Section Summary

The real point of this section is not so much how to find roots — you already know how to do that with your calculator or a computer algebra system. Rather, the point is the continuing theme of local linearity of the well-behaved functions that we use for most of our models. We used that concept to define the derivative in Chapter 2 and to visualize differential equations via slope fields in Chapter 3. We have used it here to explain the principle of a fast, accurate, automatic root-finding method — which might be what your Solve or Root key does. We will use local linearity again in the next section to derive a new differentiation formula. And we will use it in the next chapter to generate solutions of initial value problems. That will not be your last opportunity to put this important concept to work — the essence of differential calculus is linear approximation of small segments of curves.