Humbling the Calculus Students

I have a pretty good mental model of the typical student in my calculus class. Specifically, they can compute derivatives of polynomial functions (even though I haven't taught that yet), they wish to use this skill as often as possible (even if the problem doesn't call for it), and they are absolutely helpless when faced with a graph without its equation (insert snarky comment of your choice about graphing calculators).

If you are teaching Calc 1 out of the same textbook that I am, then I highly recommend devoting a class period to the graph problems from the groupwork (found in the Big Binder -- ask your book rep if you don't have one) from section 2.8.

In one activity there are graphs of eight functions, and the students are supposed to sketch the graph of the derivative of each function. It's amazing how resistant they are to following the directions, estimating the slope of the function at several points, and visualizing the derivative as the slope of the tangent line. Instead, they try to guess the equations of the graphs and to calculate the derivative and then graph the result with their calculators. Unfortunately, my students can't recognize classic graphs. They misidentified the graph of natural log as square root; for them everything U-shaped is a parabola. This is one activity where the students that haven't had calculus before do much better than the ones who have had calculus before.

Tomorrow we'll be doing the companion activity, in which there is a graph of a function and its first and second derivatives, and the students will have to identify which graph is which.

The theme of these lessons: Neither your graphing calculator nor your derivative-taking skilz will help you now.