Select The Correct Answer From Each Drop-down Menu.Consider The Following Equation: 3 X = 2 − X + 4 3^x = 2^{-x} + 4 3 X = 2 − X + 4 If The Equation Is Solved Using Successive Approximation, Select The Value Of X X X After The Following Number Of Iterations.1

Mar 13, 2025 by ADMIN 260 views

**Solving the Equation Using Successive Approximation**

Introduction

In this article, we will explore the concept of successive approximation and its application in solving equations. We will use the given equation $3^x = 2^{-x} + 4$ and determine the value of $x$ after a specified number of iterations.

Understanding Successive Approximation

Successive approximation is a numerical method used to find the roots of an equation. It involves making an initial guess and then iteratively improving the estimate until the desired level of accuracy is achieved. The method is based on the idea of repeatedly applying a function to the current estimate and using the result as the new estimate.

The Given Equation

The equation we will be working with is $3^x = 2^{-x} + 4$ . This equation is a transcendental equation, meaning that it cannot be solved using algebraic methods. We will use successive approximation to find the value of $x$ that satisfies this equation.

Initial Guess

To start the process of successive approximation, we need to make an initial guess for the value of $x$ . Let's assume that our initial guess is $x_0 = 1$ .

First Iteration

Using the initial guess $x_0 = 1$ , we can calculate the value of the function $f(x) = 3^x - 2^{-x} - 4$ .

$f(x_0) = 3^1 - 2^{-1} - 4 = 3 - 0.5 - 4 = -1.5$

Since the value of $f(x_0)$ is negative, we know that the root of the equation lies between $x_0$ and $x_0 + 1$ . Therefore, we can make a new estimate for the value of $x$ by taking the average of $x_0$ and $x_0 + 1$ .

$x_1 = \frac{x_0 + (x_0 + 1)}{2} = \frac{1 + 2}{2} = 1.5$

Second Iteration

Using the new estimate $x_1 = 1.5$ , we can calculate the value of the function $f(x)$ .

$f(x_1) = 3^{1.5} - 2^{-1.5} - 4 = 5.196 - 0.8409 - 4 = 0.3551$

Since the value of $f(x_1)$ is positive, we know that the root of the equation lies between $x_1$ and $x_1 + 1$ . Therefore, we can make a new estimate for the value of $x$ by taking the average of $x_1$ and $x_1 + 1$ .

$x_2 = \frac{x_1 + (x_1 + 1)}{2} = \frac{1.5 + 2.5}{2} = 2$

Third Iteration

Using the new estimate $x_2 = 2$ , we can calculate the value of the function $f(x)$ .

$f(x_2) = 3^2 - 2^{-2} - 4 = 9 - 0.25 - 4 = 4.75$

Since the value of $f(x_2)$ is positive, we know that the root of the equation lies between $x_2$ and $x_2 + 1$ . Therefore, we can make a new estimate for the value of $x$ by taking the average of $x_2$ and $x_2 + 1$ .

$x_3 = \frac{x_2 + (x_2 + 1)}{2} = \frac{2 + 3}{2} = 2.5$

Fourth Iteration

Using the new estimate $x_3 = 2.5$ , we can calculate the value of the function $f(x)$ .

$f(x_3) = 3^{2.5} - 2^{-2.5} - 4 = 15.588 - 0.447 - 4 = 11.141$

Since the value of $f(x_3)$ is positive, we know that the root of the equation lies between $x_3$ and $x_3 + 1$ . Therefore, we can make a new estimate for the value of $x$ by taking the average of $x_3$ and $x_3 + 1$ .

$x_4 = \frac{x_3 + (x_3 + 1)}{2} = \frac{2.5 + 3.5}{2} = 3$

Conclusion

After four iterations, we have found that the value of $x$ that satisfies the equation $3^x = 2^{-x} + 4$ is approximately $x = 3$ . This value is accurate to within a small margin of error.

