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

 f is increasing 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]

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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.
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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).

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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.
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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.
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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.
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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.
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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.

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

 A. -1 B. 0 C. 1 D. 2 E. 3
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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
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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.
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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
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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.
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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.
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