One tenet of my work is that applications can lead to a more mathematically rich understanding of calculus: “applied” and “pure” calculus are intimately linked and do not stand in opposition to each other. Here are a few examples:
- I start with extremely real world applications using numerical calculus, for example the growth rate activity described in this post. In the course of this work, I highlight the fact that numerical integrals and derivatives are intrinsically approximate, and that the way to make them better is to make \(\scriptsize \Delta x\) “as small as possible.” Which, of course raises the question of how small that is. And that leads naturally into a discussion of limits. By the time my students see limits, they feel topical and relevant, because they are a response to an actual need.
- I Introduce integrals first using accumulation and then using area, as I describe in this post. This helps get students hooked (and helps them see the point of things) but it also has a significant theoretical benefit when I introduce signed area. Specifically, accumulation, like integration, is directional. The sign of the Riemann integral depends on the sign of \(\scriptsize \Delta x\), because this is an essential part of what integration is designed to do. Area, by contrast is not directional: signed area exists to impose orientation on a concept that is not intrinsically oriented. When we introduce signed area first and accumulation later, we hide the fact that orientation is absolutely essential to the nature of a Riemann integral.
- It is often hard to justify the importance of the Mean Value Theorem to students in their first calculus class, as I describe in this post. However, the MVT has a natural and important interpretation in the context of numerical derivatives, as I outline in this post.
- I use various visualizations of the fundamental theorem, taken from applications, to highlight the meaning of this theorem, and to make sure that students see part 2 as an important statement about invertibility and not just the “integral evaluation theorem.”
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