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 $x_{0}=2 \text {. }$ Round your final estimate to four decimal places.
A) 1.71
B) 1.6535
C) 2.2361
D) 2.2662

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: $x_1 = 2.25$, $x_2 \approx 2.2361$, $x_3 \approx 2.2361$. Thus, the estimate is 2.2361.

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

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