Math 1920, Prelim 1
October 2, 2008, 7:30 PM to 9:00 PM
You are NOT allowed calculators, the text or any other book or notes. SHOW ALL WORK!
Write your name and Lecture/Section number on each booklet you use. You may leave when
you have finished, but if you have not handed in your exam booklet and left the room by 8:45,
please remain in your seat so as not to disturb others who are still working.
1) a) (7 points) Calculate the distance from (1
,
1
,
1) to the plane with the equation
x
+ 2
y
+ 3
z
= 0
.
b) (7 points) Find the equation of the plane through (0
,
0
,
0)
,
(1
,
1
,
0), and (1
,
2
,
3).
2) (14 points) The position vector of a particle moving in space is given by the vector function
r
(
t
) = (cos
t
)
i
+ 2
t
j
+ (sin
t
)
k
,
where
t
≥
0
.
For each
t
calculate the angle between the velocity vector and acceleration vector.
3) (14 points) The equation
3
x
2
y
+
yz
+
z
ln(2
x

1) = 0
defines
x
as a function of the two independent variables
y
and
z
. Find the partial derivative
∂x/∂z
at the point (1
,

1
,

3).
4) Let
f
(
x, y
) =
1
x
2
+
y
2

1
.
a) (5 points) Determine the domain of
f
(
x, y
).
b) (6 points) Determine and graph level curves in the
x

y
plane for
f
(
x, y
) =

2
,
1, and 2.
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 Fall '06
 PANTANO
 Math, Calculus, Derivative, 14 points

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