20E - HW 3 Solutions

# 20E - HW 3 Solutions - Math 20E Homework#4 27 January 2005...

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Unformatted text preview: Math 20E Homework #4 27 January 2005 Section 4.2, Problem 2: Find the arc length of the curve ( 1, 3 t 2 , t 3 ) over the interval ≤ t ≤ 1. Solution. The arc length is Z 1 k ( 0, 6 t , 3 t 2 ) k dt = Z 1 p 36 t 2 + 9 t 4 dt = Z 1 3 t p 4 + t 2 dt = ( 4 + t 2 ) 3/2 1 = 5 √ 5- 8. Section 4.2, Problem 10: The arc length function s ( t ) for a given path c ( t ) , defined by s ( t ) = R t a k c ( τ ) k d τ , represents the distance a particle traversing the trajectory of c will have travelled by time t if it starts out at a ; that is, it gives the length of c between c ( a ) and c ( t ) . Find the arc length functions for the curves α ( t ) = ( cosh t , sinh t , t ) and β ( t ) = ( cos t , sin t , t ) , with a = 0. Solution. For α ( t ) , we get s α ( t ) = Z t q sinh 2 τ + cosh 2 τ + 1 d τ = Z t r 2 e 2 τ + 2 e- 2 τ 4 + 1 d τ = √ 2 Z t e τ + e- τ 2 d τ = √ 2 sinh t . For β ( t ) we get s β ( t ) = Z t q (- sin τ ) 2 + cos 2 τ + 1 2 d τ = Z t √ 2 d τ = √ 2 t . Section 4.2, Problem 11: Let c ( t ) be a given path, a ≤ t ≤ b . Let s = α ( t ) be a new variable, where α is a strictly increasing C 1 function given on [ a , b ] . For each s in [ α ( a ) , α ( b )] there is a unique t with α ( t ) = s . Define a function d : [ α (...
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20E - HW 3 Solutions - Math 20E Homework#4 27 January 2005...

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