### The Mean Value Theorem

Graphs a function, with both secant lines and tangent lines for the function, to demonstrate instances of the Mean Value Theorem.
How to use   ||   Examples   ||   Other Notes

Try the Derivative Calculator.
How to use
• Enter a function f (x) in the text input field marked "f (x)=" (Example: f (x) = x2)
• Click the "Graph" button (this button also refreshes the graph)
• Enter the derivative f '(x) in the text input field marked "f '(x)=" (Example: f '(x)=2x)
• Select values for x1, x2, and c (Example: x1=0, x2=1, c=0.5)
This can be done in either of two ways:
• Use the mouse to click and drag the red points on the graph -- the points follow the graph of f (x), and the values of x1 and x2 correspond to the horizontal positions of the points on graph
• Enter the values in the text input fields marked "x1=", "x2=", and "c=", and click the "Graph" button to refresh
• To erase the graph and all input fields (setting x1, x2, and c to default values), click the "Clear" button
The text input field marked "f (x)=" can accept a wide variety of expressions to represent functions. The text input fields marked "x1=", "x2=", and "c=" can accept a real number in decimal notation. The buttons under the graph allow various manipulations of the graph coordinates.

For assistance computing the derivative f '(x), try the Derivative Calculator.

Examples
 Parabola: f (x) = x2 f '(x) = 2x x1=0, x2=1     c=0.5 Exponential: f (x) = ex f '(x) = ex x1=-1, x2=1 c=0.161439

Other Notes
The graph shows a secant line for f (x) through the points (x1,f (x1)) and (x2,f (x2)), and a tangent line through the point (c,f (c)). Labels with the text input fields also show the values of f (x1), f (x2), the slope of the secant line, and f (c). The Mean Value Theorem states that, as long as f '(x) is continuous, there must be a value of c between x1 and x2 for which the secant line and the tangent line are parallel. This applet allows the user to experiment with the secant line and tangent line, attempting to find such a value for c.