Estimate slope at point on graph

Dick Lane
Lefthand figure has two interactive components. Red dot on x-axis specifies input a to function f and an arrow points from it to point (a,f(a)) on the graph of f. Slider H chooses a number between -0.5 and 0.5; point a+H is marked on the x-axis and an arrow from it points to corresponding point on graph. Those two points on the graph determine a line; slope of this line estimates the rate-of-change for function f at input a.
The righthand figure shows a graph of possible estimates for rate-of-change for f at a; the marked dot shows the estimate produced by the current state of the lefthand figure. Get progressively better estimates for rate of f at a by choosing H to be closer to zero (using its slider in left figure). Animation, with values of H varying, can be stopped/started using the control in lower left corner --- notice the coordinated changes in the two figures. The expression for function f can be changed using the InputBox in lower-right corner of left figure, e.g., try x*(x-1)*(x-3) or x/(1+x^2) Viewing region of each figure can be shifted or zoomed. Upper-right corner of left figure has a button to reset everything.