課程名稱︰微積分甲上 課程性質︰大一必修 課程教師︰朱樺 開課學院：電資學院、工學院 開課系所︰電機系、材料系 考試日期（年月日）︰2014/10/20 考試時限（分鐘）：50 mins 試題 : CALCULUS QUIZ 2 Instructor: Professer Huah Chu Quiz date : October 20, 2014 Quiz time : 50 minutes, Total score: 40 points 1. Let f(x) = [x(x-1)(x-2)...(x-n)]/[(x+1)(x+2)...(x+n)], find f'(0). (5 points) 2.Let f(x) = lim_{n→∞} cos^{n} (x) , (a) Discuss the continuity of f(x), and nd the points of removable dis- continuity. (4 points) (b) Find f'(x). (3 points) 3.Suppose that the function f(x) satises f(x + y) = f(x)f(y) for all x,y, and f(0) ≠ 0, f'(0) = k. Show that f'(x) = kf(x). (7 points) 4.Find a line that passes through the origin and is tagent to y = x^x. (7 points) 5.Let f(x) = log_x 2, (a) Find lim_{x→∞} f(x), (2 points) (b) Find lim_{x→∞} f(x), (2 points) (c) Find f'(x). (3 points) 6.For what value of k does the equation e^{2x} = k*sqrt{x} have exactly one solution? (7 points) --

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