暨南大学数学学科2011年硕士研究生入学考试
2011考研数学(一二三)真题(含答案)
(B) 1,2 ．
(C) 1,2,3 ． (D) 2 ,3,4 ．
【解析】由 x 0 的基础解系只有一个知 r(A) 3 ，所以 r( A) 1，
f
(0),
B
2z xy
|(0,0)
f
(x)
f ( y) f (y)
|(0,0)
[
f (0)]2 f (0)
0,
C
2z y2
|(0,0)
f
(x)
f
( y) f
(y) [ f f 2(y)
( y)]2
|(0,0)
f (0) [ f (0)]2 f (0)
较强。
观察选项：(A)，(B)，(C)，(D)四个选项的收敛半径均为 1，幂级数收敛区间的中心在 x 1 处，
故(A)，(B)错误；
因为
an
单调减少，lim n
an
0 ，所以 an
0 ，所以
n1
an
为正项级数，将
x
2 代入幂级数得
n1
an
，
n
而已知 Sn ak 无界，故原幂级数在 x 2 处发散，(D)不正确． k 1
2011 年全国硕士研究生入学 统一考试
数学（一、二、三） 试题及解析
山东考研辅导专家 苏老师
1
2011 年全国硕士研究生入学统一考试
数学（一）试题
一、选择题：1～8 小题，每小题 4 分，共 32 分，下列每题给出的四个选项中，只有一个选项符合题目要 求，请将所选项前的字母填在答．题．纸．指定位置上．
2
暨南大学601高等数学2010--2014,2017,2019--2020年考研真题试卷
3．若 y5 2 y x 3x7 0 ，则 dy |x0 __________________________.
4．
lim(
n
n
1 2
1
2 n2 2
...
n ______.
5．以函数 y C2 作为通解的微分方程是_______________________. x C1
____________
(A) 充要条件 (B) 充分不必要条件 (C) 必要不充分条件 (D) 既不充分也不必要
4. 若级数 (an bn ) 收敛，那么说法正确的是___________
n1
(A) an 和 bn 中至少有一个收敛 (B) an 和 bn 有相同的敛散性
n1
n1
n1
n1
(C) an 和 bn 都收敛
D
6．求 4 ln(1 tan x)dx . 0
dx
7. 判断积分 0
(1 x)(1 x2 ) 的收敛，如果收敛，求其值.
8. 求一阶线性微分方程 dy 5y x 的通解. 并求满足初始条件 y(0) 0 的特解. dx
9．求在平面 x y z 1与柱面 x2 y2 1的交线上到 XOY 面的距离最远的点. 345
考试科目：高等数学B
共 4 页，第 3 页
4、证明题 （本题共2小题，每小题5分，共10分）
1. 设函数 f (x) 在 (,) 上可导,证明:若 f ' (x) f (x) 没有实数解，那么曲线
y f (x) 与 x 轴最多只能有一个交点.
df
1 ( dx
x)
|x3
___________
(A) 1 3
(B) 3
(C) 1
2011年研究生入学考试数学真题
2011年全国硕士研究生入学统一考试管理类专业学位联考英语试题Section I Use of EnglishDirections:Read the following text. Choose the best word(s) for each numbered blank and mark [A], [B], [C] or [D] on ANSWER SHEET. (10 points)The Internet affords anonymity to its users, a blessing to privacy and freedom of speech. But that very anonymity is also behind the explosion of cyber-crime that has 1 across the Web.Can privacy be preserved 2 bringing safety and security to a world that seems increasingly 3 ?Last month, Howard Schmidt, the nation’s cyber-czar, offered the federal government a 4 to make the Web a safer place—a “voluntary trusted identity” system that would be the high-tech 5 of a physical key, a fingerprint and a photo ID card, all rolled 6 one. The system might use a smart identity card, or a digital credential 7 to a specific computer, and would authenticate users at a range of online services.The idea is to 8 a federation of private online identity systems. User could 9 which system to join, and only registered users whose identities have been authenticated could navigate those systems. The approach contrasts with one that would require an Internet driver’s license 10 by the government.Google and Microsoft are among companies that already have these “single sign-on” systems that make it possible for users to 11 just once but use many different services.12 , the approach would create a “walled garden” in cyberspace, with safe “neighborhoods” and bright “streetlights” to establish a sense of a 13 community.Mr. Schmidt described it as a “voluntary ecosystem” in which “individuals and organizations can complete online transactions with 14 , trusting the identities of each other and the identities of the infrastructure 15 which the transaction runs.”Still, the administration’s plan has 16 privacy rights activists. Some applaud the approach; others are concerned. It seems clear that such a scheme is an initiative push toward what would 17 be a compulsoryInternet “driver’ s license” mentality.The plan has also been greeted with 18 by some computer security experts, who worry that the “voluntary ecosystem” envisioned by Mr. Schmidt would still leave much of the Internet 19 . They argue that all Internet users should be 20 to register and identify themselves, in the same way that drivers must be licensed to drive on public roads.1.[A]swept[B]skipped[C]walked[D]ridden2.[A]for[B]within[C]while[D]though3.[A]careless[B]lawless[C]pointless[D]helpless4.[A]reason[B]reminder[C]compromise[D]proposal5.[A]information[B]interference[C]entertainment[D]equivalent6.[A]by[B]into[C]from[D]over7.[A]linked[B]directed[C]chained[D]compared8.[A]dismiss[B]discover[C]create[D]improve9.[A]recall[B]suggest[C]select[D]realize10.[A]released[B]issued[C]distributed[D]delivered11.[A]carry on [B]linger on[C]set in [D]log in12.[A]In vain[B]In effect[C]In return [D]In contrast13.[A]trusted[B]modernized[C]thriving[D]competing14.[A]caution[B]delight[C]confidence[D]patience15.[A]on[B]after[C]beyond[D]across16.[A]divided[B]disappointed[C]protected[D]united17.[A]frequently[B]incidentally[C]occasionally[D]eventually18.[A]skepticism[B]tolerance[C]indifference [D]enthusiasm19.[A]manageable[B]defendable[C]vulnerable[D]invisible20.