Using differentials, find the approximate values of the following…

Question

Using differentials, find the approximate values of the following:
$(0.007)^{\frac{1}{3}}$

✓

Answer

Let $\text{x}=0.008,\\\text{x}+\triangle\text{x}=0.007$
$\triangle\text{x}=0.007-0.008$
$\triangle\text{x}=-0.001$
Let $\text{y}=\text{x}^{\frac{1}{3}}$
$\frac{\text{dy}}{\text{dx}}=\frac{1}{3(\text{x})^{\frac{2}{3}}}$
$\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\text{x}\Rightarrow0.008}=\frac{1}{3(0.008)^{\frac{2}{3}}}$
$=\frac{100}{2}$
$\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\text{x}\Rightarrow0.008}=8.333$
$\triangle\text{y}\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\text{x}\Rightarrow0.008}\times\triangle\text{x}$
$=(8.333)(-0.001)$
$\triangle\text{y}=-0.008333$
$(0.007)^{\frac{1}{3}}=\text{y}+\triangle\text{y}$
$=\text{x}^{\frac{1}{3}}-0.008333$
$=(0.008)^{\frac{1}{3}}-0.008333$
$=0.2-0.008333$
$(0.007)^{\frac{1}{3}}=0.191667$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Differentials, Errors and Approximations→Differentials, Errors and Approximations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let n be a fixed positive integer. Define a relation R on Z as follows:
$(\text{a, b})\in\text{R}\Leftrightarrow\ \text{a}-\text{b}$ is divisible by n. Show that R is an equivalence relation on Z.

→

Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l² + m²- n² = 0.

→

The probability distribution of a discrete random variable X is given as under:

$\text{X}$	$1$	$2$	$4$	$2\text{A}$	$3\text{A}$	$5\text{A}$
$\text{P}(\text{X})$	$\frac{1}{2}$	$\frac{1}{5}$	$\frac{3}{25}$	$\frac{1}{10}$	$\frac{1}{25}$	$\frac{1}{25}$