Pre-Calc Homework Solutions 439

Pre-Calc Homework Solutions 439 - Chapter 10 Review dr =(e...

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Unformatted text preview: Chapter 10 Review dr = (e t cos t dt dx (e t cos t dt 439 54. v(t) a(t) e t sin t)i e t sin t e cos t t t (e t sin t e t sin t e cos t t e t cos t )j e t cos t)i 61. (a) v(t) v(3) v(3) dx dy , dt dt 3 4 2 9 2 32 3 5 sin t, cos t , 4 4 4 4 , 5 4 2 25 2 32 , and 34 4 2 4 17 (e sin t ( 2e t sin t)i r(t) a(t) t e sin t)j (2e t cos t)j (e t sin t)(2e t cos t) 0 3.238 2 (e t cos t)( 2e t sin t) (b) x-component: for all t. The angle between r and a is always 90 . 1 1 d 2x dt 2 t 3 3 2 cos 16 4 5 16 2 3 3 y 2 5 3 16 2 5 2 d 2y y-component: 2 dt t 3 1 sin x 2 3 4 16 2 55. 0 (3 3t 2 e 6t) dt i 0 6 cos t dt j (c) 6i 6 sin t j 0 2 e e 3t 2 1 i 0 x 3 x2 9 dx dt cos y 25 2 4 t and 1. y 5 sin 4 t, so 1 or 56. e 2 ln t dt i t e 2 1 dt j t ln t e 2 62. (a) (ln 2)j C 1 dy and 2 dt 10 0 5 1 2 2 t so (5 t)2 dt, which using NINT ln2 t 57. r(t) r(0) r(t) 58. r(t) r(0) r(t) j i i e ln (ln t) e j 3i (sin t)j i Length evaluates to 10 dr dt dt (cos t)i j, so C 1)i (tan i t 1)i (sin t 1 25.874. y2 dx dt dt t(10 t) 2 dt 2 C j, and (b) Volume 1)j t2 i and 1j 10 (cos t dr dt dt 0 10 2 t)i 1j C 8 8 0 10 0 (100t 2 20t 3 5t 4 1 2 2 t 4) dt 1250 3 C 1 j, so C t2 (tan 100 3 t 3 t(10 t) 2 1 5 10 t 5 0 dr d 2r 59. dt 2tj C1, r(t) dt dt 2 dr C1 0, so r(t) t 2j dt t 0 dr dt dt (c) Area t 2j C1 t C2 C2 i, so 63. (a) dy dx dy dx t dy/dt dx/dt 0 2 (5 t)2 dt, which using NINT evaluates to e t sin t e t cos t 1 1 1040.728. cos t cos t sin t sin t C2. And r(0) r(t) 60. dr dt i t 2j ( 2t)i t 2i 2j 6t)i j C2 6t ( 2t)j t 2j C1 ( t2 3i 2)i C1t 4i, so C1 2t)j C2 2i 2)j 2j, and C1 , C2 6i 2j and e t cos t e t sin t r(t) dr dt t 1 d 2r dt dt 2 dr dt dt 1 (b) 2i ( t2 5i ( t2 r(t) r(1) r(t) dy dx e t(sin t cos t), e t(cos t sin t) dt dt dy 2 e 2t(sin2 t 2 sin t cos t cos2 t) dt e 2t(1 dx 2 dt 2 sin t cos t) 2 cos t sin t sin2 t) 3j, so C2 ( t2 2t e 2t(cos2 t e 2(1 2 cos t sin t) v(t) v(3) et 2 e3 2 3 (c) Distance 0 3 0 v(t) dt et 2 2 et (e3 3 0 1) 2 ...
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