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

Linearization (College Board AP® Calculus AB)

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Linearization

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Linearization

1
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r

(kilometers)

0

1

2

5

10

P open parentheses r close parentheses

(people per square kilometer)

10 000

8 000

7 000

4 000

2 000

The population density in a city at a distance r kilometers from the center of the city is given by a decreasing, differentiable function P where P open parentheses r close parentheses is measured in people per square kilometer. Values of P open parentheses r close parentheses for selected values of r are given in the table above.

Use the data in the table to estimate P to the power of apostrophe left parenthesis 3.5 right parenthesis. Using correct units, interpret the meaning of your answer in the context of this problem.

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2a
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Find the linear approximation to f open parentheses x close parentheses equals cube root of x at x equals 27 over 8.

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2b
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Use the linear approximation found in part (a) to approximate the value of cube root of 4. Find the percentage error of the approximation compared to the accurate value.

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3a
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The function theta open parentheses t close parentheses describes the measured temperature, theta, of a substance in degrees Celsius (degree straight C), where t is the time in minutes since the substance has been removed from a refrigerator.

The rate of change of theta with respect to time is given by the differential equation fraction numerator d theta over denominator d t end fraction equals 1 fifth open parentheses 25 minus theta close parentheses and it is known that the temperature of the substance at t equals 0 was 5 degree straight C.

Use the line tangent to the graph of theta at t equals 0 to approximate theta open parentheses 1.5 close parentheses, the temperature of the substance at time t equals 1.5 minutes.

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3b
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Write an expression for fraction numerator d squared theta over denominator d t squared end fraction in terms of theta. Use fraction numerator d squared theta over denominator d t squared end fraction to determine whether the approximation from part (a) is an underestimate or overestimate for the actual value of theta open parentheses 1.5 close parentheses. Give a reason for your answer.

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4a
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Consider the differential equation fraction numerator d y over denominator d x end fraction equals 1 third cos open parentheses pi over 3 x close parentheses square root of y plus 9 end root. Let y equals f open parentheses x close parentheses be the particular solution to the differential equation with the initial condition f open parentheses 1 close parentheses equals 7.

Write an equation for the line tangent to the graph of y equals f open parentheses x close parentheses at the point (1, 7). Use the equation to approximate f open parentheses 0.5 close parentheses.

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4b
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It is known that f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses less than 0 for 0 less or equal than x less or equal than 1. Is the approximation found in part (a) an overestimate or an underestimate for f open parentheses 0.5 close parentheses? Give a reason for your answer.

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5
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Use a linear approximation for f open parentheses x close parentheses equals ln space x at x equals e to show that a over e, where a is a constant, is an approximation for ln space 3.

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