Estimate using Linearization

Question

Estimate ln(1.07) using the linearization L(x) of f(x) = ln(1 + x) at a = 0 (Round to three decimal places)

L(0.07) =

Find the actual value of ln(1.07). (Round to three decimal places.)

Calculate the percentage error of Linear Approximation. (Round your answer to three decimal places.)

Answer 1

L(x) = f(a) + f'(a)(x - a)

f(x) = ln(1+x) and a = 0

ln(1.07) = ln(1 + 0.07) ≈L(0.07) = f(0) + f'(0)(0.07 - 0)

= ln1 + (1/1)(0.07) = 0.070

On calculator, ln(1.07) ≈ 0.0676586 ≈ 0.068

% error = [ (0.070 - 0.0676586) / (0.0676586) ] (100) = 3.461%