Use Newton's method to find the third approximation to the root of

Question

Use Newton's method to find the third approximation to the root of

x^4 - 20 = 0

with initial approximation x₁ = 2.

x_{3 = ___}

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) = x⁴ - 20

The derivative of f is given on R by:

f'(x) = 4x³

x₁ = 2

x₂ = x₁ - f(x₁) / f'(x₁)

= 2 - (2⁴ - 20) / (4 * 2³)

= 2 - (16 - 20) / (4 * 8)

= 2 - (-4) / 32

= 2 + 4 / 32

= 2 + 1 / 8

= 17 / 8

= 2.125

x₃ = x₂ - f(x₂) / f'(x₂)

= 2.125 - (2.125⁴ - 20) / (4 * 2.125³)

Just put this into the calculator :-)