Find an interval containing a solution to the equation $2x=\cos \left(x\right)$. Use a graphing utility to approximate the solution.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, it must also take on any value between those two points at least once. This theorem is essential for finding intervals where solutions to equations exist, as it guarantees that if the function changes signs over an interval, there is at least one root in that interval.

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate plane to visualize its behavior. In this context, graphing the functions y = 2x and y = cos(x) allows us to identify points of intersection, which represent the solutions to the equation 2x = cos(x). This visual approach can help approximate the solution and understand the relationship between the two functions.

Continuous Functions

A continuous function is one that does not have any breaks, jumps, or holes in its graph. Both y = 2x and y = cos(x) are continuous functions, which is crucial for applying the Intermediate Value Theorem. Understanding continuity helps in determining the behavior of functions and ensuring that solutions can be found within specified intervals.