千挑万选——十年高考试题精选句子
一、定语从句（请用正确的关系词填空）
1.That is the reason ___________ I’m not in favour of revising the plan.
2.The reason _____________ he came late was that he didn’t catch the early bus.
3.I’m looking forward to the day ___________ Tai Wan returns to China.
4.The English play in ______ my students acted at the New Year’s party was a great success.
5.I worked in a business（企业）_________ almost everyone is waiting for a great chance.
6.Most hotels have websites ___________ you can find answers to your questions.
7.________ is reported in the newspapers, talks between the two countries are making
progress.
8.Do you still remember the farm __________ we visited three months ago?
9.The journey around the world took the old sailor nine months, during __________ the
sailing time was 226 days.
10.There were dirty marks on her trousers __________ she had wiped her hands.
11.There are altogether eleven books on the shelf, of _________ five are mine.
12.Helen was much kinder to her youngest son than to the others, __________, of course,
made the others envy him.
13.________ is often the case, we have worked out the production plan.
14.There was a time _____________ I hated to go to school.
15.There are two buildings, the larger of ___________ stands nearly a hundred feet high.
16.A fast food restaurant is the place ___________, just as the name suggests, eating is
performed quickly.
17.A left-luggage office is a place ____________ bags can be left for a short time, especially
at a railway station.
18.New York, ___________ I visited last year, is a nice old city.
19.We are living in an age ______________ many things are done on computer.
20.__________ has been announced, we shall have our final exams next month.
21.The famous basketball star, _______ tired to make a comeback, attracted a lot of attention.
22.He is the only one of the students who _________ (has / have) been a winner of
scholarship for three years.（请选择正确答案）
23.Is this the reason __________ he explained at the meeting for his carelessness in his work?
24.________ is known to everybody, the moon travels round the earth once every month.
25.He has got himself into a dangerous situation ___________ he is likely to lose control over
the plane.
26.John said he’d been working in the office for an hour, ___________ was true.
27.Have you seen the film “Titanic,” _____________ leading actor is world famous?
28.This is a book _____________ cover is blue\the cover of which is blue.
29.Mary was always speaking highly of her role in the play, ___________ ,of course, made
the others unhappy.
30.The professor gave us another wonderful talk, ___________ I think is of great importance
to our English study.
31.My sister wants to work in the power plant __________ is still being built.
32.The film brought the hours back to me ___________ I was taken good care of in that far
away village.
名词性从句
2. His failure was due to （由于）the fact that he had not practised for a long time.
4. How they went to America is what I want to know.
14. It is a great honor for Beijing that the 2008 Olympic Games will be held in Beijing.
18. That most of languages are now threatened and may disappear is a serious matter to the people.
19. The fact that Great Britain is made up of three countries is still unknown to many people. 23. They came to the conclusion that not all things can be done by a computer.
42. The price has gone down , but I doubt whether it will remain so .
56. I want to know whether the thief was caught on the spot.
57. Our club is open to adults only. It seems that your children have entered without permission.
58. You are saying that everyone should be equal, and this is where I disagree.
59. A modern city has been set up in what was a wasteland ten years ago.
60. I think Father would like to know what I’ve been up to so far, so I decide to send him a quick note.
61. Parents are taught to understand how important education is to their children’s future.
62. The other day, my brother drove his car down the street at what I thought was a dangerous speed.
63. Along with the letter was his promise that he would visit me this coming Christmas.
64. There is a new problem involved in the popularity of private cars that road conditions need to be improved.
65. What made the school proud was that more than 90% of the students had been admitted to key universities.
66. Perseverance is a kind of quality and that’s what it takes to do anything well.
67. That fashion differs from country to country may reflect the cultural differences from one aspect.
68. What she couldn’t understand was why fewer and fewer students showed interest in her lessons.
69. It was a matter of who would take the position.
70. Whether we’ll go camping tomorrow depends on the weather.
79. What surprised me was not what he said but the way he said it.
92. It has been announced that candidates shall remain in their seats until all the papers have been collected.
101. It is certain that he will hand over his business to his son when he gets old.
116. Watch him and then try to copy what he does.
119. The thing that matters is not whether you fail or not, but whether you try or not.
126. I should pay Mary a visit, but I’m not sure whether I will have time this Sunday.
204.The film reminded him of what he had seen in Paris.
205.They realize that it is of great value to record and teach them to the young generation. 208.You can’t stop him doing what he wants.
219.I know nothing about the young lady except that she is from Beijing.
172.When do you think he will come?
非谓语动词（请用所给动词的正确形式填空）
1.He sat by the fire, __________ (read) a novel.
2.He kept looking at her, ______________ (wonder) whether he had seen her somewhere.
3.___________ (smoke) does great harm to our health.
4.When and where _________ (go) for the on-salary holiday has not been decided .
5.___________ (select) a mobile phone for personal use is no easy task because technology
is changing so rapidly
6.I a m sure that Laura’s latest play, once ____________ (stage), will prove a great success.
7.I don’t mind __________ (pick) up your things from the store. Besides, the walk will do
me good.
8.The flu is ____________ (believe) to be caused by viruses that like to reproduce in the
cells inside the human nose and throat.
9._____________________ (attack) by terrorists, the tall building collapsed.
10._________________ (attract) by the beauty of nature, the girl from London decided to
spend another two days on the farm.
11.The accident is ______________ (report) to have occurred on the first day in February.
12.It is so nice to hear from her. ____________ (believe) it or not, we last met more than
thirty years ago.
13.“The interest shall be di vided into five parts, according to the agreement __________
(make) by both sides,” declared the judge.
14.The visiting minister expressed his satisfaction with the talks, __________ (add) that he
had enjoyed his stay here.
15.The missing boy was last seen _____________ (play) near the river.
16.My car broke down last week and I still haven’t had it ________________ (repair).
17.We offered him our congratulations on his ___________ (pass) the college entrance
exams.
18.With two exams _________________ (worry about), I have to work really hard this
weekend.
19.__________________ (not receive) a reply, he decided to write again.
20.Father advised me not to say anything until ___________ (ask) at the meeting.
21.___________ (enjoy) the scenery, she would rather ________ (spend) long hours on the
train than ____________ (travel) by air.
22.Everyone was surprised to see the buildings ___________ (finish) so soon.
23.They were happy to hear the work already ___________ (do).
24.I prefer staying at home watching TV to ______________ (go) shopping out.
25.The doctor advised me ___________ (take) more exercise.
26.____________ (face) with a difficult situation, Tom decided to ask his boss for advice.
27.________________ (accustom) to climbing the steep mountains, he had no difficulty
reaching the top.
28.The purpose of new technology is to make life easier, not __________ (make) it more
difficult.
29.When first ____________ (introduce) to the market, these products enjoyed treat success.
30.Alice returned from the manager’s office __________ (tell) me that the boss wanted to see
me at once.
31.My teacher encouraged me to take a summer course ______________ (improve) my
writing skills.
32._____________ (wait) in the queue for half an hour, Tom suddenly realized that he had
left his wallet at home.
33.When Mark opened the door, he saw a woman ___________ (stand) there. He had never
seen her before.
34.Sarah, hurry up. I’m afraid you can’t have time to get _____________ (change) before the
party.
