Tangent Lines
Arc Length
Area
Exercises
Calculus of Polar Curves
Mathematics 54–Elementary Analysis 2
Institute of Mathematics
University of the PhilippinesDiliman
1 / 22
Tangent Lines
Arc Length
Area
Exercises
Tangent Lines to Polar Curves
Goal
: obtain slopes of tangent lines to polar curves of form
r
=
f
(
θ
)
2 / 22
Tangent Lines
Arc Length
Area
Exercises
Tangent Lines to Polar Curves
Parametrization of a Polar Curve
A polar curve
r
=
f
(
θ
) can be parametrized as
x
=
r
cos
θ
=
f
(
θ
)cos
θ
y
=
r
sin
θ
=
f
(
θ
)sin
θ
Recall.
slope of a parametric curve
dy
dx
=
dy
d
θ
dx
d
θ
dx
d
θ
=
f
0
(
θ
) cos
θ

f
(
θ
)sin
θ
,
dy
d
θ
=
f
0
(
θ
) sin
θ
+
f
(
θ
)cos
θ
Slope of a Tangent Line to a Polar Curve
Given that
dy
/
d
θ
and
dx
/
d
θ
are continuous and
dx
/
d
θ
6=
0, then the slope
of the polar curve
r
=
f
(
θ
)is
dy
dx
=
dr
d
θ
sin
θ
+
r
cos
θ
dr
d
θ
cos
θ

r
sin
θ
3 / 22
Tangent Lines
Arc Length
Area
Exercises
Tangent Lines to Polar Curves
Example
Find the (Cartesian) equation of the tangent line to the cardioid
r
=
1
+
sin
θ
at the point where
θ
=
π
3
.
4 / 22
Tangent Lines
Arc Length
Area
Exercises
Tangent Lines to Polar Curves
Example
Determine the points on the cardioid
r
=
1

cos
θ
where the tangent line is
horizontal, or vertical.
Recall.
For a parametric curve
C
:
x
=
x
(
θ
),
y
=
y
(
θ
),
dy
d
θ
=
0
=⇒
horizontal tangent line
dx
d
θ
=
0
=⇒
vertical tangent line
provided that they are not simultaneously zero
.
Note that for the above curve, we have the following parametrization:
x
=
r
cos
θ
=
(1

cos
θ
)cos
θ
=
cos
θ

cos
2
θ
y
=
r
sin
θ
=
(1

cos
θ
)sin
θ
=
sin
θ

sin
θ
cos
θ
5 / 22
Tangent Lines
Arc Length
Area
Exercises
Tangent Lines to Polar Curves