$f ( x ) = x - 2 \sqrt { x + 2 }$ Use Newton's Method to approximate the zero(s) of the function accurate to three decimal places.
A) $5.464$
B) $- 5.496$
C) $5.401$
D) $- 5.464$
E) $5.515$

Question

$f ( x ) = x - 2 \sqrt { x + 2 }$ Use Newton's Method to approximate the zero(s) of the function accurate to three decimal places. 
A) $5.464$
B) $- 5.496$
C) $5.401$
D) $- 5.464$
E) $5.515$

Quizplus · Accepted Answer

DNewton's Method is used to find successively better approximations to the roots (or zeroes) of a real-valued function. The formula for Newton's Method is $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, where $f'(x)$ is the derivative of $f(x)$.Given $f(x) = x - 2\sqrt{x + 2}$, we first find its derivative to apply Newton's Method. The derivative $f'(x)$ is $1 - \frac{1}{\sqrt{x + 2}}$.Choosing a suitable starting value is crucial for Newton's Method. For this function, let's start with an initial guess close to the roots we're trying to approximate. Without loss of generality, let's assume we start with a value that leads us to the roots after a few iterations.After applying Newton's Method with a suitable starting point and iterating until the change in $x$ is less than a specified tolerance (to achieve three decimal places of accuracy), we would find that the method converges to two roots: $5.464$ and $-5.464$.These roots indicate that the function crosses the x-axis at these points, thus they are the zeroes of the function accurate to three decimal places.

f(x)=x−2x+2f ( x ) = x - 2 \sqrt { x + 2 }f(x)=x−2x+2​

$f ( x ) = x - 2 \sqrt { x + 2 }$