exam_01 - o E01/051/01 Use T.C Multivariable Calculus 16s...

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Unformatted text preview: o. E01/051/01 . Use T.C. Multivariable Calculus 16s MATH 2451 exam_01 120 min Exam Course Duration 2016—02-24 (Wed) McCary Due Date Instructor / L. w i, u-s 2 1N r: we \ I W J \J x... ‘ 2 _. x. .. \ a.) n! \ 31.x). A J DJ J .( \wj ..1J\ «J x ®@@[email protected]®@@@@[email protected]&@@@[email protected]@@®@@®fi @@@®@®@®@®@ @@@®©®®®©®®@®@®fi @@@®©@®®Q®@®@a [email protected]®®@®@@®©®®©®0®@®@®®®®® [email protected]@@@@@[email protected]@@a UG®@@®@@[email protected] @[email protected]®@@@®@@§®@®®®@ ;@®@® [email protected]®@®@e . Net ID [email protected]@@®©@@®®@@@©®@®©®®@®®®®@®®G©®®m®n (First Name) @@@@®@@®@®@@ [email protected]®@®@@@®®®ei [email protected]©®@®®@ »[email protected]@®@®@@ (Last Name) 2016—02-17 08 : 54 Q. EOl/051/02 O (Scratch work, will not be graded.) so E01/051/0L O (Scratch work, will not be graded.) iv I. E01/051/04 Q 1. (10 points) State and prove Fermat’s theorem for the IR“ —-) R case. You may assume the R —> R casex proven. Lér Mail?” 136 0V6~,”€Fé “JAR— J EMA; LE7 us A pa/‘mvé {mam/A =7‘EWé A MITICAL POM” £12995; 1F" mam Tug A MNCMMTH Grirr’ImL F2)th— 51 OTHévZWISQ 1:36me 97: (it 9.5% 25%) v.74 0/, 1mg 112H 3 ‘5 DI?” F'EESC’RJTE 2:1,ng 6" “Zap. a Q 9f+ifik> lé DtFfiéénn'z/wig 4H? @57 H796 A mix: ATfi:O €AT1<¢16§ aim/W”. mam/w Tlféwém / £7.20 is (CmTrML POINT) .. {(0) I6 g'd-P‘/4M00TH cflmcm 17mm“) .LGraderOnlyi Q @@@@@®@®@®® O .0 E01/051/05 Q Ants) I Give the definition of a ball in IR”. (13) Give the definition of an open set in R”. V 76 “2h / , 10) OPEN ; (i? ; p’flt-iV/ __Ji X (c) Give the. Fréchet definition of derivative of a function f : R” —> Rm. I C (3,; 447-3.. V , f". ._-:—"E ‘1 . _. 4&2,” fir: “*4 i ~* h a. flu: w -' /' .- -/ a“; ’ i. ' I X ((21) Explain why the Fréchet derivative of a function f : R” —> R is a row matrix (and not column matrix)_ Skew/1w gaff; LP. " gig / ! ‘-__ “4’ fl". FflécHé-ii Dog W in on; i R i m , - - 3'1 ~ H . ‘. ‘ 3 :LL-LJ ‘3- a vii ‘M j if: 13%;; , O \Héiix" V355: W5“ P” IL": L 1'.” 5 W: 11/ 'W 1/4 -- .‘"--.L ‘xé, ft. Jr Grader Only ,L .. E01/051/06 . 3. (10 points) A pair of functions f and g are said to satisfy the Cauchy-Riemann equations if 8 f 8g 8 f 69 — z — and — = —— "‘~ A 7- I 6m 8y By 82: T 12a ".2" -I v; n ~§ And a function f is said to satisfy Laplace’s equation if {37:31:71 § r, c4, i, 62f 1:“ ".7 at? + 5372‘ — 0 Let z = m + i y be a complex number. Then 1/2 = z*/|z]2 according to the rules in C. Viewing this function 2: I~—> l/z as a function R2 »-> REL ’12" " ” it“; Fo= W 3! 