Approximations (using Differentiation)

Chapter 6 Class 12 Application of Derivatives
Serial order wise

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Question 1 Using differentials, find the approximate value of each of the following up to 3 places of decimal. (xii) γ(26.57)γ^(1/3)Let π¦=π₯^( 1/3) where π₯=27 & β π₯=β0. 43 Now, π¦=π₯^( 1/3) Differentiating w.r.t.π₯ ππ¦/ππ₯=π(π₯^( 1/3) )/ππ₯=1/3 π₯^( 1/3 β 1)=1/3 π₯^( (β 2)/( 3))=1/(3 π₯^( 2/( 3)) ) Using βπ¦=ππ¦/ππ₯ βπ₯ βπ¦=1/(3 π₯^( 2/( 3)) ) Γ βπ₯ Putting Values βπ¦=1/(3 (27)^( 2/( 3)) ) Γ (β0. 43) βπ¦=(β0. 43)/(3 Γ 3^(3 Γ 2/( 3)) ) βπ¦=(β0. 43)/(3 Γ 3^2 ) βπ¦=(β0. 43)/(3 Γ 9) βπ¦=(β0. 43)/27 βπ¦=β0. 015926 We know that βπ¦=π(π₯+βπ₯)βπ(π₯) So, βπ¦=γ(π₯+βπ₯) γ^(1/3)βπ₯^( 1/3) Putting Values β0. 015926=(27+(β0. 43))^( 1/3)β(27)^( 1/3) β0. 015926=(26. 57)^( 1/3)β(27)^( 3 Γ 1/3) β0. 015926=(26. 57)^( 1/3)β3 β0. 015926+3=(26. 53)^( 1/3) 2. 984=(26. 53)^( 1/3) Thus, the Approximate Values of (26. 53)^( 1/3) is π. πππ