[試題] 周青松 微積分甲 93上期中
課程名稱︰微積分甲
課程性質︰共同必修
課程教師︰周青松
開課系所︰
考試時間︰2004.11.8
試題 :
Ⅰ. Give necessary and sufficient conditions 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=1 but discontinuous at x=2.
Ⅱ. a. Find A and B given that the function
f(x) =｛x^3 , x≦1
｛Ax+B , x＞1
is differentiable at x=1.
b. Find f'(1) given that f(x)=｛^2 , x≦1
｛2x-1 , x＞1 .
Ⅲ. Let f and g be differentiable functions such that f'(x)=g(x) and
g'(x)=f(x), and let H(x)=[f(x)]^2-[g(x)]^2.
Find H'(x).
Ⅳ. Let f be differentiable on anopen interval O. Such that
(i) if f'(x)＞0 for all x in O, then f increases on O.
(ii) if f'(x)＜0 for all x in O, then f decreases on O.
Ⅴ. Find the critical numbers, the points of infletion (of any), claesify all
the extreme values, of the function f(x)=1/4[x^3-(3/2)x^2-6x+2],x{-2,∞].
Sketch the graph.
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