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The MeanValueTheorem(f(x), x=a..b) command returns a plot of the expression $f\left(x\right)$ from a to b and indicates the points between a and b where the derivative is equal to the slope $\frac{f\left(b\right)-f\left(a\right)}{b-a}$.

If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.

The opts argument can contain any of the Student plot options or any of the following equations that (excluding output and numeric) set plot options.

functionoptions =list

A list of options for the plot of the expression $f\left(x\right)$. By default, the expression is plotted as a solid red line. For more information on plot options, see plot/options .

lineoptions =list

A list of options for the plot of the line from $\left(a\,f\left(a\right)\right)$ to $\left(a\,f\left(b\right)\right)$. By default, this line is plotted as a dashed blue line. For more information on plot options, see plot/options .

numeric =true or false

Whether to use numeric methods (using floating-point computations) to find the points satisfying the mean value theorem. This option is ignored if the option output is set to plot. By default, the value is false.

output =points or plot

This option controls the return value of the function.

output = plot specifies that a plot, which shows the expression and the points where the mean value theorem is satisfied, is displayed. This is the default.

output = points specifies that the points that satisfy the mean value theorem are returned. Plot options are ignored if output = points.

pointoptions =list

A list of options for the plot of the end points as well as where the expression has derivative equal to $\frac{f\left(b\right)-f\left(a\right)}{b-a}$. By default, the end points are plotted as green circles. For more information on plot options, see plot/options .

showfunction =true or false

Whether the expression $f\left(x\right)$ is plotted. By default, the value is true.

showline =true or false

Whether the line from $\left(a\,f\left(a\right)\right)$ to $\left(b\,f\left(b\right)\right)$ is plotted. By default, the value is true.

showpoints =true or false

Whether the end points and points where the derivative equals $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ are marked. By default, the value is true.

showtangents =true or false

Whether the tangent lines where the derivative is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ are plotted. By default, the value is true.

showverticallines =true or false

Whether the vertical lines from the function where the derivative is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ to the horizontal axis are plotted. By default, the value is true.

tangentoptions =list

A list of options for the plot of the tangents where the expression has derivative equal to $\frac{f\left(b\right)-f\left(a\right)}{b-a}$. By default, these are plotted as solid blue lines. For more information on plot options, see plot/options .

tangentlength =half or positive

The length of the tangent lines. By default, they are half the distance from a to b.

verticallineoptions =list

A list of options for the plot of the vertical lines from the function where the expression has derivative equal to $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ to the horizontal axis. By default, these are plotted as dotted pink lines.

caption =anything

A caption for the plot.

The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting . For more information on plot options, see plot/options .

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right)\:$

$\mathrm{MeanValueTheorem}\left(x^3-x\,x=0..2\,\mathrm{output}=\mathrm{points}\right)$

$\left[\frac{2\sqrt{3}}{3}\right]$

$\mathrm{MeanValueTheorem}\left(\sin \left(x\right)\,1..5\right)$

$\mathrm{MeanValueTheorem}\left(\mathrm{sqrt}\left(x\right)\,x=0..5\right)$

$\mathrm{MeanValueTheorem}\left(x^3-x\,x=0..2\right)$