[A]invited[B]appointed[C]allowed[D]forcedSection Ⅱ Reading ComprehensionPart ADirections:Read the following four texts. Answer the questions after each text by choosing [A], [B], [C] or [D]. Mark your answers on ANSWER SHEET. (40 points)Text 1Ruth Simmons joined Goldman Sachs’s board as an outside director in January 2000; a year later she became president of Brown University. For the rest of the decade she apparently managed both roles without attracting much criticism. But by the end of 2009 Ms. Simmons was under fire for having sat on Goldman’s compensation committee; how could she have let those enormous bonus payouts pass unremarked? By February the next year Ms. Simmons had left the board. The position was just taking up too much time, she said.Outside directors are supposed to serve as helpful, yet less biased, advisers on a firm’s board. Having made their wealth and their reputations elsewhere, they presumably have enough independence to disagree with the chief executive’s proposals. If the sky, and the share price, is falling, outside directors should be able to give advice based on having weathered their own crises.The researchers from Ohio University used a database hat covered more than 10,000 firms and more than 64,000 different directors between 1989 and 2004. Then they simply checked which directors stayed from one proxy statement to the next. The most likely reason for departing a board was age, so the researchers concentrated on those “surprise” disappearances by directors under the age of 70. They found that after a surprise departure, the probability that the company will subsequently have to restate earnings increased by nearly 20%. The likelihood of being named in a federal class-action lawsuit also increases, and the stock is likely to perform worse. The effect tended to be larger for larger firms. Although a correlation between them leaving and subsequent bad performance at the firm is suggestive, it does not mean that such directors are always jumping off a sinking ship. Often they “trade up,” leaving riskier, smaller firms for larger and more stable firms.But the researchers believe that outside directors have an easier time of avoiding a blow to their reputations if they leave a firm before bad news breaks, even if a review of history shows they were on the board at the time any wrongdoing occurred. Firms who want to keep their outside directors through tough times may have to create incentives. Otherwise outside directors will follow the example of Ms. Simmons, once again very popular on campus.21. According to Paragraph 1, Ms. Simmons was criticized for______.[A] gaining excessive profits[B] failing to fulfill her duty[C] refusing to make compromises[D] leaving the board in tough times22. We learn from Paragraph 2 that outside directors are supposed to be______.[A] generous investors[B] unbiased executives[C] share price forecasters[D] independent advisers23. According to the researchers from Ohio University, after an outside director’s surprise departure, the firm is likely to______.[A] become more stable[B] report increased earnings[C] do less well in the stock market[D] perform worse in lawsuits24. It can be inferred from the last paragraph that outside directors______.[A] may stay for the attractive offers from the firm[B] have often had records of wrongdoings in the firm[C] are accustomed to stress-free work in the firm[D] will decline incentives from the firm25. The author’s attitude toward the role of outside directors is______.[A] permissive[B] positive[C] scornful[D] criticalText 2Whatever happened to the death of newspapers? A year ago the end seemed near. The recession threatened to remove the advertising and readers that had not already fled to the internet. Newspapers like the San Francisco Chronicle were chronicling their own doom. America’s Federal Trade Commission launched a round of talks about how to save newspapers. Should they become charitable corporations? Should the state subsidize them? It will hold another meeting soon. But the discussions now seem out of date.In much of the world there is little sign of crisis. German and Brazilian papers have shrugged off the recession. Even American newspapers, which inhabit the most troubled corner of the global industry, have not only survived but often returned to profit. Not the 20% profit margins that were routine a few years ago, but profit all the same.It has not been much fun. Many papers stayed afloat by pushing journalists overboard. The American Society of News Editors reckons that 13,500 newsroom jobs have gone since 2007. Readers are paying more for slimmer products. Some papers even had the nerve to refuse delivery to distant suburbs. Yet these desperate measures have proved the right ones and, sadly for many journalists, they can be pushed further.Newspapers are becoming more balanced businesses, with a healthier mix of revenues from readers and advertisers. American papers have long been highly unusual in their reliance on ads. Fully 87% of their revenues came from advertising in 2008, according to the Organization for Economic Cooperation & Development (OECD). In Japan the proportion is 35%. Not surprisingly, Japanese newspapers are much more stable.The whirlwind that swept through newsrooms harmed everybody, but much of the damage has been concentrated in areas where newspaper are least distinctive. Car and film reviewers have gone. So have science and general business reporters. Foreign bureaus have been savagely cut off. Newspapers are less complete as a result. But completeness is no longer a virtue in the newspaper business.26. By saying “Newspapers like … their own doom” (Lines 2-3, Para. 1), the author indicates that newspapers______.