I am teaching Calculus I this summer, starting yesterday. In the interest of improving my own performance as an instructor, I thought that I might try a little exercise as the course progresses. My intention is to spend a little time each morning analyzing what I taught, what worked, and what didn’t.
By way of introduction, the class is an interesting mix of students. I have about 25 students, a few of whom are returning students who have previously completed a bachelors degree and are back for more, and a few of whom have never taken a college class in their life. Most of them are engineering majors of one stripe or another, though I have a fair number of agriculture majors in the mix, and even a couple of liberal arts folk, which tickles me pink. Less than a third of these students have taken calculus before, so I get to start with a fairly clean slate.
What I Taught
After getting through the syllabus, I started in by trying to motivate the study of calculus. I presented the somewhat contrived problem of a ball being dropped from a high point, and asked how fast it was going at a particular time. This led to a discussion of average velocity versus instantaneous velocity and was meant to motivate the introduction of both limits and derivatives.
I then abstracted the problem a little more, and removed the physicality of it. Instead of talking about a ball being dropped from a tower, I talked about a parabola (given by the same function), and made the connection between an average velocity and the slope of a secant line. I then demonstrated that the two points of intersection could be brought close together through some kind of limiting process, and the result is a tangent line whose slope corresponds to the instantaneous velocity. This gives us both a geometric and physical interpretation to come back to when we get to the derivative.
I finished the class by giving the somewhat hand-wavy definition of the limit that the text book presents. This definition is essentially correct, but elides many of the important details. Rather than talking about a neighborhood around some point \(a\), it aludes to the idea of “nearness”, and the epsilons and deltas are omitted for a more prosaic description.
I defined left- and right-hand limits in a similar fashion, and tried to give meaning to the notation \(\lim_{x\to a}f(x) = \pm\infty\), both in terms of the abstract idea of a function increasing without bound, and graphically as a vertical asymptote.
What Worked
I thought that the transition from the concrete problem to the abstraction worked fairly well. I was careful to set up the notation so that the computations of average velocity would look similar to the computations of slope. Early in the lecture, I put the equation
\[
\text{average velocity} = \frac{s(t+\Delta t) – s(t)}{\Delta t}
\]
on the board. When moving to the abstract problem, I wrote
\[
\text{slope} = \frac{y_2-y_1}{x_2-x_1}
\]
on the board, and heard a few “Ah-ha!”s, which gives me some confidence that at least a few students really grokked it.
I was also quite pleased with one of the limit problems. In order to build intuition and to make sure that they can use the technology, I spent a lot of time showing them how to “guess” limits by evaluating a function at points closer and closer to the limit point. My first example was
\[
\lim_{x\to 1} \frac{x^2-1}{x-1},
\]
which works out very nicely when the guesses are made using \(x=1.1, 1.01, 1.001\) and \(x=0.9, 0.99, 0.999\). I like the example, as it gives students a tangible idea about what is happening, and because it so obviously gives the intuitively correct result. I even had a student point out that the rational expression can be simplified, then a substitution made. While those are tools that have not yet been introduced, it certainly helps with the intuition.
I then gave them the example
\[
\lim_{x\to 0} \sin\left(\frac{\pi}{x}\right).
\]
The above technique doesn’t work so well in this case—one can easily convince oneself that the limit is 0, or 1, or \(-1\). This example served two purposes for me: it clearly demonstrated that the method of evaluating the function at points with increasing proximity to the point of interest is useful for intuition but is bad mathematics, and it points out that functions don’t always have limits. Several students seemed interested in both ideas, so I am calling it a win.
What Didn’t Work
In setting up the ball dropping problem, I intentionally ignored the distinction between speed and velocity, and carefully crafted things so that all of my signs would turn out positive and we wouldn’t have to get into that can of worms right now. However, I had a few students raise the issue (I assume that they have taken calculus before and were confused), and I had other students who reversed the signs. Instead of front-loading the question and getting it out of the way in five minutes, I ended up in a ten minute discussion about speed versus velocity, and probably confused just about everyone. Bad idea. For future reference, trying to simplify things and save time by glossing over details that some students may have encountered before doesn’t work.