Question

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

Short answer

Answer: The function \(z = x^2 + y^2\) describes a paraboloid opening upwards. The level curves for various values of z are circles centered at (0, 0) with radii equal to the square root of the corresponding z-value. For example, for \(z = 1\), the level curve is a circle with a radius of 1; for \(z = 4\), the level curve is a circle with a radius of 2.

Step-by-step solution

Understand the function

For the function \(z=x^2+y^2\), we want to understand that it is describing a surface in 3-dimensional space. In this case, the surface looks like a paraboloid opening upwards.

Find the level curves

In order to find the level curves, we need to set \(z\) equal to a constant \(c\), so we have \(c = x^2 + y^2\). We can now rewrite this equation in the form of a circle, as follows: \(x^2 + y^2 = c\).

Determine the values of c

Since we are asked to graph "several" level curves and label at least two of them, we will choose four different \(c\) values: \(c = 1, 2, 4, 8\). For each of these values, we will have a circle with radius equal to the square root of the constant value.

Graph the level curves

Now that we have our level curves, we will graph them on the window \([-4,4] \times [-4,4]\). For \(c = 1\), we have a circle centered at \((0, 0)\) with a radius of \(1\). For \(c = 2\), we have a circle centered at \((0, 0)\) with a radius of \(\sqrt{2}\). For \(c = 4\), we have a circle centered at \((0, 0)\) with a radius of \(2\). For \(c = 8\), we have a circle centered at \((0, 0)\) with a radius of \(2\sqrt{2}\).

Label the level curves

After graphing the level curves, we will label at least two of them with their corresponding z-values. Label the circle for \(c = 1\) as a level curve of \(z = 1\) and the circle for \(c = 4\) as a level curve of \(z = 4\).

Key concepts

Paraboloid Surface

Imagine holding a satellite dish in your hands; its shape is similar to what mathematicians call a paraboloid surface. In calculus, this shape is the 3-dimensional graph of a specific type of equation, typically involving squared variables. The simplest example might be the equation \( z = x^2 + y^2 \). Here, the squared terms cause the graph to bend upwards away from the origin, creating a bowl-like shape that we call a paraboloid.

It is important to note that the paraboloid can open either up or down, depending on the signs of the squared terms. For instance, if we had \( z = -(x^2 + y^2) \), the paraboloid would open downwards, resembling an upside-down version of our initial shape. The curvature of the surface depicts how steeply the surface rises or falls as we move away from the origin.

Multivariable Calculus

In the wondrous world of multivariable calculus, unlike in single-variable calculus, we deal with functions that have more than one input. This means our functions can take in two variables like \(x\) and \(y\), and spit out a third variable, \(z\), creating surfaces in 3-dimensional space instead of just curves on a 2-dimensional plane. The function \(z = x^2 + y^2\) from our exercise is a classic example from multivariable calculus.

Exploring such functions gives us insight into changes in surfaces, optimization across multiple inputs, and even helps us compute areas and volumes in more complex shapes. The techniques we use are extended versions of differentiation and integration, but now with partial derivatives and multiple integrals due to the presence of more variables.

Graphing Level Curves

When graphing in 3D is a bit too mind-boggling, level curves come to the rescue. They are the 2-dimensional cross-sections of a 3D surface at various heights, basically slicing the surface horizontally at different values of \(z\). Returning to our example of \(z = x^2 + y^2\), setting \(z\) to be a constant value gives us the equation of a circle, \(x^2 + y^2 = c\), where \(c\) is the fixed height of the slice.

The radius of these circles is the square root of \(c\), so different values of \(c\) (such as 1, 2, 4, or 8) will give us circles of varying sizes but all centered at the origin. These level curves represent the bird’s-eye view of the paraboloid and are powerful tools for visualizing the 3D shape on a 2D plane.

3-Dimensional Space Representation

The leap from 2D to 3D can be both challenging and exhilarating. A 3-dimensional space introduces an entire new depth (literally!) to graphing. Instead of plotting points on a flat plane, we're placing them within a cube-like space. The key to mastering this art is understanding the coordinate system: namely, the \(x\), \(y\), and \(z\) axes.

In this space, functions like ours define a surface rather than a line. To visualize this, we often use projections, like shadow plots or contour maps (level curves). While it might seem complex, with practice, graphing in 3D can become as intuitive as sketching shapes on paper, and it's a beautiful but practical way to model real-life scenarios, from physical landscapes to electrical fields.