Apply Newton's Method using the given initial guess. (If an answer does not exist, enter DNE.) y = 2x3 - 6x2 + 6x – 1, x1 = 1 2-
Apply Newton's Method using the given initial guess. (If an answer does not exist, enter DNE.) y = 2x3 - 6x2 + 6x – 1, x1 = 1 2-
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
![**Newton's Method Application and Analysis**
**Instructions:**
Apply Newton's Method using the given initial guess. (If an answer does not exist, enter DNE.)
**Function and Initial Value:**
\[ y = 2x^3 - 6x^2 + 6x - 1, \quad x_1 = 1 \]
**Graph Explanation:**
The graph provided is a plot of the cubic function \( y = 2x^3 - 6x^2 + 6x - 1 \). It displays a curve intersecting the y-axis above the origin and the behavior around \( x_1 = 1 \). The tangent line at this point illustrates the initial step of Newton’s method.
**Tasks:**
Calculate the subsequent approximations using Newton's Method and fill in the blanks:
\[
x_1 = \quad
\]
\[
x_2 = \quad
\]
\[
x_3 = \quad
\]
\[
x_4 = \quad
\]
**Explanation of Why Newton’s Method May Fail:**
In the explanation section, discuss potential reasons for failure in convergence, such as:
- Choosing an initial guess near a local extremum or inflection point.
- The derivative near the initial guess being zero or very small, leading to large and unstable steps.
- Divergence due to the function's behavior, such as oscillations or steep slopes around the guess.
Students should use these guidelines to understand the potential pitfalls of Newton's Method and explore why specific initial values lead to failure in finding the root.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7505dca-29fe-49a5-bb2b-d73df2bb5729%2F8d267945-486c-464d-b6b6-62b8ae681ca8%2F454ndy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Newton's Method Application and Analysis**
**Instructions:**
Apply Newton's Method using the given initial guess. (If an answer does not exist, enter DNE.)
**Function and Initial Value:**
\[ y = 2x^3 - 6x^2 + 6x - 1, \quad x_1 = 1 \]
**Graph Explanation:**
The graph provided is a plot of the cubic function \( y = 2x^3 - 6x^2 + 6x - 1 \). It displays a curve intersecting the y-axis above the origin and the behavior around \( x_1 = 1 \). The tangent line at this point illustrates the initial step of Newton’s method.
**Tasks:**
Calculate the subsequent approximations using Newton's Method and fill in the blanks:
\[
x_1 = \quad
\]
\[
x_2 = \quad
\]
\[
x_3 = \quad
\]
\[
x_4 = \quad
\]
**Explanation of Why Newton’s Method May Fail:**
In the explanation section, discuss potential reasons for failure in convergence, such as:
- Choosing an initial guess near a local extremum or inflection point.
- The derivative near the initial guess being zero or very small, leading to large and unstable steps.
- Divergence due to the function's behavior, such as oscillations or steep slopes around the guess.
Students should use these guidelines to understand the potential pitfalls of Newton's Method and explore why specific initial values lead to failure in finding the root.
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