Calculus 1 – Derivatives
Calculus 1 – Derivatives

Rich questions at the right time can yield rich discussions

This week we moved from the definition of the derivative and computation of derivatives via limits of difference quotients to derivative rules and symbolic computation of derivatives of elementary functions. To wrap up the first part and emphasize again that the derivative measures the rate of change, I opened up class with the following three questions.

The initial vote was split pretty evenly between B and C. When I asked the students to discuss I reminded them that if they were identifying one of these as incorrect, then they were implicitly saying the other three are the same and that they needed to convince their partners of both of their claims. I sat in the middle of the room to listen in on what a couple of nearby groups were saying. Those in the C camp seemed to have more sway. The second vote was tilted toward C, so some minds were changed, but was not unanimous. I lead a discussion at the board about how A and B can be seen as the same via the substitution . Then we emphasized that language is important. These expressions are supposed to represent the derivative at . In C, is moving toward whereas it was supposed to be fixed. Then I mentioned the shortcoming of the notation in D which is that you cannot tell where you are investigating the limit of the slopes of secant lines. So what is that notation good for? The next question illustrated that.

In both classes, the initial vote was unanimous on C. Instead of just moving on though, I took a moment to emphasize how to read the notation and think about units. The charge is measured in Coulombs so the difference in charge is measured in Coulombs. Similarly, time is measured in seconds so the difference in time \$\Delta t\$ is measured in seconds. The difference quotient has units of Coulombs per second. The instantaneous rate of change is the limit of the difference quotient, the trend in these numbers and their units. The pattern in the units is Coulombs per second, Coulombs per second, Coulombs per second, … so the limit units are Coulombs per second. I think this discussion helped with reading the whole expression and understand its units rather than simply making a guess or reading the units of just the difference quotient itself.

The last question prompted the richest discussion in the groups and as a class. It required putting a whole bunch of ideas together. The data in the table is realistic, taken from the growth charts published by the US CDC.

In both classes, first vote was heavily tilted towards A. In one class, I found a lonely student and sat down with that student to discuss. They had voted A, based on the following reasoning: the length goes up by 4 cm, then 6cm, then 6 cm, then 9 cm, then 12 cm then 9 cm, and all of these numbers are greater than 2. So I asked the question, “Are all of those numbers of centimeters per month? Are the increments of time all the same?” At this point the student noticed the non-uniformity of the table (check-ups happen often early and less often as the child grows). I pointed out that we needed to estimate the slope between two pairs of points in order to estimate the rate of change. At this point, the student focused on the data from birth to 1 month and from 1 month to 3 months, estimating the growth as between 4 cm per month and 5/2=2.5 cm per month. Both of these numbers are still greater than 2 cm per month. “Okay, but I think we need to estimate the growth rate at 1 year of age, which is at 12 months.” The student then made those calculations and revised their answer. On the second vote, some of the class shifted to B but the vote was fairly split between A and B. A whole class discussion was required.

Right away, the students identified that one needed to focus on the rate of change at 12 months. Computing the slopes on either side and obtaining numbers between 1 and 2 seemed to convince the holdouts. But I wanted to drive home the point with the class that the growth rate at 1 year of age is the derivative of the length with respect to time at 1 year (=12 months), i.e., the limit of the difference quotient at 12 months. At the the board we made the following table and computed the decimal values of each fraction.

Emphasizing again that the limit is the trend in these numbers as , we observed that the trend appeared to be a number between 1 cm per month and 2 cm per month.

It was nice to come full circle with the concept of the limit on this problem. We don’t have a formula for how a child’s length varies with time. There are models, but it’s not like the child comes with a tattoo which says here is the formula for how my length will vary with time. Furthermore, we only observe their length at these moments, we don’t continuously monitor it, so it would not be possible to produce an accurate continuous graph of length versus for a specific child. We are left to make estimates from a table. From the table of length versus time, we can make a table of values of the difference quotient at 1 year (=12 months) and try to estimate the trend as a proxy for computing the limit.

If you would like to try using our ConcepTests in your classroom, our question bank http://www.cpp.edu/~conceptests currently contains questions for the last third of 1st year Calculus and multi-variable and vector calculus. Questions for 1st year differential and integral Calculus like those I’ve discussed here are in development and will be published to the web site later this academic year.