Question

Calculate two iterations of Newton's Method for the function using the given initial guess. (Round your answers to four decimal places.) f(x) = x2 − 8, x1 = 2

Answer 1

The first and second iteration of Newton's Method are 3 and
`(11)/(6)`.

Explanation:

The Newton's Method is a multi-step numerical method for continuous diffentiable function of the form
`f(x) = 0` based on the following formula:

`x_(i+1) = x_(i) -(f(x_(i)))/(f'(x_(i)))`

Where:

`x_(i)` - i-th Approximation, dimensionless.

`x_(i+1)` - (i+1)-th Approximation, dimensionless.

`f(x_(i))` - Function evaluated at i-th Approximation, dimensionless.

`f'(x_(i))` - First derivative evaluated at (i+1)-th Approximation, dimensionless.

Let be
`f(x) = x^(2)-8` and
`f'(x) = 2\cdot x`, the resultant expression is:

`x_(i+1) = x_(i) -(x_(i)^(2)-8)/(2\cdot x_(i))`

First iteration: (
`x_(1) = 2`)

`x_(2) = 2-(2^(2)-8)/(2\cdot (2))`

`x_(2) = 2 + (4)/(4)`

`x_(2) = 3`

Second iteration: (
`x_(2) = 3`)

`x_(3) = 3-(3^(2)-8)/(2\cdot (3))`

`x_(3) = 2 - (1)/(6)`

`x_(3) = (11)/(6)`