97年 - 97 台灣聯合大學系統(清、交、陽、中四校聯招)學士班轉學生聯合招生試題：微積分#113210

1. If f'(0) =-1, find.Answer : ________

2. Find dt.
 Answer:_________

3. Convert the integral  dzdydx to an equivalent integral in cylindrical coordinates.
Answer :__________(Do not evaluate the integral).

4. Evaluate the integral Answer : ____________

6. The derivative of f(x, y) at P0(1, 2) in the direction of i + j is and in the direction -2j is -3. What is the derivative of f in the direction of -i -2j?
Answer :__________

7.the area of cap cut from the sphere x² + y² + z² == 2 by the cone z =.
Answer :__________

8. Evaluate the line integral f where C consists of line segments joining successively the points (1,0), (I, 1), (-1, 1) and (-1,0). Answer : __________

(a) Find the limit: (6分) 

(a) A number a is called a fixed point of a function f if f(a) = a. Prove that if f'(x)≠1 for all real numbers x, then f has at most one fied point. (6 分)

(b) Show that if converges，thenconverges.(6分)