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Teaching an Introductory Course in Mathematical Modeling Using Technology. Dr.William P. Fox. Francis Marion University Florence, SC 29501 [email protected] Agenda. Mathematical Modeling Process Modeling Course and book Modeling Toolbox Explicative Models Model Fitting

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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 l.jpg

Teaching an Introductory Course in Mathematical Modeling Using Technology


Dr william p fox l.jpg

Dr.William P. Fox

Francis Marion University

Florence, SC 29501

[email protected]


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Agenda

  • Mathematical Modeling Process

  • Modeling Course and book

  • Modeling Toolbox

  • Explicative Models

  • Model Fitting

  • Empirical Models

  • Simulation Models

  • Projects and Labs


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Modeling Real World Systems


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The Modeling Loop


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Modeling Process

  • Identify the Problem

  • Assumptions/Justifications

  • Model Design/Solution

  • Verify the model

  • Strengths and Weaknesses of the model

  • Implement and Maintain Model


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Purpose of Mathematical Modeling

  • Explain behavior

  • Predict Future

  • Interpolate Information


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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


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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


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Tool #1 Explicative Modeling

  • Proportionality and Geometric Similarity Arguments


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Proportionality


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Proportionality-Definition


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Proportionality-Graphical

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


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Geometric Similarity

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


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Scale Models


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Formula’s

  • Area  characteristic dimension 2

  • Volume  characteristic dimension 3


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Archimedes

  • A object submerged displaces an equal volume to its weight.

  • So, under this assumption

  • W Volume


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Terror Bird

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


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Bird Data


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Dinosaur Data


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Assumptions

  • 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.


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Build the model and compare results


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Model Fitting

  • * Linear Regression (least squares)

  • Minimize the largest absolute deviation, Chebyshev’s Criterion

  • Minimize the sum of the absolute errors


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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.


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Errors

  • Least Squares: smallest sum of squared error

  • Chebyshev’s: minimizes the largest error

  • Minimizes the sum of the absolute errors


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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.


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Empirical Modeling (The data speaks)

  • Simple One Term Models

  • LN-LN transformations

  • High Order Polynomials

  • Low Order Smoothing

  • Cubic Splines


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Simple 1 Term Models


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1 Term Models


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LN-LN Transformation

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


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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.


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Smoothing with Low Order

  • Use Divided Difference Table for qualitative assessment.


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Goal

  • 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.


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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]


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Cubic Splines

  • Natural: end points have constant slope but we do not know the slope.

  • Clamped: end points have a known constant slope.


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Projects and Problem Sets

  • More sophisticated problems and projects with real world orientation.


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Simulation Modeling


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Simulation Modeling-A method of last resort!


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Monte Carlo Methods


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Deterministic Methods-Area under non-negative curve


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Y=x^3, 0<x<2


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Using Simulation: approximate area is 4.106


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Simulation Comparisons


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More Complicated Problem:


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Area via Simulation


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Volumes via simulation


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Example:Volume of Surface in 1st Octant


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Stochastic Simulation Labs


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Projects

  • 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,…


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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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Share our vision

Recruit students

Develop teachers

Empower faculty

Listen/learn/lead

Mentor/counsel/care

Selfless service

Blend of Excellence

Positive first derivative

We are successful since “we have a challenging, important mission, enough resources to do the job, and good people to work with.”

Philosophy


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