推 luky60311 :推S XD 08/24 18:08
推 Murasaki0110:淚推QQ 08/25 17:30
課程名稱︰微積分甲下(暑修)
課程性質︰
課程教師︰周青松
開課學院：
開課系所︰
考試日期（年月日）︰100/8/24
考試時限（分鐘）：115分鐘
是否需發放獎勵金：是~謝謝
（如未明確表示，則不予發放）
試題 :
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
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