March 2002

 

MATH 425  Introduction to Mathematical Modeling (3:3:0)

 

 

Course Syllabus

  1. Graphs and functions as models
  2. Overview of the modeling process
  3. Models using scale and proportion
  4. Models requiring optimization
  5. Experimental Modeling
  6. Monte Carlo and deterministic simulation
  7. Dynamic modeling using the derivative
  8. Discrete stochastic modeling

 

Course Outcomes:

1.       Promote the ability of students to formulate, solve, and assess a variety of mathematical models, arising from verbally described problems in fields such as agriculture, astronautics, biology, business, chemistry,  ecology, economics, forestry,  medicine,  sociology….

2.       To have students understand and unite, diverse mathematical concepts introduced in previous mathematics classes that lie latent, and awaiting discovery in an often non-mathematical context.  These include:

·         Influence diagrams for structural relationships and functional dependencies (Introduction to Computer Science and Discrete Math).

·         Symbolic optimization of univariate, differentiable functions (Calculus 1).

·         Lagrange multipliers with equality and inequality constraints for unconstrained multivariate optimization (Calculus III).

·         Newton’s method applied to univariate (Calculus I) and multivariate optimization.

·         Optimization by Linear Programming (Linear Algebra). 

·         Optimization of non-linear, non-differentiable, functions by search techniques (Discrete Mathematics & Introduction to Computer Science).

·         Recursive and eigenvalue methods in discrete time dynamical systems (Discrete Mathematics and Linear Algebra). 

·         Phase Portraits and eigenvalue methods in continuous time dynamical systems. (Calculus III and Linear Algebra)

·         Discrete Simulation ( Introduction to Computer Science), 

·         Discrete Monte Carlo Simulation (Statistics I).

·         Markov Chains (Discrete Math, Linear Algebra, and Statistics I).

·         Least Squared criteria and Linear Regression (Statistics I ,Linear Algebra, and Calculus I–III)

 

3.       To have students understand how the choice of measurement scales (linear vs. non-linear, continuous vs. discrete), affect the choice of a modeling strategy, extend the range of a model via transformations, affect the numerical stability of a model, or leave some aspect of the problem invariant.

4.       To have students employ geometric concepts for graphical visualization, for the exploration of relationships, for intuition and understanding, and for obtaining quick, approximate solutions or estimates.

5.       To have students become aware of and to gain ability in exploiting the more subtle geometric aspects of a problem that remain invariant under a group of transformations.

6.       To use computer software such as Mathematica as tool for problem solving, modeling, symbolic manipulation, number crunching, visualization, and reporting.

7.       To enable prospective high school teachers of mathematics to better appreciate the use of mathematics as an applied discipline. To enable them to better understand, help and advise their future college bound students, who do not plan to major in mathematics, of its importance, power, and relevance within their chosen field of endeavor.