2.1: Modeling Data
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 \begin{align*}L1\end{align*} to \begin{align*}L6\end{align*} will need to be loaded into the student calculators before beginning this activity.
- To download the list files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=12316 and select L1-L6.
Associated Materials
- Student Worksheet: Modeling Data http://www.ck12.org/flexr/chapter/9726
- L1.8xl, L2.8xl, L3.8xl, L4.8xl, L5.8xl, and L6.8xl
Problem 1 – Exponential Growth
In this problem, students will graph data given in \begin{align*}L1\end{align*} and \begin{align*}L2\end{align*}.
Students should determine an equation for the data using their knowledge of compound interest equations.
The equation is \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 \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 \begin{align*}L1\end{align*} and \begin{align*}L2\end{align*}. Students should determine an equation for the data using the regression capabilities of the graphing calculator.
Note: to find \begin{align*}Y1\end{align*}, press VARS, scroll over to Y-VARS, select 1:Function..., and select \begin{align*}1:Y1\end{align*}.
The equation is \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 \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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