Consider the second-order divided difference Prove the formula fla, b, c): fb, c-f[a, b] c-a f(a,b,c) =f" () for some € (a, b) assuming a
Consider the second-order divided difference Prove the formula fla, b, c): fb, c-f[a, b] c-a f(a,b,c) =f" () for some € (a, b) assuming a
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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![Consider the second-order divided difference
Prove the formula
f[a,b,c] =
f[b, c-f[a, b]
c-a
f[a,b,c] =f" (c)
for some € (a, b) assuming a <b<e and f (x) and is first two derivatives are continuous and bounded
from above and below on the interval [a, c].
Hint: Use Taylor's theorem to simplify f[a,b,c] by expanding f (2) about z = b and then use the
intermediate value theorem to simplify the remainder term from Taylor's theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F84f613f6-b2ed-4d0a-a13b-9654f4c42c71%2F591c2485-2506-4975-a207-b0fe5fde0c73%2Fo3dvb2b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the second-order divided difference
Prove the formula
f[a,b,c] =
f[b, c-f[a, b]
c-a
f[a,b,c] =f" (c)
for some € (a, b) assuming a <b<e and f (x) and is first two derivatives are continuous and bounded
from above and below on the interval [a, c].
Hint: Use Taylor's theorem to simplify f[a,b,c] by expanding f (2) about z = b and then use the
intermediate value theorem to simplify the remainder term from Taylor's theorem.
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