Math 216 - University of Michigan



Math 462/562 Mathematical Modeling Fall 2013

Course Outline

This course studies a number of mathematical models of real world phenomena. These models fall into the following categories

(1) Optimization problems

(2) Predicting the behavior of systems that vary with time

(3) Probability models

Instructor: Frank Massey

Office: 2075 CASL Building Phone: 313-593-5198

E-Mail: fmassey@umich.edu

Office Hours: M 1:45 - 2:45, 5:00 - 6:00 and after class as long as there are questions and TuTh 2:30 - 3:30. Also by appointment.

My office hours are those times I will usually be in my office. However, occasionally I have to attend a meeting during one of my regularly scheduled office hours. In this case I will leave a note on my door indicating I am unavailable. In particular, if you know in advance that you are going to come see me at a particular time, it might not be a bad idea to tell me in class just in case one of those meetings arises. Please feel free to come by to see me at times other than my office hours. I will be happy to see you.

Text Mathematical Modeling by Mark M. Meershaert, Academic Press. Either the 3rd edition (2007) ISBN 978-0-12-370857-1 or the 4th edition (2013) ISBN: 978-0-12-386912-8 would be OK. These are abbreviated by M in the schedule below.

Website: . This contains copies of this course outline, the assignments, exams that I gave in this course in the past and some notes that contain supplementary information. For the most part the notes are devoted to showing how to use Mathematica to do some of the computations that arise. They are abbreviated by Notes in the schedule below. They are written using Mathematica, and to read them you either need to use a computer on which Mathematica has been installed (many of the computers on campus have Mathematica on them) or you can use the "Mathematica Player" software that can be downloaded for free from products/player/. This software allows you to read Mathematica files, but does not allow you to execute the Mathematica operations in the file. If you haven't used Mathematica before you might read the last item in the Notes "An Introduction to Mathematica.nb" and see me for a demonstration. See me if you have trouble accessing any of the items in the website.

Grading: There will be 3 midterm exams and a final exam, each of which will count 100 points. In addition there will be some assignments which will count 100 points. You may continue submitting solutions to the problems on the assignments until you have accumulated 100 points. 100 points is the maximum that your assignments may count toward your grade. The assignments can be found on CANVAS and at umd.umich.edu/~fmassey/math462/Assignments/. In addition, students taking Math 562 should find an article in a journal where the mathematics in this course has been applied to some other subject and submit a report on the contents of this article. This will count 50 points. See me if you need help finding a journal article to read and report on.

The dates of the exams are on the attached schedule. All exams are closed book, but formulas will be provided. A copy of the formula sheet is at umd.umich.edu/~fmassey/math462/Exams/Formulas.doc. No make-up exams unless you are quite sick.

I would like you to work by yourself on the exams and the assignments. See me if you need help.

On each exam I will look at the distribution of scores and decide what scores constitute the lowest A-, B-, C-, D-. On the assignments, 90% of the maximum possible will be the lowest A-, 80% will be the lowest B-, 70% will be the lowest C- and 60% will be the lowest D-. The lowest A- on the exams and the assignments will be added up and the same for B-, C-, D-. The lowest A, B+, B, C+, D+, D will be obtained by interpolation. For example, the lowest B is 1/3 of the way between the lowest B- and the lowest A-, etc. All your points will be added up and compared with the lowest scores necessary for each grade. For example, if your total points falls between the lowest B+ and the lowest A- you would get a B+ in the course. This information is in the file YourGrade which is located in the course website at . After each exam and assignment is graded this information will be updated and you should be able to see how you stand. You can find out what scores I have recorded for you by going to CANVAS, selecting Math 462 or Math 562 and clicking on Grades on the left. If possible, check your grades after each exam and assignment to see that they were entered correctly.

The tests will have a mixture of "word" problems and "non-word" problems. The word problems are to test your ability to translate a verbal description of a problem into a mathematical description. Problem 1 on Exam 1, Fall 2007 is a typical problem of this type. In a word problem I won't ask you to find the solution of the mathematical problem. The non-word problems are to test your ability to use mathematical techniques. Problems 2, 3 and 4 on Exam 1, Fall 2007 are typical problems of this type. In the schedule below are some suggested problems for you to work on. Some of these problems are representative of what will be on the exams, while others are simply to help you fix the concepts in your mind or prepare you to do other problems. Work as many problems as time permits and ask for help (in class or out) if you can’t do them.

Learning Goals The Department of Mathematics and Statistics Learning Goals for its classes are the following.

