82年 - 82 中原大學轉學生招生考試(理、工、資電學院聯招)：微積分#125378

1. =   ① 

2. =   ② 

3.=  ③ 

4. the interval of convergence for 1+x+ + ... is   ④  .

5. The area inside the smaller loop of the limacon r=2-4cosθ is   ⑤  . 

6. the point of the curve y = lnx at which the curvature is a maximum is   ⑥  .

7. If f(x,y)= , then =   ⑦ .

8. The directional derivative of f(x,y) = x²+3xy-y² at (2, 1) in the direction toward (1, -3) is   ⑧  .

9. , where R={(x,y) | 0≤x≤√2, 0≤y≤}, is   ⑨  .

10. Let C be boundary of triangle with vertices (頂點) (0,0), (1,2), and (0,2), then 4xy²dx + 2y³dy =   ⑩  .

11. The volume in the first octant cut off from the cylinder x²+z²=4 by the plane y+z=6 is   ⑪  .

12. Suppose f(x) is continuous and strictly increasing on [0, 1] with f(0)=0 and f(1)=1. If , then =   ⑫  .

二. ①

②

 ①

② Let , where

Does converge?

四. The temperature at each point (x, y) in a semicircular plate defined by x²+y²=4, y≥0, is given by T(x,y)=16x²-2x²y+40-y². Find the hottest and the coldest points in the plate.