上海交通大学2004年硕士研究生入学考试试题

上海交通大学2004年研究生入学考试试题试题代号： 819 试题名称：信号系统与信号处理一、某因果系统其系统函数是H(S)有理的，且仅有两个极点：S 1=-2，S 2=-4。

有且仅有两个零点:Z 1=2,Z 2=4。

系统阶跃稳态响应的最大值是1。

试求： 1.系统函数H(S)，且画出零极图，判断系统的稳定性。

2.当输入为e(t)=e -4u(t)时候，求系统的零状态响应。

3.当输入为e(t)=sin(2t)u(t)时候，求稳态响应。

4.画出幅频特性图，并采用RLC 图来实现系统，标出元件值。

二、某离散时间LSI 因果系统。

当输入为x(n)=2n u(n)，完全响应为,n>0，当x(n)=u(n)时候，。

试求：1. H(Z),h(n) 以及系统的差分方程。

2. 用直接Ⅱ型画出本系统的信号流图。

3. 当时候，求系统的完全响应。

三、如下图所示，假设S c (t)是带限的，S c (j )=0,，对x c (t)进行采样，采样周期是T ，得到序列x(n)=x c (nT)。

试求：1. x c (t)的傅里叶变换和x(n)的离散傅里叶变换。

2.如下离散时间系统仿真图，试选择该离散时间系统函数H(e jw )，当输入s(n)=s c (nT)时候，输出为y(n)=x c (nT)。

3.当延时τ=T 及时候，求h(n)。

S四、如下图，。

试求：1.时，求输出y(t)。

2. x(t)=Sa(t)cos4t 时，求输出y(t)。

3. 当x(t)为如下波形时，再求y(t)。

五、实序列x(n)与其偶部及其奇部之间满足如下关系：已知x(n)离散傅里叶变换X R(e jw)。

其中为实数。

试求：1.x(n)，X(ejw)，x(z)。

2.设X(e jw)=X R(e jw)+jX1(e jw)，试导出X R(e jw)与X1(e jw)之间的关系。

六、令x(k)表示N点序列，x(n)的N点DFT，试证明：1.若x(n)满足x(n)=-x(N-1-n)，则X(0)=0。

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考研不同于高考，高考科目全部都是统编，但考研专业课的考试确实一门很大的学问，我这的资料很多都来自官方版权威的资料（网上很多电子其实都是照片，而且真题答案都是请人做的，不少答案都是错的，而我这边的材料都来自往年交大自己办考研班的时候发的资料，绝对权威），还有一部分是从历届学长学姐那收购而来，如果有必要我可以帮你找学长和你交流，最主要的是可以为你提供交大考研专业课的信息，这样就可以使你和交大的学生站在同一起跑线上，不会输在专业课的起跑线上，相对于别的同学外校考研，那你就更有优势了！考研也是一种投资，投资好了，你就会得到相当大的回报，可能你会觉花钱就会觉得不值，但当你以一两分的优势就力压群雄的话，那时你就会发现发这份钱很值了，考研每一分都对你很重要，而我所希望做的就是让你专业课的起点至少和交大本部学生一样高，绝不让考研输在专业课上！现在我已经可以提供以下科目资料，如果这下面有你要报考的专业的话，无疑这对你来说是一个最大的喜讯！如果你需要这些资料，请联系yangweitu@或QQ1449791880或上海交大考研淘宝店/!资料还是实时更新中，如果这里没有你想要的资料，也可来邮询问是否已经收集到你想要的资料！●上海交通大学<经济学>考研专业课资料（代码841）●上海交通大学/上海交大<金融学844>历年考研真题和辅导资料最近和一些同学接触后，让我感觉我很有必要写出如下这段话，如果大家有时间的话，就看看吧！大家都是怀揣梦想的有志者，都希望都过各种渠道获得最多最好的专业课资料，但这个市场鱼龙混杂，资料质量参差不齐，一份好的资料可以助你一臂之力，但一份差的资料也有可能影响你的一生。

上海理工大学2004硕士研究生入学考试试题
考试科目：教育学
一、简答题（70分）
1.古代学校教育与现代教育有何异同？
2.什么是遗传素质？遗传在人的发展中有何作用？
3.教育对社会发展有可能产生负向功能吗？请简述理由。

4.我国教育目的的精神实质是什么？
5.什么是学制？现代学校教育制度有哪几种类型？
6.理想的师生关系的基本特征是什么？
7.正式群体与非正式群体有何区别？如何对待一个班级中的非正式群体。

二、论述题（80分）
1.你认为教学与课程是一种什么样的关系？综观国内外课程改革的趋势，你认为课程
改
革对教学已经产生和将要产生的影响是什么？
2.你认为在知识经济时代学校应该怎样激发学生的创新精神和人文精神？
3.在目前我国普通中小学教育中，你认为德育、智育、体育、美育、劳动技术教育等
几
个方面教育中，最薄弱的是哪一方面的教育？请阐述理由并说明你的改进意见。

