2128 match the equation with its graph labeled iviii

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Unformatted text preview: ■ ■ ■ 15y 2 y 2 ■ ■ z 1 for 1 z 2 x 2 5z 2 100 xy ■ ■ ■ sx 2 ■ y2 2. 42. Sketch the region bounded by the paraboloids z x2 y2 2 y. 43. Find an equation for the surface obtained by rotating the parabola y y x 2 41. Sketch the region bounded by the surfaces z 2z 2 z2 II 0 ■ 5y 2 and z I ■ 0 4 4z 2y ■ 0 0 2z 2y 2x ■ 20 4z2 |||| Use a computer with three-dimensional graphing software to graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface. ■ 23. x 2 z ■ 4z 36 3z 2 ; 37–40 21–28 21. x 2 y ■ 4x y2 24z 4y 16 y z2 36. x 4x 2 4z2 ■ x 2y 2 32. 4 x 4 z2 35. x 2 100 4y 2 z 30. x 2 36 3z 2 y2 ■ 17. x 2 9y2 2y 2 33. 4 x 2 y2 16. 25 y 2 4 4x2 31. x z2 x2 14. z 29. z 2 44. Find an equation for the surface obtained by rotating the line y x x 2 about the y-axis. x 3y about the x-axis. 45. Find an equation for the surface consisting of all points that z III z IV are equidistant from the point Identify the surface. 1, 0, 0 and the plane x 1. 46. Find an equation for the surface consisting of all points P for y x which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface. y x z V y y x z VII z VI x 47. Show that if the point a, b, c lies on the hyperbolic parabo- z VIII loid z y 2 x 2, then the lines with parametric equations x a t, y b t, z c 2 b a t and x a t, y b t, z c 2 b a t both lie entirely on this paraboloid. (This shows that the hyperbolic paraboloid is what is called a ruled surface; that is, it can be generated by the motion of a straight line. In fact, this exercise shows that through each point on the hyperbolic paraboloid there are two generating lines. The only other quadric surfaces that are ruled surfaces are cylinders, cones, and hyperboloids of one sheet.) 48. Show that the curve of intersection of the surfaces x 2 2y 2 z2 lies in a plane. x x ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 1 and 2 x 2 4y 2 2z2 5y 0 x 2 y 2 and z 1 y 2 on a co...
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