Problem 38 - Calculus Early Transcendentals

Question

(a) Use a graphing utility to generate the graph of $$ f(x)=\frac{x}{x^{2}+1} $$ and use it to explain what happens if you apply Newton's Method with a starting value of \(x_{1}=2 .\) Check your conclusion by computing \(x_{2}, x_{3}, x_{4},\) and \(x_{5} .\) (b) Use the graph generated in part (a) to explain what happens if you apply Newton's Method with a starting value of \(x_{1}=0.5 .\) Check your conclusion by computing \(x_{2}, x_{3}, x_{4},\) and \(x_{5} .\)

Step-by-Step Solution

Answer

Graph the function to observe roots and behavior. Find derivative for Newton's Method.

Step 1: Understand the Function

The function is given by \( f(x) = \frac{x}{x^2+1} \). One must first understand how to find the roots of this function to apply Newton's Method. Here, we need to identify the behavior of the function and its intersection with the x-axis.

Step 2: Derive the Function

To proceed with Newton's Method, we need the derivative of \( f(x) \). Use the quotient rule to find \( f'(x) = \frac{(x^2+1) - x(2x)}{(x^2+1)^2}=\frac{1-x^2}{(x^2+1)^2} \).

Key Concepts

Graphing Utility

A graphing utility is a helpful tool that enables us to visually understand mathematical functions. By plotting the function graphically, we can see where the function crosses the x-axis, indicating the roots of the function. For the function \( f(x) = \frac{x}{x^2+1} \), using a graphing utility like a calculator or software, we can visualize the function’s behavior.

In this exercise, the graph helps to determine the starting points for Newton's Method. Specifically, for \( x_1 = 2 \) and \( x_1 = 0.5 \), the graph shows how each initial approximation will subsequently behave using iteration. The graph provides a clear depiction of the function's slope at these points, which is crucial for understanding how Newton's Method converges to a root or not.

Derivative

The derivative of a function is essential when using root-finding techniques like Newton's Method. It provides information about the function's slope at any given point. The derivative is crucial for determining the tangent line used in Newton's Method.

For the given function \( f(x) = \frac{x}{x^2+1} \), the derivative \( f'(x) = \frac{1-x^2}{(x^2+1)^2} \) is obtained by applying the quotient rule. This derivative tells us how the function is changing and helps determine the next approximations \( x_2, x_3, x_4, \) and \( x_5 \) when following the Newton's Method steps. Knowing the derivative helps students grasp how iterations move closer to, or further from, the actual root.

Root Finding

Root finding is a technique used to identify where a function crosses the x-axis. This is where the function equals zero. Newton's Method is one such iterative technique used for finding roots when a function graph alone is not specific enough.

In this exercise, Newton’s Method is used starting from \( x_1 = 2 \) and \( x_1 = 0.5 \). The goal is to find when \( f(x) = 0 \). Each step involves calculating a sequence of successive approximations that ideally converge to an actual root of the function. By checking the values \( x_2, x_3, x_4, \) and \( x_5 \), one can discover how effective the chosen starting value is for the convergence of this method.

Quotient Rule

The quotient rule is applied to derive the derivative of a ratio of two functions. It is crucial for implementing Newton’s Method when dealing with functions presented as quotients.

In the function \( f(x) = \frac{x}{x^2+1} \), the quotient rule helps find \( f'(x) \), which is necessary for Newton's iterative calculations. The rule states that if \( g(x) = \frac{u(x)}{v(x)} \), then its derivative is \( g'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} \). Applying this rule allows us to determine \( f'(x) = \frac{1-x^2}{(x^2+1)^2} \), providing the slope needed for calculating tangents and subsequent approximations in the method.