4.请举例说明21世纪世界教育发展的趋势？并请说明中国教育的发展和发达国家教育相比，有哪些经验，还存在哪些不足？。

上海交通大学2004年硕士生入学考试试题试题序号：453 试题名称：英语水平考试（答案必须写在答题纸上，写在试题纸上的一律不给分）Important: This test lasts for three hours. All your answers must be written on a separate sheet called “Answer Sheet”, Do not write anything in this test booklet.Part I (20分)In this part you are asked to complete each of the 20 sentences with one of the four words marked A, B, C and D that follow each sentence. The word you choose must fit into the sentence both in form and meaning. For every correct choice, you will get one point.1.I object to you speaking of “learning French as a second language” in Canada; French is as _______ a firstlanguage as English.A.farB.wellC.muchD.good2.For this situation，learning and using English for wider communication _______________ a country,particularly for educational, commercial, and political purposes, English can be referred to as an international language.A.outsideB.withinC.withD.of3.It reveals itself in the assumptions underlying _____________, in the planning of a course of study, in theroutines of the classroom, in value judgments about language teaching, and in the decisions that the language teacher has to make day by day.A.learningB.teachingC.theoryD.practice4.The debate on language teaching methods continued into the period between the two world wars, a periodwhich from the point of view of language pedagogy is characterized by the search for realistic solutions to the method _______________.A.controversyB.problemsC.issuesD.crises5.This conviction led to various experiments, all designed to ________ the traditional teacher-centred languageclass.A.changeB.convertC.modifyD.verify6.The communicative approach, understood in this comprehensive way, has had a ___________ on secondlanguage curriculum, on teaching methodology and materials, and also on evaluation.A.effectB.markC.bearingD.weight7.By virtue of their iconicity and their obvious formal aspects, poems are ideally suited to have learnersexperience early on the two main features of __________ experience: distance and relation.A.literaryB.socialC.aestheticD.dialectic8.Furthermore, being able to recite it from memory enables the teacher to keep eye contact with the students,to anticipate their misunderstandings and respond to their facial ____________.A.responsesB.expressionsC.performanceD.inquiries9.As translators move from word to word and from sentence to sentence through the text they produce bit bybit _______ of the original in a different language.A.replicasB.versionsC.relicsD.sediments10.Besides exploring different levels of the same text and different languages’ways of expressing the sameevent, intermediate and advanced learners can profit from _________ the same event into different literary forms.A.reproducingB.imitatingC.expressingD.recasting11.It has often been suggested that we lack an adequate analysis of the concept of analyticity and consequentlythat we lack adequate criteria for deciding whether a statement is _________.A.adequateB.realisticC.efficientD.analytic12.The tacit ideology which seems to lie behind these objections is that non-extensional explications are notexplications at all and that any concept which is not extensionally ______ is defective.A.ideologicalB.explicableC.explicitD.objectional13.The reason for concentrating on the study of speech acts is simply this: all linguistic communication involveslinguistic ____________A.devicesB.meaningC.formsD.acts14.This is because in certain institutional situations we not only ascertain the facts but we need an authority tolay down a decision as to what the ____________ are after the fact-finding procedure has been gone through.A.situationsB.assertionsC.factsD.reasons15.The simplest cases of meaning are those in which the speaker utters a sentence and means exactly and___________ what he says.A.verballyB.definitelyC.simplyD.literally16.And since meaning consists in part in the intention to produce understanding in the hearer, a large part of thatproblem is that of how it is possible for the hearer to understand the indirect speech act when the sentence he hears and understands means something ________A.trueB.elseC.falseD.indirect17.We all believe that it is the faculty of language which has enabled the human race to develop diverse cultures,each with its _________ social customs, religious observances, laws, oral traditions, patterns of trading, and so on.A.diverseB.distinctiveC.multipleD.varied18.In general, too, rhythmic and _________ features of speech are ignored in transcriptions; the rhythmicstructure which appears to bind some groups of words more closely than others, and the speeding up and slowing down of the overall pace of speech relative to the speaker’s normal pace in a given situation, are such complex variables that we have very little idea how they are exploited and to what effect.A.metricalB.mobileC.acousticD.temporal19.It seems reasonable to suggest that, whereas in daily life in a literate culture, we use ________ largely for theestablishment and maintenance of human relationships, we use written language largely for the working out of and transference of information.A.wordsB.speechC.soundsD.sentences20.The higher level of achievement is a contribution to the ___________ of the text: the linguistic analysis mayenable one to say why the text is, or is not, an effective text for its own purposes ----in what respects it succeeds and in what respects it fails, or is less successful.A.analysisB.readingC.evaluationD.interpretationPartⅡ(20分)Each of the following 20 sentences contains an error. And the error involves only one word. You are required to identify the error and correct it. Instructions on how to write your answersare given on the Answer Sheet. For each correction you make, you will get one point.21. A Spanish history of the “Indies,”read with eager curiosity (and later paraphrased) by the Englishentrepreneur Sir Walter Raleigh, told to the court splendors of a supposed ancestor of the “emperor of Guiana”22.Elizabethan merchants and ministers were second for none in their lively concern for treasure, but the realsuccess of Great Britain as a colonizing power was eventually to rest in its ability to sustain permanent, large-scale, self-supporting “plantations” in North America.23.The faith was sustained for the newcomers not only by the promises before but by the horrors left behind,across the Atlantic.24.In a sense, the seventeenth century saw the emergence of those institutions that are characteristic in themodern world: centralized and wholly sovereign nation-states; capitalism; individualism; secularism; and heroic grandeur in the arts.25.What was more, warfare, both civil and international, erupted epidemically in massive dislocations of power.26.No history of the American people--- a title after which, after all, the Indians have the most legitimate claim---- can omit the red men and women’s role.27.Even before Europe hung suspended between the rise of Roman Imperial order and the emergence offeudalism, in the so-called Dark Ages, some North American Indian’s had developed what anthropologists call the Hopewellian Culture.28.At first they called the chiefs they met after names both familiar and curious --- princes, --- emperors,caciques, and werowances.29.He pointed out that one of the first signs of adaptation to the new environment as a European’s part was tostrip off the garments of civilization, with their class and social connotations, and wear the undifferentiated skin garments of the India.30.The story began, then, with interaction among the continent’s new and old inhabitants --- the Indian“garrison” and the colonized immigrants.31.They learned to sing hymns, to pray, even to participate in the Mass, and to hold their new beliefs by a gripthat survived the vicissitudes of many years of battle between white warriors and red.32.After an unsuccessful attempt to get the Dutch to plant a new settlement on the Delaware, he traveled toSwede.33.Despite the political weakness of the Dutch, they set an impress on the life of Americans as unborn.34.Tradesmen went home, entered through brick-faced doorways and ascended to cozy rooms where, belowtiled roofs, windows with tiny panes illuminated polished delftware.35.The Church of England, for example, though firmly established, did not command the loyalties of greatCatholic families on the one hand, or on the other, of the Puritans who hoped to purge it into “Romish idolatry.”36.With chronic misgivings about the future, no wonder that some men were tempted by the prospects of secureestates and freedom of harassment across what seemed an infinity of ocean.37.Huddled into the city, the poor were helpless before the plagues that swept devastatingly into their slums andthen undiscriminatingly went on to lay down the proud and wealthy as well.38.Imperiled by pestilence and starvation, many of the able-bodied men among the poor might have looked atimpressment as an opportunity at least to eat and to be clothed.39.And