Let's Do It! - An Example Applet: SplineTest

Alexander Bogomolny

The applet following this paragraph depicts a graph of a function, a parabola at first, over a fixed interval. The graph can be dragged up and down by any of its points. For the resulting function, controls below the drawing area allow one to simultaneously display its derivative and its integral as a function of the upper limit. The function is in fact a natural cubic spline initially interpolating through 11 points evenly distributed on a parabola. Each of the points may be dragged up and down. To see the draggable points check the "Show points" box. If one tries to drag the graph away from one of the existing draggable points, a new draggable point will be created and added to the collection. Finally, if you drag the cursor while the "Show tangent" box is checked, a short tangent line either of the graph of the function or that of the integral will follow the mouse.

It is easy to see the correlation between intervals of monotonicity of the function and intervals on which its derivative preserves the sign. A perceptive student may make a few additional observations:

  1. Rolle's theorem is valid: Between any two zeros of a function there is a zero of the derivative.
  2. The derivative is to the function as the function is to its integral, and vice versa.
  3. Local changes in a function result in local changes in the derivative, but global changes in the integral.

So, perhaps the applet does have instructional value. However, not only has it not passed the screening process, the reason for its existence is to serve as a demonstration for several Java classes that I offer for public scrutiny (see the next section).