93年 - 93 學士後：微積分#86898

1. Let f be a continuous function such that , then the infinite series converges. (A)O(B)X

3. If f is integrable on [a,b] , then there exists a number c ∈[a,b] such that .(A)O(B)X

5. Let f be a function defined as follows:

 Then f ( x ) is differentiable at x = 0. (A)O(B)X

6. We already know that the series is convergent. Now, rewrite the series by combining 2ⁿ consecutively positive terms in the series, then followed by 2ⁿ consecutively negative terms, where n = 1, 2, 3, …, to get a new series as follows:
+... , then the new series is still convergent. (A)O(B)X

7. If , then =1 . (A)O(B)X

8. If ∑a_n is a converging series with a_n for any n∈N , then .(A)O(B)X

11. Let f (x) = xsin x , then = ? (A) 1 (B) (C) (D) 2 (E)

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

14. Consider the function f(x)= . The absolute minimum value of the function f on the interval  is: (A) – 4 (B) 0 (C) (D) 1 (E)

15. Find a and b so that function f(x)=ax³+bx²+1 will have a relative minimum value at a point inside the open interval (1, 3). (A) a =1, b =1 (B) a =1, b = −1 (C) a = −1, b =2 (D) a = 2, b = −4 (E) a = −1, b =3

16. Consider the sphere x²+y²+z²-2x=0 and plane + y + z = 0 . Find the angle between normal lines of these two surfaces at point (0, 0, 0). (A) (B) (C) (D) (E)

17. Let f ( x ) be a function defined by ...  
Which of the following items is the value of the integral  ? (A) 1 (B) 2 (C) 3 (D) 4 (E) ∞

18. Let g( x ) be the inverse function of the function f ( x )= xe^x , where x ≥ 0, i.e., g(f(x)) = x, and f(g(x)) = x for x ≥ 0. Which of the following items is the value of the integral ? (A) e – 1 (B) 1 (C) e (D) 1 + ln 2 (E) e ² – 1

19. Let f ( x ) be a function defined by  
Let p( x ) = a₀ + a₁ x + a₂x²+ … + a_nxⁿ+ … be the Maclaurin series of f ( x ).
Which of the following items is the value of a₃ ? (A) –1 (B) 0 (C) (D) (E)1

20. Which of the following items is the value of the integral ? (A) ln2 (B) 2 (C) ln 2e (D) 2 + ln2 (E) 2e²

21. Let f(x) = [x] be the greatest integer function, where [x] is the greatest integer less than or equal to x. Which of the following items is the value of the integral ? (A) 3 (B) 6 (C) (D) 8 (E)

22. Let a₁ = 1 and a_n+1 = for n ∈ N . Which of the following items is the value of the limit ? (A) 2 (B) 1 (C) (D) e (E) ln 2

23. Find the volume of the region E bounded by z=x²+y²,x²+y²=4 and z = 0 . (A) (B) 2π (C) (D) 4π (E) 8π

24. Find  (A) e − 2 (B) e −1 (C) (D) (E)

25. Determine whether the series converges. (A) (B) (C) (D) 

26. Find the directional derivative of  F(x,y,z)=xy+2xz-y²+z² at the point (1, －2, 1) along the curve x =t,y =t−3,z=t²  in the direction of increasing z. (A) (B) (C) 11 (D) 13 (E)