Question

Using differential, find the approximate values of the following:
$\left(xxviii\right)$. $(1.999{)}^5$

Answer 1

$31.92$

Let $y=x^5\Rightarrow \frac{dy}{dx}=\frac{1}{5x^4}$
And $x=2$ and $\Delta x=-0.001$
then,
$\Delta y=(x+\Delta x{)}^5-x^5\Rightarrow \Delta y=(2-0.001{)}^5-(2{)}^5\Rightarrow \Delta y=(1.999{)}^5-32\Rightarrow (1.999{)}^5=32+\Delta y\dots (1)$
Now, Approximate change in value of $y$ is given by
$\Delta y\approx \left(\frac{dy}{dx}\right)\times \Delta x\Rightarrow \Delta y\approx 5x^4\times \Delta x\Rightarrow \Delta y\approx 5\times 2^4\times (-0.001)$
$\Rightarrow \Delta y\approx -0.08$
From equation $\left(1\right)$
$(1.999{)}^5\approx 32-0.08\therefore (1.999{)}^5\approx 31.92$