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

Answer:

The first and second iteration of Newton's Method are 3 and $\frac{11}{6}$.

Step-by-step 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:

$x_{i+1} = x_{i} -\frac{f(x_{i})}{f'(x_{i})}$

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:

$x_{i+1} = x_{i} -\frac{x_{i}^{2}-8}{2\cdot x_{i}}$

First iteration: ($x_{1} = 2$)

$x_{2} = 2-\frac{2^{2}-8}{2\cdot (2)}$

$x_{2} = 2 + \frac{4}{4}$

$x_{2} = 3$

Second iteration: ($x_{2} = 3$)

$x_{3} = 3-\frac{3^{2}-8}{2\cdot (3)}$

$x_{3} = 2 - \frac{1}{6}$

$x_{3} = \frac{11}{6}$