2010江苏省专转本数学真题以及答案

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2006-2010年江苏省专转本高数真题集参考答案

2006年—2010年江苏省专转本真题参考答案1、计算11lim31--→x x x解：原式32)1)(1()1)(1(lim)1)(1)(1()1)(1)(1(lim332033233230=++-+-=+++-+++-=→→x x x x x x x x x x x x x x x 2、已知)21()21(lim ,2)2(lim==∞→→xxf x x f x x 则解：设ux 41=，则当x →0时，u →∞，代入已知极限得： 21)21(lim ,2)42(lim 4)42(4lim ===∞→∞→∞→u uf u uf u uf u u u 解得即21)21(lim =∞→xxf x3、求极限xx xx 3)2(lim -∞→ 解：原式6)6(2)21(lim --⋅-∞→=-=e x xx4、求极限xx x x sin lim 30-→解：6sin 6lim cos 13lim sin lim02030==-=-→→→x xx x x x x x x x 5、已知32lim22=-++→x bax x x ,则常数a,b 的值为( ) A 、a=-1,b=-2 B 、a=-2,b=0 C 、a=-1,b=0 D 、a=-2,b=-1解：2lim ,24,024)(lim 2222-++--==++=++→→x bax x a b b a b ax x x x34)2(lim 2)2()4(lim 224lim 22222=+=++=--+-=---+=→→→a a x x a ax x x a ax x x x x A=-1,b=-2 6、设2)(lim =-∞→xx cx x ，常数c= 。

解：2ln ,2)1(lim )1(lim )(lim ===-+=-+=--⋅-∞→∞→∞→c e cx c c x c c x x c c c ccx x x x x x7、计算xx x x )11(lim -+∞→解：21221)121(lim )121(lim )11(lim e x x x x x x x x x x =-+=-+=-++⋅-∞→∞→∞→8、设当x →0时，函数f(x)=x-sinx 与g(x)=a n是等价无穷小，则常数a,n 的值为（） A.4,61.4,121.3,31.3,61========n a D n a C n a B n a 解：3,61,12,21,2lim cos 1lim sin lim 120100====-=-=--→-→→n a na n nax x nax x ax x x n x n x n x 9、设423)(22-+-=x x x x f ，则x=2是f(x)的（）A 、跳跃型间断点B 、可去间断点C 、无穷型间断点D 、振荡型间断点解：4121lim 423lim 2222=+-=-+-→→x x x x x x x 10、若,)(lim 0A x f x =→且f(x)在x=x 0处有定义，则当A= f(x 0) 时f(x)在x 0处连续。

2010年江苏省专转本真题解析1-50

ns
Year
25
83.解决方案——高速缓存(cache)
cache直接制作在CPU芯片内，速度
CPU芯片
几乎与CPU一样快
CPU
cache
一般Cache容量越大越
8 4
9
14 10
3
好，CPU访问Cache的命中率就越高。但出于成本考虑，一般容量不大，一级缓存几kB～(几十)kB，二级缓存128 kB～1 MB。
使用比特来表示
1
比特在计算机中如何表示？
在计算机中表示与存储二进位的方法：
电路的高电平状态或低电平状态(CPU) 电容的充电状态或放电状态(RAM) 两种不同的磁化状态(磁盘) 光盘面上的凹凸状态(光盘)
···
2
例：硬盘中比特的表示与存储
磁盘表面微小区域中，磁性材料粒子的两种不同的
磁化状态分别表示0和1
磁头，用于写入和读出信息磁性材料粒子磁盘片
“0” “1”
旋转方向
3
光盘的信息记录原理
光盘上的螺旋形光道
凹坑的边缘用来表示 “1”，而凹坑内外的平坦部分表示“0”
4
黑白图像的表示
5
灰度图像的表示
6
彩色图像的表示
155 150 147 168 225 225 240 233
例1
A: 0110 B: ∨ 1010 F: 1110
例2
A: 0110 B: ∧ 1010 F: 0010
01001101 ∨ 00101011 01101111
(157)8
13
2009年第4题
7*64+4*8+4 =(1+2+4)*26+22*23+22 =(20+21+22)* 26+25+22

2010年江苏省专转本真题及详解

2010年江苏专转本英语真题第Ⅰ卷Part ⅠReading Comprehension (共40分)Directions: There are 4 passages in this part. Each passage is followed by some questions or unfinished statements. For each of them there are 4 choices marked A, B, C and D. You should decide on the best choice and mark the corresponding letter on the Answer Sheet with a single line through the center.Passage OneQuestions 1 to 5 are based on the following passage.Sometimes you’ll hear people say that you can’t love others until you love yourself. Sometimes you’ll hear people say that you can’t expect someone else to love you until you love yourself. Either way, you’ve got to love yourself first and this can be tricky. Sure we all know that we’re the people of our parents’ eyes, and that our Grandmas think we are great talents and our uncle Roberts think that we will go to the Olympics, but sometimes it’s a lot harder to think such nice thoughts about ourselves. If you find that believing in yourself is a challenge, it is time you built a positive self-image and learn to love yourself.Self-image is your own mind’s picture of yourself. This image includes the way you look, the way you act, the way you talk and the way you think. Interestingly, our self-images are often quite different from the images others hold about us. Unfortunately, most of these images are more negative than they should be, thus changing the way you think about yourself is the key to changing your self-image and your whole world.The best way to defeat a passive self-image is to step back and decide to stress your successes. That is, make a list if you need to, but write down all of the great things you do everyday. Don’t allow doubts to occur in it.It very well might be that you are experiencing a negative self-image because you can’t move past one flaw or weakness that you see about yourself. Well, roll up your sleeves and make a change of it as your primary task. If you think you are silly because you aren’t good at math. Find a tutor. If you think you are weak because you can’t run a mile, get to the track and practice. If you think you are dull because you don’t wear the latest trends, buy a few new clothes.The best way to get rid of a negative self-image is to realize that your image is far from objective and to actively convince yourself of your positive qualities. Changing the way you think and working on those you need to improve will go a long way towards promoting a positive self-image. When you can pat your self on the back, you’ll know you’re well on your way.1. You need to build a positive self-image when you _______.A．dare to challenge yourself B. feel it hard to change yourselfC. are unconfident about yourselfD. have a high opinion of yourself2. According to the passage, our self-images________.A. have positive effectsB. are probably untrueC. are often changeableD. have different functions3. How should you change your self-image according to the passage?A. Keep a different image of others.B. Make your life successfulC. Understand your own worldD. Change the way you think4. What is the passage mainly about?A. How to prepare for your successB. How to face challenges in your lifeC. How to build a positive self-imageD. How to develop your good qualities5. Who are the intended readers of the passage?A. parentsB. AdolescentsC. educatorsD. people in generalPassage 2Questions 6 to 10 are based on the following passage.Do you want to live with a strong sense of peacefulness, happiness, goodness, and self-respect? The collection of happiness actions broadly categorized as “honor” helps you create this life of good feelings.Here’s an example to show how honorable actions create happiness.Say a store clerk fails to charge us for an item. If we keep silent and profit fro m the clerk’s mistake, we would drive home with a sense of sneaky excitement. Later we might tell our family or friends about our good fortune. On the other hand, if we tell the clerk about the uncharged item, the clerk would be grateful and thank us for our honesty. We would leave the store with a quite sense of honor that we might never share with another soul. Then, what is it to do with our sense of happiness?In the first case, where we don’t tell the clerk, a couple of things would happen. Deep down inside we would know ourselves as a type of thief. In the process, we would lose some peace of mind and self-respect. We would also demonstrate that we cannot be trusted, since we advertise our dishonor by telling our family and friends. We damage our own reputations by telling others. In contrast, bringing the error to the clerk’s attention causes different things to happen. Immediately, the clerk knows us to be honorable. Upon leaving the store, we feel honorable and our internal rewards of goodness and a sense of nobility.There is a beautiful positive cycle that is created by living a life of honorable actions. Honorable thoughts lead to honorable actions. Honorable actions lead us to a happier experience. And it is easy to think and act honorably again when we are happy. While the positive cycle can be difficult to start, once it’s started, it’s easy to continue. Keeping on doing good deeds brings us peace of mind which is important for our happiness.6. According to the passage, the positive action in the example contributes to our ___________.A. self-respectB. financial rewardsC. advertising abilityD. friendly relationship7. The author thinks that keeping silent about the uncharged item is equal to _____.A. lyingB. stealingC. cheatingD. advertising8. The phrase “bringing the error to the clerk’s attention” means_______.A. telling the truth to the clerkB. offering advice to the clerkC. asking the clerk to be more attentiveD. reminding the clerk of the charged item9. How will we feel if we let the clerk know the mistake?A. We’ll be very excitedB. We’ll feel unfortunateC. We’ll have a sense of humorD. We’ll feel sorry for the clerk10. Which of the following can be the best title of this passage?A. How to live truthfullyB. Importance of peacefulnessC. Ways of gaining self-respectD. Happiness through honorablePassage 3Questions 11 to 15 are based on the following passage.Sesame Street has been called “the longest street in the world”. That is because the television program by that name can now be seen in so many parts of the world. That program became one of America’s exports soon after it went on the air in New York in 1969.In the United States more than 6 million children watch the program regularly. The viewers include more than half of the nation’s pre-school children. Although some educators object to certain elements in the program, parents praise it highly. Many teachers also consider it a great help, though some teachers find that problems arise when first graders who have learned from “Sesame Street” are in the same class with children who have not watched the program.The program uses songs, stories, jokes and pictures to give children a basic understanding of numbers, letters and human relationships. Tests have shown that children have benefited from watching “Sesame Street”. Those who watch five times a week learn more than the occasional viewers. In the United States the program is shown atdifferent times during the week in order to increase the number of children who can watch it regularly.Why has “Sesame Street” been so much more successful than other children’s shows? Many reasons have been suggested. People mention the educational theories of its creators, the support by the government and private businesses, and the skillful use of a variety of TV tricks. Perhaps an equally important reason is that mothers watch “Sesame Street” along with their children. This is partly because famous adult stars often appear on “Sesame Street”. But the best reason for the success of the program may be that it makes every child watching it feel able to learn. The child finds himself learning, and he wants to learn more.11. Why has Sesame Street been called “the longest street in the world”?A. the program has been shown ever since 1969.B. the program became one of America’s major exports soon after it appeared on TVC. the program is now being watched in most parts of the world.D. the program is made in the longest street in New York.12. Some educators are critical of the program because________.A. they don’t think it fit for children in every respectB. it takes the children too much time to watch itC. it causes problems between children who watched it and those who have notD. some parents attach too much importance to it13. So many children in the United States watch the program because______.A. it uses songs, stories and jokes to give them basic knowledgeB. it is arranged for most children to watch it regularlyC. tests have shown that it is beneficial to themD. both A and B14. Mothers often watch the program along with their children because______A. they enjoy the program as much as their childrenB. they find their children have benefited from watching the programC. they are attracted by some famous adult stars on the showD. they can learn some educational theories from the program15. What is the most important reason for the success of the program according to the author?A. the creators have good educational theories in making the programB. the young viewers find they can learn something from it.C. famous adult stars often appear in the programD. It gets support from the government and private businessPassage 4Questions 16 to 20 are based on the following passage.You will have no difficulty in making contact with the agent. As you enter his office, you will be greeted immediately and politely asked what you are looking for. The Estate Agent’s negotiator-as he is called-will probably check that you really know your financial position. No harm in that, but you can always tell him that you have confirmed the position with the XYZ Building Society. He will accept that.He will show you the details of a whole range of properties; many of them are not really what you are looking for at all. That does not matter. Far better turn then down than risk missing the right one.The printed details he will give you are called “particulars”. Over the years, a whole language has grown up, solely for use in Agent’s pa rticulars. It is flowery, ornate and, providing you read it carefully and discount the adjectives, it can be very accurate and helpful.Since the passing of the Trades Description Act, any trader trying to sell something has had to be very careful as to what they say about it. Estate Agents have, by now, become very competent at going as far as they dare. For instance, it is quite acceptable to say “delightfully” situated. That is an expression of his opinion. You many notagree, but he might like the idea of living next to the gasworks. If, on the other hand, he says that the house has five bedrooms when, in fact, it has only two, that is a misstatement of fact and is an offence. This has made Estate Agents and others for that matter rather more careful.Basically, all that you need to know about a house is : how many bedrooms it has; an indication of their size; whether the house has a garage; whether there is a garden and whether it is at the back or the front of the house; whether its semi-detached of terraced.16. The Estate Agent’s negotiator will ________.A. want all details of your financial circumstancesB. want to satisfy himself that you understand the financial implications of buying a house.C. check your financial position with the XYZ Building SocietyD. accept any statement you make about your financial position17. The author believes ________.A. it is better to be given information about too many properties than too fewB. you should only look at details of properties of the kind you have decided to buy.C. the agent will only show you the details of properties you have in mind.D. it doesn’t matter if you miss a few properties you may be interested in.18. The adjectives in Agent’s particulars are ______.A. accurateB. helpfulC. both accurateD. safe to ignore19. The Trades Description Act applies to ___________.A. house agents onlyB. most estate agentsC. any traderD. buyer of houses20. The phrase “going as far as they dare” pro bably means______.A. covering as wide an area as possibleB. selling houses as far from the estate agent’s housesC. telling lies about properties if nobody is likely to find out about it.D. trying every possible means to make the description of houses sound attractivePart ⅡV ocabulary and StructureDirections: In this part there are forty incomplete sentences. Each sentence is followed by four choices. Choose the one that best completes the sentence and then mark your answer on the Answer Sheet.21. Scientists will have to _____ new methods of increasing the world’s food supply in order to solve the problemof famine in some places.A. come up forB. come down withC. com down toD. come up with22．I’d like to rent a house, modern, comfortable and ______in a quiet and safe neighborhood.A. all in allB. above allC. after allD. over all23. In order to expose corruption, they have decided to set up a special team to ____the company’s accounts.A. search forB. work outC. look intoD. sum up24. Some people would like to do shopping on Sundays since they expect to pick up a lot of wonderful _____ inthe markets.A. batteriesB. basketsC. barrelsD. bargains25. The residents living in these apartments have free ____ to the swimming pool, the gym and other facilities.A. excessB. excursionC. accessD. recreation26．Reporters and photographers alike took great _____at the rude way the actors behaved during the interview.A. annoyanceB. offenceC. resentmentD. irritation27. Nothing healthful should be omitted from the meal of a child because of a ___ dislike.A. provedB. supposedC. consideredD. related28. I’d like to ____ this old car for a new model but I am afraid I cannot afford it.A. exchangeB. convertedC. replaceD. substitute29. She said she liked dancing but was not in the ______for it just then when it was so noisy in the hall.A. mannerB. intentionC. moodD. desire30. He was _____ admittance to the concert hall for not being properly dressed.A. rejectedB. deniedC. withheldD. deprived31. Most people tend to ______ a pounding heart and sweating palms with the experience of emotion.A. classifyB. identifyC. satisfyD. modify32. The company _____ many fine promises to the engineer in order to get him to work for them.A. held upB. held onC. held outD. held onto33. _____the tragic news about their president, they have cancelled the 4th of July celebration.A. In the course ofB. In spite ofC. In the event ofD. In the light of34. The worker’ demands are _______, they are asking for only a sma ll increase in their wages.A. particularB. moderateC. intermediateD. numerous35. Our explanation seemed only to have _____ his confusion. He was totally at a loss as to what to say.A. brought upB. added toC. worked outD. directed at36. With a standard bulb, only 5% of the electricity is ______ to light-the rest is wasted away as heat.A. compressedB. conformedC. convertedD. confined37. Modern technology has placed ______ every kind of music, from virtually every period in history and everycorner of the globe.A. at our requestB. at our disposalC. in our presenceD. in our sight38. Much has been written on the virtues of natural childbirth, but little research has been done to ___ these virtues.A. confirmB. consultC. confessD. convey39. Plastic bags are useful for holding many kinds of food ______their cleanness, toughness and low cost.A. by virtue ofB. at sight ofC. by means ofD. by way of40. I am trying to think of his mind, but my mind goes completely _____ I must be slipping.A. bareB. blankC. hollowD. vacant41. _____ in the regulations that you should not tell other people the password of your email account.A. What is requiredB. What requiresC. It is requiredD. It requires42. It is generally believed that gardening is ______it is a science.A. an art much asB. much an art asC. as an art much asD. as much an art as43. The indoor swimming pool seems to be great more luxurious than ______.A. is necessaryB. being necessaryC. to be necessaryD. it is necessary44. He ______ English for 8 years by the time he graduates from the university next year.A. will learnB. will be learningC. will have learntD. will have been learnt45. It was not until the sub prime loan crisis ______ great damage to the American financial system thatAmericans ______the severity of the situation.A. caused; realizedB. had caused; realizedC. caused; had realizedD. was causing; had realized46. ----The police think your brother John stole the diamond in the museum yesterday evening.-----Oh? But he stayed with me at home the whole evening; he ___ the museum.A. must have been toB. needn’t have been toC. should have been toD. couldn’t have been to47. _______ the meeting himself gave his supporters a great deal of encouragement.A. The president will attendB. The president to attendC. The president attendedD. The president’s attending48. Everything _____into consideration, the candidates ought to have another chance.A. is takenB. takenC. to be takenD. taking49. _______from heart trouble for years, Professor White has to take some medicine with him wherever he goes.A. SufferedB. SufferingC. Having sufferedD. Being suffered50. The concert will be broadcasted live to a worldwide television audience _____ thousand millionA. estimatingB. estimatedC. estimatesD. having estimated51. About half of the students expected there ______more reviewing classes before the final exams.A. isB. beingC. to beD. have been52. _____ made the school proud was _______more than 90% of the students had been admitted to key universities.A. What; becauseB. What; thatC. That; whatD. That; because53. Information has been released ______ more middle school graduates will be admitted into universities this year.A. whileB. thatC. whenD. as54. He is the only one of the students who ______ a winner of scholarship for three years.A. isB. areC. have beenD. has been55. ----What’s that newly-built building?---- ______ the students have out-of-class activities, such as drawing and singing.A. It is the building thatB. That’s whereC. It is in whichD. The building that56.With a large amount of work_____ the chief manager couldn’t spare time for a holiday.A. remained to be doneB. remaining to doC. remained to doD. remaining to be done57. Why! I have nothing to confess, ______you want me to say?A. What is it thatB. what it is thatC. How is it thatD. How it is that58. David has made great progress recently. ________, and ________.A. So he has, so you haveB. So he has; so have youC. So has he; so you haveD. So has he; so you have59. It is universally known that microscopes make small things appear larger than _____.A. really areB. are reallyC. are they reallyD. they really are60. ----- Tom, _______, but your TV is going too loud.----- Oh, I’m sorry. I will turn it down right now.A. I’d like to talk with youB. I’m really tired of thisC. I hate to say thisD. I need your helpPart ⅢClozeDirections: There are 20 blanks in the following passages. For each bank there are 4 choices marked A, B, C and D. You chose the ONE that best fits into the passage, the corresponding letter on the Answer Sheet with a single line through the center.The eyes are the most important (61) of human body that is used to (61) information. Eye contact is crucial for establishing rapport (63) others. The way we look at other people can (64) them know we are paying attention to (65) they are saying. We can also look at a person and gave the (66) we are not hearing a word. Probably all of us have been (67) of looking directly at someone and (68) hearing a word while he or she was talking (69) we were thinking about something totally (70) to what was being said.Eye contact allows you to (71) up visual clues about the other person; (72), the other person can pick up clues about you. Studies of the use of eye contact (73) communication indicate that we seek eye contact with others (74) we want to communicate with them. When we like them, when we are (75) toward them (as when two angry people (76) at each other). And when we want feedback from them. (77), we avoid eye contact when we want to (78) communication, when we dislike them, when we are (79) to deceiving them, and when we are (80) in what they have to say.61 A. unit B. part C. link D. section62 A. transfer B. translate C. transmit D. transport63 A. against B. with C. for D. to64 A. forbid B. allow C. permit D. let65 A. how B. which C. what D. that66 A. impression B. expression C. suggestion D. attention67 A. ignorant B. careless C. guilty D. innocent68 A. nor B. so C. not D. neither69 A. or B. unless C. why D. because70 A. related B. relevant C. unrelated D. indifferent71 A. tear B. pick C. size D. take72 A. likewise B. moreover C. otherwise D. therefore73 A. in B. about C. with D. of74 A. why B. where C. when D. what75 A. friendly B. hostile C. respectful D. mistrustful76 A. glance B. glare C. gaze D. stare77 A. Exactly B. Generally C. Conversely D. Interestingly78 A. hold B. establish C. avoid D. direct79 A. wanting B. tending C. forcing D. trying80 A. informed B. unconcerned C. uninterested D. unheard第Ⅱ卷（共50分）Part ⅣTranslationSection ADirections: Translate the following sentences into Chinese you may refer to the corresponding passages in Part ONE.81. The best way to get rid of a negative self-image is to realize that your image is far from objective and to actively convince yourself of your positive qualities.82. There is a beautiful positive cycle that is created by living a life of honorable actions Honorable thoughts lead to honorable actions. Honorable actions lead us to a happier existence.83. While the positive cycle can be difficult to start, once i t’s started, it’s easy to continue. Keeping on doing good deeds brings us peace of mind, which is important for your happiness.84. That is because television program by that name can not be seen in so many parts of the world. That program became one of American’s export soon after it went on the air in New York in 1969.85. For instance, it is quite acceptable to say “delightfully” situated. That is an expression of his opinion. You may not agree, but he might like the idea of living next to the gasworks.Section BDirections: translate the following sentences into English.86. 一个公司应该跟上市场的发展变化，这很重要。

