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.
So, my afternoon crowd was not to be outdone by my morning crew. I slipped in a subtle reference early in the conversation with them so that they would not be inclined to simply introduce the phase shift idea. I wanted them to have a little practice untangling the mechanics involved in dealing with developing a Taylor series. They were very quick to recognize and agree that the coefficients were based on factorials so jumping from the 5th degree polynomial to the 7th degree was pretty easy for them. When I asked for the cosine they were confident about using even powers instead of odds and came to a conclusion pretty quickly. Where life got interesting was when I showed them Michael’s solution from the morning and discussed why i preferred the symmetry generated by an even powered series instead. I also discussed how Michael’s translation idea might give better results for approximating cos x with negative values of x. That’s when they stepped up and knocked me out. They suggested that we take the 6th degree polynomial approximation we had for cos x and do the following: phase shift by pi radians and reflect over the x axis. I am linking to a GeoGebra file that we created. If you want to dig into that file – here are the explanations of the functions.
a and b are self-explanatory
f is the 7th degree Taylor for sin x
g is the phase shift of this by pi/2 to approximate cos x
h is the 6th degree approximation of cos x
m is the crazy reflection/shift to move the cos x approximation backwards to another portion of the cosine curve.
Whew – what a day
We are starting our journey through the study of Taylor Polynomials today. I started with looking at y = sin x and asked them to find a ‘simpler’ function that behaved like sin x does around the origin. I sort of purposely asked this in a pretty vague way and we had a good chat about what I was asking for. One of my students offered up y = x as an answer. This gave us the opportunity to talk about the limit of sin x / x as x approaches 0. It also gave us the opportunity to talk about L’Hopital’s rule. A pretty good start in my mind. Then life got interesting. One student suggested a cubic function but I was able to get someone to urge an extension to a quadratic function that might match the sin x graph as the next step. I’m not sure what he (Michael) saw that made him jump to a cubic. He’s a really insightful student. So, I held that off and got to working on a quadratic. We agreed that the quadratic better agree with sin (0) and that the slopes should be the same. Someone suggested that the second derivatives should match as well. This resulted in a quadratic with a leading coefficient of 0. Not so good. It would have been easy for them to give up on this process, but Michael had already suggested the cubic. We had success in finding one and a GeoGebra graph confirmed that this worked over a larger region than the simpler linear function. We jumped into a fourth degree polynomial – again with failure due to a leading coefficient of 0. Here is where things really started getting promising. I asked why this was happening and a different student remembered something about even and odd symmetries. The precise language did not arrive right away, but we were able tp get that together as well. Pretty promising… A fifth degree polynomial was found and it graphed even better than the third degree. The students were getting a little tired of this process so I very quickly convinced them of the behavior of the 7th degree approximation. Michael (he was on fire this morning!) recognized the factorial pattern unfolding so we jumped ahead to the 9th degree polynomial. We were feeling pretty good about ourselves at this point. I asked them what function we might be interested in next and, luckily, I was told that cos x would be our next target. I told them that I would be quiet for the next few minutes while they worked this out for themselves. Normally, I am not at all interested in my students – especially ones at this level – simply mimicking my solution patterns. In this case, I thought that this new process was intimidating enough that they would just try to parrot my work. I was fine with that idea, this unit will take some time. However, my best laid plans were foiled. About a minute after I sat down dramatically Michael asked ‘Why don’t we just replace each x with x + pi/2?’ I was SO HAPPY, but i tried to hide that for a moment. Luckily, he spoke pretty quietly and his classmates were still working. I went back to GeoGebra and wrote a new function in his honor. Taking our last guess of h(x) which was our 9th degree polynomial and writing m(x) = h(x + pi/2) and I displayed this graph on top of the graph of cos x. It was a fantastic match but it did not have the symmetry that we had seen for the sin x approximations. The students who had plowed ahead with the polynomial model gave me their 9th degree solution and we looked at three graphs together. The cos x graph, the shifted sin x Taylor series and the cos x Taylor series. A really terrific conversation ensued. Today is what we call a T day where we have 50 minute classes. This felt like an enormously productive 50 minutes. I hope that the afternoon goes at least half as well.