課程名稱︰微積分乙
課程性質︰必修
課程教師︰陳俊全
開課學院：醫學院
開課系所︰醫學系、牙醫系
考試日期（年月日）︰114/12/18
考試時限（分鐘）：120
試題 :
Part I: Answer all the problems (1.-3.) below
1.(10%) True or False.
(a) There is a differentiable function f such that f(-2)=-2, f(2)=6, and
f'(x)<1 for all x.
(b) If f(x) is integrable on the closed interval [a,b], then there exists a
b
number c∈[a,b] such that∫ f(x)dx= f(c)(b-a).
a
x t^2 √(1+u^4)
(c) Let f(x)=∫∫ ───── du dt with x>0 and t>0. Then f(x) has a local
1 √t u
minimum at x=1.
f'(x)
(d) If lim ─── can not be determined, according to L'Hopital's Rule,
x→∞ g'(x)
f'(x)
lim ─── does not exists.
x→∞ g'(x)
t
(e) If f(x) is continuous on [0,1] and f(0)=0, then ∫√(1+[f'(x)]^2)dx≧
√(t^2+[f(t)]^2) for 0≦t≦1. 0
2.(45%)
∞
(a)(6%) Evaluate the integral ∫x^(-2) ln(1+x^2) dx.
1
sin^2(x)
(b)(6%) Evaluate the integral ∫───── dx.
1+sin^2(x)
(c)(7%) Compute the volume of the solid obtained by rotating y=(x^2+4x+7)^(-¼)
, 0≦x≦2, about the x-axis.
(d)(6%) Find the limit
6 6*2 6*3 6n
lim (─────+─────+─────+...+─────).
n→∞ n^2+3*1^2 n^2+3*2^2 n^2+3*3^2 n^2+3*n^2
1 1
(e)(6%) Evaluate lim ( ─-────).
x→0+ x arctanx
(f)(7%) Show that |(66)^⅓ -4-(1/24)|<1/(9*256).
(g)(7%) Prove that (1+x)^p≧1+px if p≧1 and x≧0.
3.(15%) The Euler's gamma function Γ(x) is defined as
∞
Γ(x)=∫ t^(x-1) e^(-t) dt, x>0.
0
(a)(3%) Compute Γ(1).
(b)(6%) Show that Γ(x+1)=xΓ(x) for x>0
∞
(c)(6%) It is well known that ∫ e^(-x^2/2)dx=√(2π). Use this fact to compute
Γ(1/2) -∞
───────────────────────────────────────
Part Ⅱ: Choose 2 of the following 5 problems (4.~8.) and solve them.
4.(15%)
(1) Assume that f is continuous on [0,1] and n∈N. Evaluate the limit
1
lim ∫f(x)x^n dx.
x→∞ 0
(2)Assume that f' exists and is continous on [0,1] and n∈N. Evaluate the limit
1
lim n∫f(x)x^n dx.
x→∞ 0
(3) If we only assume that f is continuous on [0,1], does
1
lim n∫f(x)x^n dx exists?
x→∞ 0
5.(15%) Solve the differential equation
N'(t)=N(t)(N(t)-2)(1-N(t)/5).
Show that if N(0)>2, then lim N(t)=5.
t→∞
6.(15%) Let α∈R. Use Taylor's Theorem to show that
α(α-1) α(α-1)...(α-n+1)
(1+x)^α=1+αx+────x^2+...+──────────x^n+... for |x|<1.
2! n!
In the following problems, you can use the property that a closed interval
[a,b] is compact: every sequence in [a,b] has a subsequence which converges to
a point in [a,b].
7.(15%) Show that a continuous function f on [0,2] is uniformly continuous and
2 1 2
lim ∫ f(x)[1-─sin^2(nx)]dx=∫f(x)dx.
n→∞ 0 n 0
8.(15%) Show that a continuous function on [0,2] is integrable.
Happy Winter Vacation!!
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