Mean Value Theorem - Maple Help

Student[Calculus1]

 MeanValueTheorem
 graphically display the mean value theorem

 Calling Sequence MeanValueTheorem(f(x), x = a..b, opts) MeanValueTheorem(f(x), a..b, opts)

Parameters

 f(x) - algebraic expression in variable 'x' x - name; specify the independent variable a, b - algebraic expressions; specify the end points of the curve opts - equation(s) of the form option=value where option is one of functionoptions, lineoptions, numeric, output, pointoptions, showfunction, showline, showpoints, showtangents, showverticallines, tangentoptions, tangentlength, verticallineoptions, or Student plot options; specify plot options

Description

 • 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.

Examples

 > $\mathrm{with}\left({\mathrm{Student}}_{\mathrm{Calculus1}}\right):$
 > $\mathrm{MeanValueTheorem}\left({x}^{3}-x,x=0..2,\mathrm{output}=\mathrm{points}\right)$
 $\left[\frac{{2}{}\sqrt{{3}}}{{3}}\right]$ (1)
 > $\mathrm{MeanValueTheorem}\left(\mathrm{sin}\left(x\right),1..5\right)$
 > $\mathrm{MeanValueTheorem}\left(\sqrt{x},x=0..5\right)$

The command to create the plot from the Plotting Guide is

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