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1、电大微积分初步期末复习资料小抄 一、填空题函数的定义域是答案：函数的间断点是=答案：曲线在点的斜率是答案：若，则答案：微分方程的阶数是26.函数，答案：7函数在处连续，则=28.曲线在点的斜率是答案：9.答案：410.微分方程的阶数是答案：211.函数的定义域是答案：12.若，则答案：213.已知，则=答案：14.若答案：15.微分方程的阶数是316.函数的定义域是(-2,-1)(-1，4】17.若，则218.曲线在点处的切线方程是_y=x+1_19.020.微分方程的特解为 y=e的x次方 21.函数的定义域是 22.若函数,在处连续，则2 23.曲线在点处的斜率是24.25.微分方程满足初
2、始条件的特解为 26函数的定义域是 答案：27函数的定义域是 答案：28函数的定义域是答案：29函数，则 答案：30函数，则 答案：31.函数，则 答案： 32函数的间断点是答案： 33答案： 134若，则 答案： 235若，则答案： 36曲线在点的斜率是 37曲线在点的切线方程是 38曲线在点处的切线方程是39 40若y = x (x 1)(x 2)(x 3)，则(0) = 641已知，则=42已知，则= 43若，则244函数在区间内单调增加，则a应满足大于零45若的一个原函数为，则 。答案： （c为任意常数）46若的一个原函数为，则 。答案：47若，则答案：48若，则 答案：49若，则答案
3、：50若，则答案：51答案：52 答案：53若，则答案：10若，则答案：54 答案：55答案：256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：57若 答案：458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。59 答案：060= 答案：61微分方程的特解为 . 答案：162微分方程的通解为 .答案：63微分方程的阶数为 答案：264函数的定义域是_且 。65函数的定义域是_ _。66设，则_0_。67函数，则_ 。68_ 。 69设，则_。70曲线在点的切线方程是_ 。71函数在区间_内是单调减少的。72函数的单调增加区间是73若，则74_。 75
4、 760.77 2 .78微分方程的阶数是二阶二、单项选择题设函数，则该函数是（偶函数）若函数，则（）.函数在区间是（先减后增）下列无穷积分收敛的是（）微分方程的通解是（）6.函数的定义域（且）7.若函数，则（ 1 ）.8.函数在区间是（先减后增）9.下列无穷积分收敛的是（）10.下列微分方程中为一阶线性微分方程的是（）11.设函数，则该函数是（偶函数）12.当=（ 2 ）时，函数，在处连续.13.微分方程的通解是（）14.设函数，则该函数是（偶函数）15.当（2）时，函数，在处连续.16.下列结论中（在处不连续，则一定在处不可导. ）正确 17.下列等式中正确的是（）18.微分方程的阶数为（
5、3）19.设，则（）20.若函数f (x)在点x0处可导，则(，但 )是错误的 21.函数在区间是（先减后增）22.若，则（）.23.微分方程的阶数为（3）24设函数，则该函数是（偶函数）25设函数，则该函数是（奇函数）26函数的图形是关于（坐标原点）对称27下列函数中为奇函数是（）28函数的定义域为（且）29函数的定义域是（）30设，则（ ） 31下列各函数对中，（，）中的两个函数相等 32当时，下列变量中为无穷小量的是（ ）. 答案：C33当（ 1 ）时，函数，在处连续. 34当（ 3 ）时，函数在处连续. 35函数的间断点是（ ） 36函数在区间是（先减后增）37满足方程的点一定是函数的
6、（驻点）.38若，则=（-1） 39设是可微函数，则（ ） 40曲线在处切线的斜率是（ ） 41若，则（ ） 42若，其中是常数，则（ ） 43下列结论中（在处连续，则一定在处可微.）不正确 44若函数f (x)在点x0处可导，则(，但)是错误的 45.下列结论正确的有（x0是f (x)的极值点，且(x0)存在，则必有(x0) = 0） 46下列等式成立的是（）47若，则（）. 48若，则（ ）. 49以下计算正确的是（ ） 50（ ）51=（ ） 52如果等式，则（ ） 53在切线斜率为2x的积分曲线族中，通过点（1, 4）的曲线为（ y = x2 + 3 ） 54若= 2，则k =（ 1
7、） 55下列定积分中积分值为0的是（ ） 56设是连续的奇函数，则定积分（ 0 ）57（ ）58下列无穷积分收敛的是（） 59下列无穷积分收敛的是（） 60下列微分方程中，（ ）是线性微分方程 61微分方程的通解为（ ）62下列微分方程中为可分离变量方程的是（） 63.函数y的定义域是（(2，2）。64设，则（ ）。65函数的图形关于（轴）对称66、当时，变量（ ）是无穷小量 67函数 在x = 0处连续，则k = (-1)68曲线在点(1,0)处的切线方程是（ ）。69若，则（ ）。70函数在区间内满足（单调上升）71函数yx22x5在区间 (0，1) 上是（单调减少 ）。72下列式子中正确
8、的是（ ）。73以下计算正确的是（ ）74若，则（ ）75（ ）。76下列定积分中积分值为0的是（ ） 77微分方程的通解是（ ）。三、计算题（本题共44分，每小题11分） 计算极限 解 设，求. 解 3计算不定积分 解 4计算定积分 解 5计算极限 解 6 设，求. 解 7计算不定积分 解 =8计算定积分 解 9.计算极限 解 10.设，求. 解 11.计算不定积分 解 = 12.计算定积分 解 12.计算极限解：原式13.设，求.解： 14.计算不定积分解：= 15.计算定积分解：16.计算极限解：原式17.设，求.解： 18.计算不定积分解：= 19.计算定积分解：20.计算极限 解 2
9、1计算极限 解 22 解 23计算极限 解 24计算极限 解 25计算极限 解 26计算极限解 27.设，求 解 28设，求.解 29设，求.解30设，求.解，31设，求.解32设是由方程确定的隐函数，求. 解 33设是由方程确定的隐函数，求.解 34设，求解35 解 利用分部积分法 设，则， 36 解 利用分部积分法 设，则， 四、应用题 1用钢板焊接一个容积为4的底为正方形的无盖水箱，已知钢板每平方米10元，焊接费40元，问水箱的尺寸如何选择，可使总费最低？最低总费是多少？解：设水箱的底边长为，高为，表面积为，且有所以 令，得， 因为本问题存在最小值，且函数的驻点唯一，所以，当时水箱的表面
10、积最小. 此时的费用为 （元）.2欲做一个底为正方形，容积为62.5立方米的长方体开口容器，怎样做法用料最省？解：设底边的边长为，高为，容器的表面积为，由已知， 令，解得是唯一驻点，易知是函数的极小值点，此时有，所以当，时用料最省3. 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） 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.4. 欲做一个底为正方形，容积为108立方米的长方体开口容器
11、，怎样做法用料最省？解：设底边的边长为，高为，用材料为，由已知 令，解得是唯一驻点， 且，说明是函数的极小值点，所以当，时用料最省。5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）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，O(_)O谢谢！【Chinas 10 must-see animations】The Chinese
