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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

1
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bold italic x

0

4

7

9

12

bold italic f to the power of bold apostrophe stretchy left parenthesis x stretchy right parenthesis

-6

-3

1

2

-5

f is a twice-differentiable function. The table above gives selected values of f to the power of apostrophe open parentheses x close parentheses over the time interval 0 less or equal than x less or equal than 12.

Is there a value of x, 4 less or equal than x less or equal than 7, for which f to the power of apostrophe open parentheses x close parentheses equals negative 1? Justify your answer.

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2
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Let f be the function defined by

f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator pi open parentheses x plus 2 close parentheses over denominator 2 end fraction space space space space for space x less than negative 1 end cell row cell arcsin open parentheses x squared close parentheses space space space for space minus 1 less or equal than x less or equal than 1 end cell end table close

Is f continuous at x equals negative 1? Use the definition of continuity to explain your answer.

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3a
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Let f be the function defined by f open parentheses x close parentheses equals fraction numerator x squared minus 3 x over denominator x squared minus 2 x minus 3 end fraction.

Show that f has a removable discontinuity at x equals 3.

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3b
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Explain how the discontinuity identified in part (a) can be removed.

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4
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Let f be the function defined by f open parentheses x close parentheses equals open curly brackets table row cell 3 plus fraction numerator 7 over denominator x minus 4 end fraction space space space for space x less than 4 end cell row cell x minus 2 space space space space space space space space space space space for space x greater or equal than 4 end cell end table close.

Show that f has an essential discontinuity at x equals 4.

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5
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Functions f and g are differentiable functions with f open parentheses 1 close parentheses equals g open parentheses 1 close parentheses equals 7. It is known that f open parentheses x close parentheses less or equal than g open parentheses x close parentheses for 0 less than x less than 2.

Let h be a function satisfying f open parentheses x close parentheses less or equal than h open parentheses x close parentheses less or equal than g open parentheses x close parentheses for 0 less than x less than 2. Is h continuous at x equals 1? Justify your answer.

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