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 x0 such that 0 < x0 < 1. |
B. f has a maximum at some x0 such that 0 < x0 < 1. |
C. f has a minimum at some x0 such that 0 < x0 < 1. |
D. f(x0) = 0 at every x0 such that 0 < x0 < 1. |
E. f(x0) = 0 at some x0 such that 0 < x0 < 1. |
8. Let f be a real-valued 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 < a1 < b1 < b, and f(a1) < 0 < f(b1), then
there is a number c such that a1 < c < b1 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 real-valued 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 {an}, {bn}, and {cn} be sequences of positive numbers such that
Which of the following must be true?
A. |
B. |
C. |
D. |
E. |
14. Let {an} be a sequence of real numbers. Which of the following statements must be true?
I. If {an} is unbounded, then every subsequence of {an} diverges.
II. If {an} is diverges, then every subsequence of {an} also diverges.
III.If {an} is converges, then every subsequence of {an} 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 < x3 + 1 < 9}, then g.l.b. (S) =
A. 1/3 |
B. 1 |
C. 2 |
D. 9 |
E. does not exist |
16. Let f(x) = x1/2 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 = 2x2 - 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) = x3 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. x1/2 is rational.
II. x2 and x5 are rational.
III. x2 and x4 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. |