Thomas Calculus 13th [Solutions]

966 Chapter 13 Vector-Valued Functions and Motion in Space
19.
1 2t 2t 4
1 1 4
2 1 t
r ti e j v i e j v e T i e j T (ln 2) i j ;
4t
4t
1 e 1 e
17 17
d T
4t
2t
2e
2e
d T
i j (ln 2) 32 i 8 j N(ln 2) 4 i 1 j;
dt 4t
3 2
4t
3 2 dt 17 17 17 17 17 17
1 e 1 e
2t
i j k
1 4
2t
B(ln 2) T(ln 2) N(ln 2) 0 k ; a 2 e j a(ln 2) 8j and v(ln 2) i 4j
17 17
4 1 0
17 17
i j k
v(ln 2) a(ln 2) 1 4 0 8k v a 8 and v(ln 2) 17 (ln 2) 8 ;
17 17
0 8 0
1 4 0
0 8 0
a
2t
0 16 0
4 e j a(ln 2) 16 j (ln 2)
2
v a
0
20. r 3cosh 2t i 3sinh 2t j 6tk v 6sinh 2t i 6cosh 2t
j 6k
2 2
v 36sinh 2t 36cosh 2t 36 6 2 cosh 2t T v 1 tanh 2t i 1 j 1 sech 2t
k
v 2 2 2
T(ln 2) 5 i 1 j 8 k ;
17 2 2 17 2
d T
dt
2 2
sech 2t i 2 sech 2t tanh 2t
k
2 2
2
2 2
d T
2 8 2 8 15 128 240 d T
128 240 8 2
(ln 2) i k i k (ln 2)
dt
2 17 2 17 17 289 2 289 2 dt
289 2 289 2 17
i j k
N(ln 2) 8 i 15 k ; B(ln 2) T(ln 2) N(ln 2) 15 1 8 15 i 1 j 8 k ;
17 17
17 2 2 17 2 17 2 2 17 2
8 0 15
17 17
a 12cosh 2t
i 12sinh 2 t j a(ln 2) 12 17 i 12 15 j 51 i 45 j and v(ln 2) 6 15 i 6 17
8 8 2 2
8 8
j 6k
i j k
45 i 51 j 6 k v(ln 2) a(ln 2) 45 51 6 135i 153j 72k v a 153 2 and v(ln 2) 51 2
4 4 4 4
4
51 45 0
2 2
(ln 2) 153 2 32
3 867
;
2
51
4
a 24sinh 2t
i 24cosh 2 t j a(ln 2) 45i 51 j (ln 2)
45 51
4 4
51 45
2 2
6
0
45 51 0
v a
2
32
867
21.
2 2
r 2 3t 3t i 4t 4t j 6cost k v 3 6t i 4 8t j 6sin t k
v
d v
dt
1
2
2 2 2 2 2
3 6t 4 8t 6sin t 25 100t 100t 36sin t
2 2
1 2
d v
25 100t 100t 36sin t 100 200t 72sin t cos t aT
(0) (0) 10;
dt
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