高等數(shù)學大一上學期試題（卷）

1、 高等數(shù)學（上）模擬試卷一一、 填空題（每空3分，共42分）1、函數(shù)的定義域是 ；2、設函數(shù)在點連續(xù)，則 ；3、曲線在（-1，-4）處的切線方程是 ；4、已知，則 ；5、= ；6、函數(shù)的極大點是 ；7、設，則 ；8、曲線的拐點是 ；9、= ；10、設，且，則= ；11、，則 ， ；12、= ；13、設可微，則= 。二、 計算下列各題（每題5分，共20分）1、2、，求；3、設函數(shù)由方程所確定，求；4、已知，求。三、 求解下列各題（每題5分，共20分） 1、2、3、4、四、 求解下列各題（共18分）：1、求證：當時， （本題8分）2、求由所圍成的圖形的面積，并求該圖形繞軸旋轉一周所形成的旋轉體的體

2、積。（本題10分）高等數(shù)學（上）模擬試卷二一、填空題（每空3分，共42分）1、函數(shù)的定義域是 ；2、設函數(shù)在點連續(xù)，則 ；3、曲線在處的切線方程是 ；4、已知，則 ；5、= ；6、函數(shù)的極大點是 ；7、設，則 ；8、曲線的拐點是 ；9、= ；10、設，且，則= ；11、，則 ， ；12、= ；13、設可微，則= 。二、計算下列各題（每題5分，共20分）1、2、，求；3、設函數(shù)由方程所確定，求；4、已知，求。三、求解下列各題（每題5分，共20分） 1、2、3、4、四、求解下列各題（共18分）：1、求證：當時， （本題8分）2、求由所圍成的圖形的面積，并求該圖形繞軸旋轉一周所形成的旋轉體的體積。（

3、本題10分）習題4-2 1. 在下列各式等號右端的空白處填入適當?shù)南禂?shù), 使等式成立(例如: : (1) dx= d(ax); 解dx= d(ax). (2) dx= d(7x-3); 解dx= d(7x-3). (3) xdx= d(x2); 解xdx= d(x2). (4) xdx= d(5x2); 解xdx= d(5x2). (5); 解 . (6)x3dx= d(3x4-2); 解x3dx= d(3x4-2). (7)e 2x dx= d(e2x); 解e 2x dx= d(e2x). (8); 解 . (9); 解 . (10); 解 . (11); 解 . (12); 解 . (1

4、3); 解 . (14). 解 . 2. 求下列不定積分(其中a, b, w, j均為常數(shù)): (1); 解 . (2); 解 . (3); 解 . (4); 解 . (5); 解 . (6); 解 . (7); 解 . (8); 解 . (9); 解 . (10); 解 . (11); 解 . (12); 解 (13); 解 . (14); 解 . (15); 解 . (16); 解 . (17); 解 . (18); 解 . (19); 解 . (20); 解 . (21); 解 . (22); 解 . (23); 解 . (24); 解 . (25); 解 . (26); 解 . (27

5、); 解 . (28); 解 . (29); 解 . (30); 解 . (31); 解 . (32); 解 . (33); 解 . (34)(0); 解 , . (35); 解 . 或 . (36); 解 . (37); 解 . (38); 解 . (39); 解 . (40). 解 . 習題5-1 1. 利用定積分定義計算由拋物線y=x2+1, 兩直線x=a、x=b(ba)及橫軸所圍成的圖形的面積. 解 第一步: 在區(qū)間a, b內插入n-1個分點(i=1, 2, , n-1), 把區(qū)間a, b分成n個長度相等的小區(qū)間, 各個小區(qū)間的長度為: (i=1, 2, , n). 第二步: 在第i個

