fa07hw8

# fa07hw8 - 11 x = cos 2 t dx dt = 2 sin 2 t y = sin t dy dt...

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Unformatted text preview: 11. x = cos ( 2 t ) dx dt = - 2 sin ( 2 t ) y = sin t dy dt = cos t At t = π 6 ; dy dx = cos (π/ 6 )- 2 sin (π/ 3 ) = - 1 2 . 15. x = t 3- t , y = t 2 is at ( , 1 ) at t = - 1 and t = 1. Since dy dx = 2 t 3 t 2- 1 = ± 2 2 = ± 1 , the tangents at ( , 1 ) at t = ± 1 have slopes ± 1. 1. x = 3 t 2 dx dt = 6 t y = 2 t 3 ( ≤ t ≤ 1 ) dy dt = 6 t 2 Length = integraldisplay 1 radicalbig ( 6 t ) 2 + ( 6 t 2 ) 2 dt = 6 integraldisplay 1 t radicalbig 1 + t 2 dt Let u = 1 + t 2 du = 2 t dt = 3 integraldisplay 2 1 √ u du = 2 u 3 / 2 vextendsingle vextendsingle vextendsingle vextendsingle 2 1 = 4 √ 2- 2 units . 5. x = t 2 sin t , y = t 2 cos t , ( ≤ t ≤ 2 π) . dx dt = 2 t sin t + t 2 cos t dy dt = 2 t cos t- t 2 sin t parenleftbigg ds dt parenrightbigg 2 = t 2 bracketleftBig 4 sin 2 t + 4 t sin t cos t + t 2 cos 2 t + 4 cos 2 t- 4 t sin t cos t + t 2 sin 2 t bracketrightBig = t 2 ( 4 + t 2 )....
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fa07hw8 - 11 x = cos 2 t dx dt = 2 sin 2 t y = sin t dy dt...

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