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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise

Transcript

Misc 1 Using differentials, find the approximate value of each of the following: (a) (17/81)^(1/4) (17/81)^(1/4) = (17)^(1/4)/(81)^(1/4) = (17)^(1/4)/3 Let x = 16 and โˆ†๐‘ฅ=1 Since y = ๐‘ฅ^(1/4) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(๐‘ฅ^(1/4) )/๐‘‘๐‘ฅ = 1/4 ๐‘ฅ^((โˆ’3)/4) = 1/(4๐‘ฅ^(3/4) ) Now, โˆ†๐‘ฆ = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ โˆ†๐‘ฅ โˆ†๐‘ฆ = (1/(4๐‘ฅ^(3/4) ))โˆ†๐‘ฅ Putting values โˆ†๐‘ฆ = 1/(4(16)^(3/4) ) (1) = 1/(4((16)^(1/4) )^3 ) = 1/(4(2)^3 ) = 1/32 Also, โˆ†๐‘ฆ = f(x + โˆ†๐‘ฅ) โˆ’ f(x) โˆ†๐‘ฆ = ("x + " โˆ†๐‘ฅ)^(1/4) โˆ’ ๐‘ฅ^(1/4) โˆ†๐‘ฆ = ("16 + " 1)^(1/4) โˆ’ ใ€–(16)ใ€—^(1/4) โˆ†๐‘ฆ = (17)^(1/4) โˆ’ ใ€–(16)ใ€—^(1/4) (17)^(1/4) = โˆ†๐‘ฆ+(16)^(1/4) (17)^(1/4) = โˆ†๐‘ฆ+2 (17)^(1/4) = 1/32+2 (17)^(1/4) = 0.03125 + 2 (17)^(1/4) = 2.03125 Now, Approximate value of (17/81)^(1/4) (17/81)^(1/4) = ((17)1/4)/3 = 2.03125/3 = 0.677 Hence approximate value of (17/81)^(1/4) is 0.677 (As approximate value of (17)^(1/4) = 2.03125) Misc 1 Using differentials, find the approximate value of each of the following: (b) ใ€–(33)ใ€—^(โˆ’ 1/5) ใ€–(33)ใ€—^(โˆ’ 1/5) = 1/(33)^(1/5) Let y = ๐‘ฅ^(1/5) & also let ๐‘ฅ = 32 & โˆ† ๐‘ฅ = 1 Now, ๐‘ฆ = ๐‘ฅ^(1/5) (As (32)^(1/5)=2) Diff w.r.t x ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ= (๐‘‘ (๐‘ฅ^(1/5)))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ= 1/5 ๐‘ฅ^((1 )/5 โˆ’1) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1/5 ๐‘ฅ^((โˆ’4)/5) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ= 1/ใ€–5๐‘ฅใ€—^(4/5) Using โˆ†y = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ โˆ†๐‘ฅ โˆ†y = 1/ใ€–5๐‘ฅใ€—^(4/5) โˆ†๐‘ฅ Putting values โˆ†y = 1/ใ€–5(32)ใ€—^(4/5) ร— (1) โˆ†y = 1/ใ€–5(32)ใ€—^(4/5) โˆ†y = 1/ใ€–5(2)ใ€—^(5 ร— 4/5) โˆ†y = 1/5(2^4 ) โˆ†y = 1/(5 ร— 16) โˆ†y = 1/80 We know that โˆ†y = f(x + โˆ†x) โ€“ f(x) So, โˆ†y = ใ€–(x+ ฮ”x)ใ€—^(1/5) ใ€–โˆ’๐‘ฅใ€—^(1/5) Putting values 1/80= ใ€–(32+1)ใ€—^(1/5) โˆ’ (32)^(1/5) 1/80= ใ€–(33)ใ€—^(1/5) โˆ’ ใ€–(2) ใ€—^(5 ร— 1/5) 1/80= ใ€–(33)ใ€—^(1/5) โˆ’ 2 1/80+2= ใ€–(33)ใ€—^(1/5) (1 + 160)/80=ใ€–(33)ใ€—^(1/5) 161/80= ใ€–(33)ใ€—^(1/5) ใ€–(33)ใ€—^(1/5) = 161/80 But we need 1/(33)^(1/5) So 1/(33)^(1/5) = 1/(161/80) 1/(33)^(1/5) = 80/161 1/(33)^(1/5) = 0.497 (33)^((โˆ’1)/5)= 0.497 Thus, the approximate value of (33)^((โˆ’1)/5) ๐‘–๐‘  ๐ŸŽ.๐Ÿ’๐Ÿ—๐Ÿ•

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.