电大《微积分初步》（15春）期末复习考试试题及参考答案.doc

电大微积分初步（15春）期末复习考试试题及参考答案电大微积分初步（15春）期末复习考试试题及参考答案 一、 填空题 函数，则 若，则1 曲线在处的切线斜率是 若是的一个原函数，则 为 4 阶微分方程. 6.函数，则 7.若函数,在处连续，则3 8.曲线在点处的切线方程是 1 9. 10.微分方程的阶数为 3 11.函数的定义域是 12.若，则2 13.曲线在点处的切线方程是 14. 0 15.微分方程的特解为 16.函数的定义域是 17.函数的间断点是= 18.函数的单调增加区间是 19.若，则= 20.微分方程的阶数为 21.函数的定义域是 23.若函数,在处连续，则2 24.曲线在点处的斜率是 25. 26.微分方程满足初始条件的特解为 27.函数，则 28.当0时，为无穷小量.29.若y = ( 1)( 2)( 3)，则(1) = 30. 31.微分方程的特解为 . 32.函数的定义域是 33.若函数,在处连续，则1 34.曲线在点处的切线方程是 35. 36.微分方程的阶数为 37函数，则 38. 39.曲线在点处的切线方程是 40.若，则 51.微分方程的阶数为 5 二、单项选择题 函数的图形关于（B）对称 A。坐标原点B。轴 C轴D。 当（ C ）时，函数在处连续.A0 B1 CD 函数在区间是（ D ） A单调减少 B单调增加 C先减后增 D先增后减 下列等式成立的是（A） AB C D 微分方程的通解为（B） A.； B.； C.； D. 6.设函数，则该函数是（A） A奇函数 B偶函数C非奇非偶函数 D既奇又偶函数 7.已知，当（ D ）时，为无穷小量.A B C D 8.函数在区间是（ C ） A单调增加 B单调减少 C先增后减 D先减后增 9.以下等式成立的是（ A ） A B C D 10.下列微分方程中为可分离变量方程的是（B） A.； B.； C.； D. 11.设函数，则该函数是（A） A偶函数B奇函数C非奇非偶函数 D既奇又偶函数 12.当（ C ）时，函数，在处连续.A0 B1 CD 14.下列结论中（ C ）正确 A在处连续，则一定在处可微. B函数的极值点一定发生在其驻点上.C在处不连续，则一定在处不可导. D函数的极值点一定发生在不可导点上.15.下列等式中正确的是（D） A .B.C.D. 16.微分方程的阶数为（B） A.2； B.3； C.4； D.5 设函数，则该函数是（B） A奇函数 B偶函数C非奇非偶函数 D既奇又偶函数 17.当时，下列变量中为无穷小量的是（C ）.A B C D 18.设，则（D ） A B C D 19.在切线斜率为2的积分曲线族中，通过点（1, 4）的曲线为（ C ） A B Cy = 2 + 3 D y = 2 + 4 20.微分方程的通解是（A） A.；B.；C.； D. 21.设，则（ D ） A B CD 22.若函数f ()在点处可导，则( B )是错误的 A函数f ()在点处有定义 B，但 C函数f ()在点处连续 D函数f ()在点处可微 33.函数在区间是（ A ） A先减后增 B先增后减 C单调减少 D单调增加 34.若，则（ B ）. A. B.C.D. 35.微分方程的阶数为（C） A.1 B.2 C.3 D.5 36.函数的定义域是（C） A B C D 37.曲线在处切线的斜率是（ D ） A B C D 38.下列结论正确的有（ B ） A若() = 0，则必是f ()的极值点 B是f ()的极值点，且()存在，则必有() = 0 C是f ()的极值点，则必是f ()的驻点 D使不存在的点，一定是f ()的极值点 39.下列无穷积分收敛的是（A） AB C D 40.微分方程的阶数为（D） A.1；B.2；C.3； D.4 41.设，则（ C ） A B CD 42.若函数f ()在点处可导，则( B)是错误的 A函数f ()在点处有定义 B，但 C函数f ()在点处连续 D函数f ()在点处可微 43.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 44.（ A ） A.B.C. D. 45.下列微分方程中为可分离变量方程的是（B） A.； B.； C.； D. 46.设函数，则该函数是（d） A非奇非偶函数B既奇又偶函数C偶函数 D奇函数 47.当时，下列变量中为无穷小量的是（ c ）.A B C D 48.下列函数在指定区间上单调减少的是（ b ） A B C D 49.设，则（c ） A. B. C. D. 50.下列微分方程中，（a ）是线性微分方程 A B C D 三、计算题 计算极限 设，求. 计算不定积分 计算定积分 5.计算极限 6.设，求. 7.计算不定积分 8.计算定积分 9.计算极限 解：原式 10.设，求.解： 11.计算不定积分 解：= 12.计算定积分 解： 13.计算极限 解：原式 14.设，求.解： 9分 15.计算不定积分 解：= 16.计算定积分 解： 17.计算极限 解：原式 18.设，求.解： 19.计算不定积分 解：= 20.计算定积分 解： 21.计算极限 解： 22.设，求.解： 23.计算不定积分 解：= 24.计算定积分 解： 25.计算极限 解：原式 26.设，求.解： 9分 27.计算不定积分 解：= 28.计算定积分 解： 29.计算极限 解：原式 30.设，求.解： 31.计算不定积分 解：= 32.计算定积分 解： 四、应用题 1.某制罐厂要生产一种体积为V的有盖圆柱形容器，问容器的底半径与高各为多少时可使用料最省？ 2.欲做一个底为正方形，容积为32立方米的长方体开口容器，怎样做法用料最省？ 3.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。 4.欲做一个底为正方形，容积为32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，解得是惟一驻点，易知是函数的极小值点，此时有，所以当，时用料最省 16分 5.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，当时水箱的面积最小. 此时的费用为 （元） 6.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） 7.欲做一个底为正方形，容积为32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，解得是惟一驻点，易知是函数的极小值点，此时有，所以当，时用料最省 16分 8.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：设长方体底边的边长为，高为，用材料为，由已知 令，解得是唯一驻点， 因为问题存在最小值，且驻点唯一，所以是函数的极小值点，即当，时用料最省. 微积分初步考试仿真试题 一、填空题（每小题4分，本题共20分） 函数，则 当时，为无穷小量.若y = ( 1)( 2)( 3)，则(1) = 微分方程的特解为 . 二、单项选择题（每小题4分，本题共20分） 函数的定义域是（） A B C D 曲线在处切线的斜率是（ ） A B C D 下列结论正确的有（ ） A若() = 0，则必是f ()的极值点 B是f ()的极值点，且()存在，则必有() = 0 C是f ()的极值点，则必是f ()的驻点 D使不存在的点，一定是f ()的极值点 下列无穷积分收敛的是（） AB C D 微分方程的阶数为（） A.1；B.2；C.3； D.4 三、计算题（本题共44分，每小题11分） 计算极限 设，求. 