课程名称︰微积分甲上 课程性质︰必修 课程教师︰周青松 开课学院：管理学院、生农学院、理学院 开课系所︰工管系科管组、生工系、生机系、地质系 考试日期（年月日）︰2011/1/10 考试时限（分钟）：110分钟 是否需发放奖励金：是 试题 : Warning : Each part from I - V is of 20 points. Please write down your answer in the answer sheets in details as well as possible. I. A. Let f be a continuous function on [a,b]. If G is any antiderivative for f on [a,b], show that b ∫ f(t)dt=G(b)-G(a). a B. Prove that for all positive numbers a and all rational numbers p/q, p/q a a ∫ dt/t = p/q∫ dt/t. 1 1 II. A. Compare the following two statements, d/dx[∫f(x)dx] and ∫d/dx[f(x)]dx. B. Evaluate the integral 1/2 3 1/2 ∫x^5(x^2+1) dx. 0 III. -x^2/2 A. Let f(x)=e for all x. (a)Determine the symmetry of the graph and find the asymptotes. (b)On what intervals does f increases?decrases? (c)Find the extreme values. (d)Determine the concavity of the graph and find the points of inflection. (e)Sketch the graph. B. Find 3x d/dx[(x^2+1) ]. IV. A. Show that for a>0, 1/2 ∫dx/(a^2-x^2) =arcsin(x/a)+c. B.Show that arctanhx=1/2ln(1+x/1-x),-1 < x < 1. V.Prove the following equations. n n-1 n-2 A. For n属於N, ∫sin xdx = (-1/n)sin x cosx + (n-1/n)∫sin xdx. n n-2 n-2 B. For n=>2, ∫sec xdx = (1/n-1)sec x tanx + (n-2/n-1)∫sec xdx. --

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