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.

solution

 

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.

solution

 

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.

solution

 

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.

solution

 

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

solution

 

6. Let f(x) be differentiable for a < x < b. Which of the following statements must be true?

  1. f is increasing
  2. on (a,b)

B. f is continuous

on (a,b)

  • C. f is bounded
  • on [a,b]
  • D. f is continuous
  • on [a,b]
  • E. f is decreasing

    on [a,b]

    solution

     

    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.

    solution

     

    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).

    solution

     

    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.

    solution

     

    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.

    solution

     

    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.

    solution

     

    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.

    solution

     

    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.

    solution

     

    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.

    solution

     

    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

    solution

     

    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

    solution

     

    17. Let f be differentiable at x = 0 and f(0) = 2. Then

    A. -1

    B. 0

    C. 1

    D. 2

    E. 3

    solution

     

    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

    solution

     

    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.

    solution

     

    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

    solution

     

    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.

    solution

     

    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.

    solution