Problem 51 - Calculus: One and Several Variables

Question

Solve the equation \(f(x)=y_{0}\) for \(x\) in \([0,2 \pi]\) by using a graphing utility. Display the graph of \(f\) and the line \(y=y_{0}\) in one figure; then use the trace function to find the point(s) of intersection. $$f(x)=\sin 3 x ; \quad y_{0}=-1 / \sqrt{2}$$

Step-by-Step Solution

Answer

The solutions to the equation \(f(x)=y_{0}\) are approximately at \(0.38\pi\), \(1.12\pi\), \(1.62\pi\), \(2.38\pi\), \(2.88\pi\), and \(3.62\pi\) within the range [0, \(2\pi\)].

Step 1: Graph the Function

First, graph the function \(f(x) = \sin 3x\) using a graphing calculator. Make sure the x-axis range is set from 0 to \(2\pi\).

Step 2: Plot the Line \(y = y_{0}\)

Now, plot the horizontal line \(y=y_{0}=-1 / \sqrt{2}\) on the same graph. The graphing utility should allow the addition of this constant value line.

Step 3: Identify Intersection Points

Identify the point(s) of intersection between the function \(f(x) = \sin 3x\) and the line \(y=-1 / \sqrt{2}\) by using the trace function offered by the graphing utility. These points are the solutions to the equation \(f(x) = y_{0}\). In this case there should be six intersection points in interval [0, \(2\pi\)].

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that can aid in visualizing and solving complex mathematical problems. Most graphing calculators allow users to input functions, like trigonometric equations, and visualize their plots on a coordinate grid. To use a graphing calculator effectively, start by entering the equation you wish to graph. In the case of trigonometric functions, it is vital to ensure that the calculator is set to the correct mode (radians or degrees) based on the function's input requirements.

For example, when solving the equation given in the exercise, you would first enter \(f(x) = \sin 3x\) into the calculator. It's important to adjust the viewing window to include the x-axis range from 0 to \(2\pi\), affording a complete view of the function's behavior within the interval. Once the function is graphed, you can then proceed to input the horizontal line \(y = y_0\) for comparison. Understanding how to navigate and input data into a graphing calculator is key to analyzing functions and their properties.

Sine Function

The sine function, usually written as \(\sin(x)\), is one of the fundamental trigonometric functions derived from a right-angled triangle or unit circle. It relates the angle of a triangle to the ratio of the length of the opposite side to the hypotenuse. When graphed, the sine function produces a wave-like pattern known as a sinusoid, which repeats every \(2\pi\) radians, or 360 degrees.

In the context of the exercise, the sine function has been modified to \(f(x) = \sin 3x\), which indicates that the function oscillates three times faster than the basic sine wave. This means it will complete three full cycles within the interval \(0, 2\pi\). The amplitude (height of the wave) of the sine function is one, but it can be scaled up or down by multiplying the function by a coefficient. The result is a function that oscillates above and below the x-axis, intersecting it at regular intervals that correspond to multiples of \(\pi/3\) in this particular case.

Points of Intersection

Points of intersection are the points at which two or more graphs coincide on the coordinate plane. In the process of solving trigonometric equations graphically, points of intersection are the x-values where the graph of the trigonometric function crosses a particular y-value (or another function). These points represent the solutions to the equation, revealing the x-values that satisfy the equation for the given y-value.

For the exercise, the points of intersection between the graph of \(f(x) = \sin 3x\) and the line \(y = -1 / \sqrt{2}\) are of interest. These points occur where the value of \(\sin 3x\) equals \( -1 / \sqrt{2}\), and there can be multiple such points within a given interval. Finding these x-values is equivalent to solving the equation \(f(x) = y_0\) within the specified range of \(x\). To visualize these solutions, the intersection points can be located using a graphing calculator's trace or intersection-finding feature.

Tracing Function

The tracing function on a graphing calculator is a useful feature that enables a user to 'trace' along the graph of a function to observe the coordinates of various points on the graph, including the points of intersection. To use this feature, you would typically move a cursor along the curve of the graph, and the calculator will display the x and y coordinates of the cursor's location in real-time.

When tracing the function \(f(x) = \sin 3x\), the y-coordinate on the calculator will match the actual value of \(f(x)\) at the given x-position. By tracing along the function until the y-coordinate matches \(y_0 = -1 / \sqrt{2}\), the x-coordinate at that point is one of the solutions to the given equation. It's crucial to trace all possible points where the function meets the line \(y = y_0\) to identify every solution. In the current exercise, there are six distinct solutions to \(f(x) = \sin 3x\) that correspond to the x-values at which \(f(x)\) intersects the line \(y = -1 / \sqrt{2}\) within the interval \(0, 2\pi\).