Problem 25 - Calculus Early Transcendentals

Question

The mechanic's rule for approximating square roots states that \(\sqrt{a} \approx x_{n+1},\) where $$ x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right), \quad n=1,2,3, \ldots $$ and \(x_{1}\) is any positive approximation to \(\sqrt{a} .\) (a) Apply Newton's Method to $$ f(x)=x^{2}-a $$ to derive the mechanic's rule. (b) Use the mechanic's rule to approximate \(\sqrt{10} .\)

Step-by-Step Solution

Answer

The mechanic's rule is derived using Newton's Method and approximates \( \sqrt{10} \) as \( 3.1623 \).

Step 1: Define the Function

Given the function for which we want to find the square root, we define it as \( f(x) = x^2 - a \). This function will help us in both deriving the mechanic's rule and applying Newton's Method.

Step 2: Apply Newton's Method Formula

For Newton's Method, we start with the iteration formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). Evaluate \( f(x) \) and its derivative \( f'(x) \): \( f(x) = x^2 - a \) and \( f'(x) = 2x \).

Step 3: Substitute into Newton's Method

Substitute \( f(x) = x^2 - a \) and \( f'(x) = 2x \) into the Newton's Method formula:\[x_{n+1} = x_n - \frac{x_n^2 - a}{2x_n} = x_n - \frac{x_n}{2} + \frac{a}{2x_n} = \frac{2x_n - x_n + \frac{a}{x_n}}{2} = \frac{x_n + \frac{a}{x_n}}{2}.\]This matches the mechanic's rule.

Step 4: Approximate \( \sqrt{10} \) Using Mechanic's Rule

Choose an initial approximation. Let's take \( x_1 = 3 \), as \( 3^2 = 9 \) is close to 10.Calculate subsequent values using:\[ x_{n+1} = \frac{1}{2} \left( x_n + \frac{10}{x_n} \right) \]- \( x_2 = \frac{1}{2} (3 + \frac{10}{3}) \approx 3.1667 \)- \( x_3 = \frac{1}{2} (3.1667 + \frac{10}{3.1667}) \approx 3.1623 \) Continue until desired accuracy.

Key Concepts

Newton's Method

Newton's Method, also known as Newton-Raphson, is a powerful technique for finding successively better approximations to the roots (or zeros) of a real-valued function. It's like a toolset for solving equations where traditional algebra might be tricky. The method starts with a guess, then refines that guess by utilizing the function and its derivative.

The iteration formula for Newton's Method is:

• \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]

Here, \( x_n \) represents the current estimate, \( f(x_n) \) is the value of the function at that estimate, and \( f'(x_n) \) is the derivative of the function at that point. By plugging these into the formula, you adjust \( x_n \) to be \( x_{n+1} \), a new, often closer estimate.

Newton's Method speeds up the convergence to the actual root, making it an efficient choice in numerical methods, especially for well-behaved functions where derivatives are easy to calculate.

Iterative Methods

Iterative methods, like the mechanic's rule for approximating square roots, are processes where a sequence of improving approximations of a value is generated. The key goal is to reach a solution by repetition, honing in gradually with each step until the desired proximity is achieved.

The idea is to use each approximation to get to the next, more accurate approximation. This is typically done by following a simpler formula derived from the original problem, such as in the mechanics rule:

• \[ x_{n+1} = \frac{1}{2}\left(x_n + \frac{a}{x_n}\right) \]

Iterative methods are incredibly useful in computer algorithms since they break down complex calculations into repetitive steps that a computer can easily handle. Plus, they are highly valuable for scenarios where direct calculation isn't convenient or possible. Starting from an initial guess and then improving from there allows us to arrive at a satisfying approximation without needing exact arithmetic.

Mechanic's Rule

The mechanic's rule, often used for approximating square roots, is a simplified version of Newton's Method. It specifically addresses the root-finding iteration for equations of the form \( x^2 = a \). The formula is beautifully simple, yet effective:

• \[ x_{n+1} = \frac{1}{2}\left(x_n + \frac{a}{x_n}\right) \]

This rule originates from the same principles governing Newton's Method but tailored for squaring. By using an initial guess \( x_1 \), you compute a series of successive approximations \( x_2, x_3, \ldots \), each bringing you closer to \( \sqrt{a} \).

For example, to approximate \( \sqrt{10} \), we might start with \( x_1 = 3 \). Applying the mechanic's rule iteratively gives us refinements like \( x_2 \approx 3.1667 \) and \( x_3 \approx 3.1623 \). Continue this cycle, and you'll rapidly zero in on the accurate square root. The beauty of this method lies in its simplicity—using basic arithmetic to achieve impressive precision.