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Teaching by using Mathematica notebooks
*To*: mathgroup at christensen.Cybernetics.NET
*Subject*: [mg408] Teaching by using Mathematica notebooks
*From*: herrmann at siam.math.tamu.edu (Joseph M. Herrmann)
*Date*: Sun, 15 Jan 1995 11:36:41 -0600
For those interested in using Mathematica in teaching
Mathematical modeling at Texas A&M University has been taught
by having the students work through Mathematica notebooks which
assign readings in the textbook "A First Course in Mathematical
Modeling" by Frank R. Giordano and Maurice D. Weir, contain
interactive examples and explanations, and assign exercises in which
students create and analyze a mathematical model. Students complete
an individual and group project during the class.
The objective of the course is
1. Illustrate the broad range of problems which can be modeled
mathematically.
2. Synthesize mathematical models from non-mathematical descriptions
of problems.
3. Interpret the results of models and evaluate their implications.
4. Show the necessity of simplification and approximation in models
and identify their effects.
5. Work cooperatively in groups.
Currently there are 11 Mathematica notebooks. Eight Mathematica
notebooks cover the first 9 topics of the syllabus below. The
remaining 3 additional Mathematica notebooks create new Mathematica
functions and illustrate their use with examples.
1. Simplex: Creates a tableau for a standard linear programming
maximization problem. These tableaus are useful for solving linear
programming problems by the standard maximization method, dual
method, Big M method, or for achieving a sensitivity analysis.
2. Natural Spline: Determines the equations for the natural cubic
spline functions which fit the data and includes examples of how to
graph these functions automatically.
3. Clamped Spline: Determines the equations for the clamped cubic
spline functions which fit the data for specified derivatives at the
endpoints and includes examples of how to graph these functions
automatically.
Handouts:
Theory of the Simplex Method
Theory of the Dual Simplex Method
Dimensional Analysis
Syllabus:
1. Nuclear Arms Race
Develop a simple probabilistic model
Observe an equilibrium point
Changing assumptions affects parameter values
Sensitivity of the equilibrium point to change in parameters
2. The Modeling Process-identifying a problem
Vehicular Stopping Distance
Automobile Gas Mileage
Elevator Service during the morning rush hour
3. Using Geometric Similarity in the Modeling Process
Overall winner in weight lifting across weight classes
Volume of lumber from the diameter of a tree at waist level
Predicting Pulse rate from body weight
Vehicular Stopping distance
4. Model Fitting--identifying the optimal parameters for a model
Error criteria--Least square and Chebyshev
Identifying Kepler's third law from observational data
5. Models requiring optimization--Linear programming/Critical points
Linear programming economic models
Inventory problem--minimize delivery and storage cost
6. Experimental Modeling
Problems with using high order polynomials to interpolate
data
Splines
7. Project 1: Find the flow rate and water use of a small town from
.5in height measurements of the water tank
Numerical differentiation of data
Fitting curves to transformed data
Error analysis
8. Dimensional Analysis and Similitude
Range of a cannonball
Damped pendulum
Terminal speed of a raindrop
Turkey cooking times
Model design for determining the drag force on a submarine
9. Simulation Modeling
Monte Carlo simulation of area and volume
Simulation of gas station delivery and storage cost for a
stochastic demand
Simulation of harbor waiting times for stochastic arrival and
unloading times
10. Project 2: The projects for Spring 93 were
A: Identifying a near optimal ratio of slurry, greens, and paper
for composting.
B: Identifying a near optimal work schedule for a coal tipple.
11. Differential equation models
Population models
Drug dosage
Managing the fishing industry
The notebooks on the Simplex and the splines are available from
Mathsource. From Mosaic you could access it by
http://www.wri.com/MathSource.html/ Simplex #0207-447 and Natural
and Clamped Cubic spline coefficients #0207-436. If you are
interested in additional information, discussion of the advantages or
disadvantages of teaching in a format where students work in groups
or independently at a computer while the teacher acts a coach and
answers questions and stimulates inquiry, or any of the other
materials, please contact me.
Joseph Herrmann
Joseph M. Herrmann
Department of Mathematics, Texas A&M University
College Station, Texas 77843-3368
(409) 845-1474
herrmann at math.tamu.edu
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