The TaylorApproximation command returns the Taylor approximation of f of a specified degree around a specified point. By using options, you can specify that the command returns a plot or animation.  This feature is only available if one or two variables are specified.

The opts argument can contain any of the following equations that set output options.  Plotting is only available for univariate or bivariate functions.

The TaylorApproximationTutor routine offers equivalent capabilities to TaylorApproximation in a tutor interface.  See Student[MultivariateCalculus][TaylorApproximationTutor] help page.

centeroptions = list

Specifies the plot options for plotting the point $x,f\left(x\right)$ or $x,y,f\left(x\,y\right)$.  For more information on plotting options, see plot3d/options.

functionoptions = list

Specifies the plot options for plotting the function $f$. For more information on plotting options, see plot3d/options.

output = value, plot, or animation

This option controls the return value of the function.

* output = value specifies that the value of the approximation is returned. Plot options are ignored if output = value.  The default is output = value.

* output = plot specifies that a plot displays, which shows the expression and the Taylor approximation to the function.

* output = animation specifies that an animation displays, which shows the Taylor approximation approaching the function as the degree of the Taylor series increases with each frame.

showcenter = true or false

Determines whether the point around which the series is expanded is plotted. The default is true.

showfunction = true or false

Determines whether the function f is plotted. The default is true.

tayloroptions = list

Specifies the plot options for plotting the Taylor approximation to f. The default is $\left['\mathrm{color}'='\mathrm{red}'\,'\mathrm{transparency}'=0.5\right]$. For more information on plotting options, see plot3d/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.

For information on how to change the default colors, see the Student[SetColors] help page.

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

$\mathrm{TaylorApproximation}\left(\sin \left(x+y\right)\,\left[x\,y\right]=\left[1\,0\right]\,5\right)$

$\frac{\cos \left(1\right)y^5}{120}+\frac{\sin \left(1\right)y^4}{24}+\frac{\cos \left(1\right)\left(x-1\right)y^4}{24}-\frac{\cos \left(1\right)y^3}{6}+\frac{\sin \left(1\right)\left(x-1\right)y^3}{6}+\frac{\cos \left(1\right){\left(x-1\right)}^2{y}^3}{12}-\frac{\sin \left(1\right)y^2}{2}-\frac{\cos \left(1\right)\left(x-1\right)y^2}{2}+\frac{\sin \left(1\right){\left(x-1\right)}^2{y}^2}{4}+\frac{\cos \left(1\right){\left(x-1\right)}^3{y}^2}{12}+\cos \left(1\right)y-\sin \left(1\right)\left(x-1\right)y-\frac{\cos \left(1\right){\left(x-1\right)}^{2}y}{2}+\frac{\sin \left(1\right){\left(x-1\right)}^{3}y}{6}+\frac{\cos \left(1\right){\left(x-1\right)}^{4}y}{24}+\sin \left(1\right)+\cos \left(1\right)\left(x-1\right)-\frac{\sin \left(1\right){\left(x-1\right)}^2}{2}-\frac{\cos \left(1\right){\left(x-1\right)}^3}{6}+\frac{\sin \left(1\right){\left(x-1\right)}^4}{24}+\frac{\cos \left(1\right){\left(x-1\right)}^5}{120}$

$\mathrm{TaylorApproximation}\left(\sin \left(x+y\right)\,\left[x\,y\right]=\left[1\,0\right]\,5\,\mathrm{output}=\mathrm{animation}\right)$

$\mathrm{TaylorApproximation}\left(\sin \left(x\right)\,\left[x\,y\right]=\left[0\,0\right]\,5\,x=-3..3\,y=-3..3\,\mathrm{output}=\mathrm{plot}\right)$