thq_06 - O AO6/01L/01 Use T.C 16S MATH 2451 thqa06 Exam...

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

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

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

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

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

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

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

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

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: O. AO6/01L/01 . Use T.C. 16S MATH 2451 thqa06 Exam Mult' - 0 Course ble Calculus 1var1a 2016-02-11 (Thu)* McCary Due Date u . Instructor 0 Net ID 0% , \i/ , f . \. . /l\ ! Saw [email protected]@@®@®@@ @@@@®@ .0,“ fi‘m @®@@ isemeeeeeen maieeeeeeeeees [email protected]@@@®@ @®@@[email protected]®@® eeea eee / \11 @ ‘ ‘ a \ relax %@@§®@®@@Q®@ “wteeeeeeeeeeee @ Mwfiw rw®© my. 4/ 5. ® h D d ear ee [email protected]@@ J. i If ®@@ x ZN .M/H AH ff @ ©Q @ @. ./ m e \. rd ,@ ®@® \:/ miwfi .\ mm \. ‘ "Kxa ., , \Wn. we e eeee [email protected]@ ‘ ® 1. u A r a ® @5 ® a; e? @®@? ® \hv cw. W}, @ @@ m my ck \‘J 1: ea, , R .fi. PAACKENEH? e @GA ®®®@e @ (First Name) J®m eeeea b®®®$ \J @@ we AM a. \IJ e m N t S a u. 41 2016-02-06 01 O. A06/01L/02 . 1. (10points) LetUCR”beopen,EEU,f:U—>]Rm,ands,t€R. ,4 I ‘4 fr P125? " wffi'rf/l, if (a) Show the Gateam derivative (1 f is homogeneous in the direction. That is, show (1 f(a;tv) = id f (a; v). 93 (W) 311$th (i_f__§>fl(v>fljp(Z+/£M7) 4“ 9'0 i r - if, 7 , iafaflilwfi't’éfit 69§( . cf .. X if“; 2X 0 : Mfléi/{IW givéx 90 ,. _ S. 7. é" . {2? £7 Jig??? 4‘1»? I. ‘._' 3;" W = '1 ,5/ r _. .. i a : 0&4} .t‘ 2'9". i 3”, (an 7}“)wa “‘5 fri'fli é—‘ao , / (b) Show that the Gateaux derivative (if is linear in the direction when f is Fréchet- differentiable. That is, Show: ' éi" 1:" V‘s: V. Drew e} >;)(_§fi)+Tt/;4 a Df(fi.')=>df(&’;sfi+tfi)=sd a. f(£i;fi’) ’3 We He.» Wee =éfififi’)+flffw¢; I a X“ 2 i r W [W + 1:6?) ' fébfrm yew/5: 5 42?: + i 2? O. A06/OlL/OL Q '(10 points) Let f be the following function. 3 x —. (9)?” 0 otherwise (:1) Compute dflfi; '3), Where {7' = # (b) Show dflfi; 17) is not linear in 17. / (c) What does this say about D Ha)? ' w 5‘ P 5“ “we GA m ’ A [NHGQC/ V: .‘ 3 ' ’X: _ ' 1”?“ v<3<rPJFEPfiILCj > ‘30: n ‘ a '70 13-6 LiwéAfl/Tltgzgfiqgg . {If}: '15: \P ' I: 1"} :5 1— | cl f {\U; (all W/ / FA ILéTa 13—6 FIZECHEF up? 1’:er LDJFE—Klé? 2V1? L’Né’l’glj xgv NUNLIMCM an, IN V J {a w v ally/0:? \%' / *_¢&_7 _ _ l9; H 3 (a W X Q 7'} 651g}? 170:2 (dab/7&0 / J 1/ '02 . 