Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of \(y=x^{1 / 3} .\) Obviously, its only \(x\) intercept is zero. Convince yourself that Newton's Method fails to converge to the root of \(x^{1 / 3}=0 .\) Explain this failure.

Step-by-Step Solution

Answer

Newton's Method fails due to steep slopes near \(x = 0\), causing oscillation.

Step 1: Understanding the Function

The function given is \(y = x^{1/3}\). This function represents the cube root of \(x\). The cube root is an odd root, meaning it is defined for all real numbers \(x\). It is symmetric with respect to the origin, which makes it an odd function.

Step 2: Sketching the Graph

To sketch the graph of \(y = x^{1/3}\), we should consider a few key points:- At \(x = 0\), \(y = 0\), so it crosses the origin.- As \(x\) increases, \(y\) also increases slowly.- As \(x\) decreases, \(y\) decreases in the negative direction. - The graph is a gentle curve that passes through the origin.Thus, the graph resembles an 'S' shaped curve rotated 90 degrees counterclockwise.

Step 3: Evaluating Newton's Method for Convergence

Newton's Method uses the formula:\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]For \(f(x) = x^{1/3}\), the derivative is \(f'(x) = \frac{1}{3}x^{-2/3}\). Newton’s method iteration becomes:\[x_{n+1} = x_n - \frac{x_n^{1/3}}{\frac{1}{3}x_n^{-2/3}} = x_n - 3x_n\]This simplifies to:\[x_{n+1} = -2x_n\] This results in the sequence oscillating and never converging to the root \(x = 0\). Instead, it doubles its distance from 0 in each iteration for nonzero \(x_n\).

Step 4: Explaining the Failure

The failure arises because the derivative \(f'(x) = \frac{1}{3}x^{-2/3}\) becomes undefined or very large as \(x\) approaches 0. Since the iteration formula causes the value to oscillate instead of stabilizing, Newton's Method fails to converge. The problem is the sharpness of the slope near the root which causes instability in the iteration process.

Key Concepts

Cube Root Function

The cube root function, expressed as \(y = x^{1/3}\), is an interesting mathematical function due to its properties. Unlike square root functions, cube root functions are defined for all real numbers. This means that you can input any real number, and it will have a well-defined cube root. Unlike even-powered roots that are restricted to non-negative numbers, the cube root does not face such limitations.

Some key characteristics of the cube root function include its symmetry with respect to the origin, making it an odd function. This symmetry implies that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, -y)\). The function also gradually increases, which results in a smooth, gentle curve through the origin. The nature of this curve is what gives the cube root function its distinct visual appearance, which can be described as a 90-degree counterclockwise rotated letter 'S'.

This function is particularly interesting in calculus and mathematical analysis due to its behavior as \(x\) approaches zero, possessing both a derivative and an indefinite integral that need to be handled carefully.

Graph Sketching

Sketching the graph of the cube root function requires understanding its unique behavior. At first glance, the graph of \(y = x^{1/3}\) may seem simple, yet it harbors important nuances that should be noted.

• The graph passes through the origin \((0,0)\), which serves as both the x-intercept and y-intercept.
• As \(x\) increases from zero, \(y\) also increases but at a decreasing rate, producing a gentle upward curve to the right.
• As \(x\) decreases to negative values, \(y\) decreases into the negative y values, creating a symmetric downward curve to the left.
• The transition through the origin is smooth without any sharp corners or undefined points.

The characteristic 'S' shape is due to the way the cube root function maintains its growth behavior throughout both positive and negative domains. Understanding this visualization helps in interpreting behavior in real-world applications and anticipating issues that might arise in computational methods.

Convergence Failure

Newton's Method is a popular root-finding algorithm that can solve equations numerically. However, when applied to the cube root function \(f(x) = x^{1/3}\), it results in a failure to converge to the root. Here's why:

Newton's Method relies on the formula:\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]For the cube root function, this becomes:\[x_{n+1} = x_n - 3x_n\]This simplifies to:\[x_{n+1} = -2x_n\]

Instead of converging towards zero, the sequence oscillates between positive and negative values, increasing instead of decreasing. It doubles its distance from zero with each iteration, therefore never stabilizing at the root. This behavior highlights the limitations of Newton's Method under certain conditions. The oscillation and divergence result from how the derivative behaves near zero, which impacts the step size and direction unpredictably.

Derivative Instability

The derivative of the cube root function is crucial to understanding why Newton’s Method fails in this scenario. For \(f(x) = x^{1/3}\), the derivative is:\[f'(x) = \frac{1}{3}x^{-2/3}\]

This derivative becomes problematic as \(x\) approaches zero. Here are some important points to consider:

• As \(x\) gets very close to zero, \(x^{-2/3}\) results in very large values.
• The derivative becomes undefined exactly at \(x = 0\), which is problematic for Newton's Method because it requires evaluating \(f'(x)\).
• This instability near the root results in large, erratic steps in the Newton’s Method iteration.

Consequently, any small perturbation or initial guess that is not carefully chosen can lead to failure and divergence rather than convergence. This limitation highlights the importance of considering alternative numerical methods or adjustments when dealing with functions exhibiting derivative instability.