Linear Approximation

Question

Find the Linear approximation L(x) of the function f(x)=ln(x) at a=1 and use it to approximate Ln 0.9.

Answer 1

f(x) = ln(x)

f(1) = 0 -> (1,0)

f'(x) = 1/x

f'(1) = 1

The slope m = 1.

y-y₀ = m(x-x₀)

y - 0 = 1(x-1)

y = x-1

y(.9) = .9-1 = -0.1

Actual ln(.9) ≈ -0.1054