nothing short for a spectacular peice of luck or royal preferment seemed likely to improve the situation.40.Farther from the social scale, the yeoman might also try to enhance the value of his lands or the prospects ofhis children by taking fliers in New World ventures such as fishing and trading companies.PartⅢ (30分)In this part you will be asked to read five passages, each followed by six questions. Read the passages carefully and then answer all the questions by choosing the correct options marked A, B, C, and D. Answer one question correctly and you will get one point.Passage1We know that Poe fought a continuous battle against the demon of plagiarism and the twisted perversion of influence. He even declared war on his fellow-writer Longfellow, accusing him of plagiarism of which he was himself not entirely innocent. Passion and influence have their dark sides not only manifest in literary plagiarism --- which we note in Baudelaire’s translations of Poe --- but also in what may be deemed a confusion of identity or quest for an alter ego. Translating Poe became for Baudelaire a real search for the definition of his own personality and even his understanding of gender. Baudelaire’s text is a mixed entity, a complex unity like most of Poe’s characters, a unity composed of scattered elements. The “Flowers of Evil,” are filled with Poe’s own experience of despair and doubt about the world and about human beings, blended with Bauelaire’s spleen and bouts of ideal. Both writers were divided into forces of Good and Evil, love and hate, masculine and feminine, they were like two images reflected in the mirrors of their creations so perfectly inverted that the reader does not know who inspired whom. Alter egos of each other, these two monsters of selfishness and misanthropy would probably have hated each other if they had had the opportunity to meet. Looking at oneself in a mirror can be very upsetting as the hero of William Wilson discovers in the last lines of this eponymous tale. Baudelaire chose to exalt Poe’s character as Griswold presented it because he had many features in common with this portrait. Baudelaire identified with Poe in a very self-centered egotistical way. Both had a strain of masochism and a taste for self-destruction certainly provoked by parental rejection. Baudelaire’s most palpable self-destructive action was the translation of Poe’s works. From this peculiar and unique encounter of two geniuses was born a new universal poet, we could name Poedelaire. Half European, half American, the writings of this desexualized creator are tinged with black humour, sensationalism, and sprinkled with a touch of French preciosity.Questions:41. The author implies that _______________.A. Longfellow was guilty of plagiarismB. Longfellow was not guilty of plagiarismC. Poe was guilty of plagiarismD. Poe was not guilty of plagiarism42. What, according to the author, causes plagiarism?A. Passion and influenceB. Search and questC. identity and egoD. Translation43. The author’s purpose of mentioning Baudelaire’s translations of Poe is _________.A. to show how the two writers hate each other.B. to show how the two writers love each other.C. to prove that plagiarism is pardonable.D. to prove that influence may result in a search for an alter ego.44. It can be inferred that Poe’s writing _____________.A. favors the theme of evil.B. tends to describe flowers.C. reveals a vague personalityD. contains the image of mirror45. Why does the author think that Baudelaire’s translation of Poe’s works was a self-destructive action?A. Because it made Baudelaire even sadder.B. Because he allowed Poe to invade his own identity.C. Because it incurred his parents’ contempt.D. Because it ruined his reputation as a good translator.46. Which of the following words can best describe Poedelaire?A. romanticB. sentimentalC. pessimisticD. revolutionaryPassage2Baudelaire first purchased Poe’s works in London in 1851. This was his first encounter with the American, and he immediately fell in love with the tone, style and content of these texts. He never wrote anything about the theoretical concepts of literary influence and plagiarism whereas Poe had spent a lot of energy attempting to prove his originality. Baudelaire, inversely, although acknowledging that he felt an intimacy with Poe, always refused to admit that he recreated this intimacy in the works he wrote after his translations of Poe, that is to say, after 1856. He was obviously deeply influenced by Poe’s essay Eureka presenting the human condition as a simultaneous movement of attraction and repulsion. This phenomenon of unconscious reappropriation is another clear manifestation of Harald Bloom’s Anxiety of Influence. Instead of fighting against the influence of the first writer, the second writer, moved by passion, prefers to vampirize him, to suck out his creative substance like the painter absorbs his bride’s life in Poe’s The Oval Portrait. This absorption that Bloom calls a tessera, both completes and betrays at the same time. Like physical possession, it satisfies temporarily the one who possesses, while stealing some independence from the one who is possessed. This symbolic betrayal linked to the linguistics possession of Poe by Baudelaire is quite relevant when one observes the mistakes made by the French poet in his translations, Baudelaire loved the English language and used it in an instinctive way, whereas translation requires technicity and precision, a full understanding of both the source and target language which he certainly lacked. In a letter written to Maria Clemm, Poe’s mother-in-law, and published in France in 1854 in the newspaper Le Pays, as a preface to one of his first translation, “Souvenirs de M. Auguste Bedloe,” we can read the following lines: “Adieu, madame; parmi les diffěrents saluts et les formules de complementation qui ne peuvent conclure une missive d’une ame àune ame, je n’en connais qu’une aux sentiments que m’inspire votre personne: goodness, godness”. It is not my purpose to translate the whole letter but we will concentrate on the two concluding words “goodness, godness” that Baudelaire adds in English at the end of his friendly message. His desire to play upon words and to show his mastery of the English language results in a poor lexical association that Mrs Clemm must have had some problems in understanding! Goodness is an exclamation, quite inappropriate in such a context and godness is a neologism, probably used here instead of godliness which would not have been correct either.Questions:47. The author seems to imply that Baudelaire___________.A. had no idea of literary influence.B. never thought of literary influence.C. never admitted that he was influenced by Poe.D. never appreciated the writings by Poe.48. The word “intimacy” in lines 4 probably means ___________.A. friendlinessB. sympathyC. loveD. privacy49. “Anxiety of influence” means the ______________.A. the second writer is influenced by the first writer, but he does not acknowledge it.B. the second writer does not want to be influenced, but he has to.C. the second writer purposely imitates the first writer, then he feels guilty of it.D. the second writer is not influenced by the first writer, but is accused of it.50. The nationality of Baudelaire is _____________.A. EnglishB. FrenchC. AmericanD. German51. This passage mainly discusses _______________.A. translationB. misunderstandingC. plagiarismD. influence52. According to Poe, attraction and repulsion are _______________.A. simultaneousB. unconsciousC. contradictoryD. both A and C.Passage3As a literary critic, surely my best source of information on “globalization” is literature and I hardly need to say that this subject is thematic in a great many works of contemporary Latin American fiction. In fact, Latin American literature includes a long tradition of cultural theorizing that addresses the nature and effects of cultural contact, and thus the processes of globalization avant la lettre. Since the first decades of the twentieth century, indigenista movements considered cultural (and racial) difference and contested the cultural homogeneity imposed by European and U.S. colonialism; indigenismo valorized indigenous traditions and practices, and reconstituted the question of cultural inclusiveness. The movement was led by the Peruvian intellectuals JoséCarlos Mariátegui and JoséMaria Arguedas, with related discussions of transculturation and national identity by Ezequiel Martinez Estrada in Argentina, Gilberto Freyre in Brazil, and Fernando Ortiz in Cuba, JoséVasconcelos, more than his contemporaries, celebrated the process of cultural contact: racial mestizaje had its apotheosis in the 1920s in Vasconcelos’s nationalistic concept of la raza cásmica (“the cosmic race”). Alejo Carpentier dramatizes this discussion: from his first novel in 1933 he recommends not that cultures struggle against colonialism to remain discrete in their differences, but, rather, that that they recognized cultural