James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

854 chapter 13 Vector Functions
5. lim
tl `K 1 1 t 2
6. lim
tl `Kte 2t ,
1 2 t , 2 tan21 t, 1 2 e22t
t
t 3 1 t
2t 3 2 1 tL , t sin 1
7–14 Sketch the curve with the given vector equation. Indicate
with an arrow the direction in which t increases.
7. rstd − ksin t, tl 8. rstd − kt 2 2 1, tl
9. rstd − kt, 2 2 t, 2tl 10. rstd − ksin t, t, cos tl
11. rstd − k3, t, 2 2 t 2 l
12. rstd − 2 cos t i 1 2 sin t j 1 k
13. rstd − t 2 i 1 t 4 j 1 t 6 k
14. rstd − cos t i 2 cos t j 1 sin t k
15–16 Draw the projections of the curve on the three coordinate
planes. Use these projections to help sketch the curve.
15. rstd − kt, sin t, 2 cos tl
L
16. rstd − kt, t, t 2 l
17–20 Find a vector equation and parametric equations for the
line segment that joins P to Q.
17. Ps2, 0, 0d, Qs6, 2, 22d
19. Ps0, 21, 1d, Q( 1 2 , 1 3 , 1 4)
18. Ps21, 2, 22d, Qs23, 5, 1d
20. Psa, b, cd, Qsu, v, wd
21–26 Match the parametric equations with the graphs
(labeled I–VI). Give reasons for your choices.
I
III
x
V
x
z
z
z
y
y
II
x
IV
VI
x
z
z
z
y
y
;
;
21. x − t cos t, y − t, z − t sin t, t > 0
22. x − cos t, y − sin t, z − 1ys1 1 t 2 d
23. x − t, y − 1ys1 1 t 2 d, z − t 2
24. x − cos t, y − sin t, z − cos 2t
25. x − cos 8t, y − sin 8t, z − e 0.8t , t > 0
26. x − cos 2 t, y − sin 2 t, z − t
27. Show that the curve with parametric equations x − t cos t,
y − t sin t, z − t lies on the cone z 2 − x 2 1 y 2 , and use this
fact to help sketch the curve.
28. Show that the curve with parametric equations x − sin t,
y − cos t, z − sin 2 t is the curve of intersection of the surfaces
z − x 2 and x 2 1 y 2 − 1. Use this fact to help sketch the curve.
29. Find three different surfaces that contain the curve
rstd − 2t i 1 e t j 1 e 2t k.
30. Find three different surfaces that contain the curve
rstd − t 2 i 1 ln t j 1 s1ytd k.
31. At what points does the curve rstd − t i 1 s2t 2 t 2 d k intersect
the paraboloid z − x 2 1 y 2 ?
32. At what points does the helix rstd − ksin t, cos t, tl intersect
the sphere x 2 1 y 2 1 z 2 − 5?
33–37 Use a computer to graph the curve with the given vector
equation. Make sure you choose a parameter domain and viewpoints
that reveal the true nature of the curve.
33. rstd − kcos t sin 2t, sin t sin 2t, cos 2tl
34. rstd − kte t , e 2t , tl
35. rstd − ksin 3t cos t, 1 4 t, sin 3t sin tl
36. rstd − kcoss8 cos td sin t, sins8 cos td sin t, cos tl
37. rstd − kcos 2t, cos 3t, cos 4tl
38. Graph the curve with parametric equations x − sin t,
y − sin 2t, z − cos 4t. Explain its shape by graphing its
projections onto the three coordinate planes.
; 39. Graph the curve with parametric equations
x − s1 1 cos 16td cos t
y − s1 1 cos 16td sin t
z − 1 1 cos 16t
Explain the appearance of the graph by showing that it lies on
a cone.
; 40. Graph the curve with parametric equations
x − s1 2 0.25 cos 2 10t cos t
x
y
x
y
y − s1 2 0.25 cos 2 10t sin t
z − 0.5 cos 10t
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