Table of Contents
Rationale
p3
Real World Applications
p5
Mathematician of Fame
p6
Analytical Examples
p7
Multiple Choice Example
p8
Conceptual Example
p9
Calculator Example
p10
Free Responses Walkthrough
p11
Analytical Exercises
p12
AP Multiple Choice
P14
AP Free Response
P16
Work Cited
p17
Rationale
Wouldn’t it be wonderful if there was a way to model situations that are not dependent on
one another but by time, such a thrown basketball? The height of a basketball is not determined
by its position but by the amount of time that has passed. Luckily, parametric curves are exactly
that, allowing time to become a variable of its own that determines both the x and y position on a
graph. Parametric problems provide an accurate example of real life situations such as projectile
motion of a cannonball in flight or the movement of an animal through the woods where thanks
to the addition of the variable of time you can cross a point more than once. It’s application is
very much so far reaching as it helps explain concepts of other sciences such as physics
involving the change of the x and y directions independently. Even better, parametric equations
can be used for the purpose of simulating real-life situations and even providing a convincing
simulation for fantastical situations. This textbook should help teach students this wonderful new
topic by making it simple yet keeping the topics of the AP level, in a way making complex topics
simple by explaining them in a way that all students with an IQ of three or higher will be able to
understand. With help from textbooks of the past, Disney, Teja, the Larson Hosteter Edwards
eight edition Calculus textbook and from the vast amount of information from internet resources,
we plan on creating a textbook with the ability to pique the interest of the common student into
learning the interesting topic of parametric equations. We think that parametric curves are
amazing because they are relatively simple even though they can explain highly complex
situations, allowing them to be easily used in everyday life. Through the upbeat presentation of
our ideas we hope to convey our own excitement about the topic to the reader and ignite their
imagination in order to create new ideas with which parametric equations could be used. Along
with creating excitement among the reader we hope that our text also provides a welcomed
escape from the conventional teaching methods that are presented through most other textbooks.
Both through providing example problems with quick easy step by step answers coupled with a
conceptual analysis of each step and through examples of situations that are well known(such as
Disney© stories) we hope to create an easy teaching model that will appeal to and be
remembered by those who read it.
Real World Applicability with MulanBecause Parametric is reliant on time and not on a single variable its uses in the real
world are all encompassing. Whether you are tracing the trajectory of a cannonball in flight or
the path of a fox in the forest,
parametric equations can help
create an accurate analytical
example of these situations
because the x and y direction
are interdependent. You can
cross paths while on a graph
allowing for a whole new type
of visual. It is also very helpful in dire situations such as when facing a large group of angry
Huns. You can use parametric equations to plot the trajectory of a rocket so you can maximize
your output of pressure on the enemy. Whatever way you may look at it, parametric equations
are extremely useful in everyday life and deserve you respect.
A Special Mathematician- Jacob Bernoulli (1654-1705)
Jacob Bernoulli is sometimes called Jacques Bernoulli. His
father was Nicolaus Bernoulli, a mathematician. During the time
that Jacob Bernoulli was taking his university degree in theology
he was studying mathematics and astronomy against the wishes of
his parents. Kind of like a Cinderella story of calculus in which by
going against his parents he found his true path in life.
In 1676, finishing his degree, Bernoulli moved to Geneva
where he worked as a tutor. He then travelled to France spending 2
years studying with the followers of Descartes. Continuing his
studies with the leading mathematicians and scientists of Europe he
went to England where, among others, he met Boyle and Hooke.
As a result of his travels, Bernoulli began a correspondence with many mathematicians which he
carried on over many years.
Jacob's younger brother, Johann Bernoulli, began to work on mathematical topics as well,
and asked him to teach him mathematics. Jacob Bernoulli
was appointed professor of mathematics in Basel in 1687
and the two brothers began to study the calculus as
presented by Leibniz in his 1684 paper on the differential
calculus. It must be understood that Leibniz's publications
on the calculus were very obscure to mathematicians of
that time and the Bernoullis were the first to try to
understand and apply Leibniz's theories. Although the
brothers were good partners, they also had several
disagreements on some math topics over the years like
the brothers of the lion king.
By 1689 he had published important work on infinite series and published his law of
large numbers in probability theory. Jacob Bernoulli published five treatises on infinite series
between 1682 and 1704 increasing his calculus credibility. Jacob Bernoulli also discovered a
general method to determine evolutes of a curve as the envelope of its circles of curvature. He
also investigated caustic curves and in particular he studied these associated curves of the
parabola, the logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli was
first conceived by Jacob Bernoulli in 1694. In 1695 he investigated the drawbridge problem
which seeks the curve required so that a weight sliding along the cable always keeps the
drawbridge balanced. It was this study in particular that added to his credibility on parametric
topic.
Examples:
Analytical:
Parametrics are not, in their simplest form, that different from regular equations. For
example, observe the equation 𝑦 = 90𝑡 + 7. How would you take its derivative? How would
you find the integral?
-in the exact same way you would for the equation 𝑦 = 90𝑥 + 7! There is essentially no
difference in each part of the differential equation. You can think of them as regular equations,
with a T substituted in for one of the variables.
Still, the equations have different meanings. In parametrics, each equations only denotes
one part of how the particle moves. 𝑦 = 90𝑡 + 7 models how the particle moves in solely the ydirection. 𝑥 = 28𝑡 models how the particle moves in solely the x-direction.
The differences between the equations start when you need to combine both the x and y
equations.
More difficult is the new way velocity is found. You can find the velocity in the X and Y
directions in respect to time. But adding those velocities does not result in the velocity of the
particle. Take the equation below for example.
𝑥 ′ = 𝑡, 𝑦 ′ = 𝑡
The velocity of the particle is the distance from the origin to the point (X, Y) on the graph.
After you find the X and Y velocities at the given time, simply find the distance from that point to
the origin to find velocity- through Pythagorean equations of course! The X and Y coordinates
form a triangle with the origin that can be used to find the origin… such that the (𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦)2 =
𝑦 2 + 𝑥 2 ! And what about acceleration? Acceleration can be solved exactly the same way,
through the Pythagorean triangle.
However, in graphs in terms of X and Y, the variable of time seems to play no part. The
graph is made by plotting the X and Y values at time t- but the result will be the same as if you
were to graph the equation in terms of X and Y. The graph of 𝑦 = 𝑡, 𝑥 = 𝑡 looks the same as the
graph of 𝑦 = 𝑥.
In truth, the part it plays here is only in forming the graph. Though the result may look
the same, the variable time shows how fast the particle moves along that graph. While 𝑥 = 𝑡,
𝑦 = 𝑡 and 𝑥 = 2𝑡, 𝑦 = 2𝑡 are both equivalent to the equation 𝑦 = 𝑥, the latter equation travels
along the line faster.
This confusing quality has led to multiple ways to show a parametric equation. You may
be asked to solve it as a simple graph- to put it in terms of X and Y by solving for T in one
equation and substituting it into the other.
The other way is called the Vector form. It looks like the form used to plot a point: (X,Y).
You simply replace parenthesis with arrows and points with functions in reference to time:
<X(T), Y(T)>. This form is also used for velocity and acceleration, by inputting the functions of
velocity and acceleration: <X’(T), Y’(T)>.
Finding the distance the particle travels along the graph is comparatively easier, though. The
𝑑𝑦 2
𝑥
𝑑𝑦
𝑑𝑦
𝑑𝑥
original equation, as we know, is∫𝑥 𝑓 √1 + (𝑑𝑥 ) 𝑑𝑥. Well, (𝑑𝑥 ) = ( 𝑑𝑡 )/( 𝑑𝑡 ) and, consequently,
0
𝑡
𝑑𝑥 2
𝑑𝑦 2
the distance is also equal to is∫𝑡 𝑓 √( 𝑑𝑡 ) + ( 𝑑𝑡 ) 𝑑𝑡
0
Multiple Choice:
A curve is described by the parametric equations x  t 2  4t and y  t 3  t 2  1 . An equation of
the line tangent to the curve at time t  1 is
(A) 6 y  5 x  43
(B) 5 y  6 x  15
(C) 6 y  5 x  7
(D) y  x  1
(E) 10 x  17 y  0
y '  3t 2  2t  5
x'  2t  4  6
dy 5

dx 6
x5
at t  1
y3
5
( x  5)
6
6 y  18  5 x  25
6 y  5 x  7
y 3
Answer is C
To get answer A, add -18 to -25 instead of adding
To get answer B use
dx
dy
instead of
dx
dy
To get answer D incorrectly integrate y '
Answer E is completely incorrect
Conceptual:
A curve representing a travel to beasts castle and back is described by the parametric equations
x  2 cost  and y  sin 2t  on the time interval 0  t  2 . Using this data, which of the
following is true?
I.
II.
III.
(A)
(B)
(C)
(D)
(E)
There are vertical tangents at the points t  0 and t   .

There is a horizontal tangent at t  .
2

3
There are points of inflection at t  and t 
2
2
I only
I & II
II & III
I & III
I, II & III
Test of Case I
Vertical tangents happen at points where
dy
 2 cos2t 
dt
dx
 2 sin t 
dt
dx
dy
 0 and
 0.
dt
dt
y2
y  2
at t  0 and
at t  
x0
x0
Therefore, case I is true.
Test of Case II
Horizontal tangents happen at points where
dy
 2 cos2t 

y  2
dt
at t 
2
dx
x  2
 2 sin t 
dt
dy
dx
 0 and
 0.
dt
dt
Therefore, case II is false
Test of Case III
d2y
Points of inflection happen where
changes signs.
dx 2
dy
dy
2 cos2t 
 dt  
dx dx
2 sin t 
dt
  8 sin 2t sin t   4 cos( 2t ) cos(t ) 
d2y
cos(t )
 = 2 cost  
=  
2
2
sin 2 (t )
dx
4 sin (t )


Answer is C
Calculator:
Graph the curve, indicate the direction and identify any values for  0    2 where the curve is not
smooth
x  2 cos 3  , y  2 sin 3 
The graph shows that the curve is not
smooth at (0,2) and (2,0)
To solve for  :
x  2 cos 3   2
cos 3   1
  0,  ,2
y  2 cos 3   0
cos 3   0
 3
 ,
2 2
The curve is not smooth at

3
  0, ,  , ,2
2
2
Free Response:
Tramp runs around in a circular shape three times before going to bed each night. His path can
be modeled by the parametric equation
𝑥 = cos(𝜋𝑡) , 𝑦 = 2sin(𝜋𝑡)
1) At what time will tramp finish running around in a circle three times?
2) Why can’t this function be modeled by a single, simplified X=Y type function?
3) Show the integral that can be used to find the distance Tramp runs to complete one
circle?
1) Tramp will have finished running in a circle when he comes back to the point that he
started. This can be found by finding the X-position and Y-position at t = 0, then solving
for t. The first time t that works for both equations will be the time it takes for Tramp to
return to his starting place once. This occurs at time t = 2 seconds. From there, you can
either find the third time this happens or multiply the time by three, as both equations are
cyclical. The correct answer is: Tramp will run his three circles in 6 seconds.
2) This equation cannot be modeled by a single X=Y equation because such a function
cannot contain two different Y-coordinates at a single X-coordinate. This occurs in this
parametric equation, so it cannot be modeled by a single X=Y function, though it is
possible to model it with two such functions.
3) This problem is just a basic “do you know this?” question- there are no tricks here.
2
Simply use the formula for distance. ∫0 √(cos(𝜋𝑡))2 + (2 sin(𝜋𝑡))2
Exercises:
Analytical:
1.Eliminate the parameter:
xt4
y  2t
2.Eliminate the parameter
x  t2 9
y
t
3
3.Eliminate the parameter
x  t 2 1
y  t 2 1
4.Find a set of parametric equations for the rectangular equation y  2 x  5
a) when x  t
b) when x  t  5
5.Find a set of parametric equations for the rectangular equation y  x 2  4
a) when x  t  2
b) when x  t 2  1
6. Find the velocity vector at time t  2 for the particle described by the parametric equations
x  t 2  2t  4
y  t 2  4t  2
7. Find the position, velocity and acceleration vectors at time t  3 for the particle described by
the parametric equations
x  t 3  3t 2  5
y  4t 2  8t  9
8. Find the speed of the particle described by the parametric equations
x  2t 2  3t  4
y  t 2  2t
9. Find the distance traveled from time t  0 to t  10 by the particle described by the parametric
equations
x  3t 2  4t  5
y  t 2  3t  2
10) These equations represent the position of Jasmine, Rapunzel, Tiana. Sketch a graph of
these equations, from t = 0 onward.
A. Jasmine:
𝑥 = cos(𝜋𝑡/2) /𝑡
𝑦 = sin(𝜋𝑡/2) /𝑡
B. Rapunzel:
𝑥 = cos( 𝜋𝑡/2) − 1.5
𝜋𝑡
sin ( 2 )
𝑦 = 𝑎𝑏𝑠 (
)
𝑡
C. Tiana:
𝑥 = e−t
𝑦 = 5𝑒 −𝑡 − 5
Find if and when (assuming each starts walking at time t=0):
1. Jasmine and Rapunzel meet
2. Jasmine and Tiana meet
3. Tiana and Rapunzel meet
AP Multiple Choice:
1. In the xy-plane, the graph of the parametric equations x  3t  7 and y  4t , for  3  t  3, is
a line segment with slope
A) 3
B) 4
4
3
C) 3
D) 4
E) 12
2. A particle moves on a plane curve so that at any time t  0 its x-coordinate is 2t 3  t 2 and its y
3
coordinate is 3t  1 . The acceleration vector of the particle at time t  1is
A) 4,36
B) 1,8
C) 4,8
D) 10,108
E) 10,36
3. The length of the path described by the parametric equations x 
0  t  2, is given by
2
A)

t 3  t dt
0
2
B)

t 3  t dt
0
2
C)

0
t 6  t 2 dt
1 4
1
t and y  t 2 , where
4
2
2
D)

t 6  t 2 dt
0
2
1
E)  8  t dt
20


4. If f is a vector-valued function defined by f t   e t , cos t . Then f
'''
t  
A)  e  t  sin t
B) e t  cos t

C)  e t , sin t


D)  e t , sin t

E) e t , cos t


The equation of the line tangent to the curve with parametric equations xt   2t  1;
yt   3  t 3 at t  1is
5.
A) 2 x  3 y  12
B) 3x  2 y  13
C) 6 x  y  20
D) 3x  2 y  5
E) None of the above
AP Free Response:
Two particles move in the xy -plane. For time t  0 , the position of particle A is given by
x  t  4 and y  t  3 , and the position of particle B is given by x 
2
4t
2t
 1 and y  .
5
5
(A) Find the velocity vector for each particle at time t  2 .
(B) Set up an integral expression that gives the distance traveled by particle B from t  0 to
t  3.
(C) Determine the exact time at which the particles collide; that is, when the particles are at
the same point at the same time. Justify your answer.
(D) In the viewing window provided below, sketch the paths of particles A and B from t  0
until they collide. Indicate the direction of each particle along its path.
Internet Resources:
1: http://colalg.math.csusb.edu/~devel/precalcdemo/param/src/param.html
2: http://archives.math.utk.edu/visual.calculus/0/parametric.6/
3: http://www.math.hmc.edu/calculus/tutorials/parametric_eq/
4: https://apstudent.collegeboard.org/apcourse/ap-calculus-bc/exam-practice
5: http://online.math.uh.edu/apcalculus/exams/