otherness and embrace it. His formulation of the neobarroco or New World Baroque provides an overarching structure to incorporate European, African, and indigenous cultures into a shared Latin American identity. In his 1975 essay “Lo barroco y lo real maravilloso” (“The Baroque and the Marvelous Real”), Carpentier asks: “And why is Latin America the chosen territory of the baroque? Because all symbiosis, all mestizaje, engenders the baroque. The American baroque develops along with… the awareness of being Other, of being new, of being symbiotic, of being criollo; and the criollo spirit is itself a baroque spirit”. Carpentier, and following him the Cuba writers JoséLezama Lima and Severo Sarduy, understood the irony of engaging the Baroque forms of the Spanish colonizers to construct a post-colonial identity and they turned effectively the neobarroco, or New World Baroque, into an instrument of contraconquista (counterconquest). The Neobaroque is an aesthetics and ideology of inclusion by which Latin American and Latino artists have defined themselves against colonizing structures, and continue to do so. Questions:53. The word “addresses” in line 3 probably means __________A. includesB. concernsC. relatesD. talks54. Indigenista movements most probably voiced the feeling of __________A. the colonizingB. the colonizedC. the EuropeanD. the American55. According to the author, minor nations and races ___________A. welcome globalizationB. fear globalizationC. resent cultural contactD. needs cultural contact56. The term “cultural otherness” probably means _____________A. difference in cultural identityB. cultural separationC. hostility among nationsD. cultural misunderstandings57. “The cosmic race” probably refers to ______________A. the incorporation of racesB. the communication among racesC. marriage among racesD. creation of a new race58. “Baroque spirit” means the willingness to ___________A. recognize and embrace differencesB. study foreign cultures with cautionC. reject foreign influenceD. protect local integrityPassage4Having said all of this, I should, perhaps, locate myself. I teach and write about a loose and baggy territory called las Américas, the Americas, and most often about the part of that category referred to as Latin America. This latter space includes nations, of course, but the demarcation is far more flexible because of its plural referent. The writers who inhabit this territory possess dual citizenship, for they are self-avowed “Latin American” writers at the same time that they are also Mexican, Argentine, Peruvian, or Cuban. In fact, they are often engaged deeply in describing their own national cultures and are far from ready to throw out the baby with the globalizing bathwater. Mexico is a particularly interesting case of the use of nation as a defense against the leveling pressures of globalization --- a nationalism of resistance, in Wallerstein’s terms, rather than a nationalism of domination. For example, the much debated NAFTA, the North American Free Trade Agreement --- or the TLC, Tratado de Libre Comercio ---- opened Mexico’s borders to American commercial onslaughts in the early 1990s, but in cultural matters, the treaty encodes a very different attitude. The Free Trade Agreement contains an Annex that provides special protection to Mexico’s cultural industries. Some of its provisions are as follows: 1) The use of the Spanish language is required for the broadcast, cable or multipoint distribution system of radio and television, except when the Secretaria de Gobernacián authorizes the use of another language; 2) A majority of the time of each day’s live broadcast programs must feature Mexican nationals；3）The use of the Spanish language or Spanish subtitles is required for advertising that is broadcast or otherwise distributed in the territory of Mexico; and 4) Thirty percent of screen time of every theatre, assessed on an annual basis, may be reserved for films produced by Mexican persons either within or outside the territory of Mexico. I should also like to mention that it was Canada that insisted on cultural industry protection clauses in the North American Free Trade Agreement originally and the Canadian government achieved partial success, at best. In comparison, protections of cultural industries are common throughout the European Union: France passed recently legislation requiring that French radio stations devote forty percent of airtime to French music, and Spain also passed a law requiring that one-fourth to one-third of all movies shown in Spanish theaters to be of Spanish origin. English has long protected its movie industry: the great film director Michael Powell got his start, as did other British directors during the 1930s, making what were called quota quickies. So, even as I suggest that comparatists may want to review our nationalist institutional and disciplinary structures in the light of global mobilities, nations continue to protect their cultures against those same forces.Questions:59. the phrase “ plural referent ” in line 3 refers to ____________.A. the nationsB. the writersC. the AmericasD. the cultures60. The phrase “throwing out the baby with the bathwater” probably means ____________.A. embracing the globalization forceB. discarding whatever is contaminated by globalizationC. taking advantage of globalization to foster national culturesD. no discrimination should be made between national and international cultures61. It can be inferred from the passage that Mexico is a country that _____________.A. rejects foreign culturesB. is afraid of foreign cultureC. protects national cultureD. protects national commerce62. Cultural industries include _________.A. radio and televisionB. newspapers and magazinesC. movies and musicD. all of them63. The provisions contained in the Annex to the Free Trade Agreement seem to focus on ______.A. LanguageB. territoryC. cultureD. citizenship64. Which of the following statements is not true?tin American countries protect their national industries.B.North American countries protect their national industries.C.European countries protect their national industries.D.Western superpowers are not afraid of being globalized.Passage 5Once the presence of these characteristics has been recognized, most discussions of globalization move directly to comparative cultural questions. Anthropologists, economists, ecologists, and political scientists all become cultural comparatists, weighing cultural differences against what is generally considered to be the inevitable function of globalization: the leveling of cultural difference. This comparative quotient runs inexorably, it seems, through discussions of globalization, and it should interest us as a profession, since our own most basic disciplinary methods are, of course, designed to recognize and interpret difference. I think of my work in comparative American cultures, for example, as moving along a spectrum between assumptions of basic cultural difference on the one hand and literary examples of shared attitudes and expressive structures on the other. I look for common contexts in order to ground my comparisons, but it is the differences that will matter most to my analysis. So, a mirror image begins to emerge, whereas the literary comparatist may be said to value significant differences and to study literature for what we may learn from those differences, the processes of globalization would seem to work in ways that are something like the reverse--- toward a leveling of significant difference in favor of insignificant sameness. But this comparison, too, will need to be complicated, for homogeneity and heterogeneity are not necessarily antithetical, and in fact may operate in dialectical relationship. Consider, for example, my third characteristics of globalization--- unprecedented levels of immigration ---- a circumstance that suggests the following paradox: the processes of globalization may homogenize tastes and habits by means of new information technologies and global markets, but at the same time they may also generate configuration of striking difference, as immigrants occupy new cultural and linguistics spaces. Nowhere is this more true than in the U.S., where we are experiencing the greatest migratory influx of our history. Certain regions of the country are more illustrative of this than others, of course, but let me say simply that my classes at the University of Houston are far more diverse culturally, linguistically, and ethnically than they were ten years ago--- a comparative cultural opportunity that I feel, frankly, I have not yet fully engaged in my own teaching and that our curricular and departmental structures have not yet fully responded to, either.Questions:65. The author implies that the inevitable function of globalization is ________.A. maintenance of differencesB. reduction of differencesC. promotion of cooperationD. exaltation of competition66. According to the passage, the main objective of comparison is to ________.。

