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1、专业好文档数学分析专题研究试题一、填空题1集合中的关系同时为反身的、对称的、（ ），则称关系为等价关系。2一个集合若不能与其一个真子集建立一个（ ），则称该集合为有限集。3函数在点的邻域内有定义，若（ ），则称函数在点处连续。4设是从到上的连续函数，满足：1）（ ）；，2）对于有,则是以为底的对数。5若函数是定义在上的连续函数，且满足：1）（ ）；2），当时，；3）,则分别称是正弦函数与余弦函数。6设为从集合到集合中的关系，若,有唯一的，使（ ），则称为（从到中的）映射。二、单项选择题1A= B C D2实数集是（ ）A有限集 B可列集 C不可列集 D空集3是从到的映射，且，则A= B C D
2、4函数在点处（ ）A间断 B连续 C 可导 D取得极小值5函数与在上有界，且，则在上（ ）。A有界 B无界 C有下界而无上界 D结论不定6下面结论（ ）是正确的。A若是单调函数，也是单调函数，则是单调函数。B若在数集上可导，且有界，则在上有界C若是周期函数，则是周期函数D若在数集上有界且可导，则在上有界三、计算题1求过抛物线上的点的切线方程。2已知，求。3已知，求的最小值。4若，求。四、证明题1设有映射，证明：（1）若是满射，则是满射（2）若是满射，且是单射，则是满射2若在点处连续，则在点处也连续3证明：方程在区间内有且仅有一个实根。4证明不是周期函数。参考答案：一、填空题（每小题3分，共18
3、分）1.传递的；2.双射；3.；4.1）;5.1）， 6. 。二、单项选择题（每小题3分，共18分）1.C；2.C；3.D；4.B；5. D；6. A三、计算题（每小题8分，共32分）1.解 首先计算过点的切线的斜率 4分所求的切线方程为 即 8分2.解 已知 （1）将代替，得 （2） 4分得 8分3.解 已知在内，是上凸函数，由上凸函数的定义有 5分即 而且当时，故是的最小值。 8分4.解 设，则 3分因，故 8分四、证明题（每小题8分，共32分）1.证明（1）因是满射，即,进一步有，故是满射。 4分（2）采用反证法。假设不是满射，即，则存在，但。设，使，由于是单射，故，即，这与是满射矛盾。
4、说明假设矛盾，即是满射。 8分2.证明 ，因为在点连续，故存在，当时，有 由绝对值不等式的 4分故对任意的，当时，有 即在点连续。 8分3.证明：设，则是上的连续函数，且 由介值定理，至少存在一点，使。 4分 由得，当时，。即在内严格单调增加。故有且仅有一点，使，即方程在内有且仅有一实根。 8分4.证明 采用反证法。假设是周期函数，因是连续函数且不是常值，故具有最小正周期，设为。选取自然数，使得。故存在使 4分另一方面，对于，有 这与式矛盾。故不是周期函数。 8分数学分析专题研究期末试题（2002.1）一、 一、填空题1.若，则.2.若，则.3.设，若，则称为从到上的 .4.若复数是某个整系数
5、多项式方程的根，则称是 数.5.设，则.6.设函数定义在开区间内，对于，有，则称是内的 函数.二、 二、单项选择题（每小题3分，共18分）1.设，有.A. B. C. D. 2.若，且，则A. B. C. D. 3.若在内连续，则在内（ ）A. 可导 B. 单调 C. 有界 D. 对称4.设是超越数，则是（ ）A.有理数 B.代数数 C. 无理数 D. 超越数5.与都是以为周期的周期函数，且，则（ ）A. A. 不是周期函数 B. B. 是以为周期的周期函数C. C. 是周期函数，但周期大于或等于D. 是周期函数，但周期小于或等于6.设是内充分光滑的严格下凸函数，则（ ）A. 在内必取到最小值
6、 B. 在内必取到最大值C. 在内有 D. 前三个结论都不对三、 三、计算题（每小题8分，共32分）1.设，求2.设，求3.求函数的极值4.已知重根号），求四、 四、证明题（每小题8分，共32分）1.证明（1）（4分） （2）（4分）2.证明 设数集与均有上界，则集合有上界，且3.证明 设，有4.证明 设是从到的连续函数，则存在点,使得.参考答案一、 一、填空题（每小题3分，共18分）1.，2.，3. 满射，4.代数数，5.，6.下凸二、单项选择题（每小题3分，共18分）1.，2.，3.，4. ，5.，6.三、计算题（每小题8分，共32分）1解 ， 2分， 7分故 8分2设，则， 3分代入得
7、8分3解 3分令，得，易验证是极大值点，是极小值点， 6分极大值，极小值 8分4.解 显然，且，即数列，单调增加且有上界，故存在，设，由可得， 5分即，解得 8分四、证明题（每小题8分，共32分）1. 证明：（1）若设表示的补集，则有 4分（2） 8分 2. 证明：，有，故，即是的一个上界.，使得，即存在，使得故 8分3.证明：设，则，即是严格下凸，根据 有 8分4.证明：令，则是上的连续函数.若，则选取结论得证.若，则选取结论得证. 4分否则有，则，由介值定理，存在，使得，即. 8分If we dont do that it will go on and go on. We have to
8、stop it; we need the courage to do it.His comments came hours after Fifa vice-president Jeffrey Webb - also in London for the FAs celebrations - said he wanted to meet Ivory Coast international Toure to discuss his complaint.CSKA general director Roman Babaev says the matter has been exaggerated by
9、the Ivorian and the British media.Blatter, 77, said: It has been decided by the Fifa congress that it is a nonsense for racism to be dealt with with fines. You can always find money from somebody to pay them.It is a nonsense to have matches played without spectators because it is against the spirit