1. Increase students' command of problem-solving tools and facility is using problem-solving strategies, through classroom exposure and through experience with problems within and outside mathematics.

2. Increase students' ability to communicate and work cooperatively.

3. Increase students' ability to use technology and to learn from the use of technology, including improving their ability to make calculations and appropriate decisions about the type of calculations to make.

4. Increase student's knowledge of the history and nature of mathematics. Provide students with an understanding of how mathematics is done and learned so that students become self-reliant learners and effective users of mathematics.

Math 462/562 studies a number of mathematical models of real world phenomena. These topics relate to the first learning goal and the fourth learning goal. Computer labs illustrating how to do the computations relating to these topics with mathematical software relates to the third learning goal.

The University of Michigan – Dearborn values academic honesty and integrity. Each student has a responsibility to understand, accept, and comply with the University’s standards of academic conduct as set forth by the code of Academic Conduct, as well as policies established by the schools and colleges. Cheating, collusion, misconduct, fabrication, and plagiarism are considered serious offenses. Violations will not be tolerated and may result in penalties up to and including expulsion from the University.

The University will make reasonable accommodations for persons with documented disabilities. These students need to register with Disability Resource Services (DRS) every semester they are taking classes. DRS is located in Counseling and Support Services, 2157 UC. To be assured of having services when they are needed, student should register no later than the end of the add/drop deadline of each term.

Reminder: Tuesday, November 5 is the last day to drop the course. is the last day to drop the course.

Tentative Schedule

|Dates |Section(s) |Topics and Suggested Problems |

|9/9 |M: ch 1 |Part 1. Optimization. |

| |Notes 1.1 | |

| | |One variable optimization: analytical techniques, sensitivity of the solution to changes in |

| | |parameters. |

| | |Ex 1, F 12 #1 |

| | |M: Section 1.4 #1(a), 5(a), 6(a), 7(a), 9(a). On these problems just translate the problem |

| | |description into a formula for what is to be maximized or minimized along with the relevant |

| | |range of values for the independent variable. |

| | |Notes: Section 1.1 #1 |

|9/9 |M: 3.1 |One variable optimization: Newton’s method for the numerical solution. |

| |Notes 1.2 |Notes: Section 1.2 #1 |

|9/9, 16 |M: 2.1, 2.3 |Several variable optimization without constraints: analytical techniques, sensitivity of the |

| |Notes 1.3 |solution to changes in parameters. |

| | |Ex 1, F 12 #2 |

| | |Ex 1, F 08 #2 |

| | |Notes: Section 1.3 #1 |

|9/16 |M: 3.2 |Several variable optimization: Newton’s method for the numerical solution. |

| |Notes 1.4 |Ex 1, F 12 #3 |

| | |Ex 1, F 08 #3 |

| | |Notes: Section 1.4 #1 |

|9/16, 23 |M: 2.2, 2.3 |Several variable optimization with constraints: theory, Lagrange multipliers. |

| |Notes: 1.5.1 |Ex 1, F 12 #4 |

| | |Ex 1, F 08 #4 |

| | |Notes: 1.5.1 #1, 2, 3, 4 |

|9/23 |M: 2.2, 2.3 |Several variable optimization with constraints: applied problems. |

| |Notes: 1.5.2 |Ex 1, F 08 #1 |

| | |M: Section 2.4 #6(a), (d), 7(a), 9(a), (b). On these problems just translate the problem |

| | |description into a formula for what is to be maximized or minimized along with the constraints |

| | |for the independent variables. |

|9/23, 30 |M: 3.3 |Linear programming – the simplex method. |

| |Notes 1.6, 1.10 |Ex 2, F 12 #1, 4 |

| | |Ex 2, F 08 #1, 4 |

| | |M: Section 3.5 #12-22. On these problems just translate the problem description into a |

| | |formula for what is to be maximized or minimized along with the constraints for the independent|

| | |variables. |

| | |Use the simplex method to solve the linear programming problem in section 3.3 of Meerschaert by|

| | |hand. Check that you get the same answer as the book in Figure 3.15. |

| | |Notes: Section 1.6 #1 |

|9/30 |Notes 1.7, 1.9, |Linear programming – finding a feasible vertex. |

| |1.10 |Notes: Section 1.7 #1 |

|9/30 |M: 3.3 |Linear programming – the dual problem. |

| |Notes 1.8 |Ex 2, F 12 #4g |

| | |Ex 2, F 08 #4g |

| | |Notes: Section 1.8 #1 |

|9/30 | |Review |

|10/7 |Notes 2.1 |Part 2. Dynamic Modelling. |

| | | |

| | |Models involving a single differential equation. Steady state solutions, equilibrium points |

