The other day in my maths class the students were asked to find the average gradient for two points on a parabola. What they did was use the derivative to find the gradient at each point and then average those gradients. Guess what? They all got the answer that I expected but none of them did it correctly (simple rise over run using two points). Well, I was a little surprised, but of course I wondered if the two "averages" were the same for parabolas. The drawing below simply shows two points, [i]A[/i] and [i]B[/i], that you can move around on the parabola (which can also be moved), and the average gradient (from [i]A[/i] to [i]B[/i]) always equals the average of the two gradients at [i]A[/i] and [i]B[/i]. With a little algebra it can be shown that for a function of the form [math]f(x) = ax^2 + bx + c[/math] and two points with [i]x[/i]-values of [i]h[/i] and [i]k[/i], the average gradient = the average of the gradients = [math]ah + ak + b[/math].
So the task, which can be done by someone with a knowledge of basic calculus, is to prove that both "averages" will work out to the same result.