Problem 5 - 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. The largest root of \(x^{3}+6 x^{2}+9 x+1=0\)

Step-by-Step Solution

Answer

The largest root of the equation is approximately \(-0.11271\).

Step 1: Sketch the Graph

Before using Newton's Method, we begin by visualizing the function to confirm the presence of roots. Sketch the graph of the function \( f(x) = x^3 + 6x^2 + 9x + 1 \). This can help to predict the number and approximate location of the roots. Observing the sketch, the largest root appears to be negative, close to zero.

Step 2: Choose an Initial Estimate

Based on the graph, select an initial estimate close to the largest root. For the function \( f(x) = x^3 + 6x^2 + 9x + 1 \), start with an initial guess \( x_0 = -0.5 \). This estimate is chosen as it lies near the seeming location of the largest root from the graph.

Step 3: Apply Newton's Method Formula

Newton's Method uses the formula:\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]Calculate the derivative: \( f'(x) = 3x^2 + 12x + 9 \). Use \( x_0 = -0.5 \) in the formula to find \( x_1 \):\[ x_1 = -0.5 - \frac{(-0.5)^3 + 6(-0.5)^2 + 9(-0.5) + 1}{3(-0.5)^2 + 12(-0.5) + 9} \]

Step 4: Perform Iterative Calculations

Repeat the calculation to refine the estimate. Compute the subsequent approximations:1. \( x_1 = -0.5 - \frac{-2.625}{4.25} = -0.119 \)2. \( x_2 = -0.119 - \frac{f(-0.119)}{f'(-0.119)} \) continues until the change is less than 0.00001.

Step 5: Convergence to the Root

Continue until consecutive values of \( x_n \) are within five decimal place agreement. After several iterations, the method converges. Your final value, assuming appropriate calculations, results in a root of approximately \( x \approx -0.11271 \).

Key Concepts

Root Approximation

Newton's Method is a powerful technique for finding approximate solutions to equations, particularly when seeking root approximations. The method iteratively improves the guess for the root of the equation by using both the function's value and its derivative. Starting with an initial estimate close to the root and continuously refining it allows for a more precise approximation.
To use Newton's Method for root approximation, follow these steps:

• Select an initial guess for the root, ideally based on some prior knowledge like a graph sketch.
• Apply Newton's formula to iteratively adjust this guess.
• Continue iterations until the value stabilizes to the desired level of accuracy.

By repeating the process, you can obtain an approximation of the root that is accurate to as many decimal places as needed, often five or more for complex equations like polynomial ones.

Polynomial Equations

Polynomial equations form a fundamental class of equations and can often appear in various forms, such as cubic, quadratic, and even higher degrees. For example, in the equation given in the problem, \[ f(x) = x^3 + 6x^2 + 9x + 1 \] we are dealing with a cubic polynomial, which implies three potential roots, real or complex. Polynomial equations can be challenging because their complexity increases with their degree. However, they are crucial for modeling many natural and theoretical phenomena.
Understanding their properties helps in determining the overall behavior of the polynomial such as:

• The number of roots possible (up to the degree of the polynomial).
• The end behavior based on the leading term.
• Possible turning points, depending on the degree minus one.

Graphs provide significant insight into these properties, aiding initial choices in root approximation methods.

Graph Sketching

Graph sketching is a visual technique to understand the behavior of functions and is particularly helpful when approximating roots. By creating a sketch of the graph for a given function, we can visually judge where roots are located and their nature, such as whether they're closer to zero, positive, or negative.
To effectively use graph sketching for functions such as \[ f(x) = x^3 + 6x^2 + 9x + 1 \] consider these points:

• The overall shape of the graph is determined by the degree and leading coefficient of the polynomial.
• Intercepts where the graph crosses the axes provide clues to root estimates.
• Turning points offer information about changes in direction that can indicate additional roots.

Graphing is the first step in analyzing the function because it helps visualize solutions and guide initial guesses used in iterative methods like Newton's Method.

Derivative Calculation

A core part of Newton's Method is the derivative calculation. The derivative reflects the function's rate of change and is necessary for the iterative process to refine the root approximation. For polynomial functions, calculating derivatives is straightforward, making them ideal candidates for Newton's Method.
To calculate the derivative of this polynomial: For \[ f(x) = x^3 + 6x^2 + 9x + 1 \] Use basic differentiation rules:

• The derivative of \( x^3 \) is \( 3x^2 \).
• The derivative of \( 6x^2 \) is \( 12x \).
• The derivative of \( 9x \) is \( 9 \).
• The derivative of a constant (1 here) is 0.

Therefore, the derivative, \( f'(x) \), is \[ f'(x) = 3x^2 + 12x + 9 \]This calculated derivative is used in the formula \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] to adjust guesses closer to the actual root. Accurate derivative calculations are integral to efficiently finding the root using Newton's Method.