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
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.