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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

withStudent[NumericalAnalysis]:

TaylorPolynomialsinx,x

x16x3+1120x5

(1)

TaylorPolynomialsinx,x,errorboundvar='ξ'

x16x3+1120x5,15040cosξx7

(2)

TaylorPolynomialsinx,x,errorboundvar='ξ',extrapolate=1.3

x16x3+1120x5,15040cosξx7,1.3,0.9647744166,0.9635581854,0.001245010258

(3)

TaylorPolynomialⅇx,x,order=3

1+x+12x2+16x3

(4)

TaylorPolynomiallnx,x=2,order=3,errorboundvar='ξ'

ln2+12x118x22+124x23,14x24ξ4

(5)

See Also

Student[NumericalAnalysis], Student[NumericalAnalysis][ComputationOverview], taylor


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