use newtons method

Question

Using Newton's Method to solve , with an initial approximation of , the value of  (to 4 decimal places) is:

Answer 1

You can use the equation:

to estimate the zeros using Newton's Method we need f '(x):

f '(x) = 3 + sin(x)

For x₀ = 5, calculate x₁:

x₁ = x₀ - f(x₀)/f '(x₀)

f(5) = 3(5) - cos(5) = 15 - 0.283662 = 14.71634

f '(5) = 3 + sin(5) = 3 + -0.95892 = 2.04108

So x₁ = 5 - 14.71634/2.04108 = -2.21009

For x₁ = -2.21009, calculate x₂:

x₂ = x₁ - f(x₁)/f '(x₁)

f(-2.21009) = 3(-2.210090) - cos(-2.21009) = -6.03364

f '(-2.21009) = 3 + sin(-2.21009) = 3 + -0.802518 = 2.19748

So x₂ = -2.21009 - -6.03364/2.19748 = 0.535617

For x₂ = 0.535617, calculate x₃:

x₃ = x₂ - f(x₂)/f '(x₂)

f(0.535617) = 3(0.535617) - cos(0.535617) = 0.746896

f '(0.535617) = 3 + sin(0.535617) = 3.51037

So x₃ = 0.535617 - 0.746896/3.51037 = 0.322848 then round this as instructed.