Calculus

My real love in math is for calculus, because I think its one of the most beautiful insights to be found, and its core is simple and intuitive.  If you have taken calculus, and you had a really awesome teacher, you probably agree with me on this, but there's also a good chance you walked away with the impression that calculus is a technical mess involving Riemann sums, limits, differentials and lots of detailed rules.  There is a technical side to calculus, but since I'm not an engineer or mathematician, I'll leave those parts for them--the underlying insights are both more important, and more fun.

The central idea of calculus is this: "you can talk about how quickly something is changing at any given instant, and if you add up all of those little changes, you get the total amount of change."

This is something we're already comfortable with.  When we talk about speed in a car we can talk about average speed, saying "I traveled 70 miles in an hour," but we're more likely to say "at 2:08 pm, as I crossed the state line into Texas, I was in traffic, so I was only going 25 miles per hour."  The important thing to make explicit here is that in the second case, we're not talking about a specific number of miles in a specific amount of time, but instead we're talking about our speed at one instant.

Perhaps our overall travel looks like this, with total distance from home on the Y axis, and time since you started on the X axis.

Distance Traveled

It's worthwhile to talk about how far you went overall and how long it took, but it's also interesting to figure out how fast you were going exactly where that arrow is pointed.  To do that, we can zoom in:

Distance Traveled

and again:

Distance Traveled

Notice that now we have a part of the graph that looks straight--the slope, which here represents the speed of your car, is constant, so it's easy to measure.  You measure distance, divide that by time, and you're done.  Sort of.  This isn't quite the speed at any one moment, instead its the average speed over a very short period of time.  This might not matter practically, but mathematically this is important.  We really want the speed at that instant.  To do this, we imagine what the slope we would measure if we zoomed infinitely close in--this is the derivative, or "instantaneous slope."

Now lets make a new graph--again time is on the X axis, but this time the Y axis is your speed, or the "instantaneous slope" at each moment:

Speed

Notice how the speed starts out high, which matches the steep slope we see at the beginning of the "Distance Traveled" graph above, and then it almost goes to zero where you're stuck in traffic, and then comes back up as the traffic clears.  

That's it for derivatives--at some point I'll write about integrals and how these connect together, but this is more than enough for one sitting.