92年 - 92 學士後：微積分#85633

1. . (A)O(B)X(C)(D)

2. . (A)O(B)X(C)(D)

3.   (A)O(B)X(C)(D)

5. . (A)O(B)X(C)(D)

6. . (A)O(B)X(C)(D)

7. Suppose f is integrable on [ a,b] , define F(x)=, then  F is differentiable on (a,b). (A)O(B)X(C)(D)

8. Let f be a continuous function defined on a closed interval [1, 3] and 3 f (x) ≤ for all ] x ∈[1,3 . Define (A)O(B)X(C)(D)

9. Let f be a function defined on the set D = {(x, y) | −1 ≤ x ≤ 1,−1 ≤ y ≤ 1}. If (0,0) and (0,0)  exist, then f is differentiable at (0,0). (A)O(B)X(C)(D)

11. . (A) (B) (C) (D) (E)

12. The area between the curve y = and the lines y = 0 , x = 0 and x = 1 is . (A) (B) (C) (D) (E)

13. Find the equation of the curve that satisfies the differential equation yy′ + 2x = 0 and that passes through the point (3,−1). (A) (B) (C) (D)(E)

14. If f (u, v, w) is differentiable and u = x − y , v = y − z and w = z − x , then =？ (A) − 3 (B) 0 (C) 3 (D)-  (E)

15. If  f (x, y) = xe^y , then the rate of change of f at the point P(2,0) in the direction from P to , is： (A) − (B) − (C)1 (D) (E)

16. =_____ . (A) 1 (B) 2 (C) 3 (D) –1 (E)

17. Suppose a, b＞0, then =_____ . (A) (B) (C) (D) (E)

18. =_____  . (A) (B) (C) (D) (E)

19. Let x f x xe^−x and p(x) = ... be the Maclaurin series of f(x), then  a₄ = . (A) (B) (C) (D) (E)

20. The slope of the tangent line to the polar curve cosθ  r = +cosθ at the point(r ,θ) =(√2, ) = is： (A) –1 (B)  − (C)  − (D)− (E) -

21. The volume of the solid bounded by z = is ： (A) π (B) π  (C) π  (D) π  (E) π

22. Let = _____. (A) 1 (B) 2 (C) 4 (D) 16 (E)

23. Define the function . Which of the following statement is correct? (A) Function f derives its absolute maximum at point x = −1 (B) Function f derives its absolute maximum at point x = 0 (C) Function f derives its absolute maximum at point x = 1 (D) Function f derives its absolute minimum at point x = −1 (E) Function f does not have absolute maximum or minimum value

24. Define ∫  . Which of the following statement is false? 

(A) f is a strictly increasing function 

(B)  f ′(x) = (cos x)⁴ 

(C) ) f (x + 2π ) − f (x is constant 

(D) 0 f (x) ≥ for all real number x (E) 0

25. Let D = {(x, y)〡 0 ≤ x ≤ 2 , 1 ≤ y ≤ 2} π , then ∫∫ _D x cos(xy)dA =_____ . (A) − (B) −1 (C) 0 (D)  (E) 1

26. = _____ . (A) (B) (C) (D) (E)