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Student[Calculus1]

 RollesTheorem
 illustrate Rolle's theorem

 Calling Sequence RollesTheorem(f(x), x = a..b, opts) RollesTheorem(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 epsilon, functionoptions, lineoptions, numeric, output, pointoptions, showfunction, showline, showpoints, showtangents, tangentlength, tangentoptions, or Student plot options; specify plot options

Description

 • The RollesTheorem(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 zero.
 • 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.
 epsilon = positive
 A non-negative value used to determine whether the values of the function f(x) at the end points of the interval are equal. By default, this value is $0.01$.
 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 Rolle's 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 Rolle's theorem is satisfied, is displayed. This is the default.
 – output = points specifies that the points that satisfy Rolle's 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 zero derivative.  By default, the end points are plotted as blue circles and the points of zero derivative as black 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 of zero derivative are marked.  By default, the value is true.
 showtangents = true or false
 Whether the tangent lines where the function is zero are plotted. By default, the value is true.
 tangentlength = half or positive
 The length of the tangent lines.  By default, they are half the distance from a to b.
 tangentoptions = list
 A list of options for the plot of the tangents where the expression has zero derivative.  By default, these are plotted as solid black lines. For more information on plot options, see plot/options.
 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.
 title = anything
 A title for the plot.
 The default title is constructed from the parameters and the command options. title = "" disables the default title. For more information about specifying a title, see plot/typesetting.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Calculus1}]\right):$
 > $\mathrm{position}≔-0.5\cdot 9.8{t}^{2}+12t$
 ${\mathrm{position}}{:=}{-}{4.90}{}{{t}}^{{2}}{+}{12}{}{t}$ (1)
 > $\mathrm{RollesTheorem}\left(\mathrm{position},t=0..2.448979592,\mathrm{output}=\mathrm{points}\right)$
 $\left[{1.224489796}\right]$ (2)
 > $\mathrm{RollesTheorem}\left(\mathrm{position},t=0..2.448979592,\mathrm{labels}=\left["time","distance"\right]\right)$
 > $\mathrm{RollesTheorem}\left({x}^{2}-3x+1,-\frac{1}{2}..\frac{7}{2}\right)$
 > $\mathrm{RollesTheorem}\left(\mathrm{sin}\left(x\right),1..5\mathrm{π}-1\right)$

The command to create the plot from the Plotting Guide is

 > $\mathrm{RollesTheorem}\left({x}^{4}-3{x}^{2}+1,x=-2..2\right)$