課程名稱︰微積分上
課程性質︰必修
課程教師︰黃維信
開課學院：工學院
開課系所︰工程科學與海洋工程學系
考試時間︰95/11/21
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Give an ε,δ proof for lim x^3= 1. (10%)
x→1
2. Use the pinching theorem to prove lim sinx/x=1. (10%)
x→0
3. Let f(x)={x, x rational and g(x)={x^2, x irrational
{0, x irrational {0, x irrational
(a) Show both functions are continuous at 0 and are discontinuous
at each x≠0.
(b) Can either function be differentiable at a point x≠0.
(c) Show that f is not differentiable at 0. (20%)
(d) Show that g is differentiable at 0 and give g'(0). (20%)
4. Find equations for the line tangent to the ellipse 4x^2+y^2=72
that are perpendicular to the line x+2y+3=0. (10%)
5. Show that the equation x^3+ax+b=0 has exactly one real root if a ≧0
and at most one real root between -√(|a|/3) and √(|a|/3)
if a＜0. (10%)
6. Sketch the graph of f(x)=x^1/3(x+4), and indicate the extrema,
inflection points, concavity, and asymptotes (if any). (20%)
7. Assume that f is a continuous function and that
x
∫tf(t)dt= sinx-xcosx.
0
(a) Determine f(π/2).
(b) Find f'(x). (10%)
x
8. Let f be a continuous function and set F(x)=∫xf(t)dt.
0
Find F'(x). (10%)
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