Using differentials, find the approximate value of $\sqrt {0.037} $.

Hint: The most important step in the calculation is to identify the correct function which we can use to find the given value or approximate value of the required function. Then we need to establish the approximate value of any number with respect to the given number and then we will use the definition of the differential to find the required value.

Complete step-by-step answer:
It is given that we have to find the value of $\sqrt {0.037} $.
It can be observed that the function is expressed as a square root so we will select the main function accordingly.
We will select the function $y$ as follows:
$ \Rightarrow y = \sqrt x $
On differentiating with respect to $x$ we get the following,
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2\sqrt x }}$
Now using the definition of the differential, we will right the value of differential of $y$ as follows:
$ \Rightarrow \Delta y = \dfrac{{dy}}{{dx}}\Delta x$
Substitute the values,
$ \Rightarrow \Delta y = \dfrac{1}{{2\sqrt x }}\Delta x$ ….. (1)
Now the differential with respect to the variable $x$ that is $\Delta x$ the infinitesimal change in the variable $x$. We will have to find the nearest value to the number $0.037$ which square root we already know.
So, we will consider $x = 0.04$ that will imply that the small change is just $0.003$.
So, $\Delta x = - 0.003$.
Substitute the values in equation (1),
$ \Rightarrow \Delta y = \dfrac{1}{{2\sqrt {0.04} }} \times - 0.003$
Simplify the terms,
$ \Rightarrow \Delta y = \dfrac{1}{{2 \times 0.2}} \times - 0.003$
Multiply the terms in denominator,
$ \Rightarrow \Delta y = \dfrac{{ - 0.003}}{{0.4}}$
Divide numerator by denominator,
$ \Rightarrow \Delta y = - 0.0075$
Also,
$\Delta y = f\left( {x + \Delta x} \right) - f\left( x \right)$
Substitute the values,
$ \Rightarrow - 0.0075 = \sqrt {0.04 - 0.003} - \sqrt {0.04} $
Simplify the terms,
$ \Rightarrow - 0.0075 = \sqrt {0.037} - 0.2$
Move 0.2 on another side,
$ \Rightarrow \sqrt {0.037} = 0.2 - 0.0075$
Subtract the values,
$\therefore \sqrt {0.037} = 0.1925$

Hence, the approximate value of $\sqrt {0.037} $ is $0.1925$.

Note: The differentials always give an approximate value so we have used the approximate value of the dependent variable as well. We differentiate the given function and then substitute the known values to simplify the given values and reach the final answer.