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Unformatted text preview: MATH 251, Fall 2006
RAICH EXAM 1 — VERSION B Name (printed): Section #: “On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic
work.” 1 Signature: UIN: DIRECTIONS: l. The use of a calculator, laptop or computer is prohibited. 2. Please present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your ﬁnal answer. You will be graded not merely on the ﬁnal answer, but
also on the quality and correctness of the work leading up to it. 3. There are questions on BOTH SIDES of the paper. Please be aware of this. DO NOT WRITE BELOW! 1Disputes about grades on this quiz must be handled the day the quiz is handed back and must be discussed before
you leave the room. If the quiz leaves the room, it will not be reevaluated (except for possible adding mistakes). §
 1. (a) (6 pts) For what values of c is the angle between the vectors (1, 2, 1) and (1, 0, c) equal to g? +\
+1 (b) (6 pts) Find two unit vectors orthogonal to both (1,1,—2) and (—1,3, 1). mml 1 \) “431+th A \c A J +1 7 S __.. 1
m,m, 067. > (c) (8 pts) For P(0,1,1), Q(2,4,2), R(—3,0, 1), and 5(4, —1,3), ﬁnd the volume of the paral
lelepiped with adjacent edges PQ, PR, and PS. (3&2 (23.0
FR =<—3,.\,o> 93: (LL—12,7 A k A \24 35 V‘\=<\\—3.‘7> +3 "5 \ o \ (\HBHW “é‘dp'hlﬁk : L“.qu .vm‘l v y \t
kn: “ ° =<—1)(o,\c>7
“'1 —'L 2 \ <~1‘e\\o3‘<13.\3\= \~q+\9+w\ Jim 2 ’3 \
:5 \ o : Aim mm m H (mm
q‘ ‘7’ Z = L\+\‘& Ho 2 7A 2. (a) (8 pts) Find an equation of the line passing through (5, —2, 3) and parallel to the line m—3zﬂzi.
3 —2
X25+T If X‘s: 7+2_2:2 +3 pilok
V‘ﬂEﬁ ° “P? ‘ ‘1 PS V’id‘or “C
2= 34+ +2 [Cm/MU. UK (b) (12 pts) Find an equation of the plane containing the points (0,1,0), (0,3, 0), and (2,2,1). Q v a R
\3 Q i (o \zto7 +2 in?“
pic um +1 ix “2‘ O X—12=O 3. (a) (8 pts) Describe the level curves of the function f (ac, y, z) = —a:2 + y2 — Z2. You do not have
to draw pictures. \Lvo \L= ~X2+v1~21 o \xYWr‘oobW} 0% 1 sheets 4'3 \L=o W x122 — CDWQ +1 \L40 ~\(= X3 {12"  M oeriootov‘éx 09 l ﬂeet 2 (b) (12 pts) Find (may132070) 2m4m+y3y2, if it exists, or show the limit does not exist.
\12QX}. PM+L7
“ 0.: O 2‘) L :0 L: $230 OX H t a 7. . +3). l‘iil‘m. OK
2 "+ 1 1 3a _ q a) V 3a X >r a=\ :7 L ~ x15 » 1 \M 1&1 a. mu: +2 Cemlumbh (W139 Lem 4. (a) (8 pts) For f(:c,y,z) : (3922 cos :10, ﬁnd fwyz(m,y,z). 1
ﬁx [X\\1‘2’\ '= » 8Y2 sv‘mc ‘3' Z
mea = — 216% ﬁrm 6’3 2 £sz [ME = '22 3% sMx — QWSM 0L3 (b) (8 pts) Find gi and 3—,: if z : f(a3,y) and satisﬁes 3223/ — 332 + y3z2 : y  z. — = Z —. 3 1—
me XV X2 w 2 w? I W“ 3R 9 3 35 h
lat[3X = 1me H m—z 453 PM 2 a7 = ‘ 3,; +Z
AP = X13 1224 H c>_§_ 211—2
IN V or ax  Xézqszq (JP/c); = ~X +2432 %\ b
wVe X
(32)!) : _g:’ix ' — 2XV~Z 1 Y6} :2 z 5% 35
X Flag ﬂaw?“ x 43; +3. 2 +2¥ Egy =1~ a?! +7.
1 Z Z 1 Z 2
(92/ .__ ‘QWV _ X+3yz—\ at X3 2—\
a 3— ~ ~ Wm  A 7,
v V 913* X+2V32 +\ W x ~Zv32 \ + 5. (a) (6 pts) Find the directional derivative of f (93,y, z) = 332 + yz3 at (2,1,3) in the direction of
(—2, 1,1). V4=<1><,2”,3y22> ziZ
VHMstiamm) M (—de7 A L
‘L'Zx\s\V\ : <_ —3
UV .4 +\
Daimm = VHZ‘zPohu = <l—\\z7tz7j o 43%“ Effi» : —3?3r17+27 : ﬁg: W in; (b) (6 pts) What is the maximum rate of change of f at (2,1,3) and in What direction does it in m Wis 04% Mg 13 \W—(ZATSW; 5 wvmzvi“ W i
. (‘i‘e Mom \3  W27. 1‘77
We. 5H 4.. Wﬁmlmﬂz 4 +2 (c) (12 pts) Find the equation of the tangent plane to the surface :12 — 2y4 + Z3 + myz 2 —12 at
the point (2,1,—2). Hwﬂ = XZW +23 ext/E
\7Fﬂmy,a~\= <\HZ,~%\13HZ,322+XY> VF(Z\\‘~Z\ 2 \‘2— j ‘Y‘q, B‘ﬁ *2>
= <~\,—\1,\q> 1% M — (HS 42%) Metamo °r X~\Z\]+1L\? =*L\Z W XMZV —\L\2 =L\Z
2 M2 +2? ='—t7_ ' ...
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This note was uploaded on 07/22/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.
 Spring '08
 Skrypka

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