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Chapter 15: Problem 43

Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. $$z=3 \cos (2 x+y) ;[-2,2] \times[-2,2]$$

Short Answer

Expert verified
Question: Sketch the level curves of the given function $$z=3 \cos (2 x+y)$$ in the window \([-2, 2] \times [-2, 2]\) and label at least two of the level curves with their z-values. Answer: When plotting the level curves for the given function and window, we find three level curves represented by the equations: \(y = -2x + \frac{\pi}{2}\) (z = 0), \(y = -2x\) (z = 3), and \(y = -2x + \pi\) (z = -3). When labeling at least two of the curves, we can label the lines for z = 0 and z = 3: the line \(y = -2x + \frac{\pi}{2}\) with "z = 0" and the line \(y = -2x\) with "z = 3".

Step by step solution

01

1. Identify the function and the window

We are given the function: $$z=3 \cos (2 x+y)$$ and the window is \([-2, 2] \times [-2, 2]\), meaning that the x values range from \(-2\) to \(2\) and the y values range from \(-2\) to \(2\).
02

2. Set z to different constant values

To create level curves, we'll set z to different constant values and solve for y in terms of x. Let's set z to \(0\), \(3\), and \(-3\). For the three z-values, we have the equations: 1. \(0 = 3 \cos(2x + y)\) 2. \(3 = 3 \cos(2x + y)\) 3. \(-3 = 3 \cos(2x + y)\)
03

3. Solve for y in terms of x for each z-value

We'll solve for y in terms of x for each equation: 1. \(0 = 3 \cos(2x + y) \Rightarrow y = -2x + \arccos(0) = -2x + \frac{\pi}{2}\) 2. \(3 = 3 \cos(2x + y) \Rightarrow y = -2x + \arccos(1) = -2x\) 3. \(-3 = 3 \cos(2x + y) \Rightarrow y = -2x + \arccos(-1) = -2x + \pi\)
04

4. Plot the level curves on the window

Now that we have the equations for the level curves, we'll plot them on the window with x-values ranging from \(-2\) to \(2\) and y-values ranging from \(-2\) to \(2\): 1. \(y = -2x + \frac{\pi}{2}\) (z = 0) 2. \(y = -2x\) (z = 3) 3. \(y = -2x + \pi\) (z = -3) We should see three lines on the graph, each line representing a specific z-value.
05

5. Label the level curves with their z-values

Finally, we'll label at least two of the level curves with their corresponding z-values. For example, we can label the lines for z = 0 and z = 3: 1. Label the line \(y = -2x + \frac{\pi}{2}\) with "z = 0" 2. Label the line \(y = -2x\) with "z = 3" With the level curves plotted and labeled, we have completed the exercise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
Graphing functions is a way to visually represent equations or relationships between variables. A graph shows how one variable changes in response to another. In this exercise, you use the concept of level curves to represent the function in a specific region.Level curves are lines on a graph where the function value, or "z-value," is constant. For example, if the function is given by \( z = 3 \cos(2x + y) \), you can visualize the curves on a graph where points have the same output value of \( z \).
Let's simplify further with some tips:
  • Identify the axes: The x and y axes are used to represent input variables, and the output, \( z \), is the height or value for the function.
  • Select the window: You have a range of x and y values. In this case, they both range from \(-2\) to \(2\).
  • Plot points: For each chosen z-value, solve the corresponding equation to draw the level curve.
Using these steps helps you understand the spatial structure of mathematical functions.
Trigonometric Functions
Trigonometric functions like sine and cosine are vital in mathematics and graphing. They describe the relationships between angles and sides of triangles, but, as seen here, they can also model periodic behavior and oscillations.In the equation \( z = 3 \cos(2x + y) \), the cosine function appears. Here鈥檚 how it works:
  • Cosine cycles between -1 and 1, making the function periodic. The graph repeats itself in regular intervals.
  • The angles \( (2x + y) \) determine the phase shift, modifying where the cycle starts.
  • Multiplying by 3 changes the amplitude, which scales the height of the wave, resulting in z-values from -3 to 3.
Understanding these characteristics allows you to predict and plot behaviors of functions involving trig functions.
Solving Equations
When faced with an equation, it's crucial to isolate the variable you are solving for. In level curve plotting, you set \( z \) to fixed values and solve for \( y \) in terms of \( x \).Consider these steps for solving:
  • Set \( z \) values like 0, 3, or -3. These are our target outputs.
  • Use trigonometric properties to isolate \( y \). For example, use \( \cos^{-1} \) or \( \arccos \), if applicable, to find specific angles.
  • Your goal is to express \( y \) using \( x \) such as \( y = -2x + \arccos(0) \).
Every solution corresponds to a particular level curve on your plot. Successfully solving the equations reveals the paths of these curves on the graph.

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