# Stacks

I teach integrals early in Calculus, super early really. Roughly two months ago I introduced the concept. It sounds a little crazy, but it's very handy when it comes time to do curve sketching. Somewhere along this way I coined the term "the stack" for the relationship between f, f', and f'' with derivatives and integrals serving at the means of transport between layers in the stack.

Previously, I've had students graph f, f', and f'' on separate axes, this year I changed it up. I don't really know if it's different or better, but it is more space efficient. Super bonus fact, I introduced curve sketching with a review of sketching polynomials (of the factored variety).

Here was the arts and crafts project of the week, given a graph of varying points in the stack, finish the stack:

Introducing integrals prior to this is helpful when having students determine if their stack is reasonable: if I start with a linear function of f'', I know integrating that function should yield a quadratic, and integrating again should give me a cubic. The followup activity involves interpreting the sketches: validating minimums, maximums, and points of inflection. A common narrative in Calculus is starting at an arbitrary point in the stack and asking students to interpret information about a different function entirely (given f', what's up with f?).

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