Homework_4 - x = 1 y 2 and x = y 2 + z 2 (c) The...

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MATH 230 HOMEWORK 4 - DUE IN CLASS ON 9/21/09 PAUL SIEGEL, INSTRUCTOR Instructions: Solve all problems completely. The use of calculators or computer aids is permitted but not required; full credit will be given only if all work is shown. You are encouraged to work with other students, but the solutions that you hand in must be written by you in your own words and they should be a re±ection of your own thinking. If you do choose to work with other students, please hand in a list of all group members with your solutions. Problem 1. Section 13.6 For each of the following quadratic equations: rewrite it in standard form, classify the quartic surface it describes, and identify its axis of symmetry (if applicable). (a) x 2 = ± 2 x + 4 z + 8 (b) x 2 ± y 2 ± 9 z 2 = x + 2 y ± 9 z + 4 (c) z 2 ± 4 y 2 = x + 2 z ± 16 y ± 10 (d) 2 x 2 ± y 2 ± z 2 + 4 x + 3 y ± 5 z = 1 Problem 2. Section 14.1 Find parametric equations for each of the following curves. (a) The intersection of the surfaces y = 2 x 2 + 3 z 2 and y = 5 ± 3 x 2 ± 2 z 2 (b) The intersection of the surfaces
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Unformatted text preview: x = 1 y 2 and x = y 2 + z 2 (c) The intersection of the plane x + z = 1 and the ellipsoid x 2 + 4 y 2 + 4 z 2 = 1 Problem 3. Section 14.2 Find the equation of the tangent line to each of the following curves at the given point. (a) Curve: r 1 ( t ) = e t ;e 2 t ;e 3 t ; Point: r 1 (1) (b) Curve: r 2 ( t ) = e 2 t 4 ; sin( 4 t ) ;t 2 ; Point: (1 ; 1 ; 4) (c) Curve: r 3 ( t ) = r 1 ( t ) r 2 ( t + 1); Point: r 3 (1) Problem 4. Section 14.2 Let r ( t ) be a dierentiable curve in R 3 which does not pass through the origin. (a) Show that d dt j r ( t ) j = 1 j r ( t ) j r ( t ) r ( t ). (b) Show that if r ( t ) is perpendicular to r ( t ) for every t then the curve lies on the sphere x 2 + y 2 + z 2 = 1. Problem 5. Challenge Problem (will not be graded): Classify the curve x 2 + 4 xy + y 2 = 0 . 1...
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