One of the strengths of calculus is its ability to describe real-world phenomena. We have seen hints of this in our discussion of the applications of derivatives and integrals in the previous chapters. The process of formulating an equation or multiple equations to describe a physical phenomenon is called mathematical modeling. As a simple example, populations of bacteria are often described as “growing exponentially.” Looking in a biology text, we might see \(P(t) = P_0e^{kt}\text{,}\) where \(P(t)\) is the bacteria population at time \(t\text{,}\)\(P_0\) is the initial population at time \(t=0\text{,}\) and the constant \(k\) describes how quickly the population grows. This equation for exponential growth arises from the assumption that the population of bacteria grows at a rate proportional to its size. Recalling that the derivative gives the rate of change of a function, we can describe the growth assumption precisely using the equation \(P' = kP\text{.}\) This equation is called a differential equation, and these equations are the subject of the current chapter.
