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# 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

【非選題】
1.

1. Suppose that f(x) satisfes the equation f(x +y)= f(x)+ f(y) + xy - x2y - xy2 for all

【題組】(a)(6 pts) Find f(0) and f'(x).

【非選題】
2.【題組】(b) (6 pts) Sketch the graph of f(x), indicating intervals of increasing/ decreasing, and concavity.

【非選題】
3.
2. A man runs twice as fast as he swins. He is at point A on the edge of a circular pool with radius 20 meters and he wants to get to the diametrically opposite point B as quickly as possible. He can run around the edge to a point C and then swim directly from C to B.

【題組】(a) (10 pts) How should he choose the point C to minimize the total time ?

【非選題】
4.【題組】

(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) ?

【非選題】
5.

3. Investigate the integral

【題組】

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

【非選題】
6.【題組】

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

【非選題】
7.【題組】

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

【非選題】
8.

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

【非選題】
9.

5.

【題組】 (a) (5 ps), the gradient of f.

【非選題】
10.【題組】

(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).

【非選題】
11.6. (10 pts) Find the critical points of f(x, y), where z = f(x, y) satisfies the equation yz+x Iny = z2. Are these critical points local maxinum, local minimum, or saddle points?

【非選題】
12.
7. Evaluate the following multiple integrals.

【題組】

(a) (7 pts)

【非選題】
13.【題組】

(b) (8 pts)

【非選題】
14.

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

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