83年 - 83 中原大學轉學生招生考試(理、工、資電學院聯招)：微積分#124697

1.=_______

2.=_______

3. Let y³+ y = 2 cos x. Then =_______

4. =_______([ ]表高斯符號).

5.=_______

6. The area of the region inside the large loop and outside the small loop of the limacon r = 1+2 cos θ is _______

7. The equation of the tangent plane to the surface z = 3x²+4y² at (2, 0, 12) is _______

8.=_______

9.=_______

10. The area of the surface z = x²+ y² below the plane z = 9 is _______

11. Let R be the region bounded by the lines x = 0, y = 0 and x + y = 1.Then =_______.

12. Let S be the sphere x²+ y²+ z²= 4. Then
=_______

二、計算題
1. Find the convergence interval for the power series

2. Find the area A under one arch of the cycloid
x = t - sin t, y = 1 - cos t (0 ≤ t ≤ 2π).
Also, find the length L of this arch.

3. Find the minimum and maximum values of f(x, y, z) = x + 2y + 3z on the ellipse that is the intersection of the cylinder x2+ y2= 2 and the plane y + z = 1.