35.I fell asleep while _____________ (read) the English textbook. Luckily, my roommate
woke me up in time!
36.____________ (meet) my uncle after all these years was an unforgettable moment, one I
will always treasure.
状语从句
16. Although he thought he was helping us with the work, he was actually in the way.
20. Now that you’ve got a chance, you might as well make full use of it.
26. Come and see me whenever it is convenient of you.
33. Rainforests are being cut and burned at such a speed that they will disappear from the earth in the near future .
81. Don’t leave the water running while you brush your teeth.
85. The man insisted on finding a taxi for me even though I told him I lived nearby.
90. He got to the station early for fear of missing his train.
104. Once a decision has been made, all of us should stick to it.
105. Helen always helps her mother even though going to school takes up most of her day. 108. The idea puzzled me so much that I stopped for a few seconds to try to make it out. 113. As we joined the big crowd, I got separated from my friends.
141.It is almost five years since we saw each other last time.
142.Don’t leave the sharp knife where our little Jane can get it.
143.I do every single bit of housework, while my husband Bob just does the dishes now and then.
144.Roses need special care so that they can live through winter.
145.Several weeks had gone by before I realized the painting was missing.
146.Mary was holidaying with her family in a wildlife park when she was bitten on the leg by a lion.
147.Parents should take seriously their children’s requests for sunglasses because eye protection is necessary in sunny weather.
148.While I accept that he is not perfect, I do actually like the person.
149.Scientists say it may be five or six years before it is possible to test this medicine on human patients.
150.You should try to get a good night’s sleep however much work you have to do.
151.You can eat food free in my restaurant whenever you like.
152. As far as I can see, there is only one possible way to keep away from the danger.
153.However amusing the story is, I have to put it away and focus my attention on study this week.
154.Don’t be afraid of asking for help when it is needed.
155.He made a mistake, but then he corrected the situation before it got worse.
156.As long as I know the money is safe, I shall not worry about it.
157.He was about to tell me the secret when someone patted him on the shoulder.
159.The men will have to wait all day unless the doctor works faster.
160.I gave the money to Mary the moment I saw her.
216.You will succeed in the end unless you give up half way.
217.someone called me up in the middle of the night, but they hung up before I could answer the phone.
218.She found her calculator where she lost it.
220.As far as I know, they live in the same neighborhood.
情态动词和虚拟语气
30. I wonder why Jenny hasn’t written us recently. We should have heard from her by now. 41. Try some of the juice-----perhaps you will like it.
52. Lose one hour in the morning and you will be looking for it in the rest of the day.
91. My English-Chinese dictionary has disappeared. Who could have taken it?
93. Oh, I am not feeling well in the stomach. I should not have eaten so much fried chicken just now.
94. I was really anxious about you. You should not have left home without a word.
95. You can not imagine that a well-behaved gentleman should be so rude to a lady.
97. My sister met him at the Grand Theater yesterday afternoon, so he could not have attended your lecture.
98. There was a lot of fun at yesterday's party. You ought to have come, but why didn't you?
99. The fire spread through the hotel very quickly but everyone was able to get out.
100. When he was there, he would go to that coffee shop at the corner after work every day. 121. I have lost one of my gloves. I must have dropped it somewhere.
122. According to the local regulations, anyone who intends to get a driver's license must get an eye test.
123. I often see lights in that empty house. Do you think I should report it to the police?
124.Mr White should have arrived at 8:30 for the meeting, but he didn't show up.
125.Children under 12 years of age in that country must be under adult supervision when in a public library.
128.You might just as well tell the manufacturer that male customers may not like the design of the furniture.
130.How can you say that you really understand the whole story if you have covered only part of the article?
161. With so much work on hand, you shouldn’t have gone to see the game last night.
162.Tom could have passed the exam, but he wasn’t careful enough.
163.The road was muddy. It must have rained last night.
164.I would have liked to give you a valuable present for your birthday, but I was short of money.
165.If it should snow tomorrow, they couldn’t go out.
168.When a pencil is partly in a glass of water, it looks as if it were broken.
181.Tom ought not to have told me your secret, but he meant no harm.
186.I’m so sorry that he should be so careless.
191.If the dictionary is not yours, who else’s can it be?
192.We thought of selling this old furniture, but we have decided to hold on to it. It might be valuable.
247.How can you possibly miss the news? It has been on TV all day long.
It句型
3. It is likely that it will rain in the evening.
5. It occurred to him that he had left his key at home.
10. It will not be long before we meet again.
34. It will not be long before we know the result of the experiment .
强调和省略
21. It was only when I reread his poems recently that I begin to appreciate their beauty.
32. I feel it is your husband who is to blame for the spoiled child .
88. Why! I have nothing to confess. What is it that you want to say?
96. Mr Bush is on time for everything. How can it be that he was late for the opening ceremony?
196.It is the ability to do the job that matters not where you come from or what you are.
45. An awful accident did, however, occur the other day .
111. Be careful when you cross this very busy street. If not, you may get run over by a car. 188.While you are reading, it is a good idea to make a mark where necessary.
倒装
25. Only then did he realize his mistakes.
43. No sooner had he got to the lab than he set out to do the experiment.
158. Much as I have traveled, I have never seen anyone who is as capable as John.
167.Not until all the fish died in the river did the villagers realize how serious the pollution was.
173.Only when you realized the importance of English can you learn it well.
187.Not until I began to work did I realize how much time I had wasted.
221.On top of the books is the photo album you’re looking fo r.
222.Maybe you have been to many countries, but nowhere else can you find such a beautiful place.
223.I failed in the final examination last term and only then did I realize the importance of studies.
224.The old couple have been married for 40 years and never once have they quarreled with each other.
225.Only when your identity has been checked will you be allowed in.
226.Not only is the teacher himself interested in football but all his students are beginning to show an interest in it.
227.So difficult did I feel it to live in an English-speaking country that I determined to learn English.
228.Not a single song did she sing at yesterday’s party.
229.I finally got the job I dreamed about. Never in all my life had I felt so happy!
时态语态
15. Timmy ________(fine) a large sum of money by the judge for drunken driving.
27. I thought Jim ________ (say) something about his school report , but he didn’t mention it .
28. By the end of last year , another new gymnasium __________ (complete) in Beijing.
31. He __________ (learn) English for eight years by the time he graduates from the university next year .
36. In such dry weather , the flowers will have to be watered if they __________(survive) .
37. Hundreds of jobs will be lost if the factory __________(close) .
38. A new cinema __________ (build) here .They hope to finish it next month .
39. The reporter said that the UFO __________ (travel) east to west when he saw it .
40. All the preparations for the task __________ (complete) , and we are ready to start.
44. Helen __________ (leave) her key in the office, so she had to wait until her husband came back home .
50. John was late for the business meeting because his flight __________ (delay) by a heavy storm.
117. She looked up his number in the phone book to make sure that she __________ (get) it right.
176.The telephone __________ (ring) for two minutes before it was answered.
179.I __________ (wash) my face when suddenly someone knocked at the door.
180.I __________ (finish) my work by the end of next week.
183.His father sent him to study law, but instead he __________ (develop) his love for the theater.
233.Telephone messages for the manager __________ (leave) on her desk but she didn’t notice them.