江苏专升本高等数学真题(附答案)

江苏专转本高数考纲及重点总结一、函数、极限和连续（一）函数（1）理解函数的概念：函数的定义，函数的表示法，分段函数。

（2）理解和把握函数的简单性质：单调性，奇偶性，有界性，周期性。

（3）了解反函数：反函数的定义，反函数的图象。

（4）把握函数的四则运算与复合运算。

（5）理解和把握基本初等函数：幂函数，指数函数，对数函数，三角函数，反三角函数。

（6）了解初等函数的概念。

重点：函数的单调性、周期性、奇偶性，分段函数和隐函数（二）极限（1）理解数列极限的概念：数列，数列极限的定义，能根据极限概念分析函数的变化趋势。

会求函数在一点处的左极限与右极限，了解函数在一点处极限存在的充分必要条件。

（2）了解数列极限的性质：唯一性，有界性，四则运算定理，夹逼定理，单调有界数列，极限存在定理，把握极限的四则运算法则。

（3）理解函数极限的概念：函数在一点处极限的定义，左、右极限及其与极限的关系，x趋于无穷（x→∞，x→+∞，x→-∞）时函数的极限。

（4）把握函数极限的定理：唯一性定理，夹逼定理，四则运算定理。

（5）理解无穷小量和无穷大量：无穷小量与无穷大量的定义，无穷小量与无穷大量的关系，无穷小量与无穷大量的性质，两个无穷小量阶的比较。

（6）熟练把握用两个重要极限求极限的方法。

重点：会用左、右极限求解分段函数的极限，把握极限的四则运算法则、利用两个重要极限求极限以及利用等价无穷小求解极限。

（三）连续（1）理解函数连续的概念：函数在一点连续的定义，左连续和右连续，函数在一点连续的充分必要条件，函数的中断点及其分类。

（2）把握函数在一点处连续的性质：连续函数的四则运算，复合函数的连续性，反函数的连续性，会求函数的中断点及确定其类型。

（3）把握闭区间上连续函数的性质：有界性定理，最大值和最小值定理，介值定理（包括零点定理），会运用介值定理推证一些简单命题。

（4）理解初等函数在其定义区间上连续，并会利用连续性求极限。

重点：理解函数（左、右连续）性的概念，会判别函数的中断点。

01—10年江苏专转本数学真题(附答案)

2001年江苏省普通高校“专转本”统一考试高等数学一、选择题（本大题共5小题，每小题3分，共15分）1、下列各极限正确的是（）A 、e xxx =+→)11(lim 0B 、e xx x =+∞→1)11(limC 、11sinlim =∞→x x x D 、11sin lim 0=→xx x2、不定积分=-⎰dx x211 （）A 、211x-B 、c x+-211C 、x arcsinD 、c x +arcsin3、若)()(x f x f -=，且在[)+∞,0内0)('>x f 、0)(''>x f ，则在)0,(-∞内必有（）A 、0)('<x f ,0)(''<x f B 、0)('<x f ,0)(''>x f C 、0)('>x f ,0)(''<x f D 、0)('>x f ,0)(''>x f4、=-⎰dx x 21 （）A 、0B 、2C 、－1D 、15、方程x y x 422=+在空间直角坐标系中表示（） A 、圆柱面B 、点C 、圆D 、旋转抛物面二、填空题（本大题共5小题，每小题3分，共15分）6、设⎩⎨⎧+==22tt y te x t ，则==0t dx dy7、0136'''=+-y y y 的通解为 8、交换积分次序=⎰⎰dy y x f dx x x220),(9、函数yx z =的全微分=dz10、设)(x f 为连续函数，则=+-+⎰-dx x x x f x f 311])()([三、计算题（本大题共10小题，每小题4分，共40分） 11、已知5cos)21ln(arctan π+++=xx y ，求dy .12、计算xx dte x xt x sin lim202⎰-→.等价无穷小，洛必达13、求)1(sin )1()(2--=x x xx x f 的间断点，并说明其类型.x 分别为0，1，-1时化简求极限14、已知x y x y ln 2+=，求1,1==y x dxdy.15、计算dx ee xx⎰+12. 16、已知⎰∞-=+02211dx x k ，求k 的值. 17、求x x y y sec tan '=-满足00==x y 的特解.18、计算⎰⎰Ddxdy y2sin ，D 是1=x 、2=y 、1-=x y 围成的区域.19、已知)(x f y =过坐标原点，并且在原点处的切线平行于直线032=-+y x ，若b ax x f +=2'3)(，且)(x f 在1=x 处取得极值，试确定a 、b 的值，并求出)(x f y =的表达式.20、设),(2y x x f z =，其中f 具有二阶连续偏导数，求x z∂∂、yx z ∂∂∂2.四、综合题（本大题共4小题，第21小题10分，第22小题8分，第23、24小题各6分，共30分） 21、过)0,1(P 作抛物线2-=x y 的切线，求（1）切线方程；（2）由2-=x y ，切线及x 轴围成的平面图形面积；（3）该平面图形分别绕x 轴、y 轴旋转一周的体积。

