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
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
can accept a real number in decimal
buttons under the graph
allow various manipulations of
the graph coordinates.
- Enter a function f (x) in the text input field
marked "f (x)="
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)="
- Select values for x1, x2,
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
and click the
"Graph" button to refresh
- To erase the graph and all input fields (setting
and c to default values), click the
For assistance computing the derivative f '(x), try
the Derivative Calculator.
f (x) = x2
f '(x) = 2x
f (x) = ex
f '(x) = ex
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
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