To approximate the solution to the equation [tex]\( f(x) = g(x) \)[/tex] where:
[tex]\[ f(x) = \sqrt{7 - 2x} \][/tex]
[tex]\[ g(x) = \sqrt[3]{x + 2} \][/tex]
we can use the method of successive approximation. This involves iteratively improving our guess for the value of [tex]\( x \)[/tex] until we converge to an approximate solution. Here's the step-by-step process:
1. Initial Guess:
Start with an initial guess for [tex]\( x \)[/tex], say [tex]\( x_0 = 0 \)[/tex].
2. Define the Intermediate Function:
Define a function to combine [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. One approach is to average [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to get the next approximation.
[tex]\[ x_{n+1} = \frac{f(x_n) + g(x_n)}{2} \][/tex]
3. First Iteration:
- Calculate [tex]\( f(x_0) \)[/tex]:
[tex]\[ f(0) = \sqrt{7 - 2 \cdot 0} = \sqrt{7} \approx 2.645751 \][/tex]
- Calculate [tex]\( g(x_0) \)[/tex]:
[tex]\[ g(0) = \sqrt[3]{0 + 2} = \sqrt[3]{2} \approx 1.259921 \][/tex]
- Compute the average:
[tex]\[ x_1 = \frac{2.645751 + 1.259921}{2} \approx 1.952836 \][/tex]
4. Second Iteration:
- Calculate [tex]\( f(x_1) \)[/tex]:
[tex]\[ f(1.952836) = \sqrt{7 - 2 \cdot 1.952836} \approx \sqrt{7 - 3.905672} = \sqrt{3.094328} \approx 1.758517 \][/tex]
- Calculate [tex]\( g(x_1) \)[/tex]:
[tex]\[ g(1.952836) = \sqrt[3]{1.952836 + 2} = \sqrt[3]{3.952836} \approx 1.580268 \][/tex]
- Compute the average:
[tex]\[ x_2 = \frac{1.758517 + 1.580268}{2} \approx 1.669392 \][/tex]
5. Third Iteration:
- Calculate [tex]\( f(x_2) \)[/tex]:
[tex]\[ f(1.669392) = \sqrt{7 - 2 \cdot 1.669392} \approx \sqrt{7 - 3.338784} = \sqrt{3.661216} \approx 1.913446 \][/tex]
- Calculate [tex]\( g(x_2) \)[/tex]:
[tex]\[ g(1.669392) = \sqrt[3]{1.669392 + 2} = \sqrt[3]{3.669392} \approx 1.542115 \][/tex]
- Compute the average:
[tex]\[ x_3 = \frac{1.913446 + 1.542115}{2} \approx 1.727781 \][/tex]
After the third iteration, the approximate solution to the equation [tex]\( f(x) = g(x) \)[/tex] is [tex]\( x \approx 1.727781 \)[/tex].
