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Magazine Chapter 3: Problem 38
Use the Newton-Raphson method to find an approximate solution of the given equation in the given interval. Use the method until successive approximations obtained by calculator are identical. $$ x^{2}+4 x^{6}=2 ;[0,1] $$
Short Answer
Expert verified
Approximately 0.537.
Step by step solution
01
Understand the Problem
Given the function \( f(x) = x^2 + 4x^6 - 2 \), we are tasked with using the Newton-Raphson method to find the roots of this equation in the interval [0, 1].
02
Derive the Function
The first derivative of the function \( f(x) = x^2 + 4x^6 - 2 \) is \( f'(x) = 2x + 24x^5 \). This will be used in the Newton-Raphson formula.
03
Newton-Raphson Formula
The Newton-Raphson iterative formula is given by \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). We will apply this formula to determine successive values of \( x \) until they become identical.
04
Initial Guess
We begin with an initial guess, \( x_0 = 0.5 \), which is within the interval [0,1].
05
First Iteration
Calculate \( f(0.5) = 0.5^2 + 4(0.5)^6 - 2 = -1.90625 \). Calculate \( f'(0.5) = 2(0.5) + 24(0.5)^5 = 1.75 \). Apply \( x_1 = 0.5 - \frac{-1.90625}{1.75} = 1.5875 \).
06
Second Iteration
With the new value \( x_1 = 1.5875 \), compute \( f(1.5875) = (1.5875)^2 + 4*(1.5875)^6 - 2 \), and \( f'(1.5875) = 2*(1.5875) + 24*(1.5875)^5 \). Apply the Newton-Raphson formula to get \( x_2 \).
07
Repeat Iterations
Continue with the iterations until \( x_n \) and \( x_{n+1} \) are sufficiently close or identical. Compute \( x_n \) using the approximations and check for convergence in each step.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Approximate Solution
The Newton-Raphson method is a powerful technique used to find an approximate solution to equations where analytical solutions might be difficult or impossible to obtain. In mathematics, an approximate solution refers to a value close enough to the actual solution for practical purposes.
Instead of solving an equation exactly, the goal is to get an estimate that satisfies the equation to a reasonable degree of accuracy. Here's how it works:
Instead of solving an equation exactly, the goal is to get an estimate that satisfies the equation to a reasonable degree of accuracy. Here's how it works:
- Start with an initial guess that's close to the expected solution.
- Use algorithms, such as the Newton-Raphson method, to refine this guess.
- Iteratively adjust the guess until consecutive estimates are the same.
Iterative Method
An iterative method is a mathematical process that iteratively repeats calculations to move closer to the desired solution. The Newton-Raphson method is a classic example of such a method.
Each iteration in the Newton-Raphson method involves computing an estimate for the root of a function using its derivative. The process requires the following:
Each iteration in the Newton-Raphson method involves computing an estimate for the root of a function using its derivative. The process requires the following:
- Choosing an initial guess.
- Applying the Newton-Raphson formula, which revises the guess by accounting for the function's curvature.
- Repeating this revision until convergence, when subsequent iterations yield nearly the same value.
Derivative
The derivative is a fundamental concept in calculus, representing the rate of change of a function with respect to one of its variables. In the context of the Newton-Raphson method, the derivative plays a crucial role.
For a given function \( f(x) \), its derivative \( f'(x) \) provides information about the slope of the function at any point \( x \). This slope information is key in adjusting estimates during each iteration:
For a given function \( f(x) \), its derivative \( f'(x) \) provides information about the slope of the function at any point \( x \). This slope information is key in adjusting estimates during each iteration:
- The most recent guess \( x_n \) is updated based on the value of \( f'(x_n) \).
- For the Newton-Raphson method, the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \) is used.
- This formula incorporates the slope \( f'(x_n) \) to modify the guess intelligently.
Root Finding
Root finding is the process of determining where a function equals zero, i.e., finding the values of \( x \) for which \( f(x) = 0 \). The Newton-Raphson method is specifically designed as a root-finding algorithm.
The ultimate goal is to establish the value of \( x \) that satisfies the equation under investigation. This involves:
The ultimate goal is to establish the value of \( x \) that satisfies the equation under investigation. This involves:
- Identifying an initial interval where the root likely exists, as in \([0, 1]\) for this problem.
- Executing the Newton-Raphson iterations to narrow down the root's location.
- Continuing this process until the difference between successive approximations is minimal.