12、animation industry has seen considerable growth in the last several years. It went through a golden age in the late 1970s and 1980s when successively brilliant animation work was produced. Here are 10 must-see classics from Chinas animation outpouring that are not to be missed. Lets recall these col
13、orful images that brought the country great joy. Calabash Brothers Calabash Brothers (Chinese: 葫芦娃) is a Chinese animation TV series produced byShanghaiAnimationFilmStudio. In the 1980s the series was one of the most popular animations in China. It was released at a point when the Chinese animation
14、industry was in a relatively downed state compared to the rest of the international community. Still, the series was translated into 7 different languages. The episodes were produced with a vast amount of paper-cut animations. Black Cat Detective Black Cat Detective (Chinese: 黑猫警长) is a Chinese anim
15、ation television series produced by the Shanghai Animation Film Studio. It is sometimes known as Mr. Black. The series was originally aired from 1984 to 1987. In June 2006, a rebroadcasting of the original series was announced. Critics bemoan the series violence, and lack of suitability for children
16、s education. Proponents of the show claim that it is merely for entertainment. Effendi Effendi, meaning sir andteacher in Turkish, is the respectful name for people who own wisdom and knowledge. The heros real name was Nasreddin. He was wise and witty and, more importantly, he had the courage to res
17、ist the exploitation of noblemen. He was also full of compassion and tried his best to help poor people. Adventure of Shuke and Beita【舒克与贝塔】 Adventure of Shuke and Beita (Chinese: 舒克和贝塔) is a classic animation by Zheng Yuanjie, who is known as King of Fairy Tales in China. Shuke and Beita are two mi
18、ce who dont want to steal food like other mice. Shuke became a pilot and Beita became a tank driver, and the pair met accidentally and became good friends. Then they befriended a boy named Pipilu. With the help of PiPilu, they co-founded an airline named Shuke Beita Airlines to help other animals. A
19、lthough there are only 13 episodes in this series, the content is very compact and attractive. The animation shows the preciousness of friendship and how people should be brave when facing difficulties. Even adults recalling this animation today can still feel touched by some scenes. Secrets of the
20、Heavenly Book Secrets of the Heavenly Book, (Chinese: 天书奇谈)also referred to as Legend of the Sealed Book or Tales about the Heavenly Book, was released in 1983. The film was produced with rigorous dubbing and fluid combination of music and vivid animations. The story is based on the classic literatu
21、re Ping Yao Zhuan, meaning The Suppression of the Demons by Feng Menglong. Yuangong, the deacon, opened the shrine and exposed the holy book to the human world. He carved the books contents on the stone wall of a white cloud cave in the mountains. He was then punished with guarding the book for life
22、 by the jade emperor for breaking heavens law. In order to pass this holy book to human beings, he would have to get by the antagonist fox. The whole animation is characterized by charming Chinesepainting, including pavilions, ancient architecture, rippling streams and crowded markets, which fully d
23、emonstrate the unique beauty of Chinas natural scenery. Pleasant Goat and Big Big Wolf【喜洋洋与灰太狼】 Pleasant Goat and Big Big Wolf (Chinese:喜羊羊与灰太狼) is a Chinese animated television series. The show is about a group of goats living on the Green Pasture, and the story revolves around a clumsy wolf who wa
