The big event of the day was returning exams. I’m not sure why, but I always feel kind of uncomfortable when I hand back exams to students that performed poorly. I’m note sure what to do with that…
Anyway, the exam is over and done. On to the next thing!
What I Taught
The goal of the lecture was to determine the derivatives of trigonometric functions. We started with the sine function, from which the rest follow without too much difficulty.
Computing the derivatie of \(\sin(\theta)\) requires a bit of work. Working directly from the definition and using some algebra and trig identities, the problem can quickly be reduced to
\[
\frac{d}{d\theta}\sin(\theta)
= \lim_{h\to 0} \sin(\theta) \cdot \lim_{h\to 0}\frac{\cos(h)-1}{h} + \lim_{h\to 0}\cos(\theta) \cdot \lim_{h\to 0}\frac{\sin(h)}{h}.
\]
The \(\sin(\theta)\) and \(\cos(\theta)\) terms are fairly easy—both are constant with respect to \(h\), so the limit is essentially irrelevant. The remaining two terms are a bit more problematic.
We tackled the \(\sin(h)/h\) term first. The trick here is to go way back to the definition of sine in terms of both the unit circle and right triangles. After a fair amount of work (most of which is outlined in any introductory calculus text), we can bound the expression \(\sin(h)/h\) from above by 1, and from below by \(\cos(h)\). Applying the Squeeze Theorem to those bounds, we can show that
\[
\lim_{h\to 0}\frac{\sin(h)}{h} = 1.
\]
While we were motivated by a desire to differentiate the sine function, this fact proves to be rather interesting and useful all by itself, so I encouraged the students to highlight it, draw a circle around it, and otherwise keep it in mind for the future.
It is possible to use the Pythagorean Theorem (i.e. \(\sin(\theta)^2 + \cos(\theta)^2 = 1\)) and our newly discovered limit to establish that
\[
\lim_{h\to 0}\frac{\cos(h)-1}{h} = 0.
\]
Combining all of these, we finally have that
\[
\frac{d}{d\theta} \sin(\theta) = \cos(\theta).
\]
With both the derivative of the sine function and the derivation of that derivative in place, the rest of the trig functions fall pretty quickly. By a similar line of reasoning,
\[
\frac{d}{d\theta}\cos(\theta) = -\sin(\theta).
\]
As the remaining trig functions (\(\tan(\theta)\), \(\csc(\theta)\), \(\sec(\theta)\), and \(\cot(\theta)\)) can all be written in terms of sines and cosines, the rest of the derivatives can be reduced to applications of the quotient rule. I quickly worked through the details for the tangent function, but left the rest to the students, should they feel so inclined, and simply gave them a table of derivatives.
We worked one example of a limit problem for the rest of class, then went home for the night.
What Worked
While the argument is convoluted, the basic ideas are pretty easy. The whole derivation requires the use of a trig identity (the angle addition formula), the Pythagorean Theorem, some simple algebra, the definitions of some trig functions (remember “SOH CAH TOA,” anyone?), and the Squeeze Theorem. All of these are things that students have seen before (in principle), and with which they should be familiar.
The only issue is getting them all to fall into place correctly. Overall, I feel like I got that part of the task to work. I think that the argument was well-made and well-presented. Perhaps not perfect, but pretty damn good. I am proud of the presentation that I gave.
That said, it really was just a presentation…
What Didn’t Work
This was not a super-dynamic lecture. The arguments were pretty complicated, and many of the steps seemed unmotivated. Given enough time, I think that it would be possible to motivate each step—in fact, I saw it done last year in an honors calculus class, where it took three days—but I really wanted to get to the punch-line in a timely manner. This means that the lecture was almost entirely my talking, with little opportunity for questions or feedback.
I don’t like lecturing like that. I have difficulty determining what students are understanding, and I know that they are scrambling madly to get everything copied down into their notes, probably without comprehending half of what is going on. I planned the lecture this way, and I don’t see a better way of structuring this particular argument in this particular environment (yay for fast moving summer classes!), but it is still somewhat frustrating.
Aside from that, my only real issue was board space. I have three blackboards at the front of the room, and was really hoping to get the entire derivation of the derivative of sine onto those boards all at once. Unfortunately, even though I had planned out how to do it, my handwriting was too large and the figures that I drew were poorly placed and took up too much space. I ended up (barely) establishing the first limit on all three boards (squishing the result into the corner), then erasing the middle board in order to finish the derivative.
Clearly, I need to work on writing smaller on the board. The math didn’t fit today, and my thesis advisor railed against my large handwriting during my defense (well, prior to, at any rate). On some level, I think that I am worried that no one will be able to read what I have written, but I know that I am overcompensating. I mean, it is a small classroom, and when I check my own work from the back of the room, I can clearly read what I have written, but there is still that little bit of irrational worry that it isn’t big enough. Bah!