At the end of last term, I was working on two ideas that are seemingly not related – population growth and the mathematics of music. The population equation was for the MTH 111 College Algebra class and the mathematics of music was for a presentation I plan to give sometime this spring.
What caught my attention about these two subjects was that the equations which describe the most important behaviors in each case – the population equation and logistic equation in the case of population and the wave equation in the case of music – start out as differential equations. That is, they are concerned with change. Calculus allows us to take an equation describing change and transform it into an equation that tells us about the behavior of the system itself.
Why do the equations for these two situations (and many others) begin as differential equations?
Because – change is generally what scientists and researchers can MEASURE and thus have information about. Once the change is measured and described, then the other pieces of the puzzle can be put together.