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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