**the a role f(x,y) is the collection of every points (x,y,z) in an are such that (x,y) is in the domain the f and also z=f(x,y). Here are a couple of examples.**

### Wednesday, February 25

Graphs and also Level CurvesThe graphYou are watching: Z=x^2-y^2

**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 role C(x,y) = 1200/x+1200/y+3xy. Its graph is shown below.

**From the next view, it appears that the minimum value of this duty 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**that a duty f(x,y) is a set of point out (x,y) in the aircraft such that f(x,y)=c for a resolved value c.

**Example 5.**The level curve of f(x,y) = x2+y2 room curves of the type x2+y2=c for different choices of c. These space circles the radius square root of c. Several of castle are displayed below.

**One have the right to think of the level curve f(x,y)=c together the horizontal cross-section of the graph at elevation z=c. Once each level curve f(x,y)=c is plotted in ~ a height of c units above the xy-plane, we obtain the number below. Example 6.**The level curve of f(x,y) = x2-y2 are hyperbolas.

**Here they are presented at the appropriate heights.Example 7.**The level curve of f(x,y)=2e-(x+1)2-y2+3e-(x-2)2-(y-1)2-2e-(x-1)2-(y+2)2 are presented below. Comparing v the graph in instance 4, we view that the point out (x,y) in ~ which f has maxima and minima are at the centers of one level curves.

**Example 8.**

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right here are the level curves because that the cost role C(x,y) = 1200/x+1200/y+3xy. Comparing through the graph in example 3, it appears that the minimum worth of C(x,y) is attained at about (7.5,7.5).

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