# 41 Summary

This is the end of our modeling discussion. We started with linear programming problems and discussed both the graphical and tabular simplex methods for finding optimal solutions. We then looked at how to find polynomials that fit a collection of data points so that we could make predictions. We first saw the Lagrangian polynomial, but because this polynomial does not work well on data sets that are not small, we discussed another method, that of finding splines. We saw that we can use linear or cubic splines depending on the data set. We then saw how a Markov chain can be used to predict long-term behavior using probabilities. We also looked at how to use decision trees and expected value to make the best decision when given different options. We concluded with game theory and saw how the outcome of a total conflict can vary depending on the restrictions placed on the game.

Our next topic is on numerical methods. In essence, these are methods that have been devised to approximate answers when exact ones are difficult to find. We will see how to approximate the output of a function and how to approximate an integral. We will also see how to approximate the roots of a nonlinear equation and how to approximate the solution to a system of linear equations.