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Unformatted text preview: come to another equation where it doesn’t equal 0. 4) Hyperboloid of One Sheet Important to Note here that at z=0 we have an ellipse or circle trace curve
5) Hyperboloid of Two Sheets Important to Note that at z=0 we have NO SOLUTION. There is no graph on the xyaxis. Good Hint that you’re looking at
a TWO sheet hyperbola.
6) Hyperbolic Paraboloid 7) Cylinders
a) Plane b) Elliptic c) Parabolic d) Hyperbolic NOTES ON THE 7 SURFACES:
2 are easy – The ellipsoid and the sphere
All three variables are squared and all are positive when on the same side of an equation
All three traces are ellipses or circles
Paraboloids  2 types
Elliptic Paraboloid Hyperbolic Paraboloid Both have parabolas parallel to two of the coordinate planes
The third plane either has an Ellipse or a Hyperbola – given it its name
2 variables are squared and One is not z = (if both x and y positive then elliptic paraboloid
If one negative then hyperbolic paraboloid)
Hyperboloids – 3 types
All variables squared and ONE of them is Negative when all on same side of equation
= 1 (1 sheet) = 0 (Cone)
Elliptic Hyperboloids = 1 (2 sheets) The traces have hyperbolas for two of the coordinate planes and ellipses for the other Are all variables
squared? No
Yes then you have a
Paraboloid. (Solve for
the nonsquared
variable) Are all the variables
positive? Yes No then you have an
ellipsoid then you have a
hyperboloid Are the squared
variables both positive? When all three variables
are on one side of the
equation, what does the
equation equal =1 , then you have a
hyperboloid of 1 sheet Yes No then you have an Elliptic
Parabolid then you have a
Hyperbolic Paraboloid =0 then you have a
Cone = 1 then you have a
hyperboloid of 2 sheets 12.2: Graphs and Level Curves
Graph a surface by hand using Trace Curves
Interpreting Level Curves
Examples of Functions of Several Variables:
Chemistry: PV = nRT P(pressure) V(volume) n(no. of moles of gas present) R(constant) T(temperature) Economics: R = Px R(revenue) P(price $/item)...
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 Spring '08
 Loukianoff,V
 Calculus

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