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# EVNREV11 - 298 Chapter 11 Vector-Valued Functions GmM r r3...

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86. Since r is a constant multiple of a , they are parallel. Since is parallel to Also, Thus, is a constant vector which we will denote by L . r r d dt r r r r r r 0 0 0 . r r 0 . r , a r a GM r 3 r F m a m a GmM r 3 r 88. Thus, is a constant vector which we will denote by e . r GM L r r 1 r 3 r r r r r r 0 r r 3 r r 1 r 3 r r r 1 GM 0 GM r r 3 r r 1 r 3 r r r d dt r GM L r r 1 GM r 0 r L 1 r 3 r r r 90. Let: Then: r 2 d dt k and L r r r 2 d dt . r r i r cos r sin d dt j r sin r cos d dt k 0 0 d r dt d r d d dt r r sin i cos j d dt r r cos i sin j L r r 92. Let P denote the period. Then Also, the area of an ellipse is where 2 a and 2 b are the lengths of the major and minor axes. 4 2 GM a 3 Ka 3 4 2 L 2 GM L 2 a 3 4 2 ed L 2 a 3 4 2 a 4 L 2 ed a P 2 4 2 a 2 L 2 a 2 c 2 4 2 a 2 L 2 a 2 1 e 2 P 2 ab L ab 1 2 L P ab A P 0 dA dt dt 1 2 L P . Review Exercises for Chapter 11 2. (a) Domain: and (b) Continuous except at t 4 4, 0, 4 r t t i 1 t 4 j k 4. (a) Domain: (b) Continuous for all t , r t 2 t 1 i t 2 j t k 298 Chapter 11 Vector-Valued Functions

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6. (a) (b) (c) (d) 3 cos t 3 i sin t t k r t r 3 cos t i 1 sin t j t k 3 i j k r s 3 cos s i 1 sin s j s k r 2 2 k r 0 3 i j 8. y x x 1 x t t , y t t t 1 x 2 2 3 4 4 1 1 3 1 2 2 y r t t i t t 1 j 10. z y 2 y 1 2 x , z t 2 , y t , x 2 t , x y 3 4 5 z r t 2 t i t j t 2 k t 0 1 2 x 0 2 4 y 0 1 2 z 0 1 1 4 1 2 1 12. x 2 z 2 4 z 2 sin t y t , x 2 cos t , x y 2 3 2 π z r t 2 cos t i t j 2 sin t k t 0 x 2 0 0 y 0 z 0 2 0 2 3 2 2 2 3 2 2 14. y x 1 1 1 1 2 2 2 3 3 3 4 5 6 1 2 2 3 3 z r t 1 2 t i t j 1 4 t 3 k 16. One possible answer is: r 3 t 4 t j , 0 t 4 r 2 t 4 cos t i 4 sin t j , 0 t 2 r 1 t 4 t i , 0 t 1 18. The x- and y- components are 2 cos t and 2 sin t. At the staircase has made of a revolution and is 2 meters high. Thus, one answer is r t 2 cos t i 2 sin t j 4 3 t k . 3 4 t 3 2 , 20. r t t i t j 4 t 2 k r t t i t j 4 t 2 k x y 3 4 5 z z ± 4 t 2 y t , x t , t x x y 0, x 2 z 2 4, 22. 2 i j k lim t 0 sin 2 t t i e t j e t k lim t 0 2 cos 2 t 1 i j k Review Exercises for Chapter 11 299
24. (a) (c) (e) (f) D t r t u t 1 t sin t 1 t 2 cos t t sin t cos t i 1 t cos t 1 t 2 sin t t cos t sin t j r t u t 1 t cos t t cos t i 1 t sin t t sin t j D t r t t 1 t 2 r t 1 t 2 D t r t u t 0 r t u t 2 r t cos t i sin t j k u t sin t i cos t j 1 t k r t sin t i cos t j t k , (b) (d) D t u t 2 r t cos t i sin t j 1 t 2 2 k u t 2

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