In this chapter, we will study methods for some of the more practical problems engineers face to optimize use of resources, performance characteristics, or financial gain. In addition, some of the techniques and ideas are relevant to the subject of calculus of variations in Chapter 14. Companies spend millions of dollars on research to save 1% or 2% of the weight of an

automobile or get a 1% or 2% improvement of the fuel efficiency of an airplane engine. Often there is an interplay between numerical methods for modeling and solution of equations and the optimization of a performance characteristic(s).