12. Let xo < x < < x, and let f be continuously differentiable. Show that a f[xo, x₁, x₂] = f[xo, X₁, X₁, X₁, Xi+l..., Xn] axi
12. Let xo < x < < x, and let f be continuously differentiable. Show that a f[xo, x₁, x₂] = f[xo, X₁, X₁, X₁, Xi+l..., Xn] axi
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.4: Definition Of Function
Problem 52E
Related questions
Question
![12. Let xo < x < < x, and let f be continuously differentiable. Show that
a
axi
f[xo, x₁, x₂] = f[x0, x₁, xXi, Xi, Xi+l, Xn]
****](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F029aee4e-f627-4455-8e77-68da777572fa%2Fcj6r35s_processed.png&w=3840&q=75)
Transcribed Image Text:12. Let xo < x < < x, and let f be continuously differentiable. Show that
a
axi
f[xo, x₁, x₂] = f[x0, x₁, xXi, Xi, Xi+l, Xn]
****
![Theorem 1 gives us these particular formulas:
f[x0, x₁] =
f[xo, x1, x2]:
f[xi, Xi+1,…, Xi+j]
f[x]- f[xo]
x1 - x0
In these formulas, xo, x1, x2,... can be interpreted as independent variables. Be-
cause of that, we also have equations such as
f[Xi+1, Xi+2, …, Xi+j]- f[xi, Xi+h, …, Xi+j-1]
Xi+jXi
=
f[x₁, x2]- f[x0, x₁]
X2 - Xo
(13)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F029aee4e-f627-4455-8e77-68da777572fa%2Fw0o2oe6_processed.png&w=3840&q=75)
Transcribed Image Text:Theorem 1 gives us these particular formulas:
f[x0, x₁] =
f[xo, x1, x2]:
f[xi, Xi+1,…, Xi+j]
f[x]- f[xo]
x1 - x0
In these formulas, xo, x1, x2,... can be interpreted as independent variables. Be-
cause of that, we also have equations such as
f[Xi+1, Xi+2, …, Xi+j]- f[xi, Xi+h, …, Xi+j-1]
Xi+jXi
=
f[x₁, x2]- f[x0, x₁]
X2 - Xo
(13)
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