14.4%20Ex%2025-30

14.4 Ex 25-30 - S E C T I O N 14.4 Curvature(ET Section 13.4 511 25 In the notation of Exercise 23 assume that a b Show that b/a 2(t a/b2 for all t

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SECTION 14.4 Curvature (ET Section 13.4) 511 25. In the notation of Exercise 23, assume that a b . Show that b / a 2 κ ( t ) a / b 2 for all t . SOLUTION In Exercise 23 we showed that the curvature of the ellipse r ( t ) = h a cos t , b sin t i is the following function: ( t ) = ab ( b 2 cos 2 t + a 2 sin 2 t ) 3 / 2 Since a b > 0 the quotient becomes greater if we replace a by b in the denominator, and it becomes smaller if we replace b by a in the denominator. We use the identity cos 2 t + sin 2 t = 1 to obtain: ( a 2 cos 2 t + a 2 sin 2 t ) 3 / 2 ( t ) ( b 2 cos 2 t + b 2 sin 2 t ) 3 / 2 ( a 2 ( cos 2 t + sin 2 t )) 3 / 2 ( t ) ( b 2 ( cos 2 t + sin 2 t )) 3 / 2 a 3 = ( a 2 ) 3 / 2 ( t ) ( b 2 ) 3 / 2 = b 3 b a 2 ( t ) a b 2 26. Use Eq. (3) to prove that for a plane curve r ( t ) = h x ( t ), y ( t ) i , ( t ) = | x 0 ( t ) y 00 ( t ) x 00 ( t ) y 0 ( t ) | ( x 0 ( t ) 2 + y 0 ( t ) 2 ) 3 / 2 9 By the formula for curvature we have ( t ) = k r 0 ( t ) × r 00 ( t ) k k r 0 ( t ) k 3 (1) We compute the cross product of r 0 ( t ) = - x 0 ( t ), y 0 ( t ) ® and r 00 ( t ) = - x 00 ( t ), y 00 ( t ) ® . Actually, since the cross product is
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This homework help was uploaded on 04/22/2008 for the course MATH 32A taught by Professor Gangliu during the Winter '08 term at UCLA.

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14.4 Ex 25-30 - S E C T I O N 14.4 Curvature(ET Section 13.4 511 25 In the notation of Exercise 23 assume that a b Show that b/a 2(t a/b2 for all t

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