105年 - 105 台灣聯合大學系統(清華、交通、陽明、中央四校聯招)學士班轉學生考試_A3、A4、A6：微積分#112759

(b) Evaluate I if C is given by x² + y² = 1.

3. (10 points): Compute

4. (10 points): For what positive x does the following series converge?

5.(10 points): Let B=.Evaluate the integral

6. (10 points): The n-th derivative ofwhere P_n(x) is a polynomial. Pn(x) Find P_n(1) for all n ≥0.

(a) Prove that

(b) Evaluate the improper integral

8. (10 points): For each continuous function f: [0,1] →R, let I(f)=dx. Find the maximum value of I(f) over all such functions f.

9. (10 points): Evaluate