08win-m2sols - Math 51— Winter 2008 — Midterm Exam II...

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Unformatted text preview: Math 51— Winter 2008 — Midterm Exam II Please circle the name of your TA: Zachary Cohn José Perea Nikola PeneV Man Chun Li Daniel Mathews Theodora Bourni Anssi Lahtinen Isidora Milin Circle the time your TTh section meets: 10:00 11:00 1:15 2:15 Your name (print): 6 OLUT "ON £3 E Student ID: Sign to indicate that you accept the honor code: Instructions: Circle your TA’S name and the time that you attend the TTh section. Read each question carefully, and show all your work. You have 90 minutes to do all the problems. During the test, you may NOT use any notes, books, or calculators. 6 Total Question 1 2 3 4 5 | Maximum 20 E 16 12 22 . 16 E 14 100 Score Problem 1. (20 points total) Consider the matrix A : (3)2 32) (a)(10 points) find the eigenvalues and the corresponding eigenvectors of A. cm Q1 \ "Ev-“A 2. \50 .. Z a, Z,“ {5 JAM-”2%) + ‘I t O “*2. '1 b . {I 1 < Ci 1 W Z ab 5" 2. Qw&gawb%% (A\ 3:. 2:7 ,I/ 1 2“}; a ”\1 1 E a: 5 f f: G: L Z EC) 2; if: \»z w /\b/ M f ,,,,,,,,,,,,,,,,,,,,,,, ; 111111 g t; ...... W: wwwwwwwww l v; « < ) L1 2 1° 1 WWW“ (b) (3pts) What are the eigenvalues of A99? (A is the matrix given above) \Si 7\ {’3 0‘“ kw vK B 9;»th Av '23? v gt»; Sorm V750 WW“ Riv a Amt") :Mvwiav mg, (C) (3pts) is A99 diagonalizable? fimnfik A {’3 ("Eula ‘f‘mfikmw (Lick. 392‘?“ij E’WQWZQ ) View“ if“ ac ksfiswjtmkfie ( «imam AW am (we, (ax-twat W) /,1\ /A,,L,,\ :r n 5.; , rings“ :_- 11132 if r___r._ A ___‘L_AJ_ :N, 1. pour Ar :Lfl, :_MAMA "MAMA L‘IA,‘ 1:.AA_A« KG) Vipbb) 11 IL lb CL 1681011 111 RR UL area Li, Wildm 1b bile area U1 lbb 1111:1336 Lulutil bub 111 Cal transformation With associated matrix A? luméix (3‘ imam flax/w» Tfla) "2 ~ )C ) Codeq gccgjlgfl L) imn’i‘ 01mm L T (RY) A H E: 2 § 2“ 73 so (mafia Z\; -2, 3a " Z, ‘2, MGR kT(R)> 2: \”Z\mu1‘; 2,“L/ : 8 Problem 2. (16 pts total) Consider the linear transformation T that reflects vectors in R2 across the line y = 2x. (a )(8 pts) find the matrix A corresponding to this linear transformation. (we Nuxflta 30% vtxomi \ \/17'////é” m V (iyhm meta ,‘ (b)(8 pts) What are the eigenvalues andieigenvgctors of A? mm “W 11 = T (All) 17. b \l U Problem 3. (12 points) Evaluate the following limit, or explain Why the limit fails to exist. \UQ q/waJk O CLQD\<XQJM :WX\ X10; 1 ‘ “Q’s-m A—~ _, X W11?” 