課程名稱︰微積分甲下 課程性質︰物理系必帶 課程教師︰陳榮凱 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2018/04/26 考試時限（分鐘）：120分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : The total is 105 points. 1.(20pts)Test the following series for absolute convergence, conditional conv- ergence or divergence. (a)∞ Σ sin(1/n) n=1 (b)∞ (-1)^n Σ --------- n=2 n ln(n) 2.(15pts)Find the radius of convergence of the series Σ[n=0,∞](n+2)x^n and compute its sum inside the interval of convergence. 3.(10pts)Find T,N and the curvature κfor the curve r(t)=(t^3/3)i+(t^2/2)j. 4.(10pts)The existence of least upper bound for every non-empty bounded above set implies that every Cauchy sequence converges. 5.(10pts)Determine the tangent plane of z=xy at (2,1/2,1) 6.(20pts)Consider f(x,y)=x^3+2y^2+x+6. Find its extreme values on the region R :{x^2+y^2≦16}. (Hint:Find local maxima or minima in {x^2+y^2<16} and find maxima or minima in {x^2+y^2=16}.) 7.(10pts)Prove that if f(x,y) satisfies Laplace's equation fxx+fyy=0, so does φ(x,y):=f(x/(x^2+y^2),y/(x^2+y^2)). 8.(10pts)Give an example of a function f(x,y) such that it has directional de- rivatives in any direction at (0,0) but is not differentiable at (0,0). --

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