Accelerated Calculus
Name______________________
Parametric Equations (Paths & Motion)
11/9/11
The Calculus of Parametric Equations
When talking about motion of objects – cars, planets, bugs, etc., the path an object carves out is a
function of time. Since the trajectory an object takes involves both horizontal and vertical components
that can be independent of each other, we create a mathematical system that allow us to study these both
as a function of time.
We can examine the path of an object by looking at a set of parametric equations. For example, the
following is a set of parametric equations:
x  t   t 3  3t
y  t   t 2  2t
where  x  t  , y  t   is a point on the path -- where the horizontal/vertical positions are functions of time
So that we gain familiarity working with parametric equations and our graphing calculator, let’s try
looking at a graphical representation of this curve, defined parametrically.
Graphing Curves (using Parametric Mode)
Step 1: Press the MODE button.
Step 2: Go down and make sure Par is selected (instead of Func)
Step 3: Press Y= editor and type in the equations above
Step 4: Press ZOOM and go to ZStandard – notice that you are able to see a curve, but that it is difficult
to see the path and also difficult to follow the precise direction of the trajectory. So, let’s play around
with the window settings so that you know how to do this for other exercises.
Step 5: Press WINDOW and type the following:
Also set your Ymax = 4 and Yscl = 1.
Step 6: Now, press GRAPH and observe how you can better “see” the path and follow its direction! Play
with different Tstep-values so you can see its effect.
Now, let’s embark on an exploration that will lead us to consider questions about parametric equations
and the calculus that can be used to successfully answer these questions!
Accelerated Calculus
Observing Different Paths
Task 1: Suppose we are given the following set of parametric equations where x & y are measured in
meters and t is measured in seconds:
x(t )  4  3t
y (t )  1  6t
Complete the following:
1) Graph the parametric curve on a calculator (use ZStandard and play with window if needed).
2) Convert the set of parametric equations to a Cartesian equation (your result here should match
what you see in your graph!).
3) Find
dy
dx
given your result above.
4) Calculate dx
and dydt . Using these computations, find an expression for the slope of a set of
dt
parametric equations.
Task 2: Suppose we are given the following set of parametric equations where x & y are measured in
meters and t is measured in seconds:
x(t )  cos(t )
y (t )  sin(t )
Complete the following:
1) Graph the parametric curve on a calculator (use ZStandard and play with window if needed). Set
Tmin = 0. What should the minimum Tmax be to obtain a complete graph? Does this value make
sense?
2) Convert the set of parametric equations to a Cartesian equation (your result here should match
what you see in your graph!).
3) Find
dy
dx
given your result above.
dy
4) Calculate dx
dt and dt . Using these computations, find an expression for the slope of a set of
parametric equations.
Summary:
What do you believe you can safely conclude (given the results on these last two tasks) concerning the
dy
dy
dy
relationship between dx
dt , dt , and dx ? In other words, how do we find the slope ( dx ) of a curve, defined
parametrically?
Accelerated Calculus
Observing Different Motions
Task 3: Suppose we are given the following set of parametric equations, which defines the path of an
object, where x & y are measured in meters and t is measured in seconds:
x(t )  cos(3t )
y (t )  sin(3t )
Complete the following:
1) Find the minimum Tmax be to obtain a complete graph?
2) Find the instantaneous horizontal velocity (in the x-direction) of this object. Find the
instantaneous vertical velocity (in the y-direction) of this object. Check your units.
3) Explain why
dy
dx
does not give us the instantaneous speed of this object?
4) Using your intuition and a bit of extra thought, find an expression that will produce the
instantaneous speed of this object. Check your answer using a quick “unit analysis”.
Observing Different Motions
Task 4: Suppose we are given the following set of parametric equations, which defines the path of an
object, where x & y are measured in meters and t is measured in seconds:
x(t )  cos( 3t )
y (t )  sin( 3t )
Complete the following:
1) Describe the path the object takes. Be specific so as to address the start point, the end point, and
the direction of this trajectory.
2) Using this latest set of parametric equations as an example, find a set of parametric equations that
define a circle centered at origin w/radius 4, carved-out counter-clockwise, starting at (-4,0),
traced only once.
x (t ) 
on the interval
y (t ) 


Accelerated Calculus
Observing Restricted Parts of a Path
Task 5: Suppose we are given the following set of parametric equations (same as in Task 1), which
defines the path of an object where x & y are measured in meters and t is measured in seconds:
x(t )  4  3t
y (t )  1  6t
Complete the following:
1) Graph this path on your calculator
2) Briefly describe how you would construct a line segment (as opposed to a line), with endpoints
(1, -7) and (-5, 17).
Task 6: Suppose we are given the following set of parametric equations (same as in Task 1), which
defines the path of an obejct:
x(t )  2 cos(t )
y (t )  sin(t )
Complete the following:
1) Graph this path on your calculator
2) You should see an ellipse. Briefly describe how you would construct the top half of this ellipse.
Interpreting Derivatives
Task 7: Suppose we are given the following set of parametric equations (same as in Task 1), which
defines the path of an object where x & y are measured in meters and t is measured in seconds:
x(t )  t 3  3t
y (t )  t 2
, on the interval [-3, 3]
Complete the following:
dy
, the slope of this parametrically defined curve.
dx
2) At what point(s) are there horizontal tangents? vertical tangents?
d 2x
d2y
3) Show that 2  6t and that 2  2 and interpret what each of these quantities tell us in this
dt
dt
context. Use units to support your answer.
6 t2 1
d2y

4) Show that
. What is the meaning of this expression? What are the units of
3
dx 2
27 t 2 1
1) Find


this quantity?


Accelerated Calculus
Numerical Approximations (using Excel)
Task 8: Suppose we are given the following set of parametric equations (same as in Task 7), which
defines the path of an object where x & y are measured in meters and t is measured in seconds:
x(t )  t 3  3t
y (t )  t 2
, on the interval [-3, 3]
Complete the following:
1) In Excel, create a column that provides times, starting at t  3 , with t  0.1 and ending at
t  3 (clearly label this column).
2) Now create a column for x-values and y-values that correspond to each time.
3) Create a scatter-plot of y vs x.
4) Create columns for
dx
dt
,
dy
dt
, and
dy
dx
.
5) Using one of your results from Task 7, calculate
dy
dx
t 1
and
dy
dx
t 2
. Compare your results with
what you see in your columns you created.
6) Create a column for
d2y
.
dx 2




6 t2 1
d2y

7) You were given
. Where does this second derivative change sign? Does the
3
dx 2
27 t 2 1
column for the second derivative you created support this?