电大微积分初步形成性考核册作业答案小抄参考资料【精编版】 .doc

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1、微积分初步形成性考核作业(一)解答函数,极限和连续一、填空题(每小题2分,共20分)1函数的定义域是解:, 所以函数的定义域是2函数的定义域是 解:, 所以函数的定义域是3函数的定义域是解: , 所以函数的定义域是4函数,则 解: 所以5函数,则 解:6函数,则 解:,7函数的间断点是 解:因为当,即时函数无意义 所以函数的间断点是8解:9若,则解: 因为 所以10若,则解:因为 所以二、单项选择题(每小题2分,共24分)1设函数,则该函数是()A奇函数 B偶函数C非奇非偶函数 D既奇又偶函数解:因为 所以函数是偶函数。故应选B2设函数,则该函数是()A奇函数 B偶函数C非奇非偶函数 D既奇又

2、偶函数解:因为 所以函数是奇函数。故应选A3函数的图形是关于()对称A B轴C轴 D坐标原点解:因为 所以函数是奇函数 从而函数的图形是关于坐标原点对称的 因此应选D4下列函数中为奇函数是()A B C D解:应选C5函数的定义域为()A B C且 D且解:,所以应选D 6函数的定义域是()A BC D解:,函数的定义域是,故应选D 7设,则( )A B C D 解: ,故应选C8下列各函数对中,()中的两个函数相等 A, B, C, D, 解:两个函数相等必须满足定义域相同函数表达式相同,所以应选D9当时,下列变量中为无穷小量的是( ).A B C D解:因为,所以当时,为无穷小量,所以应选

3、C10当( )时,函数,在处连续.A0 B1 C D 解:因为, 若函数,在处连续,则,因此。故应选B11当( )时,函数在处连续.A0 B1 C D 解:,所以应选D12函数的间断点是( )AB C D无间断点解:当时分母为零,因此是间断点,故应选A三、解答题(每小题7分,共56分)计算极限 解:2计算极限解:3 解: 4计算极限 解:5计算极限解:6计算极限 解: 7计算极限解: 8计算极限 解: 微积分初步形成性考核作业(二)解答(除选择题)导数、微分及应用一、填空题(每小题2分,共20分)1曲线在点的斜率是 解:,斜率2曲线在点的切线方程是 解: ,斜率 所以曲线在点的切线方程是:3曲

4、线在点处的切线方程是解:,斜率 所以曲线在点处的切线方程是:,即:4 解:5若y = x (x 1)(x 2)(x 3),则(0) = 解:6已知,则=解:,7已知,则=解:,8若,则 解:, 9函数的单调增加区间是 解:,所以函数的单调增加区间是10函数在区间内单调增加,则a应满足 解:,而,所以二、单项选择题(每小题2分,共24分)1函数在区间是( D )A单调增加 B单调减少C先增后减 D先减后增2满足方程的点一定是函数的( C ).A极值点B最值点 C驻点D 间断点3若,则=(C ) A. 2 B. 1 C. -1 D. -24设,则(B ) A B C D5设是可微函数,则( D )

5、 A B C D 6曲线在处切线的斜率是( C ) A B C D 7若,则( C ) A B C D 8若,其中是常数,则( C ) A B C D9下列结论中( B )不正确 A在处连续,则一定在处可微. B在处不连续,则一定在处不可导. C可导函数的极值点一定发生在其驻点上. D若在a,b内恒有,则在a,b内函数是单调下降的.10若函数f (x)在点x0处可导,则( B )是错误的 A函数f (x)在点x0处有定义 B,但 C函数f (x)在点x0处连续 D函数f (x)在点x0处可微 11下列函数在指定区间上单调增加的是( B) Asinx Be x Cx 2 D3 - x12.下列结

6、论正确的有( A ) Ax0是f (x)的极值点,且(x0)存在,则必有(x0) = 0 Bx0是f (x)的极值点,则x0必是f (x)的驻点 C若(x0) = 0,则x0必是f (x)的极值点 D使不存在的点x0,一定是f (x)的极值点三、解答题(每小题7分,共56分)设,求 解:2设,求.解:3设,求.解:4设,求. 解:5设是由方程确定的隐函数,求.解:两边微分: 6设是由方程确定的隐函数,求. 解:两边对求导,得: , 7设是由方程确定的隐函数,求.解:两边微分,得: ,8设,求解:两边对求导,得: 微积分初步形成性考核作业(三)解答(填空题除外)不定积分,极值应用问题一、填空题(

