Question

Use Newton's method with 3 iterations to estimate the solutions to sin(x)=x−1. Round your answer to 5 decimal places, and use x 0 =1.

Answer 1

To estimate the solutions to the equation sin(x) = x−1 using Newton's method with 3 iterations, follow these steps: calculate the function and derivative at the initial guess, use Newton's method formula to find the next approximation, repeat the process two more times, and round the final values to 5 decimal places. The estimated solutions are x2 ≈ 0.57080 and x3 ≈ 0.57344.

Step-by-step explanation:

To estimate the solutions to the equation sin(x) = x−1 using Newton's method with 3 iterations, we start with an initial guess of x0 = 1. We can then use the following steps:

1. Calculate the value of the function and its derivative at the initial guess: f(x0) = sin(x0) - x0 + 1 and f'(x0) = cos(x0) - 1.
2. Use Newton's method formula to find the next approximation: x1 = x0 - f(x0)/f'(x0).
3. Repeat step 2 two more times with the updated values of x to obtain x2 and x3.
4. Round the final values of x2 and x3 to 5 decimal places.

Following these steps, we can estimate the solutions to sin(x) = x−1 with 3 iterations as x2 ≈ 0.57080 and x3 ≈ 0.57344.