Problem 37
Question
Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=3 \cos (2 x+y) ;[-2,2] \times[-2,2].$$
Step-by-Step Solution
Verified Answer
Question: Graph the given function \(z=3\cos(2x+y)\) in the window \([-2,2] \times [-2,2]\) using level curves and label at least two level curves with their corresponding z-values.
Answer: Graph the following two level curves: \(y = -2x\) with \(z=3\) and \(y = \pi - 2x\) with \(z=-3\).
1Step 1: Set up the level curve equation
Set the function equal to a constant \(c\) to create the level curve equation:
$$c=3\cos(2x+y).$$
We'll need to solve this equation for \(y\) to determine the relationship between \(x\) and \(y\).
2Step 2: Solve for y
Isolate \(y\) from the level curve equation:
$$y=\frac{\arccos(\frac{c}{3})}{2}-2x$$
Now we have the equation for the level curves. In order to graph these curves, we need to replace \(c\) with different values corresponding to the \(z\)-values.
3Step 3: Choose z-values for level curves and find their equations
Since we need two labeled level curves, let's choose two z-values: The maximum and minimum of the function \(z=3\cos(2x+y).\) The maximum value of \(3\cos(2x+y)\) occurs when \(\cos(2x+y) = 1\), giving \(z_{max} = 3(1) = 3\). The minimum value occurs when \(\cos(2x+y)=-1\), giving \(z_{min} = 3(-1)=-3\). By plugging \(c\) as these values into the equation from step 2, we obtain the following level curve equations for \(z=3\) and \(z=-3\):
For \(z=3\):
$$y=\frac{\arccos(1)}{2}-2x= -2x$$
For \(z=-3\):
$$y=\frac{\arccos(-1)}{2}-2x= \pi - 2x$$
4Step 4: Graph the level curves
Now that we have our level curve equations, we can graph them in the given window \([-2,2] \times [-2,2]\).
1. For \(z=3\), our level curve equation is \(y = -2x\). You can draw this straight line with a slope of -2 and a y-intercept of 0.
2. For \(z=-3\), our level curve equation is \(y = \pi - 2x\). You can draw this straight line with a slope of -2 and a y-intercept of \(\pi\).
Additionally, you can also graph more level curves by choosing other \(z\)-values and using the same approach.
Lastly, don't forget to label the level curves with their corresponding z-values.
Key Concepts
Trigonometric FunctionsGraphingFunctions
Trigonometric Functions
Trigonometric functions are a foundational concept in mathematics, primarily dealing with the relationships between the angles and sides of triangles. They are used in a wide range of applications, from modeling periodic phenomena such as sound and light waves to solving problems in physics and engineering. Here, we'll focus on the cosine function, which is part of the set of primary trigonometric functions such as sine and tangent.
The cosine function, denoted as \( \cos(\theta) \), measures the cosine of an angle \( \theta \). It is a periodic function, which means it repeats its values at regular intervals, specifically every \( 2\pi \) radians or 360 degrees. In our exercise, we have the function \( z = 3 \cos(2x + y) \). This indicates that the cosine function has a frequency that is scaled by a factor of 2, effectively doubling the rate at which it completes its cycle. This can significantly affect how its level curves are distributed in the graph.
Understanding the properties of the cosine function, such as its maximum and minimum values of 1 and -1 respectively, helps in solving problems involving level curves. When the cosine function is multiplied by 3, like in our function, the range extends to achieve maximum and minimum values of 3 and -3. This is crucial for identifying and labeling the level curves correctly.
The cosine function, denoted as \( \cos(\theta) \), measures the cosine of an angle \( \theta \). It is a periodic function, which means it repeats its values at regular intervals, specifically every \( 2\pi \) radians or 360 degrees. In our exercise, we have the function \( z = 3 \cos(2x + y) \). This indicates that the cosine function has a frequency that is scaled by a factor of 2, effectively doubling the rate at which it completes its cycle. This can significantly affect how its level curves are distributed in the graph.
