97年 - 97 國立暨南國際大學_轉學生入學考試試題_資工系二:微積分#124585

(a) (10%) Please give an ε and δ and show that .
(Hint: ||a|-|b||＜|a-b|)

(b) (10%) Let g be a function defined at least on some open interval containing the number 0. We say that g is of smaller order than h, or that g(h) is little-o(h) and write g(h) = o(h), if and only if g(h) is small enough compared with h that  = 0. Show that h³ = o(h).

2. (15%) Show that ln(1+x) ≈ x as x→0. (Hint: (ln(x))'=1/x)

(a) (10%) Find the critical numbers of F and determine the
intervals on which F is increasing and the intervals on
which F is decreasing.

(b) (10%) Determine the concavity of the graph of F and find the
points of inflection (if any).

(a) (10%) Verify that the region below the graph of f(x) = 1/x, x≥1 has infinite area.

(b) (10%) Suppose that the region (a) is revolved about the x-axis, what is the volume of the resulting solid ?

(a) (5%) At the point (1,-2), in what direction does the voltage increase most rapidly ?

(b) (5%) In what direction does the voltage decrease most
rapidly ?

(c) (5%) What is the magnitude of this increase or decrease ?

(d) (10%) Describe the path of a particle that starts at the point (1,-2) and moves in the direction of greatest voltage increase.