Question

While finding the roots of $f\left(x\right)=x^2-4=0$ 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.

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Answer 1

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.
$x_{n+1}=x_n-\frac{f\left(x_n\right)}{f^{\prime }\left(x_n\right)}$
Here, x_n is the current known x-value, (f(x_n)) represents the value of the function at xn, and f'(xn) is the derivative (slope) at x_n. (x_n+1) represents the next x-value that you are trying to find.

Here, we have $f\left(x\right)=x^2-2$ and (xn = 1)\) $f^{\prime }\left(x\right)=2x$ $\Rightarrow f\left(xn\right)=f\left(1\right)=-1)$ $\left(f^{\prime }\right(x_n)=f^{\prime }(1)=2)$ $\Rightarrow x_2=x_1-\frac{f\left(x_1\right)}{f^{\prime }\left(x_1\right)}$ $=1-\frac{-1}{2}=1.5$, $100.x_2=150$ ([150] = 150)