To approximate the solution to the equation [tex]\( f(x) = g(x) \)[/tex] using three iterations of successive approximation, let's go step by step through the given iterations and see if we can match them with the values in the table.
1. Step 1: Initial Guess ([tex]\(x_0 = -\frac{63}{16}\)[/tex])
Calculate [tex]\( f(x_0) \)[/tex] and [tex]\( g(x_0) \)[/tex]:
[tex]\[
f \left( -\frac{63}{16} \right) \approx 1.40096
\][/tex]
[tex]\[
g \left( -\frac{63}{16} \right) \approx 1.25397
\][/tex]
Compute the difference:
[tex]\[
\left| f \left( -\frac{63}{16} \right) - g \left( -\frac{63}{16} \right) \right| \approx 0.147
\][/tex]
2. Step 2: Second Iteration ([tex]\( x_1 = -\frac{61}{16} \)[/tex])
Calculate [tex]\( f(x_1) \)[/tex] and [tex]\( g(x_1) \)[/tex]:
[tex]\[
f \left( -\frac{61}{16} \right) \approx 1.21735
\][/tex]
[tex]\[
g \left( -\frac{61}{16} \right) \approx 1.26230
\][/tex]
Compute the difference:
[tex]\[
\left| f \left( -\frac{61}{16} \right) - g \left( -\frac{61}{16} \right) \right| \approx 0.045
\][/tex]
3. Step 3: Third Iteration ([tex]\( x_2 = -\frac{59}{16} \)[/tex])
Calculate [tex]\( f(x_2) \)[/tex] and [tex]\( g(x_2) \)[/tex]:
[tex]\[
f \left( -\frac{59}{16} \right) \approx 1.05163
\][/tex]
[tex]\[
g \left( -\frac{59}{16} \right) \approx 1.27119
\][/tex]
Compute the difference:
[tex]\[
\left| f \left( -\frac{59}{16} \right) - g \left( -\frac{59}{16} \right) \right| \approx 0.220
\][/tex]
From these calculations, we can see that for the three iterations, the values used were [tex]\( x = -\frac{63}{16} \)[/tex], [tex]\( x = -\frac{61}{16} \)[/tex], and [tex]\( x = -\frac{59}{16} \)[/tex]. These values match the first column of the table provided. Therefore, the correct location on the table corresponding to the successive approximation iterations is:
- [tex]\( \begin{array}{|c|c|}
\hline \text{Successive Approximation Intersection Values} \\
\hline -\frac{63}{16} & -\frac{31}{8} \\
\hline -\frac{61}{16} & -\frac{15}{4} \\
\hline -\frac{59}{16} & -\frac{29}{8} \\
\hline
\end{array} \)[/tex]
