If f is C2 on an interval prove that f(2 + k) – 2/(2) + {(z =A) = r"(2). - lim h-0 h2 = f"(x). The expression f(x+h) – 2f(x) +f(x-h) is called the symmetric second difference.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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**.
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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