237.The house could fall down soon if no one __________ (do) some quick repair work. 238.More patients __________ (treat) in hospital this year than last year.
239.Sales of CDs __________ (increase) since the early 1990s, when people began to enjoy the advantages of this new technology.
245.The crazy fans __________ (waite) patiently for two hours, and they would wait till the movie star arrived.
248.I arrived late. I __________ (expect) the road to be so icy.
255.The news came as no surprise to me. I __________(know) for some time that the factory was going to shut down.
253.At this time tomorrow we__________ (fly) over the Atlantic.
198.E-mail, as well as telephones, __________ (play) an important part in daily communication
244.All the employees except the manager __________ (encourage) to work online at home. 243.Go straight on and you__________ (see) a church. You won’t miss it.
250.Although he __________ (live) with us for years, he hasn’t left us much impression. 251.All morning as she __________ (grow) for the medical report from the doctor, her nervousness __________ (grow).
252.Why don’t you put the meat in the fridge? It __________ (stay) fresh for several days.
并列句
86. Stand over there and you will be able to see it better.
107. Don't mention that at the beginning of the story, or it may give away the shocking ending. 120. Have a good rest, and you need to save your energy for the tennis match this afternoon. 89. There are many kinds of sports, but my favorite is swimming.
106. If anybody calls, tell them I am out, and ask them to leave their names and address. 207.Work harder and you will find it not difficult to learn.
230.Don’t be discouraged. Take things as t hey are and you will enjoy every day of your life. 234.Let’s keep to the point or we will never reach any decisions.
词辩
46. In some western countries, demand for graduates from MBA courses has fallen down .
47. The evening news comes on at seven o’clock and lasts only thirty minutes.
48. It was not a serious illness, and she soon got over it.
49. The final examination is coming up soon. It is time for us to get down to our studies.
51. The Internet has brought about big changes in the way we work.
102. On hearing the news of the accident in the coal mine, she went pale.
103. We have to get in the wheat as soon as possible because a storm is on the way.
109. The taxi driver often reminds passengers to take their belongings when they leave the car. 112. His mother had thought it would be good for his character to get away from home and earn some money on his own.
114. Would you slow down a bit please? I can not keep up with you.
115. Nick is looking for another job because nothing he does satisfies his boss.
118. It is wise to have some money put away for old age.
166.Many people agree that a knowledge of English is a must in international trade today.
194.News reports say peace talks between the two countries have broken down with no agreement reached.
197.My name is Robert, but most of my friends call me Bob for short.
200.The car accident was similar to the one that happened in 1973
240.The discussion came alive when an interesting topic was brought in.
242.She has set a new record, that is, the sales of her latest book have reached 50 million.
18数学分析-1复习题参考答案 一、选择题 1.函数1 ()ln(2) f x x = -的连续区间是 （ B ） A. (2,)+∞ ; B. (2,3)(3,)?+∞; C. (,2)-∞ ; D. (3,)+∞. 2.若函数x x x f = )(，则=→)(lim 0 x f x （ D ）. A.0 ; B.1- ; C.1 ; D.不存在. 3.下列变量中，是无穷小量的为（ C ）. A.1ln (0)x x +→; B.cos (0)x x →;C.ln (1)x x → ;D.22(2)4 x x x -→-. 4. 1lim(1)1 n n n →∞ + =+（ B ）. 1 2.1 ...-A B e C e D e 5.1lim(1)1 →∞ + =-n n n （ B ）. 12.1...-A B e C e D e 6.下列两个函数是同一函数的是 （ C ） A. ()3,()f x x x ?=+=41 ()ln ,()ln 4 f x x x x ?== ; C. 2 2 ()sin cos ,()1f x x x x ?=+= ; D. 2 (1)(),()11 x f x x x x ?-= =-- . 7.22 39 lim 712 x x x x →-=-+ ( C ) A.0 ; B.25- ; C.6- ; D. 7 6 . 8．0sin 2lim →=x x x （ D ） A. 0 ; B. 1 ; C. 3 ; D . 2 . 9.=→x x x 1 sin lim 2 （ C ）. 1 1A B C D ∞-
一、填 空 题 1.将函数展开为麦克劳林级数，则=-+x x 11ln ______________________ 。 2.x x x f sin )(= 在( - π,π )上展开的傅里叶级数为________ ______ 。 3.已知方程 z e z y x =++可以确定隐函数，那么 =???y x z 2________________________ __。 二、单项选择题 1、幂级数∑∞ =-112n n x n 的收敛域与和函数分别是___________ 。 A 、 [ - 1 , 1 ] ，2)1(1x x -+； B 、( - 1, 1 ) ，3 )1(1x x -+； C 、(- 1 , 1 ) ，)1(1x x -+； D 、[ - 1 , 1 ] ，4) 1(1x x -+。 2、 22)(y x x f +=在( 0 , 0 )满足 ________ 。 A 、连续且偏导数存在； B 、不连续但偏导数存在； C 、连续但偏导数不存在； D 、不连续且偏导数不存在。 4、函数222z y x u -+=在点A(b,0,0)及B(0,b,0)两点的梯度方向夹 角 。 A 、2π； B 、3 π； C 、4 π； D 、6π。 三、计算题 1、设),(y x z z =是由隐函数0),(=++ x z y y z x F 确定，求表达式y z y x z x ??+??，并要求简化之
3、设函数),(v u x x =满足方程组???==0 )),(,(0)),(,(v x g y G u y f x F ，其中g f G F ,,,均为连续可微函 数，且x y g f G F G F 2211≠，记1F 为F 对第一个变量的偏导数，其他类推，求v x u x ????,。
1 北京理工大学2012-2013学年第一学期 工科数学分析期末试题(A 卷) 一. 填空题（每小题2分, 共10分） 1. 设?????<≥++=01arctan 01)(x x x x a x f 是连续函数，则=a ___________. 2. 曲线θρe 2=上0=θ的点处的切线方程为_______________________________. 3. 已知),(cos 4422x o bx ax e x x ++=- 则_,__________=a .______________=b 4. 微分方程1cos 2=+y dx dy x 的通解为=y __________________________________. 5. 质量为m 的质点从液面由静止开始在液体中下降, 假定液体的阻力与速度v 成正比, 则质点下降的速度)(t v v =所满足的微分方程为_______________________________. 二. (9分) 求极限 21 0)sin (cos lim x x x x x +→. 三. (9分) 求不定积分?+dx e x x x x )1arctan (12. 四. (9分) 求322)2()(x x x f -=在区间]3,1[-上的最大值和最小值. 五. (8分) 判断2 12arcsin arctan )(x x x x f ++= )1(≥x 是否恒为常数. 六. (9分) 设)ln(21arctan 22y x x y +=确定函数)(x y y =, 求22,dx y d dx dy . 七. (10分) 求下列反常积分. (1);)1(1 22?--∞+x x dx (2) .1)2(1 0?--x x dx 八. (8分) 一垂直立于水中的等腰梯形闸门, 其上底为3m, 下底为2m, 高为2m, 梯形的上底与水面齐平, 求此闸门所受 到的水压力. (要求画出带有坐标系的图形) 九. (10分) 求微分方程x e x y y y 3)1(96+=+'-''的通解. 十. (10分) 设)(x f 可导, 且满足方程a dt t f x x x f x a +=+?)())((2 ()0(>a , 求)(x f 的表达式. 又若曲线 )(x f y =与直线0,1,0===y x x 所围成的图形绕x 轴旋转一周所得旋转体的体积为,6 7π 求a 的值. 十一. (8分) 设)(x f 在]2,0[上可导, 且,0)2()0(==f f ,1sin )(1 21 =?xdx x f 证明在)2,0(内存在ξ 使 .1)(='ξf
数学分析（1）复习题（一） 一、按要求写出下列定义的数学描述（4?/5=20/） 1、A x f x ≠+∞ →)(lim 的X -ε正面描述为 2、由Cauchy 收敛准则，若数列{}n x 收敛，则 3、η为非空数集S 的下确界即 4、a 为无限集合S 的聚点即 5、区间套[]{}n n b a ,的定义为 二、计算题（8?/6=48/） 1、求2 1 0)sin (lim x x x x →. 2、求)sin 2 sin 1(sin lim 2 2 2 n n n n n +???++++∞ →π π π . 3、确定x x x f sin )(=的间断点并判断其类型. 4、设x x x x f x x sin )(sin +=，求)(x f '. 5、x y 3sin =，求)(n y . 6、求x e x x f 2)(=带有Lagrange 余项的n 阶Maclaurin 展式. 7、设)7ln 12(4-=x x y ，试确定其凹凸区间及拐点. 8、确定,,b a 使函数???≥++<+=0,10,2)(2x bx x x a e x f x 在0=x 处连续. 三、证明题（4?/8=32/） 1、用δε-定义证明.10 3 1lim 2 3 =+→x x x 2、设)(x f 在[]b a ,上连续，在()b a ,内可导，证明至少存在(),,b a ∈ξ使得下式成 立： .ln )()()(a b f a f b f ξξ'=- 3、证明：若f 在[]b a ,上连续，)(lim x f x +∞ →存在且有限，则f 在[)+∞,a 上一致连续.
