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Calculus
Stewart
Dr. Berg
Summer ‘09
Page 1
11.1
11.1
Curves Defined by Parametric Equations
Parametric Equations
Some curves are most easily defined by giving the
x
and
y
coordinates as
functions of a third parameter
t
.
Definition
A
parametric curve
(in
R
2
) is a curve with coordinates given by
parametric
equations
x
=
f
(
t
)
and
y
=
g
(
t
)
.
Example A
What curve is represented by
x
=
2cos
t
and
y
=
2sin
t
? Notice that
x
2
+
y
2
=
(2cos
t
)
2
+
(2sin
t
)
2
=
4
defines a circle of radius 2 centered at the origin.
Example B
What curve is represented by
x
=
t
2
+
t
and
y
=
2
t
−
1
? If we plot some points and
connect the dots, we get this curve:
The arrows on the graph indicate the direction of motion of the point
(
x
(
t
),
y
(
t
))
as
t
increases.
To find the equation in
x
and
y
, we solve
y
=
2
t
−
1
for
t
and substitute into
x
=
t
2
+
t
. Indeed
t
=
1
2
(
y
+
1)
and
x
=
t
2
+
t
=
1
2
(
y
+
1)
[ ]
2
+
1
2
(
y
+
1)
=
1
4
y
2
+
y
+
3
4
.
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Stewart
Dr. Berg
Summer ‘09
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This note was uploaded on 06/06/2010 for the course M 408 taught by Professor Hodges during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Hodges

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