Based on the provided image, let’s examine each part of the question:

Part (a)

This part involves using Newton's Method to approximate the solution $s$ of the equation $f(x) = 0$ with an initial approximation of $x_1 = 6$.

1. Newton's Method Recap: Newton's Method iteratively approximates roots of a function $f(x)$ by finding the tangent line at each approximation $x_n$ and determining where this tangent line intersects the x-axis. The formula for Newton’s Method is: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

2. Steps for Estimation:

   • The tangent line at $x_1 = 6$ should be drawn to estimate $x_2$, where it intersects the x-axis. From the image, it appears that $x_2 \approx 7.3$.
   • Similarly, the tangent line at $x_2 \approx 7.3$ should be drawn to estimate $x_3$, leading to $x_3 \approx 6.8$.

These tangent lines visually represent the iterative process of Newton’s Method, where each intersection point with the x-axis provides the next approximation.

Part (b)

For this part, we need to evaluate if starting with $x_1 = 8$ instead of $x_1 = 6$ would lead to a better approximation.

1. Reasoning: Starting with $x_1 = 8$ could potentially be closer to the actual root $s$ as per the graph. From the visual assessment, it seems that $x_1 = 8$ is on the closer side of the root $s$ on the graph, making it likely that the Newton iteration will converge faster to the solution if started closer to the root.

Thus, the answer is Yes, $x_1 = 8$ would likely be a better first approximation since it is closer to the actual solution, minimizing the required iterations for convergence.

Would you like further details or explanations on any part?

Related Questions:

1. How does the choice of the initial approximation $x_1$ affect the convergence of Newton's Method?
2. Can Newton's Method fail to converge? Under what circumstances does this happen?
3. What is the geometric interpretation of Newton's Method?
4. How can one determine the function's derivative if it's not given explicitly?
5. What are alternative methods to Newton's Method for finding roots of equations?

Tip:

In Newton's Method, a closer initial approximation to the actual root often results in faster convergence, but care must be taken as poor initial guesses can lead to divergence or convergence to a different root.