Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider the equation \(x=x-f(x) / f^{\prime}(x)\) and suppose that \(f^{\prime}(x) \neq 0\) in an interval \([a, b]\). (a) Show that if \(r\) is in \([a, b]\) then \(r\) is a root of the equation \(x=x-f(x) / f^{\prime}(x)\) if and only if \(f(r)=0\) (b) Show that Newton's Method is a special case of the FixedPoint Algorithm. in which \(g^{\prime}(r)=0\)

Step-by-Step Solution

Answer

(a) If \( r \) is a root, then \( f(r) = 0 \). (b) Newton's method corresponds to a fixed-point iteration with \( g^{\prime}(r) = 0 \).

Step 1: Understanding the Equation

We start by analyzing the equation given in the problem: \[ x = x - \frac{f(x)}{f^{\prime}(x)} \]This is an iteration step used in Newton's Method for finding roots of the function \( f(x) \). By rewriting it, we get: \[ x = x - \frac{f(x)}{f^{\prime}(x)} \Rightarrow f(x) = 0 \]This suggests that \( x \) is the root of \( f(x) \). The condition \( f^{\prime}(x) eq 0 \) ensures the division is valid.

Step 2: Part (a): Show (r \\ in \\ [a, b]) is a root of the equation if and only if f(r)=0

Given that \( r \) is in the interval \([a, b]\), we consider \( r = r - \frac{f(r)}{f^{\prime}(r)} \).Rearranging gives:\[ f(r) = 0 \]Thus, \( r \) is a root of the equation. Conversely, if \( f(r) = 0 \), then clearly the expression \( r - \frac{f(r)}{f^{\prime}(r)} = r \), implying \( r \) as a fixed point of the equation.

Step 3: Part (b): Demonstrate Newton's Method as a Fixed-Point Algorithm

In Fixed-Point iteration, we usually have \( x = g(x) \). In the context of our equation, it becomes\[ x = g(x) = x - \frac{f(x)}{f^{\prime}(x)} \]Newton's method iteratively updates via\[ x_{n+1} = x_n - \frac{f(x_n)}{f^{\prime}(x_n)} \]fitting the fixed-point format. Now calculate the derivative,\[ g^{\prime}(x) = 1 - \frac{f^{\prime}(x) f^{\prime}(x) - f(x) f^{\prime\prime}(x)}{(f^{\prime}(x))^2} \]at a root \( r \), \( f(r) = 0 \) gives us \( g^{\prime}(r) = 0 \). Thus, Newton's method is indeed a special fixed-point iteration where \( g^{\prime}(r) = 0 \).

Key Concepts

Fixed-Point Iteration

Fixed-point iteration is a method for finding a point (or, more technically, an argument of a function) for which the function does not change its value, known as a fixed point. Imagine it like finding a place where bouncing a ball in a certain manner will make it stop bouncing—it stays fixed.

Often represented in the form of an equation, a simple fixed-point iteration looks like this: \( x = g(x) \). In order to apply this iteration to find the root of a function, we can reformulate the function into fixed-point form. This involves strategically choosing or transforming another function, \( g(x) \), so that the original problem becomes looking for an \( x \) such that \( x = g(x) \). When you repeatedly apply \( g \) to a value, it should ideally converge to a solution.

• It requires the process of refining a solution through iteration.
• Suitable for functions that can be expressed in fixed-point form.

Fixed-point iteration is foundational for other methods, including Newton's Method, a powerful tool in approximating roots.

Root Finding

Root finding refers to the process of identifying values (roots) where a given function equals zero. When we say a number \( r \) is a root of a function \( f(x) \), we mean \( f(r) = 0 \). We are essentially searching for intercepts of the function with the x-axis on a graph. For those unfamiliar, think of it as finding spots where a line or curve touches or crosses the middle baseline horizontally.

Newton's Method, commonly used for root finding, involves using an initial guess followed by iterative calculations to "hone in" on the actual root. The efficiency of Newton's Method makes it particularly useful when dealing with complicated functions where other methods might be slower or less accurate.

• Root finding is crucial for solving equations and mathematical problems analytically and numerically.
• Finding accurate roots often requires several iterative methods to ensure precision.

Understanding the root finding process is key to unlocking complex mathematical equations and has practical applications in science and engineering.

Calculus

Calculus is a branch of mathematics that deals with change and motion. In the context of Newton's Method and fixed-point iteration, calculus is used to analyze how functions behave near potential roots.

At its core, calculus is concerned with two main concepts:

• **Differentiation**, which is used to understand rates of change, such as speed or growth.
• **Integration**, which is used to compute areas under curves or total accumulations.

Differentiation plays a critical role in derivative analysis used in Newton's Method. It allows us to approximate a function linearly at any point, which, in simpler terms, is like finding a flat tangent that just "touches" the curve drawn at a point. The derivative tells us the slope of this tangent, helping us understand better how the curve rises or falls near a given value.

Both differentiation and calculus as a whole are essential for analyzing and solving problems involving dynamic systems where everything keeps changing.

Derivative Analysis

Derivative analysis involves examining derivatives to understand the behavior of functions. Think of derivatives as tools to measure how a function's output value changes as its input changes. In Newton's Method, derivative analysis is key because it uses the first derivative of a function to estimate its roots. The derivative of a function at a point tells us the slope of the tangent at that point. This is crucial for determining where the function crosses the x-axis. For our scenario, we often need to find where the derivative is nonzero, \( f^{\prime}(x) eq 0 \), ensuring valid division in calculations.

Analyzing higher order derivatives can give even more insight, such as determining concavity and the behavior of the function in terms of rate of change. Proper derivative analysis often reveals the nature of roots, such as whether they are simple crossings of the axis or touchpoints, showing higher multiplicity.