Answer

Verified

373.2k+ views

**Hint:**We know that Newton- Raphson is a root finding algorithm which produces successively better approximations for roots or zero of real valued functions. We know that the iterative equation of Newton Raphson method is ${x_{n + 1}} = {x_n} - \dfrac{{f({x_n})}}{{f'({x_n})}}$. We should know that the first iteration is ${x_0}$, thus the formula is ${x_1} = {x_0} - \dfrac{{f({x_0})}}{{f'({x_0})}}$.

**Complete step by step solution:**

Here we have to find the value of $\sqrt {12} $ by Newton- Raphson’s method after the first iteration.

Let us assume $x = \sqrt {12} $. By squaring both the sides we have ${x^2} = {(\sqrt {12} )^2}$.

So we have ${x^2} = 12$. By taking the constant term to the left side of the equation we have

${x^2} - 12 = 0$.

Now the formula of the iterative equation of Newton Raphson method is ${x_{n + 1}} = {x_n} - \dfrac{{f({x_n})}}{{f'({x_n})}}$.

If $f(x) = {x^2} - 12$, then we need the derivative so we have $f'(x) = 2x$.

We will substitute $f(x) = {x^2} - 12$ in the formula, and it can be written as ${x_{n + 1}} = {x_n} - \dfrac{{f({x^2}_n - 12)}}{{2{x_n}}}$.

Now we have to find the first iteration i.e. $n = 0$, so we have ${x_1} = {x_0} - \dfrac{{f({x^2}_0 - 12)}}{{2{x_0}}} = \dfrac{{x_0^2 + 12}}{{2{x_0}}}$.

We know that $3 < \sqrt {12} < 4$ or it can be written as $\sqrt 9 < \sqrt {12} < \sqrt {16} $.

From this our first iteration i.e. ${x_0} = 3$. By putting this in the formula we have $\dfrac{{{3^2} + 12}}{{2 \times 3}} = \dfrac{{9 + 12}}{6}$.

It gives us the value $3.5$.

Now we put $3.5$ in the equation i.e. ${x_1} = 3.5 - \dfrac{{{{(3.5)}^2} - 12}}{{2 \times 3.5}}$.

On simplifying we have $3.5 - \dfrac{{12.25 - 12}}{7} \Rightarrow 3.5 - 0.0357 = 3.4643$.

**Hence the required value is ${x_1} = 3.4643$.**

**Note:**

We should note that the formula of first iteration is ${x_1} = {x_0} - \dfrac{{x_0^2 - A}}{{2{x_0}}} = \dfrac{{x_0^2 + A}}{{2{x_0}}}$, so if we have a guess ${x_0}$. We can get the more accurate value by calculating $\dfrac{{x_0^2 + A}}{{2{x_0}}}$, where $A$ is the number which square root we are trying to find as we did in the above solution. We should know that ${x_0}$ is our initial guess and ${x_1}$ is a more accurate one.

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

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

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

At which age domestication of animals started A Neolithic class 11 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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