江苏省专转本高等数学试题题型分类整理

江苏省普通高校“专转本”统一考试高等数学专转本高数试卷结构知识分类与历年真题●函数、极限和连续●一元函数微分学●一元函数积分学●向量代数与空间解析几何●多元函数微积分●无穷级数●常微分方程时间排序与参考答案◆2004年高等数学真题参考答案◆2005年高等数学真题参考答案◆2006年高等数学真题参考答案◆2007年高等数学真题参考答案◆2008年高等数学真题参考答案◆2009年高等数学真题参考答案◆2010年高等数学真题参考答案◆2011年高等数学真题参考答案◆2012年高等数学真题参考答案◆2013年高等数学真题参考答案江苏省普通高校“专转本”统一考试高等数学试卷结构全卷满分150分一、单选题（本大题共6小题，每小题4分，满分24分）二、填空题（本大题共6小题，每小题4分，满分24分）三、解答题（本大题共8小题，每小题8分，满分64分）四、综合题（本大题共2小题，每小题10分，满分20分）五、证明题（本大题共2小题，每小题9分，满分18分）知识分类与历年真题一、函数、极限和连续（一）函数（0401）[](]333,0()0,2x x f x x x ⎧∈-⎪=⎨-∈⎪⎩是（） A.有界函数 B.奇函数 C.偶函数 D.周期函数（0801）设函数)(x f 在),(+∞-∞上有定义，下列函数中必为奇函数的是（）A.()y f x =-B.)(43x f x y = C.()y f x =-- D.)()(x f x f y -+= （二）极限（0402）当0→x 时，x x sin 2-是关于x 的（）A.高阶无穷小B.同阶无穷小C.低阶无穷小D.等价无穷小（0407）设xx x x f ⎪⎭⎫⎝⎛++=32)(，则=∞→)(lim x f x ．（0601）若012lim 2x x f x →⎛⎫ ⎪⎝⎭=，则0lim 3x xx f →=⎛⎫ ⎪⎝⎭（） A.21 B.2C.3D.31 （0607）已知0→x 时，(1cos )a x ⋅-与x x sin 是等价无穷小，则=a ．（0613)计算x →．（0701）若0(2)lim2x f x x→=，则1lim 2x xf x →∞⎛⎫= ⎪⎝⎭（） A.41B.21 C.2D.4（0702）已知当0→x 时，)1ln(22x x +是x n sin 的高阶无穷小，而x nsin 又是x cos 1-的高阶无穷小，则正整数=n （） A.1B.2C.3D.4（0813）求极限：32lim xx x x →∞-⎛⎫⎪⎝⎭．（0901）已知22lim32x x ax bx →++=-，则常数b a ,的取值分别为（） A.2,1-=-=b a B.0,2=-=b aC.0,1=-=b aD.1,2-=-=b a（0907）已知lim 2xx x x C →∞⎛⎫= ⎪-⎝⎭，则常数=C ．（1001）设当0x →时，()sin f x x x =-与()ng x ax =是等价无穷小，则常数,a n 的值为 ( ) A.1,36a n == B.1,33a n == C.1,412a n == D.1,46a n == （1007） 1lim 1xx x x →∞+⎛⎫= ⎪-⎝⎭．（1101）当0→x 时，函数1)(--=x e x f x是函数2)(x x g =的( )A.高阶无穷小B.低阶无穷小C.同阶无穷小D.等价无穷小（1107）已知22lim kxx x e x →∞-⎛⎫= ⎪⎝⎭，则=k _________．（1201）极限1sin 3lim 2sinx x x x x →∞⎛⎫+= ⎪⎝⎭（） A.0 B.2 C.3D.5（1301）当0x →时，函数()ln(1)f x x x =+-是函数2()g x x =的（） A.高阶无穷小 B.低阶无穷小C.同阶无穷小D.等价无穷小（1310）设10lim xx a x e a x →+⎛⎫=⎪-⎝⎭，则常数a = ．（三）连续（0413）求函数xxx f sin )(=的间断点，并判断其类型．（0501）0=x 是xx x f 1sin )(=的（） A.可去间断点B.跳跃间断点C.第二类间断点D.连续点（0513）设()2sin 0()0f x xx F x xa x +⎧≠⎪=⎨⎪=⎩在R 内连续，并满足0)0(=f ，(0)6f '=，求a ．（0602）函数21sin 0()00x x f x xx ⎧≠⎪=⎨⎪=⎩在0x =处（） A.连续但不可导B.连续且可导C.不连续也不可导D.可导但不连续（0608）若A x f x x =→)(lim 0，且)(x f 在0x x =处有定义，则当=A 时，)(x f 在0x x =处连续．（0707）设函数1(1)0()20x kx x f x x ⎧⎪+≠=⎨⎪=⎩，在点0=x 处连续，则常数=k ．（0807）设函数21()(1)x f x x x -=-，则其第一类间断点为．（0808）设函数0()tan 30a x x f x x x x+≥⎧⎪=⎨<⎪⎩在点0=x 处连续，则a ＝．（0902）已知函数423)(22-+-=x x x x f ，则2=x 为)(x f 的（）A.跳跃间断点B.可去间断点C.无穷间断点D.震荡间断点（1123）设210arctan ()1010sin 2ax axe x ax x x xf x x e x x ⎧---<⎪⎪⎪==⎨⎪-⎪>⎪⎩，问常数为何值时：（1）0=x 是函数)(x f 的连续点？（2）0=x 是函数)(x f 的可去间断点？（3）0=x 是函数)(x f 的跳跃间断点？（1202）设()2(2)sin ()4x xf x x x -⋅=⋅-，则函数)(x f 的第一类间断点的个数为（） A.0 B.1C.2D.3（1207）要使函数()1()12xf x x =-在点0=x 处连续，则需补充定义(0)f =_________．（1303）设sin 20()0xx x f x x ⎧<⎪⎪=⎨>，这点0x =是函数()f x 的（）A.跳跃间断点B.可去间断点C.无穷间断点D.连续点（1307）设1sin0()0x x f x xa x ⎧≠⎪=⎨⎪=⎩在点0x =处连续，则常数a = ．二、一元函数微分学（一）导数与微分（0403）直线L 与x 轴平行且与曲线xe x y -=相切，则切点的坐标是（） A.()1,1B.()1,1-C.()0,1-D.()0,1（0409）设()(1)(2)()f x x x x x n =+++，N n ∈，则=)0('f ．（0415）设函数)(x y y =由方程1=-yxe y 所确定，求22d d x yx=的值．（0502）若2=x 是函数1ln 2y x ax ⎛⎫=-+ ⎪⎝⎭的可导极值点，则常数=a （） A.1-B.21C.21- D.1 （0514）设函数)(x y y =由方程cos sin cos x t y t t t =⎧⎨=-⎩所确定，求d d y x 、22d d yx ．（0614）若函数)(x y y =是由参数方程2ln (1)arctan x t y t t⎧=+⎨=-⎩所确定，求d d y x 、22d d yx ．（0708）若直线m x y +=5是曲线232++=x x y 的一条切线，则常数=m ．（0714）设函数)(x y y =由方程xy e e yx=-确定，求d d x yx=、22d d x y x =．（0802）设函数)(x f 可导，则下列式子中正确的是（） A.0(0)()lim(0)x f f x f x →-'=- B.000(2)()lim ()x f x x f x f x x→+-'=C.0000()()lim ()x f x x f x x f x x ∆→+∆--∆'=∆D.0000()()lim 2()x f x x f x x f x x∆→-∆-+∆'=∆ （0814）设函数)(x y y =由参数方程sin 1cos x t t y t =-⎧⎨=-⎩（2t n π≠，n Z ∈）所决定，求d d y x 、22d d y x ．（0903）设函数00()1sin 0x f x x x x α≤⎧⎪=⎨>⎪⎩在点0=x 处可导，则常数α的取值范围为（） A.10<<αB.10≤<αC.1>αD.1≥α（0914）设函数)(x y y =由参数方程2ln (1)23x t y t t =+⎧⎨=+-⎩所确定，d d y x 、22d d yx ．（0923）已知函数0()10x e x f x x x -⎧<=⎨+≥⎩，证明函数)(x f 在点0=x 处连续但不可导．（1008）.若(0)1f '=，则0()()limx f x f x x→--= ．（1014）设函数()y y x =由方程2x yy ex ++=所确定，求d d y x 、22d d yx ．（1022）设()0()1x x f x xx ϕ⎧≠⎪=⎨⎪=⎩，其中函数()x ϕ在0x =处具有二阶连续导数，且(0)0ϕ=，(0)1ϕ'=，证明：函数()f x 在0x =处连续且可导．（1102）设函数)(x f 在点0x 处可导，且4)()(lim 000=+--→hh x f h x f h ，则=')(0x f ( )A.4-B.2-C.2D.4（1110）设函数x y arctan=，则1d x y==_____________．（1114）设函数)(x y y =由参数方程⎪⎩⎪⎨⎧=++=22ty e tt x y 所确定，求d d y x ．（1208）设函数()22221x y x x x e =⋅+++，则=)0()7(y________．（1209）设xy x =（0x >），则函数y 的微分=dy ___________．（1214）设函数)(x y y =由参数方程⎪⎩⎪⎨⎧+=-=tt y tt x ln 212所确定，求d d y x 、22d d y x ．（1304）设1y f x ⎛⎫= ⎪⎝⎭，其中f 具有二阶导数，则22d d y x =（）A.231121f f x x x x ⎛⎫⎛⎫'''-+ ⎪ ⎪⎝⎭⎝⎭ B.231121f f x x x x ⎛⎫⎛⎫'''+ ⎪ ⎪⎝⎭⎝⎭ C.231121f f x x x x ⎛⎫⎛⎫'''--⎪ ⎪⎝⎭⎝⎭D.231121f f x x x x ⎛⎫⎛⎫'''-⎪ ⎪⎝⎭⎝⎭（1306）已知函数()f x 在点1x =处连续，且21()1lim 12x f x x →=-，则曲线()f x 在点()1,()f x 处切线方程为（） A.1y x =-B.22y x =-C.33y x =-D.44y x =-（1309）设函数由参数方程2211x t y t ⎧=+⎨=-⎩所确定，则221d d t yx == ．（二）中值定理及导数的应用（0423）甲、乙二城位于一直线形河流的同一侧，甲城位于岸边，乙城离河岸40公里，乙城在河岸的垂足与甲城相距50公里，两城计划在河岸上合建一个污水处理厂，已知从污水处理厂到甲乙二城铺设排污管道的费用分别为每公里500、700元．问污水处理厂建在何处，才能使铺设排污管道的费用最省？（0507）02limsin x x x e e xx x-→--=- ．（0508）函数x x f ln )(=在区间[]1,e 上满足拉格郎日中值定理的=ξ ．（0521）证明方程：0133=+-x x 在[]1,1-上有且仅有一根．（0603）下列函数在[]1,1-上满足罗尔定理条件的是（） A.xe y =B.1y x =+C.21x y -=D.xy 11-= （0621）证明：当2x ≤时，332x x -≤．（0703）设函数()(1)(2)(3)f x x x x x =---，则方程()0f x '=的实根个数为（） A.1B.2C.3D.4（0713）求极限01lim tan x x e x x x→--．（0722）设函数9)(23-++=cx bx ax x f 具有如下性质：（1）在点1-=x 的左侧临近单调减少；（2）在点1-=x 的右侧临近单调增加；（3）其图形在点(1,2)的两侧凹凸性发生改变．试确定a ，b ，c 的值．（0724）求证：当0>x 时，22(1)ln (1)x x x -⋅≥-．（0809）已知曲线543223++-=x x x y ，则其拐点为．（0821）求曲线1y x=（0x >）的切线，使其在两坐标轴上的截距之和最小，并求此最小值．（0823）设函数)(x f 在闭区间[]0,2a （0a >）上连续，且)()2()0(a f a f f ≠=，证明：在开区间(0,)a 上至少存在一点ξ，使得()()f f a ξξ=+．（0824）对任意实数x ，证明不等式：(1)1xx e -⋅≤．（0904）曲线221(1)x y x +=-的渐近线的条数为（）A.1B.2C.3D.4（0913）求极限30lim sin x x x x→-．（0921）已知函数13)(3+-=x x x f ，试求：（1）函数)(x f 的单调区间与极值；（2）曲线)(x f y =的凹凸区间与拐点；（3）函数)(x f 在闭区间[2,3]-上的最大值与最小值．（0924）证明：当12x <<时，24ln 23x x x x >+-．（1002）曲线223456x x y x x -+=-+的渐近线共有 ( )A.1条B.2条C.3条D.4条（1006）设3()3f x x x =-，则在区间(0,1)内 ( ) A.函数()f x 单调增加且其图形是凹的 B.函数()f x 单调增加且其图形是凸的 C.函数()f x 单调减少且其图形是凹的 D.函数()f x 单调减少且其图形是凸的（1013）求极限2|011lim tan x x x x →⎛⎫-⎪⎝⎭．（1021）证明：当1x >时，121122x e x ->+．（1103）若点(1,2)-是曲线23bx ax y -=的拐点，则（） A.3,1==b aB.1,3-=-=b aC.3,1-=-=b aD.6,4==b a（1113）求极限()()22limln 1xx x eex -→-+．（1121）证明：方程()2ln 12x x ⋅+=有且仅有一个小于2的正实根．（1122）证明：当0>x 时，x x201120102011≥+．（1203）设232152)(x x x f -=，则函数)(x f ( ) A.只有一个最大值 B.只有一个极小值 C.既有极大值又有极小值D.没有极值（1213）求极限()2302cos 2lim ln 1x x x x x →+-+．（1223）证明：当10<<x 时，361arcsin x x x +>．（1302）曲线22232x xy x x +=-+的渐近线共有（）A.1条B.2条C.3条D.4条（1313）求极限01lim ln (1)x x e x x →⎡⎤-⎢⎥+⎣⎦．（1323）证明：当1x >时，2(1ln )21x x +<-．三、一元函数积分学（一）不定积分（0410）求不定积分3x = ．（0416）设)(x f 的一个原函数为xe x，计算(2)d x f x x '⎰．（0503）若()d ()f x x F x C =+⎰，则sin (cos )d x f x x =⎰（）A.C x F +)(sinB.C x F +-)(sinC.C F +(cos)D.C x F +-)(cos（0515）计算3tan sec d x x x ⎰．（0522）设函数)(x f y =的图形上有一拐点(2,4)P ，在拐点处的切线斜率为3-，又知该函数的二阶导数6y x a ''=+，求)(x f ．（0604）已知2()d x f x x e C =+⎰，则()d f x x '-=⎰（）A.C ex+-22B.C e x +-221 C.C e x +--22 D.C e x +--221（0615）计算x ．（0622）已知曲线)(x f y =过原点且在点),(y x 处的切线斜率等于y x +2，求此曲线方程．（0704）设函数)(x f 的一个原函数为x 2sin ，则(2)d f x x '=⎰（）A.C x +4cosB.C x +4cos 21C.C x +4cos 2D.C x +4sin（0715）求不定积分2d x x e x -⎰．（0810）设函数)(x f 的导数为x cos ，且21)0(=f ，则不定积分()d f x x =⎰ ．（0815）求不定积分3d 1x x x +⎰．（0905）设()ln (31)F x x =+是函数)(x f 的一个原函数，则(21)d f x x '+=⎰（）A.C x ++461B.C x ++463C.C x ++8121D.C x ++8123（0915）求不定积分x ⎰．（1015）求不定积分arctan d x x x ⎰．（1115）设)(x f 的一个原函数为x x sin 2，求不定积分()d f x x x⎰．（1215）求不定积分sin 2d x x x ⎰．（1315）求不定积分sin 2d x x x ⎰．（二）定积分（0404）2228R y x =+设所围的面积为S ，则0x ⎰的值为（）A.SB.4S C.2S D.S 2（0421）证明：0(sin )d (sin )d 2x f x x f x x πππ=⎰⎰，并利用此式求20sin d 1cos xxx xπ+⎰．（0509）1211d 1x x x π-+=+⎰．（0516）计算10arctan d x x ⎰．（0609）设)(x f 在[]0,1上有连续的导数且(1)2f =，10()d 3f x x =⎰，则1()d x f x x '=⎰ ．（0616）计算22cos d x x x π⎰．（0709）定积分)231cos d x x x -+⎰的值为．（0716）计算定积分x ．（0811）定积分1212sin d 1xx x -++⎰的值为．（0816）求定积分10d x ⎰．（0916）求定积分：210⎰．（1009）定积分31211d 1x x x -++⎰的值为．（1016）计算定积分40x ⎰．（1111）定积分()32221sin d xx x ππ-+⋅⎰的值为____________．（1116）计算定积分3⎰　．（1216）计算定积分21⎰．（1316）计算定积分20⎰（1324）设函数()f x 在[,]a b 上连续，证明：[]2()d ()()d a b b aaf x x f x f a b x x +=++-⎰⎰．（三）变限积分与广义积分（0417）计算广义积分2+∞⎰（0422）设函数)(x f 可导，且满足方程20()d 1()x t f t t x f x =++⎰，求)(x f ．（0705）设221()sin d x f x t t =⎰，则()f x '=（）A.4sin x B.2sin 2x xC.2cos 2x xD.4sin 2x x（0803）设函数)(x f 122sin d xt t t =⎰，则()f x '等于（）A.x x 2sin 42B.x x 2sin 82C.x x 2sin 42-D.x x 2sin 82-（0908）设函数20()d x t x te t ϕ=⎰，则()x ϕ'＝．（1003）设函数22()cos d t xx e t t Φ=⎰，则函数()x Φ的导数()x 'Φ等于 ( )A.222cos x xe x B.222cos x xe x - C.2cos xxe x - D.22cos x e x - （1108）设函数2()ln (1)d x x t t Φ=+⎰　，则=Φ'')1(____________．（1211）设反常积分1d 2x ae x +∞-=⎰，则常数=a ______．（1222）已知定义在(),-∞+∞上的可导函数)(x f 满足方程31()4()d 3xx f x f t t x -=-⎰，试求：（1）函数()f x 的表达式；（2）函数)(x f 的单调区间与极值；（3）曲线()y f x =的凹凸区间与拐点．（1224）设0()d 0()(0)0x g t t x f x g x ⎧≠⎪=⎨⎪=⎩⎰，其中函数)(x g 在(,)-∞+∞上连续，且3cos 1)(lim 0=-→xx g x ．证明：函数)(x f 在0=x 处可导，且1(0)2f '=．（1322）已知251320()95d x F x t t t ⎛⎫=- ⎪⎝⎭⎰是()f x 的一个原函数，求曲线()y f x =的凹凸区间、拐点．（四）定积分的几何应用（0523）已知曲边三角形由x y 22=、0=x 、1=y 所围成，求：（1）曲边三角形的面积；（2）曲边三角形绕x 轴旋转一周的旋转体体积．（0623）已知一平面图形由抛物线2x y =、82+-=x y 围成．（1）求此平面图形的面积；（2）求此平面图形绕y 轴旋转一周所得的旋转体的体积．（0721）设平面图形由曲线21x y -=（0≥x ）及两坐标轴围成．（1）求该平面图形绕x 轴旋转所形成的旋转体的体积；（2）求常数a 的值，使直线a y =将该平面图形分成面积相等的两部分．（0822）设平面图形由曲线2x y =，22x y =与直线1=x 所围成．（1）求该平面图形绕x 轴旋转一周所得的旋转体的体积；（2）求常数a ，使直线a x =将该平面图形分成面积相等的两部分．（0922）设1D 是由抛物线22x y =和直线x a =，0y =所围成的平面封闭区域，2D 是由抛物线22x y =和直线x a =，2x =及0=y 所围成的平面封闭区域，其中20<<a ．试求：（1）1D 绕y 轴旋转所成的旋转体的体积1V ，以及2D 绕x 轴旋转所成的旋转体的体积2V ；（2）求常数a 的值，使得1D 的面积与2D 的面积相等．（1023）设由抛物线2y x =（0x ≥），直线2y a =（01a <<）与y 轴所围成的平面图形绕x 轴旋转一周所形成的旋转体的体积记为1()V a ，由抛物线2y x =（0x ≥），直线2y a =（01a <<）与直线1x =所围成的平面图形绕x 轴旋转一周所形成的旋转体的体积记为2()V a ，另12()()()V a V a V a =+，试求常数a 的值，使()V a 取得最小值．（1024）设函数()f x 满足方程()()2xf x f x e '+=，且(0)2f =，记由曲线'()()f x y f x =与直线1y =，x t =（0t >）及y 轴所围平面图形的面积为()A t ，试求lim ()t A t →+∞．（1124）设函数)(x f 满足微分方程()2()(1)x f x f x a x '-=-+（其中a 为正常数），且1)1(=f ，由曲线()y f x =（1x ≤）与直线1x =，0y =所围成的平面图形记为D ．已知D 的面积为32．（1）求函数)(x f 的表达式；（2）求平面图形D 绕x 轴旋转一周所形成的旋转体的体积x V ；（3）求平面图形D 绕y 轴旋转一周所形成的旋转体的体积y V ．（1221）在抛物线2y x =（0x >）上求一点P ，使该抛物线与其在点P 处的切线及x 轴所围成的平面图形的面积为32，并求该平面图形绕x 轴旋转一周所形成的旋转体的体积．（1321）设平面图形D 是由曲线x =y =1y =所围成，试求：（1）平面图形D 的面积；（2）平面图形D 绕x 轴旋转一周所形成的旋转体的体积．四、向量代数与空间解析几何（一）向量代数（0510）设向量{}3,4,2=-a 、{}2,1,k =b ；a 、b 互相垂直，则=k ．（0610）设1=a ，⊥a b ，则()⋅+=a a b ．（0710）已知a 、b 均为单位向量，且12⋅=a b ，则以a 、b 为邻边的平行四边形面积为．（0804）设向量(1,2,3)=a ，(3,2,4)=b ，则⨯a b 等于（）A.(2,5,4)B.(2,5,4)--C.(2,5,4)-D.(2,5,4)--（0909）已知向量{}1,0,1=-a ，{}1,2,1=-b ，则+a b 与a 的夹角为．（1010）设{}1,2,3=a ，{}2,5,k=b ，若a 与b 垂直，则常数k = ．（1109）若1=a ，4=b ，2⋅=a b ，则⨯=a b ____________．（1210）设向量a 、b 互相垂直，且3=a ，2=b ，则2+=a b ________．（1308）已知空间三点(1,1,1)A ，(2,3,4)B ，(3,4,5)C ，则ABC ∆的面积为．（二）平面与直线（0518）求过点(3,1,2)A -且通过直线L ：43521x y z-+==的平面方程．（0619）求过点(3,1,2)M -且与二平面07=-+-z y x 、0634=-+-z y x 都平行的直线方程．