Final answer:
Using the Newton-Raphson method with the function f(x) = 2x - cos(x) and an starting guess of x₀ = 1, the value of x after the third iteration can be found. The correct procedure involves keeping at least five decimal places during calculations and rounding off only the final answer to ensure accuracy.
Step-by-step explanation:
The Newton-Raphson method is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. In this case, we are given f(x) = 2x - cos(x) with an initial guess x₀ = 1, and we want to find the value of x after the 3rd iteration, rounded to five decimal places.
To perform the Newton-Raphson method, we need the derivative of f(x), which is f'(x) = 2 + sin(x). The iteration formula is given by x₁ = x₀ - (f(x₀) / f'(x₀)). Each iteration should improve the value of x and should be carried out with at least five decimals before rounding, to maintain accuracy. When the third iteration value is obtained, it should be rounded off to five decimal places, according to the rules of indicating whether to round up or down based on the digit following the last significant figure.
Rounding off is crucial in reporting the final answer but should be applied only at the end of the calculation to avoid significant numerical errors. If the digit right after the last significant figure is greater than or equal to 5, we round up the last significant digit; if it's lower than 5, we leave it as it is. Hence, the correct 3rd iteration value of x would be obtained by following the Newton-Raphson iteration formula with the correct rounding at the end.