Discussion

The method of successive approximation is a powerful tool for solving equations numerically. By making an initial guess and then iteratively improving the estimate, we can find the roots of equations with high accuracy. In this article, we have used the method of successive approximation to solve the equation $3^x = 2^{-x} + 4$ and found that the value of $x$ that satisfies this equation is approximately $x = 3$ .

References

[1] "Numerical Methods for Solving Equations" by John R. Rice
[2] "Successive Approximation" by Wikipedia

Select the Correct Answer

Based on the calculations above, the value of $x$ after four iterations is approximately $x = 3$ . Therefore, the correct answer is:

Introduction

In our previous article, we explored the concept of successive approximation and its application in solving equations. We used the equation $3^x = 2^{-x} + 4$ and determined the value of $x$ after a specified number of iterations. In this article, we will answer some frequently asked questions about successive approximation and solving equations.

Q: What is successive approximation?

A: Successive approximation is a numerical method used to find the roots of an equation. It involves making an initial guess and then iteratively improving the estimate until the desired level of accuracy is achieved.

Q: How does successive approximation work?

A: Successive approximation works by repeatedly applying a function to the current estimate and using the result as the new estimate. The process is repeated until the desired level of accuracy is achieved.

Q: What are the advantages of successive approximation?

A: The advantages of successive approximation include:

It is a simple and efficient method for solving equations numerically.
It can be used to solve equations with high accuracy.
It is a flexible method that can be used to solve a wide range of equations.

Q: What are the disadvantages of successive approximation?

A: The disadvantages of successive approximation include:

It requires an initial guess, which can affect the accuracy of the solution.
It can be slow to converge to the solution, especially for complex equations.
It may not be suitable for equations with multiple roots.

Q: How do I choose the initial guess for successive approximation?

A: Choosing the initial guess for successive approximation can be a challenging task. However, here are some tips to help you choose a good initial guess:

Start with a simple guess, such as 0 or 1.
Use a graphical method, such as plotting the function, to estimate the location of the root.
Use a numerical method, such as the bisection method, to estimate the location of the root.

Q: How do I know when to stop the successive approximation process?

A: There are several ways to determine when to stop the successive approximation process, including:

Set a maximum number of iterations.
Set a tolerance for the error.
Monitor the convergence of the solution.

Q: Can successive approximation be used to solve equations with multiple roots?

A: Successive approximation can be used to solve equations with multiple roots, but it may not be the most efficient method. In such cases, other numerical methods, such as the Newton-Raphson method, may be more suitable.

Q: Can successive approximation be used to solve equations with complex roots?

A: Successive approximation can be used to solve equations with complex roots, but it may require special handling to avoid numerical instability.

Q: What are some common applications of successive approximation?

A: Successive approximation has a wide range of applications, including:

Solving equations in physics and engineering.
Solving equations in economics and finance.
Solving equations in computer science and data analysis.

Conclusion

Successive approximation is a powerful tool for solving equations numerically. By understanding the basics of successive approximation and its applications, you can use this method to solve a wide range of equations. In this article, we have answered some frequently asked questions about successive approximation and solving equations.

References

[1] "Numerical Methods for Solving Equations" by John R. Rice
[2] "Successive Approximation" by Wikipedia

Select the Correct Answer

Based on the questions and answers above, the correct answers are:

Successive approximation is a numerical method used to find the roots of an equation.
Successive approximation works by repeatedly applying a function to the current estimate and using the result as the new estimate.
The advantages of successive approximation include its simplicity, efficiency, and flexibility.
The disadvantages of successive approximation include the need for an initial guess, the potential for slow convergence, and the possibility of multiple roots.
Choosing the initial guess for successive approximation can be a challenging task, but starting with a simple guess, using a graphical method, or using a numerical method can help.
Determining when to stop the successive approximation process can be done by setting a maximum number of iterations, setting a tolerance for the error, or monitoring the convergence of the solution.
Successive approximation can be used to solve equations with multiple roots, but other numerical methods may be more suitable.
Successive approximation can be used to solve equations with complex roots, but special handling may be required to avoid numerical instability.
Successive approximation has a wide range of applications, including solving equations in physics and engineering, economics and finance, and computer science and data analysis.