Question

A. In your own words, what is Newton's method used for? B. Use Newton's method with 3 iterations to estimate the solutions to sin(x)=x−1.

use x 0 =1

Answer 1

Newton's method is used for finding approximate solutions to equations where an analytical solution is impractical. For the equation sin(x) = x - 1 with an initial guess x_0 = 1, three iterations of Newton's method are applied, each time using the function and its derivative to find a better approximation of the root.

Step-by-step explanation:

Newton's method, also known as Newton-Raphson method, is a powerful technique used for finding successive approximations to the zeros (or roots) of a real-valued function. It is used in cases where no analytical solution to the equation is available and an approximate solution is sufficient. The method starts with an initial guess which is reasonably close to the true root and then iteratively improves the guess by applying the formula:

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

Here, 'f' represents the function for which we are trying to find the root, and 'f'' is the first derivative of that function.

To find a solution to the equation sin(x) = x - 1 using Newton's method with x_0 = 1, follow these steps:

1. First, express the problem as f(x) = sin(x) - x + 1. Our goal is to find x such that f(x) = 0.
2. The derivative of the function is f'(x) = cos(x) - 1.
3. Apply the Newton's method formula iteratively for 3 iterations:
   1. x_1 = 1 - \frac{sin(1) - (1 - 1)}{cos(1) - 1}
   2. x_2 = x_1 - \frac{sin(x_1) - (x_1 - 1)}{cos(x_1) - 1}
   3. x_3 = x_2 - \frac{sin(x_2) - (x_2 - 1)}{cos(x_2) - 1}
4. After each iteration, you will get a better approximation of the root.

Each iteration should provide you with a number that is closer to the actual solution, where sin(x) = x - 1.

It is also important to check the answers for reasonableness and ensure that the iterative process is converging towards a solution.