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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 1: Limits & ContinuityContinuity

Continuity (College Board AP® Calculus AB)

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Continuity

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Continuity

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1
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A graph labeled "Graph of f" plotting points (1,3), (3,3), (3,5), (5,4), and (5,6) on a grid, with x-axis from 0 to 6, y-axis from 0 to 6.

The graph of the function f is shown above. Which of the following statements is false?

  • The function is continuous at x equals 2.

  • The function is continuous at x equals 4.

  • The function is continuous at x equals 5.

  • limit as x rightwards arrow 5 of f open parentheses x close parentheses exists.

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2
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The graph of a curve with a 'hole' at x=a, where a>0

The graph of the function f is shown above. Which of the following statements is true?

  • a is in the domain of f

  • f has a non-removable discontinuity at x equals a

  • f has a removable discontinuity at x equals a

  • f is continuous at x equals a

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3
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A graph of a function with a non-removable jump discontinuity at x=a, where a>0

The graph of the function f is shown above. Which of the following statements is true?

  • f has an essential discontinuity at at x equals a

  • f has a jump discontinuity at at x equals a

  • f has a removable discontinuity at x equals a

  • f is continuous at x equals a

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41 mark

f is a continuous function such that f open parentheses 1 close parentheses equals 4 and f open parentheses 3 close parentheses equals 7.

Which one of the following results allows you to determine that there is some value c between 1 and 3 where f open parentheses c close parentheses equals 5?

  • extreme value theorem

  • fundamental theorem of calculus

  • intermediate value theorem

  • mean value theorem

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5
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Of the following types of functions, which are continuous on the domain of all real numbers?

I. exponential functions

II. polynomial functions

III. rational functions

  • II only

  • I and II only

  • I and III only

  • I, II and III

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1
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Graph of a curve with a discontinuity 'hole' at x=a (where a>0), and a single filled-in point above the curve also at x=a

The graph of a function f is shown on the diagram above. Which of the following statements about f is false?

  • f has a relative maximum at x equals a.

  • f is continuous at x equals a.

  • x equals a is in the domain of f.

  • limit as x rightwards arrow a of f open parentheses x close parentheses exists.

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2
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f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator open parentheses 3 x plus 4 close parentheses open parentheses x plus 1 close parentheses over denominator x plus 1 end fraction space space for space x not equal to negative 1 end cell row cell k space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space x equals negative 1 end cell end table close

Let f be the function defined above. For what value of k is f continuous at x equals negative 1?

  • 0

  • 1

  • 2

  • 3

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3
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Let f be a function that is continuous on the closed interval open square brackets 3 comma space 7 close square brackets with f open parentheses 3 close parentheses equals 12 and f open parentheses 7 close parentheses equals 18. Which of the following is guaranteed by the Intermediate Value Theorem?

  • f attains a maximum on the open interval open parentheses 3 comma space 7 close parentheses.

  • f to the power of apostrophe open parentheses x close parentheses greater than 0 for all x in the open interval open parentheses 3 comma space 7 close parentheses.

  • f open parentheses x close parentheses equals 16 has at least one solution in the open interval open parentheses 3 comma space 7 close parentheses.

  • f to the power of apostrophe open parentheses x close parentheses equals 3 over 2 has at least one solution in the open interval open parentheses 3 comma space 7 close parentheses.

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4
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x

-1

0

1

f(x)

7

k

5

The function f is continuous on the closed interval open square brackets negative 1 comma space 1 close square brackets and has values that are given in the table above. The equation f open parentheses x close parentheses equals 8 must have at least two solutions in the interval open square brackets negative 1 comma space 1 close square brackets if kequals

  • 5

  • 7

  • 8

  • 9

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5
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f open parentheses x close parentheses equals open curly brackets table row cell 3 space space space space space space space space space space space space space space space space space space for space x less or equal than a end cell row cell fraction numerator 1 over denominator x minus a end fraction space space space space space space space space space space for space x greater than a end cell end table close

Let f be the function defined above. Which of the following statements is true?

  • f has an essential discontinuity at at x equals a

  • f has a jump discontinuity at at x equals a

  • f has a removable discontinuity at at x equals a

  • f is continuous at at x equals a

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1
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f open parentheses x close parentheses equals open curly brackets table row cell 2 x squared minus 3 x plus 1 space space space space space space space for space x less or equal than a end cell row cell x squared plus 3 x minus 8 space space space space space space space space space for space x greater than a end cell end table close

The function f defined above is a continuous function. What is the value of a?

  • -3

  • -1

  • 1

  • 3

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2
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f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator x squared minus 7 x plus 10 over denominator x squared plus 5 x minus 14 end fraction space space space space space space space space space for space x greater or equal than negative 5 comma space space x not equal to 2 end cell row cell k space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space x equals 2 end cell end table close

Let f be the function defined above. For what value of k is f continuous at x equals 2?

  • negative 4 over 3

  • negative 2 over 3

  • negative 1 third

  • 0

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3
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f open parentheses x close parentheses equals open curly brackets table row cell open parentheses x minus a close parentheses squared space space space space space space space space space space space space space space space space space space space space for space x less or equal than a end cell row cell cos open parentheses fraction numerator 1 over denominator x squared minus a squared end fraction close parentheses space space space space space space space space space space for space x greater than a end cell end table close

Let f be the function defined above. Which of the following statements is true?

  • limit as x rightwards arrow a to the power of plus of f open parentheses x close parentheses equals 0

  • limit as x rightwards arrow a to the power of plus of f open parentheses x close parentheses equals infinity

  • f has an essential discontinuity at at x equals a

  • f has a jump discontinuity at at x equals a

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4
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Let f be a function that is differentiable on the open interval open parentheses 0 comma space 11 close parentheses. If f open parentheses 1 close parentheses equals 4, f open parentheses 7 close parentheses equals negative 3, and f open parentheses 10 close parentheses equals 9, which of the following must be true?

I.  f has at least 2 zeros.

II.  The graph of f has at least one horizontal tangent.

III.  For some c, space 7 less than c less than 10, space f open parentheses c close parentheses equals negative 2.

  • None

  • I only

  • I and III only

  • I, II and III

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5
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Let f be the function defined by f open parentheses x close parentheses equals tan x. To 3 decimal places, f open parentheses 1 close parentheses equals 1.557 and f open parentheses 2 close parentheses equals negative 2.185. A student claims that, according to the intermediate value theorem, there is some value c between 1 and 2 such that f open parentheses c close parentheses equals 0.

Why is the student's claim not valid?

  • The values of f open parentheses 1 close parentheses and f open parentheses 2 close parentheses have been rounded.

  • 1 and 2 have not been expressed in terms of pi.

  • f is not continuous on the closed interval open square brackets 1 comma space 2 close square brackets.

  • f is not differentiable on the open interval open parentheses 1 comma space 2 close parentheses.

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