# To Accumulate a Rate — Integrate!

Teaching High School Mathematics

# Quadratic slopes without a derivative

Note in the graph above that the slope of the tangent line at x = 1 is the same as the slope of the line that goes through the points on the parabola that are both the same distance away from x = 1. It doesn’t matter what distance as long as you go the same distance in each direction.

I did my introductory derivative lessons a little bit differently this year. We manually calculated slopes of two points that are closer and closer to the point where we wanted the slope for more problems than I normally would. We also did it in three ways including choosing a point to the left and right an equal distance from the point in question.

When students used two points on opposite sides of an x value to find the slope of quadratics they would get the correct slope on the first set of points and didn’t need to make the points get closer and closer together. I quickly gave them $f(x)=x^3$ to show them that it doesn’t always work. I was completely intrigued and was wondering if it always works with parabolas. It seemed reasonable, after all the slope of a straight line is the slope between any two of it’s points. Could the slope of a parabola at any point be the slope between two points whose x values are equidistant from the x value of the point you want to find the slope at?

I think that I might have noticed this before while doing the mean value theorem with quadratics, but didn’t think much of it. This week however my students were noticing it and I felt compelled to investigate more.

I was wondering if there is a method for cubics… At first thought my guess is no, but I haven’t proven that it can’t be done. Any thoughts?