Question

Use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. The root of \(x \ln x=2\)

Short answer

The approximate root is 3.09163.

Step-by-step solution

Sketch the Graph

To understand where the root might be, let's start by sketching the graph of the function. The function we are dealing with is \( f(x) = x \ln x - 2 \). Use graphing software or a graphing calculator to sketch the curve. Observe where it crosses the x-axis, which indicates potential roots.

Define the Function and its Derivative

The equation is given as \( x \ln x = 2 \). Define the function \( f(x) = x \ln x - 2 \). To apply Newton's Method, we also need the derivative \( f'(x) \). Calculate \( f'(x) = \ln x + 1 \).

Choose an Initial Guess

From the graph, estimate an initial guess close to where the graph crosses the x-axis. From the plot, it looks like a reasonable starting point is \( x_0 = 3 \), as it is nearby where the curve crosses the x-axis.

Apply Newton's Method

Newton's Method formula is \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). Start with \( x_0 = 3 \). Compute * \( x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \)* Continue iterating until the change \( |x_{n+1} - x_n| \) is less than \( 0.00001 \).

Perform Iterations

Calculate each iteration:1. \( x_0 = 3 \)2. \( x_1 = 3 - \frac{3 \ln 3 - 2}{\ln 3 + 1} \approx 3.09731 \)3. Continue the process: * \( x_2 = 3.09163 \) after the next iteration * Repeat until convergenceStop once \( |x_n - x_{n-1}| < 0.00001 \).

Verify the Result

After performing several iterations, you will find that the values stabilize around \( x \approx 3.09163 \). Substitute this back into the original equation \( x \ln x = 2 \) to check the accuracy: Find that \( 3.09163 \ln 3.09163 \approx 2 \), which verifies that the approximation is accurate to five decimal places.

Key concepts

Approximating Roots

Newton's Method is a powerful technique for approximating the roots of a function. A "root" refers to a value of \( x \) where the function equals zero. Imagine you have a plot of a curve and you want to find where it touches or crosses the x-axis. This point of crossing is the root.
Newton's Method helps in finding this root by making an educated guess and then refining that guess until we get a very close approximation. It's particularly useful when analytical solutions are hard to find or when dealing with complex equations.
To start approximating a root using Newton's Method, you first need to understand the behavior of the graph visually or using software. Once you place a guess close to the actual root based on this graph, Newton's Method uses calculus to refine this guess continually. This refinement process is what makes it highly effective for finding accurate solutions.

Calculus Techniques

To apply Newton’s Method for approximating roots, one of the primary calculus techniques used is finding derivatives. Derivatives are essential because Newton's Method utilizes them to adjust approximations of the root.
The derivative of a function gives us the slope of the function at any given point. This slope is crucial for determining how to adjust our initial guess towards the root. For example, in the exercise above, the function \( f(x) = x \ln x - 2 \) needed its derivative calculated to apply Newton's formula.
Calculus plays a critical role in analyzing the behavior of functions. When you want to find approximate roots, understanding changes in the function through derivatives and slopes provide the guidance necessary for precision. The application of these techniques ensures your approximation quickly converges to the true root.

Function Derivatives

In Newton’s Method, the computation of function derivatives is vital to its effectiveness. The derivative represents how a function changes at any given point, providing the slope of the tangent line at that point.
For the function \( f(x) = x \ln x - 2 \), its derivative is \( f'(x) = \ln x + 1 \). This function derivative is used within the iterative process of Newton's Method.
Derivatives allow us to better understand the behavior of functions, enabling the precise adjustment of initial guesses towards the root. Essentially, they help predict where the root of a function lies and guide the iterations in narrowing down the exact value.
By calculating the derivative as part of Newton’s Method, we ensure smoother convergence towards the root, leveraging mathematical accuracy and efficiency.

Iterative Methods

Newton's Method falls under the category of iterative methods, which involve making a sequence of calculations to gradually arrive at a solution.
An iterative method starts with an initial guess and repeatedly applies a formula to refine that guess. In Newton's Method, the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \) is used. This formula leverages both the function and its derivative to adjust the guess closer to the root with each step.
The beauty of iterative methods like Newton’s is their efficiency. With only a few iterations, you can often reach a very accurate approximation of the root. In our example, starting with an initial guess of \( x_0 = 3 \), the iterations quickly zeroed in on \( x \approx 3.09163 \).
Iterative methods provide a systematic approach to solving equations that might otherwise seem complex or unsolvable, enhancing problem-solving capabilities in mathematics.