How to use
Enter a function f (x) in the text input field
marked "f(x)=".
Example:
f (x) = x^{2}
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 x_{1}, x_{2},
and c.
Example:
x_{1}=0, x_{2}=1,
c=0.5
This can be done in either of two ways:
- Click and drag the
red points on the graph using the mouse --
the points follow the graph of f (x), and the
values of x_{1} and x_{2}
correspond to the horizontal positions of the points on graph.
- Enter the values in the text input fields marked
"x_{1}=",
"x_{2}=", and
"c=",
and click the
"Graph" button to refresh.
To erase the graph and all input fields (setting
x_{1}, x_{2},
and c to default values), click the
"Clear" button.
Examples
Parabola:
f (x) = x^{2},
f '(x) = 2x,
x_{1}=0, x_{2}=1,
c=0.5
Exponential:
f (x) = e^{x},
f '(x) = e^{x},
x_{1}=-1, x_{2}=1,
c=0.161439
Other Notes
The graph shows a secant line for f (x) through
the points (x_{1},f (x_{1}))
and (x_{2},f (x_{2})),
and a tangent line through the point
(c,f (c)).
Labels with the text input fields also show the values of
f (x_{1}), f (x_{2}),
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 x_{1} and x_{2}
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.