課程名稱︰微積分甲上
課程性質︰暑修
課程教師︰周青松
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010/7/14
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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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