Section Summary

In this section we introduced the difference equation $\frac{\Delta P}{\Delta t}=kP$ for modeling natural growth over discrete time intervals and looked at what happens as $\Delta t$ approaches \(0\). In this case we obtain the differential equation as a model for continuous natural growth.

In general, infinitely many functions are solutions of a given differential equation. We can visualize these solutions, even without formulas, by using the slope field. If we specify the value of a solution at one point, that also specifies a unique function in the family of solutions. Our knowledge of derivatives of exponential functions enables us to find formulas for solutions of all natural growth problems.