（0719）求过点(1,2,3)且垂直于直线20210x y z x y z +++=⎧⎨-++=⎩的平面方程．（0817）设平面∏经过点(2,0,0)A ，(0,3,0)B ，(0,0,5)C ，求经过点(1,2,1)P 且与平面∏垂直的直线方程．（0917）求通过直线12213-=-=z y x 且垂直于平面02=+++z y x 的平面方程．（1017）求通过点(1,1,1)，且与直线23253x ty t z t =+⎧⎪=+⎨⎪=+⎩垂直，又与平面250x z --=平行的直线的方程．（1117）求通过x 轴与直线132zy x ==的平面方程．（1217）已知平面∏通过(1,2,3)M 与x 轴，求通过(1,1,1)N 且与平面∏平行，又与x 轴垂直的直线方程．（1318）已知直线10330x y z x y z -+-=⎧⎨--+=⎩在平面∏上，又知直线23132x ty t z t=-⎧⎪=+⎨⎪=+⎩与平面∏平行，求平面∏的方程．五、多元函数微积分（一）多元函数微分学（0418）设(,)z f x y xy =-，且具有二阶连续的偏导数，求x z ∂∂、yx z∂∂∂2．（0505）设yxy x u arctan),(=，(,)v x y =，则下列等式成立的是（）A.yv x u ∂∂=∂∂ B.xvx u ∂∂=∂∂ C.x v y u ∂∂=∂∂ D.y v y u ∂∂=∂∂ （0517）已知函数2(sin ,)z f x y =，其中),(v u f 有二阶连续偏导数，求x z ∂∂、yx z∂∂∂2．（0611）设x e u xysin =，=∂∂xu．（0620）设2(,)z x f x xy =⋅其中(,)f u v 的二阶偏导数存在，求y z ∂∂、xy z∂∂∂2．（0711）设yxz =，则全微分d z = ．（0717）设(23,)z f x y xy =+其中f 具有二阶连续偏导数，求yx z∂∂∂2．（0805）函数xyz ln =在点(2,2)处的全微分d z 为（）A.11d d 22x y -+B.11d d 22x y +C.11d d 22x y -D.11d d 22x y --（0818）设函数,y z f x y x ⎛⎫=+ ⎪⎝⎭，其中)(x f 具有二阶连续偏导数，求y x z ∂∂∂2．（0910）设函数(,)z z x y =由方程12=+yz xz 所确定，则xz∂∂＝．（0919）设函数(sin ,)z f x xy =，其中)(x f 具有二阶连续偏导数，求yx z∂∂∂2．（1011）设函数z =，则10d x y z=== ．（1018）设()2,xz y f xy e =⋅，其中函数f 具有二阶连续偏导数，求2zx y∂∂∂．（1104）设),(y x f z =为由方程8333=+-x yz z 所确定的函数，则=∂∂==00y x yz （）A.21-B.21C.2-D.2（1118）设)(y xyxf z ，=，其中函数f 具有二阶连续偏导数，求y x z ∂∂∂2．（1204）设3ln 2z x y=+在点()1,1处的全微分为 ( )A.d 3d x y -B.d 3d x y +C.1d 3d 2x y +D.1d 3d 2x y -（1218）设函数22(,)()z f x xy x y ϕ=++，其中函数f 具有二阶连续偏导数，函数()x ϕ具有二阶连续导数，求yx z∂∂∂2．（1314）设函数(,)z z x y =由方程3331z xy z +-=所确定，求d z 及22zx∂∂．（1317）设()223,x yz fx e+=，其中函数f 具有二阶连续偏导数，求2zy x∂∂∂．（二）二重积分（0411）交换二次积分的次序2120d (,)d x x x f x y y -=⎰⎰．（0419）计算二重积分sin d d Dyx y y ⎰⎰，其中D 由曲线x y =及x y =2所围成．（0504）设区域D 是xoy 平面上以点(1,1)A 、(1,1)B -、(1,1)C --为顶点的三角形区域，区域1D 是D 在第一象限的部分，则(cos sin )d d Dxy x y x y +=⎰⎰（）A.⎰⎰1)sin (cos 2D dxdy y xB.⎰⎰12D xydxdyC.⎰⎰+1)sin cos (4D dxdy y x xyD. 0（0511）交换二次积分的次序11d (,)d x x f x y y -+=⎰；（0524）设)(x f 为连续函数，且1)2(=f ，1()d ()d uuyF u y f x x =⎰⎰（1u >）．（1）交换)(u F 的积分次序；（2）求(2)F '．（0606）设对一切x 有(,)(,)f x y f x y -=-，22{(,)|1,0}D x y x y y =+≤≥，=1D 22{(,)|1,0,0}x y x y x y +≤≥≥，则(,)d d Df x y x y =⎰⎰（）A. 0B.1(,)d d D f x y x y ⎰⎰C.21(,)d d D f x y x y ⎰⎰D.41(,)d d D f x y x y ⎰⎰（0612）D 为以点(0,0)O 、(1,0)A 、(0,2)B 为顶点的三角形区域，d d Dx y =⎰⎰ ．（0624）设⎪⎩⎪⎨⎧=≠=⎰⎰00)(1)(t a t dxdy x f t t g tD ，其中t D 是由t x =、t y =以及坐标轴围成的正方形区域，函数)(x f 连续．（1）求a 的值使得)(t g连续；（2）求)('t g ．（0720）计算二重积分d Dx y ，其中{}22(,)|2,0D x y x y x y =+≤≥．（0723）设0>>a b ，证明：()232d ()d ()d b b b x y xx a ayay f x e x ee f x x ++⋅=-⎰⎰⎰．（0819）计算二重积分2d d Dx x y ⎰⎰，其中D 是由曲线xy 1=，直线y x =，2x =及0=y 所围成的平面区域．（0918）计算二重积分d Dy σ⎰⎰，其中22{(,)02,2,2}D x y x x y x y =≤≤≤≤+≥．（1005）二次积分111d (,)d y y f x y x +⎰⎰交换积分次序后得 ( )A.1101d (,)d x x f x y y +⎰⎰B.2110d (,)d x x f x y y -⎰⎰C.2111d (,)d x x f x y y -⎰⎰D.2111(,)d x dx f x y y -⎰⎰（1019）计算d d Dx x y ⎰⎰，其中D 是由曲线x =y x =及x 轴所围成的闭区域．（1105）若(,)d d Df x y x y ⎰⎰可转化为二次积分1201d (,)d y y f x y x +⎰⎰ ，则积分域D 可表示为（） A.{}(,)01,11x y x x y ≤≤-≤≤ B.{}(,)12,11x y x x y ≤≤-≤≤C.{}(,)01,10x y x x y ≤≤-≤≤D.{}(,)12,01x y x y x ≤≤≤≤-（1119）计算二重积分d d Dy x y ⎰⎰，其中D 是由曲线y =直线x y -=及y 轴所围成的平面闭区域．（1205）二次积分dx y x f dy y),(11⎰⎰ 在极坐标系下可化为（）A.sec 40d (cos ,sin )d f πθθρθρθρ⎰⎰ B.sec 40d (cos ,sin )d f πθθρθρθρρ⎰⎰C.sec 24d (cos ,sin )d f πθπθρθρθρ⎰⎰D .sec 24d (cos ,sin )d f πθπθρθρθρρ⎰⎰ （1220）计算二重积分d d Dy x y ⎰⎰，其中D 是由曲线y =2xy =及x 轴所围成的平面闭区域．（1320）计算二重积分d d Dx x y ⎰⎰，其中D 是由曲线y =0x >）与三条直线y x =，3x =，0y =所围成的平面闭区域．六、无穷级数（一）数项级数（0506）正项级数（1）∑∞=1n n u 、（2）∑∞=13n n u ，则下列说法正确的是（） A.若（1）发散、则（2）必发散 B.若（2）收敛、则（1）必收敛C.若（1）发散、则（2）不确定D.（1）、（2）敛散性相同（0605）设∑∞=1n nu为正项级数，如下说法正确的是（）A.若0lim 0=→n n u ，则∑∞=1n nu必收敛 B.若l u u nn n =+∞→1lim )0(∞≤≤l ，则∑∞=1n n u 必收敛C.若∑∞=1n nu收敛，则∑∞=12n nu必定收敛D.若∑∞=-1)1(n n nu 收敛，则∑∞=1n n u 必定收敛（0706）下列级数收敛的是（）A.∑∞=122n nnB.∑∞=+11n n n C.∑∞=-+1)1(1n nnD.∑∞=-1)1(n nn（0906）设α为非零常数，则数项级数∑∞=+12n n n α（）A.条件收敛B.绝对收敛C.发散D.敛散性与α有关（1004）下列级数收敛的是（）A.11n nn ∞=+∑B.2121n n n n ∞=++∑C.nn ∞= D.212n n n ∞=∑（1206）下列级数中条件收敛的是（）A.1(1)21nn nn ∞=-+∑B.13(1)2nnn ∞=⎛⎫- ⎪⎝⎭∑C.21(1)nn n ∞=-∑D.1nn ∞=（1305）下列级数中收敛的是（）A.211n n n∞=+∑ B.11nn n n ∞=⎛⎫ ⎪+⎝⎭∑C.1!2n n n ∞=∑D.1n ∞= （二）幂级数（0412）幂级数∑∞=-12)1(n nnx 的收敛区间为．（0420）把函数21)(+=x x f 展开为2-x 的幂级数，并写出它的收敛区间．（0512）幂级数1(21)nn n x∞=-∑的收敛区间为．（0519）把函数222)(x x x x f --=展开为x 的幂级数，并写出它的收敛区间．（0618）将函数()ln (1)f x x x =+展开为x 的幂函数（要求指出收敛区间）．（0812）幂函数12n nn x n ∞=⋅∑的收敛域为．（0911）若幂函数21n nn a x n∞=∑（0a >）的收敛半径为21，则常数=a ．（1012）幂级数0(1)n nn x n ∞=-∑的收敛域为．（1106）若x x f +=21)(的幂级数展开式为0()nn n f x a x ∞==∑（22x -<<），则系数=n a ( )A.n 21B.121+n C.(1)2nn -D.1(1)2nn +-（1112）幂级数0nn ∞=的收敛域为_ _ _________．（1212）幂级数1(1)(3)3nn nn x n ∞=--⋅∑的收敛域为____________．（1312）幂级数1n nn ∞=的收敛域为．七、常微分方程（一）一阶微分方程（0520）求微分方程0'=-+xe y xy 满足1x ye ==的特解．（0617）求微分方程22x y xy y '=-的通解．（0718）求微分方程22007xy y x '-=满足初始条件12008x y==的特解．（0820）求微分方程22xy y x '=+的通解．（0912）微分方程2(1)d (2)d 0x y x y x y +--=的通解为．（1311）微分方程d d y x y x x+=的通解为．（二）二阶线性微分方程（0406）微分方程232xy y y xe '''-+=的特解*y 的形式应为（）A.xAxe 2B.xe B Ax 2)(+C.xeAx 22D.xeB Ax x 2)(+（0712）设x xe C eC y 3221+=为某二阶常系数齐次线性微分方程的通解，则该微分方程为．（0806）微分方程321y y y '''++=的通解为（）A.1221++=--x xe c e c yB.21221++=--x xe c ec yC.1221++=-xxec e c yD.21221++=-xxec e c y （0920）求微分方程y y x ''-=的通解．（1020）已知函数xy e =和2xy e-=是二阶常系数齐次线性微分方程0y py qy '''++=的两个解，试确定常数p 、q 的值，并求微分方程xy py qy e '''++=的通解．（1120）已知函数(1)xy x e =+⋅是一阶线性微分方程2()y y f x '+=的解，求二阶常系数线性微分方程)(23x f y y y =+'+''的通解．（1219）已知函数)(x f 的一个原函数为xxe ，求微分方程)(44x f y y y =+'+''的通解．（1319）已知函数()y f x =是一阶微分方程d d yy x=满足初始条件(0)1y =的特解，求二阶常系数非齐次线性微分方程32()y y y f x '''-+=的通解．时间排序与参考答案2004年高等数学真题参考答案1、A ．2、B ．3、C ．4、B ．5、A ．6、D ．7、1-e ． 8、32241-+==-z y x ． 9、!n ． 10、C x +4arcsin 41． 11、12201d (,)d d (,)d y y f x y x y f x y x -+⎰⎰⎰．12、()3,1-．13、解：间断点为πk x =（Z k ∈），当0=x 时，1sin lim)(lim 00==→→xxx f x x ，为可去间断点；当πk x =（0≠k ，Z k ∈）时，∞=→xxx sin lim0，为第二类间断点．14、解：原式0430(tan sin )d tan sin limlim312xx x t t tx xx x →→--==⎰233001tan (1cos )12lim lim 121224x x x x x x x x →→⋅-===． 15、解：0=x 代入原方程得1)0(=y ，对原方程求导得0''=--y xe e y yy，对上式求导并将0=x 、1=y 代入，解得：22''e y =．16、解：因为)(x f 的一个原函数为x e x，所以2')1()(x e x x e x f xx -=⎪⎪⎭⎫ ⎝⎛=，原式11(2)d(2)d (2)22xf x x x f x '==⎰⎰11(2)(2)d 22x f x f x x =-⎰ 222211(21)1(2)(2)d(2)24884x x xx x e e x x f x f x x C e C x x x--=-=-+=+⎰． 17211122d d 22arctan (1)12t tt tt t t π+∞∞+∞+===++⎰．18、解：12zf f y x∂''=+⋅∂； []21112221221112222(1)(1)()zf f x f y f f x f x y f xy f f x y∂''''''''''''''''=⋅-+⋅++⋅-+⋅=-+-⋅+⋅+∂∂．19、解：原式21100sin sin d d d d (1)sin d y y Dyy x y y x y y y y y ===-⎰⎰⎰⎰⎰ 1100(1)cos cos d 1sin1y y y y =--=-⎰．20、解：01111(2)()(1)24244414n n nn x f x x x ∞=-==⋅=--+-+∑)62(<<-x ． 21、证：00(sin )d ()[sin ()]d ()(sin )d t xx f x xt f t t t f t I t πππππππ=-=---=-⎰⎰⎰(sin )d (sin )d (sin )d f x x x f x x f x x I πππππ=-=-⎰⎰⎰解得： 0(sin )d (sin )d 2f x x f x x I x πππ==⎰⎰，原命题证毕．222000sin sin d d arctan (cos )1cos 21cos 24x x x x x x x x ππππππ⋅==-=++⎰⎰． 22、解：等式两边求导得()2()x f x x f x '=+，即()()2f x x f x x '-=-，且(0)1f =-，x p -=，x q 2-=，而2()d 2x x xe e --⎰=，由公式求得通解：222222()2d 2x x x f x e xq x C C e -⎡⎤⎛⎫=-+=+⎢⎥ ⎪ ⎪⎢⎥⎝⎭⎣⎦⎰，将初始条件(0)1f =-代入通解，解得：3-=C ，故22()23x f x e =-．23、解：设污水厂建在河岸离甲城x 公里处，则()500M x x =+500≤≤x ），由150070002M '=+⨯=解得：650050-=x （公里），唯一驻点，即为所求．2005年高等数学真题参考答案1、A ．2、C ．3、D ．4、A ．5、A ．6、C ．7、2． 8、1-e ． 9、2π． 10、5． 11、11d (,)d y y f x y x -⎰⎰．12、(1,1)-．13、解：因为)(x F 在0=x 处连续，所以)0()(lim 0F x F x =→，'00()2sin ()(0)lim ()limlim 2(0)28x x x f x x f x f F x f x x→→→+-==+=+=，解得：a F =)0(，故8=a ．14、解：d d cos cos sin d d d sin d yy t t t t t t x x t t-+===--，22d ()csc d (cos )y t t x t '-=='．15、解：原式22tan tan sec d (sec1)d(sec )x x x xx x =⋅-⎰⎰积进去231sec d(sec )d(sec )sec sec 3x x x x x C =-=-+⎰⎰．16、解：原式211120002d 1d(1)arctan 1421x x x x x x x π+=--++⎰⎰积进去 ()12011ln 1ln 24242x ππ⎡⎤=-+=-⎣⎦．17、解：1cos zx f x∂'=⋅∂，()21212cos 22cos z x f y y x f x y ∂''''=⋅⋅=⋅∂∂． 18、解：直线L 的方向向量{}5,2,1=s ，过点()4,3,0B -，{}1,4,2AB =-；所求平面的法向量{}5218,9,22142AB =⨯==---ij kn s ，点法式为8(3)9(1)22(2)0x y z ----+=，即592298=--z y x ．19、解：2222101111(1)()13216313212n nn n x x x x f x x x x x x ∞+=⎡⎤-⎛⎫=+=⋅+⋅=+⋅ ⎪⎢⎥+--⎝⎭⎣⎦+∑，收敛域为：11<<-x ．20、解：1x e y y x x '+⋅=，即1p x=，x e q x =，而1d 1x x e x -⎰=；故通解为1d xx e e C y x x C x x x ⎛⎫+=+= ⎪⎝⎭⎰．把初始条件1x y e ==解得：0=C ；故所求特解为：xe y x=．21、证：令13)(3+-=x x x f ，[]1,1x ∈-，且(1)30f -=>，(1)10f =-<，(1)(1)0f f -⋅<；由连续函数零点定理知：)(x f 在(1,1)-内至少有一实根；对于()1,1x ∈-恒有()22()33310f x x x '=-=-<，即)(x f 在(1,1)-内单调递减，故方程0133=+-x x 在[]1,1-上有且仅有一根；原命题获证．22、解：设所求函数为)(x f y =，则有4)2(=f ，(2)3f '=-，(2)0f ''=；由()6f x x a ''=+和(2)0f ''=解得：12-=a ，即()612f x x ''=-，故21()312f x x x C '=-+，由(2)3f '=-解得：91=C ，故22396C x x x y ++-=，由(2)4f =解得：22=C ；所求函数为：29623++-=x x x y ．23、解：（1）112300111d 266S y y y ===⎰；（如图1所示）（2）()()112222012d 4x V x x x xπππ=-=-=⎰．24、解：积分区域D 为：u y ≤≤1，u x y ≤≤；（1）111()()d d ()d (1)()d u xuDF u f x x f x y x f x x σ===-⎰⎰⎰⎰⎰；（2）()(1)()F u u f u '=-，(2)(21)(2)(2)1F f f '=-==．2006年高等数学真题参考答案1、C ．2、B ．3、C ．4、C ．5、C ．6、A ．7、2． 8、)(0x f ． 9、1-． 10、1． 11、(sin cos )xye y x x +． 12、1．13、解：原式322131lim 21341==--→x xx ．图114、解：2211d 12d 21t t y y t t t x x t-'+==='+，2222d 1d d 122d 41ty x y t t x x t t '⎛⎫ ⎪+⎝⎭==='+． 15、解：原式322ln )(1ln )3x x C =+=++．16、解：原式()2222220d(sin )sin 2sin d x x x xx x πππ=-⎰⎰积进去222220sin 2sin d 2d(cos )4x xx x xx x ππππ-+⎰⎰积进去导出来2222002cos 2cos d 244x x x x ππππ=+-=-⎰．17、解：方程变形为2y y y x x ⎛⎫'=- ⎪⎝⎭，即得到了形如d d y y f x x ⎛⎫= ⎪⎝⎭齐次方程；令y u x =，则d d d d y u u x x x =+，代入得：2d d u x u x =-，分离变量得：211d d u x u x-=；两边积分，得：211d d u x u x -=⎰⎰，1ln x C u=+，故ln x y x C =+． 18、解：令()ln (1)g x x =+，则(0)0g =；由于01()(1)1n n n g x x x ∞='==-+∑（(]1,1x ∈-），所以01(1)((1))d x n n n g x n x g t t ∞+='=+=-∑⎰（(]1,1x ∈-），故20(1)()1n n n f x x n ∞+=-=+∑，收敛域为：11x -<≤．19、解：由题意知：{}11,1,1=-n ，{}24,3,1=-n ；{}12311232,3,1431=⨯=-=++=-i j ks n n i j k ，故所求直线方程的对称式方程为：123123+=-=-z y x ．20、解：22z x f x∂'=∂，2'2'''''3''2''22122221222(2)22z x f x f x f y x f x f x y f y x ∂=+⋅+⋅=++∂∂． 21、证：令33)(x x x f -=，[]2,2x ∈-，由2()330f x x '=-=解得驻点：1±=x ，比较以下函数值的大小：(1)2f -=-，(1)2f =，(2)2f =-，(2)2f -=；所以2min -=f ，2m ax =f ，故2)(2≤≤-x f ，即332x x -≤，原命题获证．22、解：0)0(=y ，2y x y '=+，通解为：xCe x y +--=)22(；将0)0(=y 代入通解解得：2=C ，故所求特解为：xe x y 222+--=．23、解：（1）()2222648d 3S x x x -=--=⎰；（2）224804d d 16y V y y πππ=+=⎰⎰．24、解：()d d d ()d ()d tt tt D f x x y x f x y t f x x ==⎰⎰⎰⎰⎰，0()d 0()0t f x x t g t a t ⎧≠⎪=⎨⎪=⎩⎰；（1）00lim ()lim()d 0t t t g t f x x →→==⎰，由)(t g 的连续性可知：0)(lim )0(0===→t g g a t ；（2）当0≠t 时，()()g t f t '=，当0=t 时，0000()d ()(0)(0)limlim lim ()(0)hh h h f x x g h g g f h f h h→→→-'====⎰；综上，()()g t f t '=．2007年高等数学真题参考答案1、B ．2、C ．3、C ．4、A ．5、D ．6、D ．7、2ln ． 8、1． 9、π2． 10、23． 11、21d d xx y y y-． 12、06'5''=+-y y y ． 13、解：212lim 21lim 1lim tan 1lim00200==-=--=--→→→→x x x x x x x x e x e x x e x x x e ． 14、解：当0=x 时，0=y ；。

