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

Find the linear approximation to at .

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

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

Where is the temperature of the beverage in degrees Fahrenheit, and is the time in minutes since the beverage was left to cool.

For , the linear approximation to at is a better model for the temperature of the beverage.

Use 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.

Consider the differential equation .

Let be a particular solution to the differential equation with .

Use the tangent line equation at to approximate .

Solutions to the differential equation in part (a) also satisfy .

Determine whether the approximation for from part (a) is an overestimate or an underestimate. Explain your reasoning.

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 , where 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, , of the population over the time interval . The population is 40 000 birds when .

The graph of is known to be concave down for .

Estimate the population of the bird species at using the tangent line approximation at . Is your estimate greater than or less than the true value of ? Justify your answer.

Let be a function that is differentiable for all real numbers. The table below gives the values of and its derivative for selected values of in the closed interval . The second derivative of has the property that for .

Write an equation of the line tangent to the graph of at the point where . Use this line to approximate the value of . Is this approximation greater than or less than the actual value of ? Give a reason for your answer.

Consider the differential equation . Let be the particular solution to the differential equation with the initial condition .

Write an equation for the line tangent to the graph of at the point (1, 7). Use the equation to approximate .

It is known that for . Is the approximation found in part (a) an overestimate or an underestimate for ? Give a reason for your answer.

Use a linear approximation for at to show that , where is a constant, is an approximation for .

The function describes the measured temperature, , of a substance in degrees Celsius (), where is the time in minutes since the substance has been removed from a refrigerator.

The rate of change of with respect to time is given by the differential equation and it is known that the temperature of the substance at was .

Use the line tangent to the graph of at to approximate , the temperature of the substance at time minutes.

Write an expression for in terms of . Use to determine whether the approximation from part (a) is an underestimate or overestimate for the actual value of . Give a reason for your answer.