110 年 - 110 國立臺灣大學_碩士班招生考試_大氣科學研究所乙組:微積分(A)#102182　

(b) (6 pts) If he runs m tines as fast as he swims, how will his best strategy be modified as m varies (m ≥ 1) ?

(a) (7 pts) Show that the improper integral dx converges. Show that

(b) (4 pts) Write as the sum of a power series

(c) (6 pts) Write Ina dx as the sum of a series. Thus, we
can use its partial sums to estim mate the integral.

4. (10 pts) Find the twice differentiable function f(x) such that

(b) for (x, y)≠ (0,0) and f(0,0) = 0. Compute the directional derivative of f along u = (cosθ, sin θ) at (0,0).

(a) (7 pts)

(b) (8 pts)

8. (10 pts) Let S be the part of the cylinder x²+y² = 2y that lies in the sphere x² +y² +z² = 4 and inside the first quadrant. Compute