Problem 6 - 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 real root of \(7 x^{3}+x-5=0\)

Step-by-Step Solution

Answer

Use Newton's Method with the initial guess \( x_0 = 1 \) to find the root \( x \approx 0.8619 \) accurate to five decimal places.

Step 1: Understand the Problem

We need to find the real root of the equation \( f(x) = 7x^3 + x - 5 = 0 \) using Newton's Method. The goal is to approximate this root to five decimal places.

Step 2: Newton's Method Formula

Newton's Method approximates roots using the iterative formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] where \( f'(x) \) is the derivative of \( f(x) \).

Step 3: Find the Derivative

Compute the derivative of \( f(x) \): \[ f'(x) = \frac{d}{dx}(7x^3 + x - 5) = 21x^2 + 1 \]

Step 4: Choose an Initial Guess

By evaluating the function or sketching the graph, determine a reasonable starting point. Let's choose \( x_0 = 1 \) based on the graph's behaviour near the root.

Step 5: Apply Newton's Method Iteratively

Using the initial guess \( x_0 = 1 \), apply Newton's Method:- Compute \( x_1 \): \[ x_1 = 1 - \frac{f(1)}{f'(1)} = 1 - \frac{7(1)^3 + 1 - 5}{21(1)^2 + 1} = 1 - \frac{3}{22} \approx 0.8636 \]- Compute \( x_2 \): \[ x_2 = 0.8636 - \frac{f(0.8636)}{f'(0.8636)} \approx 0.8619 \]- Repeat until the result stabilizes to five decimal places. Continue this process to find \( x_3 \), \( x_4 \), etc.

Step 6: Check for Convergence

Continue iterations until \( |x_{n+1} - x_n| \) is less than \( 0.00001 \). This ensures that we have accurate approximation to five decimal places.

Step 7: Verify the Root

Verify that the calculated root satisfies the equation \( f(x) \approx 0 \) to ensure accuracy.

Key Concepts

Iterative Methods

Iterative methods are techniques for solving mathematical problems by successively refining approximations to the solution. Rather than finding an exact answer in one step, these methods involve repeating a specific calculation sequence many times.
Newton's Method is a common iterative method used for finding the roots of a function. It's particularly useful because it can rapidly converge to an accurate solution.

• It starts with an initial guess at the root.
• Each subsequent approximation is refined using the derivative of the function.
• The iterations continue until the result is sufficiently accurate.

Newton's Method demonstrates how iterative methods can turn a complex calculus problem into a manageable sequence of calculations.

Root Approximation

Root approximation techniques are important for finding solutions to equations where the root cannot be determined algebraically. In calculus and numerical analysis, these methods are used to find the values of variables that satisfy an equation like \(7x^3 + x - 5 = 0\).
Newton's Method improves the approximation of a root by:

• Using the function's derivative to estimate where the function crosses the x-axis.
• Correcting the previous guess by subtracting the ratio of the function value and its derivative at that guess.

This method is highly effective because it leverages the tangent line at each approximation to home in on the true root.

Calculus

Calculus forms the backbone of methods like Newton's Method. It involves studying how functions change and using derivatives to understand a function's slope at any given point. This is crucial for root approximation.

• The derivative tells us how steep the function is at a point and in which direction it is moving.
• In Newton's Method, derivatives help in forming the tangent line that guides the root-finding process.

Understanding calculus concepts like differentiation not only helps in solving math problems but also enhances problem-solving skills in various scientific disciplines.

Numerical Analysis

Numerical analysis is the field of mathematics that deals with finding approximate solutions to complex mathematical problems. Unlike traditional algebraic methods, numerical analysis focuses on creating algorithms and methods, like Newton's Method, that yield solutions close to the true values.

Why Use Numerical Methods?

• Many equations do not have closed-form solutions.
• Numerical methods provide practical solutions to real-world problems where exact answers are often unnecessary.
• They are essential in fields such as engineering, physics, and computer science.

Newton's Method is a prime example of numerical analysis because it combines the accuracy of calculus with iterative processes to find solutions efficiently.