52%: and denoting F = (that is, f is the first coordinate function of F and g the second). (a) Compute (b) Show f and g satisfy the Cauchy—Riemann equations. (0) Show f and g both satisfy Laplace’s equation. J, Grader Only 3, . . O. E01/051/07 Q Vic-x \/J «y N 3.. 2¥ Iiufvj, __ 2\ 1"! 732—; J }?\_E/ 4575 i 7’ I) ' ' / f); if, H "L‘- ( 55f 1} I k k 5 hr- —-. f7“ 5.. “$— 1 I j "7/ >43? T? "(f/TI f 74 J' g; . f _ * 27-7 “(07(7/4" , A? i; r I'm-{r 1) _¥ r O. E01/051/08 Q 4. (10 points (bonusD State and prove Clajraut’s theorem. Skip this question until the end and try it if you . 5Wfl9§€ «P 14 DéFiNéD ow A may: 1; "THAT comm mg POINT (5 1’: THC FRI-JCT!ng 3197!?! AND 4 Mg CoNTIMA 01A; ON 17/ THE/7‘" ‘ 1P1)! (4J9) 2' 10%: (DP-01.6 QwALL MALL/(63 GF _/ “75/0: 1 g @QML/ngflmbfl—Q afwwl?(a,b) (2)11: Wé Lé-‘r' @j-:-=P(>c,b+h)—:PC><,W gwn)- (an) [email protected]¢ @757 m.v.’r./‘n+é1éé 14A 41m [5?an a AND 92m ,0}. @[Aemyaégr 3641“ : hgéw, gaming @127 MVT THépélg A #4 ETWN 10 Amp lam Ali l/prfléL, 0H. LOPfxépflifl I y (fig @‘ MMBININC—y éQmA‘WoNg/ A Alazh’ig (gal) ‘ an (a, n: mng .n 7 x . <9 .j’kdde)—;?(aj|9) to Lil?” Ah: --— «9W giyéri) Z 4&5)” HAHA 3 m4! (5,4) 9&1?) (’3‘ Sin/ll LAW I, _ Ah Irij'o’t , [0+ L—fiijtgéaf 14/10) —vp(a\1‘ofl MOM? Mva TWfCé AND Camwwry or? 5%. I, £22"%: 3775 (kilo) 1m; cWQALL‘rS we“ 39mm 22M CW5 firms L949: fie yaw W31 1' A j 10W Warp” ( Ar) FUW’LON uni/gap] @ Q” Did A W909 0F you CAN mm M'V'T" TWICE: an WWW/S memo? Edgy (E53). .1, Grader Only J, O . 0® @@®©@@o®@@@ GD T'Héfléxjabfigl/ .0 E01/051/09 . .. EOl/OSl/lO . "‘\ If 5. (10 pointy/Belt U C R“ be open and f : U —> IR. The function f is said to be homogeneous of degree I p f(}\ :2) eye) for every A 6 JR and :3 e U for which A e e U. {a) Asqune f has the above property and is differentiable. Show [D f :E = p Hint: define g()\) = J and compute g’ (I J- r a / ?.;239(3\:T' .\ / / 49/ a??? L (b) Find p and verify the equation in the previous part for the following function. $11234? “2 f(y) =CC—2y—x/IE on $z>0 Z J, Grader Only J, '6‘ .9 E01/051/11 . .nts) ) Let A = J; 3] and B = Find all values of a and 32 such that AB = BA. ’ fir” r; j” "7 Ass-:4 «5 ‘ UMHMO M {A 0N0 U a m 61:“ L, .. .4 x 7 «r r t r” "3 W.~rliojm_l 0L ‘ a. .3 é "‘ . ,J J “MA “by EI—MI ‘, r 31+ng A. :14: (fl m V; #1 r 7 g at: Eta. ‘0?