Calculus I—Day 6

Before the term began, I was essentially handed a calendar of sections from the text that needed to be covered on which exams. Since my credentials are newly minted and I have not taught the class before in a university setting, I opted to follow this calendar fairly rigorously. This means that we are coming up on the first exam. At this moment, it feels like a fairly disjointed affair—most of the exam deals with limits (both on their own, and as they are connected to derivatives). However, two sections from the following chapter also appear on the exam, namely those dealing with some rules for computing derivatives. I’m not super excited about this, and will have to remember to adjust things in the future.

Anywho, onto the daily naval gazing:

What I Taught

Having given formal definitions of the derivative in the previous lecture, we began figuring out some rule that would make computing derivatives easier. Because high school algebra focusses so heavily on polynomials (and because polynomials are such a rich topic anyway), I tried to motivated the lecture in terms of finding the derivatives of polynomials (this was a mixed bag, as noted below).

I started by asking what the derivative of a constant should be. There were a couple of wild guesses (some correct, some not—many of my students have seen the material before, so it is unsurprising that there would be correct answers), but no one knew what the derivative of a constant was, nor could anyone justify their answer. So we found an answer by going back to the definition (a theme for the evening).

Using the definition of the derivative, it is pretty easy to see that \(\frac{d}{dx} c = 0\) for any constant \(c\), though there was one detail that tripped a few people up. In the derivation, the following identity appears:
\lim_{h\to 0} \frac{0}{h} = 0.
Many students felt that this was incorrect, and that this limit should be undefined. Clearly, as \(h\) tends to zero, the fraction becomes \(0/0\). We discussed the fact that if a limit looks like it is going to be undefined, we should find a way to do some algebraic simplification, first. This was the way that the topic was originally presented, but the current limit is simpler than most of the ones we were working with before, and that was only last week—the students haven’t really had time to completely internalize anything.

Next, I had my students work out \(\frac{d}{dx}x\) and \(\frac{d}{dx}x^2\) by hand while I wandered around the room and offered advice. I then found the derivative of \(x^3\) on the board (the computation is similar, but more tedious, and I didn’t want to give too much time to it), and asked if anyone though they saw a pattern. Of course, most of the class immediately cottoned-on to the Power Rule (again, several probably already knew it), which I then set about to prove.

This turned out to be far more time consuming than I anticipated. When students enroll in calculus, they are expected to have passed algebra and precalculus classes, which generally include at least a discussion of the Binomial Theorem. I had assumed that my students would have at least passing familiarity with the theorem and notation. This assumption was met with a lot of blank stares.

So I gave an short, impromptu introduction to the Binomial Theorem. I very quickly wrote down the result, explained the notation as best I could, gave a quick example, and demonstrated how the binomial coefficients could be found using either factorials (anther bit of notation that I didn’t expect to have to explain) or Pascal’s triangle (which I would not have mentioned, except that one of the students was familiar with the connection, and really likes the triangle).

After finishing this tangential discussion, we went back to the Power Rule, which we were able to prove for integer powers. I waved my hands a little, and asserted that the Power Rule could be extended to rational powers fairly easily, but that real powers required some more machinery. This led to a fairly nice little back-and-forth about the “rules” being “theorems” in their own right, which have hypotheses and proofs.

The last two differentiation rules for the day (multiplication by a constant, and summing functions) went much faster, as the proofs are relatively easy. At this point, we had all of the tools necessary to compute the derivatives of polynomials, which we did for a few cases.

I finished the lecture by attempting to find the derivative of a power function (i.e. one of the form \(f(x) = a^x\) for some constant \(a\)). By the end of class, I had gotten as far as showing that, with \(f\) as above, \(f'(x) = a^xf'(0)\). I had wanted to use this as an opportunity to properly define the constant \(e\) and show why \(\frac{d}{dx}e^x = e^x\), but there was literally no time left, so I simply asserted the fact, and promised to prove it next time.

Oh, yeah. There was a quiz that didn’t get taken. No major loss there.

What Worked

As much as it bothered me to spend 20 minutes on the Binomial Theorem (thus costing me part of my lecture on exponential functions and a quiz), I don’t think that it was a waste of time. In fact, I think that it was good to go “off-script” for a bit and talk about something more extemporaneously. I make more mistakes and am probably a bit less coherent, but I think that I am probably more interesting to watch when I’m BSing my way through interesting material, and I think that the process of doing mathematics on the fly is an important one to observe and learn. It wasn’t what I wanted to do, but I think that it was (mostly) time well-spent.

I am also really glad that I made the time to have students do some work on their own. I know that this is good pedagogy—one learns better through doing than observing. While the students were working, I managed to catch a lot of issues that would have plagued people as they tried to work more complicated problems down the line, and I am sure that the students benefitted from the hands-on approach.

What Didn’t Work

There are lots of things that I could have done better. Some of them are minor, and fit into a nice little list:

  • The graph of a constant is a horizontal line, which has a slope of zero everywhere. The derivative is very easily understood geometrically. I should have mentioned this.
  • Never assume that student know anything. Ever. I wish that I had prepared a better lecture on the Binomial Theorem (or, at least, had been prepared to assert it without discussion). We spent too much time on it. Not entirely unproductively, I think, but it wasn’t what I wanted to do. Had I thought it was going to be an issue, I could have been more efficient about it.
  • Don’t get hung up on notation. I started using the Leibniz notation when it became inconvenient to “prime” things. I probably could have done this without explanation—we’ve already seen the notation and worked with it a little—but I blabbed on about it, anyway, probably leading to some confusion.
  • Slow down, slow down, slow down. The silence after I ask a question is at least as disconcerting for the students as it is for me. And it really isn’t as long as it feels. Slow down.

A little more important way my (failed, I think) attempt to motivate the lecture. I wanted to focus on polynomials because I find them interesting, and because I had assumed that they would be familar to my students. Ultimately, I projected my interest onto my students, which led to the problem not really feeling authentically motivated for them.

Next time I present the topic, the motivation needs to be better. Perhaps I can open with a problem? For instance, “What is the derivative of the function \(f(x) = 3x^2-5x+10\)?” Such a problem is possible to work from the definition, but tedious enough to motivate the introduction of rules for finding derivatives.

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