Tuesday, March 29, 2011

How to do Calculus: The Derivative

Not everyone knows calculus and it's too bad because it's good stuff (as far as math goes). This is my attempt to explain one of the main concepts of calculus in one blog post. Watch this video...it's really boring but only takes a few seconds.

The point makes a bunch of random movements. It moves up, slows down, turns around, slows down, turns around, slows down turns around, et cetera and then it stops at its original position. The graph moving to the left describes the point's position as time changes. Here it is sitting still:

The position increases, flattens out, decreases, flattens out, increases, flattens out, decreases, goes negative, comes back positive, et cetera. It changes just like the point in the video. Let's say that the position is in miles and time is in hours. So in the first two hours the point goes one mile.

Hopefully everything above here makes sense. So far we haven't done any calculus.

The first interesting concept you learn in calculus is the derivative. If you take the derivative of a graph like the one I plotted above, you measure the slope of the graph. In the following video, you would be measuring the steepness of the maroon line as time changes. Notice that the maroon line is always tangent to the position graph.


When you measure the slope you measure how much the line rises divided by how much the line moves forward. You would be dividing vertical distance by time, so the units of the derivative would be miles per hour. So taking the derivative of a position vs. time graph gives you the speed (mph) at all times. You could plot the derivative as a function of time and get this:
If you compare the speed graph with the position graph, you can see that when the position graph is steep upwards the speed graph is high. When the position graph is flat, the speed graph is at zero. When the position graph is steeply sloped downward, the speed graph is very negative.

To convince you that the derivative is speed, here's a video of the position graph with a speedometer!


When the point speeds up, the speedometer needle goes up just like you would hope. When the point goes backward, the speedometer needle goes past 0mph.

Hopefully now you understand what a derivative is and that one application of the derivative is to measure the speed of something.

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