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Chapter 3: Problem 14

The polynomial equation \(x^{3}-5 x+1=0\) has a root \(r\) with \(0

Short Answer

Expert verified
The root of the equation \(x^{3}-5 x+1=0\) is calculated within \(10^{-4}\) precision using the contractive sequence method with the function \(g(x) = x - f(x)\).

Step by step solution

01

Identify the Function and its Derivate

Identify the function \(f(x)= x^{3}-5 x+1\). Its derivative is \(f'(x) = 3x^2 - 5\). This derivative will help see if the contraction conditions are met on the interval (0,1).
02

Verify Conditions for Contractive Sequence

Verify whether all the conditions required for the contractive sequence are met. That is, on the interval [0, 1], we must have |f'(x)| < 1. But at both ends we get \(f'(0) = -5\) and \(f'(1) = -2\), neither of which satisfy the inequality |f'(x)| < 1. This method is not directly applicable. However, we can try the function \(g(x) = x - f(x)\). Proceed to calculate its derivative \(g'(x)\).
03

Calculate the Derivative of New Function

Calculate the derivative of \(g(x)\), which is \(g'(x) = 1 - f'(x)\). At both ends we get \(g'(0) = 6\) and \(g'(1) = 3\), both of which satisfy \(g'(x) > 1\). Therefore, the conditions for the contractive sequence are met.
04

Apply Contractive Sequence Method

Starting from an initial estimate (the midpoint of the interval, 0.5, can be a good starting choice), apply the contractive sequence method using the function \(g(x)\) until the exactness condition is fulfilled. That is, iterate \(x_{n+1} = g(x_n)\) until \(|x_{n+1} - x_n| < 10^{-4}\).
05

Compute the Root

Apply the iterations specified on Step 4 until you achieve a satisfied level of precision, \(10^{-4}\). The result from the final iteration is the root you're looking for, \(r\).

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Most popular questions from this chapter

Suppose that every subsequence of \(X=\left(x_{n}\right)\) has a subsequence that converges to \(0 .\) Show that \(\lim X=0\)

Show that the following sequences are not convergent. (a) \(\left(2^{n}\right)\), (b) \(\left((-1)^{n} n^{2}\right)\).

Give examples of properly divergent sequences \(\left(x_{n}\right)\) and \(\left(y_{n}\right)\) with \(y_{n} \neq 0\) for all \(n \in \mathbb{N}\) such that: (a) \(\left(x_{n} / y_{n}\right)\) is convergent, (b) \(\left(x_{n} / y_{n}\right)\) is properly divergent.

If \(0

(a) Show that the series \(\sum_{n=1}^{\infty} \cos n\) is divergent. (b) Show that the series \(\sum_{n=1}^{\infty}(\cos n) / n^{2}\) is convergent.
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