Mathematical Models

Exam #1

Wednesday, Feb. 26, 5:30 - 7:00, SWORDS 302

The first exam covers Sections 1.1, 1.2, 1.3, 1.4, 2.1, 2.2, 2.3 and the material on dynamical systems from the Blanchard, Devaney, Hall text, Sections 8.1, 8.2, 8.4. You should go over homework problems, computer projects and your class notes. The exam will be designed to take 60 minutes although you will have 90 minutes to complete it.

Note: You will be given a scientific calculator for the exam which does NOT have graphing capabilities so be prepared to answer questions without your personal calculator or computer. The only numerical computations asked of you will be doable on a basic scientific calculator.

Exam Review: We will review for the exam on Monday, Feb. 24th from 7:00 - 8:15 pm in SWORDS 359. Please come prepared with specific questions. We will have a regularly scheduled class on the day of the exam.

The following concepts, definitions and models are crucial material for the exam:

1. Difference equations, numerical solutions, qualitative analysis
2. Population models --- unlimited growth model, logistic population model, carrying capacity
3. Solving basic linear difference equations (know your formulas), equilibrium values, stability, long-term behavior
4. Systems of difference equations, predator-prey models, solving linear systems with eigenvalues and eigenvectors, invariance, relationship between eigenvalues and stability of equilibria
5. Discrete dynamical systems, orbit of x_0, fixed points, periodic points, eventually fixed and periodic points, attracting and repelling fixed (periodic) points, finding fixed (periodic) points graphically
6. Graphical iteration, web diagram, classifying fixed (periodic) points via the derivative
7. Chaos, chaotic dynamical system, density of periodic points, dense orbits, sensitive dependence on initial conditions
8. Constructing models, defining variables, assumptions --- proportionality, geometric similarity, characteristic dimension, estimating slope to obtain proportionality constants

Some Practice Problems:

Chapter 1

• Section 1.1 : 4, 9
• Section 1.2 : 4, 8
• Section 1.3 : 1c, 2f, 2g, 9 (you can do this without a computer), 12
• Section 1.4 : 7 (don't forget to review eigenvalues and eigenvectors as well)

Chapter 8 Dynamical Systems

• Section 8.1 : 4, 5, 12, 14, 22
• Section 8.2 : 6, 14, 17, 24, 28
• Section 8.4 : 6, 7, 9
Chapter 2
• Section 2.1 : 4, 12
• Section 2.2 : 3, 6
• Section 2.3 : 3, 8