MATH 231
Group Work
Quiz Related
1. Classify and sketch the following surfaces:
(a)
z
= 2
x
2
+
y
2
(b)
z
2
=
x
2
+
1
3
y
2
(c) 1 + 5
z
2
= 7
x
2

y
2
(d) 4 +
z
2
=
x
2
+
y
2
Answer:
(a) Elliptic paraboloid.
(b) Cone.
(c) Hyperboloid of two sheets.
(d) Hyperboloid of one sheet.
1
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2. What is the domain of the vector function
r
(
t
) =
D
√
2

t,
ln
t,
t
2

1
t

1
E
?
3. At what point(s) do the curves
r
(
t
) =
t
2
, t
2
+ 1
,
4
t
and
s
(
t
) =
t

4
,
1
2
t,

√
t
2

4
intersect?
4. Find a vector function that represents the curve of intersection of
z
2
=
x
2
+
y
2
and 2
x
+ 3
y

z
= 1.
Answer:
This one is the really ugly one of the bunch. The concepts involved in solving it are more important
here than the specific details. For similar problems, look at 13.1 #3638.
We know that there are two pieces to the curve of intersection between these two things. One for
z >
0
and one for
z <
0. Substituting
z
= 2
x
+ 3
y

1 into
z
2
=
x
2
+
y
2
and using the quadratic formula, we
find that
x
=
1
3
2

6
y
±
p
1

6
y
+ 12
y
2
Here the + option represents where
z >
0 and the

value represents where
z <
0. Thus, letting
y
=
t
,
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 Spring '08
 BOCIU
 Calculus, Vector Calculus, Cone, vector function, Parametric equation, The Headhunters, Group Work Quiz

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