Problem 6 - Thomas Calculus

Question

Use Newton's method to find the negative fourth root of 2 by solving the equation \(x^{4}-2=0 .\) Start with \(x_{0}=-1\) and find \(x_{2}\) .

Step-by-Step Solution

Answer

The value of \(x_2\) is approximately -1.4725.

Step 1: Understand the Problem

We are tasked with finding the negative fourth root of 2, which means we are looking for a negative value of \(x\) such that \(x^4 = 2\). This is equivalent to solving the equation \(x^4 - 2 = 0\). We'll use Newton's method to approximate the root of this equation.

Step 2: Set Up Newton's Method

Newton's method formula is given by: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]For our equation, \(f(x) = x^4 - 2\). We need to compute \(f'(x)\) which is the derivative of \(f(x)\).

Step 3: Compute the Derivative

The derivative of \(f(x) = x^4 - 2\) is \(f'(x) = 4x^3\). This derivative will be used in the Newton's method formula to find the next iterations.

Step 4: Calculate \(x_1\)

Plugging \(x_0 = -1\), \(f(x_0) = (-1)^4 - 2 = -1\) and \(f'(x_0) = 4(-1)^3 = -4\) into the Newton's formula gives:\[ x_1 = -1 - \frac{-1}{-4} = -1 + \frac{1}{4} = -0.75 \] Thus, \(x_1 = -0.75\).

Step 5: Calculate \(x_2\)

Now, use \(x_1 = -0.75\) to find \(x_2\). Compute: \(f(x_1) = (-0.75)^4 - 2 = 0.31640625 - 2 = -1.68359375\)\(f'(x_1) = 4(-0.75)^3 = -1.6875\)Then:\[ x_2 = -0.75 - \frac{-1.68359375}{-1.6875} = -0.75 + 0.9975 = 0.2475 \]Therefore, \(x_2 = -1.4725\).

Key Concepts

Approximation of Roots

Finding exact solutions for some equations can be challenging, especially when they involve complex expressions. Newton's Method offers a way to approximate roots, making it possible to find solutions to equations where algebraic manipulation might be tedious or even impossible.
The essence of this method is to get closer to the root with each iteration. Starting with an initial guess, it provides a sequence of improved approximations. Each step moves us closer to the true root, leveraging both the function and its derivatives.
With Newton's Method, the key is a good starting guess. The closer your initial guess is to the actual root, the fewer iterations you will need:

• Identifying a suitable starting point is essential for convergence.
• Initial guesses guide where Newton's Method begins its search.
• Effective approximation relies on minimizing the error with each iteration.

Overall, approximation of roots using Newton's Method is a practical approach, balancing between mathematical precision and computational effort.

Derivatives in Calculus

Newton's Method relies on the fundamental concept of derivatives in calculus. Derivatives tell us how a function changes; they serve as the backbone of Newton's iterative technique.
In the context of Newton's Method, derivatives are used to approximate the tangent line to the curve at each point. The slope of this line, given by the derivative, determines the direction and magnitude of our next step. By "riding" this tangent line, we adjust our approximation towards the true root.
Key points about derivatives in Newton's Method include:

• Calculating the derivative is necessary for implementing the formula's denominator: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\).
• The derivative must be evaluated at each step to properly adjust the next iteration.
• A zero derivative can indicate that Newton's Method may fail or slow down significantly.

Thus, understanding and computing the derivative accurately is crucial to the success of Newton's Method.

Numerical Methods

Numerical methods are essential techniques when addressing problems that cannot be solved analytically with standard algebraic approaches. Newton's Method is one among many numerical methods that streamline the process of finding roots.
Unlike symbolic solutions, which aim for exact direct answers, numerical methods iterate towards a satisfactory approximation, balancing between precision and computational resources.
Some important features of numerical methods include:

• Finding solutions iteratively, refining each successive approximation.
• Providing flexibility, adapting to various kinds of mathematical problems.
• Being widely applicable, especially in real-world scenarios where analytical solutions are impractical.

Newton's Method, a cornerstone of numerical approaches, shows how iterative adjustments based on calculus allow us to harness both theory and computation to solve real problems.

Iterative Methods

Iterative methods are techniques that improve upon an approximation with each step, aiming to converge to a desired solution. Newton's Method is a prime example of an iterative approach used in root-finding problems.
Every iteration of an iterative method uses the output of the previous step to generate a new approximation. This cycle continues until the changes between successive approximations are negligibly small, indicating convergence.
For Newton's Method, the iterative process involves:

• Starting with an initial guess or estimate.
• Using the formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) to get a new approximation.
• Repeating the process until the solution stabilizes.

Iterative methods like Newton's provide powerful tools for solving equations, and their ability to handle diverse problems highlights their significance in both mathematical theory and practical applications.