To approximate the root of the function $f(x) = \cos(x) - x$ using the Newton-Raphson method, we follow these steps:

Steps in Newton-Raphson Method:

1. Formula: The iterative formula for Newton-Raphson is given by: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ Where $f'(x)$ is the derivative of $f(x)$.

2. Function and Derivative:

   • $f(x) = \cos(x) - x$
   • The derivative of $f(x)$, $f'(x)$, is: $f'(x) = -\sin(x) - 1$

3. Initial Guess: Choose an initial guess for $x_0$. Let's use $x_0 = 0.5$ as a reasonable starting point because the root is near $x = 0.74$ (known from prior analysis).

4. Iterate using the Newton-Raphson formula until the value converges to a satisfactory approximation.

Let me compute the root using these steps.The approximate root of the function $f(x) = \cos(x) - x$ using the Newton-Raphson method, starting with an initial guess of $x_0 = 0.5$, is $x \approx 0.739085$. The method converged in 3 iterations.

Would you like further details or have any questions?

Here are five related questions to expand on this topic:

1. What happens if you choose a different initial guess in the Newton-Raphson method?
2. How does the Newton-Raphson method perform on functions with multiple roots?
3. Can you explain why the Newton-Raphson method sometimes fails to converge?
4. How does the convergence of Newton-Raphson compare to other root-finding methods like the bisection method?
5. What are some practical applications of root-finding methods in real-world problems?

Tip: Choosing a good initial guess is important in the Newton-Raphson method, as a poor choice may lead to divergence or slow convergence.