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Chapter 6: Problem 24

Approximate the solution of the equation \(x=\cos x\), accurate to within six decimals.

Short Answer

Expert verified
Yes, the short answer would be the final numerical value you get for \(x\) following the steps above; this value will approximate the equation \(x = \cos{x}\) to six decimal places. Unless using a computational aid, the exact numerical value can't be given in this format.

Step by step solution

01

Initialize the solution

Start by initializing with an initial value. This initial value is often a guess. We can take \(x_{0}=0\) as our initial value because the cos of 0 is 1, which will be a positive start towards our solution.
02

Apply the Iterative Method

Start applying the iterative method, that is, calculate new \(x\) as \(x_{n+1} = \cos{x_{n}}\). You will need to calculate this repetitively, increasing \(n\) by 1 each time you calculate, so for the first calculation \(n\) will be 0.
03

Repeat the Iteration till the Desired Accuracy is Reached

Continue to find each new \(x_{n+1} = \cos{x_{n}}\) until the absolute difference between two consecutive \(x\) is less than or equal to the desired error, for which six decimal places of accuracy has been specified. This means you keep calculating until \(\left|x_{n+1} - x_{n}\right| \le 0.000001\). The newer calculated value of \(x\) will be your approximate solution for the equation \(x = \cos{x}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Iteration Methods
Iteration methods are a powerful tool in numerical analysis, helping us solve problems when a direct solution is challenging or impossible. These methods involve improving an initial guess to approach an exact solution progressively. Here, we focus on an important iteration method for equations: Fixed-Point Iteration.
  • **Starting Point**: Iteration requires beginning with an initial approximation, often referred to as a 'guess'.
  • **Iteration Process**: Apply a function repeatedly to refine the guess. In fixed-point iteration to solve an equation like \( x = \cos x \), we calculate successive values: \( x_{n+1} = \cos x_n \).
  • **Convergence**: This is key in iteration methods. It refers to the process where successive iterations get closer to the actual solution.
  • **Stopping Criteria**: Choose a threshold for acceptable iteration error to decide when to stop. In this example, the goal is to have a six decimal point accuracy.
Iteration methods can be computationally efficient and important in real-world applications, such as solving scientific and engineering problems. They are appreciated for their simplicity and practicality.
Equation Solving
Solving equations is a fundamental aspect of mathematics. It involves finding the value of a variable that makes the equation true. With non-linear equations, like what's seen here with \( x = \cos x \), direct algebraic solutions aren't available. Thus, numerical methods, like iteration, come into play.
  • **Types**: Equations can be linear or non-linear. Non-linear equations often require numerical approaches due to their complexity.
  • **Numerical Methods**: Techniques like the Newton-Raphson method, Secant method, and Fixed-Point iteration provide ways to find approximate solutions efficiently.
  • **Importance**: Numerically solving equations is crucial in fields where exact solutions are necessary but complex functions are involved, such as physics, engineering, and economics.
This exercise exemplifies using a numerical approach for a trigonometric non-linear equation. It highlights the necessity and practicality of these methods when algebraic ones are not viable.
Cosine Function
The cosine function is a principal part of trigonometry, frequently occurring in various scientific and engineering fields. It is periodic, smooth, and ranges from -1 to 1.
  • **Properties**: Key features include being an even function \( \cos(-x) = \cos(x)\), having a fundamental period of \(2\pi\), and amplitude 1.
  • **Applications**: The cosine function models wave patterns, oscillations, and other phenomena in physics like light and sound waves.
  • **Graph Behavior**: Its graph is a wave starting at \(1\) at \(x=0\) and descending to zero, reaching -1 at \(x=\pi\).
In the exercise \(x = \cos x\), it shows the intersection of the cosine curve with a linear function, a practical example of applying cosine in solving non-linear equations through numerical methods. Understanding the behavior of this function aids in controlling and predicting iteration outcomes.

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Most popular questions from this chapter

Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable at \(c\) and that \(f(c)=0 .\) Show that \(g(x):=|f(x)|\) is differentiable at \(c\) if and only if \(f^{\prime}(c)=0\).

If \(r>0\) is a rational number, let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x):=x^{r} \sin (1 / x)\) for \(x \neq 0\), and \(f(0):=0 .\) Determine those values of \(r\) for which \(f^{\prime}(0)\) exists.

Let \(g: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(g(x):=x+2 x^{2} \sin (1 / x)\) for \(x \neq 0\) and \(g(0):=0 .\) Show that \(g^{\prime}(0)=1\), but in every neighborhood of 0 the derivative \(g^{\prime}(x)\) takes on both positive and negative values. Thus \(g\) is not monotonic in any neighborhood of \(0 .\)

Approximate the real zeros of \(g(x):=x^{4}-x-3\)

Suppose that \(I \subseteq \mathbb{R}\) is an open interval and that \(f^{\prime \prime}(x) \geq 0\) for all \(x \in I\). If \(c \in I\), show that the part of the graph of \(f\) on \(I\) is never below the tangent line to the graph at \((c, f(c))\)
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