4、设f 在()+∞,a 内可微并且,0)(lim ='+∞ →x f x 证明0) (lim =+∞ →x x f x . 数学分析（1）复习题（二） 一、单项选择题（5?/3=15/） 1、=∞→n n n 2lim （ ） A.0；B 、2 1；C 、1；D 、2. 2、设函数是n 次多项式，则=+)()1(x f n （ ） A 、n ；B 、n+1；C 、0；D 、1. 3、如果当0→x 时,)(x f 是x 的高价无穷小量,则=→x x f x sin ) (lim 0 ( ). A. 2 1 ； B 、0； C 、2； D 、1. 4、设f 在x 的某邻域内有有定义，则下列命题哪一个为假？（ ） A.f 在点x 可微，则f 在点x 连续； B 、f 在点x 不连续，则f 在点x 一定不可导； C 、f 在点x 连续，则f 在点x 可微； D 、f 在点x 可导当且仅当f 在点x 可微. 5、函数2)(x x f =与x x g =)(定义在[)∞,0上，它们在定义区间上是一致连续的 吗？（ ） A.两个都是一致连续的； B 、两个都不是一致连续的； C 、f 是一致连续的，g 不是一致连续的； D 、f 不是一致连续的，g 是一致连续的. 二、填空题（5?/3=15/） 1、如果要使函数x x x f 1 sin )(=在点0=x 连续，需重新定义=)0(f 2、设1)(0='x f ，则=--+→h h x f h x f h ) ()(lim 000 3、函数???≤>+=,1,, 1,)(2x x x b ax x f 在1=x 处可导，则=+2013b a 4、设)(x y y =由方程e xy e y =+确定，则=')0(y 5、设???-=-=t y t t x cos 1sin ，则 == 2 π t dx dy
1. 掌握用线段图法来分析题中的年龄关系. 2. 利用已经学习的和差、和倍、差倍的方法求解年龄问题． 知识点说明： 一、年龄问题变化关系的三个基本规律： 1. 两人年龄的倍数关系是变化的量. 2. 每个人的年龄随着时间的增加都增加相等的量； 3. 两个人之间的年龄差不变 二、年龄问题的解题要点是： 1．入手：分析题意从表示年龄间倍数关系的条件入手理解数量关系． 2．关键：抓住“年龄差”不变． 3．解法：应用“差倍”、“和倍”或“和差”问题数量关系式． 4．陷阱：求过去、现在、将来。 年龄问题变化关系的三个基本规律： 1．两人年龄的差是不变的量； 2.两个人的年龄增加量是不变的； 3．两人年龄的倍数关系是变化的量； 年龄问题的解题正确率保证：验算！ 年龄差不变 【例 1】 小卉今年6岁，妈妈今年36岁，再过6年，小卉读初中时，妈妈比小卉大多少 岁？ 【考点】年龄问题 【难度】1星 【题型】解答 【解析】 这道题有两种解答方法： 方法一：解答这道题，一般同学会想到，小卉今年6岁，再过6年6612+=（岁）；妈妈例题精讲 知识精讲 教学目标 6-1-8.年龄问题（一）
今年36岁，再过6年是（366 -=（岁）． +）岁，也就是42岁，那时，妈妈比小卉大421230 方法二：聪明的同学会想，虽然小卉和妈妈的岁数都在不断变大，但她们两人相差的岁数永远不变．今年妈妈比小卉大（366 -）岁，不管过多少年，妈妈比小卉都大这么多岁．通过比较第二种方法更简便．列式：36630 -=（岁），再过6年，小卉读初中时，妈妈比小卉大30岁． 【答案】30岁 【例2】爸爸妈妈现在的年龄和是72岁；五年后，爸爸比妈妈大6岁．今年爸爸妈妈二人各多少岁？ 【考点】年龄问题【难度】1星【题型】解答 【解析】五年后，爸爸比妈妈大6岁，即爸妈的年龄差是6岁．它是一个不变量．所以爸爸、妈妈现在的年龄差仍然是6岁．这样原问题就归结成“已知爸爸、妈妈的年 龄和是72岁，他们的年龄差是6岁，求二人各是几岁”的和差问题．爸爸的年龄：726239 （）(岁) +÷= 妈妈的年龄：39633 -=(岁) 【答案】爸爸39岁，妈妈33岁 【例3】姐姐今年13岁，弟弟今年9岁，当姐弟俩岁数和是40岁时，两人各应该多少岁？ 【考点】年龄问题【难度】2星【题型】解答 【解析】用线段图显示数量关系，可以看出这道题实际上就是前面总结过的和差问题.姐弟俩的年龄差总是1394 -=（岁），不管经过多少年，姐弟年龄的差仍是4岁，由图可见，如果从40岁中减去姐弟年龄的差，再除以2就得到所求的弟弟的年 龄，也就可以求出姐姐的年龄了. 弟弟的年龄：(404)218 +=（岁）． -÷=（岁），姐姐的年龄：18422 【答案】弟弟年龄18岁，姐姐22岁 【例4】欢欢对乐乐说：“我比你大8岁，2年后，我的年龄是你的年龄的3倍。”欢欢现在岁？ 【考点】年龄问题【难度】2星【题型】填空
人教版新课标实验教科书小学一年级下册 数学第三单元《图形的拼组》教学设计 灯盏小学余文田设计并执教 第一课时教学设计 教学内容：图形的拼组（一） 教学目标： 1、通过直观使学生知道长方形、正方形的形状和边的特点,并能用自己的语言描述长方形、正方形的特征。 2、通过折一折、摆一摆、剪一剪、拼一拼，加深对长方形和正方形的认识，能辨别、区分这两种图形。 3、使学生在丰富的活动中感受学习数学的乐趣，从中获得积极的情感体验。教学重点：在操作和观察中让学生明白长方形和正方形的特征。 教学难点：在摆一摆中用多种方法正确地摆出长方形和正方形，体验图形的分解与组合。 教学过程： 一、认一认，说一说。 出示长方形，请学生说一说长方形的边有什么特点。（两条长边相等，两条短边相等） 再出示正方形，也请学生说一说正方形的边有什么特点。（四条边长度都相等） 二、折一折。 1、拿出每人事先准备好的长方形、正方纸，师生共同操作。 （1）引导学生先看正方形，先上下对折，边要对齐，看上下两部分是不是完全合在一起，上下两条边是不是完全合在一起；再左右对折，方法同上。然后把正方形纸的两个斜对着的角对齐，折后观察折痕两旁的部分是不是完全合在一起；再继续对折一次，观察折出的几部分是不是完全合在一起，四条边是不是完
全合在一起。（学生自己动手操作，得出结论） （2）用长方形纸折一折，看一看长方形的边长怎么样。 要求学生先思考：怎样折长方形的纸，就能使分成的两部分完全合在一起？然后，自己动手折一折，以四人一小组进行讨论，再翻开课本进行核对。 （3）区分长方形和正方形。 拿出事先准备好的长方形和正方形（长方形的一边与正方形边长相等）先将两个图形重叠在一起，让学生观察：两个图形的边有什么关系？如图： 2、小结：今天我们学了什么？大家有什么收获？ 3、学做风车。 （1）先出示一个风车，将风车展开，让学生观察风车是由什么图形剪拼成的。（2）拿出准备的长方形纸，同桌互相商量，想一想要折一个风车该怎么做。学生动手操作。（先将长方形纸剪成一个正方形，再动手做成一个风车）如图：书P27风车图。 第二课时教学设计 教学内容： 图形的拼组（二） 教学目标： 1、让学生通过剪一剪、拼一拼、摆一摆等方法，加深对正方形、长方形、三角形和圆的感性认识。 2、初步认识这些图形之间的关系，同时通过对图形的分解与组合，初步发展学生的想象力和创造力。 教学重点： 通过各种方法弄清正方形、长方形、三角形和圆的特征，并能进行判断 教学难点： 图形的分解与组合 教学过程： 一、复习。 1、把下列图形的题号填入相应的括号内。 长方形（）正方形（） 三角形（）圆（）
