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Homework_5

# Homework_5 - i(b r t = ± e ± t p 2 t;e t ²(c r t = h e t...

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MATH 230 HOMEWORK 5 - DUE IN CLASS ON 9/28/09 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 14.3 Compute the length of each of the following curves. (a) r ( t ) = D t; t 2 p 2 ; t 3 3 E from (0 ; 0 ; 0) to (1 ; 1 p 2 ; 1 3 ) (b) r ( t ) = h sin e t ; cos e t ;e t i from (0 ; ± 1 ) to ( ± 1 ; 0 ; 3 ± 2 ) Problem 2. Section 14.3 Find an arclength parameterization for each of the following curves. (a) r ( t ) = h 2 t; 1 ± 2 t;t i (b) r ( t ) = h e t cos t;e t sin t;e t i Problem 3. Section 14.3 Compute the curvature ² ( t ) of each of the following curves. (a) r ( t ) = h t; sin t; cos t
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Unformatted text preview: i (b) r ( t ) = ± e ± t ; p 2 t;e t ² (c) r ( t ) = h e t cos t;e t sin t;e t i Problem 4. Section 14.4 For each of the following problems, you are given the accelleration function of a particle moving in R 3 as well as its initial position and velocity. Find its position at time t . (a) a ( t ) = h ; ; 1 i , r (0) = h 2 ; 1 ; 1 i , v (0) = h 3 ; 1 ; 1 i (b) a ( t ) = h e t ; sin t; cos t i , r (0) = h 1 ; ; 1 i , v (0) = h ; 2 ; 2 i Problem 5. Challenge Problem (will not be graded): Given a curve r ( t ) parameterized by arclength, de±ne the binormal vector of r ( t ) to be B ( t ) = N ( t ) ² T(t) where T is the unit tangent and N is the unit normal. Show that B is a multiple N at every point on the curve, and show that the curve lies on a plane in R 3 if and only if B = 0 . 1...
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