2004年上海交通大学 数学分析一（14）设lim n n a a →∞=，证明22lim221anna a a n n =+++∞→ 证 因2n x n =∞ ，故利用Stolz 公式，11limlim n n n n n n n ny y yx x x +→∞→∞+-=-，得12112222(1)1limlim lim lim (1)212n n n n n n n a a na n a n aa n n n n ++→∞→∞→∞→∞+++++===+-+ 二（14）证明2sin()x 在[)+∞,0上不一致连续.证因n x =n y =22sin sin 1n n x y -=，0n n x y -=-=→，故2sin()x 在[)+∞,0上不一致连续.三（14）设)(x f 在[]a 2,0上连续，且)0(f ＝)2(a f ，证明∃0x ∈[]a ,0，使)(0x f ＝)(0a x f +证 作()()()g x f x a f x =+-（[]0,x a ∈），则()g x 在[]0,a 上连续，因)0(f ＝)2(a f ，故(2)(0)g a g =-，情形1 若(0)0g =，则取00x =，则)(0x f ＝)(0a x f +， 情形2 若(0)0g ≠，则因2(2)(0)(0)0g a g g =-<，故由介值定理知，存在[]00,x a ∈，使得0()0g x =，即)(0x f ＝)(0a x f +.四（14）证明不等式x π2＜x sin ＜x ，⎪⎭⎫ ⎝⎛∈2,0πx证 作sin ()x f x x =，π0,2x ⎛⎫∈ ⎪⎝⎭，则因22cos sin cos ()(tan )0x x x xf x x x x x-'==-<，故sin ()x f x x =在π0,2⎛⎫⎪⎝⎭上严格单调减少，而0lim ()1x f x →=，π22lim ()πx f x →=， 因此，在π0,2⎛⎫ ⎪⎝⎭上，有2sin ()1πx f x x <=<，即x π2＜x sin ＜x .五 (14) 设()d af x x +∞⎰收敛,且)(x f 在[)+∞,a 上一致连续，证明)(lim x f x +∞→= 0.证 因)(x f 在[)+∞,a 上一致连续，故0ε∀>，0δ∃>，使得当[)12,,t t a ∈+∞且12t t δ-<时，有12()()2f t f t ε-<，令(1)()d a n n a n u f x x δδ++-=⎰，则由积分第一中值定理得，[](1),n x a n a n δδ∃∈+-+，使得(1)()d ()a n n n a n u f x x f x δδδ++-==⎰.因()d af x x +∞⎰收敛，故级数1n n u ∞=∑收敛，从而0n u →，即()0n f x δ→，也即()0n f x →，故对上述的ε，存在N +∈ ，使得当n N >时，()2n f x ε<.取X a N δ=+，则当x X >时，因[)[)0,(1),k x a a k a k δδ∞=∈∞=+-+故存在惟一的k +∈ ，使得[)(1),x a k a k δδ∈+-+，易见k N >，且k x x δ-<，从而()()()()22k k f x f x f x f x εεε≤+-<+=六（14）设211n x n -=，121d n n n x x x +=⎰，1,2,n = ，证明级数()∑∞=--111n nn x 收敛.解. 11211d ln |ln(1)n n n n nx x x x n ++===+⎰，因2121n nS S k+=+，故只要证 ()1211111ln(1)nnk n k k k S x kk -==⎡⎤=-=-+⎢⎥⎣⎦∑∑22111()2n k k k =⎡⎤=+⎢⎥⎣⎦∑ 收敛即可.七（14）设)(x f 在[]1,0上连续，)1(f = 0 ,n n x x f x g )()(= ，1,2,n = ， 证明)}({x g n 在[]1,0上一致收敛.八（12）设()f x 在[]1,0上连续，证明10lim ()d n n n x f x x →∞⎰＝)1(f .证 （1）（令n t x =，则10()d n n x f x x ⎰111()d n nt f t t =⎰，（2）因()f x 在[]1,0上连续，故0M ∃>，使得()f x M ≤，[]0,1x ∈，（3）0ε∀>，记3a Mε=，不妨设01a <<，则11110()d ()d d 3aa an nnnt f t t t f t t M t Ma ε≤≤==⎰⎰⎰，（4）111111111()d (1)[()(1)]d ()(1)d n nnnnnaa at f t t f tf t f t t f t f t -=-≤-⎰⎰⎰11111()(1)(1)(1)d nnnnat f t t f t f f t =-+-⎰1111()(1)d (1)1d nnaaf t f t f t t ≤-+-⎰⎰（5）因()f x 在[]1,0上连续，故()f x 在[]1,0上一致连续，故对上述的正数ε，0δ∃>，当[]12,0,1x x ∈且12x x δ-<时，有12()()3(1)f x f x a ε-<-（6）因1lim 1nn a →∞=，记min{,}3(1)M a εεδ*=-，则存在正整数N ，使得当n N >时，有11na ε*-<，（7）当(,1)t a ∈时，有111111nnnt t a -=-≤-，从而当n N >时，有1111()(1)d (1)1d 33nnaaf t f t f t t εε-+-<+⎰⎰（8）由（3）和（7）知，当n N >时，有1110()d (1)nnt f t t f -⎰1111102()d ()d (1)33an n n na t f t t t f t t f εεε≤+-<+=⎰⎰九（12）设1a >0，1+n a ＝n a ＋n a 1，证明n ＝1证 （1）要证n ＝1 ，只要证2lim 12nn a n →∞=，即只要证221lim 1(22)2n nn a a n n +→∞-=+-，即证221lim()2n n n a a +→∞-= （2）因1+n a ＝n a ＋n a 1，故110n n n a a a +-=>，1211n n na a a +=+ 2211112211()()112n n n n n n n n n n na a a a a a a a a a a +++++-=-+==++=+ 因此只要证21lim0n na →∞=，即只要证lim n n a →∞=∞ （3）由110n n na a a +-=>知，{}n a 单调增加，假如{}n a 有上界，则{}n a 必有极限a ，由1+n a ＝n a ＋n a 1知，a ＝a ＋1a，因此10a =，矛盾. 这表明{}n a 单调增加、没有上界，因此lim n n a →∞=∞. （证完）十（28）计算下述积分：1．d x y ⎰⎰，其中D 是矩形区域x 1≤，20≤≤y解 记21{(,)|1,02,0}D x y x y y x =≤≤≤-≤22{(,)|1,02,0}D x y x y y x =≤≤≤≤-，2d d d DD D x y x y x y =+⎰⎰⎰⎰⎰⎰2112221122211d ()d d ()d x x x x y y x y x y --=-+-⎰⎰⎰⎰332211221122()d (2)d 33x x x x --=+-⎰⎰ 332211220044()d (2)d 33x x x x =+-⎰⎰ π143400416d cos d 33x x t t =+⎰⎰()x t =这里 π2401161cos2d 332t t +⎛⎫=+ ⎪⎝⎭⎰ π40141cos412cos2d 332t t t +⎛⎫=+++ ⎪⎝⎭⎰ π40143sin 4sin 23328t t t ⎡⎤=+++⎢⎥⎣⎦ 143ππ5133823⎛⎫=++=+ ⎪⎝⎭ 2．22d d ()d d d d Syz y z x z y z x xy x y +++⎰⎰，其中S 是曲面224z x y +=-上0≥y 的那部分正侧.解 记22{(,,)|4,0}x y z x z y ∑=+≤=（取下侧），22{(,,)|04}V x y z y x z =≤≤--，则V S ∂=+∑，由高斯公式知，2222d d ()d d d d ()d d d 0SS Vyz y z x z y z x xy x y x z x y z +∑∑+++=-=++⎰⎰⎰⎰⎰⎰⎰⎰⎰2242222()d d d d ()d d Vx z x z x y z yx z x z +=+=+⎰⎰⎰⎰42012π(4)d 4y y =-⎰ 430π32π(4)63y ⎡⎤=--=⎣⎦。

