1. If P2(x) is the second degree polynomial that interpolates to f(x) = 1+x 0,0.1, 0.2, find a reasonable bound on the error at x = 0.15. -36 at x = (A) -B) = 3 (ao)k-a) k-0,2)| s 3 31 (1s) (0,05) (0.os) =0,00225
1. If P2(x) is the second degree polynomial that interpolates to f(x) = 1+x 0,0.1, 0.2, find a reasonable bound on the error at x = 0.15. -36 at x = (A) -B) = 3 (ao)k-a) k-0,2)| s 3 31 (1s) (0,05) (0.os) =0,00225
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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Need full explanation on the given problem and detailed steps since I don't really understand the problem. Answer is given.
![### Polynomial Interpolation Error Estimation
1. **Problem Statement**:
If \( P_2(x) \) is the second-degree polynomial that interpolates to \( f(x) = \frac{6}{1+x} \) at \( x = 0, 0.1, 0.2 \), find a reasonable bound on the error at \( x = 0.15 \).
2. **Error Formula**:
\[
| f(x) - P_2(x) | = \left| \frac{f'''(\xi)}{3!} (x - 0)(x - 0.1)(x - 0.2) \right|
\]
3. **Calculating \( f'''(\xi) \)**:
\[
f'''(x) = \frac{-36}{(1+x)^4}
\]
\[
0 \leq \xi \leq 0.2
\]
4. **Error Bound Calculation**:
Since the function \( f'''(\xi) \) is maximized within the interval, we estimate:
\[
\leq \frac{36}{6} \times (0.15)(0.05)(0.05)
\]
\[
= 0.00225
\]
5. **Conclusion**:
A reasonable bound on the error of the polynomial interpolation at \( x = 0.15 \) is approximately \( 0.00225 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2Fd3a98392-c975-4630-936e-cdfd843058c5%2Fk8s2c5_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Interpolation Error Estimation
1. **Problem Statement**:
If \( P_2(x) \) is the second-degree polynomial that interpolates to \( f(x) = \frac{6}{1+x} \) at \( x = 0, 0.1, 0.2 \), find a reasonable bound on the error at \( x = 0.15 \).
2. **Error Formula**:
\[
| f(x) - P_2(x) | = \left| \frac{f'''(\xi)}{3!} (x - 0)(x - 0.1)(x - 0.2) \right|
\]
3. **Calculating \( f'''(\xi) \)**:
\[
f'''(x) = \frac{-36}{(1+x)^4}
\]
\[
0 \leq \xi \leq 0.2
\]
4. **Error Bound Calculation**:
Since the function \( f'''(\xi) \) is maximized within the interval, we estimate:
\[
\leq \frac{36}{6} \times (0.15)(0.05)(0.05)
\]
\[
= 0.00225
\]
5. **Conclusion**:
A reasonable bound on the error of the polynomial interpolation at \( x = 0.15 \) is approximately \( 0.00225 \).
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