Estimate the value of $\sqrt { 5 }$ by using three iterations of Newton's method to solve the equation $x ^ { 2 } - 5 = 0$ with initial estimate $x _ { 0 } = 2$. Round your final estimate to four decimal places. Select the correct answer.
A) $1.71$
B) $2.2361$
C) $1.7949$
D) $2.1338$

Question

Quizplus · Accepted Answer

Newton's method formula is $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$. For $f(x) = x^2 - 5$, $f'(x) = 2x$. Starting with $x_0 = 2$, the iterations are:  1. $x_1 = 2 - \frac{2^2 - 5}{2 \times 2} = 2.25$  2. $x_2 = 2.25 - \frac{2.25^2 - 5}{2 \times 2.25} \approx 2.2361$  3. $x_3 = 2.2361 - \frac{2.2361^2 - 5}{2 \times 2.2361} \approx 2.2361$  Thus, the estimate is approximately $2.2361$.

Estimate the Value Of 5\sqrt { 5 }5​ By Using Three Iterations of Newton's Method to Solve the Three

Estimate the Value Of $\sqrt { 5 }$ By Using Three Iterations of Newton's Method to Solve the Three