{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.


y = sin x and y = x/2

Verified step by step guidance

1. Start by setting the two equations equal to each other to find the intersection points: sin(x) = x/2.
2. Rearrange the equation to form a function f(x) = sin(x) - x/2, which we will use in Newton's method.
3. Choose initial approximations for x based on a preliminary graph of the functions y = sin(x) and y = x/2 to identify where they might intersect.
4. Apply Newton's method using the formula x_{n+1} = x_n - f(x_n) / f'(x_n), where f'(x) is the derivative of f(x). Calculate f'(x) = cos(x) - 1/2.
5. Iterate the process until the values of x converge to a stable solution, indicating the intersection points.