100年 - 100 國立暨南國際大學轉學生入學考試試題_資工系二：微積分#124207

(a) (5%) Let f(x)=x², prove that f'(x)=2x by the definition
f'(x)=．

(b) (5%) Let g be a function of several variables which is defined in some neighborhood of 0. We will say that g(h) is o(h) iff
= 0.
Prove that g(h) = ||h|| is o(h).

2. (15%) Suppose that the temperature at each point of a metal plane is given
by the function T(x,y)=1+x²-y². Find the path followed by a heat-seeking
particle that originates at (-2, 1).

3. (10%) Find the directional derivative of the function g(x,y)=x²+y² at the
point (1, 2) in the direction of the vector 2i-3j.

4. (10%) Use polar coordinates to calculate the volume of a sphere of radius R.

5. (10%) Find the interval of convergence of Σ(x+2)^k.

6. (10%) Sketch the graph of the function f(x) = - 5x.

7. (10%) Evaluate: ∫.

8. (10%) Estimate f(5.4) given that f(5)=3 and f'(x) =.

9. (10%) The region bounded by the curves y = x²-2x and y = 3x is revolved
around the line y = -1. Find the volume of the solid that is generated.