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)