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Home  >>  CBSE XII  >>  Math  >>  Application of Derivatives
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Using differentials, find the approximate value of each of the following: $(33)^{-\large\frac{1}{5}}$

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Toolbox:
  • Let $y=f(x)$
  • $\Delta x$ denote a small increment in $x$
  • $\Delta y=f(x+\Delta x)-f(x)$
  • $dy=\big(\large\frac{dy}{dx}\big)$$\Delta x$
Step 1:
$F(x+\Delta x)=(33)^{\Large\frac{1}{5}}$
$\qquad\;\qquad=f(x)+\Delta y$
$\qquad\;\qquad=2+\large\frac{dy}{dx}$$\times \Delta x$
$\qquad\;\qquad=2+\large\frac{1}{5x^{\Large\frac{4}{5}}}$$\times 1$
$\qquad\;\qquad=2+\large\frac{1}{5(32)^{\Large\frac{4}{5}}}$
$\qquad\;\qquad=2+\large\frac{1}{5\times 16}$
$2+\large\frac{1}{80}=\frac{2\times 80+1}{80}$
$\qquad\;\;\;\;=\large\frac{161}{80}$
Step 2:
$\big(\large\frac{1}{33}\big)^{\Large\frac{1}{5}}=\frac{1}{(33)^{\Large\frac{1}{5}}}$
$\qquad\quad\;=\large\frac{1}{\Large\frac{161}{80}}$
$\qquad\quad\;=\large\frac{80}{161}$
$\qquad\quad\;=0.49689$
$\qquad\quad\;=0.497$(approx)
answered Aug 9, 2013 by sreemathi.v
edited Sep 2, 2013 by sharmaaparna1
 

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