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Tag: circle

  • 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 (Part 3: Why we need limits)

    In the first post in this series, I claimed that calculus is the branch of math devoted to understanding the connection between a rate of change and an accumulation of change.

    In the second post, I connected this to the idea of area: specifically that the perimeter of a circle determines the rate of change in its area, per unit change in its radius.

    In this post, we will see why limits are important for understanding these ideas.

    First, we will change notation a bit. Before, we let dr represent a “small” change in radius, without saying how small dr actually is. Now, let us say the change in the radius is some specific number that we call \(\scriptsize\Delta r\). The number \(\scriptsize\Delta r\) may be anything, small or large: it is just some amount we are adding to the radius of a circle.

    Consider the strip formed by increasing a circle’s radius by an amount equal to \(\scriptsize\Delta r\). We previously treated this as a rectangle, but this isn’t really right, because the top and bottom have different lengths:

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