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Tag: fundamental theorem

  • What is Calculus (Part 4: Graphs and Equations)

    In the first post in this series, I said that calculus is the branch of math dedicated to understanding rate of change and accumulated change.  In that post, I Illustrated two relationships between the graphs of cumulative COVID cases and daily COVID cases.

    First, the slope on the graph of cumulative cases on a given day is the same as the corresponding value (y-value) on the graph of daily cases:

    Second, an area between two dates on the graph of the daily case rate equals the change in the value between the same two dates on the graph of cumulative cases.

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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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