Chapter 14
1.
Find the domain of the following vector function.
r (t )  t 8 , t  3, 10  t
Select the correct answer.
a.
b.
c.
d.
e.
3  t  10
3  t  10
t 3
t  10
t  3
2.
Find a vector function that represents the curve of intersection of the two surfaces.
The circular cylinder x 2  y 2  4 and the parabolic cylinder z  x 2 .
3.
Find the derivative of the vector function.
r(t )  a  tb  t 2c
4.
Find parametric equations for the tangent line to the curve with the given parametric equations at the
specified point.
x  t 8 , y  t 3 , z  t 7 ; (1, 1, 1)
Select the correct answer.
5.
a.
x  1  7t , y  1  2t , z  1  6t
b.
x  1  7t, y  1  2t, z  1  6t
c.
x  1  8t, y  1  3t, z  1  7t
d.
x  1  8t, y  1  3t, z  1  7t
e.
x  8t, y  3t, z  7t
The curves r1 (t )  t , t 5 , t 3
and r2 (t )  sin t , sin 4t, t intersect at the origin.
Find their angle of intersection correct to the nearest degree.
r(t )  t 9i  4t 3 j  t 5k and r(0) = j.
6.
Find r (t ) if
7.
If u(t )  i  3t 2 j  6t 3k and v(t )  ti  costj  sin tk, find
8.
Use Simpson's Rule with n = 10 to estimate the length of the arc of the twisted cubic
x  t, y  t 2 , z  t 3 , from the origin to the point (2, 4, 8).

