Select the correct answer from each drop-down menu. Consider the following equation: [tex]3^x=2^{-x}+4[/tex] If the equation is solved using successive approximation, select the value of [tex]x[/tex] after the following number of iterations. 1 iteration: [tex]x=[/tex] $\square$ 2 iteration: [tex]x=[/tex] $\square$ 3 iteration: [tex]x=[/tex] $\square$
Answer (2)
Start with an initial guess x 0 = 1 .
After the first iteration, calculate x 1 = lo g 3 ( 2 − x 0 + 4 ) ≈ 1.369 .
After the second iteration, calculate x 2 = lo g 3 ( 2 − x 1 + 4 ) ≈ 1.346 .
After the third iteration, calculate x 3 = lo g 3 ( 2 − x 2 + 4 ) ≈ 1.347 .
Thus, the approximate values of x after 1, 2, and 3 iterations are 1.369, 1.346, and 1.347, respectively.
Explanation
Understanding the Problem We are given the equation 3 x = 2 − x + 4 and asked to find the approximate value of x after 1, 2, and 3 iterations using successive approximation. Successive approximation is an iterative method where we start with an initial guess and repeatedly refine it.
First Iteration Let's start with an initial guess of x 0 = 1 . For the first iteration, we substitute x 0 = 1 into the equation and solve for x . We have 3 x = 2 − x 0 + 4 , so x = lo g 3 ( 2 − x 0 + 4 ) . Thus, x 1 = lo g 3 ( 2 − 1 + 4 ) = lo g 3 ( 0.5 + 4 ) = lo g 3 ( 4.5 ) .
Second Iteration For the second iteration, we substitute x 1 into the equation and solve for x . We have 3 x = 2 − x 1 + 4 , so x = lo g 3 ( 2 − x 1 + 4 ) . Thus, x 2 = lo g 3 ( 2 − l o g 3 ( 4.5 ) + 4 ) .
Third Iteration For the third iteration, we substitute x 2 into the equation and solve for x . We have 3 x = 2 − x 2 + 4 , so x = lo g 3 ( 2 − x 2 + 4 ) . Thus, x 3 = lo g 3 ( 2 − l o g 3 ( 2 − l o g 3 ( 4.5 ) + 4 ) + 4 ) .
Calculating the Approximations Now, let's calculate the values of x 1 , x 2 , and x 3 :
x 1 = lo g 3 ( 4.5 ) ≈ 1.36907 x 2 = lo g 3 ( 2 − 1.36907 + 4 ) ≈ 1.34595 x 3 = lo g 3 ( 2 − 1.34595 + 4 ) ≈ 1.34725
Therefore, after 1 iteration, x ≈ 1.36907 . After 2 iterations, x ≈ 1.34595 . After 3 iterations, x ≈ 1.34725 .
Final Answer After 1 iteration: x ≈ 1.369 After 2 iterations: x ≈ 1.346 After 3 iterations: x ≈ 1.347
So the values of x after 1, 2, and 3 iterations are approximately 1.369, 1.346, and 1.347, respectively.
Examples
Successive approximation is a powerful tool used in various fields like engineering and computer science to find approximate solutions to equations that are difficult or impossible to solve analytically. For instance, in circuit analysis, it can be used to determine the voltage or current in a complex circuit by iteratively refining an initial guess until a stable solution is reached. Similarly, in computer graphics, successive approximation can be employed to render realistic images by progressively improving the image quality through multiple iterations.
After using successive approximation on the equation 3 x = 2 − x + 4 , the approximate values of x are 1.369 after the first iteration, 1.346 after the second, and 1.347 after the third iteration.
;
