課程名稱︰微積分甲下
課程性質︰大一共同必修
課程教師︰曾琇瑱
開課學院：
開課系所︰工管、生機、生工、地質
考試日期（年月日）︰99.05.10
考試時限（分鐘）：110mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.Match the parametric equations with the graphs. Give reason for your choices
-t -t -t
(a) x = e cos10t, y = e sin10t, z = e
(b) x = cost, y = sint, z =lnt
(c) x = t, y = 1/(1+t^2), z = t^2
(fig in 統一教學課本 p823)
2.At what points does the heilx r(t) = (sint, cost, t) intersect the sphere
x^2+y^2+z^2=5?
3.Determine whatever each series is divergent, conditionally convergent, or
absolutely convergent. Use whatever tests or theorems seem most appropriate
to justify your answer.
k
∞ k 3
(a) Σ (-1) (___)
k=1 k!
∞ k
(b) Σ k(3/5)
k=1
k+1
∞ (-1)
(c) Σ _______
k=1 ln(k+1)
2 k+1
∞ (k) (-1)
(d) Σ ________
k=1 3
k +10
n+1
∞ (-1) lnn
(e) Σ __________
n=1 n
k
∞ (-1) k!
(f) Σ ________
k=1 (2k+1)!
j
∞ (-1)
(g) Σ _______
j=1 2
j +1
∞ 2*4*6*...*(2k)
(h) Σ ________________
k=1 1*4*7*...*(3k-2)
k
∞ k
(i) Σ ____
k=1 k!
k 2
∞ (-1) (k!)
(j) Σ __________
k=1 (2k!)
4.If r(t) = (t,t^2,t^3) , Find r'(t),T'(1),r"(t), and r'(t)╳r"(t)
5.Show that the curvature of a circle of radius a is 1/a
∞ n
6.If Σ Cn3 is convergent, does it follow that the following series are
n=0
convergent? (Explain your reason)
∞ n
(a) Σ Cn(-1)
n=0
∞ n
(b) Σ Cn(-3)
n=0
7.Show that the functon defined by
-(1/x^2)
f(x)= e ,if x≠0
0 ,if x= 0
is not equal to its Maclaurin series
8.Reparametrize the curve
2 2t
r(t) = ( ______ +1) i + ______ j
t^2 +1 t^2 +1
with respect to arc length measured by the point (1,0) in the direction of
increasing t. Express the reparamatrization in its simplest form.
What can you conclude about the curve?
9.Let f(x)=arctan(5x-10)
(a) Find the Taylor series for f(x) at a = 2
(b) Find the radius of convergent of the series in (a)
(c) Find the iterval of convergent of the series in (a)
x t^2
10. Find the power series expansion for ∫ e dt
0
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