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Unformatted text preview: Math 51, Winter 2007 Final Exam March 19, 2007 (1) (10 points) Find bases of the 11qu space and the column space of the matrix 1 2 0 1 2
1 2 0 2 3
A: 1 2 0 3 4
1 2 0 4 5
Fwd MUM
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1 ’ LI Page 2 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (2) (8 points) What condition(s) must b1,b2, b3 and b4 satisfy so that the following system has a solution? :0—33/ =b1
3m+y =b2
r1;+7y :b3
2:1;+4y :1»;
TM wgmjced mam” ‘9
I "3 b! ‘ V3 ‘0’
bob MRI
’5 . I ’91 W) O '0 Z l R1),
l 4 b; 0 ‘0 “3"” R ‘Q' 0 ‘0 bq’2b’ Fol” “HM 3‘333Um 46> lam o$olulmm) ll VHUL3’“
\QQ LOH$(€>\'€Jrv\’) “ll/108' i3 0 : ng ’b\ "lozl’gbl
o :— bq~gb"bg+3b\.
Swnpx'xM‘rWs, \9w‘9m \95\\°‘4 mws‘r SCAN$413 O71 a\‘9\"\’:1‘\’\93
O T'— \p\ ——‘o7_ +lgL+ Page 3 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (3) (5 points) Let 73,71}, and 7 be vectors in R" Whose magnitudes are 1, 2, and 3 respectively. Suppose that
Y is parallel to (and in the same direction as) 77, and T7? is perpendicular to 7. Find the constant(s)
c such that + T] + 7 and 7;) + c? + ‘2 are perpendicular. ; SwatX MAYO m QWaHel) (31,944,
7' §Amce§amd§ZMe “L ) X'%LO'
uni Mai RHWE W“ a ‘L ‘l
X 5
(x"+q4/é7' (wcg’rﬂ ‘0
5 §_§+7_c;.;Arie+2?"%+Z¥'LQ+Z><'%+%‘X+Z'EA
+%'?:
V'V'; :1+1C+ 0+2+Llc+0+0+0+0j
V2—
“H = \atec‘e ’9 (4) (7 points) A matrix A and its reduced row echelon form are shown below: 1 7 5 9 l O 0 l
2 ? 6 10 0 l 0 l
A: 3 7 7 11 and rref(A)= 0 0 1 1
4 7 8 13 0 0 0 0
What is the second column of A?
 . C A
W Rama/0L Low aporm .mplms MC P C: ‘0
C3
Cw
‘l’l/vUVI CH‘CLFO ‘ Mow prc'
C \
M
C} +6“! 70 CL, LOLUmnS 0.9 A (emu we wml 4'0 éolve {we V?) W avg,” 30 C(st: 60 V9." VH'VP‘Vs
q I 6 3
“Md 1 L0 _ '1' ._ 5 :_ ’%
ULQJUMA 3‘4 1; ’ Page 4 of 16 C Math 51, Winter 2007 Final Exam March 19, 2007 (5) (10 points) A box containing pennies, nickels and dimes contains 13 coins altogether, With a, total value
of 83 cents. HOW many coins of each type are in the box? bc Page 5 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 J . (6) (17 points) Let MMmu:mM a Show that 71 and 72 belon to the orthogonal complement Vi of V.
g HHH
HHH
MOH p—k V.
'42 
I a ll 3 ——1»\—\\‘\+l":o
I
a 0
1 , * J—
CC/N'L .Vl:’.+lvo+o‘1+l'lvb 1"?V\V8m\/ b Is 71,72 a basis of Vi? Explain Why or wh not.
y \{@S . SwwL \j '\S (L lob/mmgt'maQ Sulwpm
9’3 W W (MM/WSW“? W [C 0222 "/21.
)
WWSMVL)%Ma/wam W5 M 0kng PW VL. Page 6 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (c) Find an orthonormal basis of Vi. 6» mm ’ SC)» mfd'k
"" ,. V\ ._._ ———""‘ I c .\—— \
W" m WWLA ﬁx;
J . ..I  .4 _‘ _).—J m
3,: wva  v} (gm » w \
»   +o'\ /‘
2 (‘9 I 0.30 l‘HO+\Z («BX‘X
7. O
W
X 3 _
31
_, 0
y '\
‘
I
a, J o l 3 ‘ 0
W21 :9: —;_ i. —‘ /' ‘ A —.
