4. Find a formula for [ f(x) dx in terms of the functional values f(x₁) and f(x2), which is exact for linear polynomials. For what values of x₁ or x2 is the discretization formula super- accurate, that is the truncation error is of higher order than expected? What kind of discretization does this correspond to (Forward, Backward, Center etc.)?
4. Find a formula for [ f(x) dx in terms of the functional values f(x₁) and f(x2), which is exact for linear polynomials. For what values of x₁ or x2 is the discretization formula super- accurate, that is the truncation error is of higher order than expected? What kind of discretization does this correspond to (Forward, Backward, Center etc.)?
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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![### 4. Find a Formula for the Integral
Consider the integral
\[ \int_{0}^{1} f(x) \, dx \]
in terms of the functional values \( f(x_1) \) and \( f(x_2) \), which is exact for linear polynomials. For what values of \( x_1 \) or \( x_2 \) is the discretization formula super-accurate, meaning the truncation error is of higher order than expected? What kind of discretization does this correspond to (Forward, Backward, Center, etc.)?
### Explanation
The goal is to express the integral of \( f(x) \) from 0 to 1 using function values at specific points \( x_1 \) and \( x_2 \). This involves:
1. **Identifying the Points**: Determine the values of \( x_1 \) and \( x_2 \) where the integration formula holds exactly for linear polynomials.
2. **Super-Accuracy**: Analyze conditions where the formula provides super-accuracy – a truncation error smaller than typical.
3. **Discretization Type**: Categorize the type of discretization (Forward, Backward, Center) based on the chosen points and their impact on accuracy.
A detailed investigation into these factors will reveal the optimal \( x_1 \) and \( x_2 \) to use for an accurate and possibly super-accurate discretization method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F979d0aba-5428-414f-a3ba-5510f0301082%2Fd7d63c43-b1a8-40c1-b39d-ab7672a5d355%2Feskrb98_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### 4. Find a Formula for the Integral
Consider the integral
\[ \int_{0}^{1} f(x) \, dx \]
in terms of the functional values \( f(x_1) \) and \( f(x_2) \), which is exact for linear polynomials. For what values of \( x_1 \) or \( x_2 \) is the discretization formula super-accurate, meaning the truncation error is of higher order than expected? What kind of discretization does this correspond to (Forward, Backward, Center, etc.)?
### Explanation
The goal is to express the integral of \( f(x) \) from 0 to 1 using function values at specific points \( x_1 \) and \( x_2 \). This involves:
1. **Identifying the Points**: Determine the values of \( x_1 \) and \( x_2 \) where the integration formula holds exactly for linear polynomials.
2. **Super-Accuracy**: Analyze conditions where the formula provides super-accuracy – a truncation error smaller than typical.
3. **Discretization Type**: Categorize the type of discretization (Forward, Backward, Center) based on the chosen points and their impact on accuracy.
A detailed investigation into these factors will reveal the optimal \( x_1 \) and \( x_2 \) to use for an accurate and possibly super-accurate discretization method.
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