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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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