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Chapter 5: Problem 5

Locate the first nontrivial root of \(\sin x=x^{3},\) where \(x\) is in radians. Use a graphical technique and bisection with the initial interval from 0.5 to \(1 .\) Perform the computation until \(\varepsilon_{a}\) is less than \(\varepsilon_{s}=2 \% .\) Also perform an error check by substituting your final answer into the original equation.

Short Answer

Expert verified
Following the step-by-step solution, the first nontrivial root of the equation \(\sin x = x^3\) is approximately found at \(x_{root}\) using the bisection method on the interval \([0.5, 1]\) until the approximate error, \(\varepsilon_a\), is less than 2%. After obtaining \(x_{root}\), error checking is done by substituting it back into the original equation.

Step by step solution

01

Graph the equation

Start by graphing the equation \(\sin x = x^3\). You can use any graphing calculator or software to get the plot. By looking at the graph, you will have an idea of the approximate location of the root, which lies between 0.5 and 1.
02

Define the necessary functions

Define the function, \(f(x) = \sin x - x^3\). We will apply the bisection method on this function, and the root of this function will be the required root of the given equation.
03

Apply the bisection method

Start by defining the interval, \([a,b]=[0.5,1]\). Calculate the value of \(f(a)\), \(f(b)\), and check if the root lies in this interval: \(f(a) \cdot f(b) < 0\). If true, proceed to the next step.
04

Calculate bisected value and error

Calculate the bisected value, \(c=\frac{a+b}{2}\), and update the interval. If \(f(c) \cdot f(a) < 0\), then the root lies in the interval \([a,c]\), so update \(b = c\). If \(f(c) \cdot f(a) > 0\), then the root lies in the interval \([c,b]\), so update \(a = c\). Calculate the approximate error, \(\varepsilon_a = \frac{|c_{new} - c_{old}|}{c_{new}} \cdot 100\). Continue the bisection process until it meets the stopping criteria: \(\varepsilon_a < \varepsilon_s = 2\%\).
05

Check for the final answer

Let's call the final value of \(c\) obtained after the bisection process as \(x_{root}\). Substitute \(x_{root}\) into the original equation \(\sin x = x^3\), and verify if the equality approximately holds. Follow these steps to find the first nontrivial root of the given equation and perform error checking.

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