Chapter 14
Optimization and Analysis of Models¹

This chapter is designed to help you take you knowledge of building models to the next level - applying them to solve problems involving questions about optimization. In general, optimization is the process of trying to make something as efficient as possible, or as large as possible or as cheap as possible. It’s the study of minimizing or maximizing a quantity, like profit, as a function of some other quantity, like production. In order to optimize a quantity, though, we need a few things. The first is a skill you already have - the ability to create a model equation that represents how the quantity to be optimized varies as a function of some other quantity. For example, we might produce a model equation describing how the profits of a company depend on the number of items they produce, since the more you produce: (a) the more you can sell, generating more revenue but (b) the more it costs, in labor and materials. The other tool that you need is a knowledge of marginal analysis, which measures how a change in the independent variable will cause a change in the dependent variable in a model. We will focus our study on the marginal analysis and optimization of polynomial models, although this is only the tip of the iceberg.