课程名称︰微积分甲上 课程性质︰暑修 课程教师︰周青松 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2010/7/14 考试时限（分钟）：120分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : It's necessary to explain all the reasons in detail and show all of your work on the answer sheet. Or you will NOT get any credits. If you used any theorems in textbook or proved in class, state it carefully and explicitly. A. (a) Show that for each negative integer n. p(x)＝x^n has derivative p'(x)＝n x^(n - 1) (b) Show that the formula d p ── x^(p/q) ＝ ─ x^(p/q - 1) dx q whenever x^(p/q) is defined with p,q ∈ Z and q≠0. B. Find the indicated derivative. d (a) ──[ f(sin3x) ]. dx d (b) ──[ sin(f(3x)) ]. dx C. (a) Sketch the graph of the function f(x)＝{ x^3 , x ＜ 1 { x/2 + 2 , x ≧ 1 And find the intervals on which f increases and the intervals on which f decreases. (b) Find f(x) given that f'(x) ＝ 6x^2 - 7x - 5 for all real x and f(2)＝1. D. (a) Set f(x) ＝ sec^2 x and g(x) ＝ tan^2 x on the interval(-π/2 ,π/2). Show that f'(x) ＝ g'(x) for all x∈(-π/2 , π/2). (b) Find the critical points of the function f(x)＝{ x^2 + 2x + 2 , -1/2 ≦ x ＜ 0 { x^2 - 2x + 2 , 0 ≦ x ≦ 2 Then find and classify all the extreme values. E. (a) Find the inflection points of the function f(x) ＝ x^3 - 6x^2 + 9x + 1. (b) Sketch the graph of the function f(x) ＝ 2 sin^3 x + 3 sinx , x∈R. --

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