[u(t )  v (t )] .
t
Select the correct answer.
a. 9.5706
e. 10.5706
b.
19.5706
c.
7.5706
d.
8.5706
9.
Reparameterize the curve with respect to arc length measured from the point where t = 0 in the
direction of increasing t.
r(t )  8 sin ti  tj  8 cos tk
10. Find the unit tangent vector T (t ) .
r(t )  4sin t, 9t, 4cos t
11. A force with magnitude 40 N acts directly upward from the xy-plane on an object with mass 5 kg. The
object starts at the origin with initial velocity v (0) = 5i - 4j. Find its position function.
12. Formula
For a plane curve with equation y  f (x) we have k (t ) 
| f ( x) |
 1  ( f ( x)) 
2
3/ 2
.
Use Formula to find the curvature of y  x8 .
Select the correct answer.
a.
56 | x |6
(1  8 x 7 )3 / 2
b.
56 | x |6
(1  64 x14 )3 / 2
c.
8x 7
(1  64 x14 )3 / 2
d.
x7
(1  64 x14 )3 / 2
e.
8x 7
(1  64 x14 )1 / 2
13. At what point on the curve x  t 3 , y  3t , z  t 4 is the normal plane parallel to the plane
9x  9 y  12z  7 ?
14. Find the velocity of a particle with the given position function.
r(t )  13e15t i  10e18t j
15. What force is required so that a particle of mass m has the following position function?
r(t )  5t 3i  2t 2 j  7t 3k
16. Find the speed of a particle with the given position function.
r(t )  ti  6t 2 j  2t 6k
Select the correct answer.
a.
| v(t ) |  1  12t  12t 5
b.
| v(t ) |  1  144t 2  144t10
c.
| v(t ) |  1  12t  12t 5
d.
e.
| v(t ) |  1  144t 2  144t10
| v(t ) |  1  144t  144t 5
2
2
17. The position function of a particle is given by r(t )  8t , 2t , 8t  96t .
When is the speed a minimum?
18. A ball is thrown at an angle of 15  to the ground. If the ball lands 67 m away, what was the initial
speed of the ball? Let g  9.8 m/s2 .
3
2
19. A particle moves with position function r(t )  (12t  4t  2)i  12t j .
Find the tangential component of the acceleration vector.
20. A particle moves with position function r(t )  8 cos ti  8 sin tj  8tk .
Find the normal component of the acceleration vector.
1.
a
2.
r(t )  4 cos(t )i  4 sin tj  16 cos2 tk
3.
b  2tc
4.
c
5.
76 
6.
t10
t6
i  (t 4  1) j  k
10
6
7.
1  6t cos t  21t 2 sin t  6t 3 cos t
8.
a
9.
 s   s 
 s 
i  
 j  8 cos
k
r (t ( s))  4 sin
 65   65 
 65 
4
9
4
cos t ,
,
sin t
97
97
97
10.
11.
r(t )  5ti - 4tj  4t 2k
12.
b
13.
(1, 3, 1)
14.
v  195e15t i - 180e18t j
15.
F  m(30ti  4 j  42tk)
16.
d
17.
t 3
18.
36.2
19.
24 t
20.
8
1.
Find the domain of the following vector function.
r (t )  t 8 , t  1, 9  t
2.
Find the following limit:
lim arctan t , e 4t ,
t 
3.
ln t
t
A particle moves with position function r(t )  2cos ti  2sin tj  2tk .
Find the normal component of the acceleration vector.
4.
Find the unit tangent vector T (t ) .
r(t ) 
5.
4 3
t , 4t 2 , 8t
3
What force is required so that a particle of mass m has the following position function?
r(t )  3t 3i  7t 2 j  9t 3k
Select the correct answer.
a.
F(t )  9mt 2i  14mtj  27mt 2k
b.
F(t )  mt 2i  4mtj  27mt 2k
F(t )  18mt i  14 mj  54tk
F(t )  18mt i  14 mj  54 mt k
F(t )  27 mt i  14 mj  18mt k
c.
d.
e.
6.
Find a vector function that represents the curve of intersection of the two surfaces:
The top half of the ellipsoid
x 2  6 y 2  6z 2  36 and the parabolic cylinder y  x 2 .
Select the correct answer.
a.
r(t )  ti  t 2 j 
6  t 2  6t 4
k
6
b.
r(t )  ti  t 4 j 
36  t 2  6t
k
6
c.
r(t )  ti  t 2 j 
36  t 2  6t 4
k
6
d.
r(t )  ti  t 2 j 
36  t 2  6t
k
6
e.
r(t )  ti  t 2 j 
36  t 2  6t
k
6
7.
Find a vector function that represents the curve of intersection of the two surfaces:
The cylinder x 2  y 2  9 and the surface z  xy .
8.
Find a vector function that represents the curve of intersection of the two surfaces:
The circular cylinder x 2  y 2  4 and the parabolic cylinder z  x 2 .
9.
Find the derivative of the vector function.
r(t )  a  tb  t 2c
10. Find the point of intersection of the tangent lines to the curve r (t )  sin t , 5 sin t , cos t , at the
points where t = 0 and t = 0.5.
6
7
11. The curves r1 (t )  t , t , t
and r2 (t )  sin t , sin 5t, t
intersect at the origin. Find their angle of
intersection correct to the nearest degree.
12. Evaluate the integral.
 (e
6t
i  8tj  ln tk ) dt
13. Reparameterize the curve with respect to arc length measured from the point where t = 0 in the
direction of increasing t.
r(t )  8sin ti  tj  8cos t
14. The curvature of the curve given by the vector function r is
k (t ) 
| r(t )  r(t ) |
| r(t ) |3
Find the curvature of r(t ) 
15t , et , et
at the point (0, 1, 1).
15. Find the velocity of a particle with the given position function.
r(t )  10e7t i  7e12t j
Select the correct answer.
a.
v(t )  70et i  84et j
b.
v(t )  10e7t i  7e12t j
c.
v(t )  70e7t i  84e12t j
d.
v(t )  17e7t i  19e12t j
e.
v(t )  17e7t i  e12t j
16. Find equations of the normal plane to x  t, y  t 2 , z  t 3 at the point (3, 9, 27).
17. Find the acceleration of a particle with the given position function.
r(t )  2sin ti  6tj  4cos tk

18. A projectile is fired with an initial speed of 834 m/s and angle of elevation 38 . Find the range of the
projectile.
Select the correct answer.
a.
b.
c.
d.
e.
d  42 km
d  34 km
d  11 km
d  68 km
d  58 km

19. A ball is thrown at an angle of 15 to the ground. If the ball lands 126 m away, what was the initial
speed of the ball? Let g = 9.8 m/s2 .
Select the correct answer.
a.
b.
c.
d.
e.
v0
v0
v0
v0
v0





49.7 m/s
16.6 m/s
99.4 m/s
24.8 m/s
24.1 m/s
20. A particle moves with position function r(t )  (9t  3t 3  2)i  9t 2 j .
Find the tangential component of the acceleration vector.
1.
1 t  9
2.
r(t ) 
3.
2

2
i
t2
2t
2
, 2
, 2
t 2 t 2 t 2
4.
2
5.
d
6.
c
7.
r(t )  3cos ti  3sin tj  9sin t cos tk
8.
r(t )  4cos ti  4sin tj  16cos2 tk
9.
b  2tc
10.
 1, 5, 1
11. 79 
12.
e6t
i  4t 2 j  t (ln t  1)k  C
6
13.
 s   s 
 s 
i  
 j  8 cos
k
r (t ( s))  4 sin
 65   65 
 65 
14.
2
17
15. c
16.
x  6 y  27z  786  0
17.
a(t )  2sin ti  4cos tk
18. b
19. a
20. 18t