課程名稱︰微積分甲
課程性質︰數學 - 微積分
課程教師︰周青松
開課學院：(如下)
開課系所︰生機、生工、地質、地理、工管等
考試日期（年月日）︰2008/4/14 星期一
考試時限（分鐘）：1:20~3:10 遲到20分鐘不得進場
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(A) Show that lim (1+1/n)^n = e
n→∞
(B) For each real x, prove that (1+x/n)^n → e^x
n→∞
2.(A) Prove that the function
f(x)= ke^(-kx), x>=0
0 , x<0
is a probability exponential density function.
(B) Calculate the standard deviation for the exponential density function.
3.Let Σak be a series with nonnegative terms, and suppose that
(ak)^(1/k)→ρ
Show that (A) if ρ<1 , then Σak converges.
(B) if ρ>1 , then Σak diverges.
4.(A) (i) By using L'Hopital's rule ,
(ii) by using power series,
to evaluate the limit: lim e^x-1-x
x→0 ----------
x(arctanx)
(B) Find a power series representation for the improper integral
x
∫ arctant/t dt
0
5.(A) Deduce the differentiation formula d/dx coshx = sinhx from the expansion
of sinhx and coshx in powers of x.
(B) Find a numerical estimate for: 1
∫ e^(-x^2) dx
0
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