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MATH 2400: CALCULUS 3
MAY 9, 2007
FINAL EXAM
I have neither given nor received aid on this exam.
Name:
±
001
E. Kim .
. . . . . . . . . . . . . . .
(9am)
±
002
E. Angel .
. . . . . . . . . . . .
(10am)
±
003
I. Mishev .
. . . . . . . . . . .
(11am)
±
004
M. Daniel .
. . . . . . . . . .
(12am)
±
005
A. Gorokhovsky .
. . . . .
(1pm)
If you have a question raise your hand and remain seated. In order to receive full credit your answer
must be
complete
,
legible
and
correct
. Show all of your work, and give adequate explanations.
DO NOT WRITE IN THIS BOX!
Problem
Points
Score
1
15 pts
2
15 pts
3
15 pts
4
15 pts
5
20 pts
6
15 pts
7
30 pts
8
15 pts
9
15 pts
10
30 pts
11
15 pts
TOTAL
200 pts
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View Full Document 1.
(15 pt) Find the area of the region enclosed by the curve
x
= 1

t
2
, y
=
t
(1

t
2
)
,

1
≤
t
≤
1
(see the picture below)
1.
(15 pt) Find the area of the region enclosed by the curve
x
= 1

t
2
,
y
=
t
(1

t
2
)
,

1
≤
t
≤
1
(see the picture below)
0
0.5
1
0.5
0.5
2.
(15 pt) Find the equation for the plane tangent to the paraboloid
z
= 2
x
2
+ 3
y
2
that is also
parallel to the plane 4
x

3
y

z
= 10
By Green’s Theorem,
A
=
I
C

y dx
with
C
the above curve oriented counterclockwise. Thus
A
=
I
C

y dx
=
Z
1

1

t
(1

t
2
)(

2
t
)
dt
= 2
Z
1

1
(
t
2

t
4
)
dt
=
8
15
.
2.
(15 pt) Find the equation for the plane tangent to the paraboloid
z
= 2
x
2
+ 3
y
2
that is also
parallel to the plane 4
x

3
y

z
= 10
Write
G
(
x, y, z
) = 2
x
2
+ 3
y
2

z
, so
∇
G
(
x, y, z
) =
h
4
x,
6
y,

1
i
. This tells us that
h
4
x,
6
y,

1
i
is a normal vector for the tangent plane to our surface at the point (
x, y, z
). For the tangent
plane at a point to be parallel to 4
x

3
y

z
= 10, we need
h
4
x,
6
y,

1
i
to be parallel to
h
4
,

3
,

1
i
. This will certainly happen when
x
= 1 and
y
=

1
2
. Solving for
z
in the equation
for our surface, we get that the tangent plane at (1
,

1
2
,
11
4
) is parallel to 4
x

3
y

z
= 10. At
this point the equation for the tangent plane will be given by 4(
x

1)

3(
y
+
1
2
)

(
z

11
4
) = 0.
3.
(15 pt) Find the ﬂux of
F
(
x, y, z
) = (
x
+
y
)
i
+ (
y
+
z
)
j
+ (
z
+
x
)
k
across the portion of the
plane
x
+
y
+
z
= 1 in the ﬁrst octant oriented by unit normals with positive components.
We know Φ =
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This note was uploaded on 04/13/2008 for the course MATH 1012 taught by Professor N/a during the Fall '07 term at Colorado.
 Fall '07
 N/A
 Math, Calculus

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