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) = x2+y2.

Example 2. f(x,y) = x2-y2.

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) = x2+y2 are curves of the form x2+y2=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) = x2-y2 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).