o compute the Taylor series of f(x)=sin(x) about x=2π, we first need the values:
f(2π)=
fn(2π)=
f(4)(2π)=
f(6)(2π)=
noticing that the odd-orderderivatives
f⁽²ᵏ⁺¹⁾ (π/2)= for k=0,1,2.
The Taylor expansion of f about x= π/2 to n=7 terms is thus
f(x)=
If 0≤x≤ π/2, then since ∣∣ f (8) (x) ∣∣ on [0, π/2] is
a. increasing
b. decreasing
the smallest upper estimate one can obtain from this analysis for
∣∣​ f (8) (x) ∣∣ is
We can use these results to obtain the estimate