AP®MathsCollege BoardCalculus ABExam QuestionsUnit 1: Limits & ContinuityContinuity

Continuity (College Board AP® Calculus AB)

Exam Questions

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Continuity

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Continuity

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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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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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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