At the beginning of the term I made the decision / was instructed not to teach the \(\varepsilon\)-\(\delta\) definition of a limit. This class really is directed at engineering and natural sciences majors, so that is, I think, a reasonable (if somewhat disappointing) strategy. Calculus is—in the eyes of the university administration—a cookbook class that teaches a bunch of recipes to non-math majors.
Because I didn’t give a precise definition of the limit, we reached the end of what can easily be proved, and have moved into the realm of learning recipes. Mind you, I’m not complaining—simply pointing out that the tone and content of the course has made a subtle shift that will continue for the next several weeks. We are now in plug-and-chug mode!
What I Taught
We started the day by going over homework problems for a while. This was good and expected—since we are moving into a more computational curriculum, the computational examples and exercises are more important, and more time needs to be spent on them. Because the derivation of trigonometric derivatives took so much time, I had planned to devote a good portion of the day 11 lecture to examples that I didn’t get to on day 10.
The last example I gave was this problem: what is
\[
\frac{d}{d\theta} \sin(\theta^2)?
\]
At this point in the term, the only tools we have to deal with a problem like this are the definition of the derivative (in terms of limits), and a few facts about the limits of trig functions. There were two reasons to give this problem: first, we essentially worked the derivation of \(\frac{d}{d\theta}\sin(\theta)\) for a second time, but in a way that seems novel; and second, it is an interesting problem that motivates the introduction of the chain rule.
So we worked the problem from first principles and arrived at an answer. I then stated the chain rule. While I didn’t prove the chain rule—as noted above, we didn’t really develop the right tools early on—I nodded in the direction of an intuition by way of the Leibniz notation. Essentially, if \(u = g(x)\) and \(y = f(u)\), then we can write
\[
\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx},
\]
where the \(du\) terms “cancel” like they would if the terms were rational expressions. (This is a simplified version of the argument given in class, but contains the basic idea.)
We then worked several examples, culminating in a return to an earlier exercise: what is
\[
\frac{d}{dx} a^x?
\]
From the definition of the natural logarithm and the chain rule, it is possible to find this derivative, which adds one more arrow to our quiver.
What Worked
The computation of \(\frac{d}{d\theta}\sin(\theta^2)\) required two and a half boards worth of work to accomplish when done from first principles, then only a little corner of a board when done with the chain rule. I was quite please with the way that this came together, and with how well this particular example demonstrates the power of the chain rule. I got a few chuckles of agreement, so I guess there were one or two students that felt the same way.
I am also finding that we are moving into a style of give-and-take that I prefer. In an ideal world, I would have more time to teach theory in a more interactive manner, but that isn’t what this class is about, so the theory section goes fast, and is mostly a talking head exercise. Now that we are into a more computationally driven part of the course, I can slow down a little, work a lot of examples, and have the students work examples. This worked reasonably well last night, and should continue to work in the future (I hope).
What Didn’t Work
I was excited to work examples, so I didn’t spend much time explaining where the chain rule comes from nor justifying it in any way. As I said above, I waved my hands in the direction of canceling fractions (even though that isn’t really what is happening), and moved on.
I think that this was the right strategy, but I ended up working through nearly all of my examples with 10 minutes to spare. The students were clearly tired of doing these computations, and were starting to get bored, so I dismissed them. Clearly, I had more time than I thought I would, and could have spent some of it better justifying the main point of the lecture.
I also found myself really annoyed at a student, probably out of proportion to his offense. In differentiating \(\sin(\theta^2)\), we make use of the angle addition formula. I had spent extensive time on that formula in the previous lecture, but one of the students had not attended that lecture, and held up the class to ask about how we got from one step to the next using that formula.
I told him that we had used the angle addition formula, that it should have been taught in a precalculus class, and that if he was confused, he could ask to see someone’s lecture notes from the previous day. I am quite certain that I came across as somewhat hostile, and I regret that. I probably should have just said “That’s the angle addition formula. Talk to me after class,” or something similar.