Consider the following equation.
[tex]f^{(-x)}+4=3 x-1[/tex]
Approximate the solution to the equation above using three iterations of successive approximation. Use the graph below as a starting point.
A. [tex]x \approx \frac{25}{16}[/tex]
B. [tex]x \approx \frac{27}{16}[/tex]
C. [tex]z \approx \frac{7}{4}[/tex]
D. [tex]x \approx \frac{13}{8}[/tex]
Answer (2)
Rewrite the equation in the form x = g ( x ) , assuming f ( − x ) = ∣ x ∣ , we get x = 3 ∣ x ∣ + 5 .
Start with an initial guess x 0 = 1 and iterate using x n + 1 = g ( x n ) .
Calculate three iterations: x 1 ≈ 2.0 , x 2 ≈ 2.138 , x 3 ≈ 2.154 .
Compare the final approximation with the given options and select the closest one, which is 4 7 .
Explanation
Problem Analysis Let's analyze the problem. We are given the equation f ( − x ) + 4 = 3 x − 1 and asked to approximate the solution using three iterations of successive approximation. This means we need to rewrite the equation in the form x = g ( x ) , choose a starting point x 0 , and then iterate using the formula x n + 1 = g ( x n ) for n = 0 , 1 , 2 . The graph is not provided, so we will assume a starting point. We will also assume an example of f ( − x ) function to be ∣ x ∣ .
Rewriting the Equation First, we rewrite the equation as f ( − x ) = 3 x − 5 . Then, we express the equation in the form x = g ( x ) . In this case, x = 3 f ( − x ) + 5 . Let's assume f ( − x ) = ∣ x ∣ . So, x = 3 ∣ x ∣ + 5 .
First Iteration Let's assume an initial guess x 0 = 1 . Now we iterate using the formula x n + 1 = g ( x n ) .
Iteration 1: x 1 = 3 ∣ x 0 ∣ + 5 = 3 ∣1∣ + 5 = 3 1 + 5 = 3 6 = 2.0
Second Iteration Iteration 2: x 2 = 3 ∣ x 1 ∣ + 5 = 3 ∣2∣ + 5 = 3 1.414 + 5 = 3 6.414 ≈ 2.138
Third Iteration Iteration 3: x 3 = 3 ∣ x 2 ∣ + 5 = 3 ∣2.138∣ + 5 = 3 1.462 + 5 = 3 6.462 ≈ 2.154
Comparing with Options Now we compare the final approximation x 3 ≈ 2.154 with the given options:
A. x ≈ 16 25 = 1.5625 B. x ≈ 16 27 = 1.6875 C. x ≈ 4 7 = 1.75 D. x ≈ 8 13 = 1.625
The closest option to 2.154 is C. x ≈ 4 7 = 1.75 . However, given the example inverse function, the closest answer is C.
Final Answer Given the calculations and the options, the closest approximation is C. x ≈ 4 7 .
Examples
Successive approximation is a powerful tool used in many real-world applications, such as engineering and computer science. For example, engineers use iterative methods to refine designs and optimize performance. In computer graphics, iterative algorithms are used to render complex scenes by repeatedly refining an initial approximation. These methods are essential when a direct solution is difficult or impossible to obtain, providing increasingly accurate results through repeated calculations.
Using three iterations of successive approximation on the equation f ( − x ) + 4 = 3 x − 1 , we find that the closest approximation to the solution is option C: x ≈ 4 7 .
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