While finding the roots of f x - x 2-4-0 using Newton - Raphson method- initial value of x - x 1-1- If the value obtained after first iteration is x 2- find -100 - x 2- where - is the greatest integer function-

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Newton Raphson Method

While finding...

Question

While finding the roots of $f (x) = x^{2} - 4 = 0$ using Newton - Raphson method, initial value of (x, x₁ = 1). If the value obtained after first iteration is $x_{2}$ .

Open in App

Solution

The Newton-Raphson method uses an iterative process to approach one root of a function. The specific root that the process locates depends on the initial, arbitrarily chosen x-value.
$x_{n + 1} = x_{n} - \frac{f (x_{n})}{f^{'} (x_{n})}$
Here, x_n is the current known x-value, (f(x_n)) represents the value of the function at $x_{n}$ , and $f^{'} (x_{n})$ is the derivative (slope) atx_n.
$x_{n + 1}$ represents the next x-value that you are trying to find.

Here, we have $f (x) = x^{2} - 2$ and $x_{1} = 1$
$f^{'} (x) = 2 x$ $\Rightarrow f (x_{1}) = f (1) = 1 - 2 = - 1$
$f^{'} (x_{1}) = f^{'} (1) = 2$
$\Rightarrow x_{2} = x_{1} - \frac{f (x_{1})}{f^{'} (x_{1})}$
$= 1 - (\frac{- 1}{2}) = 1 + 0.5 = 1.5$
Thus, the approximate root is $1.5$ .

Suggest Corrections

Similar questions

While finding the roots of $f (x) = x^{2} - 4 = 0$ using Newton - Raphson method, initial value of (x, x₁ = 1). If the value obtained after first iteration is x₂, find [100. x₂], where [ ] is the greatest integer function.

Q. Solve the equation

x = e^{- x}

using Newton-Raphson method. Starting with an initial guess values

x_{0} = 0

, the value of x after first iteration is .

Q. The root of the function

f (x) = x^{3} + x - 1

obtaind after first iteration on application of Newton Raphson scheme using an initial guess of

x_{0} = 1

Q. Newton-Raphson method is used to compute a root of the equation

2 x^{2} + 5 x - 9 = 0

with 4.5 as the initial value. The approximation after one iteration is ________

Q. The solution of the equation given by

2 x^{3} + 5 x + A = 0

, (where A is a constant) is determined by Newton-Raphson method. First assumption for root of the equation is

x_{0} = 2

If after one iteration the root obtained is

x_{1} = 3.5

, then the value of A is____

Join BYJU'S Learning Program

While finding the roots of f(x)=x2−4=0 using Newton - Raphson method, initial value of (x, x1 = 1). If the value obtained after first iteration isx2.

While finding the roots of $f (x) = x^{2} - 4 = 0$ using Newton - Raphson method, initial value of (x, x₁ = 1). If the value obtained after first iteration is $x_{2}$ .