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
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 =
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
Consider the function y = sin(Pi*x). Find special points
where the quadratic and quartic polynomials are the same.