Our school has a two-week spring break at a silly, early time in the year. We have been back for a week now and I feel like my students and I are all getting back in the groove again. I know that the dreaded senior slump will continue to pick up momentum but at least I am still seeing some energy and engagement from most of my seniors.
I have a few posts bubbling in my brain and I suspect it’ll be a busy blogging week. Tonight I want to briefly touch on my AP Calculus BC class. We are just settling in to our last major required topic of the year, the Taylor / Maclaurin polynomials. I wrote a little GeoGebra demo (you can find it here) and I started off by showing them (without revealing the mechanics behind the scenes) a polynomial approximation of increasing degree for the trig function y = cos x. We played a little noticing and wondering and saw that at certain stages the polynomial did not change. It did not take long to deduce that this happened at the odd powers of the Taylor polynomial. This led to one student remembering something about the symmetry of cosine, another student mentioning that this was a y-axis symmetry and, finally, a third student mentioning that this is even symmetry. So the lack of development due to the odd powers of the Taylor made a little sense. We then switched to y = sin x (as in the link above) and, unsurprisingly, saw that the even powers seemed to do little or nothing here. We did a little more noticing and wondering watching the Taylor expand on GeoGebra. I should note that all of this was centered at x = 0 (or, in the Taylor notation, we had a = 0) GeoGebra’s sliders allowed us to begin shifting that value and some interesting (and ugly/scary) things started happening to the Taylor equation. My kiddos quickly saw that the equation seemed to be undergoing a simple horizontal transformation – at least in the x terms. The coefficients were changing in some mysterious ways. Finally, we looked at the Taylor series for y = e^x. One of my students asked a great question at this point. He asked – Why are there all those factorials in the bottoms? I skipped this question around the room a bit to see if anyone wanted to make a guess. They quickly observed that exponents in the numerator were clearly attached to the factorials int he denominator but – understandably – they had no solid guesses. Without giving away all the mechanics (we have plenty of time for that) I asked what the derivative of x^7/7! is. I was told it would be 7x^6/7! Correct for sure, but unsatisfying. I must have made my unsatisfied face because one of my students offered a much cleaner version of that answer as x^6/6! Again, I did not go into the mechanics at this point, but there did seem to be some sense that this was an interesting thing to note. I was pleased by the power of the graphics of the GeoGebra applet. I know that I could do something similar in Desmos but I don’t know the commands there as well as I do in GeoGebra. I will start class off tomorrow with the power series we derived for e ^ x and I’ll ask for derivatives and integrals of that. Should be fun to see them realize in this format why the derivative of e^x is itself.
Fun to be back and excited to unfold Taylor’s series’ with my students. This was one of the genuinely awe inspiring topics when I studied Calculus. I remember being amazed by this idea and it’s mechanics. I hope I can share that wonder.
6 thoughts on “Back in the Groove”
I’ve always enjoyed showing students how sine and cosine are embedded in e^(ix). It provides for a nice excursion / re-introduction of complex numbers.
I have a brief unit with the BC kids that attempts to tie together number lines, Cartesian Plane, Polar Curves, and the Argand (Complex) Plane all in one tidy bundle. I save this Taylor/DeMoivre idea for that unit.
Thank you for sharing. I have one student doing an independent study on BC topics this year, and so I’m going to send this post for her to read, too!
Please do and encourage her to look at the GeoGebra link to play with. She can enter her own f(x) and see the resulting Taylor at different degrees. She can also manipulate the center of convergence
A nice reflection Jim, coming off spring break is always a tough start, but what a great way to get moving forward. I love the Geogebra applet, that was fun, well done. We just finished our spring break last week, and it feels we have so much more school left this year.
“I must have made my unsatisfied face…” I can picture that! What a wonderfully subtle prompt. 😐