Thomas Calculus 13th [Solutions]

8. (a)
Section 13.4 Curvature and Normal Vectors of a Curve 947
1 3 2 2
r( t)
ti t j v i t j n t i j points toward the concave side of the curve when t 0 and
3
2
2
n t i j points toward the concave side when t 0 N 1 t i j for t 0 and
4
1 t
2
N 1 t i j for t 0
4
1 t
(b) From part (a),
4
1
1 t d T 2t 2t d T
v t T i j i j
4t 4t
1 t 1 t
dt
dt
1 t 1 t 1 t
d T
t dt 4 3 3
t t t t t
4 d 2 4
3 2
4
3 2
t
T t
4 4
dt
1 t 1 t t t t t
2 3 6 2
4 4 4
3 2
4
3 2
4
3
2
; N 1 2 i 2 j i j; t 0. N does not exist at t 0, where
1 1 1
d T
d T d T
the curve has a point of inflection; 0 so the curvature dt 0 at
dt
t 0
ds ds ds
0 1 d T
3 3
t N is undefined. Since x t and y
1
t y
1
x , the curve is the cubic power curve
ds
3 3
which is concave down for x t 0 and concave up for x t 0.
9.
2 2 2
r 3sin i 3cos j 4 k v 3cos i 3sin j 4k v 3cos 3sin 4 25 5
t t t t t t t
3 3 3 3 3
2
3
2
cos 4
3
5 t sin
5 t d T
d
sin cos sin cos
5 dt 5 t 5 t T
T v i j k i j
v
dt 5 t 5 t 5
dT
dt
N sin t i cos t j;
dT
dt
1 3 3
5 5 25
10.
2 2 2
r cos sin i sin cos j 3k v cos i sin j v cos sin
t t t t t t t t t t t t t t t
t t , if t 0 T v cost i sin t j, t 0 dT
sin t i cost
j
v
dt
dT 2 2
dt
sin t cos t 1 N sin t i cos t j;
1 1 1
dt
d T
t t
d T
dt
t t t t t t
11. r e cost i e sin t j 2k v e cost e sin t i e sin t e cost
j
t t
2
t t
2
2t t
v e cos t e sin t e sin t e cost 2e e 2; T v cost sin t i sin t cost
j
v 2 2
2 2
d T sin t cost cost sin t d T
i
j
sin t cost cos t sin t 1
dt 2 2 dt
2 2
d T
dt
N cost sin t i sin t cost
j;
1 d T 1 1 1
d T
t
t
2 2
v dt e 2 e 2
dt
12.
2 2 2
r 6sin 2 i 6cos 2 j 5 k v 12cos 2 i 12sin 2 j 5k v 12cos 2 12sin 2 5
t t t t t t t
169 13 12 12 5
24 24
13 cos 2 13 sin 2 d T
T v
t i t j k
13 dt 13 sin 2 t i
13
cos 2 t j
v
d T
dT 2 2
24 24 24
dt
sin 2t cos 2t N sin 2t i cos 2 t j;
1 d T 1 24 24
dt 13 13 13
d T
v dt 13 13 169 .
dt
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