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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.8.25
Chapter 4, Problem 4.8.25

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = ln x - x² + 3x - 1

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Step 1: Begin by analyzing the function f(x) = ln(x) - x² + 3x - 1. Identify the domain of the function, which is x > 0, since the natural logarithm ln(x) is only defined for positive x.
Step 2: Perform a preliminary analysis by graphing the function f(x) to visually identify where the roots might be located. Look for points where the graph crosses the x-axis, as these are potential roots.
Step 3: Choose initial approximations for the roots based on the graph. These approximations will be used as starting points for Newton's method. Ensure that these points are close to where the graph crosses the x-axis.
Step 4: Apply Newton's method, which is an iterative process. The formula for Newton's method is x_{n+1} = x_n - f(x_n) / f'(x_n), where f'(x) is the derivative of f(x). Calculate f'(x) = 1/x - 2x + 3.
Step 5: Use the initial approximations and apply the Newton's method formula iteratively until the values converge to a stable point, indicating a root. Repeat this process for each initial approximation to find all roots of the function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Method

Newton's Method is an iterative numerical technique used to approximate the roots of a real-valued function. Starting with an initial guess, the method uses the function's derivative to refine the estimate, converging towards a root. The formula is x_{n+1} = x_n - f(x_n)/f'(x_n), where f'(x_n) is the derivative of the function at x_n.
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Graphing Functions

Graphing functions involves plotting the function on a coordinate plane to visually analyze its behavior. This helps identify key features such as intercepts, turning points, and asymptotes, which are crucial for choosing good initial approximations for Newton's Method. For f(x) = ln x - x² + 3x - 1, graphing can reveal where the function crosses the x-axis, indicating potential roots.
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Preliminary Analysis

Preliminary analysis involves examining the function's properties before applying numerical methods. This includes finding the domain, identifying critical points, and understanding the function's behavior at infinity. For f(x) = ln x - x² + 3x - 1, analyzing the derivative can help locate intervals where roots are likely, aiding in selecting effective starting points for Newton's Method.
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