1.3 Fundamentals

The inverse operation to differentiation is called antidifferentiation or, more commonly, integration. In models of exponential growth we assume that the growth rate is proportional to the population size.

\[\begin{equation} \frac{dN}{dt} = kN \tag{1.1} \end{equation}\]

where \(N\) is the population size, \(t\) is time and \(k\) is a proportionality constant. We could solve for \(N\) as a function of time by integrating the derivative. However, we know that the exponential function is the only function which equals its derivative.

\[\frac{d(e^t)}{dt} = e^t\]

Thus, we can modify the function to give the solution

\[N = N_0 e^{kt}\]

since differentiating \(N_0 e^{kt}\) gives eqn. (1.1).

\[\frac{dN}{dt} = \frac{d(N_0 e^{kt})}{dt} = kN_0 e^{kt} = kN \]

If we write eqn. (1.1) as

\[\begin{equation} \frac{dN}{dt} = kN_0 e^{kt} \tag{1.2} \end{equation}\]

then the “antidifferentiation” becomes more obvious: find \(N\) so that its derivative equals \(kN_0 e^{kt}\). If we define

\[g(t) = kN_0 e^{kt}, \;\;\;\;\;f(t) = N_0 e^{kt} \]

then eqn. (1.2) becomes

\[g(t) = df/dt\]

Thus \(g(t)\) is the derivative of \(f(t)\) and, conversely, \(f(t)\) is an antiderivative of \(g(t)\). But there is a slight problem. The antiderivative is not unique. If we define

\[x(t) = N_0 e^{kt} + 10, \;\;\;\;y(t) = N_ e^{kt} +354 \]

then \(x(t)\) and \(y(t)\) are also antiderivatives of \(g(t)\), as can be checked by differentiating: \(dx/dt = g(t), dy/dt = g(t)\). One interpretation of this is that the graphs of \(x(t)\) and \(y(t)\) have the same slope (derivative) for any given value of \(t\). Since the functions differ by only a constant, we can write the general form of the antiderivative as \(f(t) + C\) where \(C\) can be any constant. We usually write the antiderivative as the indefinite integral

\[\begin{equation} \int g(t)dt = f(t) + C \tag{1.3} \end{equation}\]

and call \(C\) the integration constant. Recall that the term “\(dt\)” identifies the variable of integration just as it does the variable of differentiation in the derivative \(df/dt\).