After three weeks of instruction, we finally got to our first applications (i.e. “story problems”): exponential growth and decay models. As the problems themselves can be solved in terms of algebra techniques applied to exponential functions, it seems odd to teach this in a calculus class. On the other hand, the framework for understanding what the model says comes from differential calculus, so perhaps now is an appropriate place in the curriculum to talk about such models.
What I Taught
The previous lecture introduced a lot of new ideas, and so we spent a good deal of this lecture going over homework problems. This left only 40 minutes (give or take) to talk about exponential growth and decay.
Feeling a bit crunched for time, I skipped the exponential population growth model, and started in with radioactive decay. The mass of a radioactive material through time can be modeled by the formula \[ m(t) = m(0) e^{kt}, \] where \(m(t)\) is the mass at time \(t\), and \(k\) is a constant. Using carbon-14 as an example, we worked out the value for \(k\) from the half-life, and determined how much material would be left after 25,000 and how long it would be until only 1 gram remained.
After finishing this problem, we moved onto a compound interest problem, which took the rest of the class. Once again, I had hoped to discuss the value of \(e\) as a limit here, but simply ran out of time, and wrote the identity on the board. On the one hand, this is really frustrating, as I would like to show how \(e\) can be seen as a limit, thus uniting the definition I gave earlier with the notion that many of the students have from high school. On the other hand, the motivating problem is compound interest, which is not of interest (ha ha) to most of my students—they have shown a much greater desire to talk about physics and biology problems.
What Worked
Prior to class, I had received several emails asking for extra time to work on the homework. I have tried to make it clear that students will always have a chance to ask questions about the homework in class before it is due, so I didn’t grant any extensions before class. After class, I asked those students who requested extensions if they still needed in more time, and was pleased to find that all of their questions had been answered. I am going to call this a success, as the act of going over homework problems in class is clearly giving the students some confidence, though I am a bit curious to know how much of that will be born out on the next exam.
What Didn’t Work
In the discussion of radioactive decay, I essentially set up one problem, then worked on that problem for 20 minutes. I noted that the constant \(k\) didn’t depend on the initial mass (i.e. \(m(0)\)). However, when I asked how much material would be left after some number of years, I retained the initial mass, essentially working the same problem again, and not actually making relevant the fact that \(k\) doesn’t change when the initial mass is changed.
I am also worried that the compound interest problem took too long and was both uninteresting and unenlightening. The basics of the problem are fairly straight-forward, and I think I got as far as \[ A(t) = A(0)\left(1+\frac{r}{n}\right)^{nt} \] without difficulty or confusion, but then I “poofed” the formula \[ \lim_{n\to\infty} A(t) = A(0)e^{rt} \] into existence without much explanation (and still managed to run five minutes over). I feel like this was a failure on two fronts: (1) the problem wasn’t that intrinsically engaging, and (2) I rushed and left out key details.