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:
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 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
The text input field marked "f (x)=" can accept
a wide variety of expressions to represent functions.
The text input fields marked
"x_{1}=",
"x_{2}=", and
"c="
can accept a real number in decimal
notation. The
buttons under the graph
allow various manipulations of
the graph coordinates.
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.