Problem 54 - Precalculus with Limits

Question

In Exercises 49-58, use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \( [0, 2\pi) \). \( x \cos x - 1 = 0 \)

Step-by-Step Solution

Answer

The exact solutions of the equation can only be determined numerically. Use a graphing utility for an approximation, which will vary slightly depending on the accuracy of the utility.

Step 1: Graph the Equation

First, set up the graphing utility to display the equation. Plot the function \( x \cos x - 1 \) within the interval \( [0, 2\pi) \). The x-axis (where \( y = 0 \)) is important to visualize since we're looking for when the function intersects it.

Step 2: Identify Intersections with the X-Axis

Look for where the function intersects with the x-axis. These points represent the x-values where the given equation equals zero, and thus are the solutions. Each intersection represents one solution to the equation.

Step 3: Approximate the Solutions

Use the graphing utility to get the x-coordinates of those intersection points. Each of these values should be approximated to three decimal places as per exercise instruction.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos(x) \), is one of the fundamental trigonometric functions alongside sine and tangent. It is important in mathematics, especially in the study of periodic phenomena such as oscillations. The function describes the x-coordinate of a point on the unit circle as the angle, measured in radians, from the positive x-axis increases. Here's what to remember about cosine:

• It is periodic with a period of \(2\pi\).
• The range is between -1 and 1, meaning it stretches from -1 up to 1.
• The cosine of 0 is 1, and it forms a wave that keeps repeating itself.

When graphing an equation involving cosine, such as \( x \cos(x) \), it's crucial to consider this periodic nature and the domain given. In this exercise, the function \( x \cos(x) \) is modified by the multiplication with \( x \) so the specific behavior of the cosine function will influence how the graph changes, winding around the x-axis over the range \([0, 2\pi)\)."

Equation Solving

Equation solving involves finding the value of the variable that makes the equation well, equal. For the equation \( x \cos(x) - 1 = 0 \), you need to discover the values of \( x \) where this balance occurs. Here's a simple guide to approach such problems:

• Determine the components of the equation. Here, you have \(x\), \(\cos(x)\), and the constant term 1 modifying the equation.
• Identify the target of the equation, which means understanding that you are setting the function equal to zero.
• Use graphical or algebraic methods. For this specific case, a graphing utility was used to visually understand where the function crosses the x-axis representing the solutions.

Graphing the function helps in visually solving the equation, especially in complicated trigonometric cases like this one. The crossing points will be the x-values making the equation zero.

Interval Notation

Interval notation is a method of writing down a set of values easily and concisely, like a span of numbers. This is often used in calculus and algebra to describe the domain of a function or the solution to an equation. In mathematics, you will often see:

• Square brackets, \([\text{a, b}]\), indicating that the endpoints are included in the interval, often used for closed intervals.
• Round brackets, \((\text{a, b})\), indicate that the endpoints are not included, used for open intervals.

In this exercise, the interval \([0, 2\pi)\) indicates that the solutions should be considered from 0 to \(2\pi\), including 0 but excluding \(2\pi\). Recognizing the interval is important as it affects how solutions are interpreted and ensures they remain within a specified range.