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Mathematical Modeling Teaching an Introductory Course in Mathematical Modeling Using Technology

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Mathematical Modeling Dr.William P. Fox Francis Marion University Florence, SC 29501 wfox@fmarion.edu

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Mathematical Modeling 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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Mathematical Modeling Modeling Real World Systems

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Mathematical Modeling The Modeling Loop

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Mathematical Modeling 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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Mathematical Modeling Purpose of Mathematical Modeling Explain behavior Predict Future Interpolate Information

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

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Mathematical Modeling 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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Mathematical Modeling Tool #1 Explicative Modeling Proportionality and Geometric Similarity Arguments

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Mathematical Modeling Proportionality

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Mathematical Modeling Proportionality-Definition

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Mathematical Modeling 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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Mathematical Modeling 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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Mathematical Modeling Scale Models

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Mathematical Modeling Formula’s Area characteristic dimension 2 Volume characteristic dimension 3

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

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Mathematical Modeling 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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Mathematical Modeling Bird Data

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Mathematical Modeling Dinosaur Data

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Mathematical Modeling 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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Mathematical Modeling Build the model and compare results

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Mathematical Modeling 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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Mathematical Modeling 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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Mathematical Modeling 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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Mathematical Modeling 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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Mathematical Modeling Empirical Modeling (The data speaks) Simple One Term Models LN-LN transformations High Order Polynomials Low Order Smoothing Cubic Splines

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Mathematical Modeling Simple 1 Term Models

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Mathematical Modeling 1 Term Models

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Mathematical Modeling 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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Mathematical Modeling 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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Mathematical Modeling Smoothing with Low Order Use Divided Difference Table for qualitative assessment.

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Mathematical Modeling 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, 3 rd DD is zero. f(x) cubic, 3 rd DD is constant, 4 th DD is zero f(x) quartic, 4 th DD is constant, 5 th DD is zero Then fit with least squares and examine residuals.

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Mathematical Modeling 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+c1x 2 +d1x 3 for x [x1,x2] S2(x)=a2+b2x+c2x 2 +d2x 3 for x [x2,x3]

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Mathematical Modeling 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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Mathematical Modeling Projects and Problem Sets More sophisticated problems and projects with real world orientation.

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Mathematical Modeling Simulation Modeling

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Mathematical Modeling Simulation Modeling-A method of last resort!

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Mathematical Modeling Monte Carlo Methods

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Mathematical Modeling Deterministic Methods- Area under non-negative curve

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Mathematical Modeling Y=x^3, 0

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Mathematical Modeling Using Simulation: approximate area is 4.106

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Mathematical Modeling Simulation Comparisons

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Mathematical Modeling More Complicated Problem:

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Mathematical Modeling Area via Simulation

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Mathematical Modeling Volumes via simulation

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Mathematical Modeling Example: Volume of Surface in 1 st Octant

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Mathematical Modeling Stochastic Simulation Labs

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Mathematical Modeling 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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Mathematical Modeling 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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Mathematical Modeling Philosophy 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.”

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