So, it is AP Calculus prep time, my least favorite time of the year.
This year I am lucky to have one student, just one, who will NOT take the AP exam. So, I asked him, “Do you think it would be a boring waste of time to do the old AP exam your classmates are doing?”
“Yes,” he answered. So, I pulled down the old Calculus textbook by Paul Foerster–whose explorations are essential to my class–and thumbed through it for a good problem that would take him some time to do…
The factorial function and the Gamma function!
He set to it, doing some improper integrals, and some integration by parts and then he used his calculator to integrate the factorial function to get 0.5! = 0.886.
The book said, find a simple expression involving pi that equals 0.5! and after a little time, he found.
Moreoever, he learned that (-1)! and (-2)! were infinite but (-0.5)! and (-1.5)! were finite.
And I learned these facts, too!
Back in college, I saw the gamma function in my textbooks but spent zero minutes using it, much less investigating its definition. So having a student who was not constrained by the AP test prep (and I’m only spending a week and a half on prep) gave me the opportunity to broaden and deepen my knowledge of mathematics… as well as this one student, and after the exam is over and done with, I’ll use this experience to push the rest of the students to learn (0.5!). It will fit in with the “Integrals and Statistics” unit I’ve composed for the month & a half after May 6.
This is what studying math brings you: new discoveries, new connections, a sense of wonder at how the world of numbers makes sense.
Thank you, Paul Foerster, for your wonderful books!