Math 200 Notes

Math 200 Notes - CALCULUS 4 QUIZ #2 REVIEW / SPRING 09 (1.)...

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CALCULUS 4 QUIZ #2 REVIEW / SPRING 09 (1.) Determine the following about the given quadric surfaces . (a.) Identify & Sketch the quadric surface: z x 2 4 y 2 9 + = . In planes parallel to the xz-plane and planes parallel to the yz-plane, the traces are parabolic. Thus, this a parabaloid. In planes parallel to the xy-plane, the traces are ellipses. Therefore, this an elliptic paraboloid This "sketch" is computer-generated. z x y Elliptic Parabaloid Page 1 of 30
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This "sketch" is hand-generated. (b.) Identify & Sketch the quadric surface the quadric surface: zy 2 x 2 = . In planes parallel to the xz-plane and planes parallel to the yz-plane, the traces are parabolic. Thus, this a parabaloid. In planes parallel to the xy-plane, the traces are hyperbolas. Therefore, this a hyperbolic paraboloid This "sketch" is computer-generated. Page 2 of 30
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x y z Hyperbolic Parabaloid This "sketch" is hand-generated. Page 3 of 30
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(c.) Identify & Sketch the quadric surface the quadric surface: x 2 y 2 + z 2 16 = . In planes parallel to the xz-plane and planes parallel to the yz-plane, the traces are hyperbolic. Thus, this a hyperboloid. In planes parallel to the xy-plane, the traces are circles. This, nonetheless, is called a hyperboloid of 1 sheet. This "sketch" is computer-generated. z x y Hyperboloid of One Sheet This "sketch" is hand-generated. Page 4 of 30
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(d.) Identify & Sketch the quadric surface: x 2 y 2 + z 2 0 = . In planes parallel to the xz-plane and planes parallel to the yz-plane, the traces are straight lines. Thus, this a cone. In planes parallel to the xy-plane, the traces are circles. Therefore, this is a circular cone, which is a special case of the elliptic cone. This "sketch" is computer-generated. Page 5 of 30
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x y z Elliptic(Circular) Cone This "sketch" is hand-generated. Page 6 of 30
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(e.) Identify & Sketch the quadric surface: z 2 x 2 y 2 1 = . In planes parallel to the xz-plane and planes parallel to the yz-plane, the traces are hyperbolic. Thus, this a hyperboloid. In planes parallel to the xy-plane, the traces are circles. Since z 0 > , this a hyperboloid of 2 sheets. This "sketch" is computer-generated. x y z Hyperboloid of Two Sheets Page 7 of 30
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This "sketch" is hand-generated. (f.) Identify & Sketch the quadric surface: x 2 y 2 + z 2 4 + 1 = . In planes parallel to the xz-plane, planes parallel to the yz-plane, and planes parallel to the xy-plane the traces are all ellipses (circular in planes parallel to the xy-plane which is a specical case of ellitic). Thus, this a specicialized ellipsoid ( prolate sphereoid.). This "sketch" is computer-generated. Page 8 of 30
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x y z Ellipsoid(Prolate Sphereoid) This "sketch" is hand-generated. Page 9 of 30
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(a.1.) Find traces in the coordinate planes of the quadric surface: x 2 9 y 2 25 + z 2 4 + 1 = and sketch trace in the xyz-coordinate system.
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Math 200 Notes - CALCULUS 4 QUIZ #2 REVIEW / SPRING 09 (1.)...

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