Thursday, October 20, 2011

2nd Fundamental Theorem of Calculus

Our last calculus class looked into the 2nd Fundamental Theorem of Calculus (FTOC). We talked through the first FTOC last week, focusing on position velocity and acceleration to make sense of the result. Our interpretation was that the FTOC-1 finds the area by using the anti-derivative. How are those connected?

This week we wanted to peek at the 2nd part. (OK, it was me.) We looked up the result on Mathworld, and talked about barriers to understanding. The teachers identified that the conflation between the antiderivative and the integral (meaning area under a curve) is almost total, so that the theorem is just restating what we already think. Using this completely confusing notation and totally new way to define a function.  Perfect situation for a GeoGebra sketch to allow students to explore.

This sketch uses several GGB4 features.  It uses the integral[ ] command to find the area under a curve, which I would have had to cheat before, the input boxes to allow real freedom of entering a function, and buttons so that students don't have to know GeoGebra commands to refresh the view. This is my first sketch uploaded to GeoGebraTube, which is a huge improvement over the old webhosting at You can link to a teacher page or a student page, there's a link to download the file,  and there's easy to find embed code.  Best of all, the front page has a search, and shows recent uploads so it's fun just to check in.  And everything uploaded is CC3.0; darn near ideal resource.

If the embed code worked on blogger, I would have put it right here. (That's why only darn near ideal.)
Click here to go to the sketch on GeoGebraTube.

The teachers noticed all sorts of interesting things, recognizing anti-derivatives, seeing the +C (constant of integration) in action, and seemed to make sense of the integral definition of a function. They picked interesting functions to try, like increasing degrees of polynomials, trigonometric functions, functions without an analytic antiderivative (like cos(x^2)) and the fabulous e^x.

I've added the grid in to help students see the area more clearly, and set the grid to distance 1 to keep it unit sized.  Linda Fahlberg and John Scammell helped me with the right script for the button with quick Twitter responses. ZoomIn[1] to get a CTRL-F (refresh view) effect, and UpdateConstruction[] to get a CTRL-R effect (recompute all objects).  (Written down so I will never forget again.)

Photo credit: ajlvi @ Flickr. He (?) said he tried to capture everything he needed to know for the GRE on the board and then take a picture, but it was illegible on his phone. If he's that clever, I'm sure he did fine.


  1. >the conflation between the antiderivative and the integral (meaning area under a curve) is almost total, so that the theorem is just restating what we already think.

    When I teach the FTOC, I use a project that was developed at Hope College (small world, eh?). I've modified it a bunch since I started using it. It starts with defining F(x) as the area under f(t) from a stable left side to a right side at x. (So f(t) has to be above the x-axis...)

    We avoid the integral symbol, so there isn't that conflation. I haven't thought enough about velocity and position as ways to understand it. I should do that.

    I've put it in Dropbox, here. And a pdf version here.

  2. Excellent, Sue! Weirdly, I was at Hope when I was writing this post. Spooky and Halloween appropriate.

  3. Is FTOC2 the one that says if you go from here to there, then you gotta touch something in between? I get them mixed up.