Theorem(nth-degreeTaylorpolynomial)Givena functionf with continuousnthderivativein an openin-
tervalcontainingx* 0. Thereexistsuniquenth-degreepolynomialp(x) withp(j)(x 0 ) =f(j)(x 0 ), for 0 ≤j≤n.
*: the functionsin this text havecontinuousderivativesat the centerx 0 unlessotherwisestated.
This polynomial
is calledthenth
-degreeTaylorpolynomialoffatx 0. Ifx 0 = 0, it is calledthenth-degreeMaclaurinpolynomialoff
and denotedbyMn(x).Rn(x) =f(x) -Tn(x) is the remainderof the Taylorpolynomial.
Example 1 Let. Then and. So and
. Hence , andT 3 (x) =f(x) itself.
Example 2 Letf(x) =sin x,x 0 = 0 and taken= 3. Thenf(x) =cos x,
.So. isthethird-degreeMaclaurin
polynomialoff.
Example 3 Find the second-degreeTaylorpolynomialoff(x) =tan xat. Solution.
and. So and.