2, 1—99 X‘ 4L r X~>o ‘I ‘r L :2, (12 $3 'r’mx ‘3 X 4% x i 1 4 m." w M‘ Problem 4. (22 points total) Assume you are standing at a point P of coordinates :1: é 20, y = 10 on a hillside Whose height (in feet above sea level) is given by j H ha, y) = 500 — x2 + 2303/ + 3y2, Where at points E (east), and 3; points N (north). ‘1 “ (a) (6pts) suppose you start moving in the SW direction, do you ascent or descend? V l i WQ (ME/€301 wuz 3‘03M \3\ We (yanked OL,MI¢&4§~Q‘ cal PC“ ckkukfi‘nufi E vs i i i E § § «.23 ~1 ) . V =( ) 3.3 M vu‘h {1 .2, V 3 _L - l '1 nw v2 ~[ .— K7537” Xx) : (Qatari) > V -2320 +240 ~20 2x +6 262 ,. :5 0 arm we, ,3, 63\ M ’7 ,”’_ «'20 ‘6: M ' 93R,\J§u_< )<. 520 too4,QM\WCflMmM4 l 0 K7 ‘7'; \fl (b) (8pts) find the equation of the tangent plane to the graph of h(:1:7 y) at the point P. «62’ Olgkmmsieml‘ We.» (3 l) g, 2“ ' {“3”} ”KX‘W'M‘U‘Z’Q) + fiat wjmxfiaa) {1§)\%)'zgfiam'figamt‘lgéf'Sangm a “all if“ ‘ £3 Q: L“) g1 QSBC "gush (Q) S 12? leLeWQA ‘ ’ 2 «8620 : est-wag) 1‘» (m; (c) (8pts) use the result in (b) to approximate the change in height you experience if you move from P to the point of coordinates x = 217 y 2 9. MGMZ knx Gag/wax Cu, Rb) A“ % v’ZOL'UW‘LQ‘)+imkfii-io) raw/001423 Problem 5. (16 pts total) Consider A and B two 4 X 4 matrices with det A = 3 and det B = 2. Furthermore, denote by T : R4 —> 4 the linear transformation T(:c) : A50 for all a: E R4. (at) (3 pts) find det(AB) MLH‘E) ; M Raw TA : 3am; (b) (3 pts) find det A—1 4 $1”; 1-,, .L we) Cram 3 (C) (3 pts) find det(2A) 6CMCE g L9 9 at £1 ) \X/KKSE «A W ‘7 [W‘LRE @kC/l Seq‘fig ”I éujj Qfi €CiC/I: {am \KIE . A; i . C fir Mgr): Swain 222’s (e) (3pts) is the linear transformation T defined above invertible? sorrel ah; a 4:3 first“; or; Jam. r ”r". .Rwrugg - a W 9 ~' VJ \- “Va: ”flew )n e (/3 a]: w; . r a, w . _- .., ‘ , (d) (4pts) can we find a basis 8 of R4 in which T has matrix B? Problem 6. (14 points) Assume X(t) is the position vector at time t of an ant moving smoothly on a sphere of radius 5 centered at the origin. Prove that, at any moment, the velocity vector dX . . . . . ~— Of the ant IS perpendlcular to its p081t10n vector. wdt 7, *7, i ’Q 3KS'Qsz, “ X 4} 4721:25 i CRUZ muhfi )uQu‘i {unlit/WV ‘_ E E E E E E Mt)?" + Skid/1+ 1402:25— ~ a: A “bk “:3 (It “13% 1N) (3.: +2 Swing?) wags}: Wildcat (L. 1% §Q «2 {3:1 (17??) “ t)” 3 3 At 0 €53” 32?“) CL Cg; MW. hm mmmmm wwwmmmMW mmwwét g ‘9 WM {\WW kgékzt 3&1: \k 9 £5me ”3 lfig‘“ QKW'E erxgflam 05% F(%)J:))3C :1 X’K‘£S1,¢%L Gukclwla ,« ‘\’ M ~> \ a\ gig“ L: M“??? "r X2¢3f£J~G ”Reamefl; QfiMsQ? ERAMCXJ Ufigagé E) 3 I f n m i ‘13 at Elm“: “in“ ukgfim K“ S " M JW Nimm <7 F V) @UJZ. (Lac-ago? lAemT €1,ng L M 3% (film-‘8‘“ Mb (3%“ Qfi‘x ...
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