- John Golden
This sketch hopes to connect our tangent line idea of derivative with the epsilon-delta limit definition of the derivative. The idea is that if the derivative exists at a point, and you give me an acceptable amount of error, I can tell you how close you have to be to the point to guarantee that the secant line is closer than the specified error. In other words I can get as close to the slope of the tangent line as necessary. For these three cases, try this at a number of different points on the curve, especially any special points. Set an epsilon, and see if you can find a delta so that the secants stay within the acceptable range. Adjusting t lets you see all the secants in the delta-range you selected, and you can do that manually or by hitting play.
For the points where the functions are non-differentiable, can you put in your own words why the function doesn't meet the epsilon-delta idea? Extension 1: No function is differentiable at a point where it is not continuous. Can you use the epsilon-delta idea to explain why? Extension 2: Can you design a piecewise function with two different quadratics that is differentiable? Prove that it is differentiable at the joining point! More GeoGebra at mathhombre.blogspot.com