課程名稱︰微積分甲上
課程性質︰必修
課程教師︰王金龍
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010/12/2
考試時限（分鐘）：30
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
All vectors are denoted by bold-faced letters
A. (a) Let r be a smooth curve. A particle with mass m slides down
along r under the influence of gravity F = (0, -mg).
Show that the equation of motion in the tangential direction
is (d^2 x/ d t^2) = -g(d y/d s), where s denotes the arc length,
t denotes the time and y denote the y-component of r.
(b) Suppose the curve is the cycloid
r(θ) = (a(θ + π + sinθ), -a(1 + cosθ)), θ∈[-π,π]
where a is a given constant and the particle slides down from
r(-θo), 0 < θo < π, with initial velocity 0.
Find the times the particle takes to travel from r(-θo) to r(θo).
B. We have known that if f is n times continuously differentiable
n ┌ ┐
and f(x) = Σ │ a (x^k) + R (x)│ with lim (Rn(x)/x^n) =0,
k=0└ k n ┘ x→0
(k)
f (0)
then a = ──── for k = 0,1,2,3,4,5,...,n.
k k!
Use this property and the Taylor series of sin x to find
x
the Taylor series for ∫ (sin t /t) dt in a neighborhood of x = 0.
0
C. Evaluate lim (sin x /x)^(1/x^2).
x→0
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