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 ( ) k d , 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 ( t ) = ( cosh t , sinh t , t ) and ( t ) = ( cos t , sin t , t ) , with a = 0. Solution. For ( t ) , we get s ( t ) = Z t q sinh 2 + cosh 2 + 1 d = Z t r 2 e 2 + 2 e- 2 4 + 1 d = 2 Z t e + e- 2 d = 2 sinh t . For ( t ) we get s ( t ) = Z t q (- sin ) 2 + cos 2 + 1 2 d = Z t 2 d = 2 t . Section 4.2, Problem 11: Let c ( t ) be a given path, a t b . Let s = ( t ) be a new variable, where is a strictly increasing C 1 function given on [ a , b ] . For each s in [ ( a ) , ( b )] there is a unique t with ( t ) = s . Define a function d : [ (...
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20E - HW 3 Solutions - Math 20E Homework #4 27 January 2005...

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