How I Teach Calculus: A Comedy (Natural Exponent)

This post is a part a larger series doc­u­ment­ing the changes I am mak­ing to my cal­cu­lus course. My goals are to imple­ment standards-based grad­ing and to intro­duce gen­uine appli­ca­tions of the con­cepts being taught. I’m not suf­fer­ing any delu­sions that any of this is all that ground-breaking, I just want to log the com­edy that ensues:

Ah, the number e. Mystical, silly, and sublimely ubiquitous in math and science. Euler had his way with this number, and now we get to reap the benefits of the progeny. How do I teach about it to my calculus students that have absolutely no concept of this magic number and its monolithic importance? Well, I sure as hockey sticks don’t start off with the mysticism. That’s a sure fire way to make something seem inaccessible.

My goal is to present exponential functions as a separate beast from the polynomials these kids already know and love. The difference really stems from the following differential equation. I’m going to work the whole thing out for those of you who are here to learn content:


The solution to this equation is the natural exponential function. What does this mean? There’s more to that original diff eq than meets the eye. It’s actually a robot! No, wait, it says something very important in English, actually. It says, “The more of something I have, the faster that something gets bigger (or smaller).” Or, every politicians favorite phrase, “A slippery slope.” That’s quite different from any polynomial. That simple sentence describes a whole slough of physical phenomena. It is also the hardest for kids to grasp. This is where I must begin.

They want to say that anything that is increasing and concave up is “exponential.” This is just not the case. Things that go faster because there are more of them are exponential. Things like bacteria, things like radioactive nuclei, things like rabbits. There’s a fundamental connection between this idea and the real world here that cannot be ignored in a calculus course (but often is, sadly).

Do I begin with that derivation I just presented? No.  Hell no. I believe you’d get your ass kicked for sayin’ something like that.

I believe in hooks. I believe in motivation (link). I believe in comedy. The process of teaching about e is pretty messy, and I’m having a hard time organizing it into a blog post. So, I’m just going to shotgun it at you guys and you can pick out what you like, or just keep making fun of me from your mom’s basement, whatev.

Ball Drop, Yo!

Here’s the first thing we did. This was inspired from the cover of an old physics book that looked like this:

tennis ball gets lower each bounce

The kid asked, “How high is each bounce?” I tucked this one away for the moment that my HIGH SPEED VIDEO CAMERA arrived:

We spent a decent amount of time doing some video manipulation. We put grids on things. We measured fingers for scale. I have another version of the video with a meter stick and a stopwatch in frame, but I think that it takes away from the thought necessary to extract data from the video. An exponential function can fit the max heights of the bounces in time. Why? I’m not totally sure, but I bet it has something to do with the fact that the amount of energy lost per bounce is directly related to how much energy is in system at any given time (see diff eq above).

How Dead Is It?

I spent a week in the high Wyoming desert plateaus a few summers ago, and I found this:

Carbon dating experiment ensues. I’m perfectly aware that getting a good read on the amount of C14 left in an object requires a giant pile of lab equipment. But all we’ve got is a Geiger counter and some bones. C14’s main decay mode is through beta decay which creates inert N14. Setting up this experiment and learning about Carbon dating took about 3 days.

Taking data from our bones was even worse. We set up a giant metal box and put as much lead and bricks around it as possible. We measured for a few days in order to try and figure out the number of counts coming from our bones. The data was fairly worthless, but the exercise itself was quite valuable. I’m sure you’re already teaching Carbon dating in calculus, so I won’t belabor this point. I would suggest picking up a Geiger counter on the cheap via ebay.

In the end I teach the natural exponent just like everyone else. (… the only base that gives a slope of 1 at x=0…) Some good solid board work and comedy. The calculus comes in when we attempt to discuss how the rate of a change of these phenomena behave. The answer can take you a couple of different places:

The Graphical

This is probably the easiest way for kids to get this. Draw Exp[x] and put some tangent lines on it:

Now make a graph of those slopes as a function of x:

The same graph! Therefore,

Infinite Series:

If you teach calculus, you’ve probably taught about infinite series. Here’s the most fun one:

Carry that out for a few terms. Now take the derivative of it. Yup, the same thing. This must be Exp(x). (OMG!)1 This one blows their minds a bit. It’s a bit Mr. Wizardy, but remember I’m an opportunist when it comes to instruction. There’s no pedagogical dogma here. Kids need direct instruction? Do it. Kids need inquiry that day? Do it. Whatever grabs a kid and makes their learning extensible is what I’m going to do.

Cut Your Teeth:

I use these as bell-ringers <digress>I wish there were a less teachery word than that. “Warm Up” maybe? “Stretches”, “Exposition”</digress> These get at the concept of lograithmic differentiation, too. Which is a separate concept I leave until later so as not to muddle the tykes.

Find the extrema of the following functions (these aren’t new; they are under the sun after all):2

1. This also results in a sweet discussion about factorials. They represent how many times you can arrange that many unique objects. That explains why 0! = 1. You can order zero things in precisely one way! I didn’t know this until embarrassingly late into my education.

2. Answers: x=1/e. x=e.

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