5 3 J” J‘\ g 4; 42f (Gilly? ;) JEzo I? ‘flflitlf/‘p T \ l2) V5) Vvfi vjé my” VI: L w}: (I 0) fl J i / 293i) f ‘33: ” O 0149(0wiwg} all UN ‘-I£F{Ofié} x“ $%??i é ~'0+g _1_ 5.4:? 1,. i 74 i / . V 2a. r- 3 Hwe’é “W arr-mi DIFFMMW A?“ 0 |§ Ccf‘é‘lh“e’ J, Grader 01:11},r J, E 7i NOT . O® . \31g‘ 0. A06/01L/0‘4 Q 3. (10 points) (a) Let z Egg, Egg/let 2* represent conjugation. Show: @332 = (z+ z*)/2 We 9 a. 7 33h fi‘?’$£i}ifi%§20 Mai Io “' 11 z is purely real exactly when 2 = 2*. 2 'iv 2 is purely imaginaxy exactly when 2: = —z if 0” - r a— (2/? J_— In _ I” f A Mai-Eb 1”“ 1.5; riamjfg/mgfij) . hf _ 7 ' 7 ' \— '7. a :7 a r *1?“ ~ 4 x - ’ (A . Pr < ’ [wi- -__,{_;f;-f, @ 2:: r (13) Let 1%1 (\ienote a. real, square matrix. Show: ‘ "" h 7—7“ 3 9:7" S=(M+MT)/2issymmetric. fr) 1\ ‘ 7, _ A_ ,. K = (M— MT)/2 is skew-symmetric‘f‘A : —/\l “"13? ‘5“ 3: M : M=S+K. “ ~r -' 313%: w: 2,“ [5f e‘ Ira 9+? :: f r .: — 2h ' __W q: if '0 :42 / 2 fl ' J, H “k ’ L ")j V A O {f y ~| $31 — ‘ I I "T- _ A #- l . 15:;- 7;. ,1 A? . «L36» 0 x PH fl L” r /‘ ' Grader Only i 0 0-69 o 00' A06/01L/05 Q (10 points) Let A denote a real, square matrix and let f = A :3. (a) Show [D f (53)] it = [14]}; using the coordinate definition of matrix multiplication and computing [J f (5)]. re : are :‘fl ’%tfi)r€57\flj/U¢?iréffigfi /waflrfi:;/%fip H {[lJ =7 :4 [Ln/.fi Ki xi” \ 1 rt: (LA {7/}. 'l' H {Aijm //i tm/59AQ% 2%“ (b) NOW compute, flag] it using the Fréchet definition of derivative directly, Without resorting to a coordinate argument Qéfi ' in, f3: T Li {wearer W TAM/W Amflkjfiflt%yx I ’0 V , -' A?” I i _ I; t ‘ _ J —Amteh:earhufinAa A L. m f’ ,e 9 9f * thw%flu%gfl — A; ' 3 t Grader Only J, 5 .. A06/01L/06 . r K . . ‘ Ae-LfAiA",nz‘-- A'/‘- -' ‘ he. _7_ , . (10 points) Let A denote a real, squar mafiix and let f = _'T AE. "U'T‘v/‘M’A a 6"— ‘I k" mm (a) Show [D f [(1’)] H =/2/&'T (W) 5 using the coordinate definition of matrix multiplication and computinL [J flail]- [ “WW " /\i’1('\ :Alrfiji/lp} «(:1 .. .A} : aw); A at zA-mm ~ «E h 1- , W W"- . a Airs __ tw- wa— 1,: r. / KR [lg—5i. .T T- «I I J JT_/’ 1*, :r a ,1 + 9674 :. 7: :_/\1+/i\. j s. “\F T ~— Cffiififfi ’XT'A’X"(_71A’7¢) : 1A 74 w r/"-/ ‘13—.— x it . _/ , ‘ _{" / ' R L x‘ .1" /,.‘/&t.) l‘nh tween-2 Apt/- ‘13 w: seine-2, A(”“/' J ant—'1‘“; # lid—ill [:9 (b) New compute [D f (63)] I; using the Fréchet definition of derivative directly, vvithout resorting to a coordinate . . __—T——__' argument. Hmt: f is scalar—valued and y = y When 31 is a scalar. 429*" l“ ) 2' in )T A "i f: ",1, M In Eri‘f/iecirea‘n was irir/lii%T-/éifi ‘ ’ w r; " 9.57%) ear/3x: (pm fit 3-1:: 5: i 233 A??? “:53 Al Ester (1 MM l” 4‘ l3?- 5? 2a {5? 