Estimate the value of by using three iterations of Newton's method to solve the equation with initial estimate Round your final estimate to four decimal places.
A)1.71
B)2.2662
C)2.2361
D)1.6535

Question

Estimate the value of  by using three iterations of Newton's method to solve the equation  with initial estimate  Round your final estimate to four decimal places.
A)1.71 
B)2.2662 
C)2.2361 
D)1.6535

Quizplus · Accepted Answer

Using Newton's method, we have:

$f(x) = 11eaa3e5_{55b6\ 2502\ 9f8f\ 936c68f9ec5c\ TB5971_{11}}$
$f'(x) = 5.5ba1bf9_{ab5b\ 4a01\ bd3f\ 936c68f9ec5c\ TB5971_{11}}$

With an initial estimate of $x_0 = 11eaa3e5_{55b6\ 4c13\ 9f8f\ f7e91c900f4e\ TB5971_{11}}$, we iterate the following:

$x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}$

Iteration 1:
$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 2.2361$

Iteration 2:
$x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} = 2.2361$

Iteration 3:
$x_3 = x_2 - \frac{f(x_2)}{f'(x_2)} = 2.2361$

We can see that the estimation has converged to $2.2361$, rounded to four decimal places. Therefore, the best choice is C.

Estimate the Value of by Using Three Iterations of Newton's