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Chapter 5: Problem 10

If the Bisection Method is used on an interval of length 1 to find \(p_{n}\) with error \(\left|p_{n}-c\right|<10^{-5}\), determine the least value of \(n\) that will assure this accuracy.

Short Answer

Expert verified
The least value of \(n\) that assures the accuracy of \(|p_n - c| < 10^{-5}\) is 17.

Step by step solution

01

Definition of the Bisection Method's Error

To begin solving this exercise, it's important to recall the formula that estimates the error \(E\) of the Bisection Method. If \(a\) and \(b\) are the endpoints of the starting interval, with \(b > a\), and \(c\) is the actual root that lies between \(a\) and \(b\), then the error at the \(n\)th step is approximately \(E = |p_n - c| = \frac{|b - a|}{2^n}\), where \(p_n\) is the root approximated at the \(n\)th step.
02

Set Up the Error Inequality

Given in the problem that \(|p_n - c| < 10^{-5}\), we can set up an inequality using the formula from step 1: \(\frac{|b - a|}{2^n} < 10^{-5}\). Additionally, we are told that the length of the interval \(|b - a|\) is 1.
03

Solve the Inequality for \(n\)

Substitute \(|b - a|\) with 1 into the inequality to get \(\frac{1}{2^n} < 10^{-5}\). Then, take the natural logarithm of both sides to solve for \(n\). This gives: \({n > log_{2}(10^{5})}\)
04

Find the Least Value of \(n\)

By calculating the logarithm, we get \(n > 16.60964047443\). As \(n\) must be an integer, round it up to get the least value, so \(n = 17\) is the minimum number of iterations needed.

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Most popular questions from this chapter

Let \(\delta\) be a gauge on \(I:=[a, b]\) and suppose that \(I\) does not have a \(\delta\) -fine partition. (a) Let \(c:=\frac{1}{2}(a+b)\). Show that at least one of the intervals \([a, c]\) and \([c, b]\) does not have a \(\delta\) -fine partition. (b) Construct a nested sequence \(\left(I_{n}\right)\) of subintervals with the length of \(I_{n}\) equal to \((b-a) / 2^{n}\) such that \(I_{n}\) does not have a \(\delta\) -fine partition. (c) Let \(\xi \in \cap_{n=1}^{\infty} I_{n}\) and let \(p \in \mathbb{N}\) be such that \((b-a) / 2^{p}<\delta(\xi)\). Show that \(I_{p} \subseteq[\xi-\delta(\xi), \xi+\delta(\xi)]\), so the pair \(\left(I_{p}, \xi\right)\) is a \(\delta\) -fine partition of \(I_{p}\)

Show that if \(f: A \rightarrow \mathbb{R}\) is continuous on \(A \subseteq \mathbb{R}\) and if \(n \in \mathbb{N}\), then the function \(f^{n}\) defined by \(f^{n}(x)=(f(x))^{n}\) for \(x \in A\), is continuous on \(A\).

Show that the function \(f(x):=1 /\left(1+x^{2}\right)\) for \(x \in \mathbb{R}\) is uniformly continuous on \(\mathbb{R}\).

Show that the equation \(x=\cos x\) has a solution in the interval \([0, \pi / 2]\). Use the Bisection Method and a calculator to find an approximate solution of this equation, with error less than \(10^{-3} .\)

If \(g(x):=\sqrt{x}\) for \(x \in[0,1]\), show that there does not exist a constant \(K\) such that \(|g(x)| \leq K|x|\) for all \(x \in[0,1]\). Conclude that the uniformiy continuous \(g\) is not a Lipschitz function on \([0,1]\).
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