BC 1- 2
Parametric Equations
Name:
Remember vectors from MI 4? We may view a parametrically-defined curve
x(t )  cos(t ), y (t )  sin(3t ) for 0 ≤ t ≤ 2 as a vector-valued function r (t )  x(t )i  y (t ) j . We often
show a vector r (t1 ) as an arrow from (0, 0) to the point  x(t1 ), y(t1 )  .
When we consider the tips of all such
vectors r (t ) as t takes on different
values, we see a parametrically-defined
curve. The velocity vector will be the
vector v (t ) that gives the instantaneous
rate of change in the x-direction and the
instantaneous rate of change in the ydirection. That is,
 dx   dy 
v (t )  vx i  v y j    i    j .
 dt   dt 
Most often, the velocity vector at t = t1
is drawn with its tail at the specific point
on the curve.
(1)
If a particle moves along a curve r (t )  x(t )i  y (t ) j , under what conditions will the particle come
to a stop?
(2)
State a formula for the length v (t ) of the velocity vector? What does this represent in terms of
the motion of the particle?
(3)
Consider the cycloid given by x(t )  t  sin(t ), y (t )  1  cos(t ) . Will a particle traveling along this
curve come to a stop? If so, for what values of t?
Parametrics 2.1
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BC 1- 2
Parametric Equations
Name:
Reminder:
2
The speed of an object traveling along a parametric curve is given by:
(4)
 dx   dy 
   
 dt   dt 
2
Let x  cos  t  and y  sin  2t  for 0  t  2.
B)
Find
dy
.
dx
a)
Sketch this parametric curve.
c)
Find the ordered pairs (x, y) of all points where the tangent line is
i)
Horizontal
ii) Vertical
d)
Assuming these equations represent the position in meters of an object at time t seconds, find


the speed of the object at time t  and then again at time t  .
4
2
e)
Eliminate the parameter t and write the Cartesian (x-y) equation of the curve. (Try!)
Parametrics 2.2
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(5)
Let x  8cos3  t  and y  8sin 3  t 
a)
Find each of the following.
dx
i)
dt
iii)
ii)
dy
dx
b)
Write the equation of the tangent line when t 
c)
Sketch this parametric curve in the space at the
right and label the value of t at each cusp.
d)
At each cusp, what is true about the value
dx
dy
of both
and
?
dt
dt
e)
Find each of the following:
dy
lim
i.
t 0 dx
f)
dy
dt

6
ii.
.
lim
t


dy
dx
2
If these equations represent the position in meters of a particle in motion at time t seconds,
find the speed of the particle at time:

t
i)
ii) t  2
4
Parametrics 2.3
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(6)
Let x  2cos  t  +cos  2t  and y  2sin t   sin  2t  . (Note: This curve is called a deltoid.)
a)
(7)
Sketch this parametric curve.
b)
Find
dy
dx
A balloon is filled with air and then released. As the air escapes, the balloon flies around the room
for 5 seconds following the path given by the parametric equations x  0.7t  3sin  2t  and
y  0.5t  4  4cos 3t  , where the position is measured in feet.
a)
Sketch the graph of this path for time 0 ≤ t ≤ 5 seconds.
b)
Find the speed of the balloon at time t = 1.5 seconds.
c)
Find the slope of the tangent line to the path at t = 1.5 seconds.
d)
Find the first time that the balloon is moving horizontally and find its speed at this time.
e)
Find the first time that the balloon is moving vertically and find its speed at this time.
Parametrics 2.4
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(8)
To find the second derivative
d2y
with parametric equations, we can use the following formula:
dx 2
d  dy 
2
d y dt  dx 

dx
dx 2
dt
Let's return to a simple ellipse from Parametrics 1: x = 5cos(t) and y = 2sin(t)
d2y
a) Find
.
dx 2
b)
Find
d2y
. What does this value normally tell us about the concavity of the path?
dx 2 t 
6
c.
Find
d2y
. What does this value normally tell us about the concavity of the path?
dx 2 t 
6
d)
By considering the graph of this ellipse and your answers to parts b and c, explain why we
often avoid talking about concavity and inflection points on parametrically-defined curves.
Parametrics 2.5
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