SME
Calculus II · Problem type
Slope of a polar curve
Use this for “Find the slope of the tangent to r = 1 + cos θ at θ = π/2.”
Diagram
r = 1 + 1 cos θ: a cardioid, θ from 0 to 2π.
(r, θ) = (1, π/2)
Tangent
at
θ
=
π
2
:
r′
=
−1,
dy/dx
=
r′ sin θ + r cos θ
r′ cos θ − r sin θ
=
1
This is an example. Type your own values to start a new problem.
r
=
a
+
b
cos
θ
dy
dx
at
θ
=
θ (rad)
=
dy/dx
r
Distance from the pole
calculated
r′
Rate dr/dθ
calculated
dy/dθ
Rate of y per radian
worked out (not typed)
dx/dθ
Rate of x per radian
worked out (not typed)
Formulas
r
=
a
+
b
×
cos(
θ
)
1
=
1
+
1
×
cos(
π
2
)
r
′
=
−
b
×
sin(
θ
)
−1
=
−1
×
sin(
π
2
)
dy/dθ
=
r
′
×
sin(
θ
)
+
r
×
cos(
θ
)
−1
=
−1
×
sin(
π
2
)
+
1
×
cos(
π
2
)
dx/dθ
=
r
′
×
cos(
θ
)
−
r
×
sin(
θ
)
−1
=
−1
×
cos(
π
2
)
−
1
×
sin(
π
2
)
dy/dx
=
dy/dθ
dx/dθ
1
=
−
1
−1
Assumptions
•
The point is x = r cos θ, y = r sin θ, so dy/dx = (dy/dθ) ÷ (dx/dθ) wherever dx/dθ is not 0.
•
θ is in radians, as calculus takes it (90° is π/2), so r′ = dr/dθ is per radian.
•
Where dx/dθ = 0 and dy/dθ is not, the tangent is vertical and has no slope.
Step-by-step
WE KNOW
a = 1, b = 1, θ = π/2 rad
FIND
distance from the pole (r), rate dr/dθ (r′), rate of y per radian (dy/dθ), rate of x per radian (dx/dθ), slope of the tangent (dy/dx)
Step 1 · Find distance from the pole (r)
Use r = a + b × cos(θ)
Put
θ
into
the
curve’s
equation.
r
=
1
+
1
×
cos(
π
2
)
r
=
1
+
cos(
π
2
)
r
=
1
Step 2 · Find rate dr/dθ (r′)
Use r′ = −b × sin(θ)
The
derivative
of
cos
θ
is
−sin
θ
,
with
θ
in
radians;
the
constant
a
drops
out.
r
′
=
−1
×
sin(
π
2
)
r
′
=
−1
Step 3 · Find rate of y per radian (dy/dθ)
Use dy/dθ = r′ × sin(θ) + r × cos(θ)
y
=
r
sin
θ
,
so
by
the
product
rule
y
changes
at
r
′
sin
θ
+
r
cos
θ
.
dy/dθ
=
−1
×
sin(
π
2
)
+
1
×
cos(
π
2
)
dy/dθ
=
−1
×
sin(
π
2
)
+
cos(
π
2
)
dy/dθ = −1
Step 4 · Find rate of x per radian (dx/dθ)
Use dx/dθ = r′ × cos(θ) − r × sin(θ)
x
=
r
cos
θ
,
so
by
the
product
rule
x
changes
at
r
′
cos
θ
−
r
sin
θ
.
dx/dθ
=
−1
×
cos(
π
2
)
−
1
×
sin(
π
2
)
dx/dθ
=
−1
×
cos(
π
2
)
−
sin(
π
2
)
dx/dθ = −1 × 0 − 1
dx/dθ = 0 − 1
dx/dθ = −1
Step 5 · Find slope of the tangent (dy/dx)
Use dy/dx = (dy/dθ) ÷ (dx/dθ)
Chain rule: divide the rate of y by the rate of x.
dy/dx
=
−
1
−1
dy/dx = 1
Check
Put the numbers back in. Both sides should match.
1
=
1
+
1
×
cos(
π
2
)
✓
−1
=
−1
×
sin(
π
2
)
✓
−1
=
−1
×
sin(
π
2
)
+
1
×
cos(
π
2
)
✓
−1
=
−1
×
cos(
π
2
)
−
1
×
sin(
π
2
)
✓
1
=
−
1
−1
✓
Topic check
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