MAC 2313, Fall 2010 — Midterm 2 Review Problems
1
We will discuss these problems in class on Friday 10/1 and Monday 10/4. Solutions will be posted
on the course webpage over the weekend.
1
Find a vector function that represents the curve of intersection of the cylinder
x
2
+
y
2
= 16
and the plane
x
+
z
= 5.
2
A particle moves with position function
r
(
t
) =
t
ln
t
i
+
t
j
+
e
−
t
k
.
Find the velocity,
acceleration, and speed of the particle.
3
Compute the position vector for a particle which passes through the origin at time
t
= 0 and
has velocity vector
v
(
t
) = 2
t
i
+ sin
t
j
+ cos
t
k
.
4
Find the arc length of the curve
r
(
t
) = cos
3
t
j
+ sin
3
t
k
from
t
= 0 to
t
= 1.
5
Consider the curve defined by
r
(
t
) =
(
4 sin
ct,
3
ct,
4 cos
ct
)
.
What value of
c
makes the arc
length of the curve traced by
r
(
t
), for 0
≤
t
≤
1, equal to 10?
6
Show that if a particle moves at constant speed, then its velocity and acceleration vectors
are orthogonal.
Note that moving at constant speed does
not
mean that the velocity is 0,
because the particle could be turning.
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 Fall '08
 Keeran
 Calculus, Derivative, Laplace, Velocity, twodimensional Laplace equation

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