※ [本文轉錄自 NTU-Exam 看板] 作者: jeff3301 (彬) 站內: NTU-Exam 標題: [試題] 95上 周青松 微積分甲上 期末考 時間: Thu Jan 18 21:25:35 2007 課程名稱︰微積分甲上 課程性質︰數學微積分 課程教師︰周青松 開課學院： 開課系所︰限生機、生工、地質、地理、工管等系學生修習 考試日期（年月日）︰1月12日 考試時限（分鐘）：120分鐘 8:10~10:10 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Ⅰ A) Let f be a function such that f' is continuous on [a,b] Prove that b ∫ f'(t)dt = f(b) - f(a) a B) Calculate ∫ [f(x)g''(x) - f''(x)g(x)] Ⅱ A) Find f from the information given : f''(x) = sinx f'(0)= - 2 f(0) = 1 B) Calculate the derivative : d/dx 2x 2 ( ∫ t √1+t dt ) tanx Ⅲ A) The base of a solid is the region between the parabolas 2 2 x = y and x = 3 - 2y Find the volume of the solid given that the cross section perpendicular to the x - axis are squares . 2/3 B) Let Ω be the region bounded below by the curve y = x + 1 bounded to the left by the y-axis , and bounded above by the line y = 5 Find the volume of the solid generated by revolving Ω about the y-xis . Ⅳ A) If a is positive and p/q is rational , prove that p/q a a ∫ 1/t dt = p/q∫ 1/t dt 1 1 -2x -2x B) Calculate the following indefinite integration ∫sin e / e dx Ⅴ 2 2 2 -1 A) Show that F(x) = x/2√a - x + a /2 sin (x/a) a > 0 is an 2 2 a 2 2 antiderivative of f(x) = √a - x amd to calculate ∫ √a - x dx -a -1 2 B) Prove that d/dx tanh x = 1/1-x -1 < x < 1 (每大題均20分) --

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