電大數(shù)學(xué)分析專題研究試題小抄參考

1、專業(yè)好文檔數(shù)學(xué)分析專題研究試題一、填空題1集合中的關(guān)系同時為反身的、對稱的、（ ），則稱關(guān)系為等價關(guān)系。2一個集合若不能與其一個真子集建立一個（ ），則稱該集合為有限集。3函數(shù)在點(diǎn)的鄰域內(nèi)有定義，若（ ），則稱函數(shù)在點(diǎn)處連續(xù)。4設(shè)是從到上的連續(xù)函數(shù)，滿足：1）（ ）；，2）對于有,則是以為底的對數(shù)。5若函數(shù)是定義在上的連續(xù)函數(shù)，且滿足：1）（ ）；2），當(dāng)時，；3）,則分別稱是正弦函數(shù)與余弦函數(shù)。6設(shè)為從集合到集合中的關(guān)系，若,有唯一的，使（ ），則稱為（從到中的）映射。二、單項(xiàng)選擇題1a= b c d2實(shí)數(shù)集是（ ）a有限集 b可列集 c不可列集 d空集3是從到的映射，且，則a= b c d

2、4函數(shù)在點(diǎn)處（ ）a間斷 b連續(xù) c 可導(dǎo) d取得極小值5函數(shù)與在上有界，且，則在上（ ）。a有界 b無界 c有下界而無上界 d結(jié)論不定6下面結(jié)論（ ）是正確的。a若是單調(diào)函數(shù)，也是單調(diào)函數(shù)，則是單調(diào)函數(shù)。b若在數(shù)集上可導(dǎo)，且有界，則在上有界c若是周期函數(shù)，則是周期函數(shù)d若在數(shù)集上有界且可導(dǎo)，則在上有界三、計算題1求過拋物線上的點(diǎn)的切線方程。2已知，求。3已知，求的最小值。4若，求。四、證明題1設(shè)有映射，證明：（1）若是滿射，則是滿射（2）若是滿射，且是單射，則是滿射2若在點(diǎn)處連續(xù)，則在點(diǎn)處也連續(xù)3證明：方程在區(qū)間內(nèi)有且僅有一個實(shí)根。4證明不是周期函數(shù)。參考答案：一、填空題（每小題3分，共18

3、分）1.傳遞的；2.雙射；3.；4.1）;5.1）， 6. 。二、單項(xiàng)選擇題（每小題3分，共18分）1.c；2.c；3.d；4.b；5. d；6. a三、計算題（每小題8分，共32分）1.解 首先計算過點(diǎn)的切線的斜率 4分所求的切線方程為 即 8分2.解 已知 （1）將代替，得 （2） 4分得 8分3.解 已知在內(nèi)，是上凸函數(shù)，由上凸函數(shù)的定義有 5分即 而且當(dāng)時，故是的最小值。 8分4.解 設(shè)，則 3分因，故 8分四、證明題（每小題8分，共32分）1.證明（1）因是滿射，即,進(jìn)一步有，故是滿射。 4分（2）采用反證法。假設(shè)不是滿射，即，則存在，但。設(shè)，使，由于是單射，故，即，這與是滿射矛盾。

4、說明假設(shè)矛盾，即是滿射。 8分2.證明 ，因?yàn)樵邳c(diǎn)連續(xù)，故存在，當(dāng)時，有 由絕對值不等式的 4分故對任意的，當(dāng)時，有 即在點(diǎn)連續(xù)。 8分3.證明：設(shè)，則是上的連續(xù)函數(shù)，且 由介值定理，至少存在一點(diǎn)，使。 4分 由得，當(dāng)時，。即在內(nèi)嚴(yán)格單調(diào)增加。故有且僅有一點(diǎn)，使，即方程在內(nèi)有且僅有一實(shí)根。 8分4.證明 采用反證法。假設(shè)是周期函數(shù)，因是連續(xù)函數(shù)且不是常值，故具有最小正周期，設(shè)為。選取自然數(shù)，使得。故存在使 4分另一方面，對于，有 這與式矛盾。故不是周期函數(shù)。 8分?jǐn)?shù)學(xué)分析專題研究期末試題（2002.1）一、 一、填空題1.若，則.2.若，則.3.設(shè)，若，則稱為從到上的 .4.若復(fù)數(shù)是某個整系數(shù)

5、多項(xiàng)式方程的根，則稱是 數(shù).5.設(shè)，則.6.設(shè)函數(shù)定義在開區(qū)間內(nèi)，對于，有，則稱是內(nèi)的 函數(shù).二、 二、單項(xiàng)選擇題（每小題3分，共18分）1.設(shè)，有.a. b. c. d. 2.若，且，則a. b. c. d. 3.若在內(nèi)連續(xù)，則在內(nèi)（ ）a. 可導(dǎo) b. 單調(diào) c. 有界 d. 對稱4.設(shè)是超越數(shù)，則是（ ）a.有理數(shù) b.代數(shù)數(shù) c. 無理數(shù) d. 超越數(shù)5.與都是以為周期的周期函數(shù)，且，則（ ）a. a. 不是周期函數(shù) b. b. 是以為周期的周期函數(shù)c. c. 是周期函數(shù)，但周期大于或等于d. 是周期函數(shù)，但周期小于或等于6.設(shè)是內(nèi)充分光滑的嚴(yán)格下凸函數(shù)，則（ ）a. 在內(nèi)必取到最小值

6、 b. 在內(nèi)必取到最大值c. 在內(nèi)有 d. 前三個結(jié)論都不對三、 三、計算題（每小題8分，共32分）1.設(shè)，求2.設(shè)，求3.求函數(shù)的極值4.已知重根號），求四、 四、證明題（每小題8分，共32分）1.證明（1）（4分） （2）（4分）2.證明 設(shè)數(shù)集與均有上界，則集合有上界，且3.證明 設(shè)，有4.證明 設(shè)是從到的連續(xù)函數(shù)，則存在點(diǎn),使得.參考答案一、 一、填空題（每小題3分，共18分）1.，2.，3. 滿射，4.代數(shù)數(shù)，5.，6.下凸二、單項(xiàng)選擇題（每小題3分，共18分）1.，2.，3.，4. ，5.，6.三、計算題（每小題8分，共32分）1解 ， 2分， 7分故 8分2設(shè)，則， 3分代入得

7、8分3解 3分令，得，易驗(yàn)證是極大值點(diǎn)，是極小值點(diǎn)， 6分極大值，極小值 8分4.解 顯然，且，即數(shù)列，單調(diào)增加且有上界，故存在，設(shè)，由可得， 5分即，解得 8分四、證明題（每小題8分，共32分）1. 證明：（1）若設(shè)表示的補(bǔ)集，則有 4分（2） 8分 2. 證明：，有，故，即是的一個上界.，使得，即存在，使得故 8分3.證明：設(shè)，則，即是嚴(yán)格下凸，根據(jù) 有 8分4.證明：令，則是上的連續(xù)函數(shù).若，則選取結(jié)論得證.若，則選取結(jié)論得證. 4分否則有，則，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 down. 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 p