Question

Question #8c423

Answer 1

`97/72`

Explanation:

The Newton's Method is a method that approximates the solution of an equation.

It states that the zero of a function `P` can be approximated as `x_1=x_0-(P(x_0))/(P'(x_0))`, where `x_0` is an initial guess and `P'` is the derivative of `P` with respect to `x`. This can be used recursively to find more accurate solutions, using `x_1` as the second guess.

The derivative of `x^4-3` is `4x^3`. Then, our recursive equation is `x_1=x_0-(x_0^4-3)/(4x_0^3)`.

The question instructs us to start with `x_0=1`. Then, `x_1=1-(1^4-3)/(4*1^3)=3/2`.

Repeat the process with `x_0=3/2`: `x_1=3/2-((3/2)^4-3)/(4*(3/2)^3)=97/72~~1.35`. This is relatively close to one of the actual solutions: `root(4)(3)~~1.32`.