Example 43 - Examples

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Example 43 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

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Example 43 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

  1. Chapter 5 Class 12 Continuity and Differentiability (Term 1)
  2. Serial order wise

Transcript

Example 43 Verify Mean Value Theorem for the function ๐‘“(๐‘ฅ) = ๐‘ฅ2 in the interval [2, 4]. ๐‘“(๐‘ฅ) = ๐‘ฅ2 in interval [2, 4]. Checking conditions for Mean value Theorem Condition 1 Since ๐‘“(๐‘ฅ) is polynomial . it is continuous โˆด ๐‘“(๐‘ฅ) is continuous at (2, 4) Conditions of Mean value theorem ๐‘“(๐‘ฅ) is continuous at (๐‘Ž, ๐‘) ๐‘“(๐‘ฅ) is differentiable at (๐‘Ž , ๐‘) If both conditions satisfied, then there exist some c in (๐‘Ž , ๐‘) such that ๐‘“โ€ฒ(๐‘) = (๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž))/(๐‘ โˆ’ ๐‘Ž)Condition 2 Since ๐‘“(๐‘ฅ) is a polynomial . it is Differentiable โˆด ๐‘“(๐‘ฅ) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c โˆˆ (2, 4) such that, ๐‘“^โ€ฒ (๐‘) = (๐‘“(4) โˆ’ ๐‘“(2))/(4 โˆ’ 2) Conditions of Mean value theorem ๐‘“(๐‘ฅ) is continuous at (๐‘Ž, ๐‘) ๐‘“(๐‘ฅ) is differentiable at (๐‘Ž , ๐‘) If both conditions satisfied, then there exist some c in (๐‘Ž , ๐‘) such that ๐‘“โ€ฒ(๐‘) = (๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž))/(๐‘ โˆ’ ๐‘Ž) Condition 2 Since ๐‘“(๐‘ฅ) is a polynomial . it is Differentiable โˆด ๐‘“(๐‘ฅ) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c โˆˆ (2, 4) such that, ๐‘“^โ€ฒ (๐‘) = (๐‘“(4) โˆ’ ๐‘“(2))/(4 โˆ’ 2) 2๐‘= (4^2 โˆ’ 2^2)/2 2๐‘ = 12/2 2๐‘ = 6 ๐’„ = ๐Ÿ‘ Hence c = 3 โˆˆ(๐Ÿ, ๐Ÿ’) Hence, Mean value Theorem is satisfied .

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.