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7. (10 points) Can you find a C¹function 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.

其他申論題

3. (15 points) Is uniformly continuous on R? Prove your assertion.

4. (15 points) Show that is a differentiable function on R. Is x =0 the derivative f':R→R a continuous function?

5. (20 points) Show thatdx converges conditionally.

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.

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).

1. (10 %) Prove or disprove that.

2. (5%) Letdt, find derivative of F(x) .

3. (10%) Use Riemann sum to find the definite integraldx.

4. (10 %) Find , where a, b are constant real numbers.

(a) (5%) Findxdx

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