2010年江苏专转本高等数学真题(附答案)

2010年江苏专转本高等数学真题(附答案)2010年江苏省普通高校“专转本”统一考试高等数学一、单项选择题（本大题共6小题，每小题4分，满分24分）1.设当0x →时，函数()sin f x x x =-与()ng x ax =是等价无穷小，则常数,a n 的值为 ( )A. 1,36a n ==B. 1,33a n ==C. 1,412a n == D. 1,46a n == 2.曲线223456x x y x x -+=-+的渐近线共有( )A. 1条B. 2条C. 3条D. 4条 3.设函数22()cos t xx e tdtΦ=⎰，则函数()x Φ的导数()x 'Φ等于( ) A.222cos x xe x B.222cos x xe x - C. 2cos xxex-D. 22cos x e x - 4.下列级数收敛的是( ) A. 11n n n ∞=+∑ B.2121n n n n∞=++∑ C.1n n n ∞=D.212n n n ∞=∑5.二次积分111(,)y dy f x y dx+⎰⎰交换积分次序后得( ) A. 1101(,)x dx f x y dy+⎰⎰B.211(,)x dx f x y dy-⎰⎰C. 2111(,)x dx f x y dy-⎰⎰D.2111(,)x dx f x y dy-⎰⎰6.设3()3f x x x=-，则在区间(0,1)内( )A. 函数()f x 单调增加且其图形是凹的B. 函数()f x 单调增加且其图形是凸的C. 函数()f x 单调减少且其图形是凹的D. 函数()f x 单调减少且其图形是凸的二、填空题（本大题共6小题，每小题4分，满分24分）7. 1lim()1xx x x →∞+=-8. 若(0)1f '=，则0()()lim x f x f x x →--=9. 定积分312111x dxx -++⎰的值为10. 设(1,2,3),(2,5,)a b k ==，若a 与b 垂直，则常数k = 11. 设函数24z x y=+，则10x y dz===12. 幂级数0(1)n nn x n ∞=-∑的收敛域为三、计算题（本大题共8小题，每小题8分，满分64分）13、求极限211lim()tanx x x x→- 14、设函数()y y x =由方程2x yy e x++=所确定，求22,dy d ydx dx15、求不定积分arctan x xdx ⎰ 16、计算定积分4021dx x +⎰17、求通过点(1,1,1)，且与直线23253x t y t z t =+⎧⎪=+⎨⎪=+⎩垂直，又与平面250x z --=平行的直线的方程。

2010年江苏专转本(高等数学)真题试卷(题后含答案及解析)

2010年江苏专转本（高等数学）真题试卷(题后含答案及解析)题型有：1. 选择题 2. 填空题 4. 解答题 5. 综合题 6. 证明题选择题在每小题给出的四个选项中，只有一项是符合要求的。

1．设当x→0时，函数f(x)=x-sinx与g(x)=axn是等价无穷小，则常数a，n 的值为( )A．a=，n=3B．a=，n=3C．a=，n=4D．a=，n=4正确答案：A解析：由题意，n=3．2．曲线的渐近线共有( )A．1条B．2条C．3条D．4条正确答案：C解析：3．设函数ψ(x)=∫x22etcostdt，则函数ψ(x)的导数ψ’(x)等于( ) A．2xex2 cos x2B．-2xex2cosx2C．-2xexcosxD．-ex2cosx2正确答案：B解析：φ’(x)=(-∫2x2etcostdt)’=-ex2cosx2.2x=-2xex2cosx24．下列级数收敛的是( )A．B．C．D．正确答案：D解析：选项D通项公式为分子是幂函数，分母是指数函数，因为指数函数的增长速度高于幂函数的增长速度，而幂函数又大于对数函数的增长速度，故D是一个收敛级数，其余都发散．5．二次积分∫01dy∫1y+1f(x，y)如交换积分次序后得( )A．∫01dx∫1x+1f(x，y)dyB．∫12dx∫0x-1f(x，y)dyC．∫12dx∫1x-1f(x，y)dyD．∫12dx∫x-11f(x，y)dy正确答案：D解析：画出积分区域，若先对y后对x积分，则积分变为∫01dy∫1y+1f(x，y)dx=∫12dx∫x-11f(x，y)dy．6．设f(x)=x3-3x，则在区间(0，1)内( )A．函数f(x)单调增加且其图形是凹的B．函数f(x)单调增加且其图形是凸的C．函数f(x)单调减少且其图形是凹的D．函数f(x)单调减少且其图形是凸的正确答案：C解析：利用导数性质，当x∈(0，1)时，有f’(x)=3x2-3＜0，f”(x)=6x＞0，故在区间(0，1)内，函数f(x)单调减少且其图形是凹的．填空题7．_______．正确答案：e2解析：这是“12”型未定式，根据两个重要极限，8．若f’(0)=1，则_______．正确答案：2解析：由已知，根据导数定义，9．定积分的值为_____．正确答案：解析：由定积分的对称性质，10．设a=(1，2，3)，b=(2，5，k)，若a与b垂直，则常数k=______．正确答案：-4解析：由题意a.b=1×2+2×5+3×k=0，解得k=-4．11．设函数=______．正确答案：dx+2dy解析：12．幂级数的收敛域为______．正确答案：(-1，1]解析：因为，收敛半径为R==1,当x=1时，收敛(莱布尼兹级数)，当x=-1时，发散(调和级数)，故收敛域为(-1，1]．解答题解答时应写出推理、演算步骤。

