{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) = cos x - x/7

Verified step by step guidance

1. Step 1: Understand the function and the problem. We need to find the roots of the function \( f(x) = \cos x - \frac{x}{7} \). A root of the function is a value of \( x \) for which \( f(x) = 0 \).
2. Step 2: Perform a preliminary analysis. Analyze the behavior of the function by considering its components: \( \cos x \) oscillates between -1 and 1, and \( -\frac{x}{7} \) is a linear function. This suggests that the function will have intersections (roots) where these two components balance each other out.
3. Step 3: Graph the function \( f(x) = \cos x - \frac{x}{7} \) to visually identify approximate locations of the roots. Look for points where the graph crosses the x-axis. These points will serve as initial guesses for Newton's method.
4. Step 4: Apply Newton's method to refine the root approximations. Newton's method uses the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \), where \( f'(x) \) is the derivative of \( f(x) \). Calculate \( f'(x) = -\sin x - \frac{1}{7} \).
5. Step 5: Choose initial approximations from the graph and iteratively apply Newton's method. For each initial guess, compute successive approximations until the values converge to a stable root. Repeat this process for each root identified in the graph.