Sequences -- Chapter Guide

Contents -- Section Summaries

• Quickstart

  This module introduces some general terminology for sequences using bank accounts and other financial examples.

• Population Models -- Exponential Models

  This module introduces discrete dynamical systems, population models, and exponential models.

  • Exponential Models with Immigration -- Linear Dynamical Systems

    This module introduces linear dynamical systems with population models based on exponential models with immigration as the principal example. These models serve as important examples throughout this chapter.

  • Discrete Logistic Models

    This module introduces discrete logistic models. This is a rich source of examples that are one of the central themes of this chapter.

  • Pack-Hunters

    This module introduces models of species that cooperate in some way -- for example, by hunting in packs. This is another rich source of examples.

• Discrete Dynamical Systems and Difference Equations

  This module introduces some machinery and terminology for working with discrete dynamical systems. It links to the first appearance of one of the case studies -- Newton's Model of Cooling

  • A Case Study -- Newton's Model of Cooling

    Newton's Model of Cooling is a rich case study that appears several times in Calculus and in Differential Equations. This first appearance illustrates the application of exponential models with real data. You can either use your own data, collected using the Texas Instrument TI-CBL (calculator-based laboratory) or data supplied in these files. In this first appearance of this case study we learn something about the "wind chill" factor.

• Equilibrium Points

  This module studies equilibrium points and introduces algebraic, graphic, and numeric techniques for studying them. One of the principal themes of this chapter is the classification of equilibrium points.

• Cobweb Diagrams

  This module introduces cobweb diagrams. These diagrams provide a visual and geometric perspective on dynamical systems. They will give us insight into when an equilibrium point is attracting.

• Longterm Behavior

  This module provides an overview on the different kinds of longterm behavior exhibited by discrete dynamical systems.

  • Limits

    This module looks at models that approach a limit. One of the most important questions we study is when a model approaches a limit.

  • Cycles

    Ths module looks at models that settle down into periodic behavior, repeating the same sequence of numbers over and over again.

  • Chaos

    This module looks at a topic that is much in the news -- chaos. We see how very simple mathematical models give rise to sequences that look random.

• Classifying Equilibrium Points

  This module looks at two theorems for classifying equilibrium points.

  • The Linear Classification Theorem for classifying the equilibrium point of a linear discrete dynamical system.

  • The Nonlinear Classification Theorem for classifying the equilibrium points of a nonlinear discrete dynamical system.

  These theorems are important for themselves and also as an illustration of the connections and differences between a linear situation and a nonlinear situation.

  This module can be used as a discovery module with students discovering the two classification theorems or it can be used as a reference module.

• Classifying Cycles

  This module uses the machinery for classifying equilibrium points to classify 2-cycles.

• Period-Doubling and the Road to Chaos

  This module studies period-doubling from a geometric (graphical) perspective.