Math 2263
Name (Print):
Spring 2008
Student ID:
Midterm 1, WITH SOLUTIONS
Section Number:
February 21, 2008
Teaching Assistant:
Time Limit: 50 minutes
Signature:
1
15 pts
2
15 pts
3
15 pts
4
15 pts
5
10 pts
6
15 pts
7
15 pts
TOTAL
100 pts
1. (15 points) Find an equation for the
plane
passing through all three points
x, y, z
=
3
,

2
,

2
,
2
,
0
,
1
and
1
,
0
,
0 .
SOLUTION:
A normal vector
v
is the cross product of
3
,

2
,

2

1
,
0
,
0
and
2
,
0
,
1

1
,
0
,
0 . So
v
=
i
j
k
2

2

2
1
0
1
=

2
i

4
j
+ 2
k.
We can divide
v
by

2, so an equation for the plane is (
x

1) + 2(
y

0)

(
z

0) = 0,
equivalently
x
+ 2
y

z
= 1
.
2. (15 points) Find an equation for the surface in (
x, y, z
)space obtained by
rotating
the hyper
bola
x
2

4
z
2
= 1 of the (
x, z
)plane
about the
x
axis.
SOLUTION:

z

is the distance form the
x
axis in the (
x, z
)plane; we want to replace it with
the distance to the
x
axis in space, namely
y
2
+
z
2
. The equation of the surface of revolution
is
x
2

4
y
2

4
z
2
= 1
.
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Math 2263 Spring 2008
Midterm 1, WITH SOLUTIONS Page 2 of 3
February 21, 2008
3. (15 points) The lines given paramerically by
(
x, y, z
) = (7 + 2
t,

1

t,

2
t
)
,
∞
< t <
∞
and
(
x, y, z
) = (4

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 Fall '08
 Staff
 Math, Derivative, Multivariable Calculus, Vector Calculus, Gradient

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