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研究所、轉學考(插大)-高等微積分
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110年 - 110 國立中山大學碩士暨碩士專班招生考試_應數系碩士班/丙組:高等微積分#104293
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5. (20 points) Show that
dx converges conditionally.
其他申論題
1. (10 points) Is =R? State your reason.
#441513
2. (10 points) Show that = 1.
#441514
3. (15 points) Is uniformly continuous on R? Prove your assertion.
#441515
4. (15 points) Show that is a differentiable function on R. Is x =0 the derivative f':R→R a continuous function?
#441516
6. (10 points) Let fn:[0,1] →R, , be a sequence of increasing functions, i.e., fn(x) ≤fn(y) for all and n N. Assume that fn Sfnt1 and Ifn(x)I s 1 for all xe [0,1] and neN. Show that fn converges (pointwisely) to an increasing function.
#441518
7. (10 points) Can you find a C1 function f:R2 →IR such that Vf(x,y)= (-y,x) for all (x,y)? Find such a function or prove that it does not exist.
#441519
8. (10 points) Construct a function f:R2 →IR such that fx and fy exist at (0,0) but f is not differentiable at (0,0).
#441520
1. (10 %) Prove or disprove that.
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2. (5%) Letdt, find derivative of F(x) .
#441522
3. (10%) Use Riemann sum to find the definite integraldx.
#441523
dx converges conditionally.
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dx converges conditionally.