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Student[Calculus1][TaylorApproximation] - demonstrate Taylor approximations

Calling Sequence

TaylorApproximation(f(x), x = c, a..b, opts)

TaylorApproximation(f(x), c, a..b, opts)

Parameters

f(x)

-

algebraic expression in variable 'x'

x

-

name; specify the independent variable

c

-

algebraic expression; specify initial point

a, b

-

algebraic expressions; specify the plot range

opts

-

equation(s) of the form option=value where option is one of functionoptions, iterations, order, output, pointoptions, showfunction, showpoint, showtaylor, tayloroptions, view, or Student plot options; specify output options

Description

• 

The TaylorApproximation(f(x), x=c) command returns the 1st order Taylor approximation of the expression fx 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 c1,c+1 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, order, and output) set plot options.

  

 

  

functionoptions = list

  

A list of options for the plot of the expression fx.  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 order. This option is ignored if the option order is a range.

  

 

  

order = posint or posint..posint

  

The order 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 order), is displayed.

  

 

  

pointoptions = list

  

A list of options for the plot of the point c,fc. 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 fx is plotted.  By default, the value is true.

  

 

  

showpoint = true or false

  

Whether the point c,fc 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.

Examples

withStudent[Calculus1]:

TaylorApproximationsinx,x=1

cos1xcos1+sin1

(1)

TaylorApproximationⅇxx,x=2,order=1..3

ⅇ2xⅇ2x,ⅇ2ⅇ2xx+12ⅇ2x2,13ⅇ2+ⅇ2xx12ⅇ2x2+16ⅇ2x3

(2)

TaylorApproximationcoshx,x=2,output=plot,order=4

TaylorApproximationcoshx,2,output=plot,order=1..5

The command to create the plot from the Plotting Guide is

TaylorApproximationsinx,view=π..π,2..2,output=plot

See Also

plot/options, plot/typesetting, Student, Student plot options, Student[Calculus1], Student[Calculus1][TaylorApproximationTutor], Student[Calculus1][VisualizationOverview]


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