## Back in the Groove

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.

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.

## Fantastic Afternoon from BC Calc

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