Ex 6.4, 1 (xii) - Find approximate value upto 3 decimals - (26.57)^1/3

Ex 6.4, 1 (xii) - Chapter 6 Class 12 Application of Derivatives - Part 2
Ex 6.4, 1 (xii) - Chapter 6 Class 12 Application of Derivatives - Part 3

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Question 1 Using differentials, find the approximate value of each of the following up to 3 places of decimal. (xii) 〖(26.57)〗^(1/3)Let 𝑦=𝑥^( 1/3) where 𝑥=27 & ∆ 𝑥=−0. 43 Now, 𝑦=𝑥^( 1/3) Differentiating w.r.t.𝑥 𝑑𝑦/𝑑𝑥=𝑑(𝑥^( 1/3) )/𝑑𝑥=1/3 𝑥^( 1/3 − 1)=1/3 𝑥^( (− 2)/( 3))=1/(3 𝑥^( 2/( 3)) ) Using ∆𝑦=𝑑𝑦/𝑑𝑥 ∆𝑥 ∆𝑦=1/(3 𝑥^( 2/( 3)) ) × ∆𝑥 Putting Values ∆𝑦=1/(3 (27)^( 2/( 3)) ) × (−0. 43) ∆𝑦=(−0. 43)/(3 × 3^(3 × 2/( 3)) ) ∆𝑦=(−0. 43)/(3 × 3^2 ) ∆𝑦=(−0. 43)/(3 × 9) ∆𝑦=(−0. 43)/27 ∆𝑦=−0. 015926 We know that ∆𝑦=𝑓(𝑥+∆𝑥)−𝑓(𝑥) So, ∆𝑦=〖(𝑥+∆𝑥) 〗^(1/3)−𝑥^( 1/3) Putting Values −0. 015926=(27+(−0. 43))^( 1/3)−(27)^( 1/3) −0. 015926=(26. 57)^( 1/3)−(27)^( 3 × 1/3) −0. 015926=(26. 57)^( 1/3)−3 −0. 015926+3=(26. 53)^( 1/3) 2. 984=(26. 53)^( 1/3) Thus, the Approximate Values of (26. 53)^( 1/3) is 𝟐. 𝟗𝟖𝟒

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo