課程名稱︰微積分乙上
課程性質︰必修
課程教師︰張樹成
開課學院：生農學院 醫學院
開課系所︰牙醫系 藥學系 醫檢 農化 物治 職治 公衛
考試日期（年月日）︰2012/1/10
考試時限（分鐘）：二小時
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
I.(40%) Compute the followings:
(i) lim (1+2/n)^n
n→∞
(ii) lim 1/n (sin(π/n)+sin(2π/n)+...+sin(n-1π)/n+sinπ)
n→∞
(iii) ∫ln(x^2+1)dx from 0 to 1
(iv) ∫1/(4+x^2)dx
(v) ∫e^(x^0.5) dx from 0 to 4
(vi) ∫x arctanx dx
(vii) ∫cos^2(x)sinx dx
(viii)∫1/x^3 dx from -1 to 1
II. Let f(x) be a continuous function defined by
f(x)= 2 + ∫f(t)/〔(t+2)(t+3)〕dt x≧0
Find the followings:
(i) f(0)
(ii)df(x)/dx
(iii)f(1)
III.(20%)
(i) Find the length of a circle of radius r.
(ii)Find the area of reigon bounded by
y=x^0.5 , y=0 , y=x-2
IV.(20%) the disk x^2+y^2≦1 is revolved about the line x=4 to generate
a tours.
(i) Find the surface area of the torus.
(ii)Find the volume of the solid torus.
V.(bonus) Compute the following limit:
1 t 1/t
lim (∫ (1+x) dt )
0
t→0+
∫∫∫Happy New Year∫∫∫
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