Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To solve the equation [tex]\( f(x) = g(x) \)[/tex] using three iterations of successive approximation, we will iterate our initial guess through the function [tex]\( f(x) \)[/tex]. The initial guess is determined from the intersection point's approximate location on the graph.
Given the functions:
[tex]\[ f(x) = \frac{x^2 + 3x + 2}{x + 8} \][/tex]
[tex]\[ g(x) = \frac{x - 1}{x} \][/tex]
Starting with the initial guess [tex]\( x_0 = 1.5 \)[/tex]:
1. Calculate [tex]\( f(x_0) \)[/tex]:
[tex]\[ f(1.5) = \frac{(1.5)^2 + 3 \cdot 1.5 + 2}{1.5 + 8} = \frac{2.25 + 4.5 + 2}{9.5} = \frac{8.75}{9.5} \approx 0.921 \][/tex]
(The value is approximate for this calculation to aid understanding).
Next, use this as the new input to the function [tex]\( f \)[/tex]:
2. Calculate [tex]\( f(0.921) \)[/tex]:
[tex]\[ f(0.921) = \frac{(0.921)^2 + 3 \cdot 0.921 + 2}{0.921 + 8} = \frac{0.848 + 2.763 + 2}{8.921} = \frac{5.611}{8.921} \approx 0.629 \][/tex]
Continue the process:
3. Calculate [tex]\( f(0.629) \)[/tex]:
[tex]\[ f(0.629) = \frac{(0.629)^2 + 3 \cdot 0.629 + 2}{0.629 + 8} = \frac{0.396 + 1.887 + 2}{8.629} = \frac{4.283}{8.629} \approx 0.496 \][/tex]
After three iterations, the approximate solution converges to [tex]\( x \approx 0.496 \)[/tex]. Therefore, the approximate solution for the equation [tex]\( f(x) = g(x) \)[/tex] after three iterations of successive approximation is:
[tex]\[ x \approx 0.496 \][/tex]
Given the functions:
[tex]\[ f(x) = \frac{x^2 + 3x + 2}{x + 8} \][/tex]
[tex]\[ g(x) = \frac{x - 1}{x} \][/tex]
Starting with the initial guess [tex]\( x_0 = 1.5 \)[/tex]:
1. Calculate [tex]\( f(x_0) \)[/tex]:
[tex]\[ f(1.5) = \frac{(1.5)^2 + 3 \cdot 1.5 + 2}{1.5 + 8} = \frac{2.25 + 4.5 + 2}{9.5} = \frac{8.75}{9.5} \approx 0.921 \][/tex]
(The value is approximate for this calculation to aid understanding).
Next, use this as the new input to the function [tex]\( f \)[/tex]:
2. Calculate [tex]\( f(0.921) \)[/tex]:
[tex]\[ f(0.921) = \frac{(0.921)^2 + 3 \cdot 0.921 + 2}{0.921 + 8} = \frac{0.848 + 2.763 + 2}{8.921} = \frac{5.611}{8.921} \approx 0.629 \][/tex]
Continue the process:
3. Calculate [tex]\( f(0.629) \)[/tex]:
[tex]\[ f(0.629) = \frac{(0.629)^2 + 3 \cdot 0.629 + 2}{0.629 + 8} = \frac{0.396 + 1.887 + 2}{8.629} = \frac{4.283}{8.629} \approx 0.496 \][/tex]
After three iterations, the approximate solution converges to [tex]\( x \approx 0.496 \)[/tex]. Therefore, the approximate solution for the equation [tex]\( f(x) = g(x) \)[/tex] after three iterations of successive approximation is:
[tex]\[ x \approx 0.496 \][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.