{Use of Tech} 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) = cos x

Verified step by step guidance

1. Understand the concept of a fixed point: A fixed point of a function f(x) is a value x such that f(x) = x. For the function f(x) = cos(x), we need to find x such that cos(x) = x.
2. Graph the function y = cos(x) and the line y = x on the same set of axes. The points where these two graphs intersect are the fixed points of the function.
3. Perform a preliminary analysis by considering the behavior of the cosine function. Note that cos(x) oscillates between -1 and 1, while the line y = x is a straight line with a slope of 1. This suggests that fixed points, if they exist, will be within the interval [-1, 1].
4. Use a numerical method, such as the fixed-point iteration method, to approximate the fixed points. Start with an initial guess within the interval [-1, 1], such as x_0 = 0.5, and iterate using the formula x_{n+1} = cos(x_n) until the values converge to a stable point.
5. Verify the solution by checking if the value obtained satisfies the equation cos(x) = x to a desired level of accuracy. This can be done by substituting the value back into the equation and ensuring the left and right sides are approximately equal.

Key Concepts

Fixed Points

A fixed point of a function f is a value x such that f(x) = x. This means that when the function is applied to this value, it returns the same value. Graphically, fixed points are where the graph of the function intersects the line y = x. Identifying fixed points is crucial in various applications, including iterative methods and stability analysis.

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate system to visualize its behavior. This technique helps in identifying key features such as intercepts, maxima, minima, and fixed points. For the function f(x) = cos x, graphing allows us to see where the curve intersects the line y = x, providing insights into the location of fixed points.

Preliminary Analysis

Preliminary analysis refers to the initial examination of a function to gather information about its behavior before performing detailed calculations. This can include evaluating the function at specific points, analyzing its derivatives, and understanding its general shape. For f(x) = cos x, preliminary analysis can help estimate where fixed points might be located, guiding further investigation.