Question

Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of \(f\) is a value of \(x\) that satisfies the equation \(f(x)=x ;\) it corresponds to a point at which the graph of \(f\) intersects the line \(y=x .\) Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=2 x \cos x \text { on }[0,2]$$

Short answer

Question: Determine the approximate fixed points of the function \(f(x) = 2x \cos x\) on the interval \([0, 2]\). Answer: The approximate fixed points are \(x = 0\) and \(x = \pi\).

Step-by-step solution

Set up the equation

First, we'll set up the equation \(f(x) = x\). The given function is \(f(x) = 2x \cos x\). Therefore, we need to find the values of \(x\) that satisfy the equation \(2x \cos x = x\).

Solve the equation for x

To solve the equation from Step 1, i.e., \(2x \cos x = x\), we can try to isolate \(x\): $$2x \cos x = x$$ $$x(2 \cos x - 1) = 0$$ Now, observe that this is a transcendental equation and therefore we cannot solve it analytically. Instead, we have to use numerical methods to find where it has roots within the given interval. Graphing the function \(g(x) = 2 \cos x - 1\) can give us a good initial approximation of the roots to refine.

Check if the solutions are in the given interval

After graphing the function \(g(x) = 2 \cos x - 1\), we can see that there are two roots in the interval \([0, 2]\). One root is close to \(x = 0\), and the other is close to \(x = \pi\). We can further analyze these roots using numerical methods such as the bisection method or Newton-Raphson method to find more accurate approximations of the fixed points. To summarize, the fixed points of the function \(f(x) = 2x \cos x\) on the interval \([0, 2]\) are approximately \(x = 0\) and \(x = \pi\).

Key concepts

Transcendental Equations

Transcendental equations may sound complex, but they are simply equations involving a transcendental function. A transcendental function includes non-algebraic operations, such as exponential, logarithmic, or trigonometric functions. In our exercise, the function is given as \(f(x) = 2x \cos x\), where the trigonometric part \(\cos x\) makes it transcendental.

Solving transcendental equations analytically is typically difficult or impossible. Unlike polynomial equations, which have well-defined algebraic solutions, transcendental equations often require alternative approaches. This is because it's hard to isolate \(x\) on one side due to the nature of transcendental components. In our case, the equation \(2x \cos x = x\) simplifies to \(x(2 \cos x - 1) = 0\). This setup guides us to look for solutions where either the \(x\) part or \(2 \cos x - 1\) becomes zero. However, solving it purely analytically is not feasible, hence we rely on other methods to find solutions.

Graphical Analysis

Graphical analysis is a powerful tool to visually assess the behavior of functions, especially when dealing with transcendental equations. It helps in identifying fixed points where the function \(f(x) = x\).

In our example, we're looking for intersections between the graph of \(f(x) = 2x \cos x\) and the line \(y = x\). Drawing both these graphs within the interval \([0, 2]\) aids in visually detecting intersecting points. Here's why graphical analysis is valuable:

• Quickly identifies approximate solutions or roots.
• Easy to visualize where the functions meet, making initial guesses clearer.
• Facilitates interpretation of how the functions behave near intersections, guiding numerical methods.

After graphing, the fixed points roughly appear around \(x = 0\) and \(x = \pi\). Such visual analysis can then be paired with numerical methods for refinement.

Numerical Methods

When analytical solutions are hard to derive, numerical methods come to the rescue to solve transcendental equations. They offer techniques for finding approximate solutions to equations that are difficult or impossible to solve exactly.

Common numerical methods include:

• The Bisection Method: This involves bracketing a root between two values and repeatedly bisecting the interval until it converges to a solution.
• Newton-Raphson Method: A more advanced technique using tangent lines at a guessed point to find successively better approximations.

For the equation \(x(2 \cos x - 1) = 0\), these methods help zero in on the fixed points. Numerical methods excel because:

• They provide a systematic approach to refining initial approximations.
• They adapt well to computer programming, making them efficient and scalable.
• They often have a very high degree of precision when configured and applied correctly.

By applying these methods, we ascertain our fixed points at approximately \(x = 0\) and \(x = \pi\), reaffirming what was initially observed through graphical analysis.