Advanced Calculus (The Derivative and Mean Value Theorem) B.A. / B.Sc. Semester III Syllabus (Unit I)
Taylor's theorem with Lagrange's form of remainder
Statement. If a function 𝒇 defined on [𝒂,𝒂+𝒉] is such that
(i) 𝒇,𝒇′,𝒇′′,…,𝒇^(𝒏−𝟏) are continuous functions of 𝒙 on [𝒂,𝒂+𝒉]
(ii) 𝒇^𝒏 (𝒙) exists in (𝒂,𝒂+𝒉), then there exists atleast one real number 𝜽 lying between 0 and 1 such that
𝒇(𝒂+𝒉) = 𝒇(𝒂) + 𝒉 𝒇′ (𝒂) + 𝒉^𝟐 / 𝟐! 𝒇′′ (𝒂)+….+𝒉^(𝒏−𝟏)/((𝒏−𝟏)!) 𝒇^(𝒏−𝟏) (𝒂)+𝒉^𝒏/𝒏! 𝒇^𝒏 (𝒂+𝜽𝒉)
Maclaurin's theorem with Lagrange's form of remainder
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Негізгі бет Mean Value Theorem || Taylor's Theorem || B.A./ B. Sc. Mathematics || Value For Time || Kamal Kumar
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