Question

Find the approximate value of $(1.999{)}^5$

Answer 1

Let $y=x^5$
where, $x=2$ & $\Delta x=-0.001$
We know
$\Delta y=(x+\Delta x{)}^5-(x{)}^5$
$\Rightarrow \Delta y=(1.999{)}^5-(2{)}^5$
$\Rightarrow \Delta y=(1.999{)}^5-32\cdots (1)$
Now $\Delta y\approx \left(\frac{dy}{dx}\right)\Delta x$
$\Rightarrow \Delta y\approx \left(\frac{d\left(x^5\right)}{dx}\right)(-0.001)$
$\Rightarrow \Delta y\approx \left(5x{)}^4\right(-0.001)$
$\Rightarrow \Delta y\approx \left(5\right(2{)}^4(-0.001)$
$\Rightarrow \Delta y\approx \left(80\right)(-0.001)$
$\Rightarrow \Delta y\approx -0.080$
Substituting value of $\Delta y$in equation (1),
$\Rightarrow -0.080\approx (1.999{)}^5-32$
$\Rightarrow 32-0.080\approx (1.999{)}^5$
$\Rightarrow (1.999{)}^5\approx 31.920$
Hence ,approximate value of $(1.999{)}^5$ is $31.920$