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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