电大微积分初步专科期末复习题及答案资料小抄精华打印版.doc
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电大微积分初步期末复习资料小抄 一、填空题 ⒈函数的定义域是 . 答案: ⒉函数的间断点是= .答案: ⒊曲线在点的斜率是 .答案: ⒋若,则 . 答案: ⒌微分方程的阶数是 2 . 6.函数, .答案: 7.函数在处连续,则= 2 . 8.曲线在点的斜率是 .答案: 9. .答案:4 10.微分方程的阶数是 .答案:2 11.函数的定义域是 .答案: 12.若,则 .答案:2 13.已知,则= .答案: 14.若 .答案: 15.微分方程的阶数是 3 . 16.函数的定义域是(-2,-1)∪(-1,4】. 17.若,则 2. 18.曲线在点处的切线方程是_y=x+1__. 19. 0 . 20.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: 1 34.若,则 . 答案: 2 35.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) = -6 . 41.已知,则=. 42.已知,则=. 43.若,则-2 . 44.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: 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:2 56.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:4 58.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:0 60.= . 答案: 61.微分方程的特解为 . 答案:1 62.微分方程的通解为 .答案: 63.微分方程的阶数为 . 答案:2 64.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. . 76. 0 . 77. 2 . 78.微分方程的阶数是 二阶 . 二、单项选择题 ⒈设函数,则该函数是(偶函数). ⒉若函数,则(). ⒊函数在区间是(先减后增) ⒋下列无穷积分收敛的是(). ⒌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(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:C 33.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(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 ). 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y=x2-2x+5在区间 (0,1) 上是(单调减少 )。 72.下列式子中正确的是( )。 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) ⒈ 计算极限. 解 ⒉ 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 21.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,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, 请您删除一下内容,O(∩_∩)O谢谢!!!【China's 10 must-see animations】The Chinese 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 China's animation outpouring that are not to be missed. Let's recall these colorful images that brought the country great joy. Calabash Brothers Calabash Brothers (Chinese: 葫芦娃) is a Chinese animation TV series produced by Shanghai Animation Film Studio. In the 1980s the series was one of the most popular animations in China. It was released at a point when the Chinese animation 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 animation 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's education. Proponents of the show claim that it is merely for entertainment. Effendi "Effendi", meaning sir and teacher in Turkish, is the respectful name for people who own wisdom and knowledge. The hero's real name was Nasreddin. He was wise and witty and, more importantly, he had the courage to resist 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 mice who don't 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. Although 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 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 literature "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 book's contents on the stone wall of a white cloud cave in the mountains. He was then punished with guarding the book for life by the jade emperor for breaking heaven's 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 Chinese painting, including pavilions, ancient architecture, rippling streams and crowded markets, which fully demonstrate the unique beauty of China's 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 wants to eat them. It is a popular domestic animation series and has been adapted into movies. 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 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 story primarily revolves around the Dragon King's feud with Nezha over his son's 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 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 story Journey to the West. The main character is Sun Wukong, aka the Monkey King, who rebels against the Jade Emperor of heaven. The stylized animation and drums and percussion accompaniment used in this film are heavily influenced by Beijing Opera traditions. The name of the movie became a colloquialism in the Chinese language to describe someone making a mess. Regardless that it was an 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 Demon【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 followed Monk Xuan Zang's 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 Zang 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 Festival Award and the Chicago International Children's Film Festival Award in 1989. McDull【麦兜】 McDull is a cartoon pig character that was created in Hong Kong by Alice Mak and Brian Tse. Although McDull made his first appearances as a supporting character in the McMug comics, McDull has since become a central 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..- 配套讲稿:
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