Exam 3
—
Math 132 1—001 Accelerated Calculus for Engineers II
Spring 2013
• There are 100 total points in this exam.
• You have 50 minutes to complete the exam.
• Show all work, all steps, clearly and neatly, whenever possible.
• A half sheet, back and front, of notes is permitted.
• Please staple your note sheet to the exam before turning it in.
• The use of laptops, cell phones, and any other wireless devices are not allowed during
this exam.
• Good luck!
Name:
1
2a.) ( points) In considering the surface area of a pyramid with height H and square base
of length L, we write the surface area function as S(L, H) = L
2 + LV4H
2+L
. Find the
2
linearization of S around the length L = 2 in and height H = ‘/ in. Reasonably simplify
your coefficients.
s
-
H
S
(L, H)
L
2
1
(L
4
i:I
L
E*F
2b.) ( points) With a length of 2 in and height of \/ in, we believe the error in our
measurement is 0.1 in and 0.2 in, respectively. Using differentials, estimate the maximum
error in the surface area calculation. Reasonably simplify your answer and feel free to refer
to your calculations in part a.
s
3
3a.)
(* points)
function f(x. y)
I observe that the topography around me obeys the
= exp (—(x
(y 2)2), where f(r, y) is the height in meters from sea
level in feet and the positive y-axis can be interpreted as due north. If I proceed 30 degrees
south of due east, what is my instantaneous rate of change? In layman’s terms, what does
your result mean? Am I going up or down the hill?
Standing at the
—
origin.
1)2
—
—
<
())> =
—
Lkfl4
5
<
MJ\:riQ
-(-) --)
-
-k— 4
)
-c
e
1
(
O•5 e
<1
=
i>
C
e-2€
<
D
6
+
)
.
acxs
3b.) ( points) If I wanted to ascend the hill as quickly as possible, what direction should
I head in (at least initially)?
kQ
ai
b < e:
4
A
4.) (15 points) Use a double integral to find the volume of the solid that lies under the
graph z = 1 + e
T and above the rectangle given by 0
2 and 1
y <4. Does the order
of integration matter (i.e. does Fubini’s theorem apply)? Why or why not?
dj
Jo
c€
3
x
DC
6
W
Qço*tC\
cL (sv
5
Cv
+ê-i
1
5
jrl
5.) (15 points) CHOOSE ONE:
2 y
ftFor f(x y) = ex(r
) find the local maximum and minimuni values and saddle points
2
using the Second Derivative Test.
OR
+4y+z
x
‘Use Lagrange multipliers to find the maximum and minimum values of f(x, y, z) = 2
x2
16.
given the constraint
2
+y
—
—
ft)
(x*
*
-
-
-e
0
O- XD
-
cc (o
C-2,.
2 e(4c-
-2e
= 5(
Loo)L
(p,o
(-D
LO
U
(La2
-
>0
L2o
0
-1
6
‘I’
a
N
N
0
NJ5
\
A
o
1
—
(c
—
—
:_-
-4
)_
--
0
cD
L
r-
-
__i
0
-
JcC
—p
-h
r
L
I
-
-
r
I
c