24、nts to eat them. It is a popular domestic animation series and has been adapted intomovies. Nezha Conquers the Dragon King（Chinese: 哪吒闹海）is an outstanding animation issued by the Ministry of Culture in 1979 and is based on an episode from the Chinese mythological novel Fengshen Yanyi. A mother gave
25、birth to a ball of flesh shaped like a lotus bud. The father, Li Jing, chopped open the ball, and beautiful boy, Nezha, sprung out. One day, when Nezha was seven years old, he went to the nearby seashore for a swim and killed the third son of the Dragon King who was persecuting local residents. The
26、story primarily revolves around the Dragon Kings feud with Nezha over his sons death. Through bravery and wit, Nezha finally broke into the underwater palace and successfully defeated him. The film shows various kinds of attractive sceneries and the traditional culture of China, such as spectacular
27、mountains, elegant sea waves and exquisite ancient Chinese clothes. It has received a variety of awards. Havoc in Heaven The story of Havoc in Heaven（Chinese: 大闹天宫）is based on the earliest chapters of the classic storyJourney to the West. The main character is Sun Wukong, aka the Monkey King, who re
28、bels against the Jade Emperor of heaven. The stylized animation and drums and percussion accompaniment used in this film are heavily influenced byBeijingOpera traditions. The name of the movie became a colloquialism in the Chinese language to describe someone making a mess. Regardless that it was an
29、 animated film, it still became one of the most influential films in all of Asia. Countless cartoon adaptations that followed have reused the same classic story Journey to the West, yet many consider this 1964 iteration to be the most original, fitting and memorable, The Golden Monkey Defeats a Demo
30、n【金猴降妖】 The Golden Monkey Defeats a Demon (Chinese: 金猴降妖), also referred as The Monkey King Conquers the Demon, is adapted from chapters of the Chinese classics Journey to the West, or Monkey in the Western world. The five-episode animation series tells the story of Monkey King Sun Wukong, who follo
31、wed Monk Xuan Zangs trip to the West to take the Buddhistic sutra. They met a white bone evil, and the evil transformed human appearances three times to seduce the monk. Twice Monkey King recognized it and brought it down. The monk was unable to recognize the monster and expelled Sun Wukong. Xuan Za
32、ng was then captured by the monster. Fortunately Bajie, another apprentice of Xuan Zang, escaped and persuaded the Monkey King to come rescue the monk. Finally, Sun kills the evil and saves Xuan Zang. The outstanding animation has received a variety of awards, including the 6th Hundred Flowers Festi
33、val Award and the Chicago International Childrens Film Festival Award in 1989. McDull【麦兜】 McDull is a cartoon pig character that was created inHong Kongby Alice Mak and Brian Tse. Although McDull made his first appearances as a supporting character in the McMug comics, McDull has since become a cent
34、ral character in his own right, attracting a huge following in Hong Kong. The first McDull movie McMug Story My Life as McDull documented his life and the relationship between him and his mother.The McMug Story My Life as McDull is also being translated into French and shown in France. In this version, Mak Bing is the mother of McDull, not his father.
