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Chapter 4: Problem 22

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.

Short Answer

Expert verified
Newton's Method fails because the slope \( f'(x) \) becomes undefined or infinite near the root at \( x = 0 \).

Step by step solution

01

Identifying the function

We start with the function given in the problem: \( f(x) = x^{1/3} \). Our goal is to understand the behavior of this function, particularly around the point \(x = 0\).
02

Analyzing the derivative

The next step involves computing the derivative \( f'(x) \) of the function \( f(x) = x^{1/3} \). We use the power rule for derivatives: \( f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3} \cdot \frac{1}{x^{2/3}} \). Notice that as \( x \) approaches zero, \( f'(x) \) becomes undefined (or tends towards infinity), indicating a vertical tangent line at \( x = 0 \).
03

Sketching the graph

The graph of \( y = x^{1/3} \) resembles a curve that flattens and approaches a vertical tangent near the origin (0,0). It is symmetric about the origin due to its odd nature. The key features are the flattening slope near the x-axis intercept.
04

Explaining Newton's Method

Newton's Method is an iterative root-finding algorithm defined by \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). The process depends heavily on \( f'(x_n) \) being defined and not zero.
05

Identifying the failure

Newton's Method fails for \( f(x) = x^{1/3} \) at \( x = 0 \) because the derivative \( f'(x) \) becomes undefined at this point. The update rule \( x_{n+1} = x_n - \frac{x_n^{1/3}}{\frac{1}{3}x_n^{-2/3}} \) depends on dividing by zero or an undefined value, preventing the convergence.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Root-finding algorithm
Root-finding algorithms are techniques used to find the roots or solutions of a given equation. In our case, we are dealing with Newton's Method. Newton's Method is an iterative approach that
  • starts with an initial guess,
  • calculates a better guess using a specific formula,
  • continues the process until a satisfactory level of accuracy is achieved.
To be more precise, it uses the formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]This formula helps refine the guess by subtracting the ratio of function value and its derivative at the current estimate.
This method requires the derivative to be well-defined and non-zero. Therefore, if the derivative at a guessed point is zero or undefined, Newton's Method may not work well or at all. This can lead to convergence failure, which we'll explore further in another section.
Graph of x^(1/3)
The function given by \(f(x) = x^{1/3}\) represents a cube root, leading to a curve with certain interesting features. The key
  • It is symmetric about the origin owing to its odd function characteristic, meaning \(f(-x) = -f(x)\).
  • The graph tends to flatten out as \(x\) approaches zero and again after moving away from zero.
This flattening effect is due to how roots and exponents below one behave.
It's worthwhile to sketch this graph to better understand it:
  • Observe that the graph passes through the origin (0,0) since \(x^{1/3} = 0\) at \(x = 0\).
  • The graph appears to have a vertical tangent at the origin due to the quick changing slope near \(x = 0\).
Analyzing these visual elements helps us anticipate challenges like the derivative issues encountered in Newton's Method.
Derivative analysis
Understanding the derivative of \(f(x) = x^{1/3}\) is crucial for solving this problem.
Using the power rule for derivatives, we derive:\[ f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3} \cdot \frac{1}{x^{2/3}} \]This result shows the derivative becomes undefined as \(x\) approaches zero because \(x^{-2/3}\) heads towards infinity.
Here are some important points regarding this analysis:
  • The derivative indicates the slope of the tangent line of the function at a given point.
  • An undefined or infinite derivative suggests a vertical tangent line at that point.
Therefore, with \(x=0\), the derivative is not usable in Newton's Method, posing a potential issue with the algorithm.
Convergence failure
Convergence failure refers to a situation where a method does not arrive at the expected result. This problem occurs with Newton's Method applied to \(f(x) = x^{1/3}\) because
  • the derivative \( f'(x) \) becomes undefined at \( x = 0 \), which is where we need it to compute the algorithm's iteration steps, and
  • the formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) breaks down as dividing by an undefined or infinite quantity does not yield a meaningful new approximation.
The key takeaway is recognizing situations that can lead to convergence failure, such as
  • derivatives being zero or undefined at critical points,
  • choice of poor initial guesses that are close to where derivatives fail.
Being aware of these conditions helps in selecting alternative methods or handling such issues during analysis.

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