Q2: Consider the following table: x 0.800 1.000 1.200 1.400 f(x) 1.040 1.151 1.213 1.228 a. Estimate f(1.23) as accurately as possible, b. Estimate f' (1.20) using interpolating polynomial, c. Estimate f'(1.20) using the central difference expression,
Q2: Consider the following table: x 0.800 1.000 1.200 1.400 f(x) 1.040 1.151 1.213 1.228 a. Estimate f(1.23) as accurately as possible, b. Estimate f' (1.20) using interpolating polynomial, c. Estimate f'(1.20) using the central difference expression,
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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![**Q2: Table Analysis and Derivative Estimations**
Consider the following table:
| **x** | 0.800 | 1.000 | 1.200 | 1.400 |
|-------|-------|-------|-------|-------|
| **f(x)** | 1.040 | 1.151 | 1.213 | 1.228 |
**Tasks:**
a. **Estimate \( f(1.23) \) as accurately as possible.**
b. **Estimate \( f'(1.20) \) using an interpolating polynomial:**
This involves finding a polynomial that passes through the given points and then differentiating it to find the derivative at \( x = 1.20 \).
c. **Estimate \( f'(1.20) \) using the central difference expression:**
The central difference formula is a numerical method to approximate the derivative of a function. It uses values from both sides of the point where the derivative is being estimated.
\[ f'(1.20) \approx \frac{f(1.20 + h) - f(1.20 - h)}{2h} \]
In this context, suitable values for \( h \) should be considered.
d. **Estimate \( f''(1.20) \) using the method of your choice:**
This could involve using finite difference formulas, such as the central difference method for second derivatives or constructing a higher-order polynomial and differentiating it twice.
**Explanation of Concepts and Methods:**
1. **Interpolating Polynomial:**
- This is a polynomial that goes exactly through a set of data points. For small datasets, the Lagrange interpolation formula is often used.
2. **Central Difference Expression:**
- To estimate \( f'(x) \), this method uses the average rate of change around \( x \).
- For \( f''(x) \), it uses the second-order differences between points.
3. **Second Derivative Estimation:**
- A common method includes using finite difference formulas which rely on the function values at various points around the point of interest.
Each method has its own advantages and preferred use cases, with accuracy depending on the nature of the function and the distribution of the data points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ddf1e52-17d0-47a9-956c-00bd0ce93716%2F166c78a3-3337-43e3-80bd-c213d3bccaf3%2F33wh7ec_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Q2: Table Analysis and Derivative Estimations**
Consider the following table:
| **x** | 0.800 | 1.000 | 1.200 | 1.400 |
|-------|-------|-------|-------|-------|
| **f(x)** | 1.040 | 1.151 | 1.213 | 1.228 |
**Tasks:**
a. **Estimate \( f(1.23) \) as accurately as possible.**
b. **Estimate \( f'(1.20) \) using an interpolating polynomial:**
This involves finding a polynomial that passes through the given points and then differentiating it to find the derivative at \( x = 1.20 \).
c. **Estimate \( f'(1.20) \) using the central difference expression:**
The central difference formula is a numerical method to approximate the derivative of a function. It uses values from both sides of the point where the derivative is being estimated.
\[ f'(1.20) \approx \frac{f(1.20 + h) - f(1.20 - h)}{2h} \]
In this context, suitable values for \( h \) should be considered.
d. **Estimate \( f''(1.20) \) using the method of your choice:**
This could involve using finite difference formulas, such as the central difference method for second derivatives or constructing a higher-order polynomial and differentiating it twice.
**Explanation of Concepts and Methods:**
1. **Interpolating Polynomial:**
- This is a polynomial that goes exactly through a set of data points. For small datasets, the Lagrange interpolation formula is often used.
2. **Central Difference Expression:**
- To estimate \( f'(x) \), this method uses the average rate of change around \( x \).
- For \( f''(x) \), it uses the second-order differences between points.
3. **Second Derivative Estimation:**
- A common method includes using finite difference formulas which rely on the function values at various points around the point of interest.
Each method has its own advantages and preferred use cases, with accuracy depending on the nature of the function and the distribution of the data points.
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