Understanding the properties of the cosine function, such as its maximum and minimum values of 1 and -1 respectively, helps in solving problems involving level curves. When the cosine function is multiplied by 3, like in our function, the range extends to achieve maximum and minimum values of 3 and -3. This is crucial for identifying and labeling the level curves correctly.
Graphing
Graphing is a crucial skill in visualizing mathematical phenomena, enabling us to understand how equations relate to real-world contexts or abstract concepts. For the given function \( z = 3\cos(2x + y) \), graphing helps us visualize level curves, which represent points on a surface where the function's output is constant.
To graph the level curves, we start by finding the relationships between \( x \) and \( y \) that satisfy the equation for a constant \( z \)-value. In our case, we determined that two key level curves occur at \( z = 3 \) and \( z = -3 \) due to the cosine function's range. For these specific values, the derived equations are \( y = -2x \) and \( y = \pi - 2x \) respectively.
These equations describe straight lines on the Cartesian plane within the window of \([-2,2] \times [-2,2]\). Each line has a slope of -2, meaning for every unit increase in \( x \), \( y \) decreases by 2 units. The line \( y = -2x \) passes through the origin, while \( y = \pi - 2x \) indicates a parallel line shifted vertically by \( \pi \). By plotting these lines, we can effectively showcase the level curves and visually label them with their corresponding \( z \)-values, thus enhancing our understanding of the function's behavior.
To graph the level curves, we start by finding the relationships between \( x \) and \( y \) that satisfy the equation for a constant \( z \)-value. In our case, we determined that two key level curves occur at \( z = 3 \) and \( z = -3 \) due to the cosine function's range. For these specific values, the derived equations are \( y = -2x \) and \( y = \pi - 2x \) respectively.
These equations describe straight lines on the Cartesian plane within the window of \([-2,2] \times [-2,2]\). Each line has a slope of -2, meaning for every unit increase in \( x \), \( y \) decreases by 2 units. The line \( y = -2x \) passes through the origin, while \( y = \pi - 2x \) indicates a parallel line shifted vertically by \( \pi \). By plotting these lines, we can effectively showcase the level curves and visually label them with their corresponding \( z \)-values, thus enhancing our understanding of the function's behavior.
Functions
In mathematics, a function is a relation between inputs and outputs, where each input is related to exactly one output. This concept is pivotal in describing mathematical models and solving equations. In our exercise, the function we are dealing with is \( z = 3 \cos(2x + y) \), where \( z \) is the output depending on the inputs \( x \) and \( y \).
When we talk about functions, we often discuss their domains (the set of all possible inputs) and ranges (the set of possible outputs). Here, the domain is defined by the window \([-2,2] \times [-2,2]\), while the range is determined by the multiplier 3 of the cosine function, giving outputs between -3 and 3. This makes it a bounded function, as its outputs do not extend beyond these values.
Level curves of a function like \( z = 3 \cos(2x + y) \) are particularly useful in understanding the function's behavior across a surface. They allow us to visualize sets of points where the function produces the same output, helping us to identify patterns and relationships between variables. By setting the function equal to constants, we derive simple equations representing these curves, making complex relationships easier to interpret and analyze.
When we talk about functions, we often discuss their domains (the set of all possible inputs) and ranges (the set of possible outputs). Here, the domain is defined by the window \([-2,2] \times [-2,2]\), while the range is determined by the multiplier 3 of the cosine function, giving outputs between -3 and 3. This makes it a bounded function, as its outputs do not extend beyond these values.
Level curves of a function like \( z = 3 \cos(2x + y) \) are particularly useful in understanding the function's behavior across a surface. They allow us to visualize sets of points where the function produces the same output, helping us to identify patterns and relationships between variables. By setting the function equal to constants, we derive simple equations representing these curves, making complex relationships easier to interpret and analyze.
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