6、小區(qū)間xi-1, xi (i=1, 2, , n)上取右端點, 作和 . 第三步: 令l=maxDx1, Dx2, , Dxn, 取極限得所求面積 . 2. 利用定積分定義計算下列積分: (1)(a0, 所以函數(shù)f(x)=x arctan x在區(qū)間上單調增加. 于是 , . 因此 , 即 . (4)先求函數(shù)在區(qū)間0, 2上的最大值M與最小值m. , 駐點為. 比較f(0)=1, f(2)=e 2, ,得, M=e 2. 于是 , 即 . 7. 設f(x)及g(x)在a, b上連續(xù), 證明: (1)若在a, b上, f(x)0, 且, 則在a, b上f(x)0; (2)若在a, b上, f(x)

7、0, 且f(x)0, 則; (3)若在a, b上, f(x)g(x), 且, 則在a, b上f(x)g(x). 證明 (1)假如f(x)0, 則必有f(x)0. 根據(jù)f(x)在a, b上的連續(xù)性, 在a, b上存在一點x0, 使f(x0)0, 且f(x0)為f(x)在a, b上的最大值. 再由連續(xù)性, 存在c, da, b, 且x0c, d, 使當xc, d時, . 于是 . 這與條件相矛盾. 因此在a, b上f(x)0. (2)證法一 因為f(x)在a, b上連續(xù), 所以在a, b上存在一點x0, 使f(x0)0, 且f(x0)為f(x)在a, b上的最大值. 再由連續(xù)性, 存在c, da,

8、 b, 且x0c, d, 使當xc, d時, . 于是 . 證法二 因為f(x)0, 所以. 假如不成立. 則只有, 根據(jù)結論(1), f(x)0, 矛盾. 因此. (3)令F(x)=g(x)-f(x), 則在a, b上F(x)0且 , 由結論(1), 在a, b上F(x)0, 即f(x)g(x). 4. 根據(jù)定積分的性質及第7題的結論, 說明下列積分哪一個的值較大: (1)還是？ (2)還是？ (3)還是？ (4)還是？ (5)還是？ 解 (1)因為當0x1時, x2x3, 所以. 又當0xx3, 所以. (2)因為當1x2時, x2x3, 所以. 又因為當1x2時, x2x3, 所以. (

9、3)因為當1x2時, 0ln x1, ln x(ln x)2, 所以. 又因為當1x2時, 0ln x(ln x)2, 所以. (4)因為當0x1時, xln(1+x), 所以. 又因為當0ln(1+x), 所以. (5)設f(x)=ex-1-x, 則當0x1時f (x) =ex-10, f(x)=ex-1-x是單調增加的. 因此當0x1時, f(x)f(0)=0, 即ex1+x, 所以. 又因為當01+x, 所以. Part I Vocabulary and StructureDirections： There are 30 incomplete sentences in this part

10、. For each sentence there are four choices marked A）, B）, C） and D）。 Choose the ONE answer that best completes the sentence. Then write the corresponding letter on the space given.1. Until then, his family _ from him for six months.A） didnt hearB） hasnt been hearingC） hasnt heardD） hadnt heard2. The

11、 conference _ a full week by the time it ends.A） must have lastedB） will have lastedC） would lastD） has lasted3. Students or teachers can participate in excursions to lovely beaches around the island at regular _.A） gapsB） rateC） lengthD） intervals4. Physics is _ to the science which was called natu

12、ral philosophy in history.A） alikeB） equivalentC） likelyD） uniform5. Theres a man at the reception desk who seems very angry and I think he means trouble.A） makingB） to makeC） to have madeD） having made6. After the A rob states won independence, great emphasis was laid on expanding education, with g

13、irls as well as boys to go to school.A） to be encouragedB） been encouragedC） being encouragedD） be encouraged7. The new appointment of our president _ from the very beginning of next semester.A） takes effectB） takes partC） takes placeD） takes turns8. The president made a _ speech at the opening cere

14、mony of the sports meeting, which encouraged the sportsmen greatly.A） vigorousB） tediousC） flatD） harsh9. It is useful to be able to predict the extent _ which a price change will affect supply and demand.A） fromB） withC） toD） for10. Finding a job in such a big company has always been _ his wildest

15、dreams.A） underB） overC） aboveD） beyond11. It is not easy to learn English well but if you _, you will succeed in the end.A） hang upB） hang aboutC） hang inD） hang onto12. It is reported that _ adopted children want to know who their natural parents are.A） the mostB） most ofC） mostD） the most of13. L

16、ast year the advertising rate _ by 20 percent.A） raisedB） arousedC） aroseD） rose14. _ before we depart the day after tomorrow, we should have a wonderful dinner party.A） Had they arrivedB） Would they arriveC） Were they arrivingD） Were they to arrive15. The strong storm did a lot of damage to the coa

17、stal villages： several fishing boats were _ and many houses collapsed.A） wreckedB） spoiledC） tornD） injured16. The little was _ one meter fifty high.A） almost more thanB） hardly more thanC） nearly more thanD） as much as17. As _ announced in todays papers, the Shanghai Export Commodities Fair is also

18、 open on Sunday.A） beingB） isC） to beD） been18. You see the lightning _ it happens, but you hear the thunder later.A） the instantB） for an instantC） on the instantD） in an instant19. The manager lost his _ just because his secretary was ten minutes late.A） moodB） temperC） mindD） passion20. Great as

19、Newton was, many of his ideas _ today and are being modified by thework of scientists of our time.A） are to challengeB） may be challengedC） have been challengedD） are challenging21. Please be careful when you are drinking coffee in case you _ the new carpet.A） crashB） polluteC） spotD） stain22. Id ra

20、ther read than watch television； the programs seem _ all the time.A） to get worseB） to be getting worseC） to have got worseD） getting worse23. Convenience foods which are already prepared for cooking are _ in grocerystores.A） readyB） approachableC） probableD） available24. When I caught him _ me I st

21、opped buying things there and started dealing withanother shop.A） cheatingB） cheatC） to cheatD） to be cheating25. It is vital that enough money _ to fund the project.A） be collectedB） must be collectedC） is collectedD） can be collected26. Some old people dont like pop songs because they cant _ so mu

22、ch noise.A） resistB） sustainC） tolerateD） undergo27. If only the committee _ the regulations and put them into effect as soon aspossible.A） approveB） will approveC） can approveD） would approve28. _ one time, Manchester was the home of the most productive cotton mills inthe world.A） OnB） ByC） AtD） Of

23、29. _ it or not, his discovery has created a stir in scientific circles.A） BelieveB） To believeC） BelievingD） Believed30. Mr. Morgan can be very sad _, though in public he is extremely cheerful.A） by himselfB） in personC） in privateD） as individualPart II Reading ComprehensionDirections： There are 4

24、 passages in this part. Each passage is followed by some questions or unfinished statements. For each of them there are four choices marked A）, B）, C） and D）。 You should decide on the best choice and write the corresponding letter on the space given.Questions 31 to 35 are based on the following pass

25、age.Statuses are marvelous human inventions that enable us to get along with one another and to determine where we “fit” in society. As we about our everyday lives, we mentally attempt to place people in terms of their statuses. For example, we must judge whether the person in the library is a reade

26、r or a librarian, whether the telephone caller is a friend or a salesman, whether the unfamiliar person on our property is a thief or a meter reader, and so on.The statuses we assume often vary with the people we encounter, and change throughout life. Most of us can, at very high speed, assume the s

27、tatuses that various situations require. Much of social interaction consists of identifying and electing among appropriate statuses and allowing other people to assume their statuses in relation to us. This means that we fit our actions to those of other people based on a constant mental process of

28、appraisal and interpretation. although some of us find the task more difficult than others, most of us perform it rather effortlessly.A status has been compared to readymade clothes. Within certain limits the buyer can choose style and fabric. But an American is not free to choose the costume （服裝） o

29、f a Chinese peasani or that of a Hindu prince. We must choose from among the clothing presented by our society. Further more, our choice is limited to a size that will fit, as well as by our pocketbook （錢包）。 Having made a choice within these limits we can have certain alterations made, but apart fro

