College Math Teaching

December 21, 2013

Rant: please stop with the teaching of “gimmicks for calculation”

Filed under: calculus, editorial, integrals, student learning, Uncategorized — Tags: — collegemathteaching @ 1:05 pm

I finished teaching calculus II (our course: techniques for integration, applications of integrals and infinite sequences/series) and noticed that some of our freshmen students came in knowing how to do many of the calculations…did well on the first exam…then didn’t do so well in the rest of the course.

Evidently, they were well versed in calculation tricks learned in high school; give them \int x^3 sin(x) dx and they could whip out a table.

So here is my rant: we teach integration by parts not so much to calculate integrals like \int x^3 sin(x) dx (which can be rapidly done with a calculator) but rather so they can understand the technique of integration by parts.

Why? Well, there are many uses of integration by parts and I’ll just display a few uses of them:

1. Taylor Polynomials. How do we get these? If we assume that f has enough derivatives, we proceed in the following manner: calculate \int ^x_0 f'(t) dt in two different ways: use the Fundamental Theorem of calculus on one side (to obtain f(x) - f(0) and use integration by parts on the other side: u = f'(t), dv = dt, du = f''(t), v = t-x (yes, we are being choosy about which anti derivative of dv to use).
This means: -\int^x_0 f''(t)(t-x)dt +f'(t)(t-x)|^x_0 = f(x)-f(0) so f(x) = f(0) + f'(0)x -\int^x_0 f''(t)(t-x)dt =f(x) and one proceeds from there.

2. Differential equations: given y' + p(x)y = f(x) one seeks to find an integrating factor (which is e^{\int p(x)} so as to get:

e^{\int p(x)}y' + p(x)e^{\int p(x)}y = f(x)e^{\int p(x)} which can be written as \frac{d}{dx}(e^{\int p(x)}y) = f(x)e^{\int p(x)}. That is, the left hand side is just the product rule for derivatives, which, as you know (if you are a calculus teacher), is really all integration by parts is!

Sure, one can jazz it up (as we subtly did in the Taylor Polynomial calculation); the integration by parts formula is really \frac{d}{dx} (f(x)g(x)) = \frac{d}{dx}(f(x)+ C) g(x) + f(x)\frac{d}{dx}(g(x) + D) where C, D are arbitrary constants. But, my main point is that integration by parts should be UNDERSTOOD; short cuts to do tedious calculations are relatively unimportant, IMHO.

Now if you want to ask students “why does tabular integration work”, then….GREAT!

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