[A] neglected the sign of crisis[B] failed to get state subsidies[C] were not charitable corporations[D] were in a desperate situation27. Some newspapers refused delivery to distant suburbs probably because______.[A] readers threatened to pay less[B] newspapers wanted to reduce costs[C] journalists reported little about these areas[D] subscribers complained about slimmer products28. Compared with their American counterparts, Japanese newspapers are much more stable because they______.[A] have more sources of revenue[B] have more balanced newsrooms[C] are less dependent on advertising[D] are less affected by readership29. What can be inferred from the last paragraph about the current newspaper business? ______.[A] Distinctiveness is an essential feature of newspapers.[B] Completeness is to blame for the failure of newspaper.[C] Foreign bureaus play a crucial role in the newspaper business.[D] Readers have lost their interest in car and film reviews.30. The most appropriate title for this text would be______.[A] American Newspapers: Struggling for Survival[B] American Newspapers: Gone with the Wind[C] American Newspapers: A Thriving Business[D] American Newspapers: A Hopeless StoryText 3We tend to think of the decades immediately following World War II as a time of prosperity and growth, with soldiers returning home by the millions, going off to college on the G. I. Bill and lining up at the marriage bureaus.But when it came to their houses, it was a time of common sense and a belief that less could truly be more. During the Depression and the war, Americans had learned to live with less, and that restraint, in combination with the postwar confidence in the future, made small, efficient housing positively stylish.Economic condition was only a stimulus for the trend toward efficient living. The phrase “less is more” was actually first popularized by a German, the architect Ludwig Mies van der Rohe, who like other people associatedwith the Bauhaus, a school of design, emigrated to the United States before World War II and took up posts at American architecture schools. These designers came to exert enormous influence on the course of American architecture, but none more so than Mies.Mies’s signature phrase means that less decoration, properly organized, has more impact than a lot. Elegance, he believed, did not derive from abundance. Like other modern architects, he employed metal, glass and laminated wood-materials that we take for granted today but that in the 1940s symbolized the future. Mies’s sophisticated presentation masked the fact that the spaces he designed were small and efficient, rather than big and often empty.The apartments in the elegant towers Mies built on Chicago’s Lake Shore Drive, for example, were smaller—two-bedroom units under 1,000 square feet—than those in their older neighbors along the city’s Gold Coast. But they were popular because of their airy glass walls, the views they afforded and the elegance of the buildings’ details and proportions, the architectural equivalent of the abstract art so popular at the time.The trend toward “less” was not entirely foreign. In the 1930s Frank Lloyd Wright started building more modest and efficient houses—usually around 1,200 square feet—than the spreading two-story ones he had designed in the 1890s and the early 20th century.The “Case Study Houses” commissioned from talented modern architects by California Arts & Architecture magazine between 1945 and 1962 were yet another homegrown influence on the “less is more” trend. Aesthetic effect came from the landscape, new materials and forthright detailing. In his Case Study House, Ralph Rapson may have mispredicted just now how the mechanical revolution would impact everyday life—few American families acquired helicopters, though most eventually got clothes dryers—but his belief that self-sufficiency was both desirable and inevitable was widely shared.31. The postwar American housing style largely reflected the Americans’______.[A] prosperity and growth[B] efficiency and practicality[C] restraint and confidence[D] pride and faithfulness32. Which of the following can be inferred from Paragraph 3 about the Bauhaus?[A] It was founded by Ludwig Mies van der Rohe.[B] Its designing concept was affected by World War II.[C] Most American architects used to be associated with it.[D] It had a great influence upon American architecture.33. Mies held that elegance of architectural design______.