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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 

選擇:0題,非選:14題 我要補題 回報試卷錯誤
【非選題】
1.

1. Suppose that f(x) satisfes the equation f(x +y)= f(x)+ f(y) + xy - x2y - xy2 for all6168f2e98fae6.jpg


【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 6168f356c8c33.jpg


【題組】

(a) (7 pts) Show that the improper integral 6168f37e741a4.jpgdx converges. Show that6168f3a37dfb4.jpg



【非選題】
6.【題組】

(b) (4 pts) Write 6168f44ee7347.jpg as the sum of a power series6168f46ea96a2.jpg



【非選題】
7.【題組】

(c) (6 pts) Write6168f4b4d77bc.jpg 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 that6168f51d4bfcf.jpg



【非選題】
9.

5.


【題組】 (a) (5 ps)6168f55f3442a.jpg, the gradient of f.



【非選題】
10.【題組】

(b) 6168f591d40d2.jpg 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) 6168f65608ffb.jpg



【非選題】
13.【題組】

(b) (8 pts) 6168f683dd412.jpg



【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 6168f7a5b704c.jpg



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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-阿摩線上測驗

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