A root of the equation x3 - 3x - 5 = 0 lies between 2 and 2.5. Its value after the first iteration as obtained by using Newton-Raphson method is _____. (Answer up to two decimal places)Correct answer is '2.33'. Can you explain this answer?

Question

Answer

Newton-Raphson Method to Find Root of Equation

The Newton-Raphson method is a numerical method to find the root of an equation.

It is based on the idea of linear approximation of the function at a given point.

The method starts with an initial guess of the root and then iteratively improves the guess until a desired level of accuracy is achieved.

The formula for Newton-Raphson method is: x1 = x0 - (f(x0)/f'(x0)), where x0 is the initial guess, f(x) is the given function, and f'(x) is its derivative.

Given Equation and Initial Guess

We are given the equation x3 - 3x - 5 = 0.

We also know that the root lies between 2 and 2.5.

Therefore, we can start with an initial guess of x0 = 2.

First Iteration

To find the root using Newton-Raphson method, we need to calculate the value of x1 using the formula x1 = x0 - (f(x0)/f'(x0)).

First, we need to calculate the derivative of the given function: f'(x) = 3x2 - 3.

Then, we can substitute x0 = 2 and f(x0) = 3 into the formula to get x1 = 2 - (3/9) = 2 - 1/3 = 1.67.

Second Iteration

We can continue the process by using x1 as the new initial guess.

Substituting x0 = 1.67 and f(x0) = -0.635 into the formula, we get x1 = 1.67 - (-0.635)/(3*1.672 - 3) = 2.329.

Final Answer

The value of x after the first iteration as obtained by using Newton-Raphson method is 2.33 (rounded to two decimal places).

This value is closer to the actual root of the given equation, which is approximately 2.094.

Further iterations can be done to improve the accuracy of the result.

Answer

Let initial value of x₀ of the root be 2.

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