上海交通大学硕士研究生入学考试试题库（生物化学）一：选择题100’1、对于肌糖原下列哪个说法是错误的A.在缺氧的情况下可生成乳酸B.含量较为恒定C.能分解产生葡萄糖D.只能由葡萄糖合成E.贮存量较大2、丙酮酸羧化酶催化生成的产物是A.乳酸B.丙酮酸C.苹果酸D.丙二酸单酰CoAE.草酰乙酸3、糖有氧氧化、糖酵解和糖原合成的交叉点是在:A.1-磷酸葡萄糖B.6-磷酸果糖C.3-磷酸甘油醛D.6-磷酸葡萄糖E.5-磷酸核糖4、.糖异生过程经过下列哪条途径:A.磷酸戊糖途径B.丙酮酸羧化支路C.三羧酸循环D.乳酸循环E.糖醛酸途径5、人体内不能合成的脂肪酸是A.油酸B.亚油酸C.硬脂酸D.软脂酸E.月桂酸6、机体中以下哪种物质不足会导致脂肪肝:A.丙氨酸B.磷酸二羟丙酮C.VitAD.柠檬酸E.胆碱7、.利用酮体的酶不存在于下列哪个组织A.肝B.脑C.肾D.心肌E.骨骼肌8、.下列哪一项不是呼吸链的组成成分:A.NADHB.NADPHC.FADH2D.FMNH2E.Cytaa39、下列哪一种是磷酸化抑制剂:A.鱼藤酮B.抗霉素AC.寡霉素D.COE.NaN310、下列哪一组全部为必需氨基酸A.苏氨酸,色氨酸,苯丙氨酸,赖氨酸B.异高氨酸,丝氨酸,苏氨酸,缬氨酸C.亮氨酸,蛋氨酸,脯氨酸,苯丙氨酸D.亮氨酸,丝氨酸,缬氨酸,谷氨酸E.亮氨酸,丙氨酸,组氨酸,酪氨酸11、转氨酶的辅酶是:A.维生素B1的衍生物B.维生素B2C.维生素B12的衍生物D.生物素E.维生素B6的磷酸脂,磷酸吡哆醛12、维生素D的最高活性形式是:A.25-羟维生素D3B.维生素D3C.1,25-二羟维生素D3D.7-脱氢胆固醇E.1,24,25-三羟维生素D313、下述哪种分离方法可得到不变性的蛋白质制剂:A.苦味酸沉淀B.硫酸铜沉淀C.常温下乙醇沉淀D.硫酸钠沉淀E.加热14、某一符合米曼氏方程的酶,当[S]=2Km时,其反应速度V等于:A.VmaxB.2/3VmaxC.3/2VmaxD.2VmaxE.1/2Vmax15、酶的非竞争性抑制作用的特点是:A.抑制剂与酶的活性中心结构相似B.抑制剂与酶作用的底物结构相似C.抑制作用强弱与抑制剂浓度无关D.抑制作用不受底物浓度的影响E.动力学曲线中Vmax不变,Km值变小16、上述9肽中含有的碱性氨基酸是：A.组氨酸，精氨酸B.精氨酸，赖氨酸C.谷氨酸，赖氨酸D.谷氨酸，丝氨酸E.丝氨酸，酪氨酸17、欲获得不变性的蛋白质制剂，可采用下述哪种分离方法：A.生物碱试剂沉淀B.重金属盐沉淀C.常温乙醇沉淀D.低温盐析E.加热18、有关全酶的描述下列哪一项不正确：A.全酶由酶蛋白和辅助因子组成B.通常一种酶蛋白和特定的辅助因子结合C.而一种辅助因子则可与不同的酶蛋白结合D.酶促反应的特异性取决于辅助因子E.酶促反应的高效率取决于酶蛋白19、下列哪个过程是酶原的激活过程A.前胶原→原胶原B.胰岛素原→胰岛素C.凝血因子I→IaD.凝血因子II→IIaE.糖原合成D→I20、已知两种酶互为同工酶：A.它们的Km的值一定相同B.它们的等电点相同C.它们的分子结构一定相同D.它们所催化的化学反应相同E.它们的辅基一定相同21、抗干眼病维生素是指：A.VitAB.VitKC.VitCD.VitDE.VitE22、防止恶性贫血的维生素是：A.VitB6B.VitKC.叶酸D.VitB12E.VitB12和叶酸23、己糖激酶(HK)和葡萄糖激酶(GK)之间的差别在于：A.GK受胰岛素的抑制B.GK受G-6的抑制C.GK对葡萄糖的Km值较高D.HK对葡萄糖的亲和力较低E.HK受葡萄糖的诱导24、对于三羧酸循环，下列哪项是错误的：A.它是糖、脂肪和蛋白质在体内彻底氧化的共同代谢途径B.反应不可逆C.催化反应的酶都在线粒体内D.一分子乙酰CoA进入该循环可产生12分子ATPE.四次脱氢的受氢体均为NAD+25、磷酸戊糖通路产生的两种重要中间产物是：A.NADPH+H+和6-磷酸葡萄糖醛酸B.NADH+H+和6-磷酸葡萄糖醛酸C.NADH+H+和5-磷酸核糖D.NADPH+H+和5-磷酸核糖E.FADH2和6-磷酸果糖26、对于下列各种血浆脂蛋白，哪种描述是正确的：A.CM主要转运内源性TGB.VLDL转运外源性TGC.HDL主要将Ch从肝内转运至肝外组织D.LDL主要是将肝外的Ch转运到肝内27、酮体生成的关键酶是：A.HMG-CoA还原酶B.琥珀酰CoA转硫酶C.硫解酶D.硫激酶E.HMG-CoA合成酶28、在下列哪种情况下，可导致脂肪肝的发生：A.高糖饮食B.胆碱的缺乏C.脑磷脂缺乏D.胰岛素分泌增加E.胰高血糖素分泌增加29、Cytaa3的辅基为：A.FMNB.FADC.CoQ10D.血红素AE.血红素30、呼吸链抑制剂抗霉素A专一抑制以下哪一处的电子传递：A.NADH→CoQB.CoQ→CytbC.Cyte→Cytaa3D.Cytaa3→O2E.FAD→CoQ31、下列哪一种是氧化磷酸化的解偶联剂：A.寡霉素-C.COD.鱼藤酮E.2,4-二硝基酚(DNP)32、下列哪一种氨基酸是非必需氨基酸：A.酪氨酸B.苯丙氨酸C.亮氨酸D.异亮氨酸E.赖氨酸33、转氨酶的辅酶来自：A.维生素B1B.维生素B2C.维生素B6D.维生素B12E.维生素C34、下列哪一种氨基酸为生酮氨基酸：A.丝氨酸B.亮氨酸C.苏氨酸D.组氨酸E.蛋氨酸35、构成天然蛋白质的基本单位是:A.L-α-氨基酸B.D-α-氨基酸C.多肽D.α-螺旋E.核苷酸36维持蛋白质一级结构的键是:A.氢键B.盐键C.疏水键D.肽键E.3',5'-磷酸二酯键37.有关辅酶的正确叙述是:A.本质为蛋白质B.决定酶的特异性C.组成单位为氨基酸D.所有酶结构中都含有辅酶E.一种辅酶能与多种酶蛋白结合,形成具有不同特异性的全酶38.把酶蛋白完全水解,其产物为:A.辅酶或辅基B.多肽C.寡肽D.氨基酸E.核苷酸39.用来治疗坏血病的维生素是:A.维生素AB.维生素B族C.维生素CD.维生素DE.维生素E40.体内直接供能物质主要是:A.ATPB.CTPC.GTPD.UTPE.TTP41.使血糖降低的激素有:A.胰岛素B.胰高糖素C.肾上腺素D.生长激素E.糖皮质激素42、糖与脂肪的分解代谢不能生成:A.CO2B.H2OC.ATPD.NADH+E.H2S43、氰化物中毒是因为它阻断了呼吸链的哪个环节:A.NAD+→FMNB.FMN→CoQC.FAD→CoQD.CoQ→CytE.Cyt→O244、RNA的生物合成是指:A.DNA复制B.转录C.反转录D.修复E.翻译45、反密码子位于:A.DNAB.mRNAC.rRNAD.tRNAE.核蛋白体46、DNA半保留复制不需要:A.冈崎片段B.引物酶C.DNA聚合酶D.氨基酰tRNA合成酶E.DNA连接酶47各种蛋白质含氮量很接近，平均为：A.24%B.55%C.16%D.6.25%14、蛋白质不同于氨基酸的理化性质为：A.等电点B.两性电离C.呈色反应D.胶体性48、酶促反应速度与下列因素成正比：A.温度B.PH值C.酶浓度D.底物浓度49、缺乏维生素C将导致：A.坏血病B.夜盲症C.贫血D.癞皮病50、下列哪种激素为降糖激素：A.胰岛素B.胰高血糖素C.生长素D.肾上腺素51、乳酸转变为葡萄糖的过程属于：A.糖酵解B.糖原合成C.糖异生D.糖的有氧氧化52、一分子葡萄糖经有氧氧化可产生多少分子ATP？A.36(38)B.2(3)C.24(26)D.12(15)53、脂肪酸合成的直接原料是：A.乙酰CoAB.甘油C.丙酮酸D.草酰乙酸54、呼吸链中将电子直接传递给氧的组分是：A.NADHB.FADC.Cytaa3D.CoQ55、转运外源性脂肪的血浆脂蛋白是：A.CMB.VLDLC.LDLD.HDL56、2,4-二硝基酚是氧化磷酸化的一种：A.电子传递抑制剂B.磷酸化抑制剂C.解偶联剂D.激活剂57、肌肉在静止状态时，能量的储存形式是：A.肌酸B.磷酸肌酸C.ATPD.GTP58、关于tRNA的结构与功能正确的叙述是：A.tRNA不具有二级结构B.tRNA具有转肽酶活性C.tRNA将氨基酸携至核糖体D.反密码子环不是tRNA特有的59、核糖体的生物活性不包括：A.与mRNA模板结合B.具有P位和A位两个tRNA结合位点C.为蛋白合成提供能量D.水解酶活性60、竞争性抑制剂的动力学特点是A.Km值变大，Vmax不变B.Km值变大，Vmax增大C.Km值变小，Vmax增大D.Km值不变，Vmax变小E.Km值不变，Vmax变大61、关于磷酸果糖激酶，下列哪一个描述是错误的A.是糖有氧化过程中最主要的限速酶B.是变构酶，位于胞浆C.柠檬酸是变构激活剂，ATP为变构抑制剂D.胰岛素能诱导其生成E.催化反应时消耗ATP62、有关葡萄糖酵解的描述，下列哪项错误A.1克分子葡萄糖净生成2克分子ATPB.ATP的生成部位在胞浆C.ATP是通过底物水平磷酸化生成的D.ATP是通过H在呼吸链传递生成的E.ATP的生成不耗氧63、以下哪句话正确：A.蛋白质变性必沉淀B.蛋白质沉淀必变性C.蛋白质凝固必变性D.蛋白质变性必凝固64、蛋白质变性时不应出现的变化是：A.蛋白质的溶解度降低B.失去原有的生理功能C.蛋白的天然构象破坏D.蛋白质分子中各种次级键被破坏E.蛋白质分子个别肽键被破坏65、逆转录酶催化A.以RNA为模板的DNA合成B.以DNA为模板的RNA合成C.以mRNA为模板的蛋白质合成D.以DNA为模板的DNA合成E.以RNA为模板的RNA合成66、反密码子是位于A. DNAB. mRNAC. rRNAD. tRNAE.多肽链67、柠檬酸循环被认为是一个需氧代谢途径，是因为A. 循环中的某些反应是以氧为底物 B .CO2是该循环的一个产物C. 产生了H2OD.还原型的辅因子需通过电子传递链被氧化E.以上都不对68、Km值是指A.酶-底物复合物的解离常数B.酶促反应达到最大速度时所需底物浓度的一半C.达到1/2Vamx时所需的底物浓度D.酶促反应的底物常数E.酶与底物的亲和常数69.下列哪项叙述符合脂肪酸的β氧化：A．仅在线粒体中进行B．产生的NADPH用于合成脂肪酸C．被胞浆酶催化D．产生的NADPH用于葡萄糖转变成丙酮酸E．需要酰基载体蛋白参与70 .脂肪酸在细胞中氧化降解A．从酰基CoA开始B．产生的能量不能为细胞所利用C．被肉毒碱抑制D．主要在细胞核中进行E．在降解过程中反复脱下三碳单位使脂肪酸链变短71．下列哪些辅因子参与脂肪酸的β氧化：A ACPB FMNC 生物素D NAD+72．脂肪酸从头合成的酰基载体是：A．ACP B．CoA C．生物素 D．TPP73．下述关于从乙酰CoA合成软脂酸的说法，哪些是正确的（多选）？A．所有的氧化还原反应都以NADPH做辅助因子；B．在合成途径中涉及许多物质，其中辅酶A是唯一含有泛酰巯基乙胺的物质；C．丙二酰单酰CoA是一种“被活化的“中间物；D．反应在线粒体内进行。

