David Little
Mathematics Department
Penn State University
Eberly College of Science
University Park, PA 16802
Office: 403 McAllister
Phone: (814) 865-3329
Fax: (814) 865-3735
e-mail:dlittle@psu.edu

Tangent Circles and the Curvature of a Function

The following applet can be used to measure the curvature of the function f(x) and x=a. Simply enter the function f(x) and the values a, b and c. The applet automatically draws the unique circle through the points (a,f(a)), (b,f(b)) and (c,f(c)). As the values a, b and c approach each other, the radius of this circle approaches the radius of the circle tangent to f(x) at x=a. The curvature of f(x) at x=a is defined to be the reciprocal of the radius of this tangent circle.

The values a, b, and c can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change these values by using the up/down arrow keys or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x-axis in radians (i.e. multiples of pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.

Here is a list of functions that can be used with this applet.