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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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8. (10 points) Construct a function f:R
2
→IR such that f
x
and f
y
exist at (0,0) but f is not differentiable at (0,0).
其他申論題
4. (15 points) Show that is a differentiable function on R. Is x =0 the derivative f':R→R a continuous function?
#441516
5. (20 points) Show thatdx converges conditionally.
#441517
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
1. (10 %) Prove or disprove that.
#441521
2. (5%) Letdt, find derivative of F(x) .
#441522
3. (10%) Use Riemann sum to find the definite integraldx.
#441523
4. (10 %) Find , where a, b are constant real numbers.
#441524
(a) (5%) Findxdx
#441525
(b) (10 %) Evaluate the improper integral
#441526
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