2 1» "til : " /;fi’ ( 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 coordinatefree methods. J, Grader Only ‘If “A o a.» .0 AOB/OlL/OT Q (10 points} Recall that f : R2 —> R2 is said to [email protected] 35’ e R2 8.1:. f — E = :1: a fie) — E is linear. Let f : R2 »—> R2 be affine. , 2:. (a) Show that 5 in the above definition is H5) (the answer to this one is very short). 29(7): Lee W i (b) Show that flail + '3) = flé‘) + [D f(c"i)] ii for all (35,5 E R2. Hint: Show '3 1—} fléi + 13') — f(€i) is linear in 17 and then use the Fréchet definition of derivative- fi(--x)—=*f\tx if 'PP- fi an!“ A'MD ire-4. F.”— H L ‘ gig; 13.x / L; @{Qi vie x ‘ j) :25: g‘ ~ J, Grader Only J, .A n,T\r’r-x‘. P‘.r’"‘.f’_\"”’“r"\ . @Q!%)@K§Ak§[email protected]®oijxé . O. A06/01L/08 0 7. (10 points) Let f, ii and '5' be given below, and compute f(fi+tfi) - f0?) ~tlDf(fi)l 1'5 for t z 1,1/10, 1/100,1/1000. Does the difference scale like t’“ for some 1:? [2: {i7 1 2- ft”) 2 ($27“) , ’3: [ll tl firifii; l J +iél—TJ 1:5! H221)? 2.11:? *— 39+;th le '1 ' Isl/iii g it ’l”‘?i"7"' of at Di :7 fl 7’ I, K w 1L lt2g+3k+l f l #jq’H I; 2 l 1 r 0 i I L A L ’1 ,. will rt ,1”! r- _,_,l I. 32% i} ~ 3}; 5 exist M l “— l—J '3 u- V/lazéfifip goggle .4 i I 7425-4- Efir h ‘ £0” [0" JQ ,4 It _:_ :— / :70 ‘/ l‘ ‘f ’r' ‘1 'a ‘ / j - \JNl-l-LFJ L— k if; !09/” téaagiif’)‘ i162”— Jfll I }:-‘iI—«3 :I.AC;2_FJ—_i l/l-yfi :{0053‘3;:: l5- r; r l: g-i J / 5")?" /! figi/t“= lO “3/ J/;_-’.;« This question is asking you to quantify how good the linear approximation to f is at 13‘ in the direction ii: 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. ' .. AOG/OlL/OQ . I (10 points) “23% :7”:- I t m a (a) Given f = $2 + y2 +23:2 and = (t2) , What is |:D(§o f) (b) ? z t3 c , "-1. ’IA 23%?" Wig-Ilia ‘ m (b) Given 55 = (33:2) and g = a2+52, What is [mgofl A / ‘ \1 h .I-M’ ~- 1' . a n‘ ‘ -‘_‘ * J1 , ‘ , 3 2 1 I, n" 2: g.‘ i" . 1‘ ‘ a \’ I I! D- . | .. A06/01L/10 . 9. (10 points) Suppose that f : R2 —) R2 and it is known that [Dig = [1 _2], 13 : R —> R2, (3:113 —) R2, 1 . 0 l 3 # ~ 4 .,.-_’“£~i - , W) = (ropxt). ? A L t f. .4 p 2_ ‘"' '7'- ’ j‘ 1‘7? ' (a) Suppose it is known that = (t3 t jig Find if ’ x 7 I ‘1‘ E / 'WI-[L > a: pa? 2(0):.DH N) l/ I I) '1 L-r- ‘ . . ’ L1” . fiapfipfigfim U i /; f I (b) Now suppose that 13(t) is unknown, but that it IS that = (I) and 3T0) = Find 13 ' fix) w? (fi'+fi7) $601— MD I h V“ I /—4.\ l .3 :07 57$ (55W; <9 ‘ ”’ A“? J|f£=::- Slrj } f _ _I m ,4. -k—W If :3: w” H I‘; I J\\: ' E E» i _, 1 i, t: 7| :1 w, /—?' / :4: J :5: S l, Grader Only .1. 0 06:3 Q ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern