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# What part of (f(x)-f(a))/(x-a) don't you understand?

September 1, 2008, 8:28 pm

Why is the concept of the difference quotient so hard for beginning calculus students to handle? The idea is not as hard as some other concepts at this level that students have fewer problems with. You start with a function f and a point a. You are asked to write, and then simplify completely, the fraction

$$\frac{f(x) – f(a)}{x-a}$$ or $$\frac{f(a+h)-f(a)}{h}$$

This involves four clearly-defined steps. (1) Compute all the function values in the numerator. (2) Perform the subtraction between the two objects in the numerator and simplify. (3) Factor the result out completely, and (4) see if you can find a common factor to cancel. And there’s a step (5): Since you know that every time you’ve done or seen a problem like this, there’s a factor/cancel step at the end, you know you screwed up if there isn’t one.

But somehow, the fact that this is a totally algorithmic, almost automatic process that is the same procedure every single time — and even the slight variations among instances only consist in algebra tricks — doesn’t stop students from suffering a complete brain-freeze at the sight of them. They convince themselves they don’t know how or where to start (despite worked-out examples or even difference quotient exercises that they themselves have worked out before). They plug $$x-f(a)/(x-a)$$ in to f. They use $$f(a)+h$$ instead of $$f(a+h)$$. And so on. There’s a massive wall of intimidation that these exercises lay down, and even those who make it over that wall end up talking themselves into doing all kinds of stuff that is wrong bordering on bizarre. These exercises get inside their heads somehow. And these are young men and women smart and capable enough of getting into college, mind you — not dummies.

The cure for math intimidation is a disciplined heuristic for solving problems and a faith in your algorithms for more mechanical exercises. But with difference quotients somehow the heuristics and algorithms run fleeing like Tokyo residents before Godzilla. It’s not just difference quotients, either — there are lots of algebra components that throw calculus students for an absolute loop, and I cannot figure out why. Any ideas?

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