7、每小题2分,共20分)1若的一个原函数为,则 。2若的一个原函数为,则 。3若,则4若,则 5若,则6若,则78 9若,则10若,则二、单项选择题(每小题2分,共16分)1下列等式成立的是()A B CD解:应选A2若,则( ). A. B. C. D. 解:两边同时求导,得:,所以应选A3若,则( ). A. B. C. D. 解:应选A4以下计算正确的是( )ABCD 解:应选A5( )A. B. C. D. 解:,所以应选A6=( ) A B C D 解:应选C7如果等式,则( )A. B. C. D. 解:两边求导,得:,所以,故应选B三、计算题(每小题7分,共35分)1解: 2 解:

8、 3解:4解: 5解:四、极值应用题(每小题12分,共24分)1 设矩形的周长为120厘米,以矩形的一边为轴旋转一周得一圆柱体。试求矩形的边长为多少时,才能使圆柱体的体积最大。 解:设矩形的一边长为厘米,则另一边长为厘米,以厘米的边为轴旋转一周得一圆柱体,则体积为:,即:,令,得:(不合题意,舍去),这时由于根据实际问题,有最大体积,故当矩形的一边长为厘米、另一边长为厘米时,才能使圆柱体的体积最大。2 欲用围墙围成面积为216平方米的一成矩形的土地,并在正中用一堵墙将其隔成两块,问这块土地的长和宽选取多大尺寸,才能使所用建筑材料最省? 解:设矩形的长为米,则矩形的宽为米,从而所用建筑材料为:,

9、即:,令得:(取正值),这时由于根据实际问题,确实有最小值,故当矩形的长为米,宽为米时,才能使所用建筑材料最省五、证明题(本题5分)函数在(是单调增加的证明:因为,当(时, 所以函数在(是单调增加的微积分初步形成性考核作业(四)解答(选择题除外)定积分及应用、微分方程一、填空题(每小题2分,共20分)1 解:2解:3已知曲线在任意点处切线的斜率为,且曲线过,则该曲线的方程是 。解:由得所求的曲线方程由确定 因为曲线过,所以,解得: 因此所求的曲线方程为4若 解:5由定积分的几何意义知,= 。解:由定积分的几何意义知,就等于圆在第象限的面积,即 圆面积的,因此6 .解:07= 解: 8微分方程的

10、特解为 . 解:由得,两边同时积分,得 因为,所以,所以 从而,因此微分方程的特解为9微分方程的通解为 .解:, ,即 所以微分方程的通解为10微分方程的阶数为 解:微分方程的阶数为阶二、单项选择题(每小题2分,共20分)1在切线斜率为2x的积分曲线族中,通过点(1, 4)的曲线为( A )Ay = x2 + 3 By = x2 + 4 C D 2若= 2,则k =( A ) A1 B-1 C0 D 3下列定积分中积分值为0的是( A ) A B C D 4设是连续的奇函数,则定积分( D )AB CD 05( D )A0 B C D6下列无穷积分收敛的是(B )ABCD 7下列无穷积分收敛的

11、是(B )AB CD8下列微分方程中,( D )是线性微分方程 A B C D9微分方程的通解为( C ) A B C D10下列微分方程中为可分离变量方程的是(B )A. ;B. ; C. ; D. 三、计算题(每小题7分,共56分)1解:2 解: 3解:4解: 5解: 6求微分方程满足初始条件的特解解:微分方程的通解为 这里 , 代入得微分方程的通解为 将初始条件代入上式,解得 所以微分方程的特解为7求微分方程的通解。解:微分方程的通解为 这里, 代入得微分方程的通解为四、证明题(本题4分)证明等式。证明: 考虑积分,令,则,从而 所以 If we dont do that it will

12、 go on and go on. We have to 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 ma

13、tter has been exaggerated by 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 beca

14、use it is against the spirit 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 foot

15、ball.The (lack of) air up there 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

16、 to meet Yaya Toure, he told 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 football

17、s governing body, has also ordered 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 supe

18、rsonic descent could happen as 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 low

19、er level of the atmosphere (the 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 crosse

20、s the boundary layer (called 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 swim

21、ming pool that he wants to land 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 s

22、ound - 690 mph (1,110 kph) - 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 ca

23、n 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

24、 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 there17

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