SME
Calculus II
Parametric and polar calculus
Use this for “x = 3 cos t, y = 2 sin t. Find dy/dx at t = π/4.”
Diagram
At t = π/4 the point is (2.12, 1.41).
The arrows show the way t runs.
This is an example. Type your own values to start a new problem.
x
=
a
cos
t,
y
=
b
sin
t
dy
dx
at
t
=
t (rad)
=
dy/dx
x
x at t
calculated
y
y at t
calculated
dx/dt
Rate of x per radian
worked out (not typed)
dy/dt
Rate of y per radian
worked out (not typed)
Formulas
x
=
a
×
cos(
t
)
2.1213
=
3
×
cos(
π
4
)
y
=
b
×
sin(
t
)
1.4142
=
2
×
sin(
π
4
)
dx/dt
=
−
a
×
sin(
t
)
−2.1213
=
−3
×
sin(
π
4
)
dy/dt
=
b
×
cos(
t
)
1.4142
=
2
×
cos(
π
4
)
dy/dx
=
dy/dt
dx/dt
−0.6667
=
1.4142
−2.1213
Assumptions
•
x and y both change with t, so the chain rule gives dy/dx = (dy/dt) ÷ (dx/dt) wherever dx/dt is not 0.
•
t is in radians, as calculus takes it (45° is π/4).
•
Where dx/dt = 0 (t = 0 or π) the tangent is vertical and has no slope; where dy/dt = 0 it is flat.
Step-by-step
WE KNOW
a = 3, b = 2, t = π/4 rad
FIND
x at t (x), y at t (y), rate of x per radian (dx/dt), rate of y per radian (dy/dt), slope of the tangent (dy/dx)
Step 1 · Find x at t (x)
Use x = a × cos(t)
Put
t
into
the
first
equation.
x
=
3
×
cos(
π
4
)
x
=
3
×
0.7071
x
=
2.1213
Step 2 · Find y at t (y)
Use y = b × sin(t)
Put
t
into
the
second
equation.
y
=
2
×
sin(
π
4
)
y
=
2
×
0.7071
y
=
1.4142
Step 3 · Find rate of x per radian (dx/dt)
Use dx/dt = −a × sin(t)
The
derivative
of
cos
t
is
−sin
t
,
with
t
in
radians,
so
x
changes
at
−a
sin
t
.
dx/dt
=
−3
×
sin(
π
4
)
dx/dt = −3 × 0.7071
dx/dt = −2.1213
Step 4 · Find rate of y per radian (dy/dt)
Use dy/dt = b × cos(t)
The
derivative
of
sin
t
is
cos
t
,
so
y
changes
at
b
cos
t
.
dy/dt
=
2
×
cos(
π
4
)
dy/dt = 2 × 0.7071
dy/dt = 1.4142
Step 5 · Find slope of the tangent (dy/dx)
Use dy/dx = (dy/dt) ÷ (dx/dt)
Chain
rule:
dy/dt
=
(dy/dx)(dx/dt),
so
divide
the
rate
of
y
by
the
rate
of
x
.
That
is
−(
b
a
)
cot
t
.
dy/dx
=
1.4142
−2.1213
dy/dx = −0.6667
Check
Put the numbers back in. Both sides should match.
2.1213
=
3
×
cos(
π
4
)
✓
1.4142
=
2
×
sin(
π
4
)
✓
−2.1213
=
−3
×
sin(
π
4
)
✓
1.4142
=
2
×
cos(
π
4
)
✓
−0.6667
=
1.4142
−2.1213
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.