Reflections of a Second-career Math Teacher

One of my calculus periods continues to struggle mightily while learning new concepts, especially if they need to recall prior knowledge, or worse connect it to the new material. As an example, when starting to explore integration concepts, students learn to use finite methods to estimate areas under functions, as early mathematicians discovered.

The basic technique uses multiple rectangles to approximate the area under the curve of a function over the interval *x* = *a* to *x* = *b *as the figure below shows. Since the area of a rectangle is equal to its base times its height, we simply need to determine a specific value of *f*(*x*) for each rectangle used, where the number of rectangles is driven by *n. *In the figure below, *n* varies from 2 to 4 to 8. Note that Δ*x = *(*b-a*)* / n, *which determines each specific *x*

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