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

On the initial day of sales ($t=0$) for a new video game, there were 40 thousand units of the game sold that day. Ninety-one days later ($t=91$), 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 $G$ given by $G\left(t\right)=a+b\ln \left(t+1\right)$, where $G\left(t\right)$ is the number of units sold, in thousands, on day $t$ 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 $a$ and $b$ in the expression for $G\left(t\right)$.

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

The makers of the video game reported that daily sales of the video game decreased each day after $t=91$. Explain why the error in the model $G$ increases after $t=91$.

An exponential function is given by $y=8·5^x$.

Express $\log y$ as a linear function of $x$.

The value of an investment, in dollars, is modeled by the function $V$ given by $V\left(t\right)=1500·{\left(1.06\right)}^t$, where $t$ is the number of years since the investment was made.

Find the value of the investment after $10$ years, as a decimal approximation. Interpret the meaning of your answer in the context of the problem. Show the work that leads to your answer.

A model for the depth, in meters, of an object sinking in water is given by $D\left(t\right)=5\ln \left(t+1\right)$, where $t$ is the time, in seconds, since the object was released at the water's surface.

Find the value of $D\left(4\right)$ as a decimal approximation. Interpret the meaning of $D\left(4\right)$ in the context of the problem. Show the work that leads to your answer.

The function $f$ is increasing and is defined for $x>0$. The table gives values of $f\left(x\right)$ at selected values of $x$.

(i) Based on the table, which of the following function types best models function $f$: 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 $f$ and the change in the input values of $f$. Refer to the values in the table in your reasoning.

An electric utility company begins a phased installation of smart meters. At time $t=0$, 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 $t$ months after tracking began.

The total number of smart meters installed, in thousands, can be modeled by the function $M$ given by $M\left(t\right)=a+b{\left(1.25\right)}^{t+1}$, where $M\left(t\right)$ is measured in thousands and $t$ 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 $a$ and $b$ in the expression for $M\left(t\right)$.

(ii) Find the values for $a$ and $b$ 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 $t=0$ to $t=2$ 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 $t=1.8$ months. Show the work that leads to your answer.

(iii) Let $L_t$​ 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 $L_{1.8}$ found in part B (ii), it can be shown that $L_{1.8}>M\left(1.8\right)$. Explain why, in general, $L_t>M\left(t\right)$ for all $t$, where $0<t<2$. Your explanation should include a reference to the graph of $M$ and its relationship to $L_t$.

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 $M$.

A software company tracks the number of active users for a new application over its first year to model user retention. At the launch of the app $t=0$, there were 120 thousand active users. After 4 months $t=4$, the number of active users dropped to 102.5 thousand.

The number of active users can be modeled by the function $U$ given by$U\left(t\right)=a+b\ln \left(t+1\right)$, where $U\left(t\right)$ is the number of active users, in thousands, and $t$ is the number of months since the launch.

(i) Use the given data to write two equations that can be used to find the values for constants $a$ and $b$ in the expression for $U\left(t\right)$.

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

(i) Use the given data to find the average rate of change of the number of users, in thousands of users per month, from $t=0$ to $t=4$ months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Interpret the meaning of your answer from part B(i) in the context of the problem.

(iii) Consider the average rates of change of $U$ from $t=4$ to $t=k$ months, where $k>4$. Are these average rates of change less than or greater than the average rate of change from $t=0$ to $t=4$ months found in part B(i)? Your explanation should include a reference to the graph of $U$.

The company directors decide to use the model $U$ to make predictions about user levels beyond 12 months (1 year). For a given year, model $U$ is considered an appropriate model if the predicted drop in users over that 12-month period is at least 2 thousand users.

Based on this information, for how many years is model $U$ an appropriate model? Give a reason for your answer. (Note: The end of a year occurs at $t=12$, $t=24$, $t=36$, ...)

A bacteria population $P$, measured in thousands, is modeled by an exponential function of the form $P\left(t\right)=a·b^t$, where $t$ is the time, in hours, since the start of an experiment.

At time $t=0$ hours, the population is $P\left(0\right)=5$ thousand. At time $t=3$ hours, the population is $P\left(3\right)=70$ thousand.

Find the value of $a$, and the value of $b$ as a decimal approximation. Show the work that leads to your answer.

A linear model $M$ is given by $M\left(x\right)=2.5x+1$. The table gives the actual values of $y$ at three values of $x$.

Find the residual at each value of $x$ in the table. Then state for which value of $x$ the model predicts most accurately. Show the work that leads to your answers.

A model $P$ predicts the number of fish in a pond as a function of time. The model is $P\left(t\right)=50·{\left(1.08\right)}^t$, where $t$ is the number of years since the pond was stocked.

Explain why the model becomes inaccurate over a long time period. Your explanation should refer to the contextual situation.

The table gives values for a function $f$ at selected values of $x$.

(i) Based on the table, which of the following function types best models function $f$: 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 $f$ and the change in the input values of $f$. Refer to the values in the table in your reasoning.

The number of subscribers, in thousands, of a streaming service is modeled by the function $S$ given by $S\left(t\right)=a·b^t$, where $a$ and $b$ are constants and $t$ is the number of months since launch. Two months after launch ($t=2$), there were $30$ thousand subscribers. Six months after launch ($t=6$), there were $70$ thousand subscribers.

(i) Use the given data to write 2 equations that can be used to find the values for constants $a$ and $b$ in the expression for $S\left(t\right)$.

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

Use the given data to find the average rate of change of the number of subscribers, in thousands of subscribers per month, from $t=2$ to $t=6$ months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

The function $D$ models the depth, in meters, of an object sinking in water $t$ seconds after release and is given by $D\left(t\right)=5\ln \left(t+1\right)$.

An estimate of the depth at time $t$ can be found using the average rate of change of $D$ from $t=0$ to $t=12$, which is approximately $1.069$ meters per second. Let $A_t$ represent this estimate. For $A_5$, it can be shown that $A_5<D\left(5\right)$.

Explain why, in general, $A_t<D\left(t\right)$ for all $t$, where $0<t<12$. Your explanation should include a reference to the graph of $D$ and its relationship to $A_t$.

For a data set $\left(x,y\right)$, the graph of ${\log }_{10}y$ as a function of $x$ is a straight line that passes through the points $\left(0,2\right)$ and $\left(4,8\right)$.

The data are modeled by an exponential function of the form $f\left(x\right)=a·b^x$. Find the values of $a$ and $b$ as decimal approximations. Show the work that leads to your answer.