[A] was related to large space[B] was identified with emptiness[C] was not reliant on abundant decoration[D] was not associated with efficiency34. What is true about the apartments Mies built Chicago’s Lake Shore Drive?[A] They ignored details and proportions.[B] They were built with materials popular at that time.[C] They were more spacious than neighboring buildings.[D] They shared some characteristics of abstract art.35. What can we learn about the design of the “Case Study Houses”?[A] Mechanical devices were widely used.[B] Natural scenes were taken into account[C] Details were sacrificed for the overall effect.[D] Eco-friendly materials were employed.Text 4Will the European Union make it? The question would have sounded strange not long ago. Now even the project’s greatest cheerleaders talk of a continent facing a “Bermuda triangle” of debt, population decline and lower growth.As well as those chronic problems, the EU face an acute crisis in its economic core, the 16 countries that use the single currency. Markets have lost faith that the euro zone’s economies, weaker or stronger, will one day converge thanks to the discipline of sharing a single currency, which denies uncompetitive members the quick fix of devaluation.Yet the debate about how to save Europe’s single currency from disintegration is stuck. It is stuck because the euro zone’s dominant powers, France and Germany, agree on the need for greater harmonization within the euro zone, but disagree about what to harmonise.Germany thinks the euro must be saved by stricter rules on borrowing spending and competitiveness, backed by quasi-automatic sanctions for governments that do not obey. These might include threats to freeze EU funds for poorer regions and EU mega-projects, and even the suspension of a country’s voting rights in EU ministerial councils. It insists that economic co-ordination should involve all 27 members of the EU club, among whom there is a small majority for free-market liberalism and economic rigour; in the inner core alone, Germany fears, a small majority favour French interference.A “southern” camp headed by France wants something different: “European economic government” within an inner core of euro-zone members. Translated, that means politicians intervening in monetary policy and a system of redistribution from richer to poorer members, via cheaper borrowing for governments through common Eurobonds or complete fiscal transfers. Finally, figures close to the French government have murmured, euro-zone members should agree to some fiscal and social harmonization: e.g., curbing competition in corporate-tax rates or labour costs.It is too soon to write off the EU. It remains the world’s largest trading block. At its best, the European project is remarkably liberal: built around a single market of 27 rich and poor countries, its internal borders are far more open to goods, capital and labour than any comparable trading area. It is an ambitious attempt to blunt the sharpest edges of globalization, and make capitalism benign.36. The EU is faced with so many problems that______.[A] it has more or less lost faith in markets[B] even its supporters begin to feel concerned -[C] some of its member countries plan to abandon euro[D] it intends to deny the possibility of devaluation37. The debate over the EU’s single currency is stuck because the dominant powers______.[A] are competing for the leading position[B] are busy handling their own crises[C] fail to reach an agreement on harmonization[D] disagree on the steps towards disintegration38. To solve the euro problem, Germany proposed that______.[A] EU funds for poor regions be increased[B] stricter regulations be imposed[C] only core members be involved in economic co-ordination[D] voting rights of the EU members be guaranteed39. The French proposal of handling the crisis implies that ______.[A] poor countries are more likely to get funds[B] strict monetary policy will be applied to poor countries[C] loans will be readily available to rich countries[D] rich countries will basically control Eurobonds40. Regarding the future of the EU, the author seems to feel ______.