Calculus 2nd Edition Rogawski

ANSWERS TO ODD-NUMBERED EXERCISES
A103
17. (b)
10
5
0
−5
z
−10
−4 x
y 0 6 8 6 4 2 0 −2 −4 −6 −8
( )
19. (0, 1, 0), (0, −1, 0), 1√2
, √1
, 0 ,
(
) (
) 2
− √ 1 , − √ 1 , 0 , − √ 1 ,
1√ , 0
2
〈 2 2 2
21. r(t) = 2t 2 − 7, t,± √ 9 − t 2〉 , for − 3 ≤ t ≤ 3
〈
23. (a) r(t) = ±t √ 〉
1 − t 2 ,t 2 ,t for − 1 ≤ t ≤ 1
( )
√2 1
, − √ 1 , 0 , 2
(b) The projection is a circle in the xy-plane with radius 1 2 and
centered at the xy-point ( 0, 1 )
2 .
25. r(t) = ⟨cos t, ± sin t, sin t⟩; the projection of the curve onto the
xy-plane is traced by ⟨cos t, ± sin t, 0⟩, which is the unit circle in
this plane; the projection of the curve onto the xz-plane is traced by
⟨cos t, 0, sin t⟩, which is the unit circle in this plane; the projection
of the curve onto the yz-plane is traced by ⟨0, ± sin t, sin t⟩, which is
the two segments z = y and z = −yfor − 1 ≤ y ≤ 1.
〈
27. r(t) = cos t, sin t, 4 cos t 2〉 ,0≤ t ≤ 2π
29. Collide at the point (12, 4, 2) and intersect at the points
(4, 0, −6) and (12, 4, 2)
31. r(t) = ⟨3, 2, t⟩ , −∞