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MATH 230 Lecture Notes - Lecture 5: Paraboloid, Quadric

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6 Mar 2018

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E(cid:395)uations ha(cid:448)e only t(cid:449)o (cid:448)a(cid:396)ia(cid:271)les in a 3d plane. These e(cid:395)uations gi(cid:448)e a (cid:862)t(cid:396)a(cid:272)e(cid:863) of the curve on the coordinate plane denoted by the given variables. Curve is directed along the axis of the missing variable. The curve/trace does not change along the direction of the axis: (cid:1878)=(cid:886) (cid:1876)2 (cid:1878)= (cid:1876)2+(cid:886) There are two variables, so this is a cylinder. Direction along the z-axis (cid:891)(cid:1876)2+(cid:886)(cid:1877)2=(cid:885)(cid:888) (cid:1876)2(cid:886)+(cid:1877)2(cid:891)=(cid:883) . Have all 3 variables in a 3d plane. Traces occur on coordinate planes and/or on planes parallel to coordinate planes. Still directed along an axis, but the trace changes along the axis. Steps: determine the type of surface, determine the direction axis, find the trace on coordinate plane, find at least two other traces along direction axis. Intercepts: (cid:1876)= (cid:1853), (cid:1877)= (cid:1854), (cid:1878)= (cid:1855: (cid:891)(cid:1876)2+(cid:886)(cid:1877)2+(cid:1878)2=(cid:885)(cid:888) (cid:1876)2(cid:886)+(cid:1877)2(cid:891)+(cid:1878)2(cid:885)(cid:888)=(cid:883) (cid:1876)= (cid:884),(cid:1877)= (cid:885),(cid:1878)= (cid:888) Always directed along the axis with the negative variable.

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