[A] pessimistic[B] desperate[C] conceited[D] hopefulPart BDirections:Read the following text and answer the questions by finding information from the right column that corresponds to each of the marked details given in the left column. There are two extra choices in the right column. Mark your answers on ANSWER SHEET 1. (10 points)Leading doctors today weigh in on the debate over the government's role in promoting public health by demanding that ministers impose "fat taxes" on unhealthy food and introduce cigarette-style warnings to children about the dangers of a poor diet.The demands follow comments made last week by the health secretary, Andrew Lansley, who insisted the government could not force people to make healthy choices and promised to free business from public health regulations.But senior medical figures want to stop fast-food outlets opening near schools, restrict advertising of products high in fat, salt or sugar, and limit sponsorship of sports events by fast-food producers such as McDonald's.They argue that government action is necessary to curb Britain's addiction to unhealthy food and help halt spiraling rates of obesity, diabetes and heart disease. Professor Terence Stephenson, president of the Royal College of Paediatrics and Child Health, said that the consumption of unhealthy food should be seen to be just as damaging as smoking or excessive drinking."Thirty years ago, it would have been inconceivable to have imagined a ban on smoking in the workplace or in pubs, and yet that is what we have now. Are we willing to be just as courageous in respect of obesity? I would suggest that we should be," said the leader of the UK's children's doctors.Lansley has alarmed health campaigners by suggesting he wants industry rather than government to take the lead. He said that manufacturers of crisps and candies could play a central role in the Change4Life campaign, the centrepiece of government efforts to boost healthy eating and fitness. He has also criticized the celebrity chef Jamie Oliver’s high-profile attempt to improve school lunches in England as an example of how "lecturing" people was not the best way to change their behaviour.Stephenson suggested potential restrictions could include banning TV advertisements for foods high in fat, salt or sugar before 9 pm and limiting them on billboards or in cinemas. "If we were really bold, we might even begin to think of high-calories fast food in the same way as cigarettes—by setting strict limits on advertising, product placement and sponsorship of sports events," he said.Such a move could affect firms such as McDonald’s, which sponsors the youth coaching scheme run by the Football Association. Fast-food chains should also stop offering “inducements” such as toys, cute animals and mobile phone credit to lure young customers, Stephenson said.Professor Dinesh Bhugra, president of the Royal College of Psychiatrists, said: “If children are taught about the impact that food has on their growth, and that some things can harm, at least information is available up front.”He also urged councils to impose “fast-food-free zones” around schools and hospitals—areas within which takeaways cannot open.A Department of Health spokesperson said: “We need to create a new vision for public health where all of society works together to get healthy and live longer. This includes creating a new ‘responsibility deal’ with business, built on social responsibility, not state regulation. Later this year, we will publish a white paper setting out exactly how we will achieve this.”The food industry will be alarmed that such senior doctors back such radical moves, especially the call to use some of the tough tactics that have been deployed against smoking over the last decade.Section ⅢTranslation46. Directions:Translate the following text from English into Chinese. write your translation on the ANSWER SHEET. (15 points)Who would have thought that, globally, the IT industry produces about the same volume of greenhouse gases as the world’s airlines do—roughly 2 percent of all CO2 emissions?Many everyday tasks take a surprising toll on the environment. A Google search can leak between 0.2 and 7.0 grams of CO2 depending on how many attempts are needed to get the “right” answer. To deliver results to its users quickly, then, Google has to maintain vast data centres around the world, packed with powerful computers. While producing large quantities of CO2, these computers emit a great deal of heat, so the centres need to be well air-conditioned, which uses even more energy.However, Google and other big tech providers monitor their efficiency closely and make improvements. Monitoring is the first step on the road to reduction, but there is much to be done, and not just by big companies.Section IV WritingPart A47. Directions:Suppose your cousin Li Ming has just been admitted to a university. Write him/her a letter to1) congratulate him/her, and2) give him/her suggestions on how to get prepared for university life.You should write about 100 words on ANSWER SHEET 2.Do not sign your own name at the end of the letter. Use “Zhang Wei” instead.Do not write the address. (10 points)Part B48. Directions:Write a short essay based on the following chart.in your writing, you should1)interpret the chart and2)give your commentsYou should write at least 150 wordsWrite your essay on the ANSWER SHEET. (15points)试题答案Section ⅠUse of English1. A2. C3. B4. D5. D6. B7. A8. C9. C 10. B 11. D 12. B 13. A 14. C 15. A 16. A 17.D 18. A 19.C 20. DSection ⅡReading ComprehensionPart A21. B 22. D 23. C 24. A 25. B 26. D 27. B 28. C 29. A 30. A 31. C 32. D 33. C 34. D35. B 36. B 37. C 38. B 39. A 40. DPart B41. E 42. D 43. C 44. B 45. GSection ⅢTranslation略Section IV WritingPart A略Part B略。
