課程名稱︰微積分甲上 課程性質︰ 課程教師︰周青松 開課學院： 開課系所︰ 考試日期（年月日）︰2008/7/14 考試時限（分鐘）： 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. A). Give necessary and sufficient condition on A and B for the function { Ax-B , x≦1 f(x) ={ 3x , 1＜x＜2 {Bx^2-A , 2≦x to be continuous at x=2 but discontinuous at x=1. 2. Show that: A). Sin x lim ──── = 1 x→0 x . B). 1-cos x lim ──── = 0 x→0 x . 3. A). Find C and D given that the function f(x) ={ x^2-2 , x≦2 {D(x^2)+Cx , x＞2 is differentiable at x=2. B). Find C and D given that the derivative of f(x) ={C(x^3)+Dx+2 , x≦2 {D(x^2)-C , x＞2 is everywhere continuous. 4. Assume that f and g are differentiable on (-c,c), c＞0 and f(0)=g(0). A). Show that if f'(x)＞g'(x) for all x∈(0,c), then f(x)＞g(x) for all x∈(0,c). B). Show that if f'(x)＞g'(x) for all x∈(-c,0), then f(x)＜g(x) for all x∈(-c,0). 5. A). Determine C and D so that the curve y = C cos2x + D sin 3x π has a point of inflection at (── , 5 ) 6 B). Sketch the graph of the function 1 3 f(x) = ──(x^3 - ──x^2 - 6x + 2) on [-2,∞). 4 2 (每大題均20分) --

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