Study GuidesExam QuestionsPast Exams
AP®MathsCollege BoardCalculus ABExam QuestionsUnit 4: Contextual Applications of DifferentiationLinearization

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

41 mins22 questions

Select a download format for Linearization

Scroll for more

Select an answer set to view for
Linearization

Easy
Medium
Hard
1
Sme Calculator2 marks

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.

How did you do?

2
Sme Calculator2 marks

Consider the differential equation:

fraction numerator d y over denominator d x end fraction equals 1 half x open parentheses y minus 3 close parentheses squared

Let y equals f open parentheses x close parentheses be the particular solution to the given differential equation with the initial condition f open parentheses 2 close parentheses equals 4. Write an equation for the line tangent to the graph of y equals f open parentheses x close parentheses at x equals 2. Use your equation to approximate f open parentheses 2.1 close parentheses.

How did you do?

3
Sme Calculator2 marks

For small values of theta, when theta is measured in radians, the approximation sin space theta almost equal to theta is often used.

By finding the linear approximation to y equals sin space x at x equals 0.1 radians, explain why this is an appropriate approximation.

How did you do?

1a
Sme Calculator1 mark

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

How did you do?

1b
Sme Calculator1 mark

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.

How did you do?

2
Sme Calculator3 marks

A hot beverage is left in a room to cool, and the temperature of the beverage is modeled by:

T open parentheses t close parentheses equals 85 open parentheses 0.97 close parentheses to the power of t plus 75

Where T open parentheses t close parentheses is the temperature of the beverage in degrees Fahrenheit, and t is the time in minutes since the beverage was left to cool.

For t greater than 30, the linear approximation L open parentheses t close parentheses to T open parentheses t close parentheses at t equals 30 is a better model for the temperature of the beverage.

Use L open parentheses t close parentheses to predict the time, to the nearest second, at which the temperature of the beverage will reach 90°F. Show the work that leads to your answer.

How did you do?

3a
Sme Calculator2 marks

Consider the differential equation fraction numerator d y over denominator d x end fraction equals x squared y squared.

Let y equals f open parentheses x close parentheses be a particular solution to the differential equation with f open parentheses 2 close parentheses equals 3.

Use the tangent line equation at x equals 2 to approximate f open parentheses 1.9 close parentheses.

How did you do?

3b
Sme Calculator2 marks

Solutions to the differential equation in part (a) also satisfy fraction numerator d squared y over denominator d x squared end fraction equals 2 x y squared open parentheses 1 plus x cubed y close parentheses.

Determine whether the approximation for f open parentheses 1.9 close parentheses from part (a) is an overestimate or an underestimate. Explain your reasoning.

How did you do?

4
Sme Calculator3 marks

The population of a rare bird species in a wildlife reserve is measured in hundreds of birds and is modeled by a twice-differentiable function P open parentheses t close parentheses, where t is the number of years since the species was first introduced to the reserve.

The table below gives selected values of the rate of change, P apostrophe open parentheses t close parentheses, of the population over the time interval 0 less or equal than t less or equal than 15. The population is 40 000 birds when t equals 8.

t (years)

P apostrophe open parentheses t close parentheses (hundreds of birds per year)

0

5.2

5

3.4

8

2.1

10

1.2

12

0.8

15

0.4

The graph of P open parentheses t close parentheses is known to be concave down for 8 less or equal than t less or equal than 10.

Estimate the population of the bird species at t equals 8.5 using the tangent line approximation at t equals 8. Is your estimate greater than or less than the true value of P open parentheses t close parentheses? Justify your answer.

How did you do?

5
Sme Calculator3 marks

Let f be a function that is differentiable for all real numbers. The table below gives the values of f and its derivative f apostrophe for selected values of x in the closed interval negative 1.5 less or equal than x less or equal than 1.5. The second derivative of f has the property that f apostrophe apostrophe open parentheses x close parentheses greater than 0 for 1 less or equal than x less or equal than 1.5.

x

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

f open parentheses x close parentheses

-3

-5

-6

-7

-5

-4

-2

f apostrophe open parentheses x close parentheses

-6

-4

-2

0

2

4

6

Write an equation of the line tangent to the graph of f at the point where x equals 1. Use this line to approximate the value of f open parentheses 1.3 close parentheses. Is this approximation greater than or less than the actual value of f open parentheses 1.3 close parentheses? Give a reason for your answer.

How did you do?

1a
Sme Calculator2 marks

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.

How did you do?

1b
Sme Calculator1 mark

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.

How did you do?

2
Sme Calculator2 marks

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.

How did you do?

3a
Sme Calculator2 marks

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.

How did you do?

3b
Sme Calculator2 marks

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.

How did you do?