Thomas Calculus 13th [Solutions]

962 Chapter 13 Vector-Valued Functions and Motion in Space
CHAPTER 13 PRACTICE EXERCISES
1. r( t) 4cos t i 2 sin t j
2 y
x 4cost and y 2 sin t x 1;
16 2
v 4sin t i 2 cos t j and
a 4cos t i 2 sin t j;
2
r(0) 4 i, v(0) 2 j, a(0) 4 i;
2 2
r 2 2 i j, v 2 2 i j, a 2 2 i j;
v 16sin t 2cos t
4 4 4
d 14sin t cost
2
aT
v ; at t 0: a
dt
T 0, aN
a 0 4, a 0T 4N 4 N,
2 2
16sin t 2cos t
a N
2
v
4
2
4 2
2; at t : a 7 7 49
4 T , a 9 ,
8 1 3 N 9 3
a N
7 4 2
a T N,
3 3
2
v
4 2
27
2. r( t) 3 sect i 3 tan t j x 3 sect
and
2
v 3 sect tan t i 3 sec t j and
2 3 2
a 3 sect tan t 3 sec t i 2 3 sec t tan t j;
r(0) 3 i, v(0) 3 j, a(0) 3 i;
v
aT
2 2 4
3sec t tan t 3sec t
2 3 4
d v 6sec t tan t 18sec t tan t ;
dt
2 2 4
2 3sec t tan t 3sec t
2
at t 0: aT
0, aN
a 0 3,
a N
a 0T 3N 3 N,
2
v
3 1
3 3
2
2 y 2 2 2 2
y 3 tan t x sec t tan t 1 x y 3;
3 3
2 2
2
3 2
2
3 2
2
3 2
2
3 2
3. r 1 i t j v t 1 t i 1 t j v t 11
t 1 t
.
2 2
2
1 t 1 t
1 t
We want to maximize : d v 2t
d v
v and 0 2t
0 t 0. For 0,
dt 2
2 dt
2
2
1 t
1 t
2t
1 t
2
2
0
v occurs when t
max
0 v 1
max
2t
1 t
t
2
2
0;
for t 0,
t t t t t t
4. r e cost i e sin t j v e cos t e sin t i e sin t e cos t j
t t t t t t t t t t
a e cost e sin t e sin t e cos t i e sin t e cost e cos t e sin t j 2e sin t i 2e cos t j.
Let
be the angle between r and a. Then
cos
2t
2t
1 r a cos
1 2 e sin t cos t 2 e sin t cos t
r a
e cost e sin t 2e sin t 2e cos t
2 2 2 2
t t t t
1 1
cos 0 cos 0
2t
2e
for all t
2
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