Consider the differentiation formula Find the largest pen such that holds for any a,ber with a
Consider the differentiation formula Find the largest pen such that holds for any a,ber with 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 differentiation formula
Find the largest pen such that
holds for any a,ber with a<b, any
O
f'(x) =
a. p=0
b. p=1
O c. p=2
O d. p=3
e.
\ƒ'(x) -
h
2/17 ( f(x + h) + f(x) - 2ƒ(x −
2h
12/17 ( f ( x + h) + f(x) − 2ƒ ( x − ½/2))| ≤ 0(|h|³)
and all f in C([a,b],R).
Hints:
• You can use Corollary 4.19 to find numbers p for which the above formula holds.
• To check whether the best such p you can find is really the largest possible p, use simple functions like monomials to find a
counterexample for larger values of p.
x = (a, b),
None of the above: This formula diverges as h→0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe587adf5-17b3-4d31-bfe0-bfcde8e60070%2Fb465e9c9-c2b1-4336-8e65-6d91417e88ce%2F3zdwvg5_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the differentiation formula
Find the largest pen such that
holds for any a,ber with a<b, any
O
f'(x) =
a. p=0
b. p=1
O c. p=2
O d. p=3
e.
\ƒ'(x) -
h
2/17 ( f(x + h) + f(x) - 2ƒ(x −
2h
12/17 ( f ( x + h) + f(x) − 2ƒ ( x − ½/2))| ≤ 0(|h|³)
and all f in C([a,b],R).
Hints:
• You can use Corollary 4.19 to find numbers p for which the above formula holds.
• To check whether the best such p you can find is really the largest possible p, use simple functions like monomials to find a
counterexample for larger values of p.
x = (a, b),
None of the above: This formula diverges as h→0.
![Corollary 4.19: Taylor's theorem with Landau symbol
Let [a, b] CR be an interval with a ≤ b, let x, y € (a,b), and let ƒ €
Cn+1([a, b]). Then
n
· ƒ(k) (x) (y − x)k +O(\y − x|n+¹).
-
k!
k=0
f(y) = f(k) (x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe587adf5-17b3-4d31-bfe0-bfcde8e60070%2Fb465e9c9-c2b1-4336-8e65-6d91417e88ce%2Fbaskfhg_processed.png&w=3840&q=75)
Transcribed Image Text:Corollary 4.19: Taylor's theorem with Landau symbol
Let [a, b] CR be an interval with a ≤ b, let x, y € (a,b), and let ƒ €
Cn+1([a, b]). Then
n
· ƒ(k) (x) (y − x)k +O(\y − x|n+¹).
-
k!
k=0
f(y) = f(k) (x)
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