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Last updated at Jan. 7, 2020 by Teachoo

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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 โ๐ฅ=1 Since y = ๐ฅ^(1/4) ๐๐ฆ/๐๐ฅ = ๐(๐ฅ^(1/4) )/๐๐ฅ = 1/4 ๐ฅ^((โ3)/4) = 1/(4๐ฅ^(3/4) ) Now, โ๐ฆ = ๐๐ฆ/๐๐ฅ โ๐ฅ โ๐ฆ = (1/(4๐ฅ^(3/4) ))โ๐ฅ Putting values โ๐ฆ = 1/(4(16)^(3/4) ) (1) = 1/(4((16)^(1/4) )^3 ) = 1/(4(2)^3 ) = 1/32 Also, โ๐ฆ = f(x + โ๐ฅ) โ f(x) โ๐ฆ = ("x + " โ๐ฅ)^(1/4) โ ๐ฅ^(1/4) โ๐ฆ = ("16 + " 1)^(1/4) โ ใ(16)ใ^(1/4) โ๐ฆ = (17)^(1/4) โ ใ(16)ใ^(1/4) (17)^(1/4) = โ๐ฆ+(16)^(1/4) (17)^(1/4) = โ๐ฆ+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^(โ 1/5) ใ(33)ใ^(โ 1/5) = 1/(33)^(1/5) Let y = ๐ฅ^(1/5) & also let ๐ฅ = 32 & โ ๐ฅ = 1 Now, ๐ฆ = ๐ฅ^(1/5) (As (32)^(1/5)=2) Diff w.r.t x ๐๐ฆ/๐๐ฅ= (๐ (๐ฅ^(1/5)))/๐๐ฅ ๐๐ฆ/๐๐ฅ= 1/5 ๐ฅ^((1 )/5 โ1) ๐๐ฆ/๐๐ฅ = 1/5 ๐ฅ^((โ4)/5) ๐๐ฆ/๐๐ฅ= 1/ใ5๐ฅใ^(4/5) Using โy = ๐๐ฆ/๐๐ฅ โ๐ฅ โy = 1/ใ5๐ฅใ^(4/5) โ๐ฅ Putting values โy = 1/ใ5(32)ใ^(4/5) ร (1) โy = 1/ใ5(32)ใ^(4/5) โy = 1/ใ5(2)ใ^(5 ร 4/5) โy = 1/5(2^4 ) โy = 1/(5 ร 16) โy = 1/80 We know that โy = f(x + โx) โ f(x) So, โy = ใ(x+ ฮx)ใ^(1/5) ใโ๐ฅใ^(1/5) Putting values 1/80= ใ(32+1)ใ^(1/5) โ (32)^(1/5) 1/80= ใ(33)ใ^(1/5) โ ใ(2) ใ^(5 ร 1/5) 1/80= ใ(33)ใ^(1/5) โ 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)^((โ1)/5)= 0.497 Thus, the approximate value of (33)^((โ1)/5) ๐๐ ๐.๐๐๐

Miscellaneous

Misc 1
Important
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Misc 2 Important

Misc 3 Important

Misc 4

Misc 5 Important

Misc 6 Important

Misc 7

Misc 8 Important

Misc 9 Important

Misc 10

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18 Important

Misc. 19

Misc 20 Important

Misc 21 Important

Misc 22

Misc. 23 Important

Misc 24 Important

Chapter 6 Class 12 Application of Derivatives

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About the Author

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.