江苏省专转本高等数学试卷

江苏省2010年普通高校专转本选拔统一考试数学试题一、单项选择题（本大题共6小题，每小题4分，满分24分）1.设当0x →时，函数()sin f x x x =-与()n g x ax =是等价无穷小，则常数,a n 的值为( ) A. 1,36a n == B. 1,33a n == C. 1,412a n == D. 1,46a n == 2.曲线223456x x y x x -+=-+的渐近线共有( ) A. 1条 B. 2条 C. 3条 D. 4条3.设函数22()cos t x x e tdt Φ=⎰，则函数()x Φ的导数()x 'Φ等于( ) A. 222cos x xe x B. 222cos x xe x - C. 2cos x xe x - D. 22cos x e x -4.下列级数收敛的是( ) A. 11n n n ∞=+∑ B. 2121n n n n ∞=++∑C. 1n n ∞=D. 212n n n ∞=∑ 5.二次积分1101(,)y dy f x y dx +⎰⎰交换积分次序后得( )A.1101(,)x dx f x y dy +⎰⎰ B. 2110(,)x dx f x y dy -⎰⎰ C. 2111(,)x dx f x y dy -⎰⎰ D. 2111(,)x dx f x y dy -⎰⎰ 6.设3()3f x x x =-，则在区间(0,1)内( )A. 函数()f x 单调增加且其图形是凹的B. 函数()f x 单调增加且其图形是凸的C. 函数()f x 单调减少且其图形是凹的D. 函数()f x 单调减少且其图形是凸的二、填空题（本大题共6小题，每小题4分，满分24分） 7. 1lim()1x x x x →∞+=- 8. 若(0)1f '=，则0()()lim x f x f x x→--= 9. 定积分312111x dx x -++⎰的值为 10. 设(1,2,3),(2,5,)a b k ==r r ，若a r 与b r 垂直，则常数k =绝密★启用前11.设函数z =10x y dz=== 12. 幂级数0(1)nn n x n ∞=-∑的收敛域为三、计算题（本大题共8小题，每小题8分，满分64分）13、求极限2011lim()tan x x x x→- 14、设函数()y y x =由方程2x y y ex ++=所确定，求22,dy d y dx dx15、求不定积分arctan x xdx ⎰16、计算定积分40⎰ 17、求通过点(1,1,1)，且与直线23253x t y t z t =+⎧⎪=+⎨⎪=+⎩垂直，又与平面250x z --=平行的直线的方程。

2010年江苏专转本高等数学真题（附答案）

2010年江苏专转本⾼等数学真题（附答案）2010年江苏省普通⾼校“专转本”统⼀考试⾼等数学⼀、单项选择题（本⼤题共6⼩题，每⼩题4分，满分24分）1.设当0x →时，函数()sin f x x x =-与()n g x ax =是等价⽆穷⼩，则常数,a n 的值为 ( ) A. 1,36a n = = B. 1,33a n == C. 1,412a n == D. 1,46a n == 2.曲线223456x x y x x -+=-+的渐近线共有( )A. 1条B. 2条C. 3条D. 4条 3.设函数22()c o s txx e t d tΦ=?，则函数()x Φ的导数()x 'Φ等于 ( )A. 222cos x xe x B. 222cos x xe x - C. 2cos xxe x - D. 22cos x e x -4.下列级数收敛的是( )A. 11n n n ∞=+∑ B. 2121n n n n ∞=++∑C. 1n n ∞= D. 212n n n ∞=∑5.⼆次积分111(,)y dy f x y dx+??交换积分次序后得( ) A. 1101(,)x dx f x y dy +?? B. 2110(,)x dx f x y dy -?? C. 2111(,)x dx f x y dy -?D. 2111(,)x dx f x y dy -??6.设3()3f x x x=-，则在区间(0,1)( )A. 函数()f x 单调增加且其图形是凹的B. 函数()f x 单调增加且其图形是凸的C. 函数()f x 单调减少且其图形是凹的D. 函数()f x 单调减少且其图形是凸的⼆、填空题（本⼤题共6⼩题，每⼩题4分，满分24分） 7. 1lim( )1xx x x →∞+=- 8. 若(0)1f '=，则0()()limx f x f x x→--=9. 定积分211dx x -+?的值为 10. 设(1,2,3),(2,5,)a b k ==，若a 与b 垂直，则常数k =11.设函数lnz =10x y dz===12. 幂级数0(1)n nn x n ∞=-∑的收敛域为三、计算题（本⼤题共8⼩题，每⼩题8分，满分64分） 13、求极限2011lim()tan x x x x→-14、设函数()y y x =由⽅程2x yy e x ++=所确定，求22,dy d y15、求不定积分arctan x xdx ?16、计算定积分417、求通过点(1,1,1)，且与直线23253x t y t z t =+??=+??=+?垂直，⼜与平⾯250x z --=平⾏的直线的⽅程。

2010年江苏省普通高校专转本统一考试

2010年江苏省普通高校专转本统一考试我省2010年“专转本”统一考试于2010年4月24日举行。

考试时间具体如下：2010年江苏省普通高校“专转本”统一考试大学英语本试卷分第I卷（客观题）和第II卷（主观题）两部分。

满分150分。

考试时间120分钟。

卷中未注明做大对象的试题为英语类和非英语类学生共同作答的试题，注明作答对象的试题按规定作答。

第Ⅰ卷Part ⅠReading Comprehension (共40分)Directions: There are 4 passages in this part. Each passage is followed by some questions or unfinished statements. For each of them there are 4 choices marked A, B, C and D. You should decide on the best choice and mark the corresponding letter on the Answer Sheet with a single line through the center.Passage OneQuestions 1 to 5 are based on the following passage.Sometimes you’ll hear people say that you can’t love others until you love yourself. Sometimes you’ll hear people say that you can’t expect someone else to love you until you love yourself. Either way, you’ve got to love yourself first and this can be tricky. Sure we all know that we’re the people of our parents’eyes, and that our Grandmas think we are great talents and our uncle Roberts think that we will go to the Olympics, but sometimes it’s a lot harder to think such nice thoughts about ourselves. If you find that believing in yourself is a challenge, it is time you built a positive self-image and learn to love yourself.Self-image is your own mind’s picture of yourself. This image includes the way you look, the way you act, the way you talk and the way you think. Interestingly, our self-images are often quite different from the images others hold about us. Unfortunately, most of these images are more negative than they should be, thus changing the way you think about yourself is the key to changing your self-image and your whole world.The best way to defeat a passive self-image is to step back and decide to stress your successes. That is, make a list if you need to, but write down all of the great things you do everyday. Don’t allow doubts to occur in it.It very well might be that you are experiencing a negative self-image because you can’t move past one flaw or weakness that you see about yourself. Well, roll up your sleeves and make a change of it as your primary task. If you think you are silly because you aren’t good at math. Find a tutor. If you think you are weak because you can’t run a mile, get to the track and practice. If you think you are dull because you don’t wear the latest trends, buy a few new clothes.The best way to get rid of a negative self-image is to realize that your image is far from objective and to actively convince yourself of your positive qualities. Changing the way you think and working on those you need to improve will go a long way towards promoting a positive self-image. When you can pat your self on the back, you’ll know you’re well on your way.1. You need to build a positive self-image when you _______.A．dare to challenge yourself B. feel it hard to change yourselfC. are unconfident about yourselfD. have a high opinion of yourself2. According to the passage, our self-images________.A. have positive effectsB. are probably untrueC. are often changeableD. have different functions3. How should you change your self-image according to the passage?A. Keep a different image of others.B. Make your life successfulC. Understand your own worldD. Change the way you think4. What is the passage mainly about?A. How to prepare for your successB. How to face challenges in your lifeC. How to build a positive self-imageD. How to develop your good qualities5. Who are the intended readers of the passage?A. parentsB. AdolescentsC. educatorsD. people in general Passage 2Questions 6 to 10 are based on the following passage.Do you want to live with a strong sense of peacefulness, happiness, goodness, and self-respect? The collection of happiness actions broadly categorized as “honor”helps you create this life of good feelings.Here’s an example to show how honorable actions create happiness.Say a store clerk fails to charge us for an item. If we keep silent and profit from the clerk’s mistake, we would drive home with a sense of sneaky excitement. Later we might tell our family or friends about our good fortune. On the other hand, if we tell the clerk about the uncharged item, the clerk would be grateful and thank us for ourhonesty. We would leave the store with a quite sense of honor that we might never share with another soul.Then, what is it to do with our sense of happiness?In the first case, where we don’t tell the clerk, a couple of things would happen. Deep down inside we would know ourselves as a type of thief. In the process, we would lose some peace of mind and self-respect. We would also demonstrate that we cannot be trusted, since we advertise our dishonor by telling our family and friends. We damage our own reputations by telling others. In contrast, bringing the error to the clerk’s attention causes different things to happen. Immediately, the clerk knows us to be honorable. Upon leaving the store, we feel honorable and our internal rewards of goodness and a sense of nobility.There is a beautiful positive cycle that is created by living a life of honorable actions. Honorable thoughts lead to honorable actions. Honorable actions lead us to a happier experience. And it is easy to think and act honorably again when we are happy. While the positive cycle can be difficult to start, once it’s started, it’s easy to continue. Keeping on doing good deeds brings us peace of mind which is important for our happiness.6. According to the passage, the positive action in the example contributes to our ___________.A. self-respectB. financial rewardsC. advertising abilityD. friendly relationship7. The author thinks that keeping silent about the uncharged item is equal to _____.A. lyingB. stealingC. cheatingD. advertising8. The phrase “bringing the error to the clerk’s attention” means_______.A. telling the truth to the clerkB. offering advice to the clerkC. asking the clerk to be more attentiveD. reminding the clerk of the charged item9. How will we feel if we let the clerk know the mistake?A. We’ll be very excitedB. We’ll feel unfortunateC. We’ll have a sense of humorD. We’ll feel sorry for the clerk10. Which of the following can be the best title of this passage?A. How to live truthfullyB. Importance of peacefulnessC. Ways of gaining self-respectD. Happiness through honorablePassage 3Questions 11 to 15 are based on the following passage.Sesame Street has been called “the longest street in the world”. That is because the television program by that name can now be seen in so many parts of the world. That program became one of America’s exports soon after it went on the air in New York in 1969.In the United States more than 6 million children watch the program regularly. The viewers include more than half of the nation’s pre-school children. Although some educators object to certain elements in the program, parents praise it highly. Many teachers also consider it a great help, though some teachers find thatproblems arise when first graders who have learned from “Sesame Street” are in the same class with children who have not watched the program.The program uses songs, stories, jokes and pictures to give children a basic understanding of numbers, letters and human relationships. Tests have shown that children have benefited from watching “Sesame Street”. Those who watch five times a week learn more than the occasional viewers. In the United States the program is shown at different times during the week in order to increase the number of children who can watch it regularly.Why has “Sesame Street”been so much more successful than other children’s shows? Many reasons have been suggested. People mention the educational theories of its creators, the support by the government and private businesses, and the skillful use of a variety of TV tricks. Perhaps an equally important reason is that mothers watch “Sesame Street”along with their children. This is partly because famous adult stars often appear on “Sesame Street”. But the best reason for the success of the program may be that it makes every child watching it feel able to learn. The child finds himself learning, and he wants to learn more.11. Why has Sesame Street been called “the longest street in the world”?A. the program has been shown ever since 1969.B. the program became one of America’s major exports soon after it appeared on TVC. the program is now being watched in most parts of the world.D. the program is made in the longest street in New York.12. Some educators are critical of the program because________.A. they don’t think it fit for children in every respectB. it takes the children too much time to watch itC. it causes problems between children who watched it and those who have notD. some parents attach too much importance to it13. So many children in the United States watch the program because______.A. it uses songs, stories and jokes to give them basic knowledgeB. it is arranged for most children to watch it regularlyC. tests have shown that it is beneficial to themD. both A and B14. Mothers often watch the program along with their children because______A. they enjoy the program as much as their childrenB. they find their children have benefited from watching the programC. they are attracted by some famous adult stars on the showD. they can learn some educational theories from the program15. What is the most important reason for the success of the program according to the author?A. the creators have good educational theories in making the programB. the young viewers find they can learn something from it.C. famous adult stars often appear in the programD. It gets support from the government and private businessPassage 4Questions 16 to 20 are based on the following passage.You will have no difficulty in making contact with the agent. As you enter his office, you will be greeted immediately and politely asked what you are looking for. The Estate Agent’s negotiator-as he is called-will probably check that you really know your financial position. No harm in that, but you can always tell him that you have confirmed the position with the XYZ Building Society. He will accept that.He will show you the details of a whole range of properties; many of them are not really what you are looking for at all. That does not matter. Far better turn then down than risk missing the right one.The printed details he will give you are called “particulars”. Over the years, a whole language has grown up, solely for use in Agent’s particulars. It is flowery, ornate and, providing you read it carefully and discount the adjectives, it can be very accurate and helpful.Since the passing of the Trades Description Act, any trader trying to sell something has had to be very careful as to what they say about it. Estate Agents have, by now, become very competent at going as far as they dare. For instance, it is quite acceptable to say “delightfully”situated. That is an expression of his opinion. You many not agree, but he might like the idea of living next to the gasworks. If, on the other hand, he says that the house has five bedrooms when, in fact, it has only two, that is a misstatement of fact and is an offence. This has made Estate Agents and others for that matter rather more careful.Basically, all that you need to know about a house is : how many bedrooms it has; an indication of their size; whether the house has a garage; whether there is a garden and whether it is at the back or the front of the house; whether its semi-detached of terraced.16. The Estate Agent’s negotiator will ________.A. want all details of your financial circumstancesB. want to satisfy himself that you understand the financial implications of buying a house.C. check your financial position with the XYZ Building SocietyD. accept any statement you make about your financial position17. The author believes ________.A. it is better to be given information about too many properties than too fewB. you should only look at details of properties of the kind you have decided to buy.C. the agent will only show you the details of properties you have in mind.D. it doesn’t matter if you miss a few properties you may be interested in.18. The adjectives in Agent’s particulars are ______.A. accurateB. helpfulC. both accurateD. safe to ignore19. The Trades Description Act applies to ___________.A. house agents onlyB. most estate agentsC. any traderD. buyer of houses20. The phrase “going as far as they dare” probably means______.A. covering as wide an area as possibleB. selling houses as far from the estate agent’s housesC. telling lies about properties if nobody is likely to find out about it.D. trying every possible means to make the description of houses sound attractive Part ⅡVocabulary and StructureDirections: In this part there are forty incomplete sentences. Each sentence is followed by four choices. Choose the one that best completes the sentence and then mark your answer on the Answer Sheet.21. Scientists will have to _____ new methods of increasing the world’s food supply in order to solve the problem of famine in some places.A. come up forB. come down withC. com down toD. come up with 22．I’d like to rent a house, modern, comfortable and ______in a quiet and safe neighborhood.A. all in allB. above allC. after allD. over all23. In order to expose corruption, they have decided to set up a special team to ____the company’s accounts.A. search forB. work outC. look intoD. sum up24. Some people would like to do shopping on Sundays since they expect to pick up a lot of wonderful _____ in the markets.A. batteriesB. basketsC. barrelsD. bargains25. The residents living in these apartments have free ____ to the swimming pool, the gym and other facilities.A. excessB. excursionC. accessD. recreation 26．Reporters and photographers alike took great _____at the rude way the actors behaved during the interview.A. annoyanceB. offenceC. resentmentD. irritation27. Nothing healthful should be omitted from the meal of a child because of a ___ dislike.A. provedB. supposedC. consideredD. related28. I’d like to ____ this old car for a new model but I am afraid I cannot afford it.A. exchangeB. convertedC. replaceD. substitute29. She said she liked dancing but was not in the ______for it just then when it was so noisy in the hall.A. mannerB. intentionC. moodD. desire30. He was _____ admittance to the concert hall for not being properly dressed.A. rejectedB. deniedC. withheldD. deprived31. Most people tend to ______ a pounding heart and sweating palms with the experience of emotion.A. classifyB. identifyC. satisfyD. modify32. The company _____ many fine promises to the engineer in order to get him to work for them.A. held upB. held onC. held outD. held onto33. _____the tragic news about their president, they have cancelled the 4th of July celebration.A. In the course ofB. In spite ofC. In the event ofD. In the light of34. The worker’ demands are _______, they are asking for only a small increase in their wages.A. particularB. moderateC. intermediateD. numerous35. Our explanation seemed only to have _____ his confusion. He was totally at a loss as to what to say.A. brought upB. added toC. worked outD. directed at36. With a standard bulb, only 5% of the electricity is ______ to light-the rest is wasted away as heat.A. compressedB. conformedC. convertedD. confined37. Modern technology has placed ______ every kind of music, from virtually every period in history and every corner of the globe.A. at our requestB. at our disposalC. in our presenceD. in our sight38. Much has been written on the virtues of natural childbirth, but little research has been done to ___ these virtues.A. confirmB. consultC. confessD. convey39. Plastic bags are useful for holding many kinds of food______their cleaness, toughness and low cost.A. by virtue ofB. at sight ofC. by means ofD. by way of40. I am trying to think of his mind, but my mind goes completely _____i must be slipping.A. bareB. blankC. hollowD. vacant41. _____ in the regulations that you should not tell other people the password of your email account.A. what is requiredB. what requiresC. It is requiredD. It requires42. It is generally believed that gardening is ______it is a science.A. an art much asB. much an art asC. as an art much asD. as much an art as43. The indoor swimming pool seems to be great more luxurious than ______.A. is necessaryB. being necessaryC. to be necessaryD. it is necessary44. He ______ English for 8 years by the time he graduates from the university next year.A. will learnB. will be learningC. will have learntD. will have been learnt.45. It was not until the sub prime loan crisis ______ great damage to the American financial system that Americans ______the severity of the situation.A. caused; realizedB. had caused; realizedC. caused; had realizedD. was causing; had realized46. The police think you brother John stole the diamond in the museum yesterday evening.Oh? But he stayed with me at home the whole evening; he ___ the museum.A. must have been toB. needn’t have been toC. should have been toD. couldn’t have been to47. _______ the meeting himself gave his supporters a great deal of encouragement.A. The president will attendB. The president to attendC. The president attendedD. The president’s attending48. Everything _____into consideration, the candidates ought to have anotherchance.A. is takenB. takenC. to be takenD. taking49. _______from heart trouble for years, Professor White has to take some medicine with him wherever he goes.A. SufferedB. SufferingC. Having sufferedD. Being suffered50. The concert will be broadcasted live to a worldwide television audience_____ thousand millionA. estimatingB. estimatedC. estimatesD. having estimated51. About half of the students expected there ______more reviewing classes before the final exams.A. isB. beingC. to beD. have been52. _____ made the school proud was____more than 90% of the students had been admitted to key universities.A. What; becauseB. What; thatC. That; whatD. That; because53. Information has been released ______ more middle school graduates will be admitted into universities this year.A. whileB. thatC. whenD. as54. He is the only one of the students who ______ a winner of scholarship for three years.A. isB. areC. have beenD. has been55. What’s that newly-built building?______ the students have out-of-class activities, such as drawing and singing. A. It is the building that B. That’s whereC. It is in whichD. The building that56.With a large amount of work_____ the chief manager couldn’t spare time for a holiday.A. remained to be doneB. remaining to doC. remained to doD. remaining to be done57. Why! I have nothing to confess, ______you want me to say?A. What is it thatB. what it is thatC. How is it thatD. How is it that58. David has made great progress recently.__________, and __________________.A. So he has, so you haveB. So he has; so have youC. So has he; so you haveD. So has he; so you have59. It is universally known that microscopes make small things appear larger than _____.A. really areB. are reallyC. are they reallyD. they really are60. Tom, _______, but your TV is going too loud.Oh, I’m sorry. I will turn it down right now.A. I’d like to talk with youB. I’m really tired of thisC. I hate to say thisD. I need your helpPart ⅢClozeDirections: There are 20 blanks in the following passages. For each bank there are 4 choices marked A, B, C and D. You chose the ONE that best fits into the passage, the corresponding letter on the Answer Sheet with a single line through the center.The eyes are the most important 61 of human body that is used to 62 information. Eye contact is crucial for establishing rapport 63 others. The way we look at other people can 64 them know we are paying attention to 65 they are saying. We can also look at a person and gave the 66 we are not hearing a word. Probably all of us have been 67 of looking directly at someone and 68 hearing a word while he or she was talking 69 we were thinking about something totally 70 to what was being said.Eye contact allows you to 71 up visual clues about the other person; 72, the other person can pick up clues about you. Studies of the use of eye contact 73 communication indicate that we seek eye contact with others 74 we want to communicate with them. When we like them, when we are 75 toward them (as when two angry people 76 at each other). And when we want feedback from them. 77, we avoid eye contact when we want to 78 communication, when we dislike them, when we are 79 to deceiving them, and when we are 80 in what they have to say.61 A. unit B. part C. link D. section62 A. transfer B. translate C. transmit D. transport63 A. against B. with C. for D. to64 A. forbid B. allow C. permit D. let65 A. how B. which C. what D. that66 A. impression B. expression C. suggestion D. attention67 A. ignorant B. careless C. guilty D. innocent68 A. nor B. so C. not D. neither69 A. or B. unless C. why D. because70 A. related B. relevant C. unrelated D. indifferent71 A. tear B. pick C. size D. take72 A. likewise B. moreover C. otherwise D. therefore73 A. in B. about C. with D. of74 A. why B. where C. when D. what75 A. friendly B. hostile C. respectful D. mistrustful76 A. glance B. glare C. gaze D. stare77 A. Exactly B. Generally C. Conversely D. Interestingly78 A. hold B. establish C. avoid D. direct79 A. wanting B. tending C. forcing D. trying80 A. informed B. unconcerned C. uninterested D. unheard第Ⅱ卷（共50分）Part Ⅳ TranslationSection ADirections: Translate the following sentences into Chinese you may refer to the corresponding passages in Part ONE.81. The best way to get rid of a negative self-image is to realize that your image is far from objective and to actively convince yourself of your positive qualities.82. There is a beautiful positive cycle that is created by living a life of honorable actions Honorable thoughts lead to honorable actions. Honorable actions lead us to a happier existence.83. While the positive cycle can be difficult to start, once it’s started, it’s easy to continue. Keeping on doing good deeds brings us peace of mind, which is important for your happiness.84. That is because television program by that name can not be seen in so many parts of the world. That program became one of American’s export soon after it went on the air in New York in 1969.85. For instance, it is quite acceptable to say “delightfully”situated. That is an expression of his opinion. You may not agree, but he might like the idea of living next to the gasworks.Section BDirections: translate the following sentences into English.86. 一个公司应该跟上市场的发展变化，这很重要。

