Question

Using differentials, find the approximate value of the following up to 3 places of decimal.

$(82{)}^{\frac{1}{4}}$

Answer 1

$\left(82{)}^{\frac{1}{4}}Considerf\right(x)=x^{\frac{1}{4}}\Rightarrow f]\left(x\right)=\frac{1}{4}{x}^{-\frac{3}{4}}Letx=81and\Delta x\simeq 1Now,f(x+\Delta x)\simeq f\left(x\right)+\Delta x{f}^{\prime }\left(x\right)\Rightarrow (x+\Delta x{)}^{\frac{1}{4}}\simeq x^{\frac{1}{4}}+\frac{\Delta x}{4x^{\frac{3}{4}}}\Rightarrow (81+1{)}^{\frac{1}{4}}\simeq (81{)}^{\frac{1}{4}}+\frac{1}{4(81{)}^{\frac{3}{4}}}=3+\frac{1}{4\left\{\right(81{)}^{\frac{1}{4}}{\}}^3}=3+\frac{1}{4\times 3^3}=3+\frac{1}{108}=3+0.009=3.009\Rightarrow (82{)}^{\frac{1}{4}}\simeq \mathrm{3.009.}$