Answer:
Let \[f(x)={{x}^{1/4}}.\] Then, \[f'(x)=\frac{1}{4{{x}^{3/4}}}\] Now, \[\{f(x+\delta x)-f(x)\}=f'(x)\cdot \delta x\] \[\Rightarrow \] \[\{f(x+\delta x)-f(x)\}=\frac{1}{4{{x}^{3/4}}}\cdot \delta x\] ? (i) We may write, \[82=(81+1).\] Putting x = 81 and \[\delta x=1\] in Eq. (i), we get \[f\left( 81+1 \right)-f(81)=\frac{1}{4\times {{(81)}^{3/4}}}\cdot 1\] \[\Rightarrow \] \[f\left( 82 \right)-f(81)=\frac{1}{(4\times {{3}^{3}})}=\frac{1}{108}\] \[\Rightarrow \] \[f(82)=\left\{ f(81)+\frac{1}{108} \right\}=\left\{ {{(81)}^{1/4}}+\frac{1}{108} \right\}\] \[=3+0.009\] \[=3.009.\]
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