Unit 7 - Sections 3.8, 3.9, 4.1, 4.2

Module Overview

There is quite a bit of material in this unit, starting with a couple of new applications of the Chain Rule and then beginning to study what derivatives actually tell us about functions. This latter part, the beginning of Chapter 4, will allow us to graph functions, optimize them (that is, find maximum and minimum points), and even set up the concept of integration.

Learning Outcomes

• Section 3.8: The first of two more applications of the Chain Rule, this section explores how derivatives and logarithms interact.  We find the derivative of ln x and also emerge with a new method for integrating functions that would have been impossible at first glance.
• Section 3.9: In this final section of applications of the Chain Rule, we use the method that allowed us to find the derivative of logarithms and extend it to inverse trigonometric functions and even general inverse functions.
• Section 4.1: The idea behind maximum and minimum points is one of the most important in calculus and its applications.  Without the ideas presented here, there would be very few functions where we could figure out their optimal values.  What we begin here, we will extend over the next few sections into an understanding of how to graph functions and how to optimize real-world processes.
• Section 4.2: We begin Section 4.2 with this unit, looking at the uses of the first derivative in our analysis of functions.