电大微积分初步答案小抄

专业好文档微积分初步一、填空题（每小题2分，共20分）1函数的定义域是 答案：2函数的定义域是 答案：3函数的定义域是答案：4函数，则 答案：5函数，则 答案：6函数，则 答案： 7函数的间断点是 答案： 8答案： 19若，则 答案： 210若，则答案： 1曲线在点的斜率是 2曲线在点的切线方程是 3曲线在点处的切线方程是4 5若y = x (x 1)(x 2)(x 3)，则(0) = 66已知，则=7已知，则=（07年1月考题(4分)即与此题相似）8若，则29函数的单调增加区间是10函数在区间内单调增加，则a应满足大于零1若的一个原函数为，则 。答案： （c为任意常数）2若的一个原函数为，则 。答案：3若，则答案：4若，则 答案：5若，则答案：6若，则答案：7答案：8 答案：9若，则答案：10若，则答案：1 答案：2答案：23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 。答案：4若 答案：45由定积分的几何意义知，= 。答案：，它是1/4半径为a的圆的面积。6 .答案：07= 答案：8微分方程的特解为 . 答案：19微分方程的通解为 .答案：10微分方程的阶数为 答案：2 二、单项选择题（每小题2分，共24分）1设函数，则该函数是（）答案：BA奇函数 B偶函数C非奇非偶函数 D既奇又偶函数2设函数，则该函数是（）答案：AA奇函数 B偶函数C非奇非偶函数 D既奇又偶函数3函数的图形是关于（）对称答案：DA B轴C轴 D坐标原点4下列函数中为奇函数是（）答案：CA B C D5函数的定义域为（）答案：DA B C且 D且6函数的定义域是（）答案：DA BC D7设，则（ ）答案：CA B C D 8下列各函数对中，（）中的两个函数相等答案：D A， B， C， D，9当时，下列变量中为无穷小量的是（ ）. 答案：CA B C D10当（ ）时，函数，在处连续. 答案：BA0 B1 C D 11当（ ）时，函数在处连续. 答案：DA0 B1 C D 12函数的间断点是（ ） 答案：AA B C D无间断点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 ） A B C D 6曲线在处切线的斜率是（ C ） A B C D7若，则（ C ） A BC D 8若，其中是常数，则（ C ） A B C D9下列结论中（ A ）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1下列等式成立的是（）答案：AABC D3若，则（ ）. 答案：AA. B. C. D. 4若，则（ ）. 答案：A A. B. C. D. 5以下计算正确的是（ ）答案：AA B C D 6（ ）答案：AA. B. C. D. 7=（ ）答案：CA B C D 8如果等式，则（ ）答案：BA. B. C. D. 1在切线斜率为2x的积分曲线族中，通过点（1, 4）的曲线为（ ）答案：AAy = x2 + 3 By = x2 + 4 C D 2若= 2，则k =（ ） 答案：A A1 B-1 C0 D 3下列定积分中积分值为0的是（ ） 答案：A A B C D 4设是连续的奇函数，则定积分（ ）答案：DAB CD 05（ ）答案：DA0 B C D6下列无穷积分收敛的是（）答案：BA BC D 7下列无穷积分收敛的是（） 答案：BA BC D8下列微分方程中，（ ）是线性微分方程 答案：D A B C D 9微分方程的通解为（ ） 答案：C A B C D 10下列微分方程中为可分离变量方程的是（）答案：BA. ； B. ； C. ； D. 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 设，求 （答案）2设，求.（答案）3设，求.（答案）4设，求. （答案，）5设，求.（答案）6设是由方程确定的隐函数，求. （答案）7设是由方程确定的隐函数，求.（答案）8设，求（答案）1解 2 解 3解 4 解 利用分步积分法： 设，则， 5解 利用分步积分法： 设，则，1 解 2 解 3 解 利用分部积分法 设，则， 4 解 利用分部积分法 设，则， 5 解 利用分部积分法 设，则， 6求微分方程满足初始条件的特解 解 与一阶线性微分方程标准式比较，可知 ， 将代入上式，得，所以原微分方程满足初始条件的特解为 7求微分方程的通解。 解 与一阶线性微分方程标准式比较，可知 ， 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。60-x0xxy解 设矩形的一边边长为x厘米，则另一边边长为（60-x）厘米，边长为x厘米的边绕轴旋转得一圆柱体，其底面积为(60-x)2，旋转体的体积为 令，即 其中是极小值点，此时体积为0；是极大值点，也是最大值点，对应的圆柱体最大体积值为 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 设土地一边长为，另一边长为，共用材料为于是 =3令得唯一驻点（舍去） 10分 因为本问题存在最小值，且函数的驻点唯一，所以，当土地一边长为，另一边长为18时，所用材料最省.五、证明题（本题5分）1、函数在（是单调增加的证 只需证明当时，有 因为 当时，即有 所以，当时，是单调增加的。1、证明等式。证明：显然是偶函数，是奇函数，If we dont do that it will 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 matter 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 because 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 football.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 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 footballs 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 supersonic 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 lower 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 crosses 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 swimming 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 sound - 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 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 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 there15