Thomas Calculus 13th [Solutions]

Section 13.1 Curves in Space and Their Tangents 929
16.
2 2
v i 32t j and
2 2
2 2
a 32 j v(0)
i j and
2 2
2 2
2 2
(0) 32 | (0)| 1
2 2
a j v and
2 2 16 2 2
| a(0)| ( 32) 32; v(0) a(0) ( 32) 16 2 cos
3
2 1(32) 2 4
2
1/2
17. v
2t
i
1
j t t 1 k and
2 2
t 1 t 1
2
a 2t
2 i 2t
j 1 k v(0)
j and
2
2
2
2
2
3/2
t 1 t 1 t 1
a(0) 2 i k v (0) 1 and
2 2
a(0) 2 1 5; v(0) a(0) 0 cos 0
2
18.
19.
2 1/2 2 1/2
(1 t) (1 t)
1
3 3 3
1 1/2 1 1/2
2 2 1
3 3 3 3 3
v i j k and a (1 t) i (1 t) j v(0)
i j k and
2 2 2
(0) 1 1 (0) 2 2 1 1
3 3 3 3 3
a i j v and
cos 0
2
2
( ) (sin ) cos t
t
t t t t e ( t) (cos t) (2t sin t) e ;
2 2
1 1 2
(0) ; (0) (0) 2 2 0
3 3 3 9 9
a v a
r i j k v i j k t0 0 v( t0)
i k and
r t0 P0 0, 1,1 x 0 t t, y 1, and z 1
t are parametric equations of the tangent line
20.
2 3 2
r( t) t i (2t 1) j t k v( t) 2ti 2j 3 t k ; t0 2 v(2) 4i 2j 12k and r( t0) P0
(4, 3, 8)
x 4 4 t, y 3 2 t , and z 8 12t are parametric equations of the tangent line
21. r ( t ) (ln t ) i t 1
j
1 3
1
2 ( t ln t ) k v ( t
t ) i j
t (ln t 1) k ; t
2
0 1 v (1) i j k and
( t 2)
3
r t 1 1
0 P0 (0,0,0) x 0 t t, y 0 t t,
and z 0 t t are parametric equations of the tangent
3 3
line
22. r( t) (cos t) i (sin t) j (sin 2 t) k v( t) ( sin t) i (cos t) j (2 cos 2 t) k ; t0 v( t
2 0) i 2k and
r ( t0) P0
(0, 1, 0) x 0 t t, y 1, and z 0 2t 2t are parametric equations of the tangent line
23. (a) v( t) (sin t) i (cos t) j a( t) (cos t) i (sin t) j;
2 2
(i) v ( t) ( sin t) (cos t) 1 constant speed;
(ii) v a (sin t)(cos t) (cos t)(sin t) 0 yes, orthogonal;
(iii) counterclockwise movement;
(iv) yes, r(0) i 0j
(b) v( t) (2 sin 2 t) i (2 cos 2 t) j a( t) (4 cos 2 t) i (4 sin 2 t) j;
(c)
2 2
(i) v ( t) 4 sin 2t 4 cos 2t
2 constant speed;
(ii) v a 8 sin 2t cos 2t 8 cos 2t sin 2t
0 yes, orthogonal;
(iii) counterclockwise movement;
(iv) yes, r(0) i 0j
v i j a i j
( t) sin t cos t ( t) cos t sin t ;
2 2 2 2
2 2
(i) v ( t) sin t cos t 1 constant speed;
2 2
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