I'm teaching AP Calculus BC for the first time this year. I was sort of excited about it. I taught AB calculus last year and in the school where I teach, BC students have already taken AB. Which means that I get a bunch of (really awesome) kids for second year. Then there was the realization that there actually isn't that much new content that is BC only and the stuff that is new mostly fits into the story you already unfolded last year. At least, that's how I feel right now.
Of course, then there are Taylor Series. I only vaguely remembered these from my college courses, and somehow I managed to avoid them in grad school. My recollection is that I had no idea what they were or why they were used. And then there was the fear that I'm going to be forced to, the horror!, tell my students how to do something instead of getting them to explore the idea. So, I took an AP summer institute and it assuaged many of my fears. The instructor was good and did develop the idea of Taylor series and motivated their use. I was pretty excited about it.
Then I remembered that I pretty extensively used James Tanton's Thinking Mathematics (Vol 6) in developing the AB course that I really liked to teach. Fortunately for me he also has a Volume 7 for BC calculus! He has an excellent chapter (actually, he has lots of excellent chapters, but for the purposes of this lesson, there's only one) on motivating series. It goes something like this:
A lot of functions are difficult to evaluate, but polynomials are not one of them. Maybe, I wonder if, we could develop a polynomial that could be used to approximate the values of a not-nice function. He leads the reader through the development of the polynomial, which is really just a series). But what makes Tanton's books awesome is that you feel like he's trying to engage you in a conversation. So my job is taking that conversation apart and making it an activity for students to explore. Right now I'm thinking that I have small groups of 3 or 4. Each group gets a different function (sin, cos, e, ...) and get them thinking:
What would this mystery approximating polynomial look like if the function and the polynomial only agreed at 1 point? (Who knows? Couldn't it just be a horizontal line?)
What about the slopes? Should the slopes agree at that one point? (That makes sense... isn't that a derivative?)
How good is a tangent line at approximating the values of the function? (Good in a small range around the tangent point; pretty bad beyond that.)
What if you tried to make the function and the polynomial have the same concavity too? (Really? Second derivative? I guess I can try...)
Look at the graph of the function and the approximating polynomial. How close are they? Does the polynomial do a good job of estimating the points? By how much? When does the polynomial start to fail? What do you think you could do to get a better polynomial?
Then jigsaw the groups and get some similarities and differences going.
That's my idea right now. It'll go through many mental iterations before it goes through a few print alterations. But I'm looking forward to it.