Chapter Review

Our focus in this chapter is on calculating antiderivatives and using them to find symbolic solutions of initial value problems. We began in Section 1 by rewriting our differentiation formulas as formulas for the calculation of antiderivatives.

After that we picked up where we left off in Chapter 2 with the study of population growth. We saw there that unfettered natural growth is exponential, and we obtained simple symbolic solutions for the corresponding differential equations. For the more complicated logistic models, we could generate numerical and graphical solutions, even though we could not find symbolic solutions until now — here we developed the tools we had previously lacked.

The most important of these tools is the idea of separation of variables in a differential equation. The success of the separation-of-variables technique rests on Leibniz's powerful notation of differentials and on the Chain Rule. The Chain Rule assures us that, after separating variables, it is legitimate to antidifferentiate both sides of an equation, even though the antidifferentiation on one side is with respect to the dependent variable and on the other side with respect to the independent variable.

We closed the chapter with an application of the logistic growth differential equation to Verhulst's model for the population of the United States. In the process, we saw that we could use an intermediate step in the separation-of-variables solution to test whether data fit a logistic growth pattern.