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Final Exam
Math 20C
Winter 1997
Place all books, papers under your desk. On the front of your blue book write your
name and section number. In a column, number one to ten. Graphing calculators and a
3
×
5 note card is permitted. Turn in the note card with your ﬁnal exam. Box answers
and show all your work. Good luck.
1. (a) 10 pts. Find the equation of a plane passing through
P
=(3
,
2
,
1),
Q
=(1
,
1
,
1)
and
R
=(

1
,
2
,
4).
(b) 10 pts. Find the equation of a line through (2
,
3
,
7) parallel to the line of intersec
tion of the planes 2
x
+
y
+
z
= 0 and
x

y
+7
z
=0.
(c) 10 pts. Draw several level curves of
f
(
x, y
)=
x
y
. Compute
~
∇
f
(1
,
2) and sketch
the
~
∇
f
(1
,
2) and the level curve passing through (1
,
2).
2.
The graph of the surface
z
=
f
(
x, y
) is shown in the following ﬁgure.
(a)
10 pts. Is the directional derivative of
f
in the direction
~
i
+
~
j
positive or negative
at (2
,
5)?
(b)
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 Spring '08
 Helton
 Math

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