課程名稱︰微積分甲上 課程性質︰必修 課程教師︰李白飛 開課學院：理學院 開課系所︰物理學系 考試日期（年月日）︰101/1/10 考試時限（分鐘）：150 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.Let f(x) be a continuous function on the positive real axis such that xy x y ∫f(t)dt = y∫f(t)dt + x∫f(t)dt for all x>0 and all y>0. If f(1)=3, compute 1 1 1 f(x) for each x>0. (Simplify your answer!) 2.Show that the function f(x) = x + e^x has an inverse function on R. Let g(x) be the inverse function of f(x). Find the area of the region bounded bt y=g(x) and the lines y=0 and x=1+e. x ln(t) 3.Let f(x) = ∫ -------dt for x>0. Compute f(x)+f(1/x). (Simplify your answer!) 1 (t+1) 1 4.Obtain and use a reduction formula to evaluate I_n=∫(1-x^2)^n dx. (n=1,2...) 0 e^(2x) 5.Evalute ∫-------------dx. (e^x+1)^(1/4) (pi/2) sin(x) 6.Evalute ∫ ---------------dx. 0 1+cos(x)+sin(x) 1 7.Evalute lim ---[√(n^2+1^2)+√(n^2+2^2)+...+√(n^2+n^2)] n→∞ n^2 8.Evalute lim x[(1+1/x)^x-e]. n→∞ 9.A drinking glass is a right-circular cylinder of radius a and height h. It is tilted until the water level bisects the base and touch the rim. Find the volume of the remained water. (很難理解的題目，就是一杯水倒到杯底半圓，問剩下水的體積) 10.Find the centroid of a sector of radius a and angular width θ. 然後這張好像很難，平均不及格 --

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