Mt/CS 410 Mathematical Modeling - Fall 2002

Study Guide for Test #1

Test Date: Tuesday, October 8

In this course, the projects give you many opportunities to apply the principles of mathematical modeling creatively to various problem scenarios. In the tests you will be asked to demonstrate mastery of the various mathematical strategies and calculational techniques that are introduced in this course. This test will cover material in portions of Chapters 1 through 5 of your textbook. Specifically, this test will cover material in Sections 1.2, 2.1, 2.2, 3.1, 3.2, 3.3, 4.1, 4.4, and 5.1. You may use your textbook (but not your notes) during the test. I assume that you will bring a scientific calculator with you to use during the test.

• Given a dynamical system, a(n+1) = f(a(n)) with a(0) = value, you should be able to calculate the first several values of a(n).
• Given a problem scenario involving change such as those discussed in Section 3.2, you should be able to formulate a dynamical system of equations to model the situation.
• You should be able to discuss the long-term behavior of a dynamical system, a(n+1) = f(a(n)) with a(0) = value. That is, does the system have an equilibrium value? Does the system reach a stable or unstable equilibrium? Does the behavior approach a limit? Is the long-term behavior of the system periodic? ... or chaotic?

• You should be able to use the idea of proportionality to develop a model.
• Given data and several graphs representing various transformations of the data, you should be able to select the transformation in which proportionality is a reasonable assumption, and fit a model to the data.
• You should be able to estimate parameters (slope, intercept) for a linear model from a graph, and give an equation for the model.
• Once you have used graphical analysis to estimate parameters from the transformed data, you should be able to transform the equation so that the model is expressed in terms of the original data.
• You should be able to sketch graphs of linear (y = a x + b), quadratic (y = a x^2 + b), periodic (y = a sin(bx + c) + d), exponential (y = a^x), and logarithmetic (y = a ln (x) + b) models. You should be able to identify or give examples of situations (i.e., problem scenarios) which might be described by these models.
• Given a scatterplot of a set of data, you should be able to choose one of the above models as appropriate for the situation, and explain your choice.