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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 4: Contextual Applications of DifferentiationLinearization

Linearization (College Board AP® Calculus AB): Exam Questions

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1
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The function f is differentiable with f open parentheses 3 close parentheses equals 2 and f apostrophe open parentheses 3 close parentheses equals 5. What is the value of the approximation of f open parentheses 2.8 close parentheses using the line tangent to the graph of f at x equals 3?

  • -1.0

  • 1.0

  • 2.0

  • 4.6

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2
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The local linear approximation to the function f at x equals negative 5 is y equals 3 x plus 8. What is the value of f open parentheses negative 5 close parentheses plus f apostrophe open parentheses negative 5 close parentheses?

  • -7

  • -4

  • 3

  • 11

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3
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Using a linear approximation for f open parentheses x close parentheses equals cos space x at the point x equals 1, an approximation for cos open parentheses 1.5 close parentheses is

  • 0.071

  • 0.120

  • 0.961

  • 0.991

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1
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The function f open parentheses x close parentheses is approximated for values close to x equals negative 0.5 using a linear function, g open parentheses x close parentheses. The graph of g open parentheses x close parentheses is a tangent to the graph of f open parentheses x close parentheses at x equals negative 0.5.

It is known that f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses greater than 0 for all values of x.

Which of the statements below are true?

I. g open parentheses negative 0.6 close parentheses will be an overestimate for f open parentheses negative 0.6 close parentheses

II. g open parentheses negative 0.6 close parentheses will be an underestimate for f open parentheses negative 0.6 close parentheses

III. g open parentheses negative 0.6 close parentheses equals f open parentheses negative 0.6 close parentheses

  • Statement I

  • Statement II

  • Statement III

  • None of the statements are true

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2
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A function f is defined as f open parentheses x close parentheses equals 3 sin space x over 2.

Find the equation of a linear function which approximates f for values close to x equals 1.5.

  • y equals 2.195 x minus 1.25

  • y equals 1.098 x plus 2.045

  • y equals 1.098 x plus 0.545

  • y equals 1.098 x plus 0.399

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3
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Find the equation of the function which is a linearization of y equals square root of x at an integer value of x, which could be used to approximate the value of square root of 101.

  • y equals 5 open parentheses x minus 100 close parentheses plus 10

  • y equals 1 over 20 x

  • y equals 1 over 20 open parentheses x minus 100 close parentheses plus 10

  • y equals 1 over 20 x minus 90

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4
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Find the equation of the function which is a linearization of y equals cube root of x at an integer value of x, which could be used to approximate cube root of 8.1 end root.

  • y equals 1 over 12 open parentheses x minus 8 close parentheses plus 2

  • y equals 4 over 3 open parentheses x minus 8 close parentheses plus 2

  • y equals 1 over 12 open parentheses x minus 8 close parentheses plus 2

  • y equals 1 over 12 x minus 6

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5
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When a tangent to the graph of f open parentheses x close parentheses is drawn at x equals 0 and used to approximate f open parentheses 0.1 close parentheses, for which of the following functions will this be an overestimate?

  • f open parentheses x close parentheses equals x squared

  • f open parentheses x close parentheses equals negative x squared

  • f open parentheses x close parentheses equals e to the power of negative x end exponent minus 4

  • f open parentheses x close parentheses equals negative ln open parentheses x plus 1 close parentheses

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1
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A linear approximation for y equals cos space x is created using a tangent to y equals cos space x at the point open parentheses pi over 4 comma fraction numerator space square root of 2 over denominator 2 end fraction close parentheses. Find the percentage error when this linear approximation is used to approximate cos space fraction numerator 7 pi over denominator 32 end fraction.

  • 4.549 %

  • 0.455 %

  • 0.516 %

  • 17.5 %

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2
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A tangent to the graph of f open parentheses x close parentheses equals sin space x at x equals 0 is used to estimate the value of sin space x at values close to 0. Which of the following statements is true for the estimate of f open parentheses 0.2 close parentheses?

I. It will be an underestimate

II. It will be an overestimate

III. It will be the exact value

  • Statement I

  • Statement II

  • Statement III

  • None of the statements are true

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3
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Consider the cubic function f open parentheses x close parentheses equals x cubed minus 3 x squared minus 9 x plus 27.

Several different tangents are created as linear approximations for f open parentheses x close parentheses, and then used to approximate the value of f open parentheses x close parentheses nearby points.

Which one of the following statements is false?

  • The tangent at x equals negative 2 will produce an overestimate for f open parentheses negative 3 close parentheses

  • The tangent at x equals 3 will produce an underestimate for f open parentheses 4 close parentheses

  • The tangent at x equals 1 will produce an overestimate for f open parentheses 2 close parentheses

  • The tangent at x equals negative 1 will produce an overestimate for f open parentheses negative 2 close parentheses

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