07win-fsols

07win-fsols - Math 51, Winter 2007 Final Exam March 19,...

Info iconThis preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 14
Background image of page 15
This is the end of the preview. Sign up to access the rest of the document.

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 1201/2. Ql-R7-\;Lool 2,1—2\ 0 0 O l l O O 0 2 ! 1 Tia—m o a o 21 {245-22 0 0 o o 12+12\oo o 33 10442100 0 00 XI X1 X3 5 ’f \/ PIVd’M prO’} ,Lree ace, \6 \réohrrs 1 ~2. o .4) The vasp X1 1 O 0 x X, : X2- 0 “FXSl 1“ X3 0 Xq 0 Z ._‘ K 0 I So abafilS {war NLP‘} ‘3 ’2 o 4 I o O O i o O ) D ) __I 0 o 1 by 130619 -Cor CLPQ Corv—CCDVDnéUS +0 pix/05v vchcorg 0:? \(mfiiP‘) \00\C\Q {n 97/ W mm MMMA LWILWM&°~A‘E wbumgls owe Br 1 )1; l 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‘333-Um 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. ; Swat-X 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‘CLF-O ‘ 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 _ 3-1 _, 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 ‘ (lU-o-HHHI-IH-IH‘LP] ‘ K,9({.Hg.g+g.l+!0))\rg I + (,3 Mfg-ls 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:ir-M 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?) 3-1. )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\\‘$g>aC€/ 0% COLVAL‘A JPJrCm/l) \.C‘ NLcUiVA.‘ -+CV\A“§ 2}: 3,03 :9 Cle/JC (Col-Ln +7 Q‘ A *0 *CV‘AH) :1 OD Page 11 of 16 Math 51, Winter 2007 Final Exam March 19, 2007 (10) (5 points) Does there exist a constant c such that fox/y) : if 75 (0,0) 0 1f (00,21) = (0, 0) is continuous? Why or Why not? 12m @2321 ztmtw (yum—a (0:0) xq’v‘tf’ :mo ‘7‘ 0"“ Oil/L gum LWM 7?} 3 Swag/W : vu—io (\+m1)og7’ “rm 9% one/\xvflfi LO \0) W enigma (Li/yum vr» WW w «Rum/H OWMQLM W 00 Anna. him if CUQD ‘nC‘i {U$+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—MMJ-V: (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” Mum-M “ 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(e4-01(,98)_(1_03)3)' \\ “2.005) .42) we») ,X/ v‘(1.0063.0\%)\.0%> 2+ L(.005) + JEFF-01) 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 Hag-0: \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 MC Cd. (3, w (L “L Lawmfi W380. D) M (9m, (Lu;ng m' '. ’) ...
View Full Document

This note was uploaded on 01/12/2010 for the course MATH 51 at Stanford.

Page1 / 15

07win-fsols - Math 51, Winter 2007 Final Exam March 19,...

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online