thq_06-key - Exam 165 MATH 2451 thq_06 T-C Use Course...

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Unformatted text preview: Exam : 165 MATH 2451 thq_06 T-C- Use Course : Multivariable Calculus Due Date : 2099-02-11 (Wed) Instructor : McCary My (First Name) (Last Name) 2016-02-12 16:21 1. (10 points} Let U C R” be open, ('1' E U, f : U —> R“, and SJ 6 R. (a) Show the Gateaux derivative df is homogeneous in the direction. That is, Show df((i':_tfi} : tdfla; (Name): (Howl:- ¥m> (9:2 Name) = fiflm m0)“, t=hb ¢u+o vi» (MUM V ¥(8+u?)‘¥(3) v t _ m< “A z fit flaws) N z t 2g; (£(3+u:3-£<aw> = t iflmw) W = MINCE?) ' theme) (b) Show that the Gateaux derivative df is linear in the direction when f is Fréchet differentiable. That is, Show: E Df(é‘}:>df(a;sfi+tfi}=sdf(d';fi)+tdf(a;fi} 3 PM) => dflmfiflflmflv q> d¥(3ssi+ta): [D¥(8)](s7+ta) D a [may WH W 15 LINE/411 = S[D£(BW]\7+ t[D¥(a)]a : sdlflfifi) 4 Mini-,3») 2. (10 points} Let. f be the following function. f(y)_ O [1 otherwise (a) Compute df(fi:_ '6}, where 13' = 7Q 6. (b) Show df(fi:_ if} is not. linear in 13'. (c) What. does this say about. D fld}? @ q £ mm- (a) M (my J4» 732:2?) W (“‘7’ : L); (M) 1 /7 all”? © Tux; Emmsz 1:5 avg/«my NWLmM. E-s. 1A» .9 CHM-7v) IS NOT AWthVE IN V ,', NOT LIME/r4. (9 Ammonia» +0 PKEV EK‘éRchE) :1] 1349(5) => OWEN?) LINE“? IN \7 3 [P ‘> w] =r> [14,»ij ChFU-fl) IS NV'r AvorttvE IN {7 =[> Diha) Dogs NOT Extsr 3. (10 points} (a) Let. .3 E (C, and let. z* represent. conjugation. Show: i. Rez : (z + z*}/2 ii. Imz : (z — z*}/(2i} iii. .3 is purely real exactly when .3 = z*. iv. .3 is purely imaginary exactly when .3 = —z*. LET 2:95,.1473 G) * ~ 2 = i‘*<1=b wig: x-is 2+1 (x+i~5)+(x-w) Q, - a, :0 _ 2x 4" ‘5 =0 ‘ T 4:» z=f€w = x * :R‘q‘ @ 2=-?* <1=> wig =-(x 1% @ ii __ (MUD-(x493 «.1, 99. .0 5“ M . x :0 = 3.2 4:» z :i’bmi 1 —‘ 9 : W55} (b) Let. M denote a real, square matrix. Show: i. S = (M + MT}/2 is symmetric. ii. K = (M — MT}/2 is skew-symmetric. iii. M = S + K. @MJ M+M = a. @ KT = = MT-(M'Y MT—M T-M a [ i= :MiMT+ 5m 1. T T : MT+ (‘4’)? _ M7 + M M + M1- ~_——— - = g 1. 94 a 4. (10 points} Let A denote a real, square matriX and let = A5. (a) Show [D f(fi}] I; : [AH-1: using .the coordinate definition of matrix multiplication and computing [J LE1 [A]=i’4flg Vim) 96w (b) Now compute [D f((i'}] I; using the Frechet definition of derivative directly, without resorting to a coordinate argument. IN 1M5 rrzécm par go Fog rut; zAgZ) «Cw—L — - n , w mm (W) L()) 3 AH”) z Maw“ A 3; mm J 5mm, mm = [12mm = M» wE'LL Damn; - AH H3 “’5’” = MW) {Damn - e 30 60 Mam-— [own W3“) f mu m Lv-————' NM} LGu k Lag“ mar VJHtUi AND 7"” WNWETWES A Mkme. momma 1N (DMWYEMGr FREcPET 92mg. 5. (10 points} Let A denote a real, square matrix and let = "T A if. (a) Show [D : 25ET I; using the coordinate definition of matrix multiplication