Find the x2 obtained by using Newton's Method to estimate the real root of f(x)=x3−3x−5=0 when x1=3. x2 = ____

Question

Find the x₂ obtained by using Newton's Method to estimate the real root of f(x)=x³−3x−5=0 when x₁=3.

x_{2 = ____}

Answer 1

The iterations in Newton's method are given by:

x_{n + 1} = x_n - f(x_n) / f'(x_n)

Let f be the function given on R by f(x) = x³ - 3x - 5

The derivative of f is given on R by:

f'(x) = 3x² - 3

x₁ = 3

x₂ = x₁ - f(x₁) / f'(x₁)

= 3 - (3³ - 3 * 3 - 5) / (3 * 3² - 3)

= 3 - (27 - 9 - 5) / (3 * 9 - 3)

= 3 - 13 / (27 - 3)

= 3 - 13 / 24

= 72 / 24 - 13 / 24

= 59 / 24