Modeling with Exponential & Logarithmic Functions (College Board AP® Precalculus): Exam Questions

On the initial day of sales () for a new video game, there were 40 thousand units of the game sold that day. Ninety-one days later (), there were 76 thousand units of the game sold that day.

The number of units of the video game sold on a given day can be modeled by the function given by , where is the number of units sold, in thousands, on day since the initial day of sales.

(i) Use the given data to write two equations that can be used to find the values for constants and in the expression for .

(ii) Find the values for and as decimal approximations.

The makers of the video game reported that daily sales of the video game decreased each day after . Explain why the error in the model increases after .

The function is increasing and is defined for . The table gives values of at selected values of .

(i) Based on the table, which of the following function types best models function : linear, quadratic, exponential or logarithmic?

(ii) Give a reason for your answer in part (i) based on the relationship between the change in the output values of and the change in the input values of . Refer to the values in the table in your reasoning.

An electric utility company begins a phased installation of smart meters. At time , the company begins tracking the total number of smart meters installed, in thousands.

The table gives the total number of smart meters installed, in thousands, for selected times months after tracking began.

The total number of smart meters installed, in thousands, can be modeled by the function given by , where is measured in thousands and is the number of months after tracking began.

(i) Use the given data to write two equations that can be used to find the values for constants and in the expression for .

(ii) Find the values for and as decimal approximations.

(i) Use the given data to find the average rate of change of the total number of smart meters installed, in thousands per month, from to months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Use the average rate of change found in part B (i) to estimate the total number of smart meters installed, in thousands,for months. Show the work that leads to your answer.

(iii) Let ​ represent the estimate of the total number of smart meters installed, in thousands, using the average rate of change found in part B (i). For found in part B (ii), it can be shown that . Explain why, in general, for all , where . Your explanation should include a reference to the graph of and its relationship to .

The company expects to install at most 8 thousand smart meters in this phase of the project. Explain how this information can be used to determine a practical restriction on the domain of .