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Lesson 5*: Modeling with Linear Discrete Dynamical Systems (DDS)

Lesson 5*: Modeling with Linear Discrete Dynamical Systems (DDS). MAJ John R. Bacon Office: Thayer Hall 220 Phone: 938-2046 E-mail: john.bacon@usma.edu Website: http://www.dean.usma.edu/math/people/Bacon. *Modified. Agenda. Questions from Do Problems Example walk-through

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Lesson 5*: Modeling with Linear Discrete Dynamical Systems (DDS)

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  1. Lesson 5*: Modeling with Linear Discrete Dynamical Systems (DDS) MAJ John R. Bacon Office: Thayer Hall 220 Phone: 938-2046 E-mail: john.bacon@usma.edu Website: http://www.dean.usma.edu/math/people/Bacon *Modified

  2. Agenda • Questions from Do Problems • Example walk-through • Board Problems

  3. 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.

  4. Today’s Lesson Objectives 5a. Model situations and solve problems involving linear DDS models.

  5. What is a Mathematical Model?

  6. 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!

  7. Going Deeper… • Transform: • List reasonable and necessary assumptions • Define the variables • State the initial condition • Determine the difference or recursion equation • Solve • Interpret

  8. 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.

  9. 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?

  10. 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?

  11. 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?

  12. 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)

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