Calculus I—Day 21

I feel like I have been really productive so far this week. I have most of a study guide written for the next exam, the majority of the next exam is done, and I have lecture notes ready until the end of the week. On the other hand, I have so many balls up in the air right now (see above) that it is a little overwhelming.

What I Taught

I finished the introduction to curve sketching by discussing and defining concavity, and providing a test for concavity using the second derivative. This allowed us to graph the function \(g(x) = x+\cos(x)\), an example that we started on Monday, but didn’t have time to finish. After sketching the graph on the board, I pulled up Maple to get a more accurate picture of the curve.

Unsurprisingly, my sketch looked different from what Maple produced, which turned into a fruitful discussion about human error and the limits of human ability. I was able to emphasize the point that the sketches that we produce will highlight the important features (i.e. the value of the function at important points and the general shape of the graph), but it will necessarily contain errors.

I had meant to introduce the Second Derivative Test (for local minima and maxima), but I seem to have forgotten. I’ll have to go over that today.

I next returned to the topic of limits. I noted that if we blindly apply limit laws, we can get a host of indeterminate forms like \(0/0\) and \(1^\infty\). In many cases, we can actually evaluate such limits, using (the incredibly useful) l’Hopital’s Rule.

I stated the rule (very carefully, I might add—I really like Stewart’s book, but his statement of l’Hospital’s Rule is somewhat confusing), then worked several examples of limits that were of the forms \(0/0\) and \(\infty/\infty\). This included a non-example where applying l’Hospital’s Rule causes trouble (i.e. the problem doesn’t satisfy the hypotheses of the rule).

I finished the lecture with an example of a limit of the form \(0\cdot\infty\), and noted that the way to deal with these is to first transform it so that it looks like something that l’Hospital can eat, then apply l’Hospital’s Rule.

I did not have time to give further examples with other indeterminate forms, so I will address those tonight.

What Worked

Aside from 25 minutes spent introducing indeterminate forms and stating l’Hospital’s Rule, most of the class was spent working examples. The material has definitely moved from mostly theoretical in nature to mostly computational, which (I think) makes my students very happy and which (definitely) makes it easier to work lots of examples.

I was also immensely pleased with the discussion generated by the comment that my sketch didn’t look like what Maple produced. I had not expected to have the conversation (it hadn’t even really occurred to me), but it was a grave concern for one student. Several other students contributed, and we had a very positive discussion of what we, as people, can and should be able to do. More importantly, I think I was able to answer the question of why we even bother with curve sketching, when a computer can do it better and faster (the computer may be wrong due to floating point errors, or we might have input the wrong thing, for instance).

What Didn’t Work

I forgot to state the Second Derivative Test. It was the last thing on a page of notes, and I just skipped over it. Not a big deal. I can get to it first thing today. It will be a little out of context, but I think that should be okay. Or perhaps I can introduce it as part of the general outline for curve sketching? Something to think about…

The bit that really fell flat was my non-example. There were a couple of problems. The problem was “Find the limit \[
\lim_{x\to\pi^{-}} \frac{\sin(x)}{1-\cos(x)}.\text{”}
\] I wanted to mistakenly charge ahead with l’Hospital’s Rule without first checking the required hypotheses, and show how this led to an error. However, I think that I might have charged ahead too quickly: I wrote \[
\lim_{x\to\pi^{-}} \frac{\sin(x)}{1-\cos(x)}
= \lim_{x\to\pi^{-}} \frac{\cos{x}}{\sin{x}}
= \frac{-1}{+0}
= -\infty
\] on the board, assuming that my running commentary would keep everyone on track. However, I ended up having to go back and answer questions about every step (“Why did the sign in the denominator change?”, “How did you get the first equality?”, “Why is it \(+0\)?”, and so on). What was meant to be a quick, off-the-cuff computation turned into something really confusing.

After answering those questions, I wrote the correct computation of the limit, then asked which one was correct and why. We mostly got back on track, and I think that I made my point (i.e. check the hypotheses before applying l’Hospital’s Rule), but the entire exercise could have been better presented.

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