### Wednesday, February 25

#### Graphs and Level Curves

The **graph** of a function f(x,y) is the set of all points
(x,y,z) in space such that (x,y) is in the domain of f and z=f(x,y).
Here are a few examples.**Example 1.** f(x,y) = x^{2}+y^{2}.

**Example 2.** f(x,y) = x^{2}-y^{2}.

**Example 2.** f(x,y) = e^{-x2-y2}.

**Example 3.** In an example on Monday, we looked at the cost function
C(x,y) = 1200/x+1200/y+3xy.
Its graph is shown below.

From the side view, it appears that the minimum value of this function is around 500.

**Example 4.**f(x,y)=2e^{-(x+1)2-y2}
+3e^{-(x-2)2-(y-1)2}-2e^{-(x-1)2-(y+2)2}

A **level curve** of a function f(x,y) is a set of points (x,y)
in the plane such that f(x,y)=c for a fixed value c.

**Example 5.** The level curves of f(x,y) = x^{2}+y^{2} are curves of the
form x^{2}+y^{2}=c for different choices of c.
These are circles of radius square root of c. Several of them are shown below.

One can think of the level curve f(x,y)=c as the
horizontal cross-section of the graph at height z=c. When each level curve f(x,y)=c is
plotted at a height of c units above the xy-plane, we get the figure below.

**Example 6.** The level curves of f(x,y) = x^{2}-y^{2}
are hyperbolas.

Here they are shown at the appropriate heights.

**Example 7.**
The level curves of f(x,y)=2e^{-(x+1)2-y2}
+3e^{-(x-2)2-(y-1)2}-2e^{-(x-1)2-(y+2)2}
are shown below. Comparing with the graph in Example 4, we see that the points (x,y) at which f
has maxima and minima are at the centers of circular level curves.

**Example 8.** Here are the level curves for the cost function C(x,y) = 1200/x+1200/y+3xy.
Comparing with the graph in Example 3, it appears that the
minimum value of C(x,y) is attained at about (7.5,7.5).