Calculus I—Day 5

With the first week behind us, the class is starting to settle into something of a rhythm. Things are going by very quickly, but the students seem more comfortable with the homework system, and after two quizzes have a better idea about what to expect from me. And we are finally getting to the good stuff: derivatives!

What I Taught

I opened the class by writing a limit problem on the board and encouraged the students to work on it while I handed back quizzes and gave them study guides for the first exam. I spent a little bit of time going over the quiz, then we worked out the problem that I had written on the board. This particular problem was similar to the one discussed on day 4, with many of the same subtleties, but it went much better this time.

At this point in the course, we had pretty much exhausted the introductory study of limits, and were ready to move on to the derivative. I started by asking a question: given the graph of a function and a particular point \(P\) on that graph, can you find the slope of the line tangent to the graph through \(P\)? I illustrated the idea with a picture, and demonstrated that an approximation to a tangent line can be made by considering a secant line that passes through \(P\) and some other arbitrary point \(Q\). The closer that \(Q\) is to \(P\), the better the approximation should be. We know how to compute the slope of \(\overline{PQ}\), and we have good mathematical interpretation of what it means to choose \(Q\) “as close as possible” to \(P\). Thus combining the slope formula from high school algebra with the new concept of limits, we can compute the slope of the tangent line. And thus the derivative is defined!

Using this geometric idea, I defined the derivative of a function \(f\) at a point \(a\) to be the slope of the tangent line through \(a\), given by
\text{slope of the tangent line} = \lim_{x\to a} \frac{f(x)-f(a)}{x-a}.
This looked familiar to most of the students, since we wrote something similar on the first day of class, though without having defined the concept of a limit yet. We worked an example, and I gave an alternative definition, i.e.
\text{slope of the tangent line} = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h}.
I demonstrated how the two formulae are related, and discussed how they could be interpreted (as a slope, as an instantaneous rate of change, as an instantaneous velocity, and so on).

Next, I returned to ball-dropping problem from the first day of class. I asked the students to look at their notes and recall our solution, then used the derivative to obtain the same solution, making the point that the solution from day 1 was a numerical approximation, while the current solution was exact (but they matched, so our numerics must have been pretty good!).

I then used this problem as a jumping off point to discuss the idea of the derivative function. The above defined derivative gives the slope of a single tangent line, while the derivative function gobbles up any value \(x\) and spits out the slope of the tangent line through \(x\). I made the point that, in our example problem, we could pick any \(x\) we liked, and the computations would be identical.

One student asked, “The computations will be the same, but the numbers will be different? Ah-ha!” Yay! Right on the money, clearly stated, and obviously understood!

At this point, my plan was to define differentiability (at a point and on an interval), then go through three basic situations where things could “go wrong.” I stated the definition, but then a student asked, “So a continuous function is differentiable?”

Anther great question! While that was my “What can go wrong? (part 2),” I decided to address it immediately, and stated the theorem that differentiability implies continuity. This allowed me to talk about the contrapositive, converse, and inverse a little bit, and to note that discontinuous implies not differentiable by “reversing” the statement of the theorem. I then returned to the rest of my “What can go wrong?” speil and demonstrated corners and cusps (with \(|x\)), and mentioned the existence of vertical tangents.

Finally, I introduced higher order derivatives (with their notation), and mentioned that acceleration was the second derivative of position with respect to time, and finished the lecture with a quick mention of Leibniz’s notation (we’ll spend more time on that later).

What Worked

Honestly, I felt like the whole lecture gelled quite well. I managed to get through a lot of material (probably too much, but the pacing is the pacing) in a manner that was, I think, clear and effective. There are a couple of things that I think worked exceptionally well, however:

I started with an example that worked though a couple of tricky algebraic subtleties. One of these had to do with sign, and I made a little mistake in canceling negatives at one point. A student pointed it out, we fixed it, and all of my students spent the remainder of the lecture looking for my sign errors. Three different students stopped me at different times in the lecture to point out purported errors, indicating to me that they were engaged, which is a victory.

There were several great questions (two of which are highlighted above). I may have moved off of my intended train of arguments a few times, and probably spent more time on tangential matters that I would have liked (I had to skip proof because of this), I am quite happy that my students were both paying attention and, seemingly, understanding.

All-in-all, I feel like this was one of the better lectures that I have given. Engagement was high, participation was high, and I enjoyed the back and forth.

What Didn’t Work

Yeah, so, this one is a bit personal: I mentioned above that I made a sign error in my first example. This is not something that didn’t work—in fact, handled correctly, it is exactly what should happen. The problem was not the mistake, but my reaction to it.

When one is in front of a class of students with well prepared lecture notes, it is easy to believe that one has not made any mistakes, and to assume that the error is the student’s. The particular mistake that I made was this:
\frac{-a}{-b-c} = \frac{a}{b-c},
(though complicated by expressions with more substance than \(a\), \(b\), and \(c\)). A student pointed out the problem, but I misunderstood—I assumed that he was asking about the sign in front of \(b\), which I had correctly canceled! Because I had made a snap judgment, I didn’t wait for him to finish his question. While he eventually made his point, it was much more difficult than it should have been.

Again, it is easy to quickly jump to conclusions when you are the “expert,” but this is problematic. I know that I tend to come across as somewhat arrogant to some students (I believe that one student called me “overly smug” on an evaluation once), and I know that part of the problem is snap judgments like the one displayed above.

The key, I think, as with everything in the classroom, is to slow down. I have worked hard to give more time for students to think after I ask them questions, and I have made an effort to stop at the end of each sentence or equation that I write on the board so that students can catch up, but I really need to work on letting my students finish their questions before I start answering.

So, let’s try this: whenever a student speaks, I am going to count mentally count to five, take a deep breath.

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