Question

Use Taylor series expansions (zero through fourth order) to predict f (2) for f (x) = ln(x) with a
base point at
x = 1. Determine the true percentage relative error for each approximation.

Answer 1

The Taylor series about f(1) to fourth order is:

f(x) = f(1) + f'(1)*(x-1) + f''(1)*(x-a)^2/2! + f'''(1)*(x-a)^3/3!

The derivative of ln(x) is 1/x
f''(x) = -1/x^2
f'''(x) = 2/x^3

f(1) = 0
f'(1) = 1
f''(1) = -1
f'''(1) = 2
f(2) = 0 + 1*1 -1/2 + 2/6
= 1-1/2+1/3 = 0.833

The actual value (ln(2) found by calculator ) is 0.693

An error of (0.833-0.693)/0.693, or ~ 20%