Blog Post #4: Rate of Change

Good morning class, My name is Professor Moran and today I am going to teach you the mathematical concept of Rate of Change. Now, let us begin.


Before we can discuss Rate of Change, we must first consider what "Rate" in itself really is. Rate is the value of one amount/the value of another amount.

The definition of Rate of Change is as follows: The speed at which a variable changes over a specific period of time.

Now, rate of change can be used  to solve and explain various real life scenarios. Most commonly, rate of change is used when discussing momentum, and it generally is depicted mathematically as a ratio between a change in one variable relative to a corresponding change in another.

Rate of Change in essence, is a rate that describes how one quantity changes in relation to another quantity.

Lets make an example on a graph to illustrate this concept.

By looking at this graph even before considering Rate of Change, we can deduce that the x-axis represents time, in the unit of seconds and that the y-axis represents elevation, in the unit of feet. 

This graph represents the Rate of Change of a helicopter's elevation over the first 20 seconds of flight from initial takeoff. 

At "O" seconds, the helicopter is stationary on the ground. 

After five seconds, the graph tells us that the Helicopter has risen to an elevation of 50ft. 

At 10 seconds, the graph tells us that the Helicopter has risen to an elevation of 100 ft....And so on.

Now, here is where we see Rate of Change come into play. As we said in the beginning of the lesson, Rate of Change is the speed at which a variable changes over a specific period of time. In this example on the graph above, the Rate of Change is consistently changing by the same amount each interval on the graph. At "0 seconds" we see the helicopter is at "0ft." Then at "5 seconds" we see by the graph that the helicopter is at "50 feet." And so on. 

Now, one might ask, is calculating ROC always this easy?! Or, is there a shortcut to calculating ROC when the numbers provided aren't as pleasant? 

YES!!!

There is one simple equation that allows mathematicians to skip analyzing graphs and find the ROC quickly and easily.

The ROC equation is quite simple. In words, simply take any "Y" value on the graph (besides the original) and subtract it by the "Y" value that comes right before your original "Y" value. Then, divide this number by any "X" value subtracted by the "X" that comes right before your original "X" value it. 

Why does this equation work you might ask??

The ROC formula works because it simply determines the vertical change of a graph divided by the horizontal change of a graph. 

All together, the ROC formula looks a little like this when all things are considered. 

                               Vertical Change   = Y2-Y1  = Rate of Change

                             Horizontal Change    X2-X1

This concludes today's lesson, class! See you all tomorrow!