1. You have an elemental BCC material with an atomic weight of 59.71. Given the radius of the atom in
this material is .11nm. Calculate the density of the material
12.101
2. Pearlite
Is a two phase st
8
Math 241, Fall 1997, exam 3
PRINT Your Name:
There are 10 problems on 5 pages. Each problem is worth 10 points. SHOW your
work. CIRCLE your answer. NO CALCULATORS! CHECK your answer,
whenever possib
:;.
~
'.
2
3. Find the equations of the line through the points
Q = (!, 4, 6) . Check your answer.
'/
f<? =I
and
t +7f +J. 1;
X~  Y+3  ~l.f

I
.
t7
a
.
dt~.(. .~
P S;tli/~
~
P = (2, 3,4)
tf
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3
5. Find the area of the triangle with vertices P = (1,2,4) , Q = (2,1,2) , and
R = (2,4,6).
~
1/JPQ": PRII
 t.
~:
:.J L'
2.111 fL
~
p
I
I
1
Ilf

],
=cfw_II (2+.')L(2.+L.)1 f(l+/)t/l
:t
4
7. Consider the curve whose position vector is
"./
= 2t2cfw_
7(t)
:0.
1 1\.,
(
f
.=.;2L ~ tfl.,
fii.
\:2,lfCfJ
~

( 1:): 4 t
I(cfw_ f
+
Find the equation of the line tangent to this curve at
4
7. Find the directional
"
derivative
Dfi f at (1,2) for the function
in the direction of the unit vector ~ l'
"
~'Urll "
Du
~J.
=
Sy, t12
~
G
:;.
t
t
=
3x2y
0,;~T + H'1) lOti . ~ t 1~
Vt~'I
".
;
5
9. The temperature of a plate at the point (x, y) is T(x, y) = xy.
(a) Draw and label the level sets T = 0, T = 1, T = 1,
T= 2.
",1
'./
T = 2, and
x
'
(b) A heat seeking particle always hI
SLcfw_ I"1l1ey
PRINT Your N ame:~
There are 8 problems on 4 pages. The exam is worth 50 points. SHOW your work,
LCIRCL~
your answer. NO CALCULATORS!
Neither your exam, nor your score, will not be ava
4
7
( 6 points) Find the volume of the solid in the first octant which is bounded
x2 = 4y and 5y + gz = 45 .b
~
,
s
o
All:
V6f
':.
1
~
t,( dl1 =
bf(J, Y:t
,"
V(bJ')
r

o
7jxL
SS"'1
y
if
d'
~
S"j
b
4
8. Find the equation of the sphere whose center is (3,5,4)
to the xz plane.
"
/
r4
po;~ t
O~
tD4!cl;t!~ tz.,
tz.<
Y.
~ flf/14< j,et~ ~
Iv
~
X.:r f/~
Sft./~
~ ~ fqt,~
1Zv V4 J
~(~
icfw_~
i"~074
~
~
.
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2
4. Consider the triangle with vertices P = (1,2,4), Q = (2,1,2) , and
R = (2,4,6) . Find the angle of this triangle at the vertex Q.
P
\.J
V
aP
c,j (
0
~
~
65P
;:
~
1145f1111
( cfi ~ \
Vi

~4/
fXq~
s
J
;< (!O~
Ccfw_t/7 h cfw_!It
PRINT Your Name:
There are 10 problems on 4 pages. Each problem is worth 5 points. SHOW your
work. CIRCLE your answer. NO CALCULATORS!
'/
I
I
= y2 in 3 space.
MATH 241, FALL 2001, EXAM 4
PRINT Your Name:
There are 8 problems on 5 pages. Problems 1 through 7 are each worth 10
points. Problem 8 has three parts; each part is worth 10 points. SHOW your
work. CI
Math 241, Spring 2001, Exam 4
PRINT Your Name:
There are 10 problems on 6 pages. Each problem is worth 10 points. SHOW your
work. CIRCLE your answer. NO CALCULATORS!
1.
(a) Find
(b) Find
lim
(x,y )(0,
Math 241, Fall 2000, Exam 4
There are 10 problems on 6 pages. Each problem is worth 10 points. SHOW your
work. CIRCLE your answer. NO CALCULATORS!
1
3x
1. Find
0
0
x2 dy dx .
2. Find the volume of the
33
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7. The picture shows the contour map for a hill 70 ~t
the equation z = f(x, y) .

  
high, which we assume has





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17)
lo
125'"
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?,S
So
25"
O?s
so
1>
too
IJr
15"0
/1.5"
~c:o
y.
12
Math 241, Spring 1998, exam 4
PRINT Your Name:
There are 7 problems on 4 pages. Problems 1 and 2 are worth 15 points each. Each
of the other problems is worth 14 points. SHOW your work. CIRCLE your
11
Math 241, Fall 1997, exam 4
PRINT Your Name:
There are 10 problems on 5 pages. Each problem is worth 10 points. SHOW your
work. CIRCLE your answer. NO CALCULATORS! CHECK your answer,
whenever possi
MATH 241, FALL 2001, FINAL EXAM PRINT Your Name: Get your course grade from TIPS/VIP late on Tuesday or later. There are 18 problems on 10 pages. The exam is worth a total of 150 points. SHOW your wor
Math 241 Exam 2 Spring 2008 Solution
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Be sure to number your pages. Put your solution
to
Math 241 Exam 3 Spring 2008 Solutions Please leave room in the upper left corner for the staple. TAKE THESE QUESTIONS HOME WITH YOU WHEN YOU LEAVE. I WILL POST SOLUTIONS LATER TODAY. Write your answer
7. The temperature of a plate at the point (x, y) is T(x, y) = 20 2x2 y2 .
(a) Draw and label the level sets T = 7, T = 0, T = 10, and T = 20
(b) A heat seeking particle always moves in the directi