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) \( r \) is a root if \( f(r)=0 \). (b) Newton’s method uses \( g'(r)=0 \) in the FixedPoint Algorithm.

Step 1: Understand the Given Equation

The given equation is a variant of Newton's Method: \( x = x - \frac{f(x)}{f'(x)} \). This can also be written as \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \), representing an iterative approach to find roots of the function \( f(x) \).

Step 2: Analyze Equation for Root Condition

Consider the equation \( x = x - \frac{f(x)}{f'(x)} \). For \( r \) to be a root, it must satisfy \( f(r) = 0 \). If \( f(r) = 0 \), then \( x = x - \frac{0}{f'(x)} = x \), meaning the equation holds true. Conversely, if \( x = x - \frac{f(x)}{f'(x)} \), rearranging gives \( f(x) = 0 \). Thus, \( r \) is a root if and only if \( f(r) = 0 \).

Step 3: Link to Newton's Method

Newton's Method is \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). It's formed on the iterative process of finding fixed points such that \( x = g(x) \), where \( g(x) = x - \frac{f(x)}{f'(x)} \). Here, \( g'(r) = 1 - \frac{f'(r)}{f'(r)} = 0 \). This shows Newton's Method is a distinct subset of fixed-point algorithms where the derivative of \( g \) at the root is zero.

Key Concepts

Root-Finding Algorithms

Root-finding algorithms are methods used in numerical analysis to identify roots of a real-valued function. **Roots**, also known as zeros, are the values for which a given function equals zero. These algorithms are essential because they provide solutions to equations in various fields, including engineering and mathematics.
There are several types of root-finding algorithms, with each having its unique approach:

• Bisection Method: This simple method iteratively narrows down the interval in which a root must reside. It is reliable but can be slow.
• Newton's Method: This method uses the function's derivative and is known for its fast convergence. However, it requires a good initial estimate.
• Secant Method: Similar to Newton's method, it doesn't need the calculation of derivatives but instead uses a sequence of secants.

Understanding the uniqueness and effectiveness of these algorithms allows us to tackle complex equations efficiently. Despite differences, all aim to achieve the same goal—finding values of x such that the function evaluates to zero.

Fixed-Point Iteration

Fixed-point iteration is a numerical method used to solve equations that can be expressed in the form \( x = g(x) \), where the function is rewritten so that the variable is isolated. The method involves starting with a guess or initial value and iteratively applying the function \( g \) to find a solution.
This technique relies on the principle that if a fixed point exists, the iterations can converge to it, provided certain conditions are met:

• The function should be continuous in the interval considered.
• The derivative \( g'(x) \) should be in the range \((-1, 1)\) to guarantee convergence.

Newton's Method is intrinsically linked to fixed-point iteration. It transforms the original problem into the form \( x_{n+1} = g(x_n) \) with \( g(x) = x - \frac{f(x)}{f'(x)} \). One key feature of this method is that at the actual root, the derivative \( g'(r) = 0 \), showing how Newton's Method is a fixed-point iteration tailored for more efficient root approximation.

Differential Calculus

Differential calculus, one of the core branches of calculus, deals primarily with the concept of derivatives. The derivative of a function at a point provides the slope of the tangent line at that point, essentially measuring how the function value changes as its input changes.
Derivatives play a crucial role in Newton's Method. The iterative process to find the root of a function \( f(x) \) involves computing and using its derivative \( f'(x) \). This is because Newton's Method not only uses the value of the function but also the slope (given by the derivative) to find where a linear approximation of the function crosses the x-axis.
In simpler terms, differential calculus allows us to develop efficient algorithms for root finding by providing insights into the behavior of functions and their changes. Mastery of derivatives and differential calculus principles is vital for understanding and applying Newton's Method or any similar numerical techniques effectively.