Taylor Approximation - Maple Help

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, degree, 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 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$
 > $\mathrm{TaylorApproximation}\left(\mathrm{sin}\left(x\right),x=1\right)$
 ${\mathrm{cos}}{}\left({1}\right){}{x}{+}{\mathrm{sin}}{}\left({1}\right){-}{\mathrm{cos}}{}\left({1}\right)$ (1)
 > $\mathrm{TaylorApproximation}\left(\mathrm{exp}\left(x\right)-x,x=2,\mathrm{degree}=1..3\right)$
 ${{ⅇ}}^{{2}}{}{x}{-}{{ⅇ}}^{{2}}{-}{x}{,}{{ⅇ}}^{{2}}{-}{{ⅇ}}^{{2}}{}{x}{-}{x}{+}\frac{{{ⅇ}}^{{2}}{}{{x}}^{{2}}}{{2}}{,}{-}\frac{{{ⅇ}}^{{2}}}{{3}}{+}{{ⅇ}}^{{2}}{}{x}{-}{x}{-}\frac{{{ⅇ}}^{{2}}{}{{x}}^{{2}}}{{2}}{+}\frac{{{ⅇ}}^{{2}}{}{{x}}^{{3}}}{{6}}$ (2)
 > $\mathrm{TaylorApproximation}\left(\mathrm{cosh}\left(x\right),x=2,\mathrm{output}=\mathrm{plot},\mathrm{degree}=4\right)$
 > $\mathrm{TaylorApproximation}\left(\mathrm{cosh}\left(x\right),2,\mathrm{output}=\mathrm{plot},\mathrm{degree}=1..5\right)$

The command to create the plot from the Plotting Guide is

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