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  4. The approximate value of (3.92)2+3(2.1)4 1 / 6 is

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Q. The approximate value of $\left\{(3.92)^{2}+3(2.1)^{4}\right\}^{1 / 6}$ is

Application of Derivatives

Solution:

Let $f(x, y)=\left(x^{2}+3 y^{4}\right)^{1 / 6}$
Taking $x=4, \Delta x=-0.08$ and $y=2, \Delta y=0.1$
Differentiating (1) w.r.t. $x$, treating $y$ as constant,
$\therefore \frac{\Delta f}{\Delta x}=\frac{1}{6}\left(x^{2}+3 y^{4}\right)^{-5 / 6}(2 x) $
$=\frac{8}{6}(16+48)^{-5 / 6}=\frac{4}{3} \times 2^{-5}=\frac{1}{24} $
and differentiating (1) w.r.t. $y$ treating $x$ as constant,
$\therefore \frac{\Delta f}{\Delta y}=\frac{1}{6}\left(x^{2}+3 y^{4}\right)^{-5 / 6}\left(12 y^{3}\right)$
$=\frac{12(8)}{6}(64)^{-5 / 6}=16(2)^{-5}=\frac{1}{2}$
$\therefore df =\frac{\Delta f}{\Delta x} \cdot d x+\frac{\Delta f}{\Delta y} dy $
$=\frac{1}{24} \times-0.08+\frac{1}{2} \times 0.1 $
$=-\frac{0.01}{3}+\frac{0.1}{2}=0.466$
$\therefore \left\{(3.92)^{2}+3(2.1)^{4}\right\}^{1 / 6}=f(4,2)+d f$
$=2+0.466=2.466$