Teaching an Introductory Course in Mathematical Modeling Using Technology

1. Teaching an Introductory Course in Mathematical Modeling Using Technology

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

3. Agenda • Mathematical Modeling Process • Modeling Course and book • Modeling Toolbox • Explicative Models • Model Fitting • Empirical Models • Simulation Models • Projects and Labs

4. Modeling Real World Systems

5. The Modeling Loop

6. Modeling Process • Identify the Problem • Assumptions/Justifications • Model Design/Solution • Verify the model • Strengths and Weaknesses of the model • Implement and Maintain Model

7. Purpose of Mathematical Modeling • Explain behavior • Predict Future • Interpolate Information

8. 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

9. 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

10. Tool #1 Explicative Modeling • Proportionality and Geometric Similarity Arguments

11. Proportionality

12. Proportionality-Definition

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

14. Geometric Similarity • 1 to 1 relationship between corresponding points such that the ratio between corresponding points for all such points is constant

15. Scale Models

16. Formula’s • Area  characteristic dimension 2 • Volume  characteristic dimension 3

17. Archimedes • A object submerged displaces an equal volume to its weight. • So, under this assumption • W Volume

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

19. Bird Data

20. Dinosaur Data

21. 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.

22. Build the model and compare results

23. Model Fitting • * Linear Regression (least squares) • Minimize the largest absolute deviation, Chebyshev’s Criterion • Minimize the sum of the absolute errors

24. 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.

25. Errors • Least Squares: smallest sum of squared error • Chebyshev’s: minimizes the largest error • Minimizes the sum of the absolute errors

26. 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.

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

28. Simple 1 Term Models

29. 1 Term Models

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

31. 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.

32. Smoothing with Low Order • Use Divided Difference Table for qualitative assessment.

33. 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.

34. 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]

35. Cubic Splines • Natural: end points have constant slope but we do not know the slope. • Clamped: end points have a known constant slope.

36. Projects and Problem Sets • More sophisticated problems and projects with real world orientation.

37. Simulation Modeling

38. Simulation Modeling-A method of last resort!

39. Monte Carlo Methods

40. Deterministic Methods-Area under non-negative curve

41. Y=x^3, 0<x<2

42. Using Simulation: approximate area is 4.106

43. Simulation Comparisons

44. More Complicated Problem:

45. Area via Simulation

46. Volumes via simulation

47. Example:Volume of Surface in 1st Octant

48. Stochastic Simulation Labs

49. 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,…

50. 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.