计算不定积分 计算定积分 四、应用题（本题16分） 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？ 微积分初步模拟试题答案 （供参考） 一、填空题（每小题4分，本题共20分） 0 二、单项选择题（每小题4分，本题共20分） CDBA D 三、（本题共44分，每小题11分） 解： 解： 解：= 4解： 四、应用题（本题16分） 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） 微积分初步 期末复习指导 考试方式： 闭卷笔试，90分钟。 考核形式与考核成绩确定 考核形式：作业考核和期末考试相结合。 考核成绩：满分为100分，60分为及格，其中平时作业成绩占考核成绩的30，期末 考试成绩占考核成绩的70。 在考题试卷中为学生提供导数与积分的基本公式。 一、函数、极限与连续考核要求 1了解常量和变量的概念；理解函数的概念；了解初等函数和分段函数的概念.熟练掌握求函数的定义域、函数值的方法；掌握将复合函数分解成较简单函数的方法。 2了解极限概念，会求简单极限。 3了解函数连续的概念，会判断函数的连续性，并会求函数的间断点。 二、 导数与微分部分考核要求 1了解导数概念，会求曲线的切线方程。 2熟练掌握求导数的方法(导数基本公式、导数的四则运算法则、复合函数求导法则)，会求简单的隐函数的导数。 3了解微分的概念，掌握求微分的方法。 4了解高阶导数的概念，掌握求显函数的二阶导数的方法。 三、导数应用考核要求 1掌握函数单调性的判别方法。 2了解极值概念和极值存在的必要条件，掌握极值判别的方法。 3掌握求函数最大值和最小值的方法。 四、一元函数积分考核要求 1理解原函数与不定积分的概念、性质，掌握积分基本公式，掌握用直接积分法、第一换元积分法和分部积分法求不定积分的方法。 2了解定积分的概念、性质，会计算一些简单的定积分。 3了解广义积分的概念，会计算简单的无穷限积分。 五、积分应用考核要求 1会用定积分计算简单的平面曲线围成图形的面积（直角坐标系）和绕坐标轴旋转生成的旋转体体积。 2了解微分方程的几个概念，掌握变量可分离的微分方程和一阶线性微分方程的解法。 请您删除一下内容，O()O谢谢！20年中央电大期末复习考试小抄大全，电大期末考试必备小抄，电大考试必过小抄After earning his spurs in the kitchens of The Westin, The Sheraton, Sens on the Bund, and a sprinkling of other -notch venues, Simpson Lu fi nally got the chance to become his own boss in November 2021.Sort of.The Shanghai-born chef might not actually own California Pizza Kitchen (CPK) but he is in sole charge of both kitchen and frontof- house at this Sinan Mansionsstalwart.  “Its certainly a responsibility to be the head chef, and then to have to manage the rest of the restaurant as well,“ the 31-year-old tells Enjoy Shanghai.“In hotels, for ele, these jobs are strtly demarcated, so its a great opportunity to learn how a business operates across the board.“  It was a task that management back in sunny California evidently felt he was ready for, and a vote of confi dence from a company that, to date, has opened 250 outlets in 11 countries.And for added pressure, the Shanghai branch was also CPKs China debut.  “For sure it was a big step, and unlike all their other Asia operations that are franchises, they decided to manage it directly to ben with,“ says Simpson.  Two years ago a private franchisee took over the lease, but the links to CPK headquarters are still strong, with a mainland-based brand ambassador on hand to ensure the business adheres to its ethos of creating “innovative, hearth-baked pizzas“, a sle of PR blurb that Simpson insists lives up to the hype.  “They are very innovative,“ he says.“The problem with most fast food places is that they use the same sauce on every pizza and just change the pings.Every one of our 16 pizza sauces is a unique recipe that has been formulated to complement the pings perfectly.“  The largely local customer base evidently agrees and on Saturday and Sunday, at least, the place is teeming.The kids-eat-for-free poly at weekends is undoubtedly a big draw, as well as is the spacious second-fl oor layout overlooked by a canopy of green from Fuing Park over the road.  