一、把每组中不是同一类的圈上。 考查目的：学会把一些常见的物品进行分类。 答案：第①小题：小花； 第②小题：蛋糕； 第③小题：西瓜 解析：第①小题：小花是植物，其它三样是交通工具；第②小题：蛋糕是食品，其它三样是动物；第③小题：西瓜是水果，其它三样属于服装类。 二、我会连。 考查目的：进一步加深对分类的理解。 答案：蔬菜：辣椒、胡萝卜、白菜、南瓜、西红柿；水果：菠萝、葡萄、火龙果、草莓、香蕉。 解析：略。 三、整理卡片。
考查目的：按给定的不同标准进行分类急速的巩固练习，体会分类标准与分类结果的关系，并进行简单的计数。 答案：第（1）小题：青蛙卡片4张，小鸟卡片5账，奶牛卡片1张；第（2）小题：正方形卡片3张，圆形卡片4张，三角形卡片3张。 解析：略。
一、分一分。 （1）将这些物品分成两组，可以怎样分？把分组的结果表示出来。 （2）你能提出什么数学问题？ 考查目的：让学生自选标准将这些物品分成两类，并用简单的统计表呈现出来。 答案：这些物品一般分为两类：一类是学习用品，有橡皮、铅笔、地球仪、尺子、书、铅笔刨、文具盒；另一类是生活用品，有梳子、毛巾、吹风机、镜子、牙膏、牙刷。 解析：虽然自定义分类标准比较抽象，但由于这些物品都是学生比较熟悉的，所以难度应不大。 二、下面是动物园里集中动物的数量。 （1）动物园里，（ ）最多，（ ）最少。 （2）小猴比梅花鹿多多少只？
（3）你还能提出什么数学问题？并解答。 考查目的：让学生直接根据简单统计表中的数据进行简单的数据分析，体会统计的作用。 答案：第（1）小题：猴子最多，熊猫最少。第（2）小题：18-9 = 9（只）。第（3）小题：略。 解析：引导学生学会看简单统计图，知道上面一行的动物和下面一行相对应的数是表示它的数量。 三、下面是今年2月份的天气情况。 （1）数一数每种天气各有多少天？ （2）根据上面数出的结果涂格子。
小学数学一年级下册〈〈图形的拼组》教案教学设 计 第2课时图形的拼组 设计说明 本课时的教学目标是让学生在拼一拼、摆一摆等活动中，进一步 了解平面图形的特征，了解平面图形之间的内在联系。 基于上述目标，教学设计突出了以下几个方面： 1.注重调动学生学习的积极性。 将一组漂亮的由平面图形拼成的图案呈现给学生，让学生一边欣赏图案，一边找出图案中有哪些学过的平面图形。这样的安排，不仅使学生巩固了先前学过的知识，乂体会了图形世界的神奇和美妙，从而充分激发了学生的求知欲望，使学生积极主动地参与到学习活动中来。 2.注重培养学生的动手操作能力和创新意识。 在教学过程中，给学生提供充足的时间和空间，让学生动手操作，经历用同样的平面图形进行拼组的过程，在此基础上给学生提供自由拼组的时间，让他们充分发挥自己的想象力和聪明才智，尽情地创作。学生通过这些活动，不仅有利于提高他们动手操作的能力，更有利于培养他们的创新能力。 课前准备 教师准备PPT课件 学生准备长方形、正方形、平行四边形、三角形若干胶
水白纸板 教学过程 O复习旧知，导入新课 课件出示用学过的、同样的平面图形拼组的各种图案，请学生说一说里面有哪些图形是自己认识的。 师：这些图案漂亮吗？我们这节课就来练习拼一拼，看看用学过的、同样的平面图形还能拼出哪些图案。（板书：图形的拼组） 设计意图：在各种拼组图案中找到自己认识的平面图形，不仅巩固了学过的知识，而且使学生初步领略到图形拼组后的神奇和美妙，激发了学生的求知欲和兴趣。 O动手操作，探究新知 1.课件出示两个同样的长方形：把这两个长方形拼在一起， 你能拼出什么图形呢？ （学生用手中的学具进行拼组） 预设 生1:我用两个同样的长方形拼出了一个更大的长方形。 生2:我用两个同样的长方形拼出了一个正方形。 师：拼一拼，我们会有更多的发现，接下来大家小组合作，继续体验图形拼组的乐趣。 2.小组合作，自选图形进行拼组。 （1）在小组内，选择两个或几个同样的平面图形进行拼组，
山东师范大学2007-2008学年第一学期期末考试试题 （时间：120分钟 共100分） 课程编号： 4081101 课程名称：数学分析 适用年级： 2007 学制: 四 适用专业：数学与信息试题类别： A (A/B/C) 2分，共20分） 1. 数列{}n a 收敛的充要条件是数列{}n a 有界. ( ) 2. 若0N ?>, 当n N >时有n n n a b c ≤≤, 且lim lim n n n n a c →∞ →∞ ≠, 则lim n n b →∞ 不存在. ( ) 3. 若0 lim ()lim ()x x x x f x g x →→>, 则存在 00(;)U x δ使当00(;)x U x δ∈时,有()()f x g x >. ( ) 4. ()f x 为0x x →时的无穷大量的充分必要条件是当00(;)x U x δ∈时,()f x 为无界函数. ( ) 5. 0x =为函数 sin x x 的第一类间断点. ( ) 6. 函数()f x 在[,]a b 上的最值点必为极值点. ( ) 7. 函数21,0,()0, 0x e x f x x -?? ≠=??=?在0x =处可导. ( ) 8. 若|()|f x 在[,]a b 上连续, 则()f x 在[,]a b 上连续. ( ) 9. 设f 为区间I 上严格凸函数. 若0x I ∈为f 的极小值点，则0x 为f 在I 上唯一的极小值点. ( ) 10. 任一实系数奇次方程至少有两个实根. ( )
二、 填空题（本题共8小题，每空2分，共20分） 1. 0 lim x x x + →=_________________. 2. 设2 ,sin 2x u e v x ==，则v d u ?? = ??? __________________. 3. 设f 为可导函数，(())x y f f e =, 则 y '=_______________. 4. 已知3(1)f x x +=, 则 ()f x ''=_______________. 5. 设 ()sin ln f x x x =, 则()f π'=_______________ . 6. 设21,0, (),0; x x f x ax b x ?+≥=?+
