Calculus I—Day 16

Today represents the half-way point of the class, both in terms of the calendar and in terms of the quantity of material that we are going to cover. On the one hand, it seems like the course is going by incredibly quickly (and it is). On the other hand, that first week of instruction feels so very long ago. So today, we not only worked with new material, but integrated some of first ideas that we studied into the current section.

What I Taught

I started by presenting a doomsday device: a hedgehog is sitting on the top of a ladder that is leaning against a wall. The bottom of the ladder is being pulled away from the wall at some constant rate, thus the hedgehog is sliding down the wall toward the ground. When the hedgehog reaches the ground, it will impart its kinetic energy into the ground. The amount of energy depends on how fast the hedgehog is moving, so we want to compute that value.

This is a standard related rates problem. We developed the tools to deal with it in the previous class, so we started by trying to model the situation. The ladder is of fixed length, the bottom is firmly attached to the ground, and the top is firmly attached to the wall, so we use the Pythagorean Theorem to get the height of the hedgehog, take derivatives, and solve for the velocity of the hedgehog at any given time.

Unfortunately, when the hedgehog reaches the ground, our solution involves a division by zero, which is a problem. No worries—we have a theory of limits which allows us to evaluate functions “near” a point that is not in the domain. Remembering back to the second day of lecture, we eventually determined that the hedgehog is moving infinitely fast when it hits the ground! Thus, if we mount a hedgehog on the top of a ladder then slide the bottom of the ladder along the ground, we end up with a hedgehog slamming into the ground at infinite velocity, and destroy the universe.

Hence we have modeled a doomsday device.

There are three reasons to bring up this example: (1) it gives students practice with related rates problems (the current topic), (2) it requires students to recall facts about limits (particularly infinite limits), and (3) it demonstrates how important it is to take physical reality into account when modeling real-world phenomena.

After discussing the hedgehog of the apocalypse, we moved on to new material. Specifically, we started talking about linear approximation. I tried to motivate the idea by showing that as we zoom in on a point where a function is differentiable, the function starts to look linear. In fact, near any point where a function is differentiable, we can approximate the function by a tangent line.

We did some algebra to determine the equation of that line, then stated the idea formally. If \(f\) is differentiable near \(a\), then \[
f(x) \approx L(x) = f(a) + f'(a)(x-a),
\] where this approximation is the equation of the tangent line at \(a\), derived from the point-slope equation for a line.

I justified this kind of approximation by noting that some functions are easy to evaluate at some points but not others (for instance \(\sqrt{4}\) is easy, \sqrt{3.97} not so much), then worked several examples. I finished the class by introducing the notion of differentials and showing how the differential \(dy\) can be useful for estimating the error between an actual value and a measured value.

What Worked

The hedgehog of the apocalypse was fun. I think that the problem itself managed to maintain engagement: I advertised it as a doomsday device at the beginning of the discussion, and students wanted to see what I was on about. I had the attention of nearly every student, and got good input from one or two normally quiet students (though I did have to instruct my regulars to quiet down for a minute).

As an added bonus, we got to spend some time working with the textbook. I think that students often don’t really know how to read a textbook, and even more specifically, I don’t think they know how to read a mathematics textbook. Because we had to work with limits again, we had to familiarize ourselves with the index so that we could figure out how to find \[
\lim_{x\to 10} \frac{x}{\sqrt{100-x^2}}.
\] (and for an unrelated problem, we had to find out about the reference pages at the front and back of the text, which was an added bonus). While the lack of an \(\varepsilon\)–\(\delta\) definition for the limit was a little frustrating, the appropriate sections of the text did (eventually) prove helpful. I am moderately proud of myself for (mostly) shutting up and (mostly) letting the students run that part of the show.

What Didn’t Work

I am finding myself increasingly frustrated with a couple of students. These students probably aren’t really well prepared for calculus (they require a lot of hand holding for basic algebra problems), often arrive to class late (and expect me to catch them up while everyone else is waiting), and prove to be a constant drag on my flow. I think that I am going to have to have a word with these guys…

I was also stymied by the technology. It seems that IT has been reimaging hard drives recently, and upgrading software. Unfortunately, this means that Maple didn’t work. I had a whole document prepared and ready to go, and when it didn’t work, I got flustered. On the bright side, I managed to get back up and running again with WolframAlpha, but it was less than ideal. Of course, it is entirely my own damn fault—while I prepared a Maple document ahead of time, I did not check to make sure that Maple was actually working before I needed it.

Finally, I don’t think that I did a very good job of explaining absolute error vs relative error. To me, this seems like a really easy concept—absolute error is measured in units, and tells you the difference between an actual value and a measured value; while relative error is unitless, and gives the error as a proportion of the actual value. This seemed to be a really hard concept for the students, and I think that my attempts to explain just added to the confusion. The right thing to do would have been to stop trying to explain, and just work some more examples (not that there was much time for this, but I could have done one or two).

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