Problem 58 - Single Variable Calculus

Question

In Exercises \(55-58\), plot the graph of the function. $$ h(x)=2 \sin x+3 \cos 2 x+\sin 3 x, \quad-2 \pi

Step-by-Step Solution

Answer

The graph of the function \(h(x) = 2\sin{x} + 3\cos{2x} + \sin{3x}\) on the interval \(-2\pi < x < 2\pi\) can be drawn by analyzing the individual trigonometric functions, dividing the interval into segments of width \( \frac{\pi}{2}\), calculating function values \(h(x_i)\) at each point \(x_i\), and plotting the points. The approximate graph passes through the points \((-2\pi, 3), (-\frac{3}{2}\pi, -5), (-\pi, -3), (-\frac{1}{2}\pi, -2), (0, 3), (\frac{1}{2}\pi, 0), (\pi, 3), (\frac{3}{2}\pi, 0)\).

Step 1: Analyze the trigonometric functions

We have three trigonometric functions in the sum: \( 2\sin{x}, 3\cos{2x}, \) and \( \sin{3x} \). The first function, \( 2\sin{x} \), has an amplitude of 2 and a period of \( 2\pi \). The second function, \( 3\cos{2x} \), has an amplitude of 3 and a period of \( \pi \), due to the "2" in the argument. Finally, the third function, \( \sin{3x} \), has an amplitude of 1 and a period of \( \frac{2\pi}{3} \), again due to the "3" in the argument. The overall function's behavior will depend on the interaction of these individual functions.

Step 2: Divide the interval

Given that the smallest period among the individual functions is \( \pi \), it is a good idea to divide the interval into segments of width \( \frac{\pi}{2} \). This will ensure that we have enough points for an accurate graph. The points where we calculate the function values will be \(x_i = -2\pi + \frac{i\pi}{2}\), where \(i = 0, 1, 2, 3, 4, 5, 6, 7\).

Step 3: Calculate function values

Now let's calculate the function values, \(h(x_i)\), for each \(x_i\), using the given function: 1. \( h(-2\pi) = 2\sin{-2\pi} + 3\cos{(-4\pi)} + \sin{(-6\pi)} = 0 + 3(1) + 0 = 3 \) 2. \( h(-\frac{3}{2}\pi) = 2\sin{-\frac{3}{2}\pi} + 3\cos{(-3\pi)} + \sin{-3\pi} = -2 + (-3) + 0 = -5 \) 3. \( h(-\pi) = 2\sin{-\pi} + 3\cos{(-2\pi)} + \sin{(-3\pi)} = 0 + 3(-1) + 0 = -3 \) 4. \( h(-\frac{1}{2}\pi) = 2\sin{-\frac{1}{2}\pi} + 3\cos{(-\pi)} + \sin{-\frac{3}{2}\pi} = +2+(-3)+(-1) = -2 \) 5. \( h(0) = 2\sin{0} + 3\cos{0} + \sin{0} = 0 + 3(1) + 0 = 3 \) 6. \( h(\frac{1}{2}\pi) = 2\sin{\frac{1}{2}\pi} + 3\cos{\pi} + \sin{\frac{3}{2}\pi} = 2 + (-3) + 1 = 0 \) 7. \( h(\pi) = 2\sin{\pi} + 3\cos{2\pi} + \sin{3\pi} = 0 + 3(1) + 0 = 3 \) 8. \( h(\frac{3}{2}\pi) = 2\sin{\frac{3}{2}\pi} + 3\cos{3\pi} + \sin{\frac{9}{2}\pi} = 2 + (-3) + 1 = 0 \)

Step 4: Plot the points and draw the graph

Now, let's plot these points on a graph: \((-2\pi, 3), (-\frac{3}{2}\pi, -5), (-\pi, -3), (-\frac{1}{2}\pi, -2), (0, 3), (\frac{1}{2}\pi, 0), (\pi, 3), (\frac{3}{2}\pi, 0)\). Draw a smooth curve through these points to get an approximate graph of the function \(h(x) = 2\sin{x} + 3\cos{2x} + \sin{3x}\) on the interval \( -2\pi \lt x \lt 2\pi \).

Key Concepts

Graphing

Graphing trigonometric functions involves displaying the behavior of these functions as visual representations. It helps in understanding how these functions behave over intervals. The process usually starts by identifying key characteristics of the function such as amplitude, period, and phase shifts if present.

In the exercise, we have the function \[h(x) = 2\sin{x} + 3\cos{2x} + \sin{3x}, \]which combines three distinct trigonometric components. Before plotting, it is important to find critical points where the function values change noticeably. These points are key to accurately mapping the function's behavior across the interval from \(-2\pi\) to \(2\pi\).

By calculating values at strategic points, and then plotting these on a graph, a smooth curve that accurately represents the function's nature within the given domain can be constructed.

Amplitude and Period

Amplitude and period are crucial aspects when working with trigonometric functions. The amplitude indicates the vertical stretch or compression of the graph, essentially the 'height' from the midline, while the period determines how often the function repeats itself over a specified span.

For the function we are graphing, \(2\sin{x}\) has an amplitude of 2 and repeats every \(2\pi\). The component \(3\cos{2x}\) has an amplitude of 3, with a shorter period of \(\pi\) because the 2 in \(2x\) compresses the wave horizontally. The last part, \(\sin{3x}\), carries an amplitude of 1 and a period of \(\frac{2\pi}{3}\).

These differing amplitudes and periods result in a complex graph where each component influences the overall shape. Understanding these properties allows us to anticipate the function's peaks and troughs and the frequency at which they occur.

Function Behavior Analysis

Analyzing the behavior of combined trigonometric functions requires examining how each individual function adds to or counterbalances the others. Each function contributes oscillations that can align or interfere with one another. The task is to decipher how these oscillations interact over the interval.

In our example, the different periods of \(2\sin{x}\), \(3\cos{2x}\), and \(\sin{3x}\) mean their waves pass through peaks and valleys at different intervals. By calculating function values, you can predict points where all terms might enhance or cancel each other out, allowing a clearer picture of maximum and minimum values in the resultant wave.

Such analysis highlights how superposition of waves occurs, a principle widely seen in fields like physics and engineering.

Plotting Techniques

Using effective plotting techniques ensures a precise graphical representation of functions, especially when they are a combination of multiple trigonometric forms.

• Determine the critical points: Identify intervals based on the smallest period—ensuring enough detail is captured across the graph. For this function, we use intervals aligned with the smallest period \(\frac{\pi}{3}\).
• Calculate and plot key points: Evaluate the function at these strategic points. In our case, this is done at increments of \(\frac{\pi}{2}\), capturing critical behavior within \(-2\pi\) to \(2\pi\).
• Draw smooth curves: Connect points smoothly to reflect the rhythmic nature of trigonometric functions, avoiding sharp turns.

By following these steps, we can depict complex functions in ways that are both accurate and informative. Remember, the blend of different trigonometric terms creates dynamic graphs, and each plotting decision can greatly affect the visual clarity of the function.