课程名称︰微积分甲下(暑修) 课程性质︰ 课程教师︰周青松 开课学院： 开课系所︰ 考试日期（年月日）︰100/8/24 考试时限（分钟）：115分钟 是否需发放奖励金：是~谢谢 （如未明确表示，则不予发放） 试题 : It's necessary to explain all the reason in detail and show all of your work on the answer sheet. Or you will NOT get any credits. If you used any theorems in textbook or proved in class, state it catefully and ecplicitly. Ⅰ. A. (10%) Find lim (1+x)^1/x. x→0+ B. (10%) Determine whether the sequence An= x^100n/n! is convergent or not as n→∞. If yes, find the limit of the sequence. ∞ Ⅱ. A. (10%) Prove that ∫ dx/x^p convergent if p > 1 and diverges if 0＜p≦1. 1 B. (10%) Find the mean, denoted by μ, of the exponential density function , that is , ∞ μ＝∫ xf(x)dx -∞ where f(x)＝ke^-kx ,if x≧0 f(x)＝0 ,if x＜0 k＞0. ∞ ∞ Ⅲ. A. (10%) Show that Σ x^k =1/1-x, if |x|<1. And Σ x^k diverges if |x|≧1. k=0 k=0 ∞ B. (10%) Show that Σ 1/k^p converges if, and only if, p > 1. k=1 Ⅳ. A. (10%) Find the Taylor polynomial Pn(x) of f(x)=e^x to the order n. And show that the n-th remainder trem of the Taylor expansion,denoted by Rn(x), is convergent to 0 as n→∞. ∞ B. (10%) Show that sinh x =Σ x^2k+1/(2k+1)! for all real x. k=0 Ⅴ. A. (10%) Show that arctan x = x - x^3/3 + x^5/5 - x^7/7 + ...,for -1≦x≦1 1 B. (10%) Estimate the integral ∫ e^(-x^2) dx with error within 0.0001. 0 --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 140.112.7.214 ※ fanif:转录至看板 NTUBIME104HW 04/13 16:37