課程名稱︰微積分甲
課程性質︰
課程教師︰周青松
開課系所︰數學系
考試時間︰2006年7月14日
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
I.
A). Let f(x) = { x^2 , x＜1 Find A given that f is continuous at 1.
{Ax-3 , x≧1.
B). 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=1 but dis continuous at x=2.
II.
A). Show that, for all real numbers c,
1-cos x
lim Sin x = Sin c , lim ──── = Cos c
x→c x→c x
B). Prove that
Sin x 1-cos x
lim ──── = 1 and lim ──── = 0
x→0 x x→0 x
III.
A). Find f'(x) given that f(x) ={ x^2, x≦1
{2x-1, x＞1 .
B). Show that if f is diifferentiable at x, then f is continuous at x.
IV.
A). Prove that if P(x)=a x^n + a x^n-1 + ... + a x +a
n n-1 1 0
then P'(x)=na x^n-1 + (n-1)a x^n-2 + ... + a1
n n-1
B). Show that (i). d n n-1
─ x = nx holds for all integers.
dx
(ii).d p/q p p/q - 1
─ x = ─x , q≠ and p,q屬於Z
dx q
V.
A). d [ d ] B). d [ ]
─[t─(cos t^2)] ─[Sin(f(3x))]
dt[ dt ]. dx[ ]
(每大題均20分)
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