MATH262_FALL2010.260429562.5[1]

MATH262_FALL2010.260429562.5[1] - s r t s = i j k 6(1 pt...

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Jignesh Patel Assignment 5 MATH262, Fall 2010 due 10/24/2010 at 11:59pm EDT. 1. (1 pt) For the given position vectors r ( t ) compute the unit tangent vector T ( t ) for the given value of t . A.) If r ( t ) = ( cos3 t , sin3 t ) Then T ( π 4 ) = ( , ) C.) If r ( t ) = ( t 2 , t 3 ) Then T ( 2 ) = ( , ) C.) If r ( t ) = e 3 t i + e - 2 t j + t k . Then T ( - 5 ) = i + j + k . 2. (1 pt) Find parametric equations for the tangent line at the point ( cos ( - 1 π 6 ) , sin ( - 1 π 6 ) , - 1 π 6 )) on the curve x = cos t , y = sin t , z = t x ( t ) = y ( t ) = z ( t ) = 3. (1 pt) Find the length of the given curve: r ( t ) = ( 3 t , 2sin t , 2cos t ) where - 2 t 5. 4. (1 pt) Find the arclength s ( t ) of the curve r ( t ) = 8 t 3 i + 15 t 4 j + 15 t 5 k from r ( 0 ) to r ( t ) . You can assume that t is posi- tive. Don’t forget to submit your answer as a function of t . s ( t ) = 5. (1 pt) Starting from the point ( 5 , 4 , - 2 ) reparametrize the curve r ( t ) = ( 5 - 5 t ) i +( 4 - 2 t ) j +( - 2 - 3 t ) k in terms of arclength. Be sure that the points corresponding to t = 0 and s = 0 are the same and that t and s are increasing in the same direction. Remember, the answers are functions of
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Unformatted text preview: s . r ( t ( s )) = i + j + k 6. (1 pt) Consider the helix r ( t ) = ( cos (-5 t ) , sin (-5 t ) , 5 t ) . Compute, at t = π 6 : A. The unit tangent vector T = ( , , ) B. The unit normal vector N = ( , , ) C. The unit binormal vector B = ( , , ) D. The curvature κ = Note that all of your answers should be numbers. 7. (1 pt) Find the unit tangent, normal and binormal vectors T , N , B , and the curvature κ of the curve x = 4 t , y = 4 t 2 , z = 4 t 3 at t = 1. T = i + j + k N = i + j + k B = i + j + k κ = Note that all of the answers are numbers. Generated by the WeBWorK system c ± WeBWorK Team, Department of Mathematics, University of Rochester 1...
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This note was uploaded on 12/21/2010 for the course MATH 262 taught by Professor Faber during the Spring '08 term at McGill.

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