To approximate the solution for the equation [tex]\( f(x) = g(x) \)[/tex] using three iterations of successive approximation, let's follow a step-by-step process.
First, recall the definitions of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = \frac{x^2 + 3x + 2}{x + 8}
\][/tex]
[tex]\[
g(x) = \frac{x - 1}{\pi}
\][/tex]
### Step 1: Identify the Starting Point
From the given information, we set the initial guess for our starting point. Let's denote this starting point as [tex]\( x_0 \)[/tex]. According to the problem, we will use [tex]\( x = 1.0 \)[/tex] as our initial guess:
[tex]\[
x_0 = 1.0
\][/tex]
### Step 2: Define the Successive Approximation Method
In each iteration, we will compute a new approximation of [tex]\( x \)[/tex] by averaging [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
x_{\text{new}} = \frac{f(x) + g(x)}{2}
\][/tex]
### Step 3: Perform Successive Approximation
#### Iteration 1
- Compute [tex]\( f(x_0) \)[/tex]:
[tex]\[
f(1.0) = \frac{(1.0)^2 + 3(1.0) + 2}{1.0 + 8} = \frac{1 + 3 + 2}{9} = \frac{6}{9} = \frac{2}{3} = 0.6667
\][/tex]
- Compute [tex]\( g(x_0) \)[/tex]:
[tex]\[
g(1.0) = \frac{1.0 - 1}{\pi} = \frac{0}{\pi} = 0
\][/tex]
- Average the results:
[tex]\[
x_1 = \frac{0.6667 + 0}{2} = \frac{0.6667}{2} = 0.3333
\][/tex]
#### Iteration 2
- Compute [tex]\( f(x_1) \)[/tex]:
[tex]\[
f(0.3333) = \frac{(0.3333)^2 + 3(0.3333) + 2}{0.3333 + 8} = \frac{0.1111 + 0.9999 + 2}{8.3333} \approx \frac{3.111}{8.333} \approx 0.3732
\][/tex]
- Compute [tex]\( g(x_1) \)[/tex]:
[tex]\[
g(0.3333) = \frac{0.3333 - 1}{\pi} = \frac{-0.6667}{\pi} \approx -0.2120
\][/tex]
- Average the results:
[tex]\[
x_2 = \frac{0.3732 + (-0.2120)}{2} = \frac{0.1612}{2} = 0.0806
\][/tex]
#### Iteration 3
- Compute [tex]\( f(x_2) \)[/tex]:
[tex]\[
f(0.0806) = \frac{(0.0806)^2 + 3(0.0806) + 2}{0.0806 + 8} \approx \frac{0.0065 + 0.2418 + 2}{8.0806} \approx \frac{2.2483}{8.0806} \approx 0.2783
\][/tex]
- Compute [tex]\( g(x_2) \)[/tex]:
[tex]\[
g(0.0806) = \frac{0.0806 - 1}{\pi} = \frac{-0.9194}{\pi} \approx -0.2927
\][/tex]
- Average the results:
[tex]\[
x_3 = \frac{0.2783 + (-0.2927)}{2} = \frac{-0.0144}{2} = -0.0072
\][/tex]
### Conclusion
After three iterations, the successive approximation yields the value [tex]\(-0.0072\)[/tex]. Therefore, our approximate solution to the equation [tex]\( f(x) = g(x) \)[/tex] after three iterations is:
[tex]\[
x \approx -0.0072
\][/tex]