上海交通大学2004年硕士研究生入学考试题数学分析试题解答一.（14分）设a a n n =∞→lim ,证明22lim221an na a a n n =+++∞→ .证明法一:利用平均极限定理.记n n a a b -=,则n n a b a =+,lim 0n n b →∞=.记12n n S b b b =+++ , 则12limlim lim 0n nn n n n S b b b b n n →∞→∞→∞+++=== .11S b =,212S b b =+,……12n n S b b b =+++ .将以上n 各式子相加得: 1212(1)n n S S S nb n b b +++=+-++从而 1212(2)()n n b b nb S S S +++++++ []1212(2)(1)n n b b nb nb n b b =+++++-++12()n n n b b b nS =+++= ,所以 12122()n n n b b nb nS S S S +++=-+++故 1212222()lim lim n n n n n b b nb nS S S S n n→∞→∞+++-+++= 122()lim n n n S S S S n n →∞+++⎡⎤=-⎢⎥⎣⎦ 121lim lim n n n n S S S S n n n →∞→∞+++⎛⎫=-⋅ ⎪⎝⎭1210lim lim 0n n n S S S n n →∞→∞+++=-⋅= .1212222...()2()...()lim lim n n n n a a na b a b a n b a n n→∞→∞+++++++++= 122222lim n n b b nb a a na n n →∞++++++⎛⎫=+ ⎪⎝⎭1222(1)22lim lim 022n n n n n ab b nb a a n n →∞→∞++++=+=+= . 证明法二:利用Stolz 公式.令122n n x a a na =+++ ,2n y n =,则{}n y 严格增加且lim n n y →∞=+∞,112121(2)[2(1)]n n n n n x x a a na a a n a na ---=+++-+++-= .根据Stolz 公式得12122212 (i)lim lim lim (1)n n n n nn n n n n n n a a na x x x na n y y y n n -→∞→∞→∞→∞-+++-===--- 1lim lim lim 212122n n n n n na n a a a n n →∞→∞→∞==⋅=⋅=--. 二.（14分）证明)sin(2x 在),0[+∞上不一致连续.证明:令22ππ+=n x n ,πn y n 2=, ,2,1=n .则1)sin()sin(22≡-n ny x , 02221222→++=-+=-ππππππn n n n y x n n (∞→n )因此)sin(2x 在),0[+∞上不一致连续. 或210=∃ε,0>∀δ,∃22ππ+=n x n ,πn y n 2=,其中22111δδ>+⎥⎦⎤⎢⎣⎡=n . 尽管δππππππ<<++=-+=-nn n n n y x n n 12221222,但0221)sin()sin(ε>≡-n n y x ,因此)sin(2x 在),0[+∞上不一致连续.三.（14分）设)(x f 在]2,0[a 上连续,且)0(f ＝)2(a f .证明],0[0a x ∈∃,使得)(0x f )(0a x f +=. 证明:令)()()(x f a x f x g -+=,],0[a x ∈.则)(x g 在],0[a 上连续,且)0()()0(f a f g -=, )()2()(a f a f a g -=)0()()0(g a f f -=-=.情形1 若0)0(=g ,取00=x 或a x =0,则],0[0a x ∈且)(0x f ＝)(0a x f +.情形 2 若0)0(≠g ,则因为0)0()()0(2<-=g a g g ,故由介值定理知:),0(0a x ∈∃,使得0)(0=x g ,即)(0x f ＝)(0a x f +.四.（14分）证明不等式x π2＜x sin ＜x ,⎪⎭⎫⎝⎛∈2,0πx . 证明:令⎪⎩⎪⎨⎧=≤<=.0,1,20,sin )(x x x xx f π则)(x f 在⎥⎦⎤⎢⎣⎡2,0π上连续,在⎪⎭⎫ ⎝⎛2,0π内可导,且 0)(tan cos sin cos )(22<--=-='x x x xx x x x x f , 因此)(x f 在⎥⎦⎤⎢⎣⎡2,0π上严格减少,所以当⎪⎭⎫ ⎝⎛∈2,0πx 时,有)0()(2f x f f <<⎪⎭⎫⎝⎛π,即x x x<<sin 2π. 五. (14分) 若无穷积分⎰+∞adx x f )(收敛,且)(x f 是),[+∞a 上的一致连续,则0)(lim=+∞→x f x .证明:0>∀ε, 由于)(x f 是),[+∞a 上的一致连续,因此,0>∃δ(不妨设εδ≤),使得 当),[,21+∞∈a x x 且δ<-21x x 时,有 2)()(21ε<-x f x f ;又由⎰+∞adx x f )(收敛的Cauchy 准则知:对022>δ,a A >∃,使得当A x x >''',时,有2)(2δ<⎰'''x x dx x f ;当A x >时,取x x ''',使x x x A ''<<'<,且δ='-''x x ,可估计得:22)()()()()]()([)()(2δδεδ+<+-≤+-==⎰⎰⎰⎰⎰'''''''''''''''x x x x x x x x x x dt t f dt t f x f dt t f dt t f x f dt x f x f ,因此当A x >时,有εδε≤+<-220)(x f ,即0)(lim =+∞→x f x .六.（14分）设n x n 112=-,⎰+=121n n n dx x x , ,2,1=n .证明级数∑∞=--11)1(n n n x 收敛. 证明: 因为1212121111-++=<=<+=⎰n n n n n x ndx x x n x , ,2,1=n . 所以数列{}n x 严格减少且0lim =∞→n n x ,故交错级数∑∞=--11)1(n n n x 收敛.七.（14分）设)(x f 在]1,0[上连续,0)1(=f ,n n x x f x g )()(=, ,2,1=n .证明:)}({x g n 在]1,0[上一致收敛.证明:ⅰ>因为)(x f 在]1,0[上连续,所以)(x f 在]1,0[上有界,因此0>∃M ,使得]1,0[∈∀x ,有M x f ≤)(.0>∀ε,ⅱ>因为)(x f 在1=x 处左连续,所以0)(>=∃εδδ(不妨设1<δ),使得当11≤<-x δ时,有ε<=-)()1()(x f f x f .从而当11≤<-x δ时,有ε<≤=)()()(x f x x f x g n n .ⅲ>因为0)1(lim =-∞→nn M δ,所以)(εN N =∃,使得当N n >时,有εδ<-<n M )1(0.从而当Nn >时,]1,0[δ-∈∀x ,有εδ<-≤=nn n M x x f x g )1()()(.ⅳ>当N n >时,]1,0[∈∀x ,⑴当]1,0[δ-∈x 时, 有εδ<-≤=nn n M x x f x g )1()()(.⑵当]1,1(δ-∈x 时, 有ε<≤=)()()(x f x x f x g nn .综上所述当N n >时,]1,0[∈∀x , 都有ε<)(x g n .