华南农业大学期末考试试卷（ A 卷 ） 2009学年第1学期 考试科目：数学分析I 考试类型：（闭卷）考试 考试时间： 120 分钟 学号 姓名 年级专业 一、 填空题 (每题4分，共24分) 1. 用N ε-语言叙述数列极限的柯西准则： . 2. 用εδ-语言叙述()0lim x x f x A →=： . 3. (归结原则)设()f x 在00(U x ;)δ内有定义，()0lim x x f x →存在的充要条件是： . 4. 设0x →时，函数1(1)1x x --+与x α是同阶无穷小量，则α= . 5. 曲线221x t y t t ?=-??=-??在1t =处的切线方程为： . 6. 设函数,0sin ()3,02(1),0x ax be x x f x x a b x x ?+?? 在0x =处连续，则a =_____，b =____．
二、 计算题. （共52分） 1. 求下列极限（每题6分，共24分） (1) 7020 90(36)(85)lim (51) x x x x →+∞+--. (2) 01lim []x x x →. (3) 30tan sin lim ln(1)x x x x →-+. (4) 2132lim ()31x x x x -→+∞+- .
2. 求下列导数（每小题6分，共18分） （1）32(arctan )y x =. （2）设cos x y e x =， 求(4)y . （3）求由参数方程()()()x f t y tf t f t '=??'=-? (设()f t ''存在且不为零)所确定的函数()y f x =的二阶导数22d y dx .
习题（八）——减数分裂 一、选择题 1、下列哪一项不是成熟的生殖细胞（） A．精子细胞 B．卵细胞 C．精子 D．配子 2、细胞内没有同源染色体的是（） A．体细胞 B．精原细胞 C．初级精母细胞 D．次级精母细胞 3、联会是指（） A．同源染色体配对的过程 B．同源染色体配对时期的细胞 C．一个配对同源染色体 D．同源染色体配对时期细胞中所有的染色体 4、在减数分裂过程中，染色体的复制发生在（） A．第一次分裂前的间期 B．联会时 C．四分体时期 D．第一次分裂与第二次分裂过程之间 5、人的体细胞内有46条染色体，在减数分裂过程中的某时期出现四分体的个数是（） A.46 B．16 C．23 D．64 6、在减数分裂过程中，每一个染色体的着丝点分裂后形成二条（） A．染色单体 B．非同源染色体 C．同源染色体 D．相同的染色体 7、下列细胞中没有同源染色体的是（） A．受精卵 B．口腔上皮细胞 C．初级精母细胞 D．红细胞 8、在卵细胞的形成过程中，卵原细胞、初级卵母细胞、次级卵母细胞、卵细胞的比例为（） A．1∶1∶2∶4 B．1∶1∶1∶2 C．1∶1∶4∶4 D．1∶1∶1∶1 9、具有24条染色体的水稻植株，其生长点的一个细胞连续分裂三次后，产生的子细胞中有染色体（） A．96 B．48 C．24 D．12 10、家兔是进行有性生殖的生物，它所以能保持前后代染色体数目的恒定，是因为在生殖过程中要进行 （） A．减数分裂和有丝分裂 B．有丝分裂和受精作用 C．减数分裂和受精作用 D．染色体复制和平均分配 11、下列细胞中，不能进行有丝分裂的是（） A．精原细胞 B．卵原细胞 C．受精卵 D．初级卵母细胞 12、人体细胞内有46条染色体，次级卵母细胞中，姐妹染色单体有（） A．46 B．23 C．92 D．184 13、蟾蜍的体细胞有丝分裂后期染色体数为44条，其精子中染色体数是（） A．11个 B．22个 C．33个 D．44个 14、一雌蛙产1000粒卵，一雄蛙产100万个精子，那么形成这些卵和精子的雌蛙卵巢中的卵原细胞数目 及雄蛙精巢中的精原细胞分别为（） A．250个和250万个 B．1000个和25万个 C．1000个和100万个 D．250个和100万个 15、有关受精作用的叙述中，不正确的是（） A．受精卵中全部遗传物质的一半来自精子 B．受精后，精子的细胞核与卵细胞的核融合 C．合子中的染色体一半来自父方，一半来自母方 D．合子中染色体数与本物种体细胞染色体数一致 16、在减数分裂的第一次细胞分裂过程中，染色体变化的顺序是（）
一年级奥数练习题及答案 1、小明和小红都集邮票。小明给了小红6枚后，两人的邮票同样多，原来小明的邮票比小红的多（）枚。 2、王老师和张老师带着14个小朋友到公园去玩，他们一共要买（）张票 3、小军跟爸爸到外地旅游，爸爸买一张火车票是5元，小军买半票，他们来回一共要付（）元。 4、植树节在四边形花坛边上植树，要使每边有3棵树，那么最少需要（）树 5、小朋友排队做操，红红排在队伍的中间，无论上从前往后数还是从后往前数，都是第10个，这一队一共有（）个小朋友做操 6、校园门后摆了两排菊花，每排6盆，现在想在每两盆菊花之间插3盆玫瑰花，问需要（）盆玫瑰花 7、一次上体育课排队，从左边开始报数，明明报了“7”，林林报了“10”；从右边开始报数，明明报了“7”，林林应该报（），这一队共有（）人 8、去年，爸爸比小强大25岁，今年小强有10岁，今年爸爸（）岁 9、小朋友在玩捉迷藏的游戏，龙龙捉到了5人，还有4个人没有找到，他们一起玩的有（）人 10、小强他们班有48人，数学测试时，小强考了第15名，你知道如果倒数小强这次考试成绩应排第（） 1、小明和小红都集邮票。小明给了小红6枚后，两人的邮票同样多，原来小明的邮票比小红的多（ 12）枚。 2、王老师和张老师带着14个小朋友到公园去玩，他们一共要买（ 16 ）张票 3、小军跟爸爸到外地旅游，爸爸买一张火车票是5元，小军买半票，他们来回一共要付（ 15 ）元。 4、植树节在四边形花坛边上植树，要使每边有3棵树，那么最少需要（ 8 ）树 5、小朋友排队做操，红红排在队伍的中间，无论上从前往后数还是从后往前数，都是第10个，这一队一共有（ 19 ）个小朋友做操 6、校园门后摆了两排菊花，每排6盆，现在想在每两盆菊花之间插3盆玫瑰花，问需要（ 30 ）盆玫瑰花 7、一次上体育课排队，从左边开始报数，明明报了“7”，林林报了“10”；从右边开始报数，明明报了“7”，林林应该报（ 4 ），这一队共有（ 13 ）人 8、去年，爸爸比小强大25岁，今年小强有10岁，今年爸爸（ 35 ）岁 9、小朋友在玩捉迷藏的游戏，龙龙捉到了5人，还有4个人没有找到，他们一起玩的有（ 10 ）人
