| |
The purpose of this module is to familiarize mathematics teachers at the
high school and community college levels with discrete dynamical systems.
These systems, which can be analyzed with the tools of high school algebra,
are discrete analogs to
differential equation systems, which are based on calculus. As such,
discrete dynamical systems can be used to introduce advanced algebra
students to some of the most important ideas of calculus and differential
equations. Also, discrete dynamical systems can be used to model many
interesting problems on such diverse topics as: loans and annuities,
population growth, radioactive decay, medication dosages, water and air
pollution, and heating/cooling problems. Teachers will learn how to set
up and solve such problems and will also learn how to integrate content on
discrete dynamical systems into their advanced algebra, precalculus and
calculus courses.
A discrete dynamical system consists of a
recurrence relation (or difference equation) describing
some relationship, pure or applied, with a
given set of initial conditions.
The course examines a variety of discrete dynamical systems using
an iterative and graphical approach facilitated by the use of calculators
such as the TI-82, 83.. Algebraic methods for solving difference equations
are also developed.
Particular emphasis is given to seeing the relationship between such
discrete dynamical systems and real-world applications. Such applications
are used to motivate the mathematical ideas, and both the illustrative
examples and the homework exercises encourage
those participating in the module to apply their knowledge to interesting
and realistic settings. Through the
experiences gained via these activities, the concepts "limit" and
"solution" take on new meaning.
This course draws on the approach to Discrete Dynamical Systems employed
in the Department of Mathematical Sciences at the U.S. Military Academy at
West Point. The faculty there have been using this approach as an
introduction to the calculus strand since the late 1980s. The material
parallels the approach taken in the text Discrete Dynamical Modeling by
James Sandefur (New York, NY: Oxford University Press, 1993.
ISBN:0-19-508438-1).
This approach, which bridges the gap between precalculus and the calculus,
analyzes the sequences of values that result from
iterating difference equations under varying initial values. These
equations, by their very nature describe the shifts from one, or more,
state(s) to another. As such, they are the discrete analog to
differential equations.
This module is self-contained. You do not have to buy any additional textual
materials. You do need to set aside ample time to wrestle with the
problems and the content presented. Your TI-82 or TI-83 calculator will
become a good friend
before the module is over, in new ways that you may have never imagined
possible before. If you are not familiar with the operation of the
sequence mode giving dot output, the web format for the window, or the use
of tables, you can acquire these skills through the Interactive TI-82 and
TI-83 Tutorials available at the Math Teacher Link Course Module Program
Homepage.
The course is especially suited for Educators seeking ways to enhance
their precalculus and calculus curriculum.
|
