Thomas Calculus 13th [Solutions]

806 Chapter 11 Parametric Equations and Polar Coordinates
37. Draw line AM in the figure and note that AMO is a right angle
since it is an inscribed angle which spans the diameter of a circle.
2 2 2
Then AN MN AM . Now, OA a,
AN tan t
a , and
AM sin t . Next MN OP
a
2 2 2 2 2 2 2
OP AN AM a tan t a sin t
2
2 2 2 2 2 a sin t
OP a tan t a sin t ( a sin t) sec t 1 .
cos t
3
a sin t 2
In triangle BPO, x OP sin t a sin t tan t and
cos t
2 2
y OP cos t a sin t x a sin t tan t and
2
y a sin t.
38. Let the x -axis be the line the wheel rolls along with the y -axis through a low point of the trochoid (see the
accompanying figure).
Let denote the angle through which the wheel turns. Then h a and k a . Next introduce x y -axes
parallel to the xy -axes and having their origin at the center C of the wheel. Then x b cos and
y b sin , where 3 . It follows that x b cos 3 b sin and y b sin 3
2
2
2
b cos x h x a b sin and y k y a b cos are parametric equations of the trochoid.
39.
40.
2 2 2 2 2 2 2
2
2 4
D ( x 2) y 1 D ( x 2) y 1 ( t 2) t 1 D t 4t
17
2 2 2 4
2
d D
3
4t 4 0 t 1. The second derivative is always positive for t 0 t 1 gives a local
dt
2
minimum for D (and hence D) which is an absolute minimum since it is the only extremum the closest
point on the parabola is (1, 1).
3
2
2 2 3
2
2
D 2 cos t (sin t 0) D 2 cos t sin t
4 4
2
d D
dt
2 2 cos t 3 ( 2 sin t) 2sin t cos t ( 2 sin t) 3 cos t 3 0 2 sin t 0 or 3 cos t 3 0
4 2
2
2 2
d D
0, or , 5
2 2
d D
t t . Now 6 cos t 3 cos t 6 sin t so that (0) 3
3 3
2
2
dt
dt
2 2
2 2
d D
d D
maximum, ( ) 9 relative maximum,
9 relative minimum, and
2 2
dt
dt 3 2
2 2
relative
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