Answered: Solve f(x)=x³+4x² - 10 using the…

Introduction to Chemical Engineering Thermodynamics

8th Edition

ISBN:9781259696527

Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Chapter1: Introduction

Section: Chapter Questions

Problem 1.1P

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$**Problem Statement:** Solve $ f(x) = x^3 + 4x^2 - 10 $ using the Newton-Raphson method for a root in the interval $[1, 2]$. **Explanation:** To find the root of the given function within the specified interval using the Newton-Raphson method, follow these steps: 1. **Select an Initial Guess $ x_0 $:** Start with an initial guess within the interval, typically closer to where you anticipate the root might be. 2. **Calculate the Function's Derivative:** Compute the derivative $ f'(x) = 3x^2 + 8x $. 3. **Iterative Formula:** Use the formula \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] to get successive approximations of the root. 4. **Convergence Check:** Repeat the iteration until the difference between successive approximations is within an acceptable tolerance or the function value $ f(x_n) $ is sufficiently close to zero. 5. **Result:** The value of $ x $ at which the iterations converge is the root. This method is efficient and typically converges quickly if the initial guess is reasonable and the function behaves well.$

Transcribed Image Text:**Problem Statement:** Solve $ f(x) = x^3 + 4x^2 - 10 $ using the Newton-Raphson method for a root in the interval $[1, 2]$. **Explanation:** To find the root of the given function within the specified interval using the Newton-Raphson method, follow these steps: 1. **Select an Initial Guess $ x_0 $:** Start with an initial guess within the interval, typically closer to where you anticipate the root might be. 2. **Calculate the Function's Derivative:** Compute the derivative $ f'(x) = 3x^2 + 8x $. 3. **Iterative Formula:** Use the formula \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] to get successive approximations of the root. 4. **Convergence Check:** Repeat the iteration until the difference between successive approximations is within an acceptable tolerance or the function value $ f(x_n) $ is sufficiently close to zero. 5. **Result:** The value of $ x $ at which the iterations converge is the root. This method is efficient and typically converges quickly if the initial guess is reasonable and the function behaves well.

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