] [Backwards difference approximation of first derivative] Using Taylor's theorem, in lecture we derived the forward difference ap- oximation to the first derivative. In a similar fashion, derive the backwards fference approximation f(xo)-f(xo-h) 1 = f'(x) = ²h f"(E)
] [Backwards difference approximation of first derivative] Using Taylor's theorem, in lecture we derived the forward difference ap- oximation to the first derivative. In a similar fashion, derive the backwards fference approximation f(xo)-f(xo-h) 1 = f'(x) = ²h f"(E)
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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Taylor's theorem
![### Backwards Difference Approximation of First Derivative
Using Taylor’s theorem, in lecture we derived the forward difference approximation to the first derivative. In a similar fashion, derive the backwards difference approximation:
\[
\frac{f(x_0) - f(x_0 - h)}{h} = f'(x_0) - \frac{1}{2} h f''(\xi)
\]
assuming that \( f \in C^2([a, b]) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F14a9cec5-cfe4-41d3-9755-f52cb517a3a1%2Fx8irrsw_processed.png&w=3840&q=75)
Transcribed Image Text:### Backwards Difference Approximation of First Derivative
Using Taylor’s theorem, in lecture we derived the forward difference approximation to the first derivative. In a similar fashion, derive the backwards difference approximation:
\[
\frac{f(x_0) - f(x_0 - h)}{h} = f'(x_0) - \frac{1}{2} h f''(\xi)
\]
assuming that \( f \in C^2([a, b]) \).
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