Answer

Verified

427.8k+ views

**Hint:**$0.02=\dfrac{2}{100}\Rightarrow \sqrt{0.02}=\sqrt{\dfrac{2}{100}}=\dfrac{\sqrt{2}}{10}$ .

The nearest perfect square numbers to 2 are 1 and 4.

$\sqrt{1}<\sqrt{2}<\sqrt{4}$ ⇒ $1<\sqrt{2}<2$

We can use either the method of long division, binomial expansion or calculus to find the square root of 2.

**Complete step-by-step answer:**Since, $0.01<0.02<0.04$ , we can say that $\sqrt{0.01}<\sqrt{0.02}<\sqrt{0.04}$ or $0.1<\sqrt{0.02}<0.2$ .

Let us use differentiation (calculus) to find the value of $\sqrt{0.02}$ .

Let's say $y=f(x)=\sqrt{x}$ is a function of x.

For a change of Δx in the value of x, let's say that the value of y changes by Δy.

⇒ y + Δy = f(x + Δx)

We know that $f\left( 0.01 \right)=\sqrt{0.01}=1$ .

∴ f(0.02) = f(0.01 + 0.01) which means that the change Δx = 0.01.

We also know that for small values of Δx and Δy, $\dfrac{\Delta y}{\Delta x}\approx \dfrac{dy}{dx}$ .

Now, $\dfrac{\Delta y}{\Delta x}=\dfrac{dy}{dx}=\dfrac{d}{dx}\left( \sqrt{x} \right)$ .

Using the definition that roots are fractional powers ( $ {{x}^{\dfrac{p}{q}}}=\sqrt[q]{{{x}^{p}}}$ ):

$\dfrac{\Delta y}{\Delta x}=\dfrac{d}{dx}\left( \sqrt{x} \right)=\dfrac{d}{dx}\left( {{x}^{\dfrac{1}{2}}} \right)$

And using the formula of derivatives $\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}} $ , we get:

⇒ $\dfrac{\Delta y}{\Delta x}=\dfrac{1}{2}{{x}^{\left( \dfrac{1}{2}-1 \right)}}=\dfrac{1}{2}{{x}^{\dfrac{-1}{2}}}$

Substituting x = 0.01 and Δx = 0.01, we get:

$\dfrac{\Delta y}{0.01}=\dfrac{1}{2}{{(0.01)}^{\dfrac{-1}{2}}}$

Using the meaning of negative powers $ {{a}^{-x}}=\dfrac{1}{{{a}^{x}}}$ , we get:

⇒ $\dfrac{\Delta y}{0.01}=\dfrac{1}{2}\times \dfrac{1}{\sqrt{0.01}}$

⇒ $\Delta y=\dfrac{1}{2}\times \dfrac{1}{0.1}\times 0.01$

⇒ Δy = 0.05

Finally, since y + Δy = f(x + Δx), we can say that:

$\sqrt{0.02}=f(0.02)=f(0.01+0.01)=f(0.01)+\Delta y$

Substituting the values f(0.01) = 0.1 and Δy = 0.05, we get:

$\sqrt{0.02}=0.1+0.05=0.15$ .

**Hence, the value of the square root of 0.02 is approximately 0.15.**

**Note:**The smaller the value of Δx, the better the approximation.

This process can be repeated infinitely many times to get a closer value of the function at a given point.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE