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Test 1: Math 212
Practice Exam
This is a practice exam for Math 212 Exam I, Spring 2008. It is LONGER than
the in class exam will be, but should give you a good idea for the standard types
of problems and diﬃculty level that will be on the exam. Problems like these will
constitute roughly 80% of the exam. The rest of the exam will consist of problems
that you have not seen before. Let me know if you see any typos or incorrect
solutions!!
1
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View Full Document Problem 1
Find an equation of the plane in
R
3
which is tangent to the surface
x
3

2
y
3
+
xz
2
= 0
at the point (1
,
1
,
1)
.
The surface is the level surface given by
g
(
x,y,z
) =
x
3

2
y
3
+
xz
2
= 0. The
gradient is
∇
g
= (3
x
2
+
z
2
,

6
y,
2
xz
). Therefore,
∇
g
(1
,
1
,
1) = (4
,

6
,
2) is the
normal vector to the surface. The equation of the plane is given by
(4
,

6
,
2)
.
(
x

1
,y

1
,z

1) = 4
x

6
y
+ 2
z
= 0
.
Problem 2
Sketch the level curves for the function
f
(
x,y
) = 2

x
2

y
2
for
c
=

1
,
0
,
1.
Solution: Concentric circles of radius
√
3
,
√
2
,
1 all centered at the origin.
Problem 3
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This note was uploaded on 10/04/2010 for the course MATH 166 taught by Professor Hotlz during the Spring '08 term at HCCS.
 Spring '08
 Hotlz
 Math, Calculus

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