# Curve Relationships Project

A huge problem in AB Calc has been an understanding of a function, its first derivative, and second derivative in a lot of contexts. Students have be able to infer the behavior of a function from the graph of f’, a table of f’, or just the equation of f’. Being able to construct all three elements when given one is a path to fluency.

Prior to break, students in AB Calculus had to create a polynomial that represented a first derivative. Based on that graph, they had to identify minimums, maximums, and points of inflection for the original function f. In addition, they highlighted the differences in concavity and when f could be expected to be increasing or decreasing. They translated their findings into tables of f’ and f’’ that validated their findings, gave justifications for their findings, and sketched f based on the derivative they created.

We were working on this fluency in class through a variety of prompts. Sometimes we started with a table, other times a picture, and others the equation. Here students created an equation and built out the whole process on their own. The results are a lot more polished than the version I tried last year.

Here’s the full write up students were given:

Most impressive to me is that students didn’t shy away from tricky to analyze functions. A lot of students created derivatives that would generate “fake” maximums, minimums, or points of inflection (where f’ or f’’ has a value of 0 but doesn’t complete the required sign change) and it can be tricky to do sketches based off that. But every student who attempted one was on the right track with their thinking. A few hit a common curve sketching snag of drawing an original function with the right features, but everything was upside down.

Collectively though, this group is doing a great job with the ideas.

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