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.