Question

Find the linear approximation of f ( x ) = ln x at x = 1 and use it to estimate ln ( 1.25 ) .

Answer 1

The linear approximation to f(x) = ln(x) at x = 1 is

L(x) = f(1) + f'(1) (x - 1)

We have f(1) = ln(1) = 0, and the derivative of f is

f'(x) = 1/x

so that f'(1) = 1/1 = 1. Then

L(x) = 0 + 1 (x - 1)

L(x) = x - 1

We then approximate ln(1.25) by

ln(1.25) ≈ L(1.25) = 1.25 - 1 = 0.25