Epsilon-Delta Differentiability
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.