Introduction to Antideviratives as Applied in Biology Louis J. Gross Math 152 - University of Tennessee As we have seen in numerous examples, derivatives are useful in describing the rates of change of various biological processes. We have considered many ways of calculating derivatives of different functions using different "rules". These rules make it simple to determine how rapidly a biological process (photosythesis, pupil diameter, population size) changes as something that affects it is varied (light level for photosythesis and pupil diameter, time for population size). We then saw how to use the information about derivatives to determine whether a measurement (e.g. population size) was increasing, deacreasing, or stable and what the shape was for the graph of the function (concave up or down). The next section of our course deals with the reverse of finding derivatives. That is, given the rate of change of some biological process, how do we how measurements of that process changes through time. So if we are given an equation that expresses the derivative of population size, how can we find what the population size is at any particular time. This procedure of going from the derivative of some measurement to the measurement itself is called "antidifferentiation" or "integration". So, the idea is that if we are given a formula for dN/dt = f(t) where N(t) is population size at time t, and f(t) is a function of time that gives the population growth rate at time t, then we will use the process of antidifferentiation to find N(t). We have seen several examples of going from a differential equation to its solution already. In the case of exponential growth, we saw that if N(t) is popuation size, and dN/dt = kN for some positive constant k, then the solution is exponential growth - that is N(t) = N(0) exp(kt). Similarly, we saw that Newton's Law of Cooling, which expresses the derivative of temperature T of a body as proportional to the difference between the ambient temperatute the body's current temperature has a solution which gives T(t). In both of thee cases we did not show how to find the solution (N(t) or T(t)), but just demonstrated that the formula given was a solution. In this section of the course we will learn some rules to help us find these solutions to differential equations.