Gauthier, Joseph – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert
1
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have
17
questions.
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The due time is Central
time.
YES, homework 2 is due BEFORE
homework 1
001
(part 1 oF 1) 10 points
Determine
A
so that the curve
y
= 8
x
+ 21
can be written in parametric Form as
x
(
t
) =
t

3
,
y
(
t
) =
At

3
.
1.
A
=

7
2.
A
=

6
3.
A
=

8
4.
A
= 7
5.
A
= 6
6.
A
= 8
correct
Explanation:
We have to eliminate
t
From the parametric
equations For
x
and
y
. Now From the equation
For
x
it Follows that
t
=
x
+ 3. Thus
y
= 8
x
+ 21 =
A
(
x
+ 3)

3
.
Consequently
A
= 8
.
keywords:
Cartesian equation, parametric
equations, eliminate parameter
002
(part 1 oF 1) 10 points
Determine a Cartesian equation For the
curve given in parametric Form by
x
(
t
) = 2
e
t
,
y
(
t
) = 4
e

2
t
.
1.
x
2
y
= 16
2.
x
2
y
= 32
3.
x
2
y
= 32
4.
x
2
y
= 16
correct
5.
x
y
2
= 8
6.
xy
2
= 8
Explanation:
We have to eliminate the parameter
t
From
the equations For
x
and
y
.
Now From the
equation For
x
it Follows that
e
t
=
x
2
,
From which in turn it Follows that
y
= 4
‡
2
x
·
2
.
Consequently,
x
2
y
= 16
.
keywords:
Cartesian equation, parametric
equations, eliminate parameter
003
(part 1 oF 1) 10 points
Describe the motion oF a particle with posi
tion
P
(
x, y
) when
x
= 2 sin
t,
y
= 4 cos
t
as
t
varies in the interval 0
≤
t
≤
2
π
.
1.
Moves along the line
x
2
+
y
4
= 1
,
starting at (2
,
0) and ending at (0
,
4).
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View Full DocumentGauthier, Joseph – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert
2
2.
Moves once counterclockwise along the
ellipse
x
2
4
+
y
2
16
= 1
,
starting and ending at (0
,
4).
3.
Moves once clockwise along the ellipse
x
2
4
+
y
2
16
= 1
,
starting and ending at (0
,
4).
correct
4.
Moves once counterclockwise along the
circle
(2
x
)
2
+ (4
y
)
2
= 1
,
starting and ending at (0
,
4).
5.
Moves along the line
x
2
+
y
4
= 1
,
starting at (0
,
4) and ending at (2
,
0).
6.
Moves once clockwise along the circle
(2
x
)
2
+ (4
y
)
2
= 1
,
starting and ending at (0
,
4).
Explanation:
Since
cos
2
t
+ sin
2
t
= 1
for all
t
, the particle travels along the curve
given in Cartesian form by
x
2
4
+
y
2
16
= 1 ;
this is an ellipse centered at the origin.
At
t
= 0, the particle is at (2 sin 0
,
4 cos 0),
i.e.
,
at the point (0
,
4) on the ellipse.
Now as
t
increases from
t
= 0 to
t
=
π/
2,
x
(
t
) increases
from
x
= 0 to
x
= 2, while
y
(
t
) decreases from
y
= 4 to
y
= 0 ; in particular, the particle
moves from a point on the positive
y
axis to
a point on the positive
x
axis, so it is moving
clockwise
.
In the same way, we see that as
t
increases
from
π/
2 to
π
, the particle moves to a point
on the negative
y
axis, then to a point on the
negative
x
axis as
t
increases from
π
to 3
π/
2,
until Fnally it returns to its starting point on
the positive
y
axis as
t
increases from 3
π/
2 to
2
π
.
Consequently, the particle moves clockwise
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 Fall '07
 Gilbert
 Cos, Parametric equation, Conic section, parametric form

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