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Using differentials, find the approximate value ofeach of the following: (a)(17/81)^(1/4)""(b) (33)^(-1/5) |
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Answer» Solution :(a) Let `f(x)=x^(1/4)rArrf'(x) = 1/4 x ^((-3)/4)` Now, let`x = 16/81andDelta x = 1/81` `:.f(x+Delta x)~= f(x)+Delta xf'(x)` ` RARR(x+Delta x)^(1/4) ~= x^(1/4 + 1/(4x^(3/4)) xx Delta x` `rArr(16/81+1/81)^(1/4) ~=(16/81)^(1/4)+(1/81)/(4(16/81)^(3/4))` `rArr (17/81)^(1/4) ~= 2/3 + 1/(81 xx 4 (2/3)^(3))` `=2/3 + 1/96 = 65/96` `rArr(17/81)^(1/4) ~=0.677` (b) Let` f(x)=x^(-1/5) rArr f'(x)=(-1)/5 x ^(-6)/5)` Now, let x = 32AND`Delta x = 1` ` :.f(x+Delta x) ~= f(x) + Delta x f'(x)` ` rArr(x+ Delta x)^(-1//5) ~= x ^(-1//5) - 1/(5X^(6/5)) Delta x` `(33)^(-1/5) ~= (32)^((-1)/5) - 1/(5(32)^(6/5))` `= 1/2 - 1/(5 xx 2^(6)) = 0.5 - 1/320` ` = 0.5 - 0.003` ` rArr (33)^(-1/5) ~= 0.497` |
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