課程名稱︰ 微積分甲
課程性質︰ 數學 - 微積分
課程教師︰ 周青松
開課學院：
開課系所︰生工、地質、工管
考試日期（年月日）︰2008/11/14
考試時限（分鐘）：8:20~10:00
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Ⅰ.A) 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.
B) Let f be the Dirichlet function
f(x)={ 1 , x rational
{ 0 , x irrational
Show that lim xf(x)=0
x→0
II.
A) Find A and B given that the function
f(x) = { x^2-2 , x <= 2
{ Bx^2+Ax, x > 2
is differentiable at x=2
B) Set f(x)={ x^2-x , x<=2 Show that f is continuous at x=2,
{ 2x-2 , x>2 is f differentiable at x=2?
III Let f be a differentiable function. Use the chain rule to show that:
A) If f is even , then f' is odd.
B) If f is odd , then f' is even.
IV A) Set f(x)=(secx)^2 and g(x)=(tanx)^2 on the interval(-π/2 , π/2)
Show that f'(x)=g'(x) for all x in (-π/2 , π/2).
B) Assume that f and g are differentiable on the interval (-c , c),
c>0 , and f(0)=g(0). Show that if f'(x)>g'(x) for all 屬於(0,c),
then f(x)>g(x) for all x屬於(0,c).
V Sketch the graph of f(x)=1/4‧x^4 - 2(x^2) + 7/4 ,x屬於[ -3, 無窮大).
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