作者def2864 (def2864)
看板NTU-Exam
标题[试题] 97上 林绍雄 微积分甲 第四次小考
时间Fri Jan 9 17:27:05 2009
课程名称︰微积分甲
课程性质︰必带
课程教师︰林绍雄
开课学院:理学院
开课系所︰物理系
考试日期(年月日)︰2009.1.5
考试时限(分钟):60
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
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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