Math 128
ASSIGNMENT 6
Winter 2010
Submit all problems by
8:20 am
on
Wednesday, March 3
rd
in the drop boxes across from
MC4066, or in class depending on your instructor’s preference.
All solutions must be
clearly stated
and
fully justified
.
1. Each of the following curves represents the path of a particle moving in
R
2
. Make a
sketch of the curve, showing the direction of motion; then find the velocity and speed of
the particle, and locate the points
(
x, y
)
at which the velocity is vertical or horizontal,
or
0
. (Part d) not required).
a)
x
(
t
) = (2 cos
t,
3 sin
t
)
,
0
≤
t
≤
2
π
c)
x
(
t
) = (
e

t
cos
t, e

t
sin
t
)
,
0
≤
t
≤
4
π
b)
x
(
t
) =
t
+
1
t
, t

1
t
, t >
0
d)
x
(
t
) = (cos
3
t,
sin
3
t
)
,
0
≤
t
≤
2
π
2. For each of the following curves, find the vector equation of the tangent line at
t
0
,
and state the slope of the tangent line at
t
0
. Sketch the curve and tangent at
t
0
.
[Note that the slope of a vector
v
= (
v
1
, v
2
) is
v
2
/v
1
.]
a)
x
(
t
) = (3
t
2
, t
3
)
, t
0
=
1
2
b)
x
(
t
) = (sin
2
t,
2 cos
t
)
, t
0
=
π/
4
3. Find the total distance travelled by a particle along each of the following paths. Sketch
the paths.
a)
x
(
t
) = (3
t
2
, t
3
)
,

1
≤
t
≤
1
b)
x
(
t
) = (cos
3
t,
sin
3
t
)
,
0
≤
t
≤
2
π
4. A particle moving in a plane has position at time
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 Winter '10
 Zuberman
 Math, Calculus, Derivative, Vector Space, Velocity

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