2011年暨南大学830数据结构考研试题
暨南大学2011 年全国硕士研究生统一入学考试自命题试题*******************************************************************************学科与专业名称：计算机技术, 软件工程考试科目代码与名称：数据结构考生注意：所有答案必须写在答题纸（卷）上，写在本试题上一律不给分。
一. 选择题(每题2 分，共30 分)1. 算法分析的目的是（）。
A. 找出数据结构的合理性B. 研究算法中的输入和输出关系C. 分析算法的效率以求改进D. 分析算法的易读性和文档性2. 下列函数中渐近时间复杂度最小的是（）。
A. T1(n)=log2n+5000nB. T2(n)=n2-8000nC. T3(n)=n3+5000n D. T4(n)=2nlog2n-1000n3. 线性表的动态链表存储结构与顺序存储结构相比，优点是（）。
A. 所有的操作算法实现简单B. 便于随机存取C. 便于插入与删除D. 便于节省存储器空间4．若进栈序列为1,2,3,4,5,6, 且进栈和出栈可以穿插进行，则可能出现的出栈序列为( )。
A．3,2,6,1,4,5 B．5,6,4,2,3,1C．5,1,2,3,4,6 D．3,4,2,1,6,55. 顺序存储的线性表的第一个元素的存储地址是100，每个元素的长度为4，则第4 个元素的存储地址是（）。
A. 108B. 112C. 116D. 1206. 在任意一棵二叉树的先序序列和后序序列中，各叶子之间的相对次序关系( )。
A．不一定相同B．互为逆序C．都不相同D．都相同7. 高度为5 的二叉树至多有结点数为（）。
A. 63B. 3 2C. 31D.648. 图的邻接矩阵表示法适用于表示（）。
A．无向图B．有向图C．稠密图D．稀疏图9. 在一个单链表中,若p 所指的结点不是最后一个结点,在p 之后插入s 所指的结点, 则执行( )。
暨南大学810高等代数2010--2020年考研专业课真题
考试科目名称及代码：810高等代数（A卷）
考生注意：所有答案必须写在答题纸（卷）上，写在本试题上一律不给分。
一、（10分）设 为给定正整数， 为给定常数，计算对角线上元素均为 、其它位置元素均为1的 阶矩阵 的行列式 .
2证明 在某基下的矩阵是
六（15分）1设 ，证明秩 =秩 =秩 。
2设 是实对称矩阵， ，证明 。
七（15分）已知矩阵 是数域 上的一个 级方阵，如果存在 上的一个 级可逆方阵 ，使得 为对角矩阵，那么称 在 上可对角化。分别判断 能否在实数域上和复数域上可对角化，并给出理由。
八（16分）用 表示实数域 上次数小于4的一元多项式组成的集合，它是一个欧几里得空间，内积为 。设 是由零次多项式及零多项式组成的子空间，求 以及它上的一个基。
研究方向：各专业研究方向
考试科目名称：810高等代数
考生注意：所有答案必须写在答题纸（卷）上，写在本试题上一律不给分
一、判断下列命题的正误（只需回答“正确”或“错误”并将你的答案写在答题纸上，不需说明理由，每题2分，共20分）：
1唯一解，并求其解；
2无穷多解，给出解的表达式；
3无解。
四（15分）设
1求 的全部特征值；
2对 的每个特征值 ，求 的属于特征值 的特征子空间的维数和一组基；
3求正交矩阵 ，使 是对角矩阵，并给出此对角矩阵。
五（15分）设 是数域 上的一个n维线性空间 ，若有线性变换 与向量 使得 ，但 。
1证明 线性无关；
2020年招收攻读硕士学位研究生入学考试试题
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2011考研数学二真题及答案解析
2011年全国硕士研究生入学统一考试数学二试题一、选择题(1～8小题，每小题4分，共32分．下列每题给出的四个选项中，只有一个选项符合题目要求的，请将所选项前的字母填在答题纸．．．指定位置上．)(1)已知当0x →时，()3sin sin 3f x x x =-与kcx 是等价无穷小，则()(A)1,4k c ==．(B)1,4k c ==-．(C)3,4k c ==．(D)3,4k c ==-．(2)已知()f x 在0x =处可导，且()00f =，则()()2332limx x f x f x x →-=()(A)()20f '-．(B)()0f '-．(C)()0f '．(D)0．(3)函数()ln (1)(2)(3)f x x x x =---的驻点个数为()(A)0．(B)1．(C)2．(D)3．(4)微分方程2(0)xx y y e e λλλλ-''-=+>的特解形式为()(A)()xx a ee λλ-+．(B)()xx ax ee λλ-+．(C)()xx x aebe λλ-+．(D)2()xx x aebe λλ-+．(5)设函数(),()f x g x 均有二阶连续导数，满足(0)0,(0)0,f g ><且(0)(0)0f g ''==，则函数()()z f x g y =在点(0,0)处取得极小值的一个充分条件是()(A)(0)0,(0)0.f g ''''<>(B)(0)0,(0)0.f g ''''<<(C)(0)0,(0)0.f g ''''>>(D)(0)0,(0)0.f g ''''><(6)设40ln sin I x dx π=⎰，40ln cot J x dx π=⎰，40ln cos K x dx π=⎰，则,,I J K 的大小关系是()(A)I J K <<．(B)I K J <<．(C)J I K <<．(D)K J I <<．(7)设A 为3阶矩阵，将A 的第2列加到第1列得矩阵B ，再交换B 的第2行与第3行得单位矩阵，记1100110001P ⎛⎫ ⎪= ⎪ ⎪⎝⎭，2100001010P ⎛⎫ ⎪= ⎪ ⎪⎝⎭，则A =()(A)12PP ．(B)112P P -．(C)21P P ．(D)121P P -．(8)设1234(,,,)A αααα=是4阶矩阵，*A 为A 的伴随矩阵，若(1,0,1,0)T是方程组0Ax =的一个基础解系，则*0A x =的基础解系可为()(A)13,αα．(B)12,αα．(C)123,,ααα．(D)234,,ααα．二、填空题(9～14小题，每小题4分，共24分．请将答案写在答题纸．．．指定位置上．)(9)1012lim()2x x x →+= ．(10)微分方程'cos xy y e x -+=满足条件(0)0y =的解为 ．(11)曲线0tan (04xy tdt x π=≤≤⎰的弧长s = ．(12)设函数,0,()0,0,0,x e x f x x λλλ-⎧>=>⎨≤⎩则()xf x dx +∞-∞=⎰ ．(13)设平面区域D 由直线,y x =圆222x y y +=及y 轴围成，则二重积分Dxyd σ=⎰⎰ ．(14)二次型222123123121323(,,)3222f x x x x x x x x x x x x =+++++，则f 的正惯性指数为．三、解答题(15～23小题，共94分．请将解答写在答题纸．．．指定位置上．解答应写出文字说明、证明过程或演算步骤．)(15)(本题满分10分)已知函数20ln(1)()xat dt F x x+=⎰，设0lim ()lim ()0,x x F x F x +→+∞→==试求a 的取值范围．(16)(本题满分11分)设函数()y y x =由参数方程3311,3311,33x t t y t t ⎧=++⎪⎪⎨⎪=-+⎪⎩确定，求()y y x =的极值和曲线()y y x =的凹凸区间及拐点．(17)(本题满分9分)设函数(,())z f xy yg x =，其中函数f 具有二阶连续偏导数，函数()g x 可导且在1x =处取得极值(1)1g =，求211x y zx y==∂∂∂．(18)(本题满分10分)x设函数()y x 具有二阶导数，且曲线:()l y y x =与直线y x =相切于原点，记α为曲线l 在点(,)x y 处切线的倾角，若,d dydx dxα=求()y x 的表达式．(19)(本题满分10分)(I)证明：对任意的正整数n ，都有111ln(11n n n<+<+成立．(II)设111ln (1,2,)2n a n n n=+++-= ，证明数列{}n a 收敛．(20)(本题满分11分)一容器的内侧是由图中曲线绕y 轴旋转一周而成的曲面，该曲线由2212()2x y y y +=≥与2211()2x y y +=≤连接而成的．(I)求容器的容积；(II)若将容器内盛满的水从容器顶部全部抽出，至少需要做多少功？(长度单位：m ，重力加速度为2/gm s ，水的密度为3310/kg m )．图１(21)(本题满分11分)已知函数(,)f x y 具有二阶连续偏导数，且(1,)0f y =，(,1)0f x =，(,)Df x y dxdy a =⎰⎰，其中{}(,)|01,01D x y x y =≤≤≤≤，计算二重积分(,)xyDI xy f x y dxdy ''=⎰⎰．(22)(本题满分11分)设向量组123(1,0,1),(0,1,1),(1,3,5)T T T ααα===，不能由向量组1(1,1,1)T β=，2(1,2,3)T β=，3(3,4,)T a β=线性表示．(I)求a 的值；(II)将123,,βββ由123,,ααα线性表示．(23)(本题满分11分)A 为三阶实对称矩阵，A 的秩为2，即()2r A =，且111100001111A -⎛⎫⎛⎫ ⎪ ⎪= ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭．(I)求A的特征值与特征向量；(II)求矩阵A．2011年全国硕士研究生入学统一考试数学二试题答案一、选择题(1～8小题，每小题4分，共32分．下列每题给出的四个选项中，只有一个选项符合题目要求的，请将所选项前的字母填在答题纸．．．指定位置上．)(1)【答案】(C)．【解析】因为03sin sin 3limk x x x cx →-03sin sin cos 2cos sin 2limk x x x x x xcx →--=()20sin 3cos 22cos limkx x x x cx →--=2103cos 22cos lim k x x xcx -→--=()22132cos 12cos limk x x xcx -→---=22110044cos 4sin lim lim k k x x x x cxcx --→→-==304lim1k x cx -→==．所以4,3c k ==，故答案选(C)．(2)【答案】(B)．【解析】()()2332limx x f x f x x →-()()()()22330220limx x f x x f f x f x →--+=()()()()33000lim 2x f x f f x f x x →⎡⎤--⎢⎥=-⎢⎥⎣⎦()()()0200f f f '''=-=-．故答案选(B)．(3)【答案】(C)．