Saturday, November 26, 2011

Introducing Limits and Derivatives

I know you non-IB people probably cannot fathom this, but Calculus is only one of many units in the IB curriculum. I am introducing it to my kids now, and it's the last big topic I plan on teaching before we start to do mixed review in the spring time.

The reason why Calculus is condensed into a single unit in the IB makes sense to me, actually. Although there are so many applications of Calculus, I would be just as happy if a kid can walk away from Calculus knowing the big ideas of limits, differentiation, and integration, and to be able to do basic differentiation/integration of polynomials by hand without a formula sheet. Everything else, I'm OK with them relying on a formula sheet in order to remember the mechanics. So, it's very do-able as a single unit and to still develop the relevant concepts as a class.

To that end, I have already introduced instantaneous rates vs. average rates. Then, after that I wanted to pull in the idea of limits, which the kids had already seen a little bit of in the context of geometric series. So, I gave the kids this worksheet, and as they worked through it, we set up a huge grid of comparison charts on the board to go over when a "forbidden x value" inside a rational function will be a vertical asymptote, versus a "hole".... and to highlight the idea that a limit is what happens to the theoretical output as you approach those "impossible" x values. We are not done with the worksheet yet, but my hope is that by the start of the next class, the kids will fully grasp the idea that a function with a "hole" somewhere (as opposed to a vertical asymptote) can still have a limit at that breaking point, and many functions also have limits at extreme values of x. That (along with the previous instantaneous rate intro) will prime us for going into talking about the mechanics of differentiation in this next intro to differentiation lesson. One of the things I want to immediately tie into differentiation is that you can check your sensibility of your answers using a graph. I will also right away tie differentiation to the algebraic meaning of turning points. This way, they are immediately exposed to the key concepts in an integrated manner before we do any further practice.

Thoughts?? It's my first time teaching Calculus, so I'm still muddling my way through it while doing my best to sequence the concepts clearly. Your feedback is welcome!


  1. I'm not familiar with IB so I'm not sure if the following suggestions would be helpful.

    If you have to cover continuity (at a point) as part of unit, be sure to also to cover an example of where limit exists as you approach c even when the function is defined at c.

    If you have covered transformations, there's a connection possible in the last example. You can rewrite h(x)=(2x-8)/(x-3) as h(x)=-2/(x-3)+2 by doing long division or synthetic division, where the +2 in the end indicates an upward vertical shift of 2 units which would move the horizontal asymptote of -2/(x-3) up by 2.

  2. Thanks for the reminder. I assume you mean when f(x1) = c but the limit as x->x1 of f(x) is equal to some other constant? That'd be easy enough to pull in and illustrate using a graph.