30、m minor adjustments, we tend to be limited to what the stores have on their racks. Statuses too come ready made, and the range of choice among them is limited.31. In the first paragraph, the writer tells us that statuses can help us _.A） determine whether a person is fit for a certain jobB） behave a

31、ppropriately in relation to other peopleC） protect ourselves in unfamiliar situationsD） make friends with other people32. According to the writer, people often assume different statuses _.A） in order to identify themselves with othersB） in order to better identify othersC） as their mental processes

32、changeD） as the situation changes33. The word “appraisal” （Line 5, Para. 2） most probably means “_”。A） involvementB） appreciationC） assessmentD） presentation34. In the last sentence of the second paragraph, the pronoun “it” refers to “_”。A） fitting our actions to those of other people appropriatelyB

33、） identification of other peoples statusesC） selecting ones own statusesD） constant mental process35. By saying that “an American is not free to choose the costume of a Chinese peasantor that of a Hindu prince“ （Lines 2-3, Para. 3）, the writer means _.A） different people have different styles of clo

34、thesB） ready made clothes may need alterationsC） statuses come ready made just like clothesD） our choice of statuses is limitedQuestions 36 to 40 are based on the following passage.Many a young person tells me he wants to be a writer. I always encourage such people, but also explain that theres a bi

35、g difference between “being a writer” and writing. In most cases these individuals are dreaming of wealth and fame, not the long hours alone at a typewriter. “Youve got to want to write,” I say to them, “not want to be a writer”。The reality is that writing is a lonely, private and poorpaying affair.

36、 For every writer kissed by fortune there are thousands more whose longing is never rewarded. When I left a 20-year career in the U. S. Coast Guard to become a freelance writer （自由撰稿人）, I had no prospects at all. What I did have was a friend who found me my room in a New York apartment building. It

37、didnt even matter that it was cold and had no bathroom. I immediately bought a used manual typewriter and felt like a genuine writer.After a year or so, however, I still hadnt gotten a break and began to doubt myself. It was so hard to sell a story that barely made enough to eat. But I knew I wanted

38、 to write. I had dreamed about it for years. I wasnt going to be one of those people who die wondering. What if I would keep putting my dream to the testeven though it meant living with uncertainty and fear of failure. This is the Shadow land of hope, and anyone with a dream must learn to live there

39、.36. The passage is meant to _.A） warn young people of the hardships that a successful writer has to experienceB） advise young people to give up their idea of becoming a professional writerC） show young people its unrealistic for a writer to pursue wealth and fameD） encourage young people to pursue

40、a writing career37. What can be concluded from the passageA） Genuine writers often find their work interesting and rewardingB） A writers success depends on luck rather than on effort.C） Famous writers usually live in poverty and isolation.D） The chances for a writer to become successful are small.38

41、. Why did the author begin to doubt himself after the first year of his writing careerA） He wasnt able to produce a single book.B） He hadnt seen a change for little barter.C） He wasnt able to have a rest for a whole year.D） He found his dream would never come true.39. “ people who die wondering. Wha

42、t if ”（Line 3, para. 3） refers to “those _”。A） who think too much of the dark side of lifeB） who regret giving up their career halfwayC） who think a lot without making a decisionD） who are full of imagination even upon death40. “Shadow land” in the last sentence refers to _.A） the wonder land one of

43、ten dreams aboutB） the bright future that one is looking forward toC） the state of uncertainty before ones final goal is reachedD） a world that exists only in ones imaginationQuestions 41 to 45 are based on the following passage.It is, everyone agrees, a huge task that the child performs when he lea

44、rns to speak, and the fact that he does so in so short a period of time challenges explanation.Language learning begins with listening. Individual children vary greatly in the amount of listening they do before they start speaking, and late starters are often long listeners. Most children will “obey

45、” spoken instructions some time before they can speak, though the word obey is hardly accurate as a description of the eager and delighted cooperation usually shown by the child. Before they can speak, many children will ask questions by gesture and by making questioning noises.Any attempt to trace the development from the noises babies make to their first spoken words leads to considerable difficulties. It is agreed that they enjoy making noise