1w! \B; (/6; )ﬁll (d) Find the orthogonal projection of u on V. ._v — ‘ d J J
mjw. a : (U'WDW‘ 4’ “'W1)W2
F ~ xx . «L ‘ (lUoHHHIIHIH‘LP]
‘ K,9({.Hg.g+g.l+!0))\rg I + (,3 Mfgls
M L/——F_T———~’—J
g] 5 9/3 1
1 ' O I
l + ,l. " ... l 3 3 " us
0 I \‘lls
.J A . \x i i o.
r : d 4. 2 7’ '1 90 V00ka M WW“ 1’ '* :113
~~ " a
I "/6 rz/B’ Page 7 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (7) (10 points) Let T : R3 —) R3 be projection onto the plane P that passes through .6 and is orthogonal 1
to the line spanned by [ ] .
9 (a) Find an eigenbasis for T. T We mom Wkva 0 MA I a \ (ML x W
WW L\O)W$\7{«C€ aimed WM K? Um W] i q \ " \L
SQ M mgmhws i8 X6 AW
.47 c
1)] ﬂ Lek Ple’ 4
C1 0 l ' vx _ 0 O]
’ ‘ WWW? (13) Write down a matrix in standard coordinates which represents T. You can express your matrix as
a product of matrices and inverses of matrices. 0/4,! T ; "Qio—Io/OK 780
H” “i M iwx o RI+UH441.M: TPJ’ is SPM([::]). (ASL Frojp 7 Ig ~FmJ'pL Loy—L 8: a]. W “Ma/Mix fvf Pmdial 1'5
B<W>"B+ = MMW w w [a;§1[§‘r2:irM q Math 51, Winter 2007 Final Exam March 19, 2007 (8) (15 points) Globo—tech Marketing monitors the dollars spent each year by its customers on apples and
oranges. With representing the number of dollars spent (in millions) on apples in year is, and 0(k’)
the number of dollars spent (in millions) on oranges in year k’, they determine that (We + l) : %a(k’) + %0(k’) 0(k' + l) = {550(k) + ‘ . —> _ (1,045)
We shall write 1) k. — [ 0(k) (a) Find a matrix A so that A716 2 7H1. Notice that this will imply Ale—170 = 7k. um “he gg/Io g/IO (b) Find the eigenvalues of A, and for each eigenvalue ﬁnd a basis for the corresponding eigenspace. 32.
Mm: who "W AcHAVA):(A’z/uoNA‘ﬂol‘ﬁ 2. 5' '1 ,1?
A‘ﬁk +1/00 (00 % of 16 Math 51, Winter 2007 Final Exam March 19, 2007 2
1 [a] = 000+ “[1 (C) Express [ ] as a linear combination of the eigenvectors you just computed. 9
nl ., =“C
2.; JiCl..C,L ~) Z/ZC‘ _3 3‘1/2!
\: C"’rC2_ —7CL"l*n
C"; 2
1 number of dollars (in millions) spent on apples in year 100? What about dollars (in millions) spent
on oranges in year 100? d Suppose that 770 = . Usin our answers from above, what is a good estimate for the
g y N o
,J l LL] limin 9h maﬁa/s
QMWVOH Oh orwwbﬁ§ Page 10 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (9) (8 points) Show that if A is an n x 77, matrix then there exist scalars co,    ,cn—not all zero—so that
det(cOIn —— 01A + czA2 +    + ann) = 0.
(Hint: For a vector "77’, What can you say about linear dependence of the collection 7,1477),    ,A“?? Why might this help you?)  n :1] s9 0. \nonzovo Veal—or my “27
\J , “0) A?) 31. )Rmv \s 0» Lb\\CLho~/\ h
off (vwmw V’EG'LV‘E m m) 30 muSl/ \DZ LlMOmrlﬁ dQF/Undemlr' \\
l .J
(Cgin +C\A~F\. "l (A .
