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

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Cauchy's Mean Value Theorem For Derivatives

Cauchy's Mean Value Theorem states that if f(x) and g(x) are continuous on [a,b] and differentiable on (a,b), and g(a) is not equal to g(b) then there exists a number c between a and b such that

The following applet can be used to approximate the values of c that satisfy the conclusion of Cauchy's Mean Value Theorem. Simply enter the functions f(x) and g(x) and the values a, b and c. The applet automatically draws the secant line through the points (g(a),f(a)) and (g(b),f(b)), the line through the point (g(c),f(c)) parallel to the secant line and the line tangent to the curve (g(x),f(x)) at x=c.

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.

© 2008 David P. Little
Download this applet for off-line viewing (includes source code). The above applet uses the Java Math Expression Parser (JEP) developed by Singular Systems