The company is also focusing on increasing brand recognition and in recent years has taken part in outside events such as the regular California Week.Still, the sta are honest enough to admit that business could be better; as good, in fact, as in CPKs second outlet in the popular Kerry Parkside shopping mall in Pudong.  “Sinan Mansions has really struggled to get the number of visitors that were envisaged when it first opened, and it hasnt been easy for any of the tenants here,“ adds Simpson.“Were planning a third outlet in the city in 20, and we will probably choose a shopping mall again because of the better foot traff.“ The tearooms once frequented by Coco Chanel and Marcel Proust are upping stks and coming to Shanghai, u Junqian visits the Parisian outpost with sweet treats.One thing the century-old Parisian tearoom Angelina has shown is that legendary fashion designer Coco Chanel not only had style and glamor but also boasted great taste in food, pastries in partular.One of the most popular tearooms in Paris, Angelina is famous for having once been frequented by celebrities such as Chanel and writer Marcel Proust.Now Angelina has packed up its French ambience, effient serve, and beautiful, comforting desserts and flown them to Shanghai. At the flagship dine-in and take-out space in Shanghai, everything mims the orinal tearoom designed from the benning of the 20th century, in Paris, the height of “Belle Epoque“.The paintings on the wall, for ele, are eactly the same as the one that depts the landscape of southern France, the hometown of the owner; and the small tables are intentional imitations of the ones that Coco Chanel once sat at every afternoon for hot chocolate.The famous hot chocolate, known as LAfrain, is a luurious miture of four types of cocoa beans imported from Afra, blended in Paris and then shipped to Shanghai.Its sinfully sweet, rh and thk as if putting a bar of melting chocolate directly on the tongue and the fresh whipped cream on the side makes a light, but equally gratifying contrast.It is also sold in glass bottles as takeaway.The signature Mont-Blanc chestnut cake consists of three parts: the pureed chestnut on , the vanilla cream like stuffing, and the meringue as base.Get all three layers in one scoop, not only for the different tetures but also various flavors of sweetness.The dessert has maintained its popularity for a century, even in a country like France, perhaps the worlds most competitive place for desserts.A much overlooked pairing, is the Paris-New York chou pastry and N226 chocolate flavored tea.The chou pastry is a mouthful of airy pecan-flavored whipped cream, while the tea, a blend of black teas from China and Ceylon, cocoa and rose petals, offers a more subtle fragrance of flowers and chocolate.Ordering these two items, featuring a muted sweetness, makes it easier for you to fit into your little black dress.Breakfast, brunch, lunch and light supper are also served at the tearoom, “a hub of many cultures“ and takes in “a mi of different styles of French cuisines“, according to the management team.The semi-cooked foie gras terrine, is seductive and deceptive.Its generously served at the size and shape of a toast, while the actual brioche toast is baked into a curved sle dipped with fig chutney.The flavor, however, is