Teaching an Introductory Course in Mathematical Modeling Using Technology

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Teaching an Introductory Course in Mathematical Modeling Using Technology

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Teaching an Introductory Course in Mathematical Modeling Using Technology

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Dr.William P. Fox

Francis Marion University

Florence, SC 29501

wfox@fmarion.edu

- Mathematical Modeling Process
- Modeling Course and book
- Modeling Toolbox
- Explicative Models
- Model Fitting
- Empirical Models
- Simulation Models
- Projects and Labs

- Identify the Problem
- Assumptions/Justifications
- Model Design/Solution
- Verify the model
- Strengths and Weaknesses of the model
- Implement and Maintain Model

- Explain behavior
- Predict Future
- Interpolate Information

- “A First Course in Mathematical Modeling”, Giordano, Weir, and Fox, Brooks-Cole Publisher, 3rd Edition, 2003.
- Labs available on the course/book web site using
- MAPLE, EXCEL, TI-83 Plus calculator

- Show mathematical modeling tool
- Provide development, theory, and analysis of the tool
- Provide appropriate technology via labs
- Maple
- Excel
- Graphing Calculator
- Project to tie together the process

- Proportionality and Geometric Similarity Arguments

- We want to estimate the plot of the transformed data as a line through the origin. W versus x^3.

- 1 to 1 relationship between corresponding points such that the ratio between corresponding points for all such points is constant

- Area characteristic dimension 2
- Volume characteristic dimension 3

- A object submerged displaces an equal volume to its weight.
- So, under this assumption
- W Volume

- Problem Identification: Predict the size (weight) of the terror bird as a function of its fossilized femur bone found by scientists.

- Geometrically similar objects (terror bird is a scale model of something)
- Assume volume is proportional to weight under Archimedes Principal.
- Characteristics dimension is the femur bone.

- * Linear Regression (least squares)
- Minimize the largest absolute deviation, Chebyshev’s Criterion
- Minimize the sum of the absolute errors

- Least squares—calculus or technology
- Chebyshev’s—Linear Programming
- Minimize Sum of absolute errors—nonlinear search method like Golden Section Search.

- Least Squares: smallest sum of squared error
- Chebyshev’s: minimizes the largest error
- Minimizes the sum of the absolute errors

- Concern is model adequacy.
- Plot residuals versus model (or independent variable)—check for randomness. If there is a pattern then the model is not adequate.

- Simple One Term Models
- LN-LN transformations
- High Order Polynomials
- Low Order Smoothing
- Cubic Splines

- Try to linearize the data in a plot. This plot does not have to pass through the origin.

- N data points create a (N-1)st order polynomial.
- Problems exist with high order polynomials that provide possible strong disadvantages.
- Advantage: passes perfectly through every pair of data points.
- Disadvantages: oscillations, snaking, wild behavior between data pairs.

- Use Divided Difference Table for qualitative assessment.

- To find columns that qualitatively reveal one of the following:
- f(x) linear, 1DD is constant, 2 DD is zero.
- f(x) quadratic, 2 DD is constant, 3rd DD is zero.
- f(x) cubic, 3rd DD is constant, 4th DD is zero
- f(x) quartic, 4th DD is constant, 5th DD is zero
- Then fit with least squares and examine residuals.

- Put data for x in increasing order and carry along the y coordinate (x,y). Fit a cubic polynomial between successive pairs of data pairs. For example given three pairs of points (x1,y1),(x2,y2), and (x3,y3):
- S1(x)=a1+b1x+c1x2+d1x3 for x [x1,x2]
- S2(x)=a2+b2x+c2x2+d2x3 for x [x2,x3]

- Natural: end points have constant slope but we do not know the slope.
- Clamped: end points have a known constant slope.

- More sophisticated problems and projects with real world orientation.

- Sporting Event: Baseball game, horse race (based on odds), consecutive free throws contest, racketball games
- Gambling Events: Craps, Blackjack
- Real World: Queuing problems (emission inspections, state car inspections, traffic flow, evacuating a city for a disaster, fastpass at Disneyland,…

- Dynamic, connected modeling curriculum responsive to rapidly changing world.
- Blend of excellence: Mentoring relationship between faculty and students and between college and school faculty.
- Enthusiastic teachers empowered to motivate, challenge, and be involved.
- Effective teaching tools and supportive educational environment.
- Competent, confident problem solvers for the 21st Century.
- Continued development and student/teacher growth.

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Positive first derivative

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