FIRST PRACTICE MIDTERM
The following five questions are worth 20 points each, although many have
multiple parts. Show ALL your work in your answers to each question. You
have 75 minutes for the examination. Calculators are NOT permitted.
1.
Let
P, Q,
and
R
be points in
R
3
with coordinates
P
= (0
,
2
,
1),
Q
=
(2
,
0
,
1),
R
= (0
,
0
,

1).
1a. Give a parametrization (in terms of auxiliary variables
t
and
u
) for the
plane containing
P
,
Q
, and
R
.
1b. Give an equation of the form
ax
+
by
+
cz
=
d
for the plane containing
P
,
Q
, and
R
.
2.
Draw the level sets
f
(
x, y
) =
c
for the given values of
c
, in the given
domains.
2a.
f
(
x, y
) =

x

+

y

,

2
≤
x
≤
2,

2
≤
y
≤
2,
c
=

1
,
0
,
1
,
2
,
3
,
4.
2b.
f
(
x, y
) =
xy
,

2
≤
x
≤
2,

2
≤
y
≤
2,
c
=

2
,

1
,
0
,
1
,
2.
3. Define vectors
~u
,
~v
in
R
3
by
~u
= (

2
,
1
,
0),
~v
= (0
,
2
,

1). Let
θ
be the
angle between the two vectors.
3a. Compute the lengths of
~u
and
~v
.
3b. Compute cos
θ
.
3c. Give a unit vector orthogonal to both
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 Spring '08
 SCHURLE
 Derivative, Differential Calculus, Fréchet derivative, following ﬁve questions

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