Mt/CS 410 Mathematical Modeling - Fall 2002

Department of Mathematics & Computer Science
CARDINAL STRITCH UNIVERSITY
Sr. Barbara E. Reynolds, Ph.D.

Study Guide for Final Exam

Exam Date: Thursday, December 12, 2002 --- 8:00 - 10:00 a.m.

In this course, the projects give you many opportunities to apply the principles of mathematical modeling creatively to various problem scenarios. The cumulative final exam, which will contribute 15% to your final course grade, will give you an opportunity to demonstrate understanding and mastery of the various mathematical strategies and calculational techniques which are used in this course. You may use your text book, one 3-inch by 5-inch note card which you have prepared ahead of time, and a scientific or graphing calculator as you work on the test.

This exam will (probably) have 4 or 5 problems, and will cover material presented in Chapters 2 - 9, and the first part of Chapters 10 and 13 of our text. It will be an individual, in-class exam. After you have been working on the exam for about an hour, you will have an opportunity for a 10-minute group meeting. This will give you an opportunity to check on some concepts with your group members, but will probably not be enough time to discuss the entire test.

Topics to be covered by the exam include:

Using Graphs of Functions to Model a Problem Scenario

You should be able to:

  • Identify and discuss interesting questions, problems, and underlying assumptions suggested by a graphical representation of a particular problem scenario.
  • Sketch a qualitative graph which illustrates a given scenario.
  • Use linear, quadratic, polynomial, exponential, logarithmic, and cyclic (trigonometric) functions to model various problem scenarios.
  • Select an appropriate single-term function from the "ladder of powers" (see page 169 - 170) to model a given situation.
  • Fit a Lagrangian polynomial to a set of data points, and discuss strengths and weaknesses of such a polynomial model.
  • Use a divided difference table to determine which order polynomial would provide a reasonable fit for a given set of data.

    • Modeling with Discrete Dynamical Systems

    You should be able to:

  • Set up a dynamical system or a difference equation to model change approximately for situations such as births, deaths, population growth, spread of a contagious disease, value of an investment annuity, balance in a checking account, and other changes which occur more or less continuously.
  • Explain how divided differences relate to difference quotients --- and thus to the concept of the derivative.
  • Set up a discrete dynamical system to model change exactly for a problem scenario in which change occurs in discrete intervals.
  • Determine / discuss whether a discrete dynamical system would model a particular scenario "exactly" or "approximately" -- that is, whether the changes being modeled are occuring at discrete intervals or continuously.

    • Modeling using Proportionality

    You should be able to:

  • Formulate an appropriate mathematical proportionality for a given problem scenario in which geometric similarity is a reasonable assumption.
  • Write a mathematical equation for a given proportionality expression.
  • Determine whether or not proportionality is a reasonable assumption for a given data set by looking at a scatterplot of the data.

    • Fitting and Optimizing a Model for a Problem Scenario

    You should be able to:

  • Sketch an appropriate linear graph on a scatterplot, and estimate parameters for a (linear) function for a given data set.
  • Determine from the graphs (using "eyeball analysis") which of several proposed models is "better," and estimate parameters for a proposed model from the graph.
  • Use an appropriate method to transform a proposed relationship to a linear form.
  • Apply the least squares criterion and the methods of calculus to develop a set of equations to fit a model to a given set of data points.

    • Modeling using Simulation

    You should be able to:

  • Estimate probabilities for specified events given a set of frequency data for a particular scenario.
  • Describe a simulation experiment which would generate data to model the situation in a given problem.

    • Continuous Optimization Models and Linear Programming

    You should be able to:

  • Formulate an optimization model for a given scenario in terms of the variables, the objective funtion, and the constraints.
  • Use the criteria from page 261 to verify whether a particular optimization model is (or is not) a linear program.
  • Set up a linear program for an optimization problem, give a graphical representation for the feasible region, and use a geometric strategy to find the optimal solution.
  • Explain why/how such a geometric strategy gives an optimal solution.

    • Population Growth

    You should be able to:

  • Use methods of calculus to develop a model describing the growth of a population or the spread of a disease.
  • Solve a simple first-order differential equation.
  • Explain the difference between exponential and logistic models of change in a problem scenario.
  • Discuss the difference between unconstrained and constrained population growth.
  • Develop equations for models of exponential and logistic growth. (See equations 10.7 and 10.11, pages 350 and 353 of your text.)
  • Estimate the point where the rate of growth for a given population is a maximum from a logistic curve of the data, and use this to estimate the carrying capacity of the environment for that population.
  • Discuss assumptions implicit in a given analytic or graphical representation of a situation involving population growth, and criticize the model naming some strengths and some weaknesses.

    • Dimensional Analysis

    You should be able to:

  • Use the methods of Section 13.1 to determine whether (or not) a given equation is dimensionally compatible.
  • Find the dimensions of a particular constant in a given equation so that the equation will be dimensionally compatible.
  • Prove (or disprove) that a particular constant in a given equation is a dimensionless constant.

    Suggested Study Problems

    1. Discuss the appropriateness of fitting a higher-order polynomial to data: #4, page 187. (For an in-class exam, a problem like this would include some spreadsheet calculations and graphs to facilitate your computations.)

      • Using smoothing techniques, is there a lower-order polynomial that would "fit" this data?
      • Choose an appropriate one-term model from the ladder of powers (see pages 169 - 170) to fit this data. Then develop the model using a least-squares method to determine the coefficients.

    2. Use a difference table or a divided-difference table to determine whether or not a low-order polynomial would fit a given set of data: #1 - 4, page 198 - 199.

    3. Set up a discrete dynamical system to model a situation which changes at discrete moments in time: #2, page 67; #6 and 7, page 80.

      • Does the model have an equilibrium value? If it does, calculate the equilibrium value and determine whether it is a stable or unstable equilibrium.
      • What is the significance of the equilibrium value in the given scenario?

    4. Formulate (then solve) a differential equation to model unconstrained or constrained growth: #1, 3, 4, and 5, page 357.

    5. Describe a simulation experiment that you could use to study a particular problem: #1 and 3, page 224; #1 - 3, page 232. (On an in-class test, you might be given a set of "data" and asked to interpret the given data for the simulation experiment you devise, and relate this to the problem under investigation.)

    6. Formulate an optimization model for a given scenario, and solve the model geometrically (using the strategies of Section 9.1) or algebraically (using the strategies of Section 9.2): #4, page 272; #3, page 310.

    7. Formulate a mathematical proportionality to model a situation in which geometric similarity is a reasonable assumption: #2 and #5, page 117.

    8. For a given data set, calculate parameters to fit a model of a specified form to the data: #7, page 145; #4, page 157; #7, page 164.

      • You might be asked to estimate parameters from a graph, or to calculate parameters using a least-squares computation. (For an in-class test, a problem like this would include some spreadsheet calculations and graphs to facilitate your calculations.)

    9. For a given equation, you might be asked to perform a dimensional analysis of the equation: #1, 3, 4, 7, and 8, page 450 - 451.

      • Determine dimensional compatibility of the equation.
      • Find the dimensions of a particular constant so that the equation is dimensionally compatible.
      • Prove (or disprove) that a particular constant in a given equation is a dimensionless constant.

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    This page was updated on November 21, 2002.