Calculus 2 Taylor Polynomia

Question

Let f(x) = lnx

(a) Find the second degree Taylor Polynomial of f centered at a = 4 with the corresponding remainder.

(b) Use the Taylor polynomial to approximate ln(1.1).

(c) Use the remainder to find an upper bound for the absolute error when approximating ln(1.1).

Answer 1

(a) Second degree Taylor Polynomial P₂(x) = f(a) + f'(a)/1!(x - a) + f''(a)/2!(x - a)².

We have a = 4, f(4) = ln4, f'(x) = 1/x, f'(4) = 1/4, f''(x) = -1/x², f''(4) = - 1/16.

So, P₂(x) = ln4 + 1/4(x - 4) - 1/32(x - 4)². Reminder R₂(x) = f'''(c)/3!(x - a)³, f'''(x) = 2/x³; so, R₂(x) = 1/(3c³)(x - 4)³, where c between 4 and x

(b) Now we have to evaluate ln4.1 ≈ P₂(4.1) = ln4 + 1/4(4.1 - 4) - 1/32(4.1 - 4)² ≈ 1.410982

(c) Absolute error e ≤ 1/(3·4³)(4.1 - 4)³ ≈ 5.21·10^-6·