作者kerott (KERO)
看板NTU-Exam
标题[试题] 96下 周青松 微积分甲 期中考
时间Sat Apr 19 17:57:59 2008
课程名称︰微积分甲
课程性质︰数学 - 微积分
课程教师︰周青松
开课学院：(如下)
开课系所︰生机、生工、地质、地理、工管等
考试日期（年月日）︰2008/4/14 星期一
考试时限（分钟）：1:20~3:10 迟到20分钟不得进场
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
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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