daughter was thin and very youngD. the father punished the girl when she was youngSection BDirections: In the following article, some sentences have been removed. For Questions 1～5, choose the most suitable one from the list A～E to fit into each of the numbered blank.Where Is the News Leading Us?Not long ago I was asked to join in a public symposium on the role of the American press. Two other speakers were included on the program. The first was a distinguished TV anchorman. The other was the editor of one of the nation's leading papers, a newsman to the core-tough, aggressive, and savvy in the ways and means of solid reporting.The purpose of the symposium, as I understood it, was to scrutinize the obligations of the media and to suggest the best ways to meet those obligations.(1) . Why, he asked, are the newspapers and television news programs so disaster-prone? Why are newsmen and women so attracted to tragedy, violence, failure?The anchorman and editor reacted as though they had been blamed for the existence of bad news. Newsmen and newswomen, they said, are only responsible for reporting the news, not for creating it or modifying it.(2) . The gentleman who had asked it was not blaming them for the distortions in the world. He was just wondering why distortions are most reported. The news media seem to operate on thephilosophy that all news is bad news. Why? Could it be that the emphasis on downside news is largely the result of tradition—the way newsmen and newswomen are accustomed to respond to daily events?(3) . News is supposed to deal with happenings of the past 12 hours—24 hours at most. Anything that happens so suddenly, however, is apt to be eruptive. A sniper kills some pedestrians;a terrorist holds 250 people hostage in a plane; OPEC announces a 25 percent increase in petroleum prices; Great Britain devalues by another 10 percent; a truck conveying radioactive wastes collides with a mobile cement mixer.(4) . Civilization is a lot more than the sum total of its catastrophes. The most important ingredient in any civilization is progress. But progress doesn't happen all at once. It is not eruptive. Generally, it comes in bits and pieces, very little of it clearly visible at any given moment, but all of it involved in the making of historical change for the better.It is this aspect of living history that most news reporting reflects inadequately. The result is that we are under informed about positive developments and over informed about disasters. This, in turn, leads to a public mood of defeatism and despair, which in themselves tend to be inhibitors of progress. An unrelieved diet of eruptive news depletes the essential human energies a free society needs. (5) .I am not suggesting that "positive" news be contrived as an antidote to the disasters on page one. Nor do I define positive news as in-depth reportage of functions of the local YMCA. What I am trying to get across is the notion that the responsibility of the news media is to search out and report on important events—whether or not **e under the heading of conflict, confrontation, or catastrophe. The world is a **bination of heaven and hell, and both sectors call for attention and scrutiny.A. Focusing solely on these details, however, produces a misshapen picture.B. Perhaps it would be useful here to examine the way we define the word news, for this is where the problem begins.C. A mood of hopelessness and cynicism is hardly likely to furnish the energy needed to meet serious challenges.D. During the open-discussion period, a gentleman in the audience addressed a question to my two colleagues.E. It didn't seem to me that the newsmen had answered the question.1.2.3.4.5.Part Ⅲ Short Answer QuestionsDirections: Read the following passage and then give short answers to the five questions. Surveys have shown that most football and basketball injuries involve the knee, either throughtwisting or through application of lateral force. Surgery for such injuries has become much simpler with the invention of a thin device containing a fiber optics light that can be inserted into a thin slit in the knee. Repair can be accomplished through this narrow opening. Long-distance runners also suffer knee injuries, but a **mon problem for runners is stress fracture, which is a weakening of the front of the shinbone caused by overuse, with pain and possible bone cracking as the result. Ligament tears are **mon in gymnastics. Almost all these conditions heal with rest. Prevention of injuries depends primarily on good conditioning. Athletes are also protected by the use of better padding materials and of face masks and eye protectors in rough sports.The improper or illegal use of drugs and substances for the temporary improvement of athletic performance in competitions has been a frequent subject of inquiry since the 1960s, when drug misuse by athletes to gain an unfair advantage began to rise dramatically. Anabolic steroids supposedly improve strength and endurance, but they can also have harmful side effects as liver damage. Tests for drugs such as heroin and other stimulants were introduced at the Olympic Games in 1968. Anabolic steroids were not banned until 1974, when a suitable test was developed. The illegality of some drugs has not been accepted by a number of other international and national amateur athletic federations, for reasons including testing uncertainties, doubts about banning certain medicinal substances or common drugs such as caffeine, and simple lack of concern. Controversy has also arisen over the legality of the practice of "blood doping," in which an athlete receives a blood transfusion just before an event. The resulting increase in red blood cells apparently increases the athlete's aerobic power.1. In addition to knee injuries, what will also heal with rest?2. What does the word "slit" in Pars. 1 most probably mean?3. When were the tests for drugs used at the Olympic Games?4. According to the passage, drug misuse by athletes is considered as ______.5. Controversy arises