Use Newton's method to approximate the value of the cube root of 59.

Question

Use Newton's method to approximate the value of

as follows:

Let x₁ = 3 be the initial approximation.

The second approximation x₂ is

and the third approximation x₃ is

Answer 1

Newton's method is x_n+1 = x_n - f(x_n)/f'(x_n) in order to find the zeroes of f(x)

A useful function for this problem is x³ - 59 (this will be zero when x = (59)^1/3

f'(x) = 3x²

If x₁ = 3

x₂ = 3 - (3³-59)/(3*3²) = 4.185185

I'll leave x₃ for you.

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