Welcome to week 8 of Science Friday! Last week, we drew insight on the field of calculus by discussing derivatives, limits, and tangent slopes. The natural question that begs to be answered is what is the practical application of these abstract concepts. Calculus has a variety of applications in a number of fields—economics, physics, chemistry, and engineering to name just a few. The concept of the derivative can be used to describe how systems change and therefore predict the future condition of those systems. In algebra, any changes were constant—the slope of a line, the speed of a car. Yet, in real life, change is not constant, and most processes are not linear. Differential calculus provides a powerful tool to model these practical situations. In order to convert these abstract ideas into applications, let’s consider a very simple example. The position of a car along a road is given by a model, s(t) = 6t^2 + 2t + 1, where t is measured in minutes and s is measured in feet. Given this information, we need to determine the speed of the car precisely at the five minute mark. In algebra, we could use the formula speed = distance / time. The problem here is that this formula only provides the [i]average[/i] speed of an object over a given time interval, not the precise speed of an object at an instant. However, we do know that the rate of change of position, that is, the [i]derivative[/i] of position, is velocity. We need to differentiate the position function to obtain the velocity function and then plug in t = 5. If s(t) = 6t^2 + 2t + 1, then s’(t) = v(t) = 12t + 2. The process by which I differentiated the position function was by the power rule, a shortcut to differentiating power functions. The limit definition of the derivative could also be used, but shortcuts, as their descriptor would imply, are much faster methods of receiving the same result. Now we simply plug in t = 5 to v(t), and we get 12(5) + 2 = 62 feet per minute (a very slow car). If this example seems contrived, that is because it is [i]to some extent[/i]. In order to truly exploit the power of calculus, we must combine the ideas of derivatives with the concept of [i]integrals[/i]. While these two tools in calculus are often taught in separate classes, we will see another time how they are inextricably linked. I hope you enjoyed this short Science Friday. I will return to integrals another week. While I know this causes a break in my lineup, it is best for those readers who may not be entirely familiar with the prerequisite knowledge required to fully grasp the ideas of calculus. As a result, I will return next week to to a specific scientific discipline once more. Stay tuned!