Calculus I—Days 23 & 24

The term is rapidly drawing to a close, and I am beginning to feel just a little overwhelmed. Moreover, the class has finally hit a rhythm, and the things are mostly running smoothly, thus I have less to discuss in this space. For that reason, I am combining the posts for last Thursday and Friday.

What I Taught

On Thursday, we started by finishing up the curve sketching exercise. Since the second derivative involved some gnarly algebra, I went through the details with them after everyone had finished (more or less). The exercise ended up taking a total of 40 minutes of class time (between Wednesday and Thursday), which scares me a little with respect to the exam, on which there will be a curve sketching problem.

The rest of the class was spent on optimization problems. As with most “story problems,” the mathematics is generally straight-forward; it is the set-up that is difficult. I went through a couple of examples on the board (build a fence to enclose the largest possible area, minimize the surface area of a cylinder with a given volume), and used these to justify a slightly different statement of the First Derivative Test.

Specifically, the First Derivative Test gives a means of identifying local minima and maxima (i.e. if the derivative is continuous and changes signs at \(c\), then \(f(c)\) is a local minimum or maximum, depending on which way the change of signs goes). I stated a First Derivative Test for Absolute Extreme Values, which can be used to identify absolute minima and maxima using the first derivative (at a point to identify a local min or max, then on the rest of the domain to show that it is an absolute min or max).

On Friday, I started by answering homework questions on optimization. Since the topic is rather important and the problems themselves all require slightly different approaches, I think it was time well spent. I then gave them a quiz on L’Hospital’s Rule (two problems, one of type “\(\infty^0\)” and the other of type “\(\infty-\infty\)”), which seemed to be more difficult than I had intended. I’ll probably spend a little time on that today before doing anything else.

Finally, we finished the chapter on applications of differentiation by introducing the antiderivative. This involved a few definitions and a few examples, as well as an application to the study of motion. This lays the groundwork for the last chapter that we will cover, which introduces integrals.

What Worked

The quiz on L’Hospital’s rule proved to be quite intimidating, which (I think) is a good thing. The problems were algebraically difficult, but similar to homework problems, so I don’t think it was unfair to ask them; and they are of a type that is likely to appear on the next exam (in fact, one of them will appear word-for-word), hence I think that a bit of a shock will get them studying in the right direction.

Aside from that, the class keeps plugging along… I think that the students kind of like the optimization problems, as they have something tangible to grab onto (i.e. some familiar geometry or algebra), and they can be kind of fun to work through.

What Didn’t Work

I am not sure that my presentation of antiderivatives was all that enlightening. I think that it was adequate and at least minimally competent, but it wasn’t very deep. The last few weeks of experience with this class tells me that quite a few students are going to be on me for not going over every detail.

I may spend some time working one or two more examples (a particle in motion, perhaps? or a simple differential equation?) this evening. That should flesh out some of the details, I hope.

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