Using a calculator, pencil and paper, apply the bisection method to find a root of y=f(x)=1-2x cosx by filling in a table of the form shown below. For the “sign" columns enter + or – sign f (a) sign f (x) b sign f (b) а 0.5 0.7 In each row x=(a+b)/2 . Once the table is complete, report your best estimate of the root in the form r±d.
Using a calculator, pencil and paper, apply the bisection method to find a root of y=f(x)=1-2x cosx by filling in a table of the form shown below. For the “sign" columns enter + or – sign f (a) sign f (x) b sign f (b) а 0.5 0.7 In each row x=(a+b)/2 . Once the table is complete, report your best estimate of the root in the form r±d.
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...
Related questions
Question
![---
### Bisection Method to Find a Root of the Function \( y = f(x) = 1 - 2x \cos x \)
To apply the bisection method to find a root of the function \( y = f(x) = 1 - 2x \cos x \), follow these steps:
1. **Initial Interval Selection:**
- Choose the initial interval \([a, b]\) where \(f(a)\) and \(f(b)\) have opposite signs.
2. **Table Filling Procedure:**
- Complete the table below by calculating the midpoint \(x\) of the interval \([a, b]\) and the function value \(f(x)\).
- Based on \(f(x)\), decide the next interval for the bisection method.
#### Table Format
| \(a\) | sign \(f(a)\) | \(x\) | sign \(f(x)\) | \(b\) | sign \(f(b)\) |
|-------|----------------|-------|----------------|-------|----------------|
| 0.5 | | | | 0.7 | |
| | | | | | |
| | | | | | |
| | | | | | |
3. **Procedure for Each Row:**
- Calculate \( x = \frac{a+b}{2} \).
- Evaluate \(f(x)\).
- Enter the signs \(+\) or \(-\) for \(f(a)\), \(f(x)\), and \(f(b)\).
- Update the interval \([a, b]\) based on the sign of \(f(x)\).
4. **Estimation of Root:**
- Continue this procedure until the interval is sufficiently small.
- Report the best estimate of the root in the form \( r \pm \delta \).
**Example Interval:**
- Initial interval: \(a = 0.5\), \(b = 0.7\).
By evaluating the function at these points and their midpoint, complete the table, and follow through the bisection steps to find the root.
**Note:** To ensure accuracy in your application of the bisection method, use a calculator to determine the value of \(f(x)\) accurately at each step.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5312a15-860e-407c-ae9e-8729c86c8316%2F743866ac-85f1-4478-b4bc-e884fde06285%2Fj8g4c8p_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Bisection Method to Find a Root of the Function \( y = f(x) = 1 - 2x \cos x \)
To apply the bisection method to find a root of the function \( y = f(x) = 1 - 2x \cos x \), follow these steps:
1. **Initial Interval Selection:**
- Choose the initial interval \([a, b]\) where \(f(a)\) and \(f(b)\) have opposite signs.
2. **Table Filling Procedure:**
- Complete the table below by calculating the midpoint \(x\) of the interval \([a, b]\) and the function value \(f(x)\).
- Based on \(f(x)\), decide the next interval for the bisection method.
#### Table Format
| \(a\) | sign \(f(a)\) | \(x\) | sign \(f(x)\) | \(b\) | sign \(f(b)\) |
|-------|----------------|-------|----------------|-------|----------------|
| 0.5 | | | | 0.7 | |
| | | | | | |
| | | | | | |
| | | | | | |
3. **Procedure for Each Row:**
- Calculate \( x = \frac{a+b}{2} \).
- Evaluate \(f(x)\).
- Enter the signs \(+\) or \(-\) for \(f(a)\), \(f(x)\), and \(f(b)\).
- Update the interval \([a, b]\) based on the sign of \(f(x)\).
4. **Estimation of Root:**
- Continue this procedure until the interval is sufficiently small.
- Report the best estimate of the root in the form \( r \pm \delta \).
**Example Interval:**
- Initial interval: \(a = 0.5\), \(b = 0.7\).
By evaluating the function at these points and their midpoint, complete the table, and follow through the bisection steps to find the root.
**Note:** To ensure accuracy in your application of the bisection method, use a calculator to determine the value of \(f(x)\) accurately at each step.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning