$$\def\ans#1{\bbox[border:1px solid green,6pt]{#1}}$$

# Functions of Two Variables and Their Graphs

## Graphing

We've been graphing curves in space using parametric equations, where x, y, and z vary with respect to t. Now we're going to switch gears a bit and look at graphs of functions like $z=f(x,y),$ where z is a function of x and y. In other words, each combination of x and y corresponds to a value of z. What will these graphs look like, in general? In two dimensions, we had $$y=f(x)$$, which described a curve in the xy plane, as each value of x corresponded to a value of y. Now, in three dimensions, a function $$z=f(x,y)$$ will trace out a surface in $$\mathbb{R}^3$$, where the height of the surface at any given point $$(x,y)$$ is $$z$$.

For example, consider $z=\dfrac{\sqrt{2x+y-2}}{x-2}.$ The domain of this function depends on where the denominator is 0 and where the square root in the numerator is negative. Specifically, the domain is $\ans{D = \{(x,y)\ :\ x \neq 2,\ y \geq -2x+2\}.}$ The graph is shown below (note that Matlab goes a bit wacky at $$x=2$$, the discontinuity.

###### Ellipsoid

Graph the following equation. Note that it is not actually a function, since it isn't one-to-one. $\dfrac{x^2}{9}+\dfrac{y^2}{16}+\dfrac{z^2}{25} = 1$

#### Solution

One way to graph an equation like this is to use traces, which are curves in a plane parallel to one of the coordinate planes. For instance, if we let $$z=0$$, we get a graph in the xy plane: $\dfrac{x^2}{9}+\dfrac{y^2}{16}+\dfrac{0^2}{25} = 1 \longrightarrow \dfrac{x^2}{9}+\dfrac{y^2}{16} = 1$ This is an ellipse with major and minor axes of 4 and 3. For other values of z, we graph ellipses in planes above or below $$z=0$$, until $$z=\pm 5$$, where the ellipse shrinks to a point.

The graph looks like the following (kind of like an M&M shape):

###### Paraboloid

Graph the following function. $z=f(x,y)=x^2+y^2$

#### Solution

Consider different values of z again:

$\begin{array}{l l l} z & \textrm{Curve} & \textrm{Description}\\ \hline -1 & x^2+y^2=-1 & \textrm{Impossible}\\ 0 & x^2+y^2=0 & \textrm{Point at the origin}\\ 1 & x^2+y^2=1 & \textrm{Circle of radius 1 centered at the origin}\\ 2 & x^2+y^2=2 & \textrm{Circle of radius 2 centered at the origin}\\ 4 & x^2+y^2=4 & \textrm{Circle of radius 4 centered at the origin}\\ 9 & x^2+y^2=9 & \textrm{Circle of radius 9 centered at the origin}\\ \end{array}$

We'll use Matlab to see the graph of this function:

        	>> fsurf(@(x,y) x.^2+y.^2)


This is called a paraboloid, and as the name suggests, it looks like a parabola in both directions.

The domain of this function is $D = \{(x,y)\ :\ -\infty < x < \infty,\ -\infty < y < \infty\}$ and the range is $R = \{z\ :\ 0 \leq z < \infty\}.$

## Level Curves

We've already used level curves; when we held z constant and described the resulting curve in a horizontal plane, we were finding paths where the elevation doesn't change. Think about a topographic map; it shows the level curves of terrain.

Describe the level curves of the following function. $f(x,y) = y-x^2-1$

#### Solution

If we hold the function constant ($$f(x,y)=z=z_0$$), we get curves like

$y-x^2-1=z_0 \longrightarrow y=x^2+1+z_0$

Since $$1+z_0$$ is a constant, these are parabolas.

The graph of the function and a few of the level curves are shown below.

Describe the level curves of the following function. $f(x,y) = 2+\sin (x-y)$

#### Solution

Hold $$z$$ constant:

$2+\sin (x-y)=z_0 \longrightarrow y=x-\sin^{-1}(z_0-2)$

Note that $$\sin^{-1}(z_0-2)$$ is a constant, so these are straight lines $$y=x-k$$.

The graph of the function is shown below, and you should be able to see that the function stays constant along straight lines.

## Examples

The electric potential function for two positive charges, one at $$(0,1)$$ with twice the charge of one at $$(0,1)$$ is given by

$\phi (x,y) = \dfrac{2}{\sqrt{x^2+(y-1)^2}} + \dfrac{1}{\sqrt{x^2+(y+1)^2}}$

First, we'll graph this function using Matlab:

        	>> fsurf(@(x,y) 2./sqrt(x.^2+(y-1).^2) + 1./sqrt(x.^2+(y+1).^2))


1. For what values of $$x$$ and $$y$$ is $$\phi$$ defined?
2. Is the electric potential greater at $$(3,2)$$ or $$(2,3)$$?
3. Describe how the electric potential varies along the line $$y=x$$

#### Solution

1. Domain: where is the denominator in each term 0? \begin{align} x^2+(y-1)^2 = 0 &\longrightarrow (0,1)\\ x^2+(y+1)^2 = 0 &\longrightarrow (0,-1) \end{align}

Therefore, the domain is every point except for these two (the potential approaches infinity near these points).

$D = \{(x,y)\ :\ \mathbb{R}^2 \setminus \{(0,1) \cup (0,-1)\}\}$
2. Plug these two points into the function: \begin{align} \phi (3,2) &= \dfrac{2}{\sqrt{9+1}} + \dfrac{1}{\sqrt{9+9}} \approx 0.868\\ \phi (2,3) &= \dfrac{2}{\sqrt{4+4}} + \dfrac{1}{\sqrt{4+16}} \approx 0.931 \end{align}

Therefore, the potential is greater at $$(2,3)$$.

3. Replace $$y$$ with $$x$$: \begin{align} \phi &= \dfrac{2}{\sqrt{x^2+(x-1)^2}} + \dfrac{1}{\sqrt{x^2+(x+1)^2}}\\ &= \dfrac{2}{\sqrt{2x^2-2x+1}} + \dfrac{1}{\sqrt{2x^2+2x+1}} \end{align}
###### Economics

The output $$Q$$ of an economic system subject to two inputs, such as labor $$L$$ and capital $$K$$, can be modeled by the Cobb-Douglas production function $Q (L,K) = cL^aK^b$ where $$a$$, $$b$$, and $$c$$ are positive real numbers.

Suppose $$a=\dfrac{1}{3}$$, $$b=\dfrac{2}{3}$$, and $$c=40$$. We can graph this with Matlab:

        >> fsurf(@(x,y) 40.*x.^(1/3).*y.^(2/3))


If $$L$$ is held constant at 10, find the dependence of $$Q$$ on $$K$$.

#### Solution

$Q=40(10^{1/3})K^{2/3}$

This is a trace along a vertical plane.

#### Try it yourself:

(click on a problem to show/hide its answer)

1. Find the domain of $$f(x,y) = 2xy-3x+4y$$.
2. $$D = \{(x,y)\ :\ -\infty < x < \infty,\ -\infty < y < \infty\}$$

3. Find the domain of $$f(x,y) = \dfrac{12}{y^2-x^2}$$.
4. $$D = \{(x,y)\ :\ y \neq \pm x\}$$

5. Find the domain of $$f(x,y) = \sqrt{25-x^2-y^2}$$.
6. $$D = \{(x,y)\ :\ x^2+y^2 \leq 25\}$$

7. Find the domain of $$f(x,y) = \ln(x^2-y)$$.
8. $$D = \{(x,y)\ :\ y < x^2\}$$

9. Describe the level curves of $$z=x-y^2$$.
10. These are horizontal parabolas $$x=y^2+z_0$$.

11. Describe the level curves of $$z=2x-y$$.
12. These are straight lines $$y=2x-z_0$$.

13. Describe the level curves of $$z=3\cos(2x+y)$$.
14. These are straight lines $$y=\cos^{-1}\left(\dfrac{z_0}{3}\right)-2x$$.