Newton’s Method. The golden ratio is the positive root of the equation x^2-x-1=0

Question

Starting with x1 = 1 use Newton’s method to calculate rational approximations   and Write your estimates as rational numbers. How close to  is

Answer 1

f'(x) = 2x - 1; x₀ = 1

x₁= 1 - f(1)/f'(1) = 1 - (- 1)/1 = 2

x₂ = 2 - f(2)/f'(2) = 2 - 1/3 = 5/3

x₃ = 5/3 - f(5/3)/f'(5/3) = 5/3 - (25/9 - 5/3 - 1)/(10/3 - 1) = 5/3 - (1/9(7/3) = 5/3 - 1/21 = 34/21 ≈ 1.619

Exact value: x = (1 + √5)/2 ≈ 1.618

Difference is 0.001