課程名稱︰微積分甲
課程性質︰必帶
課程教師︰林紹雄
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2009.1.5
考試時限（分鐘）：60
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
A.Solve the following problem. Each has 20 points
(a) A 45 degree notch is cut to one quarter way (10 cm) into a cylindrical
log having radius 20cm. One plane face is perpendicular to the axis of
the log. What volume of wood is removed from the log by cutting the
notch?
(b) Find the area of the smaller of the two loops enclose by the curve
r=1+(2)^0.5*sinθ
B.(20 points) Does the improper integral ∫(0 to ∞) xsin x^3 dx converge?
C.(20 points) Determine whether the series
∞ n-1 2
summation (-1) *(n!) /(kn)!
n=1
(where k is a positive integer converges absolutely, or
conditionally, or diverges.
D.Determine which of the following statements is true. Give sufficient
reasioning to support yout answer. Each has 10 points.
(a) Let a(n) be a sequence satisfying │a(n+1)-a(n)│小於等於 sin(1/n^2)
for n = 1,2,3.... Then a(n) must converge.
(b) Assume that the power series summation (n=0 to ∞)a(n)x^n has R as its
radius of convergence (where 0 < R < ∞). If lim(n→∞)a(n)*R^n=0,
then summation (n=0 to ∞) (-1)^n*a(n)R^n must converge.
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