88年 - 88 國立中山大學轉學生招生考試_應數系：微積分（一）#118437

(a)

(b)

(a) Find the equation of the tangent line to C at the point (1,1).

(b) Find at the point (1,1).

(a) Is f continuous at x = 0? Prove your answer.

(b) Is f differentiable at x = 0 ? Prove your answer.

4. Let a and b be real numbers with a ＜ b. Let f be a function continuous on the closed interval [a,b] and differentiable on the open interval (a, b). If f'(x) =0 for all a ＜ x＜ b, prove that f is a constant function on [a,b].

5. Evaluate the following integral :

6. Find the length of the curve y = for 0 ≤ x ≤ 1.

7. Let f(x) = sinx, 0 ≤ x ≤ , and let Ω be the region bounded by the graph of f, the x-axis and the vertical line x =. Find the volume of the solid generated by revolving Ω about the x-axis.

8. Use the definition to prove :

(a) Prove that f is one-to-one.

(b) Let g be the inverse of f. Find g'(-3).

(c) Does the limit of f at ∞ exist ? Prove your answer.

88年 - 88 國立中山大學轉學生招生考試_應數系：微積分（一）#118437

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