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Tag: rate of change

  • What is Calculus (Part 2)?

    In a previous post, I wrote that Calculus is the branch of math that deals with understanding the relationship between rate of change and accumulation of change.

    The website Better Explained says that “calculus finds patterns between equations,” and specifically mentions the connection between the area of a circle \(\scriptsize (\pi r^2)\) and the perimeter of a circle \(\scriptsize( 2\pi r)\). Calculus relates these two: the perimeter is the derivative of the area.

    But what does the area of a circle have to do with rate of change and accumulation of change? How is this situation similar to the COVID cases used as an example in my previous post?

    Suppose we have a circle, and we imagine growing it slowly: starting with a radius of 0 and getting bigger. In the circle below, we have the accumulated area (the full circle) in white.  The gray strip around this circle shows the change in the area resulting from increasing the circle’s radius just a bit.

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  • What is Calculus?

    One of my all-time favorite complements was from a student who told me that he loved calculus because the class is one topic all year long: we just keep learning about the same idea in more detail. Indeed, part of what I love about teaching calculus is that I see it as being a unified class, a single multifaceted concept.

    So what is the topic that I claim is the subject of calculus?

    I claim that calculus is the branch of math dedicated to understanding the relationship between accumulated change and rate of change. This is a fairly radical claim: many would say that limits (or continuous change) are the heart and soul of calculus.1

    The graphs below give a specific example of accumulation and rate of change. On the left, we have a graph showing the new daily COVID cases in the US in the first couple years of the pandemic. On the right, we have the cumulative number of cases: total cases ever recorded in the US, up to a given day.

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