Calculus Maximus WS 7.2: Param & Vector Accumulation
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Name_________________________________________ Date________________________ Period______
Worksheet 7.2—Parametric & Vector Accumulation
Show all work. No calculator except unless specifically stated.
Short Answer/Free Response
1. If
2t
xe
and
sin 3yt
, 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
1xt t
and
42
28yt t t
have a vertical tangent?
Calculus Maximus WS 7.2: Param & Vector Accumulation
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4. Find the equation of the tangent line to the curve given by the parametric equations
2
() 3 4 2xt t t
and
at the point on the curve where
1t
.
5. If
() 1
t
xt e
and
2
2
t
ye
are the equations of the path of a particle moving in the
xy
plane, write
an equation for the path of the particle in terms of x and y.
6. (Calculator) 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
?
Calculus Maximus WS 7.2: Param & Vector Accumulation
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7. (Calculator) The position of a particle at time
0t
is given by the parametric equations
3
2
() 4
3
t
xt

and
2
() 4 4yt t t
.
(a) Find the magnitude of the velocity vector 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?
Calculus Maximus WS 7.2: Param & Vector Accumulation
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8. (Calculator) An object moving along a curve in the
xy
plane has position
(), ()xt yt
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. (Calculator) A particle moves in the
xy
plane so that the position of the particle is given by
() cosxt t t
and
() 3 2sinyt t t
,
0 t

. Find the velocity vector when the particle’s vertical
position is
5y
.
Calculus Maximus WS 7.2: Param & Vector Accumulation
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10. (Calculator) An object moving along a curve in the
xy
plane has position
(), ()xt yt
at time t with
3
2sin
dx
t
dt
and
2
cos
dy
t
dt
for
04t
. At time
1t
, the object is at the 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 object at time
2t
.
Calculus Maximus WS 7.2: Param & Vector Accumulation
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Multiple Choice:
11. (Calculator) An object moving along a curve in the
xy
plane has position
,xt yt
with
2
cos
dx
t
dt
and
3
sin
dy
t
dt
. At time
0t
, the object is at position
4, 7
. Where is the particle
when
2t
?
(A)
0.564,0.989
(B)
0.461,0.452
(C)
3.346,7.989
(D)
4.461,7.452
(E)
5.962,8.962
12. (Calculator) The path of a particle moving in the plane is defined parametrically as a function of time t
by
sin 2xt
and
cos 5yt
. What is the speed of the particle at
2t
?
(A)
1.130
(B)
3.018
(C)
1.307, 2.720
(D)
0.757, 0.839
(E)
1.307, 2.720
Calculus Maximus WS 7.2: Param & Vector Accumulation
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13. For what values of t does the curve given by the parametric equations
32
1xt t
and
42
28yt t t
have a vertical tangent?
(A) 0 only (B) 1 only (C) 0 and 2/3 only (D) 0, 2/3, and 1 (E) No value
14. The distance traveled by a particle from
0t
to
4t
whose position is given by the vector
2
,st t t
is given by
(A)
4
0
41tdt
(B)
4
2
0
21tdt
(C)
4
2
0
21tdt
(D)
4
2
0
41tdt
(E)
4
2
0
241tdt