Lucky you! Two calculus posts in one day. Mainly because I don’t want some of these ideas to disappear in my hiatus from teaching it. This one deals with our favorite topic: the formal definition of the derivative.
I see that expression and my mind goes to the following places:
- Doing a bunch of tedious algebraic calculations for a particular function in order to find the derivative.
- I “see” in the expression the slope of two points close together.
- I envision the following image, showing a secant line turning into a tangent line
And I think for many teachers and most calculus students, they think something similar.
However I asked my (non-AP) calculus kids what the stood for. Out of two sections of kids, I think only one or two kids got it with minimal prompting. (Eventually I worked on getting the rest to understand, and I think I did a decent job.) I dare you to ask your kids and see what you get as a response.
What I suspect is that kids get told the meaning of and it gets drilled into their heads that they might not fully understand what algebraically is going on with it.
It was only a few years ago that I came to the conclusion that even I myself didn’t understand it. And when I finally thought it all through, I came to the conclusion that all of differential calculus is based on the question: how do you find the height of a hole? I started seeing holes as the lynchpin to a conceptual understanding of derivatives. I never got to fully exploit this idea in my classes, but I did start doing it. It felt good to dig deep.
The big thing I realized is that I rarely looked at the formal definition of the derivative as an equation. I almost always looked at it as an expression. But if it’s an equation…
… what is it an equation of? An equation with a limit as part of it?! Let’s ignore the limit for now.
Without the limit, we have an average rate of change function, between and . And since we have removed the limit, we really have a function of two variables.
We feed an and into the function, and we get an output of a slope! It’s the slope between and !
Let’s get concrete. Check out this applet (click the image to have it open up):
On the left is the original function. We are going to calculate the “average rate of change function” with an x-input of 1.64 (the x-value the applet opens up with).We are now going to vary h and see what our average rate of change function looks like: . That’s what the yellow point is.
Before varying h, notice in the image when h is a little above 2, the yellow “Average Rate of Change” dot is negative. That’s because the slope of the secant line between the original point and a second point on the function that is a little over 2 units to the right is negative. (Look at the secant line on the graph on the left!)
Now let’s change h. Drag the point on the right graph that says “h value.” As you drag it, you’ll see the second point on the function move, and also the yellow point will change with the corresponding new slope. As you drag h, you’re populating points on the right hand graph. What’s being drawn on the right hand graph is the average rate of change graph for all these various distances h!
Here’s an image of what it looks like after you drag h for a bit.
Notice now when our h-value is almost -3 (so the second point is 3 horizontal units left of the original point of interest), we have a positive slope for the secant line… a positive average rate of change.
The left graph is an graph (those are the axes). The right graph is a $h-AvgRateOfChange$ graph (those are the axes).
Okay okay, this is all well and dandy. But who cares?
We may have generated an average rate of change function, but we wanted a derivative function. That is when h approaches 0. We want to examine our average rate of change graph near where h is 0. Recall the horizontal axis is the h-axis on the right graph. So when h is close to 0, we’re looking at the the vertical axis… Let’s look…
Oh dear missing points! Why? Let’s drag the h value to exactly h=0.
The yellow average rate of change point disappeared. And it says the average rate of change is undefined! 0/0. We have a hole! Why?
(When h=0 exactly, our average rate of change function is: which is 0/0. YIKES!
But the height of the hole is precisely the value of the derivative. Because remember the derivative is what happens as h gets super duper infinitely close to 0.
We can drag h to be close to 0. Here h is 0.02.
But that is not infinitely close. So this is a good approximation. But it isn’t perfect.
And this is why I have concluded that all of differential calculus actually reduces to the problem of finding the height of a hole.
Here are three different average rate of change applets that you might find fun to play with:
one (this is the one above) two three
In short (now that you’ve made it this far):
- Look at the formal definition of the derivative as an equation, not an expression. It yields a function.
- What kind of function does it indicate? An average rate of change function. And in fact, thinking deeply, it actually forces you to create a function with two inputs: an x-value and an h-value.
- Now to make it a derivative, and not an average rate of change, you need to bring h close to 0.
- As you do this, you will see you create a new function, but with a hole at h=0.
- It is the height of this hole that is the derivative.
PS. A random thought… This could be useful in a multivariable calculus course. Let’s look at the average rate of change function for :
Let’s convert this to a more traditional form:
Now we have a function of two variables. We want to find what happens as h (I mean y) gets closer and closer to 0 for a given x-value. So to do this, we can just visually look at what happens to the function near y=0. Even though the function will be undefined at all points where y=0, visually the intersection of the plane y=0 and the average rate of function should carve out the derivative function.
If this doesn’t make sense, I did some quick graphs on WinPlot…
This is for . And I graphed the plane where y=0. We should get the intersection to look like the line .
I did it for also… The intersection should look like .