【解析】()ln 1ln 2ln 3f x x x x =-+-+-111'()123f x x x x =++---231211(1)(2)(3)x x x x x -+=---令'()0f x =，得1,2633x ±=，故()f x 有两个不同的驻点．(4)【答案】(C)．【解析】微分方程对应的齐次方程的特征方程为220r λ-=，解得特征根12r r λλ==-,．所以非齐次方程2xy y e λλ''-=有特解1x y x a e λ=⋅⋅,非齐次方程2xy y eλλ-''-=有特解2x y x b e λ-=⋅⋅，故由微分方程解的结构可知非齐次方程2xx y y ee λλλ-''-=+可设特解().x x y x ae be λλ-=+(5)【答案】(A)．【解析】由题意有()()zf xg y x ∂'=∂，()()z f x g y y∂'=∂所以，()0,0(0)(0)0zf g x ∂'==∂，()0,0(0)(0)0z f g y ∂'==∂，即()0,0点是可能的极值点．又因为22()()zf xg y x ∂''=∂，2()()z f x g y x y ∂''=∂∂，22()()z g y f x y∂''=∂，所以，2(0,0)2|(0)(0)zA f g x ∂''==⋅∂，2(0,0)|(0)(0)0zB f g x yα''==⋅=∂∂，2(0,0)2|(0)(0)zC f g y∂''==⋅∂，根据题意由()0,0为极小值点，可得20,AC B A C -=⋅>且(0)(0)0A f g ''=⋅>，所以有(0)(0)0.C f g ''=⋅>由题意(0)0,(0)0f g ><，所以(0)0,(0)0f g ''''<>，故选(A)．(6)【答案】(B)．【解析】因为04x π<<时，0sin cos 1cot x x x <<<<，又因ln x 是单调递增的函数，所以ln sin ln cos ln cot x x x <<．故正确答案为(B)．(7)【答案】(D)．【解析】由于将A 的第2列加到第1列得矩阵B ，故100110001A B ⎛⎫ ⎪= ⎪ ⎪⎝⎭，即1AP B =，11A BP -=．由于交换B 的第2行和第3行得单位矩阵，故100001010B E ⎛⎫ ⎪= ⎪ ⎪⎝⎭，即2,P B E =故122B P P -==．因此，121A P P -=，故选(D)．(8)【答案】(D)．【解析】由于(1,0,1,0)T是方程组0Ax =的一个基础解系，所以(1,0,1,0)0TA =，且()413r A =-=，即130αα+=，且0A =．由此可得*||A A A E O ==，即*1234(,,,)A O =αααα，这说明1234,,,αααα是*0A x =的解．由于()3r A =，130αα+=，所以234,,ααα线性无关．又由于()3r A =，所以*()1r A =，因此*0A x =的基础解系中含有413-=个线性无关的解向量．而234,,ααα线性无关，且为*0A x =的解，所以234,,ααα可作为*0A x =的基础解系，故选(D)．二、填空题(9～14小题，每小题4分，共24分．请将答案写在答题纸．．．指定位置上．)(9)【答案】．【解析】原式=0121lim(1)2x x x e →+-00212ln 21limlimln 2222x x x x x eee→→-⋅====．(10)【答案】sin xy e x -=．【解析】由通解公式得(cos )dx dxx y e e x e dx C --⎰⎰=⋅+⎰(cos )x e xdx C -=+⎰(sin )x e x C -=+．由于(0)0,y =故C =0．所以sin xy ex -=．(11)【解析】选取x 为参数，则弧微元sec ds xdx ===所以440sec ln sec tan ln(1s xdx x x ππ==+=+⎰．(12)【答案】1λ．【解析】原式0x xx e dx xde λλλ+∞+∞--==-⎰⎰1lim0x x xx x x xee dx ee λλλλλ+∞-+∞--+∞→+∞=-+=-+-⎰01111limlim x x x x e e e λλλλλ→+∞→+∞⎛⎫=---= ⎪⎝⎭．(13)【答案】712．【解析】原式2sin 2sin 322044cos sin cos sin d r r rdr d r drππθθππθθθθθθ=⋅=⋅⎰⎰⎰⎰4241sin cos 16sin 4d ππθθθθ=⋅⋅⋅⎰5522444cos sin 4sin sin d d ππππθθθθθ=⋅=⎰⎰66447sin 612ππθ==．(14)【答案】2．【解析】方法1：f 的正惯性指数为所对应矩阵的特征值中正的个数．二次型f 对应矩阵为111131111A ⎛⎫ ⎪= ⎪ ⎪⎝⎭．111000131131132111111112E A λλλλλλλλλλλ-----=---=---=------------()()321412λλλλλλ--==----，故1230,1,4λλλ===．因此f 的正惯性指数为2．方法2：f 的正惯性指数为标准形中正的平方项个数．()222123123121323,,3222f x x x x x x x x x x x x =+++++()2222212322332323232x x x x x x x x x x x =++---+++()2212322x x x x =+++，令11232233,,,y x x x y x y x =++⎧⎪=⎨⎪=⎩则22122f y y =+，故f 的正惯性指数为2．三、解答题(15～23小题，共94分．请将解答写在答题纸．．．指定位置上．解答应写出文字说明、证明过程或演算步骤．)(15)(本题满分10分)【解析】如果0a ≤时，220(1)limlim ln(1)xxa ax x ln t dt x t dt x -→+∞→+∞+=⋅+=+∞⎰⎰，显然与已知矛盾，故0a >．当0a >时，又因为22230110000ln(1)ln(1)1limlim lim lim 0xaaa a x x x x t dt x x x x ax ax a++++---→→→→++===⋅=⎰．所以30a ->即3a <．又因为223201222ln(1)ln(1)210lim lim lim lim (1)(1)1xa a a a x x x x xt dt x x x x ax a a x a a x ---→+∞→+∞→+∞→+∞+++====--+⎰所以32a -<，即1a >，综合得13a <<．(16)(本题满分11分)【解析】因为221()1dyt dt y x dx t dt -'==+，2222222231(12(1)(1)2141(),(1)1(1)t d t t t t t t y x dx dt t t t dt-+--⋅+''=⋅=⋅=+++令()0y x '=得1t =±，当1t =时，53x =，13y =-，此时0y ''>，所以13y =-为极小值．当1t =-时，1x =-，1y =，此时0y ''<，所以1y =为极大值．令()0y x ''=得0t =，13x y ==．当0t <时，13x <，此时0y ''<；当0t >时，13x >，此时0y ''>．所以曲线的凸区间为13⎛⎫-∞ ⎪⎝⎭，，凹区间为13⎛⎫+∞ ⎪⎝⎭，，拐点为11(,)33．(17)(本题满分9分)【解析】[],()z f xy yg x =[][]12,(),()()zf xy yg x y f xy yg x yg x x∂'''=⋅+⋅∂[][]211112,()(,())(,())()zf xy yg x y f xy yg x x f xy yg x g x x y∂'''''=++∂∂[]{}21222(),()()[,()][,()]()g x f xy yg x yg x f xy yg x x f xy yg x g x '''''''+⋅+⋅+．因为()g x 在1x =可导，且为极值,所以(1)0g '=，则21111121|(1,1)(1,1)(1,1)x y d zf f f dxdy =='''''=++．(18)(本题满分10分)【解析】由题意可知当0x =时，0y =，'(0)1y =，由导数的几何意义得tan y α'=,即arctan y α'=，由题意()arctan d dyy dx dx '=，即21y y y '''='+．令y p '=，y p '''=，则21p p p '=+，3dpdx p p =+⎰⎰，即21dp p dp dx p p -=+⎰⎰⎰，211ln ||ln(1)2p p x c -+=+，即2211x p ce -=-．当0x =，1p =，代入得2c =，所以'y =则0()(0)t xxy x y -==⎰⎰004t t xxx d π===⎰．又因为(0)0y =，所以()arcsin 24x y x e π=-．(19)(本题满分10分)【解析】(Ⅰ)设()()1ln 1,0,f x x x n ⎡⎤=+∈⎢⎥⎣⎦显然()f x 在10,n⎡⎤⎢⎥⎣⎦上满足拉格朗日的条件，()1111110ln 1ln1ln 1,0,1f f n n n n n ξξ⎛⎫⎛⎫⎛⎫⎛⎫-=+-=+=⋅∈ ⎪ ⎪ ⎪ ⎪+⎝⎭⎝⎭⎝⎭⎝⎭所以10,n ξ⎛⎫∈ ⎪⎝⎭时，11111111101n n n n ξ⋅<⋅<⋅+++，即：111111n n n ξ<⋅<++，亦即：111ln 11n n n⎛⎫<+< ⎪+⎝⎭．结论得证．（II）设111111ln ln 23nn k a n n n k==++++-=-∑ ．先证数列{}n a 单调递减．()111111111ln 1ln ln ln 1111n n n n k k n a a n n k k n n n n ++==⎡⎤⎡⎤⎛⎫⎛⎫-=-+--==-+⎪ ⎢⎥⎢⎥+++⎝⎭⎝⎭⎣⎦⎣⎦∑∑，利用（I）的结论可以得到11ln(1)1n n <++，所以11ln 101n n ⎛⎫-+< ⎪+⎝⎭得到1n n a a +<，即数列{}n a 单调递减．再证数列{}n a 有下界．1111ln ln 1ln nnn k k a n n k k ==⎛⎫=->+- ⎪⎝⎭∑∑，()11112341ln 1ln ln ln 1123nnk k k n n k k n ==++⎛⎫⎛⎫⎛⎫+==⋅⋅=+ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭∑∏ ，()1111ln ln 1ln ln 1ln 0nn n k k a n n n n k k ==⎛⎫=->+->+-> ⎪⎝⎭∑∑．得到数列{}n a 有下界．利用单调递减数列且有下界得到{}n a 收敛．(20)(本题满分11分)【解析】(I)容器的容积即旋转体体积分为两部分12V V V =+()()1222211221y y dy y dyππ-=-+-⎰⎰232123y y π⎛⎫=- ⎪⎝⎭+13213y y π-⎛⎫- ⎪⎝⎭=π1534⎛⎫+-⎪⎝⎭=94π．(II)所做的功为22(2)(1)(2)(2)dw g y y dy g y y y dyπρπρ=--+--12222112(2)(1)(2)(2)w g y y dy g y y y dyπρπρ-=--+--⎰⎰1232322112(22)44)g y y y dy y y y dy πρ-⎛⎫=--+++-+ ⎪⎝⎭⎰⎰111224322312222221111211122242243243yy y yy g y yπρ----⎛⎫⎪=--++-+ ⎪ ⎪⎝⎭3271033758g g ππ⨯==．