Absolute Minimums and Maximums

Another function relationship project, this time from BC Calculus. Prior to the break we were working with functions defined as the integral of another, and how you could figure out absolute minimums and maximums of a given function. There was a lot of interest in this topic from the AP test last year. A number of items designed to show fluency in function relationships didn’t go well for students at large. The issues involved being very particular with your evidence and how you use notation. At TMC 18 while you were playing games at game night, me and Dave Cesa were sitting in the corner talking about this. We know how to party.

Now, we went through these ideas at the very end of the semester, so to simplify things, we made an assumption that our function would have an initial condition of f(0) = 0 and that all accumulations would happen left to right. First thing for the new semester is revisiting that idea and being a little more flexible.

Here were the student instructions:

Students used Desmos to graph their function and compute integrals at any milestone point, local minimums or maximums of F(x). Students calculated the area of their discrete pieces and kept running totals along the x-axis. In tending to precision, they collected their data into a table of x and F(x) and used a series of integrals to show their computations.

While I’ve covered this topic in years past, I have not been as particular as I need to be when it comes to notation. A real struggle in FRQ for my students is showing their knowledge of notation. Often they will cut corners by not including it, or it will be written incorrectly (integrals without dx, for example).

Like their AB counterparts, students didn’t shy away from tricky situations, including points that would register as “fakes” that they could ignore when rectifying their totals. Desmos allowed us to construct polynomials easily and get a feel for using integral notation with named functions. In parallel, when computing integrals with Desmos I will have students replicate the process on their TI-89 to verify they are proficient with both.

AB Calc will repeat the project in the coming weeks and all of this is a good sign for better efforts with precision and argument structure.

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