Thomas Calculus 13th [Solutions]

Section 13.4 Curvature and Normal Vectors of a Curve 949
18. r a cos t i bsin t j v a sin t i b cos t j a a cos t i bsin
t j
i j k
v a
v a a sin t b cos t 0 abk v a ab ab,
since a b 0; ( t)
3
v
a cos t bsin t 0
2 2 2 2
3 2
2 2 2 2
5 2
2 2
ab a sin t b cos t ; ( t) 3 ab a sin t b cos t 2a sin t cos t 2b sin t cost
2
2 2 2 2 2 2
5 2
3 ab a b sin t a sin t b cos t ; thus, ( t) 0 sin 2t 0 t 0, identifying points
2
on the major axis, or t , 3 identifying points on the minor axis. Furthermore, ( t ) 0 for 0 t and
2 2
2
for t 32 ; ( t ) 0 for t and 3 t 2 . Therefore, the points associated with t 0 and t
2
2
on the major axis give absolute maximum curvature and the points associated with t and t 3 on the
2
2
minor axis give absolute minimum curvature.
19.
2 2
a d a b 2 2
; d 0 a b 0 a b a b since a, b 0. Now, d 0
2 2 da 2 2
2
a b da
da
a b
if a
b
and d 0
da
if a b is at a maximum for a b and ( b ) b 1 is the maximum value of
2 2
b b 2b
20. (a) From Example 5, the curvature of the helix r( t) a cost i asin t j btk , a, b 0 is a ; also
2 2
a b
v a
2 b
2 . For the helix r( t) 3cost i 3sin t j tk , 0 t 4 , a 3 and b 1 3 3
2 2
3 1 10
4
4
and v 10 K 3 10 dt 3 t 12
0 10
10 0 10
2
2 2 2
(b) y x x t and y t , t r( t) ti t j v i 2t j v 1 4 t ;
2
1 2t d T
; 4t 2 d T
T i j i j ; 16t
4 2 . Thus 1 2t
2 2 2
3 2
2
3 2
2
3 2
1 4t 1 4t dt dt 2
1 4t 1 4t 1 4t
1 4t
2
1 4t
1 4t
2
1 4t
2
3
.
2 2
2
0
K 1 4t dt dt lim 2 dt lim b 2 dt
1 4t
a
b
Then
3 2 2 2
2
1 4t a 1 4t 0 1 4t
1
0
1 1 1
lim tan 2t lim tan 2t b
lim tan 2a lim tan 2b
a a b 0 a b
2 2
21.
2 2
2 2
r ti sin t j v i cost j v 1 cost 1 cos t v 1 cos 1;
2 2
i cost j d T sin
sin
sin t cost sin t d T t d T
T v
i j ; 1 1. Thus
v 1 cos
1
1 cos 1 cos
2
2 2
3 2
2
3 2 2 2
t
dt dt 1 cos dt
t t
t t 1 cos
2
2
1 1 1 1 1 and the center is
2 1 1
2
2
,0 x y 1
2 2
22.
2 2 2
r 2ln t i t 1 j v 2 i 1 1 j v 4 1 1 t 1 T v 2 t i t 1 j;
t t 2 2 2 2 2 2
t t t t v t 1 t 1
2 2
2
2
d T 2 t 1 4 t 1 16t
4t
d T
i j 2 . Thus
dt 2
2
2
2 dt 2
4 2
t 1 t 1 t 1 t 1
v
d T 2 2
t 2 2t
dt t 1 t 1 t 1
2 2
2
2
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