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# 2.1: Modeling Data

Difficulty Level: At Grade Created by: CK-12

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

ID: 12316

Time required: 20 minutes

## Activity Overview

In this activity, students will graph data modeling exponential and logarithmic growth and find equations representing the data.

Topic: Modeling Growth

• Exponential Growth and Decay
• Compound Interest
• Logarithmic Growth

Teacher Preparation and Notes

• This activity can be used with the TI-83, 84, or 89. This supplement is geared towards the 83 and 84.
• Students should be aware of the compound interest formula in order to be able to find the growth of the exponential data given in Problem 1.
• Students should be aware of basic logarithms in order to determine the base of the log function for Problem 2.
• The lists L1\begin{align*}L1\end{align*} to L6\begin{align*}L6\end{align*} will need to be loaded into the student calculators before beginning this activity.

Associated Materials

## Problem 1 – Exponential Growth

In this problem, students will graph data given in L1\begin{align*}L1\end{align*} and L2\begin{align*}L2\end{align*}.

Students should determine an equation for the data using their knowledge of compound interest equations.

The equation is y=1000(1.04)x\begin{align*}y = 1000(1.04)^x\end{align*}.

Students can plot the function to determine if it goes through the graphed points by pressing Y=\begin{align*}Y=\end{align*} and entering the function.

Discussion Questions

• What variable should be on the horizontal axis? Vertical axis?
• How can you determine the interest rate for this growth?

## Problem 2 – Logarithmic Growth

In this problem, students will graph data given on L1\begin{align*}L1\end{align*} and L2\begin{align*}L2\end{align*}. Students should determine an equation for the data using the regression capabilities of the graphing calculator.

Note: to find Y1\begin{align*}Y1\end{align*}, press VARS, scroll over to Y-VARS, select 1:Function..., and select 1:Y1\begin{align*}1:Y1\end{align*}.

The equation is y=8.6×108+0.369×lnx\begin{align*}y = 8.6 \times 10^{-8} + 0.369 \times \ln x\end{align*}.

Discussion Questions

• What variable should be on the horizontal axis? Vertical axis?

## Extension Problem – Exponential Decay

In this problem, students will continue with the process in the first two problems. The given data represents exponential decay.

The equation is y=1250(0.85)x\begin{align*}y = 1250(0.85)^x\end{align*}.

Discussion Questions

• What is the number of acres the farmer started with in year zero?
• By what percent does the amount of acres available decrease every year?

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