over the illegality of **mon drugs due to ______.Part Ⅳ English-Chinese TranslationDirections: Read the passage carefully and translate the underlined sentences into Chinese.Doing a PhD is certainly not for everybody, and I do not recommend it for most people. However, I am really glad I got my PhD rather than just getting a job after finishing my Bachelor's. The number one reason is that I learned a hell of a lot doing the PhD, and most of the things I learned I would never get exposed to in a typical software engineering job. (1) The process of doing a PhD trains you to do research, to read research papers, to run experiments, to write papers, to give talks. It also teaches you how to figure out what problem needs to be solved. You gain a very sophisticated technical background doing the PhD, and having your work subject to the intense scrutiny of the academic peer-review process—not to mention your **mittee.I think of the PhD a little like the Grand Tour, a tradition in the 16th and 17th centuries where youths would travel around Europe, getting a rich exposure to high society in France, Italy, and Germany, learning about art, architecture, language, literature, fencing, riding—all of the essential liberal arts that a gentleman was expected to have experience with to be an influential member ofsociety. Doing a PhD is similar. You get an intense exposure to every subfield of Computer Science, and have to become the leading world's expert in the area of your dissertation work. (2) The top PhD programs set an incredibly high bar. a lot of coursework, teaching experience, qualifying exams, a thesis defense, and of course making a groundbreaking research contribution in your area. Having to go through this process gives you a tremendous amount of technical breadth and depth.Some important stuff I learned doing a PhD.How to read and critique research papers. As a grad student you have to read thousands of research papers, extract their main ideas, critique the methods and presentation, and synthesize their contributions with your own research. As a result you are exposed to a wide range of CS topics, approaches for solving problems, sophisticated algorithms, and system designs. This is not just about gaining the knowledge in those papers (which is pretty important), but also about becoming conversant in the scientific literature.How to write papers and give talks. Being fluent in **munications is a really important skill for engineers. I've noticed a big gap between the software engineers I've worked with who have PhDs and those who don't in this regard. (3) PhD-trained folks tend to give clear, well-organized talks and know how to write up their work and visualize the result of experiments. As a result they can be much more influential.How to run experiments and interpret the results: I can't overstate how important this is. A systems-oriented PhD requires that you run a zillion measurements and present the results in a way that is both bullet-proof to peer-review criticism (in order to publish) and **pelling. Every aspect of your methodology will be critiqued (by your advisor, your co-authors, your paper reviewers) and you will quickly learn how to run the right experiments, and do it right.(4) How to figure out what problem to work on. This is probably the most important aspect of PhD training. Doing a PhD will force you to cast away from shore and explore the boundary of human knowledge. (Matt Might's cartoon on this is a great visualization of this. ) I think that at least 80% of making a scientific contribution is figuring out what problem to tackle, a problem that is at once interesting, open, and going to have impact if you solve it. There are lots of open problems that the **munity is not interested in (c.f., writing an operating system kernel in Haskell) . There are many interesting problems that have been solved over and over and over (c.f. , file system block layout optimization; wireless multi hop routing) . There's a real trick to picking good problems, and developing a taste for it is a key skill if you want to become a technical leader.(5) So I think it's worth having a PhD, especially if you want to work on the hardest and most interesting problems. This is true whether you want a career in academia, a research lab, or a more traditional engineering role. But as my PhD advisor was fond of saying, "doing a PhD costs you a house. " (In terms of the lost salary during the PhD years—these days it's probably more like several houses. )1.2.3.4.5.Part Ⅴ WritingDirections: You are asked to write an essay on the following topic:1. Many universities and colleges offer qualification: through some sort of distance learning using the Internet, rather than by face-to-face contact in a classroom.In your opinion, do the advantages of this development outweigh the disadvantages of learning in this way?You should write at least 250 words.You should use your own ideas, knowledge and experience and support your arguments with examples and relevant evidence.。
2016年厦门大学数学分析考研试题
厦门大学2016年招收攻读硕士学位研究生入学统一考试试题科目名称：数学分析考生须知：1.本试卷满分为150分，全部考试时间总计180分钟；2.所有答案必须写在答题纸上，写在试题纸上或草稿纸上一律无效。
————————————————————————————————————————1.(20分)已知f (x )在[0,+∞)上单调递减,且lim x →+∞f (x )=0,证明∞∑n =1f (n )收敛的充分必要条件是∫+∞0f (x )dx 收敛.2.(20分)设f ∈C 1[0,+∞],f (0)=1,f ′(x )=1x 2+f 2(x ).证明:(a)lim x →+∞f (x )存在;(b)lim x →+∞f (x )≤1+π2.3.(15分)已知lim n →∞a n n=0,证明limn →∞max {a 1···,a n }n =0.4.(20分)已知f (x )有界,且在R 上连续.设T >0,证明:存在数列{x n },使得lim n →∞x n =+∞,lim n →∞(f (x n +T )−f (x n ))=0.5.(20分)设f 在[a ,b ]上二阶可导,且∀x ∈(a ,b )有f ′′(x )>0.证明:∀x 1,x 2∈(a ,b ),有f (x 1+x 22)<12[f (x 1)+f (x 2)].6.(15分)设f 在[a ,b ]上可积,且有∫xa f (t )dt ≥0,∫b a f (x )dx =0.证明:∫ba x f (x )dx ≤0.7.(20分)设B 为单位球x 2+y 2+z 2≤1的区域,∂B 为其球面.已知f 为k 次齐次函数,即f (ax ,ay ,az )=a k f (x ,y ,z ).证明:∫∫∂B f (x ,y ,z )dS =∫∫∫B △f dxdydz ,其中△f =∂2f ∂x 2+∂2f ∂y 2+∂2f ∂z2.8.(20分)设有一张长方形纸片,要在上面涂颜色.长方形纸片内部涂颜色的面积为A cm 2，边缘有空隙:上下边宽度之和为r cm,左右宽度为h cm.意思是:在长方形纸片上给矩形求:当长方形纸片长(y cm)和宽(x cm)为多少时,长方形纸片面积最小?注：感谢数学人才小基地群(342767800)Veer 提供的真题.考试科目：数学分析第1页共1页。