Category: Taylor Formula

Taylor Polynomial

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

Maclaurin Polynomials of Common Functions

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

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