{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = x/6 - sec x on [0,8]

Verified step by step guidance

1. Begin by understanding Newton's method, which is an iterative technique to approximate the roots of a real-valued function. The formula is: x_{n+1} = x_n - f(x_n) / f'(x_n).
2. Perform a preliminary analysis of the function f(x) = x/6 - \sec(x) on the interval [0, 8]. Consider the behavior of \sec(x), which is undefined at odd multiples of \pi/2. This will help identify intervals where roots may exist.
3. Graph the function f(x) = x/6 - \sec(x) over the interval [0, 8] to visually identify where the function crosses the x-axis. These crossings are potential roots and will help in choosing initial approximations for Newton's method.
4. Calculate the derivative f'(x) = 1/6 + \sec(x)\tan(x). This derivative will be used in Newton's method to update the approximations of the roots.
5. Choose initial approximations based on the graph and preliminary analysis. Apply Newton's method iteratively for each initial approximation until the changes between successive approximations are sufficiently small, indicating convergence to a root.