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Chapter 4: Problem 33

Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=\cos 2 x-x^{2}+2 x$$

Short Answer

Expert verified
Question: Using the given function $$f(x) = \cos 2x - x^2 + 2x$$, find all of its roots. Answer: Follow the step-by-step solution provided to analyze the function, graph it, and apply the Newton-Raphson method to find the accurate roots. After obtaining the roots, double-check the values by plugging them back into the original function to ensure that they yield a result very close to 0. The roots you find are the solution to this exercise.

Step by step solution

01

Preliminary analysis of the function

First, let's analyze the given function: $$f(x)=\cos 2 x-x^{2}+2 x$$ We can observe that this function has a combination of trigonometric, quadratic, and linear components. Typically, roots of such combined functions are difficult to find algebraically, so obtaining good initial approximations and using numerical techniques becomes important.
02

Graphing the function

Using a graphing software or calculator, we can plot the function, which will help us visualize the roots and provide a rough approximation of their values. When graphing the function, it is helpful to set an appropriate domain for the x-axis, considering the cosine term oscillates between -1 and 1. After graphing the function, we should be able to see where the curve intersects the x-axis (the function's roots). Take note of these points as they will serve as our initial approximations.
03

Using numerical methods to find the roots

Now that we have the initial approximations for the roots, we can use numerical methods to find their accurate values. Some common numerical methods for finding roots are the Bisection Method, Newton-Raphson Method, and Secant Method. We'll describe the Newton-Raphson method for this exercise, but the other methods can be used as well. First, we need to derive the function's first derivative: $$f'(x)=-2\sin 2x - 2x + 2$$
04

Applying the Newton-Raphson method

To use the Newton-Raphson method, we'll iterate over the following formula until we reach the desired level of accuracy: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ Start with the initial approximations we found during the graphing step and iterate the Newton-Raphson formula until the difference between consecutive x-values is smaller than a chosen tolerance (e.g., 1e-6).
05

Obtaining the roots of the function

After applying the Newton-Raphson method (or another numerical method) and reaching the desired level of accuracy, we'll have the roots of the function. Double-check the values by plugging them back into the original function to ensure that they yield a result very close to 0. The roots found using this process are the solution to the given exercise.

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