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Math 20E Practice Midterm Solutions

Math 20E Practice Midterm Solutions - Practice Midterm...

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Unformatted text preview: Practice Midterm Examination Instructor J. Verstraete Time: 40 minutes No notes allowed All questions carry equal weight Question 1. (a) (b) Show that :5y 3 + yg). Use part and the 5-6 deﬁnition of limits to show my fa]. \$2+y2 liiri mwsmm Ki]; 0 I L; A 5’ . 44% :v Cle U 0 oiivx‘il .l'UIOll (I Cl M? < a. <——-~ 3.13.? ' {ll-~11 l {24’ ea Gr" allUMWW-l < U pul— fl“: 26 Question 2. Deﬁne What it means for a function f : R“ —> R to be differentiable at a. point. a. Then prove that the function f (:c, y) = |:13y|1/2 is not differentiable at (:13, y) : (0, U). Awaken 619m 3‘ ‘. at) C x 1 ("I-'1’ 0V" ('1 “3" - r: w , 12m arm w Jim. - E. 2am ax : «cm g 1 a cf f 5% ,c\‘\ To 502. “V1” W7! ’2 '5 wt Adfﬁﬂﬂheéfe of (0,0); E rd o0 W I r i - ' ' _ 3 P. . '. 1 j m a r {o m 34m 5.31:; ~ 0 ‘ f ' {mo n _ 51 50; M T M ﬂQH U3? “'3 6 EV) M . WM ()0 R f n lﬁJ‘fA—haﬂx (AL (0,01 “RM 2AM , if '11” (.5 (w r: 3:: \‘I X "i ‘1 gaff ,C ul‘wmx m“: 901/ VH7. ‘ '1, a r Q, (m x U r 1 V’M ? {I’M Fry? ‘Z t : ' X430 \Ex‘lfmlkr XF—3L \‘jgﬁmz Question 3. Let f : R2 -> R be deﬁned by f(u,v) = 1m and let 9 : R2 —> R2 be deﬁne by g(:L','y) = (y, 22:). If (Mm, y) : f(g(\$, 30), use the Chain rule to ﬁnd g—j. WW; WWW _ ﬂ WWW a ww lea am- ‘ «\7 L") Y 1"“ \ W i] Um .6 L sh ht] Ki?” 13.3. + as; , i r 3mg. I Q l ()1- X ‘3‘ ) I; O i E: VQP‘W‘ 5W” :2 01¢ a \ u ’ \ 3 PM: 232% Practice Midterm Examination Instructor J. Verstraete Time: 40 minutes N 0 notes allowed All questions carry equal weight Question 1. State preciser the 6—6 definition of lirnrna ﬂm) : L for a function f : R" —> R. Then prove using the 6—6 deﬁnition of limits that lim sin 3:2 +12 2 0. (WWW) ( J ) If . . :{J AJM I {V\ I: L— J’a Dix ‘ I; ll la ( b l3 0 r ’ it 6370 Elm-‘1 13 a _> o sud, ‘ M, LU? cl(?<'£a\(_(§m 53"” {m l \(2 ( ‘11“ ‘ L j, HAUL 04 dig: LW - u “Whit/Hm ‘3’“ aid C?!) 1». E56 55% a __ 2 "M . t gmvié K : {Evil} <L 6%“ ‘j\ 1° §<C S‘Lvi ism“ l Ll lw gaze , r“ “"7 r“ <’< xh L 5 x“: \ ‘1 Question 2. Find the direction of steepest increase of the function f (m, y) : (:1: +y)e\$y from the origin. What is the equation of the tangent hyperplane to the surface 3 = f (:13, y) at the origin? ﬁg“? Maw 'H-Ym‘ 6k" {0,0,} I 22+ 25M :79 W m f2)><)”2,7 -=’ ' x l 2&1 + .4" LI :6 ii We DbL-v‘ﬁex‘j) . iii \ﬂ‘ﬁ ) v (3‘ U 2 ‘4 I “he