(a) Re-write this formula using step size h/2 instead of h. (b) Multiply your answer from (a) by 4, subtract (*) from the result, and then divide everything by 3. We now should have an approximation to ƒ'(x) based on f(xo+h, f(x−h), f(xo+h/2) and f(xo-h/2). What is the error in this new approximation f'(xo)?
(a) Re-write this formula using step size h/2 instead of h. (b) Multiply your answer from (a) by 4, subtract (*) from the result, and then divide everything by 3. We now should have an approximation to ƒ'(x) based on f(xo+h, f(x−h), f(xo+h/2) and f(xo-h/2). What is the error in this new approximation f'(xo)?
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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Question
Taylor's theorem
Transcribed Image Text:(a) Re-write this formula using step size \( h/2 \) instead of \( h \).
(b) Multiply your answer from (a) by 4, subtract \( (\ast) \) from the result, and then divide everything by 3. We now should have an approximation to \( f'(x_0) \) based on \( f(x_0 + h), f(x_0 - h), f(x_0 + h/2) \) and \( f(x_0 - h/2) \). What is the error in this new approximation \( f'(x_0) \)?
![**[7] [Richardson extrapolation for a 2nd order accurate approximation]**
Using Taylor’s theorem, it can be shown that if \( f \in C^5([a, b]) \), then the centered difference approximation formula for the first derivative is given by
\[
f'(x_0) = \underbrace{\frac{f(x_0 + h) - f(x_0 - h)}{2h}}_{\text{approximation}} - \left(\underbrace{\frac{h^2}{6} f^{(3)}(x_0) + \frac{h^4}{120} f^{(5)}(\xi)}_{\text{error}}\right) \quad (*)
\]
- **True value:** \( f'(x_0) \)
- **Approximation:** \(\frac{f(x_0 + h) - f(x_0 - h)}{2h}\)
- **Error:** \(\left(\frac{h^2}{6} f^{(3)}(x_0) + \frac{h^4}{120} f^{(5)}(\xi)\right)\)
This formula provides a way to approximate the first derivative of a function using values of the function at points \( x_0 + h \) and \( x_0 - h \), corrected by a term that quantifies the error based on the higher derivatives of \( f \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2Ffdcae17f-1272-4069-866a-66100037f1c1%2Fmezfbn19_processed.png&w=3840&q=75)
Transcribed Image Text:**[7] [Richardson extrapolation for a 2nd order accurate approximation]**
Using Taylor’s theorem, it can be shown that if \( f \in C^5([a, b]) \), then the centered difference approximation formula for the first derivative is given by
\[
f'(x_0) = \underbrace{\frac{f(x_0 + h) - f(x_0 - h)}{2h}}_{\text{approximation}} - \left(\underbrace{\frac{h^2}{6} f^{(3)}(x_0) + \frac{h^4}{120} f^{(5)}(\xi)}_{\text{error}}\right) \quad (*)
\]
- **True value:** \( f'(x_0) \)
- **Approximation:** \(\frac{f(x_0 + h) - f(x_0 - h)}{2h}\)
- **Error:** \(\left(\frac{h^2}{6} f^{(3)}(x_0) + \frac{h^4}{120} f^{(5)}(\xi)\right)\)
This formula provides a way to approximate the first derivative of a function using values of the function at points \( x_0 + h \) and \( x_0 - h \), corrected by a term that quantifies the error based on the higher derivatives of \( f \).
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