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1、浙江师范大学高等数学（二）期末试卷（A卷）(20072008学年第二学期)考试形式：闭卷使用学生：初阳综合理科07级 考试时间：150 分钟出卷时间：2008年5月25日说明: 考生应将全部答案都写在答题纸上, 否则作无效处理。1 Multiple Choices (2 questions, 5.0 points in total)1 (3 points) If series and are diverse, then which of the following series must be diverse:(A)；(B)(C) ； (D)。 Answer ( )2 (2 points)As
2、suming that C circles counterclockwisely along x2+y2=R2 for a whole circle，then use Greens formula to calculate， Answer ( )2 Questions and answers (3 questions, 14.0 points in total)1 (3 points) Try to transfer the function into power series with regard to.2 (8 points) If ，then try to calculate the
3、power series of with regard to x, and get the value of .3 (3 points) Is series convergent，is it absolute convergent？3 Evaluations (12 questions, 77.0 points in total)1 (8 points)Evaluate the curvilinear integral，where L is the positive border of the region rounded by the curve |x|+|y|=1 2 (5 points)
4、Transform appropriately，and try to find the general solution of the equation .3 (6 points)Calculate the double integral Where D is the region rounded by xy=2,y=1+x2 and line x=2.4 (9 points)Calculate , whie L is 5 (6 points)Proof (2xcosyy2sinx)dx+(2ycosxx2siny)dy is the total differential of a funct
5、ion，and find one of its original function.6 (6 points)Find the solution of the differential solution with regard to the initial conditions given：7 (7 points)Find the general solution to the differential equation .8 (6 points)Suppose L is the arch from through to along ，calculate the curvilinear inte
6、gral .9 (8 points)Calculate the linear integral , while L is part of the Hyperbolic plane spiral from to 10 (2 points)Find the general solution to the differential equation 11 (6 points)Suppose f(t) is derivable at t=0, continue on 1,1，and f(0)=0, try to calculate。12 (8 points)Find the solution of t
7、he differential equation with regard to .4 Proof (1 question, 2.0 points in total)1 (2 points)Suppose series is ，prove that it is diverse.5 True or false (1 question, 2.0 points in total)1 (2 points)Use definition to prove whether series is diverse or convergent. If it is convergent, find the sum.=A
8、nswers=There are 19 questions in the sheet, with 100 points in total.i. Multiple Choices (2 questions, 5.0 points in total)(3points)1 Answer (D)(2 points)2AnswerDii. Questions and answers (3 questions, 14.0 points in total)(3 points)1 Answer Since while then (8 points)2Answer (3 points)3Answer Becau
9、se the absolute series of the original series is ，but So the absolute series of the original series is convergent, then the original series is convergent. iii. Calculations (12 questions, 77.0 points in total)(8 points)1 AnswerSol：Let L1： forward(5 points)2AnswerLet ，put it into the equation, then P
10、erform integration, then Put into the equation above，then we get the general solution to the original equation (6 points)3AnswerThe point of intersection between xy=2 and y=1+x2 is (1，2)(9 points)4Answer(6 points)5Answerso the equation is the total differential of some function.One of which is (6 po
11、ints)6AnswerThe characteristic equation is And the Eigen value is The general solution is： From the original condition, we know So the solution to the original question is： (7 points)7AnswerMultiply to both sides, then So the general equation is (6 points)8Answer(8 points)9AnswerSol: transfer Cartes
12、ian Coordinate System into Polar Coordinates, then：(2 points)10AnswerThe characteristic equation is： Eigen value is： The general solution is： (6 points)11Answer(8 points)12AnswerLet Since ，then Since ，then Thus： iv. Proof (1 question, 2.0 points in total)(2 points)1 AnswerLet Obviously and is divergent so is also divergent v. True or false (1 question, 2.0 points in total)(2 points)1 AnswerThe part sum is So Thus the series is convergent，and the sum is
