Math 117          Difference Equations and Linear Algebra (3 semester credits)

Course Syllabus for Spring Semester, 2001 – prepared by Marie Vanisko

Professor and Office Hours

Mrs. Marie Vanisko, Professor of Mathematics, 118 Science Bldg, 447-4451, mvanisko@carroll.edu

Office Hours:    MWF  (9-10, 11-11:30, 1:30-2, 3-4 (MW)), TTh  (8-9:30, 11-11:30, 2-4:30(3:30 Th))

Pre-Requisite:  3 years of high school mathematics, including Algebra II

                         

Course Description

This is an introductory college mathematics course in finite difference equations and linear algebra. Topics include sequences, differences, linear and nonlinear difference equations, systems of difference equations, numerical solutions of linear and nonlinear equations, analytical techniques for solving linear systems using linear algebra, up through eigenvalues and eigenvectors, and an introduction to projections and regression with vectors. Applications from many fields are studied and the role of mathematical modeling is a central focus. Formal computer labs will be a part of the course each week, with spreadsheets being the primary software employed. This course satisfies a general liberal arts requirement for all students and the math requirement for business majors. It is offered each semester.

                                   

Required Text

Discrete Dynamical Systems , by Arney, Giordano, and Robertson, © 2000 by McGraw-Hill

Course Objectives

·        Extend knowledge and understanding of mathematical concepts in difference equations and linear algebra.

·        Develop mathematical skills to formulate and solve difference equations and systems of equations

·        Recognize and identify mathematical patterns of change in real-world contexts, specifically, to see situations where a difference equation or system of equations can be used to describe behavior.

·        Design and interpret mathematical models using difference equations and systems to capture the essence of real-world phenomena.

·        Expand knowledge and understanding of the real world through mathematical analysis.

·        Develop skills to effectively use spreadsheets and other computing, information, and communication technologies.

Course Policies

Most Wednesday classes will meet in the computer laboratory; Excel will be the primary software used for this class. Mathematica will also be used in the last half of the course.

No late projects will be accepted and at most two late assignments will be accepted.

Writing about what is being learned will be an essential part of this class, and this is especially true for computer assignments.  Computer printouts with no comments will not be accepted.

Selected projects including Interdisciplinary Lively Applications Projects (ILAPs) will be assigned in groups.  Each member will be held accountable and certain presentations will be oral.

Exams will be given every two to three weeks and will be announced one week in advance. Some exams will be given in the computer lab or will have an component that will be done outside of class. The final exam is scheduled for Wednesday, May 9th at 3 p.m.

Grading scale used:    A (90-100),  B (80-89),  C (70-79),  D (60-69),  F (below 60)

Relative weights:            Exams (65%),  Homework & Projects (35%)

Course Outline

Chapter 1 – Discrete Dynamical Systems                                                                      1.5 weeks

Introduction to modeling with difference equations and computing numerical solutions

Sequences and differences and relation to functions and mappings

Chapter 2 – First Order Linear Discrete Dynamical Systems                                  3 weeks

Properties and applications of difference equations and motivation for obtaining analytical solutions

Solving linear homogeneous difference equations and extending to nonhomogeneous equations

Analyzing solutions for stability of equilibrium points

Applications and modeling of first order linear difference equations

Introduction of geometric series with drug dosage problem (Section 9.3.5 of text)

Chapter 7 – Nonlinear Discrete Dynamical Systems                                              1.5 weeks

Nonlinear difference equations applications investigated and solved numerically

Analysis of stable and unstable equilibrium points

Chapter 3 – Dynamical Systems of Several Variables                                            2 weeks

Modeling and classification of systems of difference equations

Numerical solutions to these difference equations

Part 1 of the Lake Pollution Project

Chapter 4 – Matrix Algebra                                                                                               2 weeks

Basic rules of matrix algebra

Markov chains analyzed with Part 4 of the Lake Pollution Project

Chapter 5 – Linear Dynamical Systems of Several Variables                                3 weeks

Systems of linear difference equations analyzed from a matrix perspective

Application of geometric series to vectors and matrices in Part 1 of the Lake

Pollution ILAP to show analogy to the drug dosage problem

Extension of the Lake Pollution ILAP, Parts 1 and 2 from the linear algebra

perspective using eigenvalues and eigenvectors

Chapter 8 – Nonlinear Systems and Their Classification                                      1 week

Modeling and numerically solving nonlinear systems like the predator-prey problem