Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

In applying Newton's Method to solve \(f(x)=0\), one can usually tell by simply looking at the numbers \(x_{1}, x_{2}, x_{3}, \ldots\) whether the sequence is converging. But even if it converges, say to \(\bar{x}\), can we be sure that \(\bar{x}\) is a solution? Show that the answer is yes provided \(f\) and \(f^{\prime}\) are continuous at \(\bar{x}\) and \(f^{\prime}(\bar{x}) \neq 0\).

Step-by-Step Solution

Answer

Yes, if both \( f \) and \( f' \) are continuous at \( \bar{x} \) and \( f'(\bar{x}) \neq 0 \), \( \bar{x} \) is a solution.

Step 1: Understand Newton's Method

Newton's Method is an iterative formula used to find successively better approximations to the roots (or zeros) of a real-valued function. The iteration formula is given by \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). This method is based on the assumption that sufficiently close approximations will lead to the exact root.

Step 2: Define the Limit of the Sequence

Assume the sequence \( x_n \) generated by Newton's Method converges to a point \( \bar{x} \). Mathematically, this is expressed as \( \lim_{n \to \infty} x_n = \bar{x} \).

Step 3: Apply Continuity of Functions

Since \( f \) and \( f' \) are continuous at \( \bar{x} \), it follows that \( f(x_n) \to f(\bar{x}) \) and \( f'(x_n) \to f'(\bar{x}) \) as \( n \to \infty \).

Step 4: Analyze the Iterative Formula Near the Limit

From the iterative formula, as the sequence \( x_n \) converges to \( \bar{x} \), it holds that \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \to \bar{x} \). This implies \( \lim_{n \to \infty} \frac{f(x_n)}{f'(x_n)} = 0 \) because the sequence converges.

Step 5: Show that \( \bar{x} \) is a Root

For \( \frac{f(x_n)}{f'(x_n)} \to 0 \), it must be the case that \( f(\bar{x}) = 0 \) if \( f'(\bar{x}) eq 0 \). Since \( f \) is continuous and the limit of a continuous function at a point equals the function's value at that point, \( f(\bar{x}) = 0 \). Thus, \( \bar{x} \) is a solution of \( f(x) = 0 \).

Key Concepts

Root-finding Algorithm

Newton's Method is a well-known root-finding algorithm designed to locate the roots of a function, also known as zeros. A root of a function is any number that makes the function equal to zero. Finding such numbers can be essential in fields like engineering, physics, and mathematics.
This method effectively navigates the complex landscape of a function's graph and homes in on its roots by using calculus techniques involving derivatives. The core idea is to refine guesses progressively. The formula used in Newton's Method is:

\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
This formula takes a current approximation \(x_n\), and updates it based on the function \(f\) and its derivative \(f'\) evaluated at that point, guiding closer approximations to the actual root.

• Start with an initial guess \(x_0\)
• Apply the formula over multiple iterations
• Each step aims to reduce the distance to the root

Iterative Methods

Newton's Method belongs to a broader category of techniques known as iterative methods. These methods use repeated approximations to eventually converge on a solution.
An iterative process like this involves the repetition of a sequence of operations. In Newton's Method, each operation refines the previous result, aiming to get closer to the root. This can be a powerful approach when analytical solutions are difficult.

• Each iteration moves us closer to the desired solution
• Key to sticking close to a preferred pathway
• Enables tackling problems that don't have straightforward solutions

The success of iterative methods often relies on having a good initial guess and ensuring that the function behaves "nicely" around the supposed root, which guarantees that each step effectively homes in on the correct solution.

Convergence Analysis

Convergence analysis is crucial in understanding whether Newton's Method will successfully find a root. It involves examining whether the sequence of approximations \(x_n\) actually approaches a true solution \(\bar{x}\).
This concept revolves around the idea that for the method to work, the sequence must converge, or become increasingly close to a specific value. In this exercise, once these sequences appear to be converging, we assume they are approaching a definite solution. This results in the mathematical expression:

\[ \lim_{n \to \infty} x_n = \bar{x} \]
Convergence is greatly influenced by the nature of the function and its derivative at \(\bar{x}\). The function should be well-behaved — smooth and continuous — to ensure uniform steps towards the root. A convergence analysis helps determine and guarantee that the process ends in a successful result, verifying that \(f(\bar{x}) = 0\) is indeed true as long as \(f'(\bar{x}) eq 0\), making \(\bar{x}\) a real solution.

Continuity of Functions

The continuity of functions is vital for the assurance that Newton's Method will correctly identify the root. In simple terms, continuity means there are no breaks, jumps, or holes in a function's graph within the range of interest.
For Newton's Method to affirm that \(\bar{x}\) is a real solution, the function \(f\) and its derivative \(f'\) must be continuous at \(\bar{x}\). When both are continuous, it implies:

• As \(x_n\) gets closer to \(\bar{x}\), \(f(x_n)\) approaches \(f(\bar{x})\)
• Similarly, \(f'(x_n)\) approaches \(f'(\bar{x})\)
• This behavior confirms \(f(\bar{x}) = 0\) when the sequence converges

Continuity ensures that changes in \(x\) result in predictable, steady changes in \(f(x)\), which is crucial for assessing whether the approximations generated by Newton's Method lead to an actual root.