Over the last couple of days, I’ve thought a lot more about the FTC than I have in a while. For those of you not entirely immersed in the wonderful world of calculus, the FTC is the Fundamental Theorem of Calculus. It beautifully and usefully ties together the worlds of differential and integral calculus, and provides a way for us to handle a huge cross-section of interesting problems.
And it’s not like I don’t think about the FTC–I teach Calculus for Euclid’s sake–I just haven’t given it much more thought than as a tool. I guess that’s more a comment on life in general–you learn something at a basic level (perhaps as a concrete application), then come back to it again (perhaps in a more abstract or symbolic manner), and finally begin to see how something is put together. The problem is, when something like the FTC is internalized in a teacher’s mind, after high school calculus, college calculus, graduate school calculus, and workshop calculus, it’s easy to forget that there’s more to discover, and harder to remember how confusing something is when first you encounter it.
Discovery/Rediscovery 1: Re-thinking point-slope form of a line. (Reflection # 1)
What if you take something as simple as the FTC:
Well, latex isn’t working in wordpress right now, so I’ll do some fun type-y stuff. int(f'(x)dx,a,b) = f(b)-f(a), and rewrite it as a starting point and a sum: f(b) = f(a) + int(f'(x)dx,a,b). You end up looking at something kind of profound. (1) we’ve arrived at a place where we’re saying that a function value is equal to an initial value, plus the accumulated changes between the two x-values. This has ramifications in terms of stats, where everything is linearized, and everything is written as a starting point (y-intercept) plus the accumulated change (slope*x); and in algebra, where we can revisit the idea of point-slope form of a line y-y1=m(x-x1) as a starting point y1, added to the accumulated change m(x – x1). You end up with something I’ve never really had a strong affinity for, though I’ve seen some students write point-slope lines as y = y1 + m(x-x1). I like that this ties to the FTC. I like more that this ties beautifully into Taylor Polynomials–a super-hard topic for students to wade into, made easier by the fact that they have been doing linear approximations with Taylor Polynomials since kindergarten (that’s when you learn to write equations in point-slope form, right?). Anything that can give someone a leg up in life (or calculus), I’ll take.
Discovery/Rediscovery 2: The Other Part of the FTC (Journal Part 2)
I figured it might be nice to combine my journal entries, since I found myself writing thematically about the fundamental theorem of calculus again. This will be shorter–getting students to not only compute, but understand the process of applying the FTC (the one showing derivatives and antiderivatives as inverses) when there’s a chain rule involved. There were two instances I’d like to share here, in hopes that I remember to discuss both later this year.
This is the method that I worked out with my students this year–certainly not what I did in school (think follow the pattern), and not what I’ve seen most math students do (follow the pattern).
Actually perform the symbolic integration, and you end up with something like d/dx(F(g(x))-F(a)); when you take the derivative, the chain rule for the first function is easy to see, f(g(x))g'(x), and the second value (a constant) goes to zero. This is nice way for students to see (1) where the chain rule part comes from, and (2) why the lower bound doesn’t affect the FTC-based result.
Let A(x) = int (f(x)dx,a,x); A'(x) = f(x); A(g(x)) = int (f(x)dx,a,g(x)), A'(g(x))g'(x) = f(g(x))g'(x).
I like the algebraic reasoning here, and can reach another chunk of students, for whom algebra comes plainly, and some of the calculus is a bit overwhelming.