Category: Taylor Formula

Taylor Polynomial

Suppose that we have n-th degree polynomial p(x)=a0+a1(xa)+a2(xa)2++an1(xa)n1+an(xa)n{p}{\left({x}\right)}={a}_{{0}}+{a}_{{1}}{\left({x}-{a}\right)}+{a}_{{2}}{{\left({x}-{a}\right)}}^{{2}}+\ldots+{a}_{{{n}-{1}}}{{\left({x}-{a}\right)}}^{{{n}-{1}}}+{a}_{{n}}{{\left({x}-{a}\right)}}^{{n}}, where a,a0,a1,a2,,an{a},{a}_{{0}},{a}_{{1}},{a}_{{2}},\ldots,{a}_{{n}} are constants.

Maclaurin Polynomials of Common Functions

When a=0a=0 we call Taylor polynomial Maclaurin polynomial. In this case formulas for polynomials are fairly simple.

Maclaurin Polynomial. For function y=f(x)y={f{{\left({x}\right)}}} Maclaurin polynomial of n-th degree is Mn(x)=f(0)+f(0)1!x+f(0)2!x2++f(n)(0)n!xn{M}_{{n}}{\left({x}\right)}={f{{\left({0}\right)}}}+\frac{{{f{'}}{\left({0}\right)}}}{{{1}!}}{x}+\frac{{{f{''}}{\left({0}\right)}}}{{{2}!}}{{x}}^{{2}}+\ldots+\frac{{{{f}}^{{{\left({n}\right)}}}{\left({0}\right)}}}{{{n}!}}{{x}}^{{n}}.