Mt/CS 410 Mathematical Modeling - Fall 2002

Study Guide for Test #2

Test Date: Tuesday, November 5

In this course, the projects give you many opportunities to apply the principles of mathematical modeling creatively to various problem scenarios. In the test you will be asked to demonstrate mastery of the various mathematical strategies and calculational techniques that are introduced in this course. You may use your text book (but not your notes) during the test. This test will cover material in Chapters 1 - 6 of our text. The relevant sections will be announced about a week prior to the test. I assume that you will bring a scientific calculator with you to use during the test.

• 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 = e^x), and logarithmetic (y = a ln (x) + b) models.
• You should be able to sketch a graph of a function given by any of the one-term models in the Ladder of Powers or Ladder of Transformations (pages 169 - 170).
• 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.
• You should be able to distinguish between a power curve (y f(x) = x^n) and an exponential curve (y = f(x) = e^x).
• 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.
• 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).
• You should be able to discuss the long-term (or "limiting") 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.
• Given data and several graphs representing various transformations of the data, you should be able to use "eyeball analysis" to select the "best" model. You should be able to explain why the one you choose is best, and estimate parameters for the model from the graph.

• 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.
• If two objects are geometrically similar and their linear measurements are related by a constant k, then you should be able to show algebraically that their area measurements are related by k^2, and their volume measurements are related by k^3.
• You should be able to use the least-squares criterion and methods of calculus to derive the equations and solve for the parameters in a model of a given form.

• You should be able to set up the Lagrangian form of a polynomial to fit a given set of data. You should be able to identify the (maximum) degree of the polynomial for a given set of data, and show that the graph of this polynomial really does go through each point in the given data set. (The question on the test will ask you to fit a polynomial to no more than six data points. You would not be asked to simplify this polynomial; you may leave it in the form given on page 183.)

• You should be able to set up a divided difference table for a given set of data. Then using this divided difference table, you should be able to determine what lower-degree polynomial (if any) would be an appropriate interpolating polynomial for a given data set.

• You should be able to estimate probabilities for specified events given a set of frequency data for a particular scenario.
• You should be able to describe a simulation experiment which would generate data to model the situation in a given problem.