Answered: 8) Describe graphically, what Newton's…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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$**Question 8: Newton's Method Graphical Description** Describe graphically what Newton's Method is doing in order to estimate a root of a function, $ f(x) $. Include both some kind of picture and a verbal description. You do not need to derive anything algebraically. **Explanation:** Newton's Method is a technique used to find successively better approximations to the roots (or zeroes) of a real-valued function. Graphically, this method involves the following steps: 1. **Initial Guess**: Begin with an initial guess, $ x_0 $, which is ideally close to the actual root. 2. **Tangent Line**: At the initial guess $ x_0 $, draw the tangent line to the function $ f(x) $. The tangent line is the linear approximation of the function at that point. 3. **Intersection with the x-axis**: The point where this tangent line intersects the x-axis is the next approximation, $ x_1 $. 4. **Iterate**: Repeat this process, using $ x_1 $ to draw another tangent line to the curve. The intersection of this new tangent line with the x-axis becomes $ x_2 $, and so on. 5. **Convergence to Root**: Continue this iterative process. Each step ideally brings the approximation closer to the actual root of the function. By visually understanding these steps, one can see how Newton's Method uses the slope of the tangent line to hone in efficiently on the root of a function.$

Transcribed Image Text:**Question 8: Newton's Method Graphical Description** Describe graphically what Newton's Method is doing in order to estimate a root of a function, $ f(x) $. Include both some kind of picture and a verbal description. You do not need to derive anything algebraically. **Explanation:** Newton's Method is a technique used to find successively better approximations to the roots (or zeroes) of a real-valued function. Graphically, this method involves the following steps: 1. **Initial Guess**: Begin with an initial guess, $ x_0 $, which is ideally close to the actual root. 2. **Tangent Line**: At the initial guess $ x_0 $, draw the tangent line to the function $ f(x) $. The tangent line is the linear approximation of the function at that point. 3. **Intersection with the x-axis**: The point where this tangent line intersects the x-axis is the next approximation, $ x_1 $. 4. **Iterate**: Repeat this process, using $ x_1 $ to draw another tangent line to the curve. The intersection of this new tangent line with the x-axis becomes $ x_2 $, and so on. 5. **Convergence to Root**: Continue this iterative process. Each step ideally brings the approximation closer to the actual root of the function. By visually understanding these steps, one can see how Newton's Method uses the slope of the tangent line to hone in efficiently on the root of a function.

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