Lesson 5*: Modeling with Linear

Discrete Dynamical Systems (DDS)

MAJ John R. BaconOffice: Thayer Hall 220

Phone: 938-2046E-mail: john.bacon@usma.edu

Website: http://www.dean.usma.edu/math/people/Bacon

*Modified

Agenda

Questions from Do Problems

Example walk-through

Board Problems

Prior Lesson Objectives

1a.  What is a mathematical model?  What is meant by mathematical modeling?

1b.  What are some of the things you should consider when solving a problem?

1c.  Why must military officers be good problem solvers?                                                         

2a.  Explain the benefits of using a problem solving process.

2b.  What is meant by mathematical modeling?

2c.  Know and be able to apply the three steps in the Mathematical Modeling Process – Transform, Solve, Interpret.

Today’s Lesson Objectives

5a. Model situations and solve problems involving linear DDS models.

What is a Mathematical Model?

What can we model?

Analysis of a New Car Purchase…will the Hybrid pay for itself?

Predict Investment Plans…How long until you’re a millionaire?

Strategies for Games…what’s your best bet?

Population Models…when will our resources run out?

Skills that will help you in your other USMA Classes and as Lifelong Leaders!

Going Deeper…

• Transform:

1. List reasonable and necessary assumptions

2. Define the variables

3. State the initial condition

4. Determine the difference or recursion equation

• Solve

• Interpret

Example Walkthrough(Example 1.5.1 Page 40 in MRCW Text)

• Suppose that your uncle gives you a gift of $50,000. You deposit it into a bank account that earns interest at the rate of 5% per year (compounded annually) and on the anniversary of the gift, after the interest is posted, you withdraw $5,000 to go on a vacation. How long will you be able to continue taking vacations using your uncle’s gift?

What information do you have and what information would you like to have? What are we trying to find?

What critical assumptions do you have to make? Are they reasonable and necessary? What is the difference?

What is your plan? Remember the Big 4.

Example Walkthrough(Example 1.5.1 Page 40 in MRCW Text)

• Suppose that your uncle gives you a gift of $50,000. You deposit it into a bank account that earns interest at the rate of 5% per year (compounded annually) and on the anniversary of the gift, after the interest is posted, you withdraw $5,000 to go on a vacation. How long will you be able to continue taking vacations using your uncle’s gift?

Board Problem #1(Board Sheet & Problem 1.5.2 Page 41 in MRCW

Text)

• Suppose you deposit $3,000 each year on your birthday in a bank account that earns interest at the rate of 4% per year. Using a discrete dynamical system and an excel spreadsheet to model this situation, answer the following questions:

How much money would you have immediately after making your 10th deposit? [Show using a Time Series Graph]

What assumptions did you make regarding your bank paid interest?

Board Problem #2(Board Sheet)

Analyze an investment of $12,000 for five years in a bank account earning 4% annual interest compounded monthly.  • How much is this investment worth at the end of five years? [Show

using a Time Series Graph].

• What is the effective annual interest rate?

• How much is the investment worth after five years if compounded daily?

• What is the effective annual interest rate?

Before you leave…

This weekend• Logon to the course website and complete assignment –

Lesson 6 (Equilibrium Values)

For Monday• Right back here

Next week• Two IT-105 drops (Tuesday and Wednesday)• Problem Solving Lab #1: MA103 Guest Speaker lecture in

Robinson Auditorium (Friday)