Taylor Approximations

This applet is designed to let students visualize how Taylor polynomials can be thought of as better and better approximations of a function.  The applet lets you look at the first 4 Taylor polynomials of a function, as well as the error terms obtained by subtracting the approximation from the original function.  Since we are looking at approximation, the applet evaluates the function and the approximations at a distance delX from the original point.

Click on the button below to launch the applet.
• The applet has a control window and 2 graphics windows.
• The Taylor Polynomials window shows a graph of the function in green, with a red dot on the axis at the hot spot (P_1, 0), and another red dot on the graph at (P_1, f(P_1).  Depending on the checkboxes in the control window, you are also shown the approximations of degree 1 (blue), degree 2 (purple or magenta), degree 3 (yellow), and degree 4 (teal or light blue).   A yellow dot is shown on the axis at (P_1 + delX, 0), with a yellow line through the point to help the user visualize the effectiveness of the approximations.
• The ErrorTerms window plots the function minus the approximation with the same color coding.
• The control window lets you specify f(x), the domain the function will be plotted over, and delX.  It lets you use checkboxes to determine which approximations will be shown.  You are also shown the values of the function and the approximating polynomials at x = (P_1 +delX)

Things to notice:
• Raising the degree of the approximation increases the size of the region where the polynomial is a very good approximation.
• The default example is a 4th degree polynomial.  The Taylor quartic approximation to this function is the original function.
• On the other hand, if we look at y=x^4, it is clear that the lower order Taylor approximations at not gotten by simply truncating f(x).
• Consider the function y = sin(Pi*x).  Find special points where the quadratic and quartic polynomials are the same.

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University.  It is used with permission.   Go to the Banchoff Applet Help page.