数学分析1 期末考试试卷（A 卷） 一、填空题（本题共5个小题，每小题3分，满分15分） 1、设 82lim =?? ? ??-+∞→x x a x a x ， 则 =a 。 2、设函数) 2(1 )(--=x x e x f x ，则函数的第一类间断点是 ，第二类间断点 是 。 3、设)1ln(2 x x y ++=，则=dy 。 4、设)(x f 是连续函数，且dt t f x x f )(2)(1 0?+=，则=)(x f 。 5、xdx arctan 1 ?= 。 二、单项选择题（本题共5个小题，每小题3分，满分15分） 1、设数列n x 与数列n y 满足0lim =∞ →n n n y x ，则下列断言正确的是（ ）。 （A ）若n x 发散，则n y 必发散。 （B ）若n x 无界，则n y 必无界。 （C ）若n x 有界，则n y 必为无穷小。 （D ）若n x 1 为无穷小，则n y 必为无穷小。 2、设函数x x x f =)(，则)0(f '为（ ）。 （A ） 1。 （B ）不存在。 （C ） 0。 （D ） -1。 3、若),() ()(+∞<<-∞=-x x f x f 在)0(，-∞内0)(,0)(<''>'x f x f ，则 )(x f 在),0(+∞内有（ ）。 （A ）0)(,0)(<''>'x f x f 。 （B ）0)(,0)(>''>'x f x f 。
（C ）0)(,0)(<''<'x f x f 。 （D ）0)(,0)(>''<'x f x f 。 4、设)(x f 是连续函数，且? -=dt t f x F x e x )()(，则)(x F '等于（ ） 。 （A ）() )(x f e f e x x ----。 （B ）() )(x f e f e x x +---。 （C ） () )(x f e f e x x --- 。 （D ）() )(x f e f e x x +--。 5、设函数x x a x f 3sin 31sin )(+=在3 π =x 处取得极值，则（ ）。 （A ）)3(,1πf a =是极小值。 （B ）)3 (,1π f a =是极大值。 （C ）)3(,2πf a =是极小值。 （D ）)3 (,2π f a =是极大值。 三、计算题（本题共7个小题，每小题6分，满分42分） 1、求 ) 1ln(sin 1tan 1lim 30x x x x ++-+→ 2、设4lim 221=-++→x x b ax x x ，求 b a 、。
第一章复习题 一.填空 1、数集,...}2,1:)1({=-n n n 的上确界为 1 ，下确界为 -1 。 2、 =∈-=E R x x x E sup ,|][{则 1 , =E inf 0 ; 3、)(lim 2 n n n n -+∞ → = _______ 1 2 ________。 4、设数列}{n a 递增且 a a n n =∞ →lim （有限）. 则有a = {}sup n a . 5. 设,2 12,21221 2n n n n n n x x +=-=- 则 =∞→n n x lim 1 二． 选择题 １、设)(x f 为实数集R 上单调增函数，)(x g 为R 上单调减函数，则函数 ))((x g f 在R 上( B )。 Ａ、是单调递增函数; Ｂ、是单调递减函数; Ｃ、既非单调增函数，也非单调减函数 ; Ｄ、其单调性无法确定. 2、在数列极限的“δε-”极限定义中,ε与δ的关系是（ B ） A 、 先给定ε后唯一确定δ; B 、 先给定ε后确定δ,但δ的值不唯一; C 、 先给定δ后确定ε; D 、 δ与ε无关. 3、设数列{}(0,1,2,...)n n a a n ≠=收敛,则下列数列收敛的是（ D ） A 、}1 { 2n a ； B 、}1{a n ； C 、 }1{a n ； D 、}{n a . 4. 若数列}{n x 有极限a ，则在a 的ε邻域之外，数列中的点（ B ） (A) 必不存在； (B) 至多只有有限多个； (C) 必定有无穷多个； (D) 可能有有限多个，也可能有无穷多个. 5．设a x n n =∞ →||lim ，则 （ D ） (A) 数列}{n x 收敛； (B) a x n n =∞ →lim ； (C) a x n n -=∞ →lim ； (D) 数列}{n x 可能收敛，也可能发散。 6. 设}{n x 是无界数列，则 （ D ） (A) ∞=∞ →n n x lim ； (B) +∞=∞ →n n x lim ；
小学一年级奥数练习题及答案 1.小明从家到学校跑步来回要10分钟.如果去时步行.回来时跑步一共需 要12分钟.那么小明来回都是步行需要几分钟? 答案与解析： “来回”包括“去时”和“回来时”共两趟.所以小明跑一趟要10÷2=5 分钟.步行一趟就是12-5=7分钟.来回都步行要14分钟。 2.白雪公主和7个小矮人一起玩游戏.过了一会儿.又来了6个小朋友跟他 们一起玩.现在一共有多少人在一起玩游戏? 答案与解析： 这道题的关键就是.我们在计算总人数的时候.不能把白雪公主给忘掉了. 原来有白雪公主和7个小矮人做游戏.一共是8个人.后来又来了6个小朋友.就 要加上后来的小朋友.一共是1+7+6=14(人)在一起玩游戏. 3.如果从甲班调一名学生到乙班.甲、乙两班人数相同。如果从乙班调一 名学生到丙班.丙班就比乙班多2人.甲班和丙班相比.哪个班人多?多几人? 答案与解析： 甲班比乙班多2人.乙班和丙班人数相同。 甲班比丙班多2人。 4.一些十位数字和个位数字相同的二位数能够由十位数字和个位数字不同 的两个二位数相加得到.如12+21=33(人们通常把12和21这样的两个数叫做一 对倒序数).问在100之内有多少对这样的倒序数? 答案与解析：十位数字和个位数字相同的二位数有：11、22、33、44、55、66、77、88、99九个.其中11和22都不能由一对倒序数相加得到.其他各数的 倒序数是： 33：12和21…………………………1对 44：13和31…………………………1对 55：14和41、23和32………………2对 66：15和51、24和42……………2对 77：16和61、25和52、34和43……3对 88：17和71、26和62、35和53……3对 99∶18和81、27和72、36和63、45和54…4对 总数=1+1+2+2+3+3+4=16对. 5.*家距学校2千米.一次他上学走了1千米.想起忘带铅笔盒.又回家去取。这次他到学校共走了多少千米? 答案与解析：由题意我们能够知道他走了1千米之后还要回去.说明他多 走了1+1=2(千米)再加上他家距学校的距离就是他这次共走的：2+2=4(千米)
