Thenwe may be able to deducea usefulTaylorSeriescenteredcloseto the givenx.
Example 2 Approximatesin(1.1) to 4 decimalplaces.
Since1.1 is closeto , we wouldtry to find a TaylorSeriesofsin xat. Letf(x) =sin x. Then
and
.This patternrepeatsand can be checkedas in the casex 0 = 0. So the TaylorSeriesis
We may also applyalgebraicmanipulationto standardTaylorSeries.
Example 3 Approximate to 4 decimalplaces.
Solution.Thereis standardTaylorSeries:
for |x| < 1 throughterm-by-termdifferentiationof the seriesfor (
is inadequateatx= 0.9).
Since1.9 is closeto 2, we consider
for |x| < 2.
So we takex= 0.1 and then
Exercise
- Approximateln0.9 to 4 decimalplaces.