Chapter Review

The unifying theme of this chapter was our discussion of models of natural population growth and the mathematical concepts and tools necessary to understand them. The first of these concepts was the notion of rate of change.

We introduced difference quotients as average rates of change and then considered the limiting values of these rates as the time interval approached zero. This led to the important concept of instantaneous rate of change. In graphical terms, this was the slope of the graph when we zoomed in far enough for the curve to appear to be a straight line. The corresponding mathematical concept was the derivative — one of the two fundamental notions of all calculus.

In Section 2.3 we developed tools for calculating derivatives and used these tools to calculate derivatives of polynomials.

Next, we investigated the family of functions most closely associated with natural growth — the exponential functions $b^t$. We saw that these functions have derivatives that are constant multiples of themselves.

Then we returned to natural population growth. We saw that the appropriate mathematical model in the continuous case was a differential equation. The family of solutions of a differential equation can be pictured using slope fields. To identify a particular solution, we need to specify an initial value. Together, the differential equation and the initial value constitute a differential-equation-with-initial-value problem.

Finally, we developed graphical tools for deciding whether we could reasonably assume that a set of data could be modeled by an exponential function or by a power function. These tools were semilog plots and log-log plots, respectively. In addition, we saw how to determine a model function in each of these two cases.