Explanation The graph of second degree equation in x , y , and z in space is said to be a quadric surface. The general form of quadric surfaces is given by A x 2 + B y 2 + C z 2 + D z = E , where A , B , C , D , and E are constants. Ellipsoids, paraboloids, elliptical cones and hyperboloids are some basic quadric surfaces. For example, consider the ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 . The cross section of the ellipsoid by the plane z = z 0 will be an ellipse as shown below. From Figure 1, it is observed that the ellipse x 2 a 2 + y 2 b 2 = 1 lie in the xy -plane, x 2 a 2 + y 2 b 2 = z c lie in the yz -plane and x 2 a 2 + z 2 c 2 = 1 lie in the xz -plane. Now consider the example of elliptical paraboloid whose equation is given by x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 . The cross section of elliptical paraboloid by the plane z = c will be parabola and ellipse as shown below. From Figure 2, it is observed that the parabola z = c a 2 x 2 lie in the xz -plane, z = c b 2 y 2 lie in the yz -plane and ellipse x 2 a 2 + y 2 b 2 = 1 lie in the plane z = c . Consider the example of elliptical cone whose equation is given by x 2 a 2 + y 2 b 2 = z 2 c 2 . The cross section of elliptical cone by the plane z = c will be line and ellipse as shown below. From Figure 3, it is observed that the line z = c b y lie in the yz -plane, z = c a x lie in the xz -plane and ellipse x 2 a 2 + y 2 b 2 = 1 lie in the plane z = c

University Calculus: Early Transcendentals Plus MyLab Math -- Access Card Package (3rd Edition) (Integrated Review Courses in MyMathLab and MyStatLab)

3rd Edition

ISBN: 9780321999573

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 11, Problem 17GYR

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To explain: The quadric surfaces and give example for ellipsoids, paraboloids, cones, and hyperboloids.

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Chapter 11, Problem 16GYR

Chapter 11, Problem 1PE

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University Calculus: Early Transcendentals Plus MyLab Math -- Access Card Package (3rd Edition) (Integrated Review Courses in MyMathLab and MyStatLab)

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