《数学分析》考试题 一、（满分10分，每小题2分）单项选择题： 1、{n a }、{n b }和{n c }是三个数列，且存在N,? n>N 时有≤n a ≤n b n c ， （ ） A. {n a }和{n b }都收敛时，{n c }收敛； B. {n a }和{n b }都发散时，{n c }发散； C. {n a }和{n b }都有界时，{n c }有界； D. {n b }有界时，{n a }和{n c }都有界； 2、=)(x f ??? ????>+=<,0 ,2.( ,0 ,0, ,sin x x k x k x x kx 为常数） 函数 )(x f 在 点00=x 必 （ ） A.左连续； B. 右连续 C. 连续 D. 不连续 3、''f （0x ）在点00=x 必 （ ） A. x x f x x f x ?-?+→?)()(lim 02020 ； B. ' 000)()(lim ??? ? ???-?+→?x x f x x f x ; C. '000)()(lim ???? ???-?+→?x x f x x f x ; D. x x f x x f x ?-?+→?)()(lim 0'0'0 ； 4、设函数)(x f 在闭区间[b a ,]上连续，在开区间（b a ,）内可微，但≠)(a f )(b f 。则 （ ） A. ∈?ξ（b a ,），使0)('=ξf ； B. ∈?ξ（b a ,），使0)('≠ξf ; C. ∈?x （b a ,），使0)('≠x f ； D.当)(b f >)(a f 时，对∈?x （b a ,），有)('x f >0 ； 5、设在区间Ⅰ上有?+=c x F dx x f )()(， ?+=c x G dx x g )()(。则在Ⅰ上有 （ ） A. ?=)()()()(x G x F dx x g x f ； B. c x G x F dx x g x f +=?)()()()( ； C. ?+=+c x G x F dx x F x g dx x G x f )()()]()()()([ ；
一 选择题(55分=25分) (A)1. 3.142和3.141分别作为π的近似数具有（）和（）为有效数字(有效数字) A. 4和3 B. 3和2 C. 3和4 D. 4和4 解,时,, m-n= -3,所以n=4,即有4位有效数字。当时,, ，m-n= -2,所以n=3,即有3位有效数字。 (A)2. 为了减少误差，在计算表达式时，应该改为计算，是属于（）来避免误差。（避免误差危害原则） A.避免两相近数相减； B.化简步骤，减少运算次数； C.避免绝对值很小的数做除数； D.防止大数吃小数 解：由于和相近，两数相减会使误差大，因此化加法为减法，用的方法是避免误差危害原则。 (B)3.下列算式中哪一个没有违背避免误差危害原则（避免误差危害原则） A.计算 B.计算 C.计算 D.计算 解:A会有大数吃掉小数的情况C中两个相近的数相减，D中两个相近的数相减也会增大误差 (D)4.若误差限为,那么近似数0.003400有()位有效数字。(有效数字) A. 5 B. 4 C. 7 D. 3 解：即m-n= -5,，m= -2，所以n=3，即有3位有效数字 (A)5.设的近似数为，如果具有3位有效数字，则的相对误差限为()（有效数字与相对误差的关系） A． B. C. D. 解：因为所以,因为有3位有效数字，所以n=3,由相对误差和有效数字的关系可得a的相对误差限为 二 填空题：(75分=35分)
1.设则有2位有效数字，若则a有3位有效数字。(有效数字) 解:,时,，，m-n= -4,所以n=2,即有2位有效数字。当时, ，m-n= -5,所以n=3,即有3位有效数字。 2.设 =2.3149541...，取5位有效数字，则所得的近似值x=2.3150(有效数字)解：一般四舍五入后得到的近似数，从第一位非零数开始直到最末位，有几位就称该近似数有几位有效数字，所以要取5位有效数字有效数字的话，第6位是5，所以要进位，得到近似数为2.3150. 3.设数据的绝对误差分别为0.0005和0.0002，那么的绝对误差约为 0.0007 。（误差的四则运算） 解：因为，， 4.算法的计算代价是由 时间复杂度 和 空间复杂度 来衡量的。(算法的复杂度) 5.设的相对误差为2%，则的相对误差为 2n% 。（函数的相对误差） 解：， 6.设>0，的相对误差为δ，则的绝对误差为 δ 。（函数的绝对误差） 解：，， 7.设，则=2时的条件数为 3/2 。(条件数) 解:， 三 计算题(220分=40分) 1.要使的近似值的相对误差限小于0.1%，要取几位有效数字？（有效数字和相对误差的关系） 解：设取n位有效数字,由定理由于知=4所以要使相对误差限小于0.1%,则，只要取n-1=3即n=4。所以的近似值取4位有效数字,其相对误差限小于0.1%。 2.已测得某场地长的值为，宽d的值为，已知试求面积的绝对误差限和
一、 判断题（每小题2分，共20分） 1.开域是非空连通开集，闭域是非空连通闭集． （ ） 2.当二元函数的重极限与两个累次极限都存在时，三者必相等． （ ） 3.连续函数的全增量等于偏增量之和． （ ） 4. xy y x f =),(在原点不可微． （ ） 5.若),(),(y x f y x f yx xy 与都存在，则),(),(y x f y x f yx xy =． （ ） 6. dy y x xy y ) 1(sin 2 1 +? +∞ 在)1,0(内不一致收敛． （ ） 7.平面图形都是可求面积的． （ ） 8.学过的各种积分都可以以一种统一的形式来定义. （ ） 9.第二型曲面积分也有与之相对应的“积分中值定理”． （ ） 10.二重积分定义中分割T 的细度 T 不能用}{max 1i n i σ?≤≤来代替． （ ） 二、 填空题（每小题3分，共15分） 1.设)sin(y x e z xy +=，则其全微分=dz ． 2.设 3 2),,(yz xy z y x f +=，则f 在点)1,1,2(0-P 处的梯度= )(0P grad ． 3.设L 为沿抛物线 22x y =,从)0,0(O 到)2,1(B 的一段，则?=+L ydx xdy ． 4.边长为a 密度为b 的立方体关于其任一棱的转动惯量等于 ． 5.曲面2732 22=-+z y x 在点（3,1,1）处的法线方程为 ． 三、计算题（每小题5分，共20分） 1．求极限 xy y x y x )(lim 22) 0,0(),(+→． 2． 设),(y x z z =是由方程z e z y x =++所确定的隐函数，求xy z ． 3．设 ]1,0[]1,0[?=A ，求??++=A y x ydxdy I 2 322)1( . 4．计算抛物线) 0()(2 >=+a ax y x 与x 轴所围的面积.