1) Use Bisection to find a root of the function f(x) = ln x-2 on the interval [7,8] to within 1 decimal place. Organize the results of all the calculations in the table below. You don't need to write down more than 5 decimal places. f(a) C₁ f(c.,) b₁ f(b₁) a₁ Half Interval Length
1) Use Bisection to find a root of the function f(x) = ln x-2 on the interval [7,8] to within 1 decimal place. Organize the results of all the calculations in the table below. You don't need to write down more than 5 decimal places. f(a) C₁ f(c.,) b₁ f(b₁) a₁ Half Interval Length
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter2: Functions
Section: Chapter Questions
Problem 28P
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![**Bisection Method for Finding a Root**
**Problem Statement:**
1. Use the Bisection Method to find a root of the function \( f(x) = \ln x - 2 \) on the interval \([7, 8]\) to within 1 decimal place.
**Instructions:**
- Organize the results of all the calculations in the table below.
- You don't need to write down more than 5 decimal places.
| \( a_i \) | \( f(a_i) \) | \( c_i \) | \( f(c_i) \) | \( b_i \) | \( f(b_i) \) | Half Interval Length |
|-----------|-------------|-----------|-------------|-----------|-------------|-----------------------|
| | | | | | | |
**Explanation of the Table Columns:**
- \( a_i \): The lower bound of the interval at the \( i \)-th step.
- \( f(a_i) \): The value of the function \( f(x) \) at \( a_i \).
- \( c_i \): The midpoint of the interval at the \( i \)-th step.
- \( f(c_i) \): The value of the function \( f(x) \) at \( c_i \).
- \( b_i \): The upper bound of the interval at the \( i \)-th step.
- \( f(b_i) \): The value of the function \( f(x) \) at \( b_i \).
- Half Interval Length: Half the length of the current interval, used to determine the stopping criterion.
**Method Overview:**
The Bisection Method is an iterative numerical technique for finding roots of a continuous function. It repeatedly bisects the interval and selects a subinterval in which the function changes sign, thereby narrowing down the interval that contains the root. This process is continued until the desired precision is achieved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10d9297c-1f71-4c91-b4fa-3fe20bb968b3%2Fb9cc086f-8396-4a0e-a8b7-18597a2b85de%2Fc4pqbr_processed.png&w=3840&q=75)
Transcribed Image Text:**Bisection Method for Finding a Root**
**Problem Statement:**
1. Use the Bisection Method to find a root of the function \( f(x) = \ln x - 2 \) on the interval \([7, 8]\) to within 1 decimal place.
**Instructions:**
- Organize the results of all the calculations in the table below.
- You don't need to write down more than 5 decimal places.
| \( a_i \) | \( f(a_i) \) | \( c_i \) | \( f(c_i) \) | \( b_i \) | \( f(b_i) \) | Half Interval Length |
|-----------|-------------|-----------|-------------|-----------|-------------|-----------------------|
| | | | | | | |
**Explanation of the Table Columns:**
- \( a_i \): The lower bound of the interval at the \( i \)-th step.
- \( f(a_i) \): The value of the function \( f(x) \) at \( a_i \).
- \( c_i \): The midpoint of the interval at the \( i \)-th step.
- \( f(c_i) \): The value of the function \( f(x) \) at \( c_i \).
- \( b_i \): The upper bound of the interval at the \( i \)-th step.
- \( f(b_i) \): The value of the function \( f(x) \) at \( b_i \).
- Half Interval Length: Half the length of the current interval, used to determine the stopping criterion.
**Method Overview:**
The Bisection Method is an iterative numerical technique for finding roots of a continuous function. It repeatedly bisects the interval and selects a subinterval in which the function changes sign, thereby narrowing down the interval that contains the root. This process is continued until the desired precision is achieved.
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