93年 - 93 國立暨南大學轉學生招生考試試題_微積分_(資工系)#125297

1.Find

 [Hint: limit of Riemann sums]

2.(a)  Show that If f is continuous on [a,b] and c is any number in [a,b], then the function F(x) = f(t)dt is continuous on [a,b], differentiable on (a,b), and satisfies F'(x) = f(x) for all x in (a,b). (Hint: f(t)dt = f(t)dt + f(t)dt )

(b)f(t)dt = √(3x²+1) - 5 , find f(x).

3.  If the circle x²+(y-b)² = a², b＞a＞0, is rotated around the x-axis, the resulting "doughnut-shaped" solid is called a torus. Find the formula for the volume of the torus.

4. (a)Prove that f(x)dx = f(a-x)dx

(b)By (a), evaluate

(a)Show that it is increasing.

(b)Find the greatest lower bound.

(c)Find the least upper bound.

(a)√(xlnx) = 0

(b)

7.Determine the path of steepest descent along the surface z = x²+3y² from the point (1,1,4) to the point (0,0,0).