2001—2010年江苏专转本高等数学真题附答案

2001—2010年江苏专转本高等数学真题附答案2009 一、单项选择题（本大题共6小题，每小题4分，满分24分）1、已知32lim 22=-++→x b ax x x ，则常数b a ,的取值分别为（） A 、2,1-=-=b a B 、0,2=-=b a C 、0,1=-=b a D 、1,2-=-=b a2、已知函数423)(22-+-=x x x x f ，则2=x 为)(x f 的 A 、跳跃间断点B 、可去间断点C 、无穷间断点D 、震荡间断点 3、设函数??>≤=0,1sin 0,0)(x x x x x f α在点0=x 处可导，则常数α的取值范围为（） A 、10<<α B 、10≤<α C 、1>α D 、1≥α4、曲线2)1(12-+=x x y 的渐近线的条数为（）A 、1B 、2C 、3D 、45、设)13ln()(+=x x F 是函数)(x f 的一个原函数，则=+?dx x f )12(' （） A 、C x ++461 B 、C x ++463 C 、C x ++8121 D 、C x ++8123 6、设α为非零常数，则数项级数∑∞=+12n n n α（）A 、条件收敛B 、绝对收敛C 、发散D 、敛散性与α有关二、填空题（本大题共6小题，每小题4分，满分24分）7、已知2)(lim =-∞→x x Cx x ，则常数=C . 8、设函数dt te x x t ?=20)(?，则)('x ?＝ . 9、已知向量)1,0,1(-=→a，)1,2,1(-=→b ，则→→+b a 与→a 的夹角为 . 10、设函数),(y x z z =由方程12=+yz xz 所确定，则x z ??＝ . 11、若幂函数)0(12>∑∞=a x na n n n 的收敛半径为21，则常数=a .12、微分方程0)2()1(2=--+xdy y ydx x 的通解为 .三、计算题（本大题共8小题，每小题8分，满分64分）13、求极限：xx x x sin lim 30-→14、设函数)(x y y =由参数方程-+=+=32)1ln(2t t y t x 所确定，，求22,dx y d dx dy .15、求不定积分：+dx x 12sin .16、求定积分：?-10222dx x x .17、求通过直线12213-=-=z y x 且垂直于平面02=+++z y x 的平面方程.18、计算二重积分Dyd σ，其中}2,2,20),{(22≥+≤≤≤≤=y x y x x y x D . 19、设函数),(sin xy x f z =，其中)(x f 具有二阶连续偏导数，求yx z 2.20、求微分方程x y y =-''的通解.四、综合题（本大题共2小题，每小题10分，满分20分）21、已知函数13)(3+-=x x x f ，试求：（1）函数)(x f 的单调区间与极值；（2）曲线)(x f y =的凹凸区间与拐点；（3）函数)(x f 在闭区间]3,2[-上的最大值与最小值.22、设1D 是由抛物线22x y =和直线0,==y a x 所围成的平面区域，2D 是由抛物线22x y =和直线2,==x a x 及0=y 所围成的平面区域，其中20<（2）求常数a 的值，使得1D 的面积与2D 的面积相等.五、证明题（本大题共2小题，每小题9分，满分18分）23、已知函数≥+<=-0,10,)(x x x e x f x ，证明函数)(x f 在点0=x 处连续但不可导. 24、证明：当21<<x 时，32ln 42-+>x x x x .。

2001—210年江苏专转本高等数学真题(附答案)

2001年江苏省普通高校“专转本”统一考试高等数学一、选择题（本大题共5小题，每小题3分，共15分）1、下列各极限正确的是（）A 、e xxx =+→)11(lim 0B 、e xx x =+∞→1)11(limC 、11sinlim =∞→x x x D 、11sin lim 0=→xx x2、不定积分=-⎰dx x211 （）A 、211x-B 、c x+-211C 、x arcsinD 、c x +arcsin3、若)()(x f x f -=，且在[)+∞,0内0)('>x f 、0)(''>x f ，则在)0,(-∞内必有（）A 、0)('<x f ,0)(''<x f B 、0)('<x f ,0)(''>x f C 、0)('>x f ,0)(''<x f D 、0)('>x f ,0)(''>x f4、=-⎰dx x 21 （）A 、0B 、2C 、－1D 、15、方程x y x 422=+在空间直角坐标系中表示（） A 、圆柱面B 、点C 、圆D 、旋转抛物面二、填空题（本大题共5小题，每小题3分，共15分）6、设⎩⎨⎧+==22tt y te x t ，则==0t dx dy7、0136'''=+-y y y 的通解为 8、交换积分次序=⎰⎰dy y x f dx x x220),(9、函数yx z =的全微分=dz10、设)(x f 为连续函数，则=+-+⎰-dx x x x f x f 311])()([三、计算题（本大题共10小题，每小题4分，共40分） 11、已知5cos )21ln(arctan π+++=xx y ，求dy .12、计算xx dte x xt x sin lim 202⎰-→.13、求)1(sin )1()(2--=x x xx x f 的间断点，并说明其类型.14、已知x y x y ln 2+=，求1,1==y x dxdy.15、计算dx ee xx⎰+12.16、已知⎰∞-=+02211dx x k ，求k 的值.17、求x x y y sec tan '=-满足00==x y 的特解. 18、计算⎰⎰Ddxdy y 2sin ，D 是1=x 、2=y 、1-=x y 围成的区域.19、已知)(x f y =过坐标原点，并且在原点处的切线平行于直线032=-+y x ，若b ax x f +=2'3)(，且)(x f 在1=x 处取得极值，试确定a 、b 的值，并求出)(x f y =的表达式.20、设),(2y x x f z =，其中f 具有二阶连续偏导数，求x z∂∂、yx z ∂∂∂2.四、综合题（本大题共4小题，第21小题10分，第22小题8分，第23、24小题各6分，共30分） 21、过)0,1(P 作抛物线2-=x y 的切线，求（1）切线方程；（2）由2-=x y ，切线及x 轴围成的平面图形面积；（3）该平面图形分别绕x 轴、y 轴旋转一周的体积。

