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APPM 2350
FINAL EXAM
FALL 2007
INSTRUCTIONS:
Electronic devices, books, and crib sheets are not permitted. Write your (1) name, (2) instructor’s
name, and (3) recitation number on the front of your bluebook. Work all problems. Show your work clearly. Note that a
correct answer with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work may
receive partial credit.
1. (30 Points) Find the absolute maximum and minimum values of the function
f
(
x,y
) = 20

16
x

4
y
+ 4
x
2
+
y
2
on the
closed region bounded by the lines
x
= 4,
y
=
x
, and
y
= 0. Be sure to clearly give both the locations and values of
the absolute extremum.
2. (30 points) Consider the curve
y
=
√
x
for 0
≤
x
≤
3. Find the point on the curve closest to, and furthest from, the
point (2
,
0).
Although this problem can be worked using simple Calculus I principles, you need to work this problem using Calculus
III principles! Clearly explain your approach as you work through the problem! (Hint: you may ﬁnd it useful to graph
the constraint curve in the
x

y
plane as well as the level curves of the objective function.)
3. (30 Points) Consider a particle moving along a path in space through a temperature ﬁeld
T
(
x,y,z
). At a particular
instant in time (and only at that instant in time) the position of the particle is
r
= 2
i
+
j
+
k
, its velocity is
v
= 2
i
+ 2
j
+
k
, and its acceleration is
a
= 2
j
. At the location (2
,
1
,
1) the value of the temperature is
T
(2
,
1
,
1) = 5
and the gradient of the temperature ﬁeld at the same location is
∇
T
(2
,
1
,
1) = 3
i
+ 2
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 Fall '07
 ADAMNORRIS

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