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

860 Chapter 13 Vector Functions
1. The figure shows a curve C given by a vector function rstd.
13. rstd − t sin t i 1 e t cos t j 1 sin t cos t k
12. rstd − 1
1 1 t i 1 t
1 1 t j 1 t 2
1 1 t k intersect at the origin. Find their angle of intersection correct
to the nearest degree.
(a) Draw the vectors rs4.5d 2 rs4d and rs4.2d 2 rs4d.
(b) Draw the vectors
14. rstd − sin 2 at i 1 te bt j 1 cos 2 ct k
15. rstd − a 1 t b 1 t
rs4.5d 2 rs4d
rs4.2d 2 rs4d
c
and
0.5
0.2
16. rstd − t a 3 sb 1 t cd
(c) Write expressions for r9s4d and the unit tangent
vector Ts4d.
17–20 Find the unit tangent vector Tstd at the point with the
(d) Draw the vector Ts4d.
given value of the parameter t.
y
C R
17. rstd − kt 2 2 2t, 1 1 3t, 1 3 t 3 1 1 2 t 2 l , t − 2
18. rstd − ktan 21 t, 2e 2t , 8te t l, t − 0
1
r(4.5)
Q
19. rstd − cos t i 1 3t j 1 2 sin 2t k, t − 0
r(4.2)
20. rstd − sin 2 t i 1 cos 2 t j 1 tan 2 t k, t − y4
P
r(4)
21. If rstd − kt, t 2 , t 3 l, find r9std, Ts1d, r0std, and r9std 3 r0std.
0 1 x
22. If rstd − ke 2t , e 22t , te 2t l, find Ts0d, r0s0d, and r9std ? r0std.
2. (a) Make a large sketch of the curve described by the vector
23–26 Find parametric equations for the tangent line to the curve
with the given parametric equations at the specified point.
function rstd − kt 2 , tl, 0 < t < 2, and draw the vectors
rs1d, rs1.1d, and rs1.1d 2 rs1d.
(b) Draw the vector r9s1d starting at (1, 1), and compare it
with the vector
rs1.1d 2 rs1d
0.1
23. x − t 2 1 1, y − 4st , z − e t 2 2t ; s2, 4, 1d
24. x − lnst 1 1d, y − t cos 2t, z − 2 t ; s0, 0, 1d
25. x − e 2t cos t, y − e 2t sin t, z − e 2t ; s1, 0, 1d
26. x − st 2 1 3 , y − lnst 2 1 3d, z − t; s2, ln 4, 1d
Explain why these vectors are so close to each other in
length and direction.
27. Find a vector equation for the tangent line to the curve of
3–8
intersection of the cylinders x 2 1 y 2 − 25 and y 2 1 z 2 − 20
(a) Sketch the plane curve with the given vector equation.
at the point s3, 4, 2d.
(b) Find r9std.
(c) Sketch the position vector rstd and the tangent vector r9std for
the given value of t.
3. rstd − kt 2 2, t 2 1 1l, t − 21
4. rstd − kt 2 , t 3 l, t − 1
28. Find the point on the curve rstd − k2 cos t, 2 sin t, e t l,
0 < t < , where the tangent line is parallel to the plane
s3 x 1 y − 1.
CAS 29–31 Find parametric equations for the tangent line to the curve
with the given parametric equations at the specified point. Illustrate
by graphing both the curve and the tangent line on a common
5. rstd − e 2t i 1 e t j, t − 0
6. rstd − e t i 1 2t j, t − 0
7. rstd − 4 sin t i 2 2 cos t j, t − 3y4
8. rstd − scos t 1 1d i 1 ssin t 2 1d j, t − 2y3
9–16 Find the derivative of the vector function.
9. rstd − kst 2 2 , 3, 1yt 2 l
10. rstd − ke 2t , t 2 t 3 , ln tl
screen.
29. x − t, y − e 2t , z − 2t 2 t 2 ; s0, 1, 0d
30. x − 2 cos t, y − 2 sin t, z − 4 cos 2t; ss3 , 1, 2d
31. x − t cos t, y − t, z − t sin t; s2, , 0d
32. (a) Find the point of intersection of the tangent lines to the
curve rstd − ksin t, 2 sin t, cos tl at the points
where t − 0 and t − 0.5.
; (b) Illustrate by graphing the curve and both tangent lines.
11. rstd − t 2 i 1 cosst 2 d j 1 sin 2 t k
33. The curves r 1std − kt, t 2 , t 3 l and r 2std − ksin t, sin 2t, tl
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