Day One of my Calculus I class: Part 1

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 begin by noting the different topics that appear in Precalculus, which they should have taken in the recent past:

• The definition of a function and an inverse function
• Graphing polynomials and rational functions
• Properties and applications of exponential and logarithmic functions
• Trigonometry
• Sequences and series

These different topics, when taught in Precalculus, really don’t talk to one another. With a couple of exceptions, it feels like five different units being squeezed into the same course. I’ll present a visual image of laying down an imaginary brick on the floor, and then laying down a second brick next to the first one, and so on. The above topics (with a couple of exceptions) really don’t build upon each other; they’re lateral to one another. In other words, these topics made the foundation necessary for the study of calculus. After all, the class was called Pre-Calculus.

Now that we’re in calculus, I tell my students, we’re going to have topics that build on this foundation, and the topics will build on each other. Continuing the building image, I’ll start laying imaginary bricks on the initial foundation, building vertically higher and higher, noting that the topics that we’ll see in Weeks 13 and 14 will ultimately be built upon the topics that we’ll talk about in Weeks 1 and 2. Unlike Precalculus, the topics in Calculus are explicitly interconnected, building up a body of thought from the foundation of Precalculus.

So the good news is that, unlike Precalculus, Calculus I will be an incrementally developed course from start to finish. The bad news, of course, is that Calculus I will be an incrementally developed course from start to finish. In Precalculus, if you didn’t particularly like one topic (say, logarithms), that really would not affect your success later on with a future topic (say, trigonometry). However, in Calculus, the whole course is put together from start to finish.

The good news is that while there are many interconnected topics in calculus, there are going to be two themes that run throughout the course:

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

And we’re going to be applying these two themes again and again throughout the semester. (I wish I could take credit for synthesizing the topics of calculus into these two themes, but I learned this idea from my own calculus professor back in the mid-1980s.)

For the remainder of this first lecture, I show how these two themes apply to two completely different problems:

1. Finding the speed of a falling object when it hits the ground.
2. Finding the area under the curve between and .

I’ll describe how I present these to new calculus students in the coming posts.