Wednesday, February 25Graphs and also Level CurvesThe graph 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.
You 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.
See more: D&Amp;D 3.5 Attack Of Opportunity, Vitamin D Toxicity: What If You Get Too Much
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).