The Difference Quotient (Illustrated)

This interactive GeoGebra applet visualizes the difference quotient (secant slope) for a continuous function f over an interval [a,b]. It demonstrates how the secant line's slope changes as points a and b are dragged, and how the difference quotient approaches the derivative as the interval shrinks. Core concepts include the difference quotient, secant lines, and the limit definition of the derivative. The construction uses Textfield and InputBox for dynamic parameters, Line and Segment to draw the secant, Slope to compute and display the slope, Polygon to shade the geometric region, FractionText to show the quotient as a fraction, and Intersect for key points. Suitable for high school or introductory college calculus students, this applet supports classroom demonstrations and self-guided exploration of the relationship between average and instantaneous rates of change.