‘—\"\r\\‘$ wehég \l \$ 0\ momzero V‘Qﬂl‘ar g V\
\m. hm: wwu\\‘$g>aC€/ 0% COLVAL‘A JPJrCm/l) \.C‘ NLcUiVA.‘ +CV\A“§ 2}: 3,03
:9 Cle/JC (ColLn +7 Q‘ A *0 *CV‘AH) :1 OD Page 11 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (10) (5 points) Does there exist a constant c such that fox/y) : if 75 (0,0)
0 1f (00,21) = (0, 0) is continuous? Why or Why not? 12m @2321 ztmtw (yum—a (0:0) xq’v‘tf’ :mo ‘7‘ 0"“ Oil/L gum LWM 7?} 3 Swag/W
: vu—io (\+m1)og7’ “rm 9% one/\xvﬂﬁ LO \0) W
enigma (Li/yum vr» WW w «Rum/H OWMQLM
W 00 Anna. him if CUQD ‘nC‘i {U$+a $0 m0
)UQJN'Q 9/5 Q \NOUMK math g, unﬂhnuoug} (11) (5 points) Let S be the surface in R3 deﬁned by
2 nag—kg? —22= 1.
What is the tangent plane to this surface at the point (1, 2, 1)?
1
. . 1 5’9” 1
XVK’VM‘OWC) " x J" L) ’76 iii—MMJV: (79% "’13 ’21“) Page 12 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (12) (12 points) Consider the function f (133/) 2 13/314. (a) Carefully draw the level curve passing through the point (17 —1). On this graph, draw the gradient
of the function f at (1, —1). 0+ C‘)"'D \ £Q‘MV‘33 : (IMF—n: “(f—ma "I
iwvs W (MFV‘Q 1% QQXAJD') ‘: ,_ I
’79 ocq’olﬁq
’33 1 c “is/)2 (b) Compute the directional derivative of f at the point (1, —1) in the direction if : [ ‘ hair 1 ‘73?th “:1
: [3113:]
9 2; MS +\a‘§ : " A 01K») w!»
L.—..._l (c) Suppose that f(:1;,y) gives the height of a mountain above (:v,y), and suppose further that you are . . . . . Ar:
stuck on the mountain at pos1tion (1, —1, f (1, —1)). In what direction A; should you take your ﬁrst step if you want to descend the mountain as quickly as possible?
ViCer is the dMCi’lOM 0( ng} MOE/ml”
MumM “ N (mm. a 4.
SO Walk Ni 2L»! Page 13 of 16 Math 51, Winter 2007 Final Exam (13) (10 points) Consider the function f(:r,y,z) :x/ln (62933123) (21) Write down the ﬁrst order Taylor polynomial centered at the point (2, 1, 1). 9‘ (3(3‘3‘33 ‘ £(%‘\') + [4% £5 '91:] [53:17]] _____. 91%;: ———‘———
* NE?) W WW")
I ,_l__. 7;; .L— e,
lazamﬂ V527! mu) alum”)
: ‘   y ’32:” 1
1% am?) W9 )5 W”? 9 2 ._.L — 2 ’
Mm» = 2 + arm» » 1:11:40: 0
(bl Find the approximate value of the number 1n(e401(,98)_(1_03)3)' \\
“2.005) .42) we») ,X/ v‘(1.0063.0\%)\.0%>
2+ L(.005) + JEFF01) l, 2’ grog) —
‘ March 19, 2007 Page 14 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (14) (10 points) Find all critical points of the function 211:3 + 63:31 + 3312 and describe their nature. 6 6
WHOM: o 6 Hag0: \a lo
6 L r 6 (a
dpo d.=\2 > o
at; 0'6“ 62='%40 dz: \zte—ez >0
Lo\o\ \s w Saddle VOWE am} is aLomQ :m'wmm. Page 15 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (15) (10 points) Use calculus to ﬁnd the point on the circle — 1)2 + (y — 2)2 = 1 which is nearest to the [YWMWWM ; MSW/WM 8cm” awa (to moat/W
mslmnt (a 472 + co «2)7—71 , 3mg)
@ lx'Mv ﬁx—1(i(“"))=o a w};
(9 ‘5’ 9‘99; ‘9 M“ MM’ZWO HAW/",1 @ g = \ (art): + Lg—2)¢: x J, 3.? 41:79'3 ,9 we
14 ’ 9’2 ’79 ‘ A ‘2 4» (91/2)?‘ l
ﬁr
u 7’ (1")71; J.
LIMA) 5’ rx,[ 5 i \ v
“"56” PO‘W’CS m ﬂing;
(\ W5 ) 1’ 2W5) —5 Cl\$km&1= E'loﬁg 5ma,\\<_¢ (H ﬁg) 1+ 2x175) ~9 Instance}: 5+ Io W5 “weer $0 Ll’ﬂ5)l’1{ﬁ5) l5 hcaxc5+ 4504446 ariglf). “ LO CQ‘Q M ‘ _ Page 16 of 16
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