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題組內容

3. Investigate the integral

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

其他申論題

(a)(6 pts) Find f(0) and f'(x).

(b) (6 pts) Sketch the graph of f(x), indicating intervals of increasing/ decreasing, and concavity.

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

(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) ?

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

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

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

(a) (5 ps), the gradient of f.

(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).

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?

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