1. Let f be a continuous function on the closed interval [0,1]. Which of the following statements about f must be true?
A. None 
B. I only 
C. II only 
D. III only 
E. The correct answer is not given by A, B, C, or D. 
2. Which of the following conditions are necessary for a function f to be Riemann integrable on the closed interval [a,b], where a < b?
I. f is bounded on [a,b].
II. f is continuous on [a,b].
III. f is differentiable on [a,b].
A. None 
B. I only 
C. II only 
D. III only 
E. The correct answer is not given by A, B, C, or D. 
3. Let the functions f, g, and h be defined as follows:
Which of these functions are differentiable at 0?
A. None 
B. f and g only 
C. f and h only 
D. g and h only 
E. The correct answer is not given by A, B, C, or D. 
4. Let f(x) = g(x)/h(x), where g and h are continuous functions on the open interval (a,b). Which of the following statements is true for a < x < b?
A. f is continuous at all x for which x is not zero. 
B. f is continuous at all x for which g(x) = 0. 
C. f is continuous at all x for which g(x) is not equal to zero. 
D. f is continuous at all x for which h(x) is not equal to zero. 
E. f is possibly discontinuous even though h(x) is not equal to zero. 
5. Let f be twice differentiable on (a,b). If g is an antiderivative of f" on (a,b), then gÆ(x) must equal
A. f(x) 
B. f(x) 
C. f"(x) 
D. f(x) + C, for some C not necessarily 0 
E. f"(x) + C, for some C not necessarily 0 
6. Let f(x) be differentiable for a < x < b. Which of the following statements must be true?

B. f is continuous on (a,b) 


E. f is decreasing on [a,b] 
7. Let f be a function that is continuous on the closed interval [0,1] and differentiable on the open interval (0,1). If f(0) = f(1), then which of the following statements must be true?
A. f has a minimum at some x_{0} such that 0 < x_{0} < 1. 
B. f has a maximum at some x_{0} such that 0 < x_{0} < 1. 
C. f has a minimum at some x_{0} such that 0 < x_{0} < 1. 
D. f(x_{0}) = 0 at every x_{0} such that 0 < x_{0} < 1. 
E. f(x_{0}) = 0 at some x_{0} such that 0 < x_{0} < 1. 
8. Let f be a realvalued function defined on the closed interval [a,b]. Which of the following conditions guarantees the existence of a number c such that a < c < b and f(c) = 0 ?
A. f is continuous on [a,b], and f(a) = f(b). 
B. f is differentiable on [a,b], and f(a) = f(b). 
C. f is continuous on [a,b], and f(a) and f(b) have opposite signs. 
D. f is differentiable on [a,b], and f(a) and f(b) have opposite signs. 
E. f(a) = f(b), and f(a) = f(b). 
9. Which of the following functions are differentiable on the interval (1,1) ?
A. I and II only 
B. I and III only 
C. II and III only 
D. I, II, and III 
E. The correct answer is not given by A, B, C, or D. 
10. Let
Which of the following properties does f have on the interval (0,6)?
I. ln f exists.
II. f is continuous.
III. f is monotonic.
A. None 
B. I only 
C. II only 
D. III only 
E. The correct answer is not given by A, B, C, or D. 
11. Let f be a differentiable function on the open interval (a,b). Which of the following statements must be true?
I. f is continuous on the closed interval [a,b].
II. f is bounded on the open interval (a,b).
III. If a < a_{1} < b_{1} < b, and f(a_{1}) < 0 < f(b_{1}), then
there is a number c such that a_{1} < c < b_{1} and f(c) = 0
A. I and II only 
B. I and III only 
C. II and III only 
D. I, II, and III 
E. The correct answer is not given by A, B, C, or D. 
12. Let a and b be real numbers, a < b, and let f be a realvalued function that is defined on the interval (a,b). Which of the following statements implies that f is continuous on (a,b)?
I. The range of f is an interval.
II. The graph of f has a highest and a lowest point.
III. The graph of f intersects any horizontal line at most once.
A. None 
B. I only 
C. II only 
D. III only 
E. The correct answer is not given by A, B, C, or D. 
13. Let {a_{n}}, {b_{n}}, and {c_{n}} be sequences of positive numbers such that
Which of the following must be true?
A. 
B. 
C. 
D. 
E. 
14. Let {a_{n}} be a sequence of real numbers. Which of the following statements must be true?
I. If {a_{n}} is unbounded, then every subsequence of {a_{n}} diverges.
II. If {a_{n}} is diverges, then every subsequence of {a_{n}} also diverges.
III.If {a_{n}} is converges, then every subsequence of {a_{n}} also converges.
A. I and II only 
B. I and III only 
C. II and III only 
D. I, II, and III 
E. The correct answer is not given by A, B, C, or D. 
15. If S = {x  2 < x^{3} + 1 < 9}, then g.l.b. (S) =
A. 1/3 
B. 1 
C. 2 
D. 9 
E. does not exist 
16. Let f(x) = x^{1/2} for x >= 0. With respect to the closed interval [1,4], what value of x satisfies the statement of the mean value theorem for derivatives?
A. 1 
B. 3/2 
C. 9/4 
D. 3 
E. 4 
17. Let f be differentiable at x = 0 and f(0) = 2. Then
A. 1 
B. 0 
C. 1 
D. 2 
E. 3 
18. Let S be a set of real numbers such that l.u.b.(S) = 5/6 and g.l.b.(S) = 1/3, and let T = {3x/2  x belongs to S}. Then l.u.b.(T) =
A. 3/2 
B. 5/4 
C. 5/9 
D. 1/2 
E. 4/9 
19. What is the greatest lower bound of the set of rational numbers whose squares are between 2 and 3?
A. 
B. 
C. 
D. 
E. 
20. What is the y coordinate of the point on the curve y = 2x^{2}  3x at which the slope of the tangent line is the same as that of the secant line between x = 1 and x = 2?
A. 1 
B. 0 
C. 1 
D. 3 
E. 9 
21. What value of x satisfies the Mean Value Theorem for derivatives with respect to the function f(x) = x^{3} on the open interval (0,1)?
A. 
B. 
C. 
D. 
E. 
22. Which of the following conditions imply that the real number x is rational?
I. x^{1/2} is rational.
II. x^{2} and x^{5} are rational.
III. x^{2} and x^{4} are rational.
A. I and II only 
B. I and III only 
C. II and III only 
D. I, II, and III 
E. The correct answer is not given by A, B, C, or D. 