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）︰2007/1/17
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. Calculate the integral. (30%)
dx 2
(a) ∫——————— (b) ∫ 2^(-x)dx
√(x√(x)+x) 1
-1 dx
(c) ∫xtan xdx (d) ∫——————
1+sinx+cosx
(x^5+2) 5
(e) ∫———— dx (f) ∫sin xdx
(x^2-1)
2. Let f be a one-to-one, twice differentiable function and let g=f^(-1).(10%)
(a) Find g^n.
(b) Suppose that the graph of f is concave up on an interval I.
What can you say about the graph of f^(-1)?
-1 1 1+x
3. Show that tanh x = — ln( —— ), -1＜x＜1, and sketch its graph.(15%)
2 1-x
4. Determine the centroid of the region between y=cosx, 0≦x≦3π/4
and the x-axis, and the volume generated by revolving the region about
x-axis. (15%)
5. Determine whether the sequence converges, and if it does, give the
limit.(10%)
(a) n^(1/n) (b) n(a^(1/n)-1), a>0
6. Plot the polar curve and find its length. r=1+cost from t=0 to
t=2π. (10%)
7. (a) Let｛ a ｝be a convergent sequence. Prove that lim ( a - a ) = 0.
n n→∞ n n-1
(b) What can you say about the converse? That is, suppose that ｛ a ｝
n
is a sequence such that lim ( a - a ) = 0. Does ｛ a ｝
n→∞ n n-1 n
necessarily converge? Prove or give a counterexample. (10%)
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