Newton’s method can be used to give an approximation close to the solution $x = w$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2018 - Paper 1

To determine which initial approximation gives the second approximation closest to $x = w$, we can analyze the graph presented.

1. Assess the Graph: The graph suggests that the function has roots at points between the intervals highlighted (a, b, c, d, w). In particular, we need to focus on the locations of points a and b relative to w.

2. Evaluate Choices:

   • If we choose $x_1 = a$: This will likely lead to an approximation that is further away from w, given that point a is located before the peak of the function.
   • If we choose $x_1 = b$: This choice, being closer to w, will lead to a better intermediate approximation, as the function will descend towards the solution at w.

3. Newton’s Method Implementation: Applying Newton’s method using $x_1 = b$ provides a better trajectory as it starts the iterations closer to w. Thus, it more quickly converges towards the solution.

Based on this analysis, the best choice for initial approximation $x_1$ is B. $x_1 = b$.