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
考試日期（年月日）︰2007/11/27
考試時限（分鐘）：120
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 │2x+5│=3. (10%)
x→-4
2.True or False? Justify your answers. (10%)
(a) If lim [f(x)+g(x)] exists but lim [f(x)] does not exist, then
x→c x→c
lim [g(x)] dose not exist.
x→c
(b) If lim [f(x)+g(x)] and lim [f(x)] exist, then it can happen
x→c
lim [g(x)] does not exist.
x→c
(c) If lim [f(x)] exists, then lim √f(x) exists.
x→c x→c
(d) If lim [f(x)] exists, then lim 1/f(x) exists.
x→c x→c
(e) If f(x)<g(x) for all x≠c, then lim f(x) < lim g(x).
x→c x→c
3. Prove that if there is a number B such that │f(x)/x│≦B for all x≠0,
then lim f(x)=0. (10%)
x→c
3
4. Find A and B given that the derivative of f(x)=╭ Ax +Bx+2, x≦2
│ is
│ 2
╰ Bx -A , x>2
everywhere continuous. (10%)
n
5. Set f(x)=x ,n is a positive integer. (10%)
(k)
(a) Find f (x) for k=n.
(k)
(b) Find f (x) for k>n.
(k)
(c) Find f (x) for k<n.
6. Let f be a differentiable functuon. Use the chain rule to show that if f
is even,then f' is odd. (10%)
7. Find the critical points, the local extreme values, and the concavity,
2 1/3
and then sketch the graph of f(x)=x (x-7) . (15%)
8. Calculate. (15%)
3 2 2 2
(a)∫(x -1)/x dx (b)∫[(t - a)(t - b)/ √t] dt
π 2
(c)∫xcosx dx
0 3
x - 4
9. Find H'(2) given that H(x)=∫ x/(1+√t) dt. (10%)
2x
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◆ From: 140.112.242.17
※ 編輯: joeyer 來自: 140.112.242.17 (11/27 18:17)
※ 編輯: joeyer 來自: 140.112.242.17 (11/27 18:18)