24. (a)
[
2
1, 5] by [
2
1, 3]
Initial point: (0, 2)
Terminal point: (4, 0)
(b)
y
5
ˇ
4
w
2
w
t
w
2
w
5
ˇ
4
w
2
w
x
w
The parametrized curve traces the right portion of the
curve defined by
y
5
ˇ
4
w
2
w
x
w
, that is, for
x
$
0.
25. (a)
[
2
3, 3] by [
2
2, 2]
No initial or terminal point, since the
t
interval has no
beginning or end. The curve is traced and retraced in
both directions.
(b)
y
5
cos 2
t
5
cos
2
t
2
sin
2
t
5
1
2
2 sin
2
t
5
1
2
2
x
2
52
2
x
2
1
1
The parametrized curve traces the portion of the
parabola defined by
y
52
2
x
2
1
1 corresponding to
2
1
#
x
#
1.
26. (a)
[
2
4, 5] by [
2
4, 2]
Initial point: None
Terminal point: (
2
3, 0)
(b)
x
5
t
2
2
3
5
y
2
2
3
The parametrized curve traces the lower half of the
parabola defined by
x
5
y
2
2
3 (or all of the curve
defined by
y
52
ˇ
x
w
1
w
3
w
).
27.
Using (
2
1,
2
3) we create the parametric equations
x
52
1
1
at
and
y
52
3
1
bt
, representing a line which
goes through (
2
1,
2
3) at
t
5
0. We determine
a
and
b
so
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 Spring '08
 GERMAN
 Parametric equation, Vectorvalued function, possible parametrization

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