CLﬁ‘ﬂC/\WQ’\ CPL ;r\ @6091:- r'iitxb (ASDUA ME EMT—nix FWKVqu : -7 were,“ I; : Question 3. State precisely the Chain rule for determining the gradient of V( f o g)(a) where a E Rm and f : R” —_> R” and g : Rm 4 R” are functions. Then determine WU" o 9M1) ﬁwhen f : R2 a R3 is deﬁned by ﬂay) = (:1;,y,:.3y) and when 9 : R —> R2 is deﬁned by 9(z)=(z,1/Z)» Question 4. Find all second order partial derivatives for the function f : R3 —~+ R deﬁned by ﬂair-w) : (1+ 1M1 +y)(1+z)v ' . «‘16.; ‘ {I - .“x- " {Q 7 Mtc‘t it We K‘ﬁu‘” “ Q “C Question 1. Prove that does not exist. "L MK - Practice Midterm Examination Instructor J. Verstraetc Time: 40 minutes No notes allowed All questions carry equal weight sin :51 lim 2 ( J) (sawmwm) Li: + y? ("Hand M 901’ {7c m=fru 3's. . \. s m xi - 15L Warm x X '3 U Ls; 5:21:53 l ‘t mm S's MN; ﬂit/M1? dam mi“ Exist. Question 2. State the deﬁnition of differentiabihty of a. function f : R” —) R. Find the derivatives fmm, 0) and [HULUJ of the function ﬂay) : sci/Bf”. State Without proof whether this function is differentiable at (0, O). Pkg-13' Pad-“£7 e 542*?- ie (J1me hog—“ef- \ ~ s "w tie—0,0: J’x (0 g 0 \ {L’mwwL‘jM-w—w-«mm—J ‘ M g. o i..7co,oy Indo h ii— 4‘ \S cMHuwimﬂaﬂ a} (0‘0); Thom we Nail—A « [ugh/3 boa 3: “ilk-"3»: 3 C) {’kiW)“}(Ugo‘ \x i“? >L W} O Mix“ '\ e! F j M NEW: ._ m ‘3 GEM/tit.) on m .3, b .whwummwrl EL S1 + I”: K ' Question 3. Let f : R2 —> R2 be a differentiable function where : ('u,(.1:,y),v(\$,y)). Let @(Iyy) : Determine a formula for 23: in terms of derivatives of f with respect to u and e and derivatives of u and v with respect to a: and y. vséhtsﬂ (“tweak/3) ‘ thmﬂ), N ‘ :‘JV ﬁght ’23 Li D 1/4 g: .v M“ ‘ V79t>qv3 ‘7 I "9521 3:511“ 55% '31.} \ if} 73\$ 2' 3 m 3v 1 new \vg f {2t l M: K a“: fl EH“ mam W‘w‘ ‘ J :1 31., 3“ 3‘1 F (3% egg 3f”; \ 3" EU / [any Q‘vf "3gb Be 2m an, "Tm 2m 2“. 4 ac; W 3. i ‘—-—m _~ + .___ ,7 W- Hm ‘ - ﬁé‘ﬂ FBU‘ '23? 23V ) 3L4 “3‘1 '3.” \ Question 4. Let f : R3 a R be a function. How many different second order partial derivatives can f have? Now suppose f E 02(R3 How many different second order partial derivatives can f have? Find all second order partial derivatives of the function f : R3 —> IR deﬁned by i 35 Z fez-m = — + a + —~ 3; z m ; la}: ‘3: ‘ - ‘ r l/x C “C. If“ “ ‘n' t a" ’ in 3 l: a” W WWW give Q ‘1'" (“‘17: git-1 f ‘1‘] frat “on \ ﬂ g ’2 g — M“ 1 l ‘1 "NM yt‘i ‘1‘ l h ' :f m V 5r“ X3 7 \ “ d—k‘ . ‘1 .1 2x , r: F” in: I i: “ f‘m 7.3 3W £1 21 ...
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