Day One of my Calculus I class: Part 3

In this series of posts, I’d like to describe what I tell my students on the very first day of Calculus I. On this first day, I try to set the table for the topics that will be discussed throughout the semester. I should emphasize that I don’t hold students immediately responsible for the content of this lecture. Instead, this introduction, which usually takes 30-45 minutes, depending on the questions I get, is meant to help my students see the forest for all of the trees. For example, when we start discussing somewhat dry topics like the definition of a continuous function and the Mean Value Theorem, I can always refer back to this initial lecture for why these concepts are ultimately important.

I’ve just told students that the topics in Calculus I build upon each other (unlike the topics of Precalculus), but that there are going to be two themes that run throughout the course:

1. Approximating curved things by straight things, and
2. Passing to limits

We are now studying the following problem.

Problem #1. A building on campus is 144 feet tall. A professor takes a particularly annoying student to the top of the building, and throws him (or her) off to his (or her) certain demise. (Usually I pick a student that I know and like as the one to throw off the building. This became a badge of honor over the years.) The distance that the student travels (in feet) after seconds is . How fast is the student going when he (or she) hits the concrete sidewalk?

At this point in the lecture, we have done some experimental numerical work with successfully smaller time intervals to find better and better approximations to the speed at impact.

• With a time interval of length seconds, the approximation is ft/s.
• With a time interval of length seconds, the approximation is ft/s.
• With a time interval of length seconds, the approximation is ft/s.
• With a time interval of length seconds, the approximation is ft/s.
• With a time interval of length seconds, the approximation is ft/s.

I’ll then tell the class that this is an example of passing to limits, the second theme of calculus. By making the time intervals smaller and smaller, we get better and better approximations to the true speed at impact.

By this point, students realize that we’re getting better and better approximations… however, we’re probably not going to get the correct answer by just plugging in numbers. And we certainly can’t just take a time interval of seconds since dividing by zero is a no-no.

Depending on my read of the class — on whether or not they’re ready for a little more abstraction — I’ll then ask the class, “How can we make these fractions without plugging in all of these numbers?” Usually students are at a loss at first. Perhaps someone will volunteer that we ought to introduce a variable… but, in my experience, even bright students at the start of calculus do not have this step of abstraction at the tips of their fingers. So I’ll lay out the fractions that we’ve studied so far, like

and ,

and ask, “How could we do this more systematically? Does anyone see a pattern in these fractions?” Hopefully someone will notice that the input of the second function call is 3 minus the denominator; if not, I’ll volunteer this observation to the class. So both of these fractions can be written as

,

where is a small positive number. Let’s now simplify this fraction:

.

The last step is permitted because is assumed to be a nonzero number. I then check to see if the previous work matches this algebraic expression:

• If , then , matching the previous answer.
• If , then , matching the previous answer.

I then ask the class, what’s the ultimate goal with ? The answer: send to zero. So we conclude that the velocity at impact is ft/s, which is the final answer.

Reviewing, the curved thing was the changing speed of the falling object, which was approximated by the straight thing, the ordinary distance-rate-time formula. Finally, we passed to limits to find the real velocity at impact.

All of the above is eventually done more systematically later in the semester after the properties of derivatives have been more fully developed. However, I think that doing this calculation on the very first day of class gives my students a taste of what’s going to be happening in the days and weeks to come. Again, I emphasize that I probably cover this material in maybe 15-20 minutes, and that I don’t hold students immediately responsible for repeating such a calculation on their own. (I do hold them responsible for this, of course, after they know how to differentiate .