作者ajenny13 (~~~)
看板NTU-Exam
标题[试题] 103-2 周青松 微积分甲下 期中考
时间Fri Jul 3 17:25:46 2015
课程名称︰微积分甲下
课程性质︰必修
课程教师︰周青松
开课学院：理学院
开课系所︰地科系 生工系 生机系 工管系科管组
考试日期（年月日）︰104/4/27
考试时限（分钟）：110分钟
试题 :
(I) (20 pts)
(A) Show that lim (1+1/n)^n =e
n→∞
(B) Show that for each real x, lim (1+x/n)^n =e^x
n→∞
(II) (20 pts)
(A) Determine whether the improper integral∫∞ r/(r^2+x^2) dx (r>0)
-∞
converges or diverges. If it converges, give the value of it.
(B) Evaluate∫1 1/ x^(4/5) dx
-2
(III) (20 pts)
Determine whether the following series converge or diverge.
∞ 2^k ∞ k^k
(A) Σ ----------- (B) Σ ---------
k=1 k^3 k=1 k!
(IV) (20 pts)
∞ (-1)^k
(A) Show that for all real x, cosx =Σ ---------- x^(2k)
k=0 (2k)!
∞ (-1)^k
(B) Show that for all│x│<1, ln(1+x)=Σ---------- x^(k+1)
k=0 k+1
(V) (20 pts)
e^x-1-x
(A) Evaluate lim －－－－－－
x→0 x arctan x
by using (i)L'Hopital's rule (ii)power series.
(B) Deduce the differentiation formulas,
d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x,
from the expansions of sinh x and cosh x in powers of x.
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