compute a Taylor polynomial approximation - Maple Help

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Student[NumericalAnalysis][TaylorPolynomial] - compute a Taylor polynomial approximation

 Calling Sequence TaylorPolynomial(f, x, opts) TaylorPolynomial(f, x = x0, opts)

Parameters

 f - algebraic; expression in the variable x x - name; the name of the independent variable in f x0 - realcons; the point about which the series is expanded opts - (optional) equation(s) of the form keyword = value where keyword is one of digits, errorboundvar, order; the options for computing the Taylor polynomial

Description

 • The TaylorPolynomial command computes the taylor series expansion of f about x0 and converts it into a polynomial.
 • If x0 is not specified, then the Taylor expansion is calculated about the point x=0.
 • The possible return values of the TaylorPolynomial command are:
 – P
 P is the n-th order Taylor polynomial of f calculated about x = x0
 – [P, R]
 P is the n-th order Taylor polynomial of f calculated about x = x0 and R is the remainder term, such that P+R=f. This output is returned when the errorboundvar option is specified and the extrapolate option is not specified.
 – [P, R, B]
 P is the n-th order Taylor polynomial of f calculated about x = x0, R is the remainder term, such that P+R=f and B is a list containing the extrapolated point(s), the value(s) of P at the extrapolated point(s), the value(s) of f at the extrapolated point(s) and the associated error bound(s). This output is returned when both the errorboundvar and the extrapolate options are specified.
 – If order is specified as a list or range the return will be an expression sequence of the output described above, with an expression sequence member for each order.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):$
 > $\mathrm{TaylorPolynomial}\left(\mathrm{sin}\left(x\right),x\right)$
 ${x}{-}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}\frac{{1}}{{120}}{}{{x}}^{{5}}$ (1)
 > $\mathrm{TaylorPolynomial}\left(\mathrm{sin}\left(x\right),x,\mathrm{errorboundvar}='\mathrm{ξ}'\right)$
 $\left[{x}{-}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}\frac{{1}}{{120}}{}{{x}}^{{5}}{,}{-}\frac{{1}}{{5040}}{}{\mathrm{cos}}{}\left({\mathrm{ξ}}\right){}{{x}}^{{7}}\right]$ (2)
 > $\mathrm{TaylorPolynomial}\left(\mathrm{sin}\left(x\right),x,\mathrm{errorboundvar}='\mathrm{ξ}',\mathrm{extrapolate}=1.3\right)$
 $\left[{x}{-}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}\frac{{1}}{{120}}{}{{x}}^{{5}}{,}{-}\frac{{1}}{{5040}}{}{\mathrm{cos}}{}\left({\mathrm{ξ}}\right){}{{x}}^{{7}}{,}\left[{1.3}{,}{0.9647744166}{,}{0.9635581854}{,}{0.001245010258}\right]\right]$ (3)
 > $\mathrm{TaylorPolynomial}\left({ⅇ}^{x},x,\mathrm{order}=3\right)$
 ${1}{+}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}$ (4)
 > $\mathrm{TaylorPolynomial}\left(\mathrm{ln}\left(x\right),x=2,\mathrm{order}=3,\mathrm{errorboundvar}='\mathrm{ξ}'\right)$
 $\left[{\mathrm{ln}}{}\left({2}\right){+}\frac{{1}}{{2}}{}{x}{-}{1}{-}\frac{{1}}{{8}}{}{\left({x}{-}{2}\right)}^{{2}}{+}\frac{{1}}{{24}}{}{\left({x}{-}{2}\right)}^{{3}}{,}{-}\frac{{1}}{{4}}{}\frac{{\left({x}{-}{2}\right)}^{{4}}}{{{\mathrm{ξ}}}^{{4}}}\right]$ (5)