Surfaces in Three-Space: Review of Conic Sections and Quadric Surfaces, Exercises of Advanced Calculus

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Surfaces in Three-Space

Quick Review of the Conic Sections a) Parabola b) Ellipse c) Hyperbola (^) = 1

Surfaces in Three-Space The graph of a 3 - variable equation which can be written in the form F(x,y,z) = 0 or sometimes z = f(x,y) (if you can solve for z ) is a surface in 3 D. One technique for graphing them is to graph cross-sections (intersections of the surface with well-chosen planes) and/or traces (intersections of the surface with the coordinate planes). We already know of two surfaces: a) plane Ax + By + Cz = D b) sphere (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2

EX 1 Sketch a graph of z = x^2 + y^2 and x = y^2 + z^2.

ELLIPSOID HYPERBOLOID OF ONE SHEET HYPERBOLOID OF TWO SHEETS Basic Quadric Surfaces ELLIPTIC PARABOLOID HYPERBOLIC PARABOLOID ELLIPTIC CONE

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EX 2 Name and sketch these graphs

• a) 9 x^2 + y^2 - z^2 = -
• b) 9 x^2 + y^2 - z^2 =
• c) x^2 + 4 y^2 - z =
• d) x^2 + y^2 =
• e) x^2 - y^2 =
• f) z = y