作者YYPPOO (阿po)
看板NTU-Exam
标题[试题] 96暑 周青松 微积分甲下期中考
时间Thu Aug 21 11:31:37 2008
课程名称︰微积分甲下
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）︰2008/08/21
考试时限（分钟）：110min
是否需发放奖励金：是！
试题 :
I.
A). Given that f is continuous, use L'Hopital's rule to determine
1 x
lim (─ ∫ f(t) dt )
x→0 x 0
B). Calculate
1 x t^2
lim (─ ∫ e dt )
x→∞ x 0
II.
A). Find
x^(-1)
lim (1+x)
x→0+
B). Show that, for each real x,
n x
lim (1+(x/n)) = e
n→∞
III.
Prove that, if f is contimuous, positive, and decreasing on [1,∞), then
∞ ∞
Σ f(k) converges iff ∫ f(x)dx converges
k=1 1
IV.
Let Σak be a series with nonnegative terms, and suppose that
k^(-1)
(ak) → ρ
Show that,
A). if ρ<1, then Σak converges.
B). if ρ>1, then Σak diverges.
V.
Find a power series representation for the improper integral:
A).
x 1 - cost
∫ ───── dt
0 t^2
B).
x cosh t
∫ ──── dt
0 t
(每大题均20分)
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