Earlier this year, when we were finding the equation of a line tangent to a function at a given point, one of my students observed the following. He found that if his function was quadratic, if he simply subtracted (x-c)2 from his function (where c was the x-value for which he was finding the equation of the tangent line), this produced his tangent line equation, and he did not need to take the derivative to find it.
For instance, f(x) = x2 + 6x ? 1. Let?s find the equation of the line tangent to the curve at x=2. We get f?(x) = 2x + 6, f?(2) = 10, f(2) = 15, and then we get the tangent line equation y = 10x ? 5. My student took x2 + 6x ? 1 ? (x-2)2 = x2 + 6x ? 1 ? (x2 -4x + 4) = 10x ? 5. (same answer)
My student and I were both able to prove it algebraically and we also considered why it always works from a graphing perspective. This is my second year teaching Calculus, and I thought it was great that my student thought outside the box and made this connection. I didn?t notice this connection on my own, and don?t remember learning it when I took Calculus. I would like some opinions on this. Is this something you teach and I just never noticed it? I?m also curious as to how other teachers would explain ?why it works.? When I talked to my student about it from a graphing perspective, we considered f(x) ? (x-c)2 = tangent line. But perhaps there?s a good graphing explanation thinking of it slightly differently as f(x) ? tangent line = (x-c)2, and considering why this specific parabola is created when you subtract the tangent line equation from the function (I wasn't sure from this perspective).
Thanks for your input! Lorraine Darwin Cabot High School Mathematics Dept. Cabot, AR