College Math Teaching

February 7, 2012

Forgotten Basic Algebra: or why we shouldn’t rely on the “conjugate trick”

Filed under: basic algebra, calculus, derivatives, elementary mathematics, how to learn calculus, pedagogy — collegemathteaching @ 7:01 pm

I’ll admit that, after 20 years of teaching at the university level, I sometimes get lazy. But…as I age, I must resist that temptation even though at times I find myself muttering “I don’t have 30 extra f*cking minutes to figure out how to do this…”

But often if I stick with it, it doesn’t take 30 “f*cking” minutes. 🙂

Here is an example: I was trying to remember how to calculate lim_{z \rightarrow w} \frac{z^{1/3} - w^{1/3}}{z - w} and was trying to remember instead of think. I looked at an old calculus book…no avail…then I was shamed into thinking. About 2-3 minutes later it struck me:
“you know how to simplify \frac{u - v}{u^3 - v^3} don’t you?”

Problem solved…shame WIN.

of course things like lim_{z \rightarrow w} \frac{z^{7/8} - w^{7/8}}{z - w} are easily converted to things like \frac{u^7 - v^7}{u^8 - v^8} , etc.

This leads to another point. Often when we teach lim_{h \rightarrow 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} we use the “conjugate trick” which only works for square roots. The above method works for the other fractional powers.

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