Monday, April 1
±
Lecture 33 :
Parametric Equations, curves and tangents
(Refers
to Section 9.1, 9.2)
After having practiced the problems associated to the concepts of this lecture the student
should be able to
: Plot simple parametric equations in the plane, find the equations of all tangent
lines to a curve in the plane which is expressed as parametric equations, compute curve lengths.
33.1
Definition
±
A curve
C
in the Cartesian plane is called a
parametric curve
if the
points (
x
,
y
) on the curve
C
can be represented as (
x
(
t
),
y
(
t
)) where
t
is a variable in an
interval [
a
,
b
] on the real line
R
.
The variable
t i
s called a
parameter
.
The point (
x
(
a
),
y
(
a
)) is called the
initial point
of the curve, while the point (
x
(
b
),
y
(
b
)) is called the
terminal point
of the curve.
The "
direction
of the curve
" is from the initial point to the terminal point.
The two equations
x
=
x
(
t
) and
y
=
y
(
t
) are called its
parametric equations
of the
curve
C
.
A simple example of this is a curve
C
which represents the trajectory of an object in
two-space during a time interval [0, 10], where (
x
(
t
),
y
(
t
)) gives the position of the
object at time
t,
t
between 0 seconds and 10 seconds.
33.1.1
Examples
±
Graphing parametric equations in a rudimentary way.
Plot the curve whose parametric equations are
x = t
±
3 sin
t
and
y
= 4
±
3 cos
t
,
0
d
t
d
10.
By drawing out a table and plotting the points we get: (The values of
t
are in radians,
values of
x
and
y
are computed with a calculator.)
t
= 0
x
= 0.0
y
= 1.0.
t
= 1
x
=
±
1.5
y
= 2.4.
t
= 2
x
=
±
0.7
y
= 5.2.
t
= 3
x
= 2.6
y
= 7.0.
t
= 4
x
= 6.3
y
= 6.0.
t
= 5
x
= 7.9
y
= 3.1.
t
= 6
x
= 6.8
y
= 1.1.
t
= 7
x
= 5.0
y
= 1.7.

33.1.2
Note on how to plot parametric curves
±
Sometimes parametric equations can
easily be transformed into and
xy
-equation (but this may be difficult or even
impossible to do).
When trying to plot a parametric curve it is a good idea, if it is possible, to
eliminate the parameter t to obtain an expression of a familiar curve in x and
y.
Example – Plot the parametric curve (
x
(
t
) ,
y
(
t
)) where
and
t
is in [0,1],
by eliminating the parameter and indicate the direction of the curve.