10、of football and against the visiting team. It is all nonsense.We can do something better to fight racism and discrimination.This is one of the villains we have today in our game. But it is only with harsh sanctions that racism and discrimination can be washed out of football.The (lack of) air up the
11、re Watch mCayman Islands-based Webb, the head of Fifas anti-racism taskforce, is in London for the Football Associations 150th anniversary celebrations and will attend Citys Premier League match at Chelsea on Sunday.I am going to be at the match tomorrow and I have asked to meet Yaya Toure, he told
12、BBC Sport.For me its about how he felt and I would like to speak to him first to find out what his experience was.Uefa hasopened disciplinary proceedings against CSKAfor the racist behaviour of their fans duringCitys 2-1 win.Michel Platini, president of European footballs governing body, has also or
13、dered an immediate investigation into the referees actions.CSKA said they were surprised and disappointed by Toures complaint. In a statement the Russian side added: We found no racist insults from fans of CSKA.Baumgartner the disappointing news: Mission aborted.The supersonic descent could happen a
14、s early as Sunda.The weather plays an important role in this mission. Starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (t
15、he troposphere) where our day-to-day weather lives. It will climb higher than the tip of Mount Everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called
16、the tropopause),e can expect a lot of turbulence.The balloon will slowly drift to the edge of space at 120,000 feet ( Then, I would assume, he will slowly step out onto something resembling an Olympic diving platform.Below, the Earth becomes the concrete bottom of a swimming pool that he wants to la
17、nd on, but not too hard. Still, hell be traveling fast, so despite the distance, it will not be like diving into the deep end of a pool. It will be like he is diving into the shallow end.Skydiver preps for the big jumpWhen he jumps, he is expected to reach the speed of sound - 690 mph (1,110 kph) -
18、in less than 40 seconds. Like hitting the top of the water, he will begin to slow as he approaches the more dense air closer to Earth. But this will not be enough to stop him completely.If he goes too fast or spins out of control, he has a stabilization parachute that can be deployed to slow him dow
19、n. His team hopes its not needed. Instead, he plans to deploy his 270-square-foot (25-square-meter) main chute at an altitude of around 5,000 feet (1,524 meters).In order to deploy this chute successfully, he will have to slow to 172 mph (277 kph). He will have a reserve parachute that will open automatically if he loses consciousness at mach speeds.Even if everything goes as planned, it wont. Baumgartner still will free fall at a speed that would cause you and me to pass out, and no parachute is guaranteed to work higher than 25,000 feet (7,620 meters).cause there
