The TaylorApproximation(f(x), x=c) command returns the 1st degree Taylor approximation of the expression $f\left(x\right)$ at the point $x=c$. By using options, you can specify that the command returns a plot instead.

The optional argument a..b gives the range of the plot.  If no range is given, the interval $\left[c-1\,c+1\right]$ is used.

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 iterations, degree, and output) 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.

iterations = posint

In an animation, by default $5$ iterations beginning with the approximation given by degree. This option is ignored if the option degree is a range.

degree = posint or posint..posint

The degree of the Taylor approximation.  If a range of values is given, all Taylor approximations in that range are returned or plotted.  By default the value is $1$.

output = polynomial, plot, or animation

This option controls the return value of the function.

output = polynomial specifies that the Taylor polynomial is returned.  Plotting options are ignored if output = polynomial. This is the default.

output = plot specifies that a plot, which shows the expression and one or more Taylor approximations at the point x=c, is displayed.

output = animation specifies that an animation, which shows the expression and the specified Taylor approximations (arranged by approximation degree), is displayed.

pointoptions = list

A list of options for the plot of the point $\left(c\,f\left(c\right)\right)$. By default, this point is plotted as a blue circle. 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.

showpoint = true or false

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

showtaylor = true or false

Whether the Taylor approximation of the expression at the point $x=c$ is plotted. By default, the value is true.

tayloroptions = list

A list of options for the plot of the Taylor approximation. By default, these are plotted as solid blue 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.

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

$\mathrm{TaylorApproximation}\left(\sin \left(x\right)\,x=1\right)$

$\sin \left(1\right)+\cos \left(1\right)\left(x-1\right)$

$\mathrm{TaylorApproximation}\left(\exp \left(x\right)-x\,x=2\,\mathrm{degree}=1..3\right)$

${\ⅇ}^2-2+\left({\ⅇ}^2-1\right)\left(x-2\right),{\ⅇ}^2-2+\left({\ⅇ}^2-1\right)\left(x-2\right)+\frac{{\ⅇ}^2{\left(x-2\right)}^2}{2},{\ⅇ}^2-2+\left({\ⅇ}^2-1\right)\left(x-2\right)+\frac{{\ⅇ}^2{\left(x-2\right)}^2}{2}+\frac{{\ⅇ}^2{\left(x-2\right)}^3}{6}$

$\mathrm{TaylorApproximation}\left(\cosh \left(x\right)\,x=2\,\mathrm{output}=\mathrm{plot}\,\mathrm{degree}=4\right)$

$\mathrm{TaylorApproximation}\left(\cosh \left(x\right)\,2\,\mathrm{output}=\mathrm{plot}\,\mathrm{degree}=1..5\right)$

$\mathrm{TaylorApproximation}\left(\sin \left(x\right)\,\mathrm{view}=\left[-\mathrm{\pi }..\mathrm{\pi }\,-2..2\right]\,\mathrm{output}=\mathrm{plot}\right)$