Question

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

Short answer

Question: Graph several level curves of the function $$z=2x-y$$ within the window $$[-2, 2] \times [-2, 2]$$ and label at least two level curves with their z-values. Answer: In the window $$[-2, 2] \times [-2, 2]$$, the level curves for the function $$z=2x-y$$ are as follows: 1. For $$c=-2$$: $$y = 2x+2$$ (level curve with $$z = -2$$) 2. For $$c=0$$: $$y = 2x$$ (level curve with $$z = 0$$) 3. For $$c=2$$: $$y = 2x-2$$ (level curve with $$z = 2$$) 4. For $$c=4$$: $$y = 2x-4$$ (level curve with $$z = 4$$) We have labeled the level curves representing $$z = 0$$ and $$z = 2$$.

Step-by-step solution

Set z equal to a constant and solve for y

For the given function $$z=2x-y$$, let's set z equal to a constant $$c$$. This gives us the equation $$c=2x-y$$. We need to solve this equation for y in terms of x and the constant c. So, we have $$y = 2x - c$$.

Choose values for c and plot the level curves

Now, we can choose some different values for c and plot the level curves in the window $$[-2,2] \times [-2,2]$$. Let's choose the following values for c: $$c = -2, 0, 2, 4$$. For each of these values, the obtained level curves equations are as follows: 1. For $$c=-2$$: $$y = 2x+2$$ 2. For $$c=0$$: $$y = 2x$$ 3. For $$c=2$$: $$y = 2x-2$$ 4. For $$c=4$$: $$y = 2x-4$$ Now, plot these lines within the given window.

Label at least two level curves with their z-values

We need to label at least two level curves with their z-values. Let's pick the level curves representing $$c = 0$$ and $$c = 2$$ and label them on the graph. For the level curve representing $$c = 0$$, the equation is $$y = 2x$$. Label the curve with its corresponding z-value: $$z = 0$$. For the level curve representing $$c = 2$$, the equation is $$y = 2x-2$$. Label the curve with its corresponding z-value: $$z = 2$$. This completes the graphing of several level curves of the function $$z=2x-y$$ within the given window and labeling at least two of them with their z-values.

Key concepts

Multivariable Calculus

Embarking on the study of multivariable calculus opens a vast landscape of mathematical concepts that extend the principles of single-variable calculus to multiple dimensions. Multivariable calculus explores functions of two or more variables, which is essential when dealing with phenomena in three-dimensional space or in any context where multiple factors influence an outcome. For instance, when considering the temperature at various points in a room, one must account for both the position within the room (defined by two variables: x and y coordinates) and time, adding a third variable to the function.

In this field, we deal with partial derivatives, multiple integrals, and more complex applications like vector fields. The function in our exercise, represented as z = 2x - y, is a simple example of a function with two input variables, x and y, and one output, z. The concept of level curves is a pivotal aspect of multivariable calculus, as it allows us to visualize functions and understand how their outputs change across different input values.

Plotting Level Curves

Plotting level curves, also known as contour lines, is a fundamental technique used to visualize functions of two variables. These curves connect points where the function has the same value, which is analogous to a topographical map illustrating elevations.

To plot level curves, you first set the output of the function (z-value) to a constant value. This transforms the equation into one involving only the two input variables (x and y). The result is a relationship that can be graphed on a 2D plane, showing a 'slice' of the function at that particular z-value. By choosing different constants, we obtain various slices, which together provide a more comprehensive picture of how the function behaves across its domain. This visualization method proves extremely useful in applications ranging from engineering to economics, where it can represent cost efficiency or the potential energy across a field.

Solving for 'y'

When working with functions of two variables in the form z = f(x, y), 'solving for y' involves rearranging the equation so that y is on one side and everything else on the other. This rearrangement is particularly useful when plotting level curves because it allows us to get a clear equation in terms of one variable with the other being a parameter.

In the context of our example, we've turned the original equation z = 2x - y into y = 2x - c by setting z to a constant value, c. Solving for y in terms of x simplifies plotting since each level curve can now be represented as a simple linear equation and graphed as a straight line in the x-y plane. This step is critical for both visualizing the function and understanding the relationship between its variables.

Graphing Functions

Graphing functions is a visual interpretation of mathematical equations that describe relationships between variables. It's an indispensable tool in mathematics, allowing us to consider abstract concepts in a more tangible form. When graphing functions with two variables, we typically use a coordinate plane, with one axis for each variable.

In the case of the function z = 2x - y, since we have solved for y, we can treat x as the independent variable and y as the dependent variable, plotting them accordingly on the x and y axes. After plotting the points that satisfy the equation for each level curve, we draw lines through those points, and these lines represent the level curves. By doing this for several different z-values, we can create a complete 'contour map' of the function, which can tell us a lot about how the output z changes in response to changes in x and y. Finally, labeling the curves with their respective z-values gives us a clear understanding of what each curve represents in terms of the function's output.