use newtons method to approximate a solution of the equation\n3 sin(x) - x = 0.\nlet x₀ = 1 be the initial…

use newtons method to approximate a solution of the equation\n3 sin(x) - x = 0.\nlet x₀ = 1 be the initial approximation, and then calculate x₁ and x₂.\n\nx₁ = 1 - (3 sin(1) - 1)/(3 cos(1) - 1)\n\nexcellent work! youve accurately applied newtons method to calculate the first iteration!\n\nx₂ = \n\nincorrect. double-check your calculation for x₂. remember to use the value you found for\nx₁ as your input in this step. verify that youve correctly evaluated the sine function and its\nderivative at this new point.\n\nhint

use newtons method to approximate a solution of the equation\n3 sin(x) - x = 0.\nlet x₀ = 1 be the initial approximation, and then calculate x₁ and x₂.\n\nx₁ = 1 - (3 sin(1) - 1)/(3 cos(1) - 1)\n\nexcellent work! youve accurately applied newtons method to calculate the first iteration!\n\nx₂ = \n\nincorrect. double-check your calculation for x₂. remember to use the value you found for\nx₁ as your input in this step. verify that youve correctly evaluated the sine function and its\nderivative at this new point.\n\nhint

Answer

Explanation:

Step1: Calculate (x_1)

First, we know (x_0 = 1). [ \begin{align*} x_1&=1-\frac{3\sin(1)-1}{3\cos(1)-1}\ &\approx1-\frac{3\times0.8415 - 1}{3\times0.5403-1}\ &\approx1-\frac{2.5245 - 1}{1.6209 - 1}\ &\approx1-\frac{1.5245}{0.6209}\ &\approx1 - 2.4553\ &\approx - 1.4553 \end{align*} ]

Step2: Calculate (x_2)

Now, use (x_1\approx - 1.4553) to calculate (x_2). [ \begin{align*} x_2&=x_1-\frac{3\sin(x_1)-x_1}{3\cos(x_1)-1}\ &\approx - 1.4553-\frac{3\sin(-1.4553)-(-1.4553)}{3\cos(-1.4553)-1}\ &\approx - 1.4553-\frac{3\times(-0.9804)+1.4553}{3\times0.1961-1}\ &\approx - 1.4553-\frac{-2.9412 + 1.4553}{0.5883-1}\ &\approx - 1.4553-\frac{-1.4859}{-0.4117}\ &\approx - 1.4553 - 3.6092\ &\approx - 5.0645 \end{align*} ]

Answer:

(x_2\approx - 5.0645)