# Boxing Pythagoras

## Infinitesimal Calculus 2: The Changes in Change

The mathematics of change are quite interesting. In a naive sense, we can often describe a change by a simple collection of data points. For example, let’s think about a little boy rolling a ball across the floor. The boy pushes the ball, and four seconds later, the ball has come to be 2 meters away from him. Given these data points, we may attempt to connect them in some meaningful analytical manner– perhaps by saying that the ball rolled at a speed of half a meter per second. But even this is a somewhat naive bit of information, as it only really tells us something about the completed journey. Mathematicians are greedy, however; they want to be able to know about every point of the ball’s travel, at any arbitrary moment in time.

We can use a function for just such a purpose. A function is a specific mathematical tool which allows us to describe an entire set of data points all at once which we symbolize as $f(x)$ (read “$f$ of $x$“). We encode the data by means of a mathematical formula. For example, our exemplary rolling ball might well have been encoded by the function $f(x)=\frac{1}{2}x$, where the $x$ represents the time, in seconds, that the ball has been rolling, and the value of the function, $f(x)$ tells us the distance in meters which the ball has traveled in that time. In this particular function, the coefficient of $x$ tells us the rate at which distance changes as time passes– that is, $\frac{1}{2}$ a meter per second. When the boy first rolls it, the ball is traveling at $\frac{1}{2}$ a meter per second; when it finishes it had been traveling at $\frac{1}{2}$ a meter per second; and at any single point during the journey the ball is traveling at $\frac{1}{2}$ a meter per second.

However, this is a very simple example. It describes a situation involving a constant velocity. Things become a bit more muddied when the rate at which a change occurs is, itself, changing.