Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 5-14, use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. All real roots of \(x^{4}-8 x^{3}+22 x^{2}-24 x+8=0\)

Step-by-Step Solution

Answer

The roots near 1 and 3 are approximately 1.41421 and 2.58579, respectively.

Step 1: Sketch the Graph

Sketch the graph of the function \( f(x) = x^4 - 8x^3 + 22x^2 - 24x + 8 \) using a graphing tool or calculator. Identify approximate locations of the real roots by observing where the graph crosses the x-axis.

Step 2: Define Newton's Method Formula

Newton's Method is used to approximate roots by iterating the formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). We need the first derivative of \( f(x) \): \( f'(x) = 4x^3 - 24x^2 + 44x - 24 \).

Step 3: Select Initial Guess

Based on the graph, select initial guesses close to where the graph crosses the x-axis. For example, the guesses could be \( x_0 = 1 \) and \( x_0 = 3 \).

Step 4: Iteration to Approximate Root at x=1

Using \( x_0 = 1 \) as the initial guess, compute the next values using Newton's Method:1. \( x_1 = 1 - \frac{f(1)}{f'(1)} = 1 - \frac{-1}{-12} = 0.9167 \ \)2. Continue iterating until the difference between consecutive \( x_n \) is less than \( 0.00001 \). This root converges very quickly.

Step 5: Iteration to Approximate Root at x=3

With \( x_0 = 3 \) as another initial guess, proceed with iterations:1. \( x_1 = 3 - \frac{f(3)}{f'(3)} = 3 - \frac{2}{-8} = 3.25 \ \)2. Continue calculations until the difference in \( x_n \) values is less than \( 0.00001 \).

Step 6: Check for Additional Roots

Review the graph to see any other intersections with the x-axis. Compute similar iterations for additional suspected locations, adjusting the initial guess if needed, until all roots are found.

Key Concepts

Polynomial Root Approximation

Polynomials can be complex when it comes to finding their roots, especially when they have high degrees, like the fourth degree polynomial in our exercise. Finding the exact roots analytically can be challenging, and that's where polynomial root approximation methods, such as Newton's Method, become invaluable.
Newton's Method allows us to focus on narrowing down the potential real roots of a polynomial through iterative calculations. We start with an initial guess close to where we suspect a root is based on a graph. From there, the method refines this guess repeatedly, honing in on the precise root. Each iteration brings the estimate closer to the actual root by reducing the margin of error until it fits our accuracy requirement, such as five decimal places.
This approach is highly useful when graphs suggest root locations, as it translates visual approximations into precise numerical solutions easily handled by a calculator or software.

Numerical Methods

Numerical methods are crucial in situations where analytical solutions to equations are difficult or impossible to derive. They offer ways to find approximate solutions within acceptable margins of error. For our polynomial, applying Newton's Method is a prominent example of such a numerical method.
Newton's Method involves iterations using the formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). This formula uses both the function value \( f(x) \) and its first derivative \( f'(x) \) at the current point \( x_n \).

• The derivative helps in predicting how changes in \( x \) affect \( f(x) \).
  
• The formula corrects the current guess \( x_n \) based on how far and in which direction \( f(x_n) \) is from zero.
  

By performing iterative updates, numerical methods bridge the gap between theoretical math and practical computations, making them indispensable in engineering, sciences, and finance.

First Derivative

The first derivative is a critical component of Newton's Method. It represents the slope of the tangent line to the function graph at any point. In other words, it shows how steeply the function is changing at that particular point.
For the polynomial \( f(x) = x^4 - 8x^3 + 22x^2 - 24x + 8 \), the first derivative is \( f'(x) = 4x^3 - 24x^2 + 44x - 24 \). This function gives us the necessary gradient information to guide our guesses towards the actual roots.
During polynomial root approximation using Newton's Method, the derivative allows us to adjust our guesses effectively. If the derivative is large, it means the function value is changing rapidly, necessitating smaller moves towards the root. Conversely, a smaller derivative indicates that the function value is not changing quickly, allowing larger moves.

Graphical Analysis

Graphical analysis is the first step in applying Newton's Method effectively. By sketching the polynomial function, we can visually discern where the graph crosses the x-axis. These intersections are possible indicators of real roots.
Our initial guesses for the roots are derived from viewing these intersections. However, these graphical estimates can be rough and need refinement, which Newton's Method provides.

• Graphing helps narrow down where to begin our numerical method.
  
• It also helps identify potential additional roots that may not be obvious without a visual reference.
  

Utilizing graphical analysis alongside numerical methods is advantageous as it combines the strengths of visual insights with precise, iterative calculations. This approach ensures a more comprehensive understanding and accurate solution to finding polynomial roots.