If f is C2 on an interval prove that f(x + h) – 2f(x) + f(x – h) lim h-0 = f" (x). h2
If f is C2 on an interval prove that f(x + h) – 2f(x) + f(x – h) lim h-0 = f" (x). h2
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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Can we prove this without Langrange's Form of Remainders?
![If \( f \) is \( C^2 \) on an interval, prove that
\[
\lim_{{h \to 0}} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2} = f''(x).
\]
The expression \( f(x+h) - 2f(x) + f(x-h) \) is called the **symmetric second difference**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5071459a-d319-4f4e-b4b5-61daae7ca4c5%2Ff85eb2f2-45a3-4e73-af63-55df04618ab4%2Fitgr0qk_processed.png&w=3840&q=75)
Transcribed Image Text:If \( f \) is \( C^2 \) on an interval, prove that
\[
\lim_{{h \to 0}} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2} = f''(x).
\]
The expression \( f(x+h) - 2f(x) + f(x-h) \) is called the **symmetric second difference**.
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