Also it is very important to notice that this is

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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 xy-axis. 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 non-squared 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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