and computing [J f(ti'}l- (b) Now compute [D f((i'}] I; using the Frechet de 1tion of derivative directly, without resorting to a coordinate argument. Hint: f is scalar-valued and yT : :1 when 39‘ is a scalar. In? m In "Ami _ a Alfifi) = (my/WM) 131% A...) 72%”- s Lillm— WARN-70 A6 R o = "M r MWQWTM-WM 6a . r x A»; [0mm QM—f)?» = air/4% +97er WAR = {WNW WET/rt =QZT<MT>R WIT/H? Q. Lfd [mam]; x’thW-l The purpose of this exercise (and the previous one} is to 1} help you get familiar with the Fréchet definition and 2} to remind you of the importance (and brevity!) of coordinate-free methods. 6. (10 points} Recall that f : R2 —> R2 is said to be afiine if 35 E R2 s. t. f—E : :iE I—> —5 is linear. Let f : R2 —> R2 be affine. {a} Show that 5 in the above definition is (the answer to this one is very short}. 3: (38—13(6) = my}; => 1 aha) (b) Show that fit? + 1'5} : firi} + [D f(d'}] 1'5 for all 65,1'5 E R2. Hint: show 1'5 I—> fir? + 1'5} — flfi} is linear in 1'5 and then use the Fréchet definition of derivative. Even)! Mums MM CAN BE WET/TTEN A5 7502) Mm? H5 Pot: W $0 ANN) = ham-1(a) = (Mafifltl-(AMB) = A? = A? +5 Afl’fi) 3/0 [919(1)] \7 - H $0 for) = 3902) + rams T. (10 points} Let fifi and if be given below, and compute f(i3+ it?) — flfi} — f [Dfufi'll'fi 7 for t: 1,1/1o,1/1oo,1/1ooo. Does the difference scale like H“ for some k? f (.35) = 5 = lil ‘3 = El Wt; :5 In (53w); Affirm = [BMW 4:033”) 9., éo MUD-s60 = WM?»- m : (gt44)(t+4) - ‘7 (WW- (“nm ‘0) : at“- 3“: \_ <q£a+4t+++k2w> g’t‘4'5'b : < giant Jc(fist9) = tam WW“ 13 M = i WWW; M1) = [2,] AND l9¥lfil= [‘5 IX] IND) YES, ruff ozvrway scALEs LIKE t”_/ 2% —Q5 [M0] = .4] [9mm] = [Z] t[D¥(‘0][i]-— [m 5:: This question is asking you to quantify how good the linear approximation to f is at 33 in the direction 1'5: the bigger k is, the better the approximation is. Ungraded: try to think about what this value of It means. You are encouraged to use a symbolic calculator on this problem. 8. (10 points} I. t a (a) Given f = 3:2 + 35:2 + 2,22 and : (f2) , what. is |:D(§o (33)] '? z t3 c ‘3” [WWW as M [Dam] +31] H. 60 [0343(2)]: [useemfiwwfl : \afltfimafll [9w M, h] 1 Z 3 (mm a?” 4 2211 40 TW’ QWULT wtw BE However: was ammmmm I; prngErlflE‘o B/c tr IS FAR $1.,”me (T'MN mNWRUchNQ HA5 3” MAwrztfl . a I. 2 a I. (b) Given f = (I V: Z) and g = a2 + 332, what. is |:D(go :| '? z ‘ z l 9. (10 points} Suppose that. f : R2 —> R2 and it. is known that. [Bf = [1 _2], 33 : R —> R2, (if : R —> R2, and em = (f o W}. . . s t2 — t + 1 . q, (a) Suppose 1t. 1s known that. pH} . Flnd q (0}. {St—29 2m = WWW» % F'<£>=(‘ifil) em = [13W] [‘3‘] W =[1'Z‘M‘a’] m (b) Now suppose that. is unknown, but. that. it. 18 known that. 33(0) : and 610} = Find 33 {(0}. film : [Damowflqu [1 ‘2‘] W = fi’m 433% mm H11] ...
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