课程名称︰微积分甲 课程性质︰系必带 课程教师︰林绍雄 开课学院：理学院 开课系所︰物理学系 考试日期（年月日）︰98/1/11 考试时限（分钟）：180分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : There are problems A to H with a total of 140 points. Please write down your computational or proof steps clearly on the answer sheets. A.Find the following anti-derivatives, or definite integrals, or determine the convergence of the improper integrals. Each has 9 points. (a)∫(0 to 1) [(x^4)(1-x)^4/(x^2)+1]dx (b)∫(1+e^x)^(1/2) (c)∫dx/x(1+x+x^2)^(1/2) (d)∫(0 to π/2) secx dx/2tanx+secx-1 B.(20 points) The curve y^2+x(x+1/3)^2=0 enclosed a bounded plane region. Find its area, its centroid, and the arclength of its boundary curve. Now we revolve this plane region around the y-axis to obtain a solid of revolution. Find the volume and the surface area of this solid. C.(10 points) Apply the Taylor expansion with remainder to estimate ∫(0 to 0.25) (1+x^2)^(1/3)dx accurately up to three decimal points. D.(12 points) A cat chase a mouse along a straight line. Their distance at time t is d(t). The cat keeps a constant velocity 1, while the mouse's velocity at time t is 1/d(t). Write down the differential equation satisfied by d(t), and find d(t). Show that if d(0)>1, then d(t)>1 for all t>0 so that the cat can never be closer to the mouse than distance 1. E. Determine which of the following infinite series converges absolutely, or converges conditionallly, or diverges. Each has 6 points. (a)Σ(0,∞) (-1)^n (n^100 *2^n / (n!)^(1/2)) (b)a - b/2 + a/3 - b/4 + a/5 - b/6 + .... a and b in (b) are two positive constants. F.(12 points) Find the radius of convergence R for the power series Σ(1,∞)[(b^n / n^2)+(a^n / n )] x^n , and discuss the convergent behaviour of the series at x=±R. Find the sum of this series. G.(13 points) Let μ(n) and υ(n) (n=1,2,3,...) be two sequences. s(n)=Σ(1,n)υ(j) is the partial sum of Σ(1,∞)υ(n). If (a)υ(n) is positive, decreasing and convergent to 0, and (b)|s(n)|≦K for all n, where K＞0 is a constant, prove that Σ(1,∞)μ(n)υ(n) converges. H.Determine which of the following statements is true. Prove your answer. Each has 5 points. (a)If ∫(a to b) |f(x)|dx exists for a given function defined on [a,b], then ∫(a to b) f(x)dx exists also. (b)Let Sn be the Simpson rule approximation to the integral ∫(0 to 1)f(x)dx using 2n equal partition on [0,1]. Define the error En=|∫(0 to 1)f(x)dx - Sn|. If f(x)=x(x-1)(x+1), then En=0 (c)Let f(x) be strictly increasing for all x 属於 R. If f(0)=0, then there must exist a real number x(0)≠0 such that the sequence x(n) (n=0,1,2,...) obtained by applying the Newton method to f(x) starting from x(0) converges to 0. (d)If the series Σ(1,∞)a(n) and Σ(1,∞)(-1)^n a(n) converges, then Σ(1,∞)a(n) converges absolutely. (e) Since lim(θ→π) 1/θ-π = ±∞, the curve r =1/θ-π on the plane has the line θ=π as its asymptotic line. 若有符号不清楚的地方，这边有图 http://www.badongo.com/pic/5049643 至於答案....不要问我Q_Q --

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