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Math 20C
Final Exam.
June 15, 2006
No calculators or any other devices are allowed on this exam.
Write your solutions clearly and legibly; no credit will be given for illegible solutions.
Read each question carefully. If any question is not clear, ask for clariFcation.
Answer each question completely, and show all your work.
1. (10 points) Find the plane through the point
P
0
= (2
,

1
,
1) which is perpendicular to
the planes 2
x

y

z
= 3 and
x
+ 2
y
+
z
= 2.
The plane is determined by its normal vector
n
and a point.
We choose the point to be
P
0
= (2
,

1
,
1). The normal vector can be computed as
n
=
n
1
×
n
2
,
n
1
=
h
2
,

1
,

1
i
,
n
1
=
h
1
,
2
,
1
i
.
where
n
1
and
n
2
are the normal vectors to the planes 2
x

y

z
= 3 and
x
+ 2
y
+
z
= 2,
respectively. Then,
n
=
¯
¯
¯
¯
¯
¯
i
j
k
2

1

1
1
2
1
¯
¯
¯
¯
¯
¯
=
h
(

1 + 2)
,

(2 + 1)
,
(4 + 1)
i
⇒
n
=
h
1
,

3
,
5
i
.
Then, the equation of the plane is
(
x

2)

3(
y
+ 1) + 5(
z

1) = 0
⇒
x

3
y
+ 5
z
= 10
.
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View Full Document 2. (8 points) Decide whether the
lim
(
x,y
)
→
(0
,
0)
y
4

x
2
y
4
+
x
2
exists. Give reasons your answer.
Consider the path given by the line
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This note was uploaded on 04/30/2008 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.
 Spring '08
 Helton
 Math

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