計算題(請依題號順序答題)

計算題 (請依題號順序答題)

x x

x π

π ) limsin(

→0 . (10%)

2. Find the power series representation of f(x) = e^x. (10%)

3. Find the maximum value of f(x) = x³ – 7x² + 16x – 12. (10%)

4. Let ^f′

( )

(a) Identify where the extrema of f occur. (5 %)

(b) Find the intervals on which f is increasing and the intervals on which f is decreasing. (5 %) (c) Find where the graph of f is concave up and where is concave down. (5 %)

5. Find the tangent line and normal line to the ellipse x²−xy+y² =7 at the point (−1, 2). (10 %) 6. Find dy/dx for y=x^x, x>0. (5 %)

7. Find the derivative of y; cos 1 sin y x

= x

− . (5 %) 8. Find

∫

1 x2dx

x . (7 %)

9. Find

∫

10. Find

∫

11. Find

∫ ∫ ∫

12. 試建立一積分式並以之求出由函數y=3x−x²與y =x所封閉之平面區域面積。(7%)