Question

Using differentials,find the approximate value of each of the following (a) $(\frac{17}{81})^\frac{1}{4}$ (b) $(33)^{-\frac{1}{5}}$

Answer 1

(a) Consider $y=x^\frac{1}{4}$.Let x=$\frac{16}{61}$ and Δx=$\frac{1}{81}$
Then,Δy=(x+Δx)1/4-x1/4
= $(\frac{17}{81})^\frac{1}{4}$- $(\frac{16}{81})^\frac{1}{4}$
= $(\frac{17}{81})^\frac{1}{4}$ -$\frac{2}{3}$
∴$(\frac{17}{81})^\frac{1}{4}$= $\frac{2}{3}$+Δy

Now,dy is approximately equal to ∆y and is given by,
dy=$(\frac{dy}{dx})$ Δx= $\frac{1}{4(x)^{\frac{3}{4}}}$(Δx)                       (as $y=x^\frac{1}{4}$)
= $\frac {1}{4(\frac{16}{81})^\frac{3}{4}}(\frac{1}{81})$= $\frac{27}{4\times8}$x $\frac{1}{81}=\frac{1}{32\times3}=\frac{1}{96}=0.010$
Hence, the approximate value of $\frac{17}{81}^\frac{1}{4}$ is  $\frac{2}{3}$+0.010=0.667+0.010=0.677.

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(b)Consider y=$x=\frac{1}{5}$ . Let x=32 and ∆x=1.

$Δy=(x+Δx)^{-\frac{1}{5}}-x^{-\frac{1}{5}}$$=33^{-  (\frac{1}{5})}  -(32)^{-  (\frac{1}{5})}  -  (\frac{1}{2})$
∴$(33)^{-  (\frac{1}{5})}$=$\frac{1}{2}$+Δy
Now,dy is approximately equal to ∆y and is given by,
$dy=(\frac{dy}{dx})$(Δx)$=-   \frac{1}{5  (x) ^{ (\frac{6}{5})}}  (Δx)$  (           as  $y=\frac{1}{5}$)
=-$\frac{1}{5(2)^6}(1)=-\frac{1}{320}=-0.003$

Hence, the approximate value of$(33)^{\frac{-1}{5}}$ is $\frac{1}{2}$+(-0.003)
=0.5−0.003=0.497.