5. (a) The series is absolutely convergent.

 (b) The series is absolutely convergent.
 (c) The series is absolutely convergent.

  Among the above three statements, how many of them are true?
(A)Only one.
(B) Only two.
(C)All of them.
(D) None of them. 

1. Let f be a smooth function on R. (a) If f" ＞ 0 on R, then f' must.be increasing on R. (b) If f" ＞ 0 on R, then f must be concave upward on R. (c) If f"(a) = 0, then the point (a, f(a)) must be an infection point. Among the above three statements, how many of them are true? (A)Only one.(B) Only two. (C)All of them. (D) None of them.

複選題2. A function f on R is odd if f(-x)"=-f(x) for all xR while it is even if f(-x) = f(x) for all x  R. Suppose that g and h are two functions on R. (a) If g is an even function, then the composite function h o g must be even. (b) If g is an odd function and , then the limit must be -3.(c) If g is an even function and , then the limit must be 3.Among the above three statements, how many of them are true? (A)Only one.(B) Only two. (C)All of them. (D) None of them. 

3. Consider the integral dA, where D is the region bounded by two concentric circles centred at the origin with radi r and 1 respectively, 0＜" ＜ 1. Let .  (a)Jpn is convergent if p ＜ 1. (b) Jp is convergent if 1 ＜p ＜2. (c) Jp is convergent if 2 ＜ p. Among the above three statements, how many of them are true? (A)Only one.(B) Only two. (C)All of them. (D) None of them.

4. Let (a)(b)(c)Among the above three staterents, how many of them are true? (A)Only one.(B) Only two. (C)All of them. (D) None of them. 

1. (A) (B)(tan π)=π (C) If f(R) = , then f(x) = x. (D) .

2. (A) If f(x) is continuous on (a,b), then there is c (a,6) such that f(c) =(B) Suppose that f(x) is continuous but not differentiable at a = 0. The function g(x) = xf(x) must be differentiable at x = 0. (C) If f(2) is a differentiablo function, then . (D) If f(x) is continuous on [a, b] and differentiable on (a,b), then there is a unique c (a,b) such that f(b)-f(a)=f(c)(b-a). 

3. Consider the function(A) f(x) does not have any horizontal asymptotes and any vertical asymptotes. (B) f(xc) has a horizontal asymptote, but f(x) does not have any vertical asymptotes. (C) f(x) has a vertical asymptote, but f(x) does not have any horizontal asymptotes. (D) f(x) has a horizontal asymptote and a vertical asymptote.

複選題4. (A) sin me sin ne de = 0 for all m,n N. (B) sin me cos na de = 0 for all m,n  N. (C) cos me cos nae do = 0 for all m,n N. (D) sinm t cosn a dx = 0 for all m,n  N.

5. Consider the improper integral , where p is a postive aumber. (A) Ip is convergent if p ＞ 1 and divergent if 0 ＜ p ≤ 1. (B) Ip is convergent if 0 ＜ p <1 and divergent if p ≥ 1. (C) Ip is convergent for all p ＞ 0. (D) Ip is divergent for all p ＞ 0.

6. The enclosed area of the curve (x2 + y2)2 = x2 - y2can be written in the integral form as(A)(B)(C) (D)