Tuesday, April 10, 2007

What You Don't Know Can't Hurt You -- Except When It Does

Yesterday there was a bumper crop of students in my office hours, asking questions about all sorts of calculus. (There's an exam coming up on Thursday.) We've returned to integration, specifically to the applications of integrals, and I assigned a problem that asked the students to find the volume formed by taking the region bounded by y = ln x, the y-axis, and two horizontal lines and rotating it about the y-axis. It's a relatively straightforward problem if you can tolerate working with x = f(y) -- and if you know the inverse function of ln x.

We drew the region, I put my pencil on a point on the y-axis within the region, and I turned to my student and asked, "How do you find the x-coordinate of the point on the curve that goes with this y?" I moved my pencil to indicate the point on y = ln x.

He didn't know.

I reminded him: "When you have a function y = f(x), and you want to find an x that goes with a given y, you want to think about inverse functions. What's the inverse function of y = ln x?"

He didn't know.

He admitted that his high school didn't teach logarithms. The first time he'd seen a logarithm was in Calc 1, when we introduced derivatives of logarithmic functions. Calculus, chemistry, the Richter scale, decibels, who needs that? It's not like college-bound students planning to major in science would ever have the need to know logarithms, would they? Normally I'd be tempted to disbelieve my student, but this is the second time that a really sharp student has told me that his high school didn't teach logarithms.

I suppose that this is an argument in favor of teaching late transcendental calculus?