The final exam is cumulative, covering all of the material from the entire semester including the subject of dynamical systems from the Blanchard, Devaney, Hall text, Sections 8.1, 8.2 and 8.4. Approximately 60% of the exam will focus on Chapters 3, 4, 7 and 10, the material covered since the midterm exam. It is highly recommended you go over homework problems, computer projects and your class notes. The exam will be designed to take 2 hours although you will have 3 hours to complete it.

**Note:** You will be allowed one "cheat sheet" 8.5 x 11 piece of paper, front and back, full of whatever formulas,
graphs, etc. you wish. 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 required will be the kind a scientific calculator can perform.

**Exam Review:** We will review for the exam on Friday, May 2nd from 1:00 - 2:30 pm
in Swords 359. Please come prepared with specific questions.

The following concepts, definitions and models are important topics
for the exam:

- Difference equations vs. differential equations, discrete problems vs. continuous problems
- Population models --- unlimited growth model, logistic population model, carrying capacity
- Linear difference equations and their solutions, equilibrium values, stability, long-term behavior
- Systems of difference equations, predator-prey models, solving linear systems with eigenvalues and eigenvectors, invariance, relationship between eigenvalues and stability of equilibria
- 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 and analytically
- Graphical iteration, web diagram, classifying fixed (periodic) points via the derivative, chaotic dynamical system, sensitive dependence on initial conditions
- Constructing models, defining variables, assumptions --- proportionality, geometric similarity, characteristic dimension
- Fitting models to data: graphical techniques, Chebyshev method, Least Squares (know the Linear Algebra formulation), comparison of fits, ladder of powers, choosing a one-term model, high-order polynomial fits (Vandermonde matrix), smoothing data via divided differences, cubic splines
- Constrained optimization problem, Linear program, solving linear programs graphically (geometric method) and using the simplex method (numerical method), sensitivity analysis (how do changes in the coefficients of a linear program effect the solution?)
- Differential equations, population models (unlimited growth, logistic), qualitative analysis --- slope field, equilibrium solutions, phase line (autonomous equations only), long-term behavior, stability of equilibria (source, sink, node), Euler's method, separate and integrate to analytically solve a differential equation

**Some Practice Problems:** These include the same problems listed
for the first midterm. Note that some of these problems require a
computer to be solved fully. The point of listing them here is to remind
you of the process which must be undertaken in order to solve them.

**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)

** Dynamical Systems (Ch. 8 handout) **

- Section 8.1 : 4, 5, 12, 14, 22
- Section 8.2 : 6, 14, 17, 24, 28
- Section 8.4 : 6, 7, 9

- Section 2.1 : 4, 12
- Section 2.2 : 3, 6
- Section 2.3 : 3, 8

- Section 3.1 : 4
- Section 3.2 : 1, 2a, 4
- Section 3.3 : 2a, 4
- Section 3.4 : 2

- Section 4.1 : 6, 8
- Section 4.2 : 1
- Section 4.3 : 2, 6
- Section 4.4 : 1c

- Section 7.1 : 3, 4
- Section 7.2 : 3, 4b
- Section 7.4 : 3
- Section 7.5 : 2

- Section 10.1 : 7
- Section 10.4 : 1c, 3, 7
- Section 10.5 : 2