# 2.1: Modeling Data

**At Grade**Created by: CK-12

*This activity is intended to supplement Calculus, Chapter 1, Lesson 3.*

## Problem 1 – Exponential Growth

When Connor was born, his parents put \begin{align*}\$1,000\end{align*} dollars into an account to give him as a present on his 21st birthday. However, his parents forgot the yearly interest rate on the account.

Enter the data to the right in \begin{align*}L1\end{align*} and \begin{align*}L2\end{align*} by pressing **STAT ENTER.**

Create a scatter plot of the data by pressing \begin{align*}2^{nd} \ [Y=]\end{align*} **ENTER** and matching the screen to the right.

To view the scatter plot, press **ZOOM** and select **9:ZoomStat**.

Using your knowledge of compound interest, study the data and the graph to determine a function for the growth of Connor’s money.

Enter your equation in \begin{align*}Y1\end{align*} and press **GRAPH** to check your result.

Note: the regressions can be found by pressing **STAT** and scrolling over to the **CALC** menu. Select ExpReg, **ENTER** \begin{align*}2^{nd}\end{align*} [1] [,] \begin{align*}2^{nd}\end{align*} [2] **ENTER**.

The equation for the data is: ______________

## Problem 2 – Logarithmic Growth

Scientists are testing the amount of greenhouse gases present at a research site near the north pole to determine the effect on polar ice melting. Enter in the results in \begin{align*}L3\end{align*} and \begin{align*}L4\end{align*} of your graphing calculator.

Create a scatter plot of the data by pressing \begin{align*}2^{nd} \ [Y=]\end{align*}, selecting **1:Plot1**, and matching the screen to the right.

To view the scatter plot, press **ZOOM** and select **9:ZoomStat**.

Note: \begin{align*}L3\end{align*} and \begin{align*}L4\end{align*} are found by pressing \begin{align*}2^{nd}\end{align*} [3] and \begin{align*}2^{nd}\end{align*} [4] respectively.

Determine a natural log equation to model the data by pressing **STAT**, scrolling over to **CALC** and selecting **9:LnReg**.

To finish the regression, enter, \begin{align*}L3, L4, Y1\end{align*} so the screen reads **LnReg** \begin{align*}L3, L4, Y1\end{align*}

Press **GRAPH** to view the scatter plot and regression equation both plotted.

The equation for the data is: ______________

## Extension – Exponential Decay

*You will need to download L5.8xl and L6.8xl from your teacher for this extension*.

Due to an environmental spill, a farmer is losing the amount of land on which he can plant crops. The data in \begin{align*}L5\end{align*} and \begin{align*}L6\end{align*} of your graphing calculator show the year and amount of useable land for each year. Determine an equation that models the amount of land the farmer can use each year.

After creating a scatter plot of the data, use your graphing calculator to perform an exponential regression that models the data and plot the function by pressing **GRAPH.**

The equation for the data is: ______________