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Teaching an Introductory Course in Mathematical Modeling Using Technology. Dr.William P. Fox. Francis Marion University Florence, SC 29501 wfox@fmarion.edu. Agenda. Mathematical Modeling Process Modeling Course and book Modeling Toolbox Explicative Models Model Fitting

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dr william p fox

Dr.William P. Fox

Francis Marion University

Florence, SC 29501


  • Mathematical Modeling Process
  • Modeling Course and book
  • Modeling Toolbox
  • Explicative Models
  • Model Fitting
  • Empirical Models
  • Simulation Models
  • Projects and Labs
modeling process
Modeling Process
  • Identify the Problem
  • Assumptions/Justifications
  • Model Design/Solution
  • Verify the model
  • Strengths and Weaknesses of the model
  • Implement and Maintain Model
purpose of mathematical modeling
Purpose of Mathematical Modeling
  • Explain behavior
  • Predict Future
  • Interpolate Information
modeling textbook
Modeling textbook
  • “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
my process
My Process
  • 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
tool 1 explicative modeling
Tool #1 Explicative Modeling
  • Proportionality and Geometric Similarity Arguments
proportionality graphical
  • We want to estimate the plot of the transformed data as a line through the origin. W versus x^3.
geometric similarity
Geometric Similarity
  • 1 to 1 relationship between corresponding points such that the ratio between corresponding points for all such points is constant
formula s
  • Area  characteristic dimension 2
  • Volume  characteristic dimension 3
  • A object submerged displaces an equal volume to its weight.
  • So, under this assumption
  • W Volume
terror bird
Terror Bird
  • 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.
model fitting
Model Fitting
  • * Linear Regression (least squares)
  • Minimize the largest absolute deviation, Chebyshev’s Criterion
  • Minimize the sum of the absolute errors
solution methods for each
Solution Methods for each
  • 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
least squares and residuals errors
Least Squares and Residuals (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.
empirical modeling the data speaks
Empirical Modeling (The data speaks)
  • Simple One Term Models
  • LN-LN transformations
  • High Order Polynomials
  • Low Order Smoothing
  • Cubic Splines
ln ln transformation
LN-LN Transformation
  • Try to linearize the data in a plot. This plot does not have to pass through the origin.
high order polynomials
High Order Polynomials
  • 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.
smoothing with low order
Smoothing with Low Order
  • 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.
cubic splines
Cubic Splines
  • 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]
cubic splines35
Cubic Splines
  • Natural: end points have constant slope but we do not know the slope.
  • Clamped: end points have a known constant slope.
projects and problem sets
Projects and Problem Sets
  • 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,…
goals of a modeling course
GOALS of a Modeling Course
  • 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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