Let's analyze this problem using Newton's method and the provided graph.

Problem Analysis

1. Newton’s Method Recap: Newton's Method approximates roots by iteratively applying: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ where $x_1$ is the initial approximation.

2. Given Information:

   • Initial approximation $x_1 = 6$.
   • We need to find $x_2$ and $x_3$ by drawing the tangent lines on the graph, starting from $x_1 = 6$, to estimate their numerical values.
   • Evaluate if starting at $x_1 = 8$ would be a better approximation.

Solution Steps

Part (a): Drawing Tangents to Find $x_2$ and $x_3$

1. Tangent at $x_1 = 6$:

   • Looking at the graph, at $x = 6$, the function value $f(x_1)$ seems positive.
   • Draw a tangent line at $x = 6$ to locate the intersection with the x-axis. This intersection will give us $x_2$, the next approximation.

2. Finding $x_3$:

   • Repeat the process by drawing another tangent line at $x = x_2$, where this new tangent intersects the x-axis will approximate $x_3$.

Since I can't draw directly here, you would use the graph and sketch the tangent lines as described to estimate $x_2$ and $x_3$.

Part (b): Checking if $x_1 = 8$ is a Better Initial Approximation

1. Starting from $x_1 = 8$ would place us closer to the root $s$ based on the graph, as it appears that $x = 8$ is closer to where $f(x) = 0$.
2. With an initial guess closer to the actual root, Newton's method generally converges faster. So, yes, $x_1 = 8$ would likely be a better initial approximation since it would reduce the number of iterations required to reach an accurate answer.

Would you like to see more detailed calculations for Newton's method steps or discuss further?

Related Questions for Deeper Understanding

1. What is the importance of the derivative in Newton's method, and what does it represent geometrically?
2. How does the convergence of Newton's method change with different starting points relative to the root?
3. What would happen if we chose an initial approximation far from any root, say $x_1 = 0$?
4. How can we determine if Newton’s method will converge or diverge for a particular function?
5. What other numerical methods could we use to approximate the roots of functions?

Tip:

When using Newton's method, always choose an initial approximation close to the root if possible, as this improves the convergence rate.