Suppose that ?(?) is differentiable on an interval centered at ? = ? and ?(?) = ?0 +?1(? − ?) + ⋯ + ??(? − ?)? is a polynomial of degree ? with constants ?0, ⋯ , ??. Let ?(?) = ?(?) − ?(?). Show that if we impose on ? conditions i) E(a)=0 ii) lim?→??(?)(?−?)? = 0 Then ?(?) = ?(?) + ?′(?)(? − ?) +?(?)2!(? − ?)2 + ⋯ +??(?)?!(? −?)? .
Suppose that ?(?) is differentiable on an interval centered at ? = ? and ?(?) = ?0 +?1(? − ?) + ⋯ + ??(? − ?)? is a polynomial of degree ? with constants ?0, ⋯ , ??. Let ?(?) = ?(?) − ?(?). Show that if we impose on ? conditions i) E(a)=0 ii) lim?→??(?)(?−?)? = 0 Then ?(?) = ?(?) + ?′(?)(? − ?) +?(?)2!(? − ?)2 + ⋯ +??(?)?!(? −?)? .
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 3E
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Question
Suppose that ?(?) is differentiable on an interval centered at ? = ? and ?(?) = ?0 +?1(? − ?) + ⋯ + ??(? − ?)?
is a polynomial of degree ? with constants ?0, ⋯ , ??. Let
?(?) = ?(?) − ?(?). Show that if we impose on ? conditions
i) E(a)=0
ii) lim?→??(?)(?−?)? = 0
Then ?(?) = ?(?) + ?′(?)(? − ?) +?(?)2!(? − ?)2 + ⋯ +??(?)?!(? −?)?
.
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