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2224-Sec10_6-HWT

# 2224-Sec10_6-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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Math 2224 Multivariable Calculus – Sec. 10.6: Cylinders and Quadratic Surfaces I. Review of Conic Sections A. Parabolas y = ± x 2 or x = ± y 2 B. Ellipses x 2 a 2 + y 2 b 2 = 1 If a=b , then we have a circle. C. Hyperbolas x 2 a 2 ! y 2 b 2 = 1 or y 2 a 2 ! x 2 b 2 = 1

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II. Cylinders A. A cylinder is the surface composed of all the lines (rulings) that lie parallel to a given line in space and pass through a given plane curve (a generating curve). B. An equation in any two Cartesian coordinates defines a cylinder parallel to the axis of the third (unrestricted) coordinate. C. Examples 1. Graph y=x 2 . 2. Graph 9y 2 +z 2 = 16 .
III. Quadratic Surfaces A. A quadratic surface is the graph in space of a second-degree equation in x, y, and z . B. Quadratic surfaces are the 3D counter part of conic sections in 2D. See page 655. C. Types 1. Ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 (all coefficients are positive) a. All three traces are ellipses. b.

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2224-Sec10_6-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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