(21)(本题满分11分)【解析】因为(,1)0f x =，(1,)0f y =，所以(,1)0x f x '=．110(,)xyI xdx yf x y dy ''=⎰⎰11(,)x xdx ydf x y '=⎰⎰()()111000,|,x x xdx yf x y f x y dy ⎡⎤''=-⎢⎥⎣⎦⎰⎰()1100(,1)(,)x x xdx f x f x y dy''=-⎰⎰1100(,)x xdx f x y dy '=-⎰⎰1100(,)x dy xf x y dx '=-⎰⎰111000(,)|(,)dy xf x y f x y dx ⎡⎤=--⎢⎥⎣⎦⎰⎰1100(1,)(,)dy f y f x y dx ⎡⎤=--⎢⎥⎣⎦⎰⎰(,)Df x y dxdy =⎰⎰a =．(22)(本题满分11分)【解析】(I)由于123,,ααα不能由123,,βββ线性表示，对123123(,,,,,)βββααα进行初等行变换：123123113101(,,,,,)12401313115a ⎛⎫⎪= ⎪⎪⎝⎭βββααα113101011112023014a ⎛⎫ ⎪→- ⎪ ⎪-⎝⎭113101011112005210a ⎛⎫ ⎪→- ⎪ ⎪--⎝⎭．当5a =时，1231231(,,)2(,,,)3r r ββββββα=≠=，此时，1α不能由123,,βββ线性表示，故123,,ααα不能由123,,βββ线性表示．(II)对123123(,,,,,)αααβββ进行初等行变换：123123101113(,,,,,)013124115135⎛⎫ ⎪= ⎪⎪⎝⎭αααβββ101113013124014022⎛⎫ ⎪→ ⎪ ⎪⎝⎭101113013124001102⎛⎫ ⎪→ ⎪ ⎪--⎝⎭1002150104210001102⎛⎫ ⎪→ ⎪ ⎪--⎝⎭，故112324βααα=+-，2122βαα=+，31235102βααα=+-．(23)(本题满分11分)【解析】(I)由于111100001111A -⎛⎫⎛⎫⎪ ⎪= ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭，设()()121,0,1,1,0,1T T αα=-=，则()()1212,,A αααα=-，即1122,A A αααα=-=，而120,0αα≠≠，知A 的特征值为121,1λλ=-=，对应的特征向量分别为()1110k k α≠，()2220k k α≠．由于()2r A =，故0A =，所以30λ=．由于A 是三阶实对称矩阵，故不同特征值对应的特征向量相互正交，设30λ=对应的特征向量为()3123,,Tx x x α=，则13230,0,T T⎧=⎨=⎩αααα即13130,0x x x x -=⎧⎨+=⎩．解此方程组，得()30,1,0Tα=，故30λ=对应的特征向量为()3330k k α≠．(II)由于不同特征值对应的特征向量已经正交，只需单位化：))()3121231231,0,1,1,0,1,0,1,0T T Tαααβββααα==-====．令()123,,Q βββ=，则110TQ AQ -⎛⎫ ⎪=Λ= ⎪ ⎪⎝⎭，T A Q Q =Λ22022012200110220010022⎛⎫-⎛⎫ ⎪ ⎪ ⎪-⎛⎫ ⎪ ⎪ ⎪= ⎪ ⎪ ⎪⎪ ⎪ ⎪⎝⎭ ⎪ ⎪- ⎪ ⎪⎝⎭⎪⎝⎭2202200012200000002210022010022⎛-⎛⎫ - ⎪ ⎪⎛⎫ ⎪ ⎪ ⎪==⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎪ ⎪⎪ ⎪⎝⎭ ⎪⎝⎭．。
2011年研究生入学统一考试数学二试题及解析
2011年全国硕士研究生入学统一考试数学二试题及解析一、选择题：1～8小题，每小题4分，共32分，下列每小题给出的四个选项中，只有一项符合题目要求，把所选项前的字母填在答题纸指定位置上.1、已知当0x →时，函数()3sin sin 3f x x x =-与k cx 等价无穷小，则（A ）1,4k c == （B ）1,4k c ==- （C ） 3,4k c == （D ）3,4k c ==- 【分析】本题考查等价无穷小的有关知识.可以利用罗必达法则或泰勒公式完成。
【详解】法一:由题设知 13sin sin 33cos 3cos 31=lim=limkk x x x xx xcxkcx-→→--233sin 9sin 33cos 27cos 3=lim=lim(1)(1)(2)k k x x x x x x k k cxk k k cx--→→-+-+---324=lim(1)(2)k x k k k cx-→--从而(1)(2)243k k k c k --=⎧⎨=⎩，故3,4k c ==。
从而应选（C ）。
法二:333333(3)()3(())(3())4()3!3!xx f x x o x x o x x o x =-+--+=+所以3,4k c ==。
，从而应选（C ）。
2、已知()f x 在0x =处可导，且(0)0f =，则233()2()limx x f x f x x→-=（A ）2'(0)f - （B ）'(0)f - （C ） '(0)f （D ）0【分析】本题考查导数的定义。
通过适当变形，凑出()f x 在0x =点导数定义形式求解。
【详解】23223333()2()()(0)()(0)limlim[2]x x x f x f x x f x x f f x f xxx→→---=-()22333()(0)()(0)lim2lim'0x x x f x x f f x f f xx→→--=-=-故应选（B ）。
2011暨南大学数电试卷
2011年全国硕士研究生统一入学考试自命题试题 A
******************************************************************************************** 学科与专业名称：光学工程光学工程（专业学位）
四、（8分）如图2所示电路中，（a）、（c）为TTL门，（b）、（d）为CMOS门。
写出各电路输出端的函数式并确定其输出状态。
（a）（b）（c）（d）考试科目：数字电子技术共 5 页，第 2
六、（14分）试分析图 4 逻辑电路的功能（要求列逻辑函数式、真值表，说明功
能）。
图4 图5
F
七、（16分）用两片2线-4线译码器（见图5 ）和少量门电路扩展为3线
器，再用扩展后的译码器和与非门设计一个一位全加器。
考试科目：数字电子技术共 5 页，第
九、（16分）试用边沿JK触发器及门电路，设计一个同步时序逻辑电路，实现如
图7所示功能，其中M为控制（输入）端，M取不同的电平时，状态转换不同。
要求：设计最简电路，列出状态转换表、卡诺图、状态方程、驱动方程（不必画电路）。
图7 图8
十、（8分）一个可编程逻辑阵列电路如图8 所示。
请写出输出端L0、L1逻
辑函数表达式。
共 5 页，第 4 页
图10
考试科目：数字电子技术共 5 页，第 5 页。
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暨南大学数学学科2011年硕士研究生入学考试自命题科目
《数学分析》
考试大纲
本《数学分析》考试大纲适用于暨南大学数学学科各专业（基础数学、概率论与数理统计、应用数学）硕士研究生入学考试。
数学分析是大学数学系本科学生的最基本课程之一，也是大多数理工科专业学生的必修基础课。
它的主要内容包括极限与连续、一元函数的微分学、一元函数的积分学、无穷级数、多元函数的微分学与积分学、含参变量积分。
要求考生熟悉基本概念、掌握基本定理、有较强的运算能力和综合分析解决问题能力。
一、考试的基本要求
要求考生比较系统地理解数学分析的基本概念，掌握数学分析的基本理论、基本思想和方法，具有一定的综合运用所学的知识分析问题和解决问题的能力，以便为以后继续学习和从事科研奠定坚实的分析基础。
二、考试内容
1．极限与连续
（1）极限的ε－δ、ε－N 定义及其证明；极限的性质及运算、无穷小量的概念及基本性质；
（2）函数的连续性及一致连续性概念，函数的不连续点类型，连续函数的性质的证明及其应用；
（3）上、下极限概念，实数集完备性的基本定理及其应用；
（4）二元函数的极限的定义及性质，重极限与累次极限概念，二元函数的连续性概念及性质；
（5）数列极限的计算，一元与二元函数极限的计算。
2．一元函数的微分学
（1）函数的导数与微分概念及其几何意义，函数的可导、可微与连续之间的关系；
（2）求函数（包括复合函数及分段函数）的各阶导数与微分；
（3）Rolle中值定理、Lagrange中值定理、Cauchy中值定理、Taylor定理及其应用；
（4）用导数研究函数的单调性、极值、最值和凸凹性；
（5）用洛必达法则求不定式极限。
3．一元函数的积分学
（1）不定积分的概念及不定积分的基本公式，换元积分法与分部积分法，求初等函数、有理函数和可化为有理函数的不定积分；
（2）定积分的概念，可积条件与可积函数类；
（3）定积分的性质，微积分学基本定理，定积分的换元积分法和分部积分法，积分第一、二中值定理及其应用；
（4）用定积分计算平面图形的面积、平面曲线的弧长、旋转体的体积、平行截面面积已知的立体体积、变力做功和物体的质量；
（5）反常积分的概念及性质，两类反常积分的比较判别法、阿贝耳判别法和狄
立克雷判别法，两类反常积分的计算。
4．无穷级数
（1）数项级数敛散性的概念及基本性质；
（2）正项级数收敛的充分必要条件、比较原则、比式判别法、根式判别法与积分判别法；
（3）一般数项级数绝对收敛与条件收敛的概念及其相互关系，绝对收敛级数的性质，交错级数的莱布尼兹判别法，一般数项级数的阿贝耳判别法和狄立
克雷判别法；
（4）函数项级数一致收敛性的概念以及判断一致收敛性的Weierstrass判别法、Cauchy判别法、Abel判别法和Dirichlet判别法；
（5）幂级数的收敛半径、收敛域的求法，幂级数的性质与运算；函数的幂级数展开及幂级数的和函数的性质与求法；
（6）周期函数的Fourier级数展开及Fourier级数收敛定理。
5．多元函数的微分学与积分学
（1）多元函数的偏导数和全微分的概念、几何意义与应用，连续、可微与可偏导之间的关系，多元函数的偏导数（包括高阶偏导）与全微分的计算，方向导数与梯度的定义与计算；
（2）多元函数的无条件极值、中值定理与泰勒公式；
（3）隐函数存在定理及求隐函数的偏导数；
（4）曲线的切线与法平面、曲面的切平面与法线的求法；
（5）重积分、曲线积分和曲面积分的概念与计算；
（6）格林公式、高斯公式和斯托克斯公式及其应用。
6．含参变量积分
（1）含参变量正常积分的概念及性质；
（2）含参变量反常积分一致收敛的概念及其判别法，一致收敛的含参变量反常积分的性质及其应用。
三、考试题型
填空题、单项选择题、计算题、证明题。
四、考试方法和考试时间
采用闭卷笔试形式，试卷满分为150分，考试时间为180分钟。
五、主要参考教材
（1）《数学分析》（上、下册），华东师范大学数学系编，2001年6月第3 版，2007年5月第17次印刷，高等教育出版社.
（2）《数学分析》（上、下册），复旦大学数学系陈传璋等编，1983年11月第2版，2003年5月第23次印刷，高等教育出版社.。