Section 10.1: Parametric equations; Tangent
lines and arc length for parametric curves
Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves
Now that we have finished chapter 9, we have completed
what is generally considered “calculus”.
For the remainder of the semester, we will discuss some topics
that will prepare you for calculus III, as well as other advanced
courses in mathematics.
First we study: Parametric Curves.
Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves
We begin with the following definition:
Definition 1
A
parametric curve
is a curve in the
xy
plane whose coordinates
(
x
,
y
) can be specified by functions
f
(
t
) and
g
(
t
) such that each
point (
x
,
y
) on the curve satisfies
x
=
f
(
t
)
and
y
=
g
(
t
)
for some
t
. The variable
t
is called a
parameter
.
By selecting a few values of
t
, we can plot points and sketch
graphs of the curves in much the same way as we do for functions.
Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves
Example 1
Sketch a graph of the curve defined by
x
=
t
+ 1, and
y
=
t
+ 2.
Solution.
First we make a table of values:
t
1
0
1
2
x
0
1
2
3
y
1
2
3
4
Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves
If we plot (
x
,
y
) for each
t
, we get the following scatter plot:
Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves
If we draw a smooth curve through these points, we obtain the
following graph:
Remarks:
Here, it is easy to see what the final curve will look like.
For more complicated curves, the resulting scatter plot may
not be so easy to sketch.
Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves
Example 2
Sketch a graph of the parametric curve defined by
x
=
t

3 sin
t
,
y
= 4

3 cos
t
.
Solution.
As before, we first make a table of values:
t
0
1
2
3
4
5
6
7
8
9
10
x
0
1.5
0.7
2.6
6.3
7.9
6.8
5.0
5.0
7.8
11.6
y
1.0
2.4
5.2
7.0
6.0
3.1
1.1
1.7
4.4
6.7
6.5
Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves
If we plot these points, we obtain the following scatter plot:
Looking at these points, it is not so clear that what the curve
should look like. If we add (many) more points to our scatter plot,
you should (hopefully) be able to see that it should look like this:
Section 10.1: Parametric equations; Tangent lines and arc length
Parametric curves
Thus, drawing sketches of parametric curves can be quite difficult;
more so than for functions
y
=
f
(
x
).