Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.
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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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