— 0 ~— flfl; __ A Page '3”) {)er Age; EA 0: 1m (13) Consider the rule associating the complex number 2 = $443} to the 2 X 2 matrix Tz = [ and 32 = $2 + 1392 be two complex numbers. Show that Tz1 32 = T21 T22. 24w)? _ r _ [if ' _ {If 2’=7<‘+“’L\/' 22%756/(fi/L7Wffl” 7-) ._ _.1,~‘_ .I Vac I 7 %—7V _ :; .1; :JJ f-Fj./‘;~: filrjziyl —} £7? ’fifl \{l it; 2 J «fig/,LHA/ij; ‘y’jylffiali r i ‘x -" ._ __r K :5 ‘- 1:— fi 1!); 7% “"é’jJF—Xl: J'- u ' I“ \-"[:/\1; J1 "i; ___ ' l "iw'k. w‘fl I , ’ fi fi (Hf/V J i Grader Only i O 0 $29 _y$ . Let Z1 = $1+iyl .. E01/051/12 . 7. 10 't ,1 , . , , . ( pom S) Algéfl a? MEL-:‘ALLE-Lgrin-Hm 1 H a. .-, (a) Compute the formula for the function A : R2 —> R which computes the area, of the parallelogram spanned b‘, and E]— ?( V2_M (b) compute [DA (3)] [g]. ,.--- {7 (c) Find a point at which A is not Fréchet di erentiable, and explain your reasons! \ A lé NUT w; (H r “D!?‘%'€:fp:6fl'l AW AT ( ‘2 /‘ B€W§é \T’ lé/Nor CONT mum/9 ' CONT INLH’T‘7‘ Liwéfifll‘rg/ €Yté_f’7 f . J, . . .0 E01/051/1L . mts} 7 SC ‘5») Determine everywhere the J acobian of f (y .1: ) = (3 y?) is invertible, or explain why this question does not make 2 x 3; sense. (I, a I V _ .. THié Jug’sfi' lo M [20 WW” MA MI" '32-:ng Fol/i A MATSQi-y! Ta Be" Wags-1- lake; Wl/fl’lQl'fiK/ rel l’l‘x' "r" (/36? " Il-iQé-f/e. l‘; Wit/fill if; inn}? 5' ' = "'IHE jACoeu‘w 15 A 3x: MAHth V’ I?” (b) Deternnne everywhere the J acob1an of f (y) = (332 H 3/2) 15 1nvert1ble, or explam Why th1s questmn does not make sense. mm” 76 E _ e w Salli/66 “Hg A Slam-Afié M/n‘VZ-bfi Wm} A ol{'l' #:0/ “me {win Vin! 1g Mag H;— _ “\x-kfida—g— r y 50 “.110 “‘/ J, Grader Only J, O r. O. EOl/OSl/14 Q 9‘ (10 points) Given f(a:): @— ifxyéO 31' 0 if$=0,y7é0 27 (a) Show that lirn f (y (z) (b) Show that lim f = |k|/e[ki along every parabola y : k562. 2: H 9' —:~D (y) (c) What is the conclusion about the limit- at 6? § Q l i W1??? _,/ m3! (,7 ) =0 along every line y = max, maé O. m and 1, pflm‘ “WK/"- fl 0W5} ““ ‘ ‘ L'H "2 x’fl' _ /.* D IFPQZém d5“ A3511? ,1“ r r _r ._:—.. P f I / KP, /'fl’r ’ "‘5’ M“ r I DH drmzém Viv a; Eflgumg LHWT— W‘LL‘ x/mé flwcflew, E DIFFQZéMIflgLe/gwr MT” ODMIWM .5er mama, . QATé'AV‘W (mwéflflcx 7H6 Wéé JCT Pazw-mwé I706 . war?" saw ‘5‘ V oewuéé mg .1 m. mg. J, GraderOnly ‘L . OED @@®@@®G®@®® . O. EOl/051/15 0 O. E01/051/16 Q 10. (10 points) [F12 “27- (a) Given f m = a: y use a linear approximation to estimate f ' A314— 3; xz—yz’ 2.2' L130“ (10) " .l, Grader. Only J, 0 0:33 @®@@%@@®©@® . ...
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