L>Graphs and also Level Curves

Wednesday, February 25

Graphs 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.
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Example 2. f(x,y) = x2-y2.
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Example 2. f(x,y) = e-x2-y2.
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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.
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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
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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.
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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.
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Example 6. The level curve of f(x,y) = x2-y2 are hyperbolas.
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Here they are presented at the appropriate heights.
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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.
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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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