Summary: Chapter 4
Applications of Derivatives
In this section we introduce higher order derivatives and study two applications of derivatives. The first, curve
sketching, permits us to draw a picture that helps us understand the shape and behavior of a curve. When this
curve is a mathematical model of a real-word situation, this can be quite useful.
Our second application is optimization, the discipline of finding the largest and smallest value a formula takes
on in a given interval. This will require us to look a bit at what types of intervals there are. This, in turn raises
questions like the following:
What is the largest number smaller than 4?
The problem with that question is that the answer is "there isn't one". Any number y < 4 has another number
(y + 4)/2 between it and 4 and so no largest number is possible.
In the fall of 1972 (United States) President Nixon announced that the
rate of increase of inflation was decreasing. This was the first time a
sitting president used the third derivative to advance his case for reelec-
Mathematics is not a careful march down a well-cleared highway, but
a journey into a strange wilderness, where the explorers often get lost.
Rigour should be a signal to the historian that the maps have been made,
and the real explorers have gone elsewhere.