honest: strong, smooth and sublime.And you dont actually need the toast for crunchiness.This is the season for high teas, with dainty cups of fine china and little pastries that eal to both visual and physal etites.But there is one high tea with a difference, and Pauline D.Loh finds out just eactly why it is special.Earl Grey tea and macarons are all very well for the crucial recuperative break in-between intensive bouts of holiday season shopping.And for those who prefer savory to sweet, there is still the selection of class Chinese snacks called dim sum to satisfy and satiate.High tea is a meal to eat with eye and mouth, an in-between indulgence that should be light enough not to spoil dinner, but suffiently robust to take the edge off the hunger that strikes hours after lunch.The afternoon tea special at Shang-i at the Four Seasons Hotel Pudong has just the right elements.It is a pering meal, with touches of luury that make the high tea session a treat in itself.Whole baby abalones are braised and then ped on a shortcrust pastry shell, a sort of Chinese version of the Western vol-au-vent, but classier.Even classier is the dim sum staple shrimp dumpling or hargow, upgraded with the addition of slivers of midnight dark truffles.This is a master touch, and chef Simon Choi, who presides unchallenged at Shang-i, has scored a ner again.Sweet prawns and aromat truffles whats not to love? His masterful craftsmanship is ehibited in yet another pastry a sweet pastry that is shaped to look like a walnut, but whh you can put straight into the mouth.It crumbles immediately, and the slightly sweet, nutty morsel is so easy to eat youll probably reach straight for another.My favorite is the dessert that goes by the name yangzhi ganlu, or ambrosia from the gods.The hotel calls it chilled mango cream with sago, pomelo and birds nest made with ingredients that resonate with every female soul.It does taste like ambrosia, with the sweet-sour fragrance of the mango forming the first layer of taste and sensation, and the pomelo sacs and sago pearls providing the tactile contrast.The birds nest its the ingredient that lds the lily, since it is supposed to nourish and nurture a perfect compleion.For those unfamiliar with this eot ingredient, the birds nest is not a bundle of twigs, but is, instead, the dried secretion that cave swifts use to build their nest.Technally, its bird saliva, but the Chinese believe that it has collagen-boosting properties essential to beauty.To me, it just tastes good, adding a slight gelatinous crunch to the back of the teeth.There is also a baked mushroom puff pastry that includes the highly prized Matsutake or pine mushroom.You also get a choe of teas, but I strongly recommend the aged Puer from Yunnan province.It is a mellow tea that will go perfectly with the rh little nuggets, and will warm you up on a cold afternoon.Incidentally, the general guide is green tea such as Longjing or Dragonwell for the hot season, semi-fermented teas like Wu-long or Iron Buddha for autumn, and more warming teas like Puer for ter.The Shang-i afternoon tea costs 228 yuan ($37) per person, and 388 yuan for the more opule 第 11 页 共 11 页