因此)}({x g n 在]1,0[上一致收敛于0. 八.（12分）设)(x f 在]1,0[上连续,证明:)1()(lim 1f dx x f x n nn =⎰∞→.证明:⑴因为)(x f 在]1,0[上连续,所以)(x f 在]1,0[上有界,因此0>∃M ,使得]1,0[∈∀x ,有M x f ≤)(.⑵令t x n=,则nt x =,dt ntt dx n=,dt t f t dx x f x n n n n)()(1010⎰⎰=.⑶0ε∀>(不妨设8M ε<),记8a Mε=,则01a <<.00(1)])(1)2243af f dt f f dt M Ma εε⎤-≤+≤≤=<⎦⎰⎰⎰, 1,2,n = .⑷1(1)](1)(1)(1)(1)d aaaf dt f dt f t -≤-=-+-⎰⎰⎰(1)d (1)1d af f t f t ≤-+⎰⎰.⑸因为()f x 在[0,1]上连续,故()f x 在[0,1]上一致连续，故对03(1)a ε>-,0δ∃>,使得12,[0,1]x x ∀∈,当12x x δ-<时,有12()()3(1)f x f x a ε-<-.⑹因为1n =,所以对min ,,03(1)3M a εεεδ*⎧⎫=>⎨⎬-⎩⎭,则N ∃,使得当n N >时,1ε*<.⑺当n N >时,有(,1]t a ∀∈,111ε*=<.从而当n N >时,有1(1)1d 1d d (1)3aaf t M t M t M a εεε**≤<<-≤⎰⎰⎰,11(1)d (1)d d 3aaf f t f f t t εεε**-≤-≤<≤⎰⎰⎰.⑻综上所述知当n N >时,有⎰⎰-=-110)]1()([)1()(dt f t f t f dt t f t nn n n ⎰⎰-+-=1)]1()([)]1()([an n n an dt f t f t dt f t f t10(1)](1)]aa f dt f dt≤-+-⎰⎰(1)](1)d (1)1d aaaf dt f f t f t ≤-+-+⎰⎰⎰333εεεε<++=.因此)1()(lim 10f dx x f x n nn =⎰∞→.九.（12分）设01>a ,n n n a a a 11+=+.证明:12lim =∞→na n n .证明:ⅰ>因为01>a ,所以01112>+=a a a ,假设0>n a ,则011>+=+n n n a a a .根据数学归纳法知: 0>n a , ,2,1=n .由011>=-+nn n a a a 知{}n a 严格增加.假如{}n a 有上界,则{}n a 收敛,记a a n n =∞→lim .在nn n a a a 11+=+中令∞→n 得:a a a 1+=,即01=a ,矛盾.因此{}n a 单调增加但无上界,故+∞=∞→n n a lim .ⅱ>因为2111nn n a a a +=+,所以 21111221121))((nn n n n n n n n n n n a a a a a a a a a a a a +=+=+=+-=-+++++, 从而2)(lim 221=-+∞→n n n a a .ⅲ>由于{}n 2严格增加且+∞=∞→n n 2lim ,利用Stolz 公式得:n n a a n a nn n n n 2)22(lim2lim 2212-+-=+∞→∞→12lim 221=-=+∞→n n n a a .因此12lim=∞→na n n .十.（28分）计算下述积分：1．⎰⎰-Ddxdy x y 2,其中D 是矩形区域x 1≤,20≤≤y .解:记{}0,20,1),(21≤-≤≤≤=x y y x y x D , {}0,20,1),(22≥-≤≤≤=x y y x y x D .=-⎰⎰Ddxdy x y 2⎰⎰⎰⎰-+-2122D D dxdy x y dxdy x y⎰⎰⎰⎰-+-=--221121122x xdy x y dx dy y x dx 22321102321122)(32)(32x x x y dx y x dx -⋅+-⋅-=⎰⎰--⎰⎰---+=11232113)2(3232dx x dx x ⎰⎰-+=1232103)2(3434dx x dx x )sin 2()sin 22(343140232104⎰-+=πt d t x ⎰+=404cos 31631πtdt ⎰⎪⎭⎫ ⎝⎛++=40222cos 131631πdt t ⎰⎪⎭⎫ ⎝⎛++++=4024cos 12cos 213431πdt t t 4084sin 2sin 233431π⎪⎭⎫ ⎝⎛+++=t t t ⎪⎭⎫ ⎝⎛++=1833431π235π+=. 2．⎰⎰+++Sxydxdy ydzdx z x yzdydz )(22,其中S 是曲面224z x y +=-上0≥y 的那部分正侧.解:记{}4),0,(221≤+=z x z x S （取左侧）,{}2240),,(z x y z y x --≤≤=Ω,则1S S =Ω∂,由高斯公式知，⎰⎰⎰⎰⎰⎰⎰⎰-≤+Ω+=+=+++yz x Vdxdz z xdydxdydz z x xydxdy ydzdx z xyzdydz 4224222222)()()(332)4(2402ππ=-=⎰dy y .而0)(122=+++⎰⎰S xydxdy ydzdx z x yzdydz ,因此332)(22π=+++⎰⎰Sxydxdy ydzdx z x yzdydz . 或曲面S:224z x y --=,{}4),(),(22≤+=∈x z x z D x z .因此z z y 2-=∂∂,x xy 2-=∂∂. ⎰⎰+++Sxydxdyydzdx z xyzdydz )(22⎰⎰⋅--+--++⋅--=Ddzdx z z x x y x z x x z z x ]2)4()4)((2)4[(22222222⎰⎰++--=Ddzdx z x xz z x )2)(4(2222⎰⎰+-=222220)cos sin 4)(4(rdr r r r d θθθπ⎰⎰-=203220)4(dr r r d πθ332622064ππ=⎪⎪⎭⎫ ⎝⎛-=r r .第二型曲面积分计算公式：ⅰ>若曲面S 的方程为:(,),(,)z f x y x y D =∈,则(,,)(,,)(,,)SP x y z dydz Q x y z dzdx R x y z dxdy ++⎰⎰[(,,(,))(,)(,,(,))(,)(,,(,))]x y DP x y f x y f x y Q x y f x y f x y R x y f x y dxdy ''=±--+⎰⎰,上侧取“+”, 下侧取“-”.ⅱ>若曲面S 的方程为:(,),(,)y g z x z x D =∈,则(,,)(,,)(,,)SP x y z dydz Q x y z dzdx R x y z dxdy ++⎰⎰[(,(,),)(,)(,(,),)(,(,),)(,)]x z DP x g z x z g z x Q x g z x z R x g z x z g z x dzdx ''=±-+-⎰⎰,右侧取“+”, 左侧取“-”.ⅲ>若曲面S 的方程为:(,),(,)x h y z y z D =∈,则(,,)(,,)(,,)SP x y z dydz Q x y z dzdx R x y z dxdy ++⎰⎰[((,),,)((,),,)(,)((,),,)(,)]y z DP h y z y z Q h y z y z h y z R h y z y z h y z dxdy ''=±--⎰⎰.前侧取“+”, 后侧取“-”.。