2010年普通高等学校招生全国统一考试数学（江苏卷）（附答案-完全）

2010年普通高等学校招生全国统一考试（江苏卷）数学Ⅰ试题参考公式：锥体的体积公式：13V Sh 锥体，其中S 是锥体的底面面积，h 是高.一、填空题：本大题共14小题，每小题5分，共计70分.请把答案填写在答题卡相应位置上．．．．．．．．.1.设集合{1,1,3}A，4,22aa B ，3BA ，则实数a 的值为____▲____.1.【答案】1.【命题意图】本题考查交集的定义，对求得的集合中的元素要进行检验.【解析】由题意得1,32a a .又由342a不符合题意.经检验得1a.2.设复数z 满足(23)64z i i （i 为虚数单位），则z 的模为____▲____.2.【答案】2.【命题意图】本题考查复数有关运算及复数模的计算.【解析】由i i z 46)32(得,2)32)(32()32)(46(3246i i i i i ii z即2,2z i z.3.盒子里共有大小相同的3只白球，1只黑球.若从中随机摸出两只球，则它们颜色不同的概率是_ ▲__.3.【答案】21.【命题意图】本题考查古典概型知识.【解析】31.62p4.某棉纺厂为了解一批棉花的质量，从中随机抽测了100根棉花纤维的长度(棉花纤维的长度是棉花质量的重要指标).所得数据均在区间[5,40]中，其频率分布直方图如图所示，则在抽测的100根中，有_ ▲__ 根棉花纤维的长度小于20mm.4.【答案】30.【命题意图】本题考查概率统计中频率分布直方图的有关运用,注意纵坐标是频率/组距.【解析】由频率分布直方图得棉花纤维长度小于mm 20的根数为(0.01+0.01+0.04)301005.5.设函数()()xxf x x eae (x R )是偶函数，则实数a 的值为____▲____.5.【答案】1.【命题意图】本题考查函数的奇偶性.【解析】设R xae e x g x x ,)(,由题意分析)(x g 应为奇函数(奇函数奇函数=偶函数),又R x ，0)0(g ，则,01a 所以1a .6.在平面直角坐标系xOy 中，已知双曲线221412xy上一点M 的横坐标为3，则点M 到此双曲线的右焦点的距离为____▲____. 6.【答案】4.【命题意图】本题考查求曲线上点的坐标、双曲线的焦点坐标、两点间距离公式的运用.【解析】由题意得点15,3(M ),双曲线的右焦点的坐标为(4,0)，2MF 22)015()43(=4.或用第二定义：2MF ed，2d，4MF .7.右图是一个算法流程图，则输出的S 的值是____▲____.7.【答案】63.【命题意图】本题考查算法流程图,由流程图得出S 的关系式,比较得出S 的值.【解析】由流程图得12345122222S =1+2+48+16+32=6333,即.63S 8.函数2(0)y x x的图象在点2(,)k ka a 处的切线与x 轴的交点的横坐标为1k a ，其中kN *.若116a ，则123a a a 的值是____▲____.8.【答案】21.【命题意图】考查函数的切线方程、数列的通项.【解析】在点2(,)k k a a 处的切线方程为22(),k k k y a a xa 当0y时，解得2k a x，所以1135,1641212kka a a a a .9.在平面直角坐标系xOy 中，已知圆224x y上有且只有四个点到直线1250x y c 的距离为1，则实数c 的取值范围是____▲____. 9.【答案】(13,13).【命题意图】本题考查直线与圆的位置关系.【解析】如图,圆422yx的半径为2,圆上有且仅有四个点到直线的距离为1,问题转化为原点(0,0)到直线0512cyx 的距离小于1,即1313,13,151222c c c .10.设定义在区间(0,)2上的函数y=6cosx 的图象与y=5tanx 的图象交于点P ，过点P 作x 轴的垂线，垂足为P 1，直线PP 1与函数y=sinx 的图像交于点P 2，则线段P 1P 2的长为____▲____.10.【答案】.32【命题意图】本题考查三角函数问题,由图象相交,即三角函数值相等,建立关系式,求出,32sin x结合图象,数形结合分析P 1P 2的值.xyO 0512cyx1 11【解析】由题意得x x tan 5cos 6，即x x xx xsin 5cos 6,cos sin 5cos 62，226(1sin )5sin ,6sin 5sin 60x x x x 得,32sin x结合图象分析得32sin 21P P x.11.已知函数21,0,()1,0,xx f x x则满足不等式2(1)(2)f x f x 的x 的取值范围是____▲____.11.【答案】).12,1(【命题意图】本题考查分段函数的单调性.【解析】2212,10,x x x解得121x ，所以x 的取值范围是).12,1(12.设x,y 为实数，满足3≤2xy ≤8，4≤2xy≤9，则34xy 的最大值是____▲____. 12.【答案】27.【命题意图】考查不等式的基本性质，等价转化思想.【解析】22()[16,81]xy，2111[,]83xy，322421()[2,27]xxyyxy，43yx 的最大值是27.13.在锐角△ABC 中,角A 、B 、C 的对边分别为a 、b 、c.若6cos b a C ab,则tan tan tan tan C C AB的值是▲.【答案】4.【解析】考查三角函数知识，三角形中的正、余弦定理的应用，等价转化思想.（方法一）考虑已知条件和所求结论对于角A 、B 和边a 、b 具有轮换性.当A=B 或a=b 时满足题意，此时有1cos 3C，21cos 1tan21cos 2C C C ，2tan22C .等腰三角形中，1tan tan 2tan2A BC ，tan tan tan tan C C AB=4.（方法二）226cos 6cos b a C ab Cab ab，2222222236,22a bccabab abab.2tan tan sin cos sin sin cos sin sin()1sin tan tan cos sin sin cos sin sin cos sin sin C C C B A B A C A B CABCA B C A B C A B,由正弦定理，得上式22222214113cos ()662cc cc C abab .14.将边长为1m 的正三角形薄铁皮沿一条平行于某边的直线剪成两块，其中一块是梯形，记2(s梯形的周长）梯形的面积,则s 的最小值是____▲____.【答案】3233.【解析】考查函数中的建模应用，等价转化思想.设剪成的小正三角形的边长为x ，则222(3)4(3)(01)1133(1)(1)22x x sx xx x .（方法一）利用导数求函数最小值.224(3)()13x S x x，22224(26)(1)(3)(2)()(1)3x x x x S x x 2242(31)(3)(1)3x x x 1()0,01,3S x xx.当1(0,]3x时，()0,S x 递减；当1[,1)3x 时，()0,S x 递增.故当13x时，S 取最小值3233.（方法二）利用函数的方法求最小值.令1113,(2,3),(,)32x t t t ，则2224418668331tS tttt .故当131,83xt时，S 取最小值3233.二、解答题：本大题共6小题，共计90分.请在答题卡指定区域．．．．．．．内作答，解答时应写出文字说明、证明过程或演算步骤.15.（本小题满分14分）在平面直角坐标系xOy 中，已知点(1,2)A ，(2,3)B ，(2,1).C (1)求以线段AB 、AC 为邻边的平行四边形的两条对角线的长；(2)设实数t 满足(OC t AB)·OC =0，求t 的值.【解析】本小题主要考查平面向量的几何意义、线性运算、数量积，考查运算求解能力.满分14分.解：（1）由题设知(3,5)AB，(1,1)AC，则(2,6)A B A C ，(4,4).AB AC所以||210AB AC ，||4 2.ABAC 故所求的两条对角线长分别为42，210.（2）由题设知(2,1)OC，(32,5).ABtOCt t由()0AB tOC OC ，得(32,5)(2,1)0t t ，从而511t ，所以11.5t16.（本小题满分14分）如图，在四棱锥P-ABCD 中，PD ⊥平面ABCD ，PD=DC=BC=1，AB=2，AB ∥DC ，∠BCD=900.(1)求证：PC ⊥BC ；(2)求点A 到平面PBC 的距离.【解析】本小题主要考查直线与平面、平面与平面的位置关系，考查几何体的体积，考查空间想象能力、推理论证能力和运算能力.满分14分. 解：（1）因为PD ⊥平面ABCD ，BC 平面ABCD ，所以PD ⊥BC. 由∠BCD=900，得BC ⊥DC. 又PD DC D ，PD平面PCD ，DC平面PCD ，所以BC ⊥平面PCD.因为PC平面PCD ，所以PC ⊥BC.（2）（方法一）分别取AB 、PC 的中点E 、F ，连DE 、DF.则易证DE ∥CB ，DE ∥平面PBC ，点D 、E 到平面PBC 的距离相等. 由（1）知BC ⊥平面PCD ，所以平面PBC ⊥平面PCD.因为PD=DC ，PF=FC ，所以DF ⊥PC ，所以DF ⊥平面PBC 于F. 易知DF=22.又点A 到平面PBC 的距离等于E 到平面PBC 的距离的2倍，故点A 到平面PBC 的距离等于2.（方法二）连结AC.设点A 到平面PBC 的距离h.因为AB ∥DC ，∠BCD=900，所以∠ABC=900. 从而由AB=2，BC=1，得ABC 的面积1ABCS.由PD ⊥平面ABCD 及PD=1，得三棱锥PABC 的体积11.33ABCVSPD因为PD ⊥平面ABCD ，DC 平面ABCD ，所以PD ⊥DC.又PD=DC=1，所以222.PC PDDC由PC ⊥BC ，BC=1，得PBC 的面积2.2PBCS 由11213323PBCVShh ，得2h .因此，点A 到平面PBC 的距离为2.17.（本小题满分14分）某兴趣小组要测量电视塔AE 的高度H(单位：m).如示意图，垂直放置的标杆BC 高度h=4m ，仰角∠ABE=，∠ADE=.（1）该小组已测得一组，的值，算出了tan =1.24，tan =1.20，请据此算出H 的值；。

江苏专升本高等数学真题(附答案)

江苏专转本高数考纲及重点总结一、函数、极限和连续（一）函数（1）理解函数的概念：函数的定义，函数的表示法，分段函数。

（2）理解和把握函数的简单性质：单调性，奇偶性，有界性，周期性。

（3）了解反函数：反函数的定义，反函数的图象。

（4）把握函数的四则运算与复合运算。

（5）理解和把握基本初等函数：幂函数，指数函数，对数函数，三角函数，反三角函数。

（6）了解初等函数的概念。

会求函数在一点处的左极限与右极限，了解函数在一点处极限存在的充分必要条件。

（2）了解数列极限的性质：唯一性，有界性，四则运算定理，夹逼定理，单调有界数列，极限存在定理，把握极限的四则运算法则。

（3）理解函数极限的概念：函数在一点处极限的定义，左、右极限及其与极限的关系，x趋于无穷（x→∞，x→+∞，x→-∞）时函数的极限。

（4）把握函数极限的定理：唯一性定理，夹逼定理，四则运算定理。

（5）理解无穷小量和无穷大量：无穷小量与无穷大量的定义，无穷小量与无穷大量的关系，无穷小量与无穷大量的性质，两个无穷小量阶的比较。

（6）熟练把握用两个重要极限求极限的方法。

重点：会用左、右极限求解分段函数的极限，把握极限的四则运算法则、利用两个重要极限求极限以及利用等价无穷小求解极限。

（三）连续（1）理解函数连续的概念：函数在一点连续的定义，左连续和右连续，函数在一点连续的充分必要条件，函数的中断点及其分类。

（2）把握函数在一点处连续的性质：连续函数的四则运算，复合函数的连续性，反函数的连续性，会求函数的中断点及确定其类型。

（3）把握闭区间上连续函数的性质：有界性定理，最大值和最小值定理，介值定理（包括零点定理），会运用介值定理推证一些简单命题。

（4）理解初等函数在其定义区间上连续，并会利用连续性求极限。

重点：理解函数（左、右连续）性的概念，会判别函数的中断点。

江苏专升本高等数学真题(附答案)

江苏专转本高数考纲及重点总结一、函数、极限和连续（一）函数（1）理解函数的概念：函数的定义，函数的表示法，分段函数。

（2）理解和把握函数的简单性质：单调性，奇偶性，有界性，周期性。

（3）了解反函数：反函数的定义，反函数的图象。

（4）把握函数的四则运算与复合运算。

（5）理解和把握基本初等函数：幂函数，指数函数，对数函数，三角函数，反三角函数。

（6）了解初等函数的概念。

会求函数在一点处的左极限与右极限，了解函数在一点处极限存在的充分必要条件。

（2）了解数列极限的性质：唯一性，有界性，四则运算定理，夹逼定理，单调有界数列，极限存在定理，把握极限的四则运算法则。

（3）理解函数极限的概念：函数在一点处极限的定义，左、右极限及其与极限的关系，x趋于无穷（x→∞，x→+∞，x→-∞）时函数的极限。

（4）把握函数极限的定理：唯一性定理，夹逼定理，四则运算定理。

（5）理解无穷小量和无穷大量：无穷小量与无穷大量的定义，无穷小量与无穷大量的关系，无穷小量与无穷大量的性质，两个无穷小量阶的比较。

（6）熟练把握用两个重要极限求极限的方法。

重点：会用左、右极限求解分段函数的极限，把握极限的四则运算法则、利用两个重要极限求极限以及利用等价无穷小求解极限。

（三）连续（1）理解函数连续的概念：函数在一点连续的定义，左连续和右连续，函数在一点连续的充分必要条件，函数的中断点及其分类。

（2）把握函数在一点处连续的性质：连续函数的四则运算，复合函数的连续性，反函数的连续性，会求函数的中断点及确定其类型。

（3）把握闭区间上连续函数的性质：有界性定理，最大值和最小值定理，介值定理（包括零点定理），会运用介值定理推证一些简单命题。

（4）理解初等函数在其定义区间上连续，并会利用连续性求极限。

重点：理解函数（左、右连续）性的概念，会判别函数的中断点。

01—10年江苏专转本数学真题(附答案)

2001年江苏省普通高校“专转本”统一考试高等数学一、选择题（本大题共5小题，每小题3分，共15分）1、下列各极限正确的是（）A 、e xx x =+→)11(lim 0B 、e xx x =+∞→1)11(limC 、11sinlim =∞→x x x D 、11sin lim 0=→xx x2、不定积分=-⎰dx x211 （）A 、211x-B 、c x+-211C 、x arcsinD 、c x +arcsin3、若)()(x f x f -=，且在[)+∞,0内0)('>x f 、0)(''>x f ，则在)0,(-∞内必有（）A 、0)('<x f ,0)(''<x f B 、0)('<x f ,0)(''>x f C 、0)('>x f ,0)(''<x f D 、0)('>x f ,0)(''>x f4、=-⎰dx x 21 （）A 、0B 、2C 、－1D 、15、方程x y x 422=+在空间直角坐标系中表示（） A 、圆柱面B 、点C 、圆D 、旋转抛物面二、填空题（本大题共5小题，每小题3分，共15分）6、设⎩⎨⎧+==22tt y te x t ，则==0t dx dy7、0136'''=+-y y y 的通解为 8、交换积分次序=⎰⎰dy y x f dx x x220),(9、函数yx z =的全微分=dz10、设)(x f 为连续函数，则=+-+⎰-dx x x x f x f 311])()([三、计算题（本大题共10小题，每小题4分，共40分） 11、已知5cos)21ln(arctan π+++=x x y ，求dy .12、计算xx dte x xt x sin lim202⎰-→.等价无穷小，洛必达13、求)1(sin )1()(2--=x x xx x f 的间断点，并说明其类型.x 分别为0，1，-1时化简求极限14、已知x y x y ln 2+=，求1,1==y x dxdy.15、计算dx ee xx⎰+12. 16、已知⎰∞-=+02211dx x k ，求k 的值. 17、求x x y y sec tan '=-满足00==x y 的特解.18、计算⎰⎰Ddxdy y2sin ，D 是1=x 、2=y 、1-=x y 围成的区域.19、已知)(x f y =过坐标原点，并且在原点处的切线平行于直线032=-+y x ，若b ax x f +=2'3)(，且)(x f 在1=x 处取得极值，试确定a 、b 的值，并求出)(x f y =的表达式.20、设),(2y x x f z =，其中f 具有二阶连续偏导数，求x z∂∂、yx z ∂∂∂2.四、综合题（本大题共4小题，第21小题10分，第22小题8分，第23、24小题各6分，共30分） 21、过)0,1(P 作抛物线2-=x y 的切线，求（1）切线方程；（2）由2-=x y ，切线及x 轴围成的平面图形面积；（3）该平面图形分别绕x 轴、y 轴旋转一周的体积。