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）︰2010/7/22
考試時限（分鐘）：50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.
1. Assume that f is continunous and 注:試卷continunous疑為continuous誤寫
x
∫ f(t)dt = sinx - x cosx
0
(a) Determine f(π/2).
(b) Find f'(x).
2. Let
{ x + 2, -2≦x≦0,
f(x) = { 2, 0＜x≦1,
{ 4 - 2x, 1＜x≦2.
x
And set g(x) = ∫ f(t)dt.
-2
(a) Carry out the integration.
(b) Where is f continuous? Where is f differentiable? Where is g
differentiable?
d 4
3. (a) Calculate ──( ∫ sint^2dt ).
dx tanx
d x^2 + x dt
(b) Calculate ──( ∫ ────── ).
dx x^(1/2) 2 + t^(1/2)
4. Evaluate the integral
π/4
∫ ( x^2 - 2x + sinx +cos2x )dx.
-π/4
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