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︰數學系
考試時間︰2005.11.11 8:10~10:00
試題 :
Ⅰ.Show that
sin x
A. lim ─── = 1
x→0 x
1-cos x
B. lim ──── = 0
x→0 x
Ⅱ.
A. Determine the discontinuites, if any , on the following function:
{ 2x+1 , x≦0
f(x) = { 1 , 0＜x≦1
{ x^2+1 , x＞1
B. Given that
{ x^2 , x≦1
f(x)={
{ 2x-1 , x＞1
find f'(1)
Ⅲ.Let f be continuous on an arbitrary interval I and differentiable on
the interior of I.
A. If f'(x)＞0 for all x in the interior of I,
then f increase on all of I
B. If f'(x) = 0 for all x in the interior of I, then f is constant
on all of I.
Ⅳ. Let f be twice differentiable on an open interval O 屬於 R
A. If f''(x)＞0 for all x in O, then the graph of f is concave up.
B. If f''(x)＜0 for all x in O, then the graph of f is concave down.
4 2
Ⅴ.Sketch the graph of the function f(x)=1/4 x - 2x + 7/4
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