| | |and qualitative behavior of solutions. |

| | |Ex 2, F 12 #2 |

| | |Ex 2, F 08 #2 |

| | | |

| | |Notes: Section 2.1 #1 |

|10/7 | |Exam 1 |

|10/14 |Notes 2.2 |Models involving a single difference equation. Steady state solutions, equilibrium points and |

| | |qualitative behavior of solutions. |

| | |Ex 2, F 12 #3 |

| | |Ex 2, F 08 #3 |

| | | |

| | |Notes: Section 2.2 #1 |

|10/14 |M: 4,1, 4.2, |Models involving systems of differential equations. Steady state solutions and equilibrium |

| |Notes 2.3 |points, numerical solution using mathematical software |

| | |Ex 3, F 07 #2b |

| | |Ex 3, F 08 #2b |

| | |Notes: Section 2.3 #1 |

|10/21 |M: 4.3, |Models involving systems of difference equations. Steady state solutions and equilibrium |

| |Notes 2.4 |points |

|10/21 |Notes 2.5 |Solving linear systems of difference equations using eigenvalues and eigenvectors of the |

| | |associated matrix. |

| | |Ex 3, F 07 #1a, b, c, d, f |

| | |Ex 3, F 08 #1a, b, c, d, f |

|10/21 |Notes 2.6 |Solving linear systems of differential equations. |

| | |Ex 3, F 07 #1a, b, c, e, g |

| | |Ex 3, F 08 #1a, b, c, e, g |

| | |Notes: Section 2.6 #1, 2. |

|10/28 |M: 5.1, 5.2 |Linearization for systems of differential equations. |

| |Notes 2.7 |Ex 3, F 12 #2a, c |

| | |Ex 3, F 08 #2a, c |

|10/28 |M: 5.1, 5.2 |Linearization for systems of difference equations. |

| |Notes 2.8 | |

|10/28 |M: 5.3 |Trajectories and phase portraits for systems of differential equations |

| |Notes 2.9 |Ex 3, F 12 #2 |

| | |Ex 3, F 08 #2 |

| | |Notes: Section 2.9 #1, 2 |

|10/28 | |Review |

|11/4 |Notes: 3.1, 3.2 |Part 3. Probability Models. |

| | | |

| | |Discrete probability: Outcomes, events and random variables, probability of outcomes and |

| | |events, probability distributions, independent events and random variables, averages (or |

| | |expected values) of random variables and the law of large numbers |

| | |Notes: Section 3.1 #1.1, 1.2, 1.3, 1.4, 1.5, 1.6 |

|11/4 | |Exam 2 |

|11/11 |M: 7.1 |Optimization problems with a probabilistic aspect. |

| | |Final Exam, F 12 #1 |

| |Notes: 3.3 |Final Exam, F 08 #1 |

| | |Notes: Section 3.2 #1 |

|11/11 |Notes: 3.4 |Single period inventory |

| | |Notes: Section 3.4 #1 |

|11/11 |Notes: 3.5 |Machine replacement |

| | |Notes: Section 3.5.1 #1 |

|11/11 |Notes: 3.6 |Binomial probabilities. |

| | |Final, F12 #2 |

| | |Final, F08 #3a |

|11/18 |M: 7.2 |Continuous probability: Continuous random variables, probability density functions, |

| |Notes: 3.7 |independent random variables, expected values of random variables, exponential random |

| | |variables. |

| | |Final, F 12 #3 |

| | |Final, F 08 #2 |

| | |Notes: Section 3.7 #1, 2 |

|11/18, 25 |Notes: 3.8 |Joint probability distributions, calculating probabilities with two continuous random variables|

| | |using double integrals, sums of independent random variables. |

| | |Final, F 08 #2 |

| | |Notes: Section 3.8 #1 |

|11/18 | |Review |

|11/25 | |Exam 3 |

|12/2 |Notes: 3.9 |Sums of exponential random variables, probabilities of the number of occurrences in a given |

| | |time, the Poisson distribution. |

| | |Final, F12 #4 |

| | |Final, F 08 #4 |

| | |Notes: Section 3.9 #1 |

|12/2, 9 |M: 7.3 |Variances and standard deviations, the normal distribution, the central limit theorem and |

| | |statistics. |

| |Notes: 3.10, 3.11 |Final, F 08 #3b |

| | |Notes: Section 3.11 #1. |

|12/9 | |Review |

|Monday, December 16, 6:30 – 9:30 p.m. Final Exam |

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