94年 - 94 國立暨南大學轉學生招生考試試題_微積分_(資工系)#125252

(a)

(b)

2)
(a) Calculate the derivative

(b) Let f be continuous, Calculate the derivative

(a) Prove that has a strict maximum at x = 0 (i.e. f(0) ＞ f(x) for all x ≠ 0).

(b) Prove that f(x) is differentiable on R.

(c) Prove that f(x) is not increasing on the interval (-ε, 0) and f(x) is not decreasing on the interval (0, ε) for any ε ＞ 0.

(a) Find the area bounded by f(x) and the x-axis.

(b) Find the volume of the solid generated by revolving the region in part (a) around the x-axis.

5) Let a ＞ 0. Give a definition of the following improper Riemann integral as
a limit of Riemann integrals:



For what values of a does this integral converge.

6) Let x ∈ R, find the interval of convergence of the following power series:

7) Given that 0 ＜ a ＜ b, find the absolute maximum value taken on by the function



on the open square {(x,y): a＜x＜b,a ＜y＜b}.