Chapter 7: Related Rates and Integration Tools

When one measurement in a situation changes, other measurements are usually forced to change. For example, when liquid is poured into a glass, not only is the volume of liquid changing, but the height of the liquid in the glass must also change in a related manner. In this chapter, we will study how various rates in a situation are related and how to use one rate to solve for another.

Up to this point, integration has been limited to polynomial functions and recognizable functions such as y=sin⁡(x). In this chapter you will develop a substitution method for integrating more complex functions.

We will also apply the use of integration to solve equations involving derivatives, called differential equations. You will learn to graph representations of these differential equations as “slope fields.” These slope fields will help us visualize the solutions of differential equations.

The chapter concludes with advanced strategies for graphing functions (Euler’s Method) and integration (integration by parts and partial fraction integration).

Chapter Goals

Express rate of change scenarios
symbolically and solve a variety of
related rate problems.

Learn the u-substitution method for
integration.

Use integration to solve differential
equations.

Use slope fields and Euler’s
Method to graph differential
equations and their solutions.

Learn to integrate using integration
by parts and partial fractions.

Chapter Outline

Section 7.1

You will describe the relationship between rates of change for different scenarios. You will write related rates statements and progress to solving problems involving related rates.

Section 7.2

You will learn the u-substitution method for integration, which will enable you to easily integrate a wider variety of functions. This involves finding patterns and undoing the Chain Rule.

Section 7.3

You will learn how to use implicit integration to solve special equations involving derivatives, called differential equations. You will graph these differential equations and their solutions using slope fields and Euler’s Method.

Section 7.4

BC Section: You will learn Euler’s Method of approximating the shape of a curve. You will learn more integration techniques: integration by parts and partial fractions.