Table of Contents
1. INTRODUCTION TO MAPLE.
The Structure of Maple. A General Introduction to Maple. Maple Quick Review. Maple Training.
2. INTRODUCTION, OVERVIEW, AND THE PROCESS OF MATHEMATICAL MODELING.
The Modeling Process. Illustrative Examples.
3. DISCRETE DYNAMICAL MODELS.
Introduction. Modeling Discrete Change. Tower of Hanoi. Drug Dosage Problem. Time Value of Money. Simple Mortgage. The Spotted Owl. Equilibrium Values and Long-term Behavior. Nonlinear Discrete Dynamical Systems. Growth of a Yeast Culture. Spread of a Contagious Disease. Systems of Discrete Dynamical Systems. Merchants. Competitive Hunter Model. Fast Food Tendencies. Modeling Predator-Prey, SIR, and Military. Predator-Prey. SIR of an Epidemic. Modeling Military Insurgencies.
4. MODEL FITTING CRITERION.
Introduction. Different Curve Fitting Criterion. Plotting the Residuals for a Least-Squares Fit. Bass Fish. Population. Bounding on Chebyshev's.
5. MODELING WITH PROPORTIONALITY AND GEOMETRIC SIMILARITY.
Introduction. Proportionality. Kepler's Law. Bass Fishing Derby. Geometric Similarity. Heart Sizes. Crew Races. Terror Bird.
6. EMPIRICAL MODEL BUILDING.
Introduction. Simple One Term Models. Bass Fishing Derby. Terror Bird Revisited. Fitting an (n − 1)st order Polynomials to N Data Points. Polynomial Smoothing. Cost of a U.S. Postage Stamp. The Cubic Spline Model. Population Fruit Flies. Vehicle Stopping Distance. Cost of a U.S. Postage Stamp.
7. LINEAR PROGRAMMING.
Introduction. Formulating Linear Programming Problems. Product Mix of New Drinks. Financial Planning. Blending. Production Planning. Graphical simplex. CPU Memory Chips. Feasible Region. Minimization Problem. Unbounded Case. Graphical Sensitivity Analysis. The Simplex Method and Tableaus. Linear Programming with Maple. Data Envelopment Analysis. Ranking Banks. Ranking Banks as an LP. Sensitivity Analysis with Maple.
8. SINGLE VARIABLE OPTIMIZATION.
Introduction. Single Variable Optimization and Basic Theory. Applications of Max-Min Theory. Chemical Company. Manufacturing. SP6 Computer Development. Applied Optimization Models. Inventory Problem. Oil Rig Location. Numerical Search Methods. Unrestricted Methods. Dichotomous Search. Golden Section Search. Fibonacci Search. Interpolation Methods.
9. MODELING USING UNCONSTRAINED OPTIMIZATION: MAXIMIZATION AND
MINIMIZATION WITH SEVERAL VARIABLES.
Introduction. Basic Theory. The Hessian Matrix. Unconstrained Optimization. Least Squares. Find the Island. Numerical Search Methods. Gradient Search. Modified Newton's Method.
10. EQUALITY AND INEQUALITY CONSTRAINED MULTIVARIABLE OPTIMIZATION.
Introduction. Equality Constraints: Method of LaGrange Multipliers. Basic Theory. Graphical Representations. Cobb Douglas. Oil Transfers. Inequality Constraints: Kuhn-Tucker (KTC Condition). Spanning Cones. Two-variable Linear. Maximize Profit from Perfume. Minimum Variance of Investment Returns.
11. MODELS WITH LINEAR ALGEBRA.
Introduction. Introduction to Systems of Equations. Models with Unique Solutions Using Systems of Equations. A Bridge Too Far. Leontief Economic Models. Least Squares Revisited. Natural Cubic Splines Revisited. Models with Infinite Solutions using Systems of Equations. Basic Chemical Balancing. Redox Equations.
12. ORDINARY FIRST ORDER DIFFERENTIAL EQUATIONS MODELS.
Introduction. Applied First Order Models. Radioactive Decay. Newton's Law of Cooling. Mixtures. Population Models. Spread of a Disease. Slope Fields and Qualitative Assessment of Autonomous ODEs. Analytical Solutions to First ODEs. Separation of Variables. Linear Equations. Numerical Methods for Solutions to First Order Differential Equations. Euler's Method. Improved Euler's Method. Runga-Kutta 4 Method.
13. SYSTEMS OF LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS.
Introduction. Applied Systems of ODEs and Models. Economic Supply and Demand. Circuits. Competition. Predator-Prey. Diffusion. Insurgencies. Phase Portraits and Qualitative Assessment of Autonomous Systems. Fish Pond. Solving Homogeneous and Non-homogeneous Systems of ODEs with Constant Coefficients. Applied Systems with Maple. Diffusion. Diffusion through Two Membranes. Electrical Circuits. Numerical Methods to Systems of ODEs with Maple. Predator-Prey Model.
14. DISCRETE PROBABILITY MODELS.
Introduction. Introduction to Classical Probability. Reliability Models in Engineering and Science. Overbooking Airlines Model. Markov Chains.
15. CONTINUOUS PROBABILITY MODELS.
Introduction. Reliability Revisited. Modeling using the Normal Distribution. Confidence Interval and Hypothesis Testing. Regression: Linear, Transformed, and Nonlinear.
16. SIMULATION MODELS.
Introduction. Introduction. Monte Carlo Simulation. Deterministic Behavior. Area Problems. Volume Problems. Probabilistic Behavior. Applied Simulation Models. Missile Attacks. Gasoline Inventory.
17. MODELING WITH GAME THEORY.
Introduction. Zero-sum Games. Predator-Prey. Hitter-Pitcher Duel. Non- Zero-sum Games. Nash Arbitration. Illustrative Example: Artist's Guild Strike.