AP®MathsCollege BoardCalculus ABExam QuestionsUnit 2: Differentiation Definition & Fundamental PropertiesDefinition of Differentiation

Definition of Differentiation (College Board AP® Calculus AB)

Exam Questions

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Definition of Differentiation

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Definition of Differentiation

1
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Let f be the function given by f open parentheses x close parentheses equals 1 over x and let g be the function given by g open parentheses x close parentheses equals negative 11 x squared.

At what value of x do the graphs of f and g have parallel tangent lines?

  • 0.213

  • -0.357

  • 0.357

  • -0.450

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2
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Let f be the function defined by f open parentheses x close parentheses equals square root of open vertical bar x minus 5 close vertical bar end root for all x. Which of the following statements is true?

  • f is continuous but not differentiable at x equals 5.

  • f is differentiable at x equals 5.

  • f is not continuous at x equals 5.

  • limit as x rightwards arrow 5 of f open parentheses x close parentheses not equal to 0

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3
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limit as h rightwards arrow 0 of fraction numerator sin open parentheses pi plus h close parentheses minus sin open parentheses pi close parentheses over denominator h end fraction is

  • negative 1

  • 0

  • 1 half

  • 1

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4
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negative quadratic from -1,2 to 1, 2 with open circle. Closed circle at 1,0. Semi circular section between 1,0 and 3,0, and a larger semicircle between 3,0 and 7,0

The graph of the function f shown in the figure above has a vertical tangent at the point (3, 0) and horizontal tangents at the points (2, 0) and (5, -2).

For what values of x, negative 1 less than x less than 7 is f not differentiable?

  • 1 only

  • 1 and 3 only

  • 2 and 5 only

  • 1, 2, and 5 only

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5
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Which of the following is an equation of the tangent line to the graph of f open parentheses x close parentheses equals 2 x to the power of 4 plus 3 x squared at the point where f to the power of apostrophe open parentheses x close parentheses equals negative 1?

  • y equals x plus 0.082

  • y equals negative x minus 0.161

  • y equals negative x plus 0.079

  • y equals negative x minus 0.082

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