Apply Newton’s method to estimate the solution of x3 − x − 1 = 0 by taking x1 = 1 and finding the least n such that xn and xn + 1 agree to three digits after the decimal point.

Solution :

Given

Let the initial approximation is

So by Newton's method, we get

are identical up to eight decimal places.

The approximate real root is x ≈ 1.32471795

∴ x = 1.32471795

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step-by-step explanation: