AP Calculus BC
Unit 8 – Parametric and Polar Equations
AP Calculus BC – Worksheet 63 Parametric Equations
1
Sketch the parametric curves. Find an equation that relates x and y directly.
a)
2 3 and 4 3 for in the interval 0,3x t y t t= + = −
b)
sin and 2cos for in the interval 0,x t y t t
==
2
Find (a)
dy
dx
and (b)
2
2
dy
dx
in terms of t.
a)
4sin , 2cosx t y t==
b)
23
3,x t t y t= − =
c)
( ) ( )
4
ln 2 , ln 3x t y t==
d)
( )
5
ln 5 ,
t
x t y e==
3
If
2
1xt=−
and
3
t
ye=
, find
dy
dx
.
4
A curve C is defined by the parametric equations
3
xt=
and
2
52y t t= − +
. Write an equation of the line that is
tangent to the graph of C at the point
( )
8, 4−
.
5
Consider the curve C given by the parametric equations
2 3cosxt=−
and
3 2sinyt=+
, for
22
t
−
.
(a) Find
dy
dx
as a function of t.
(b) Find an equation of the tangent line at the point where
4
t
=
.
(c) (Calculator) The curve C intersects the y-axis twice. Approximate the length of the curve between the two
y-intercepts.
6
A parametric curve is defined by
sinxt=
and
cscyt=
for
0
2
t
. Which of the following best describes the
curve?
(A) Increasing & concave up (B) increasing & concave down (C) decreasing and concave up
(D) Decreasing & concave down (E) decreasing with a point of inflection
AP Calculus BC – Worksheet 64 Parametric Equations
1
Determine the rectangular equation for the parametric curve defined by
lnxt=
and
yt=
for
0t
.
2
A particle moves along the curve
10xy =
. If
2x =
and
3
dy
dt
=
, what is the value of
dx
dt
?
3
Point
( )
,P x y
moves in the xy-plane in such a way that
1
1
dx
dt t
=
+
and
2
dy
t
dt
=
for
0t
.
(a) Find the coordinates of P in terms of t when
1t =
,
ln2x =
, and
0y =
.
(b) Write an equation expression y in terms of x.
(c) Find the average rate of change of y with respect to x as t varies from 0 to 4.
(d) Find the instantaneous rate of change of y with respect to x when
1t =
.
4
Given the parametric equations,
31xt=+
,
94yt=−
, find the length of the path over the interval
02t
.
5
Given the parametric equations,
2
2xt=
,
2
31yt=−
, find the length of the path over the interval
04t
.
6
Given the parametric equations,
sin3xt=
,
cos3yt=
, find the length of the path over the interval
0 t
.
7
What is the maximum height of a particle whose path has the parametric equations
9
xt=
,
2
4yt=−
?
8
Find the length of the curve that has parametric equations
3
cosxt=
,
3
sinyt=
on the interval
02t
.
9
Identify the lowest point on the curve that has parametric equations
1xt=+
,
2
y t t=+
on the interval
22t−
10
Identify the rightmost point on the curve that has parametric equations
2sinxt=
,
cosyt=
on the interval
0 t
.
11
Identify the leftmost point on the curve that has parametric equations
2
2x t t=+
,
2
23y t t= − +
on the interval
23t−
.
AP Calculus BC – Worksheet 65 Particle Motion/Vector-Valued Equations
No calculator, unless explicitly stated.
1
If a particle moves in the xy-plane so that at any time
0t
, its position vector is
( )
( )
22
ln 5 ,3s t t t t=+
, find its
velocity vector at time
2t =
.
2
A particle moves in the xy-plane so that at any time t, its coordinates are given by
( )
5
1x t t=−
,
( )
43
32y t t t=−
.
Find its acceleration vector at
1t =
3
If a particle moves in the xy-plane so that at time t, its position vector is
( )
2
sin 3 ,3
2
s t t t
=−
, find the velocity
vector at time
2
t
=
.
4
A particle moves on the curve
lnyx=
so that its x-coordinate has velocity
( )
'1x t t=+
for
0t
. At time
0t =
, the
particle is at point
( )
1,0
. Find the position of the particle at time
1t =
.
5
A particle moves in the xy-plane in such a way that its velocity vector is
( )
3
1,v t t t=+
. If the position vector at
0t =
is
5,0
, find the position of the particle at
2t =
.
6
The position of a particle in the xy-plane is given by the parametric equations
( )
32
3
18 5
2
x t t t t= − − +
and
( )
32
6 9 4y t t t t= − + +
. For what value(s) of t is the particle at rest?
7
A particle moves in the xy-plane so that the position of the particle is given by
( )
5 3sinx t t t=+
and
( ) ( )( )
8 1 cosy t t t= − −
. Find the speed of the particle at the time when the particle’s horizontal position is
25x=
8
The position of a particle at any time
0t
is given by
( )
2
3x t t=−
and
( )
3
2
3
y t t=
.
(a) Find the speed of the particle at time
5t =
.
(b) Find the total distance traveled by the particle from
0t =
to
5t =
.
(c) Find an expression that would represent the slope of the path of particle in terms of x.
9
A particle moves on the curve
2yx=
so that its x-coordinate has velocity
( )
2
' 3 1x t t=+
for
0t
. At time
0t =
, the
particle is at point
( )
2,4
. Find the position of the particle at time
1t =
.
AP Calculus BC – Worksheet 66 Particle Motion/Vector Calculus 2
1
If
( )
2t
x t e=
and
( ) ( )
sin 3y t t=
, find
dy
dx
in terms of t.
2
Write an integral expression to represent the length of the path described by the parametric equations
3
cosxt=
and
2
sinyt=
for
0
2
t
.
3
For what value(s) of t does the curve given by the parametric equations
32
1x t t=−−
and
42
28y t t t= + −
have a
vertical tangent?
4
Find the equation of the tangent line to the curve given by the parametric equations
( )
2
3 4 2x t t t= − +
and
( )
3
4y t t t=−
at the point on the curve where
1t =
.
5
If
( )
62x t t=−
and
( )
3
3y t t=+
are the equations of the path of a particle moving in the xy-plane, in which
direction is the particle moving as it passes through the point
( )
4,4
?
6
A particle moves in the xy-plane so that its position at any time t is given by
( )
cos 5xt=
and
3
yt=
. What is the
speed of the particle when
2t =
?
7
The position of a particle at time
0t
is given by the vector-valued equation
( )
( )
3
2
2
4, 4 4
3
t
s t t t
−
= + − +
.
a) Find the speed of the particle at
1t =
.
b) Find the total distance traveled by the particle from
0t =
to
1t =
.
c) When is the particle at rest? What is its position at that time?
8
An object moving along a curve in the xy-plane has position given by
( ) ( ) ( )
,s t x t y t=
at time
0t
with
( )
2
1 tan
dx
t
dt
=+
and
3
t
dy
e
dt
=
. Find the acceleration vector and the speed of the object when
5t =
.
9
A particle moves in the xy-plane so that the position of the particle is given by
( )
cotx t t t=+
and
( )
3 2siny t t t=+
for
0 t
. Find the velocity vector when the particle’s vertical position is
5y =
.
10
An object moving along a curve in the xy-plane has velocity vector
( )
( ) ( )
32
2sin ,cosv t t t=
at time t for
04t
. At time
1t =
, the object is at position
( )
3,4
.
a) Write an equation for the line tangent to the curve at
( )
3,4
.
b) Find the speed of the object at time
2t =
.
c) Find the total distance traveled by the object over the time interval
01t
.
d) Find the position of the particle at time
2t =
.
11
An object moving along a curve in the xy-plane has velocity
( )
( ) ( )
23
cos ,sinv t t t=
. At time
0t =
, the object is
at position
( )
4,7
. Where is the particle at time
2t =
?
12
The path of a particle moving in the plane is defined parametrically as a function of time t by
sin2xt=
and
cos5yt=
. What is the speed of the particle at time
2t =
?
13
Find the total distance traveled by a particle from
0t =
to
3t =
whose position is given by the vector
( )
23
4
1,
3
s t t t=+
.
AP Calculus BC - Worksheet 67 Polar Equations and Derivatives
1
Convert the following equations to polar form:
a)
22
16xy+=
b)
4 3 1 0xy+ − =
c)
7y =
2
Convert the following equations to rectangular form:
a)
3secr
=
b)
2
4 cosrr
=
c)
5
6
=
3
For each of the polar functions, find
dy
dx
for the given value of
.
a)
1 sinr
=−
,
0
=
b)
cosr
=
,
3
=
c)
( )
3 1 cosr
=−
,
2
=
4
Find the point(s) where the polar curve given
1 sinr
=+
has horizontal and vertical tangent lines.
AP Calculus BC - Worksheet 68 Polar Equations and Motion
1
For the curve
( )
3 3sin 2r
=+
, find the value of
dx
d
at
3
=
.
2
Find
dy
dx
for
( )
3cosr
=
when
4
=
.
3
Find the equation of the tangent line to the curve
1 sin when r
= − + =
.
4
A particle is moving along the curve
( )
4 sin 3r
=−
so that
3
d
dt
=
for all times
0t
. Find the value of
dr
dt
at
6
=
.
5
A particle moves along the polar curve
4 2sinr
=−
so that at time t seconds,
2
t
=
.
a) Find the position vector in terms of t. Find the velocity vector at time
1.5t =
b) Find the time t in the interval
12t
for which the x-coordinate of the particle’s position is -1.
6
For a certain polar curve,
( )
r
, it is known that
cos sin
dx
d
=−
and
sin cos
dy
d
=+
. What is the value
of
2
2
dy
dx
at
3
2
=
.
AP Calculus BC – Worksheet 69 Polar Equations –Area
1
Find the area bounded by
5sinr
=
.
2
Find the area of the shaded region of the polar curve for
1 cos2r
=−
3
Find the area of one petal of the rose curve
( )
3cos 3r
=
.
4
Find the area of the region in the plane enclosed by the cardioid
( )
4 4sinr
=+
.
5
Find the area inside the smaller loop of the limacon
1 2cosr
=+
.
6
Find the area of one petal of the rose curve defined by
( )
4sin 6r
=
.
AP Calculus BC – Worksheet 70 Polar Area
A calculator is required for all problems.
1) Find the area of the shaded region of the polar curve
4 6sinr
=−
.
2) Find the area of the shaded region of the polar curve
cos2r
=
.
3) Find the area of the shaded region bounded by the
polar curves
3r =
and
3cos3r
=
indicated in the
figure below.
4) Find the area of the shaded region for the polar curve
1 cosr
=−
.
5) Find the area of the region bound by the two polar
curves
1r =
and
1 cosr
=−
as shown in the graph
below.
6) Find the area of the common region to the polar
graphs
2r =
and
2 2sinr
=−
.
7) Find the area of common interior bounded by the
graphs of polar curves
3cosr
=
and
2 cosr
=−
.
8) Find the area of the region that is inside the polar
graph of
1 2cosr
=+
but outside the inner loop.
AP Calculus BC – Worksheet 71 Parametric and Polar Equations Review
1
The position of a particle at any time
0t
is given by
( )
2
2x t t=−
and
( )
3
2
3
y t t=
.
a) Find the speed of the particle at
2t =
.
b) Set up, but do not evaluate, an integral expression that can be used to find the total distance that the
particle traveled from
0t =
to
4t =
.
c) Find
dy
dx
as a function of x.
d) At what time t is the particle on the y-axis? Find the acceleration vector at this time.
2
A particle moving along a curve in the xy-plane has position vector
( )
2
2 3sin , 2coss t t t t t= + +
, where
0 10t
. Find the velocity vector at the time the particle’s vertical position is
7y =
.
3
An object is moving along a curve in the xy-plane has velocity vector
( )
( ) ( )
cos ,sin
tt
v t e e=
for
02t
. At
time
1t =
, the object is at the point
( )
3,2
.
a) Find the equation of the tangent line to the curve at the point where
1t =
.
b) Find the speed of the object at
1t =
.
c) Find the total distance traveled by the object over the time interval
02t
.
d) Find the position of the particle at time
2t =
.
4
Let R be the region in the first quadrant that is bounded by the polar curves
2
r
=
and
k
=
, where k is a
constant, such that
0
2
k
, as shown in the figure above. Find the area of region R in terms of k.
5
Find the slope of the tangent line to the polar curve
2cos4r
=
at the point where
4
=
.
6
For a certain polar curve,
( )
r
, it is known that
cos sin
dx
d
=−
and
sin cos
dy
d
=+
. What is the
value of
2
2
dy
dx
at
6
=
.
7
Consider the polar curve defined by the function
( )
2 cosr
=
, where
3
0
2
. The derivative of r is
given by
2cos 2 sin
dr
d
=−
. The figure above shows the graph of r for
3
0
2
.
a) Find the area of the region enclosed by the inner loop of the curve.
b) Determine if r is increasing or decreasing when
2
=
. Justify your answer.
c) For
3
0
2
, find the greatest distance from any point on the graph of r to the origin. Justify your
answer.
d) A particle is moving along the curve. There is a point on the curve at which the slope of the line tangent
to the curve is
2
2
−
. At this point,
1
2
dy
d
=
. Find
dx
d
at this point. In which direction is the particle
moving?
