92年 - 92 國立清華大學轉學生招生考試試題_微積分_(八系聯招)#113231

1. If , then a+b+c=_____

2.=_____

3.=_____

4. Suppose f(x) is a differentiable function on (0,∞) and satisfies

Then f'(4)=_______

6. The smallest value of the function f(x)=  sin x on the interval  is________ .

7.1 is the power series expansion of , then a₃=________

 二、計算與證明題
1. Does there exist a differentiable function f : R - R such that f(0) = 0 and f(f(x)) = x⁴- x for all x ε R? Give reasons for your answer.

2. Let a₁= 1 and, for n ≥ 2, let Prove that exists and find the limit.

(1) Prove that a₂ ≤ a₁≤

(2) Prove that= 0.

4. (1) Compute the area of the region
D =｛(x,y) /4x² ≤ y ≤ (4x +1)²+3 and y ≤ 4)｝.

(2) Let be the centroid of D. Find .