92年 - 92 國立暨南大學轉學生招生考試試題_微積分_(資工系)#124709

(a) (7%) Calculate g(h).

(b) (13%) Show that, of all lines that pass through (x, f(x)), the tangent line is the line that best approximates the graph of f near the point (x, f(x)) byshowing that
g(h) = o(h) iff m = f'(x).

(a) (6%) Find the points (x,y), if any, at which ∇f(x,y) = 0.

(b) (6%) Sketch the graph of the surface z = f(x,y).

(c) (8%) Determine the path of steepest descent along the surface z = f(x,y)
from the point (1, 1, 3).

(a) (12%) Prove that ln(ab) = ln(a) + ln(b) for a, b ＞ 0.

(b) (8%) By (a), to prove ln(1/b) = -ln(b) and ln(a/b) = ln(a) - ln(b).

(a) (8%) Find the area of the region R bounded by the graph of f and the
x-axis.

(b) (12%) Find the volume of the solid generated by revolving R around the
x-axis.

5. (20%) Verify that the serieshas interval of convergence [-1, 1].