ISM_T11_C13_A - ~ ~ X X ~ X ~ X ~ ~ ~ ~

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CHAPTER 13 VECTOR-VALUED FUNCTIONS AND MOTION IN SPACE 13.1 VECTOR FUNCTIONS 1. x t 1 and y t 1 y (x 1) 1 x 2x; 2t 2 2 and 2 œ œ Ê œ œ œ œ Ê œ œ Ê œ œ # # # v i j a j v i j a j d d dt dt r v at t 1 œ 2. x t 1 and y 2t 1 x x (y 1) 1; 2t 2 2 œ œ Ê œ " Ê œ œ œ Ê œ œ # # # # " ˆ y 1 4 dt dt d d v i j a i r v 2 and 2 at t Ê œ œ œ v i j a i " # 3. x e and y e y x ; e e e e 3 4 and 3 8 at t ln 3 œ œ Ê œ œ œ Ê œ Ê œ œ œ t 2t t 2t t 2t 2 2 d 4 8 9 9 dt 9 9 # v i j a i j v i j a i j r 4. x cos 2t and y 3 sin 2t x y 1; ( 2 sin 2t) (6 cos 2t) œ œ Ê œ œ œ Ê œ # # " 9 dt dt d d v i j a r v ( 4 cos 2t) ( 12 sin 2t) 6 and 4 at t 0 œ Ê œ œ œ i j v j a i 5. (cos t) (sin t) and (sin t) (cos t) v i j a i j œ œ œ œ d d dt dt r v for t , and Ê œ œ 1 1 4 4 2 2 v i j ˆ ‰ È È # # ; for t , and a i j v j ˆ ‰ ˆ ‰ 1 1 1 4 2 2 œ œ œ È È # # # # a i ˆ ‰ 1 # œ 6. 2 sin 2 cos and v i j a œ œ œ d t t d dt dt r v ˆ ˆ # # cos sin for t , ( ) 2 and œ Ê œ œ ˆ ˆ t t # # i j v i 1 1 ( ) ; for t , 2 2 and a j v i j 1 œ œ œ 3 3 1 1 # # ˆ È È a i j ˆ 3 2 2 1 # # # œ È È 7. (1 cos t) (sin t) and v i j a œ œ œ d d dt dt r v (sin t) (cos t) for t , ( ) 2 and ( ) ; œ Ê œ œ œ i j v i a j 1 1 1 for t , and œ œ œ 3 3 3 1 1 1 # # # v i j a i ˆ ˆ 8. 2t and 2 for t 1, v i j a j œ œ œ œ Ê œ d d dt dt r v ( 1) 2 and ( 1) 2 ; for t 0, (0) and v i j a j v i œ œ œ œ (0) 2 ; for t 1, (1) 2 and (1) 2 a j v i j a j œ œ œ œ
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826 Chapter 13 Vector-Valued Functions and Motion in Space 9. (t 1) t 1 2t 2t 2 2 ; Speed: (1) 1 (2(1)) 2 3; r i j k v i j k a j v œ Ê œ œ Ê œ œ œ œ a b k k È # # # # d d dt dt r r Direction: (1) 3 v i j k v (1) 2(1) 2 (1) 3 3 3 3 3 3 3 2 2 2 2 k k œ œ Ê œ " " i j k v i j k ˆ 10. (1 t) t 2t ; Speed: (1) r i j k v i j k a j k v œ Ê œ œ Ê œ œ t t d 2t d 2 2 2 2 3 dt dt È È È r r # k k 1 (1 ) 2; Direction: (1) œ œ œ œ Ê Ê Š # # # # # # # " " " 2(1) (1) 2 2 (1) (1 ) È È k k v v i j k 2(1) 2 i j k v 2 œ Š " " " # # i j k È 2 11. (2 cos t) (3 sin t) 4t ( 2 sin t) (3 cos t) 4 ( 2 cos t) (3 sin t) ; r i j k v i j k a i j œ Ê œ œ Ê œ œ d d dt dt r r Speed: 2 sin 3 cos 4 2 5; Direction: ¸ ¸ ˆ ˆ ˆ ‰ É È v 1 1 1 # # # # # œ œ 2 v v ˆ ¸ ¸ ˆ sin cos 2 5 œ œ Ê œ Š Š Š ˆ ‰ È 2 3 4 2 2 5 5 5 5 5 5 5 # # # # # # " " È È È È È È È 1 1 1 i j k i k v i k 12. (sec t) (tan t) t (sec t tan t) sec t r i j k v i j k a œ Ê œ œ Ê œ 4 d 4 d 3 dt 3 dt r r a b # sec t tan t sec t 2 sec t tan t ; Speed: sec tan sec 2; œ œ œ a b a b ¸ ¸ ˆ ˆ ˆ ‰ ˆ ‰ É # $ # # # # # i j v 1 1 1 1 6 6 6 6 3 4 Direction: 2 v i j k v ˆ ˆ ¸ ¸ ˆ 6 6 6 6 3 6 4 œ œ Ê œ sec tan sec # " " ˆ 3 3 3 6 3 3 3 2 2 2 2 i j k v i j k ˆ ‰ ˆ 1 13. (2 ln (t 1)) t 2t t 2 ; r i j k v i j k a i j k œ Ê œ œ Ê œ œ # t d 2 d 2 2 dt t 1 dt (t 1) r r ˆ Speed: (1) (2(1)) 1 6; Direction: k k Ɉ È v œ œ œ 2 1 1 (1) (1) 2(1) (1) 6 # # # v v i j k k k Š È 2 1 (1) 6 œ Ê œ " " " È È È È È È 6 6 6 6 6 6 2 1 2 i j k v i j k È Š 14. e (2 cos 3t) (2 sin 3t) e (6 sin 3t) (6 cos 3t) r i j k v i j k a œ Ê œ œ Ê œ a b a b t t d d dt dt r r e (18 cos 3t) (18 sin 3t) ; Speed: (0) e [
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