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
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Chapter3: Functions And Graphs
Section3.4: Definition Of Function
Problem 52E
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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]
****
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)
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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