Calculus I—Day 2

The course started yesterday with one very concrete example, followed by loads of abstractions. Last night’s lecture began the trip back into things that are a little bit more concrete.

What I Taught

I began the lecture by handing out a few pages from Larry Gonick’s fantastic The Cartoon Guide to Calculus. As I noted yesterday, the compressed summer schedule means that certain parts of the curriculum are giving short shrift, including the formal definition of the limit. I would like them to at least see the ideas, so I gave them something entertaining to read, and spent a couple of minutes trying to get the general gist of idea across.

With that out of the way, we dove straight into a rather overwhelming section on rules for finding limits. I asked them to essentially memorize six computational rules (things like “the limit of a sum is the sum of the limits”), as well as two special cases (for any constant \(c\) and any point \(a\), we have \(\lim_{x\to a} c = c\) and \(\lim_{x\to a} x = a). I showed how a couple of the rules could be used to derive others, and demonstrated the special cases graphically. No formal proofs were given.

Using these “limit laws,” we worked through a couple of examples involving polynomials and rational functions in detail. We noted that in these cases, it seemed like it was possible to find \(\lim_{x\to a} f(x)\) by evaluating \(f\) at \(a\). I told them that this was, in fact the case (assuming that \(f\) is defined at \(a\)), and gave a quick sketch of the proof.

I finished the lecture by relating limits to one-sided limits, stating the Squeeze Theorem, and demonstrating how the results covered in the lecture could be used to find \(\lim_{x\to 0} x\sin(\pi/x)\). In the last ten minutes of class, I opened the floor for questions about the homework.

What Worked

I think that the examples were pretty solid, and that most of the students were able to see and understand the process of applying the limit laws to compute the limits of polynomials and rational functions. One student noticed that the general idea was to use the laws to reduce all of the terms to either constant functions or the identity function, which allowed us to use the special cases. I am heartened that I didn’t have to point this out—it was “discovered” by one of the students, which I think makes it a bit real and accessible for the entire class.

What Didn’t Work

On the other hand, this was a pretty rough lecture. Because the treatment of the material is a bit less formal and much faster, the limit laws look like a bunch of arbitrary rules that need to be memorized. I tried to motivated them and show the connections, but there was neither time nor sufficient background to really do the job justice. I’m not sure what else to do in the situation—there is a certain amount of material that I am expected to cover (and not cover) in a limited amount of time—but it seems like there might be a better way. To the three or four of you reading along, any thoughts would be appreciated.

I also made a complete and utter hash of my last example. The limit
\lim_{x\to 0} \left[x\sin\left(\frac{\pi}{x}\right)\right]
was meant to highlight a lot of ideas. Because the sine term doesn’t have a limit at zero, the previous technique of reducing the problem to terms with known limits wasn’t going to work. This requires the use of other tools, and the only other theorem presented was the Squeeze Theorem. To use this theorem, it is necessary to find two functions that bound the function of interest, and have the same limit at 0. In this case, \(|x|\) and \(-|x|\) do the job, which means finding those limits at 0. To do this, we have to take the absolute value apart piecewise, find one-sided limits, and use those to obtain the limit. All in all, there is a lot to see with this example.

Unfortunately, when I first wrote the problem on the board, I was beginning to stress about having enough time to work the homework (I had about 30 minutes left), and I wrote a 3 rather than an \(x\) for the multiplication. Obviously, this didn’t work the way I wanted, and I had to backtrack. In and of itself, no big deal—chalk is cheap, and students are generally willing to forgive one mistake, especially if it is caught early, as it was here. But I let myself get flustered, which was bad.

Because I was flustered and worried about time, I rushed through the next step. I stated, without much explanation, that
-|x| \le x\sin\left(\frac{\pi}{x}\right) \le |x|.
This is true, but I lost several students here, and had to come back to it after finishing the problem, which must have been even more confusing and disorientating for the students.

Instead of properly explaining the above inequality, I trooped on ahead, and found \(\lim_{x\to 0^-} |x|\) and \(\lim_{x\to 0^+} |x|\) from the piecewise definition. This part was, I think, pretty clear, but on the opposite side of the board because I didn’t want to erase the statement of the squeeze theorem from the center board. Bad Xander. On the bright side, this was the last piece of the puzzle, and I was able to finish things off without too much more difficulty.

Except that I had to go back and explain the inequality.

Ultimately, I think that I lost a few students with the example because I got flustered, then allowed myself to compound errors upon errors. There is a truism in flying planes that one mistake doesn’t lead to a crash—crashes happen when a pilot makes a mistake, then another, then another. That’s what I allowed myself to do.

So, here’s how to fix the problem in the future:

  • Stop, breath, and relax after making a mistake. One mistake is easily corrected.
  • Slow down and don’t stress about time. The amount of backtracking required to correct for rushing quickly costs more than doing things right the first time.
  • Be willing to erase the board. For seriously.
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