Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem 6: Applying Newton's Method**
**Objective:**
Apply Newton's method twice to approximate the cube root of 3, denoted as \(3^{1/3}\).
**Instructions:**
- **Initial Approximation:** Begin with \(x_0 = 3\).
- Find the subsequent approximations \(x_1\) and \(x_2\) using Newton's method.
**Explanation:**
Newton's method is an iterative technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. The general formula for Newton’s iteration is:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
For the cube root approximation, we define the function:
\[ f(x) = x^3 - 3 \]
The derivative of this function is:
\[ f'(x) = 3x^2 \]
With these, you can apply the Newton’s iteration to find \(x_1\) and \(x_2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa149c215-2234-465a-9de5-fb0497c84054%2F23adabfc-148a-442f-9ccc-051d64a5f3d4%2F3h980w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6: Applying Newton's Method**
**Objective:**
Apply Newton's method twice to approximate the cube root of 3, denoted as \(3^{1/3}\).
**Instructions:**
- **Initial Approximation:** Begin with \(x_0 = 3\).
- Find the subsequent approximations \(x_1\) and \(x_2\) using Newton's method.
**Explanation:**
Newton's method is an iterative technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. The general formula for Newton’s iteration is:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
For the cube root approximation, we define the function:
\[ f(x) = x^3 - 3 \]
The derivative of this function is:
\[ f'(x) = 3x^2 \]
With these, you can apply the Newton’s iteration to find \(x_1\) and \(x_2\).
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