### Bisection Method to Estimate Roots of a Function #### Problem 1: Root Estimation Using Bisection Consider the Bisection Method to estimate \(\alpha = \frac{1}{2}\), the root of the function: \[ f(x) = x^2 - \frac{1}{4} \] 1. **Determine Pair (a, b):** Identify which pair of \((a, b)\) provides the fastest convergence to \(\alpha\). 2. **Starting Pairs:** State the \(x\)-value with the fastest convergence for each given pair: i. \(a = 0\), \(b = 3\) ii. \(a = -1\), \(b = 1\) iii. \(a = 1\), \(b = 2\) iv. \(a = 0\), \(b = 2\) 3. **Explanation:** Explain your reasoning for which interval \([a, b]\) you chose and why it converges faster. In Bisection Method, always ensure that the root lies within the interval \([a, b]\), and the function changes signs at the endpoints \(a\) and \(b\). ### Diagrams/Graphs This exercise does not contain specific graphs or diagrams, but typically, you would illustrate: - **The Function \( f(x) = x^2 - 1/4 \)**: A parabola opening upwards, intersecting the x-axis at \(x = \pm 1/2\). - **Interval \([a, b]\)**: Highlight intervals where \(f(a) \cdot f(b) < 0\) to ensure the root is within this range. - **Iterative Steps**: Demonstrate the midpoint calculations and how the intervals are halved in each step. By examining these pairs and using the bisection method iteratively, you can determine which interval aids in the fastest convergence to the root \(\alpha\).
### Bisection Method to Estimate Roots of a Function #### Problem 1: Root Estimation Using Bisection Consider the Bisection Method to estimate \(\alpha = \frac{1}{2}\), the root of the function: \[ f(x) = x^2 - \frac{1}{4} \] 1. **Determine Pair (a, b):** Identify which pair of \((a, b)\) provides the fastest convergence to \(\alpha\). 2. **Starting Pairs:** State the \(x\)-value with the fastest convergence for each given pair: i. \(a = 0\), \(b = 3\) ii. \(a = -1\), \(b = 1\) iii. \(a = 1\), \(b = 2\) iv. \(a = 0\), \(b = 2\) 3. **Explanation:** Explain your reasoning for which interval \([a, b]\) you chose and why it converges faster. In Bisection Method, always ensure that the root lies within the interval \([a, b]\), and the function changes signs at the endpoints \(a\) and \(b\). ### Diagrams/Graphs This exercise does not contain specific graphs or diagrams, but typically, you would illustrate: - **The Function \( f(x) = x^2 - 1/4 \)**: A parabola opening upwards, intersecting the x-axis at \(x = \pm 1/2\). - **Interval \([a, b]\)**: Highlight intervals where \(f(a) \cdot f(b) < 0\) to ensure the root is within this range. - **Iterative Steps**: Demonstrate the midpoint calculations and how the intervals are halved in each step. By examining these pairs and using the bisection method iteratively, you can determine which interval aids in the fastest convergence to the root \(\alpha\).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Bisection Method to Estimate Roots of a Function
#### Problem 1: Root Estimation Using Bisection
Consider the Bisection Method to estimate \(\alpha = \frac{1}{2}\), the root of the function:
\[ f(x) = x^2 - \frac{1}{4} \]
1. **Determine Pair (a, b):**
Identify which pair of \((a, b)\) provides the fastest convergence to \(\alpha\).
2. **Starting Pairs:**
State the \(x\)-value with the fastest convergence for each given pair:
i. \(a = 0\), \(b = 3\)
ii. \(a = -1\), \(b = 1\)
iii. \(a = 1\), \(b = 2\)
iv. \(a = 0\), \(b = 2\)
3. **Explanation:**
Explain your reasoning for which interval \([a, b]\) you chose and why it converges faster.
In Bisection Method, always ensure that the root lies within the interval \([a, b]\), and the function changes signs at the endpoints \(a\) and \(b\).
### Diagrams/Graphs
This exercise does not contain specific graphs or diagrams, but typically, you would illustrate:
- **The Function \( f(x) = x^2 - 1/4 \)**: A parabola opening upwards, intersecting the x-axis at \(x = \pm 1/2\).
- **Interval \([a, b]\)**: Highlight intervals where \(f(a) \cdot f(b) < 0\) to ensure the root is within this range.
- **Iterative Steps**: Demonstrate the midpoint calculations and how the intervals are halved in each step.
By examining these pairs and using the bisection method iteratively, you can determine which interval aids in the fastest convergence to the root \(\alpha\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc58b2eee-5b79-4c9d-923b-d4e913d44dbe%2F8483a387-8b6c-4a37-af26-006b2cff75aa%2Fo4q476e_processed.png&w=3840&q=75)
Transcribed Image Text:### Bisection Method to Estimate Roots of a Function
#### Problem 1: Root Estimation Using Bisection
Consider the Bisection Method to estimate \(\alpha = \frac{1}{2}\), the root of the function:
\[ f(x) = x^2 - \frac{1}{4} \]
1. **Determine Pair (a, b):**
Identify which pair of \((a, b)\) provides the fastest convergence to \(\alpha\).
2. **Starting Pairs:**
State the \(x\)-value with the fastest convergence for each given pair:
i. \(a = 0\), \(b = 3\)
ii. \(a = -1\), \(b = 1\)
iii. \(a = 1\), \(b = 2\)
iv. \(a = 0\), \(b = 2\)
3. **Explanation:**
Explain your reasoning for which interval \([a, b]\) you chose and why it converges faster.
In Bisection Method, always ensure that the root lies within the interval \([a, b]\), and the function changes signs at the endpoints \(a\) and \(b\).
### Diagrams/Graphs
This exercise does not contain specific graphs or diagrams, but typically, you would illustrate:
- **The Function \( f(x) = x^2 - 1/4 \)**: A parabola opening upwards, intersecting the x-axis at \(x = \pm 1/2\).
- **Interval \([a, b]\)**: Highlight intervals where \(f(a) \cdot f(b) < 0\) to ensure the root is within this range.
- **Iterative Steps**: Demonstrate the midpoint calculations and how the intervals are halved in each step.
By examining these pairs and using the bisection method iteratively, you can determine which interval aids in the fastest convergence to the root \(\alpha\).
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