Tuesday, May 8, 2007

Substitutions in Calculus - U Can Do It (Maybe Not)

With the AP Exams this week and the year coming to an end, the timing of this post is way off but I wanted to share some thoughts about using u-substitution in integration (and differentiation). Every text promotes it, I've been teaching it for years, but, like so much else in mathematics, u-substitution is really a form of 'information-hiding' that may sacrifice understanding for the sake of brevity. While I believe using u in place of f(x) is an important technique for some calculus algorithms, experience has shown me that keeping f(x) in the formulas works better during the learning phase. In this post I will show how using f(x) in the formulas for integrals may help students deal with the variety of functions they encounter when integrating using the 'Inverse Chain Rule' (I coined that phrase to help students recognize that the f'(x) factor disappears when anti-differentiating (integrating) and appears when differentiating. If your first reaction is that these formulas are more complicated, I think you might need to actually experience the difference in students' reactions. There's no substitute for that!!

If you've already tried both approaches, let me know which form worked better for a majority of your students. My impression is that many students are at first overwhelmed by the varieties of different forms that appear in the typical set of exercises following this lesson. We can say that they're basically all the same, but it doesn't appear that way to the novice!

Outline of Lesson:
The topic was introduced by reviewing several differentiation problems using the General Power Rule for d/dx(f(x)), using the Chain Rule. Each derivative was immediately followed up with the corresponding antiderivative problem. The idea that both (f(x))n and f'(x) must be present in the integrand was emphatically stressed. I repeatedly asked the question, "What is the f(x) here?" then, "What would f'(x) be?" Also, "Is the exact f'(x) in the integrand?" If not, we discussed the method of supplying a missing numerical factor and its reciprocal, using properties of antiderivatives, etc. They seemed to get this, struggling a bit with the idea of using differentiation in an integration problem, a natural source of confusion at first. We made sense of all this for the first few examples by differentiating our answer to check it. As with all new ideas, some caught on quickly, others had not yet internalized it after 3-4 examples because I too quickly moved into variations on the basic form before they had really processed it.

The following are the 'forms' that helped them adjust to the variety of trigonometric and other types of examples. This seemed to eliminate much confusion...

Some ∫ form(ula)s:

1. (General Power Rule): ∫f'(x)(f(x))ndx = (f(x))n+1 /(n+1) + C,
n ≠ -1

2. ∫cos(x)sinn(x)dx = (sinn+1(x))/(n+1) + C, n≠ -1

3. ∫f'(x)sin(f(x))dx = -cos(f(x)) + C
4. ∫f'(x)cos(f(x))dx = sin(f(x)) + C
Note: There is a critical but subtle difference between #2 and #3, even though they are both applications of the Chain Rule.

5. ∫f'(x)sec2(f(x))dx = tan(f(x)) + C
6. ∫f'(x)sec(f(x))tan(f(x))dx = sec(f(x)) + C

7. ∫f'(x)ef(x)dx = ef(x) + C

8. ∫f'(x)/f(x)dx = ln|f(x)| + C vs. ∫du/u = ln|u| + C

There are many others, but you get the idea.
If you still believe that the u-forms are so much simpler, that's fine, since I like the u-method as well and it is needed later on. I think all experienced math teachers latch on to their favorite methods or models because they have worked effectively with many students. However, I do not believe that any method will be effective for all due to learning style issues. There may be some students who will simply feel more comfortable and perform better if they use Method A vs. Method B.

I should have included some of the actual examples used, but...


Alex McFerron said...

I enjoy your blog very much. I have started my own blog and I am wondering what software you use to get the math symbols (like integration) on your blog?

Dave Marain said...

Best of luck to you! I am coming to realize there are unlimited opportunities for new bloggers in the 'blogosphere'. As long as your content is interesting to a few, your blog will catch on. Post a few more comments like this on others' blogs and don't be shy about inviting readers to visit your new site! I find it ironic that I'm somehow considered experienced at this and I've only been seriously posting since January of this year! I'm no techie, but I've learned a lot from those more knowledgeable than myself who freely share their wisdom and experience. Google 'html math symbols' and you'll find some excellent resources for unicode that allows you to enter many math symbols. Blogger does not yet allow entering formulas in LaTeX which would make for beautiful formatting but I'm surviving with a few codes. I copy the most common ones in my notebook and refer to it as needed. More complicated formulas or diagrams are entered in Word and stored as images.
Thanks for the nice words!

Eric Jablow said...

Alex, look at Brian Wilson's page on character entities.

Dave, I prefer doing the symbolic replacements as fast as possible because that lets me (suitably simplified) show how f:X→Y induces linear maps on the tangent spaces Df: TX→TY. Granted, in the one-dimensional case, locally, TX and TY are both just R. But for a multi-variable case, one sees that the chain rule is just matrix multiplication.

Alex McFerron said...

Great, thanks so much.

zac said...

Dave - thanks for this post and it is interesting that the students warm to the approach.

I think of each integral in this order:
∫(f(x))^n f'(x)dx

Then later, when introducing du/dx, it is conceptually more akin to f'(x)dx.

On the issue of typing mathematics in blogs, one way to reduce the nightmare is to use the free Windows LiveWriter (yes, amazingly I am recommending a Windows product...! No Mac version, sorry.)

It works with "Windows Live Spaces, as well as SharePoint, Blogger, LiveJournal, TypePad, WordPress, Community Server".

For equations, you can create them in Word, copy and paste into LiveWriter and you don't have to upload any images separately when you publish.