9.1 Parametric Curves
9.2 Calculus with Parametric
Curves
There are times when we need to describe motion (or a
curve) that is not a function.
We can do this by writing equations for
the x and y coordinates in terms of a
third variable (usually t or  ).
x  f t  y  g t 
These are called
parametric equations.
“t” is the parameter. (It is also the independent variable)
Circle:
If we let t = the angle, then:
t
x  cos t
y  sin t
0  t  2
sin 2 t  cos 2 t  1
y 2  x2  1
We could identify the
parametric equations
as a circle.
x2  y 2  1
Ellipse:
x  3cos t
y  4sin t
x
 cos t
3
y
 sin t
4
2
2
x  y
2
2


cos
t

sin
t
   
3  4
2
2
x  y
    1
3  4
This is the equation
of an ellipse.
Tangents
The formula for finding the slope of a parametrized
curve is:
dy
dy
 dt
dx
dx
dt
We assume that the
denominator is not
zero.
To find the second derivative of a parametrized curve,
we find the derivative of the first derivative:
dy
2
d y d
dt



y


2
dx
dx
dx
dt
Tangents
Example:
d2y
2
3
Find
as
a
function
of
t
if
x

t

t
and
y

t

t
.
2
dx
0.5
-2
-1.5
-1
-0.5
0.5
0
-0.5
-1
Example (cont.):
d2y
2
3
Find
as
a
function
of
t
if
x

t

t
and
y

t

t
.
2
dx
1. Find the first derivative (dy/dx).
dy
dy
y 
 dt
dx
dx
dt
1  3t

1  2t
2

2. Find the derivative of dy/dx with respect to t.
dy d  1  3t  2  6t  6t
 
 
2
dt dt  1  2t 
1  2t 
2
2

2  6t  6t
3. Divide by dx/dt.
dy
2
d y
dt

2
dx
dx
dt
1  2t 

2
2
1  2t

2  6t  6t
1  2t 
3
2
Areas under parametric curves
• If a curve is given by parametric equations x=f(t), y=g(t)
and is traversed once as t increases from α to β, then the
area under the curve is

A   g (t ) f (t )dt

• Examples on the board
Lengths of parametric curves
• If a curve C is described by the parametric equations
x=f(t), y=g(t), α ≤ t ≤ β, where f’ and g’ are continuous
on [α, β] and C is traversed exactly once as t increases
from α to β, then the length of the curve is
L


  
dx
dt
• Examples on the board
2
dy

dt
2
dt