Question

Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. $$z=\sqrt{x^{2}+4 y^{2}} ;[-8,8] \times[-8,8]$$

Short answer

Question: Graph the function \(z=\sqrt{x^2+4y^2}\) within the given window \([-8, 8] \times [-8, 8]\) by plotting and labeling at least two level curves. Solution: The level curves of the given function are found from the equation \(c^2 = x^2 + 4y^2\), where \(c\) is the constant z-value. We chose two z-values \(z=4\) and \(z=8\), which correspond to the level curve equations \(16 = x^2 + 4y^2\) and \(64 = x^2 + 4y^2\), both representing ellipses centered at the origin. We then plot these curves within the window \([-8, 8] \times [-8, 8]\) and label the first level curve with its z-value \(z=4\) and the second level curve with its z-value \(z=8\).

Step-by-step solution

Identify level curves

A level curve is a curve where the function has a constant z-value. In this case, we are looking for curves \(f(x,y) = z\) such that \(z = \sqrt{x^2 + 4y^2}\) is constant.

Find equations for the level curves

To find equations for the level curves, we will set \(z\) to a constant value, say \(c\). So we have: $$c=\sqrt{x^2+4y^2}$$ To simplify the equation, we can square both sides to get rid of the square root: $$c^2 = x^2 + 4y^2$$ Now we have the equation of the level curves in terms of \(x\) and \(y\).

Choose the values of \(z\) to plot

We need to choose a few z-values to plot the level curves. Let's choose \(z=4\) and \(z=8\).

Determine the equations for the chosen z-values

For the first chosen z-value \(z=4\), substitute it in the equation: $$4^2 = x^2 + 4y^2$$ Which simplifies to: $$16 = x^2 + 4y^2$$ For the second chosen z-value \(z=8\), substitute it in the equation: $$8^2 = x^2 + 4y^2$$ Which simplifies to: $$64 = x^2 + 4y^2$$

Plot the level curves

Now that we have the equations for the chosen level curves, we can plot them on the coordinate plane within the window \([-8, 8] \times [-8, 8]\). - The first level curve has the equation \(16 = x^2 + 4y^2\), which represents an ellipse centered at the origin with radii 4 and 2. - The second level curve has the equation \(64 = x^2 + 4y^2\), which represents another ellipse centered at the origin but with radii 8 and 4.

Label the level curves with their z-values

After plotting the level curves in the previous step, we need to label each level curve with its corresponding z-value. The first level curve corresponds to \(z=4\), and the second level curve corresponds to \(z=8\). Label each curve with their respective z-values.

Key concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, curves, and more. It is essential for visualizing mathematical functions and their properties.

• The plane is divided into four quadrants by two perpendicular lines called axes – the x-axis (horizontal) and the y-axis (vertical).
• The point where these axes intersect is the origin, which has coordinates (0, 0).
• Each point on the coordinate plane is defined by an ordered pair of numbers (x, y), representing its position along the x and y axes, respectively.

In the context of level curves, the coordinate plane helps us to understand the positional relationships of these curves as we change the z-value.

It provides a clear visual framework to determine where exactly the plotted curves lie with respect to each other.

Ellipses

Ellipses are closed shapes that look like stretched circles. They play a significant role when dealing with level curves like those in the given exercise.

• An ellipse has two main axes – a major axis and a minor axis. The major axis is the longest diameter, and the minor axis is the shortest.
• In the context of this exercise, when we plot level curves with equations like \(x^2 + 4y^2 = c^2\), these curves form ellipses.

The equations derived from replacing the z-value in the level curve equation give us the specific dimensions of these ellipses.

For example, when \(z=4\), we have the equation \(x^2 + 4y^2 = 16\), which forms an ellipse centered at the origin with radii 4 and 2 on the x and y axes, respectively.

Moreover, the ellipses reflect how different values of z stretch or shrink the size of the ellipse. This variability helps illustrate the function's behavior over different z-levels.

z-values

In mathematics, z-values are often used to represent the output of a function in a three-dimensional context, although in level curves these are held constant to depict a two-dimensional shape.

• The specific constant z-values determine the shape and size of the level curves plotted on a coordinate plane.
• For instance, choosing z-values such as 4 and 8 in this exercise helps illustrate how the level curve alters.”

When you plug these z-values into the level curve equation, you derive unique ellipses.

By observing how each of these ellipses corresponds to a different z-value, we better understand the function's "height" or "depth" at that point, though manifesting on a two-dimensional plane.

In diagrams, labeling these curves with their z-values elevates our comprehension, associating them directly with their mathematical representation.