Thursday, January 22, 2015

U-Substitution

When we work u-substitution problems in calc, the kids sometimes drop things like powers or a base of e while they're re-writing their integral.  Also, sometimes they don't quite see which parts of the original integral they've taken care of, and which they still need to work on.

So, I had an idea about five minutes before I was to teach u-substitution this year.  I call it the highlight-out method.  I think it's easier just to show a slide with two examples rather than try to explain in words:


I had the kids "highlight out" the du portion so they could focus on what's left.  Alternatively, you could have them highlight u in one color and du in another.

It may help some; it won't help others, but I think it's a step in the right direction for me.

*****

Another thing I get asked a lot is, "What happened to the du?"  This is a way I explain indefinite integrals that I've found helpful:
  • The indefinite integral symbol and the differential dx (or du or d-whatever) TOGETHER are a command that mean "Find the family of antiderivatives."
  • Once you have found an antiderivative, the two symbols disappear because you have completed the command.
  • You cannot have an integral symbol without a differential[1]; they're akin to a capital letter and period.
That has seems to help a little.  Nothing ground-breaking here, but just some thoughts on u-substitution.  Would love to hear other ideas!


Update:
Here's a slide that seemed to clear things up a little bit more:

One kid told me the last example actually shed a lot of light.  Hooray!



[1]  Yes, I know, technically you can; I've taken Calculus on Manifolds, but these are Calc AB kids, ok?

3 comments:

  1. Thank you! I just taught this, and my students are having the same problem and question.

    Also, I've been going back to rectangles a lot when asked about du. I remind them that the integral means add, the function is the height of a rectangle, and the dx was the width of the rectangle. So once we find the area, we don't care about the width anymore and the dx goes away. Also, once we've added, the integral sign goes away. It helped some of them, but not all.

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  2. Totally! I use the same type of logic when dealing with definite I integrals, but for indefinite integrals, the differential dx becomes a bit more hazy...

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