Updating function discrete time dynamical system
Updating function discrete time dynamical system - Free Online
This course was developed at our MTL partner institution, Illinois State University. 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.
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.Through the experiences gained via these activities, the concepts "limit" and "solution" take on new meaning. The faculty there have been using this approach as an introduction to the calculus strand since the late 1980s.This course draws on the approach to Discrete Dynamical Systems employed in the Department of Mathematical Sciences at the U. The material parallels the approach taken in the text Discrete Dynamical Modeling by James Sandefur (New York, NY: Oxford University Press, 1993. 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.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.
Discrete dynamical systems can be used to model and analyze many real-world problems including population growth, compound interest and annuities, radioactive decay, pollution control, and medication dosages.
Such problems are usually delayed until calculus; however, the discrete dynamical systems approach uses only high school advanced algebra.
Teachers who complete this module will be able analyze a variety of discrete dynamical systems using the TI-83 or TI-84 calculators (or any other calculator with similar graphing and sequence capabilities). Common Core Standards for Mathemtical Practice that are emphasized include: The purpose of this module is to familiarize mathematics teachers at the high school and community college levels with discrete dynamical systems.
They will also learn how to integrate this material into advanced algebra, precalculus, and calculus courses. 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.