return the Taylor series for a multivariate function

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Student[MultivariateCalculus][TaylorApproximation] - return the Taylor series for a multivariate function

	Calling Sequence
	TaylorApproximation(f(x,y,...), [x,y,...]=[a,b,...], order, x=xmin..xmax, y=ymin..ymax, ..., opts)

Parameters

f(x, y, ...)	-	algebraic expression
x, y, ...	-	name; specify the independent variables
a, b, ...	-	name or real constants; point around which the function is expanded
order	-	(optional) positive integer; order of taylor polynomial
xmin, xmax, ymin, ymax, ...	-	(optional) real constants specifying range over which to plot function
opts	-	(optional) equation(s) of the form option=value where option is one of centeroptions, functionoptions, output, showcenter, showfunction, tayloroptions; specify output options

Description

•	The TaylorApproximation command returns the Taylor approximation of f of a specified order 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 or . For more information on plotting options, see plot3d/options.

functionoptions = list

Specifies the plot options for plotting the function . 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 order 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 . 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.