[試題] 朱 樺 微積分甲上 第一次中考

[stacyfish (欣)]

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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