Use Newton's method to approximate the given number correct to eight decimal places. 4 sqr (76)

Answer 1

Let x = 4√76; x² = 1216. f(x ) = x² - 1216; Let x₀ = 36; f'(x) = 2x

Then x₁ = x₀ - f(x₀)/f'(x₀) = 36 - (36² - 1216)/(2·36) = 34.88888889

Then x₂ = x₁ - f(x₁)/f'(x₁) = 34.88888889 - (34.88888889² - 1216)/(2·34.88888889) = 34.87119604

Next, x₃ = x₂ - f(x₂)/f'(x₂) = 34.87119604 - (34.87119604² - 1216)/(2·34.87119604) = 34.87119155.

This is approximation match calculator result.

Answer 2

let f(x) = x²-4²(76) = x²-1216 = 0

f '(x) = 2x

x_n+1 = x_n-(x_n²-1216)/2x_n

Let x₁ = 76/2 = 38

x₂ = 38 - 228/76 = 35

x₃ = 35 - 9/70 = 34.87142857

x₄ = 34.87142857 - 0.01653051/69.74285714 = 34.8711915483254

x₅ = 34.8711915483254 - ≈0 = 34.8711915483254

Actual 4√76 = 34.87119155