Taylor Inequality

[ijd5000]

Consider the following function.f(x) = sin x, a = π/6, n = 4, 0 ≤ x ≤ π/3

(a) Approximate f by a Taylor polynomial with degree n at the number a.
T₄(x) =

(b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ T_n(x)
when x lies in the given interval. (Round your answer to six decimal places.)
|R₄(x)| ≤

i have the Taylor polynomial, for some reason it didn't paste in, just need help with the R4(x) part.
I know i have to use the 5th derivative in the Taylor series, just not sure what x i should use in the derivative and whats my (x-a)^5 should be?

 

[HallsofIvy]

That's because you can't know the "c" in the error term. The error in using the nth Taylor Polynomial to estimate f(x) is $\displaystyle |\frac{f^{(n+1)}(c)}{(n+1)!}||x- a|^{n+1}$- the "n+1" term evaluated at "c" where c is some number between a and x. We can't know what number it is because if we did we could calculate the exact error and so find the exact value of f(x) with just a simple polynomial. All we can do is find some c at which the nth derivative has some maximum value and say the error must be less than that.