review_01-key - Exam 165 MATH 2451 review_01 T-C Use Course...

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Unformatted text preview: Exam : 165 MATH 2451 review_01 T-C- Use Course : Multivariable Calculus Due Date : 2099-02-11 (Wed) Instructor : McCary E (First Name) (Last Name) 2016-02-15 02 : 58 1- (10 POiflt-S} You!“ have to prove Fermat’s theorem in the [R —> [R case in the oral exam. You‘ll have to prove Fermat‘s theorem in the R” —> R case on the written exam (which is a straightforward consequence of the R —> R case}. MUEWV L21 (1.932" 3e 0m!) 59 a) 794,913; CLAIM: [a T5 24 RELATIVE EXTIZEMA] =b~ [ Z Is ,4 crunch POINT] 3y IF 73 [Wm] rum 5 1:5 ,4 None/mm cPrvre/fu Form (,5 CLAIMED) OTueewtqs <3 [D£(z)]>) DEFIME 5 (’6) = 39(5-15 FL) For: Frxro, Bur Arfirrw RH) fi, 61;” W "W tHZ+£R rs DfiFF. Mus: (a 31 [Ram @ 3 = 79%“) EMF») rs DIFF 3/ cmn/ rem; @ 3’05) {Dflamnflh 3/ mm! mm. (a j "1‘5 RE. A? t:o 3/ “Mara/error]: 3 (0): /—'\ VgewED FfloMa 4, THE GFDE; E 2 I 0 £50 (9 AND! 3 gang”; WE wMDITIWs 0" THE $6M ngMfi 71/3322)“ .'. t = 0 1‘3 Jr 6-P. or: 5 g’(o) c o (tn; :5 6Mooru C-P.) Gomezme @ C? o 3'([email protected]=)[D¥(Z)]fi =1> 0 =[1>¥’<83]V» \(t jun we; V. “Ms fiuxrw) [WW] = [o] (w. Mali [3: 1- For: i=4,...,n.) 2. (10 points) The proof of Clairaut’s theorem will be a bonus question on the exam. 3: 3‘} (ll _ —E3Gfr'3:r ( .. 3. (10 points} Suppose G ( ) = f} and that y is an implicit function of 33:1 : Show £1 — where 633;; 7Q f}. I Hint: consider the function MI} : ), the composition (G o p}(3:} : f}, and the chain rule. ll Now: THAT 60%) o I; TME ZEW LEVEI/ ’79T or 6r: w C3 Re.) I; 6(a): acidiya-j rMEA/ 6, :0 :5 UNIT cIPcLE. This is the reason implicit diflereiitiation from scalar calculus works. 4. (10 points} In this exercise, we’ll arrange a situation where 3—2 — —1. ('33: ('33,! ('32 — 3: (a) Assume a surface 8 given as the zero level set of a function F: F 3; = f}. z (b) Assume further that each variable can be expressed in terms of the others by implicit functions: 3:;(3) WC”) “(3) (c) Now invoke the chain rule in each of the following: f(1,z} 3: 3: (Dr? Q :0 Q9 F “an :0 [email protected] P y :0 z z 9(I y} (d) Now algebraically solve for each factor in the product and establish the result. (fl Ma): 3 w) D(F°F4)(?t) = 00 [DF(E)][0P4(1)]= [0] Q5 2. 95 9: :[O] 4 o 0 4 4x3 3x; 9M 9F : F94: 9_F, _ [fifi*fi 3%«n‘oel'l0 0] Go 91 _ ‘9F 9% 92 ‘ QF/Qaz @ my on = . W m 9F 93 DlF‘Fall a; = -9F/95 11 OPS? ‘1‘ 9i W l9x 95 9mm) lan B _ 3 @ 773(9) ‘( - (I’ll emu" ‘ -1 9F 9P9 :9F [fidfific : 55* The purpose of this exercise is to serve as warning about the Leibniz notation they look like fractions, and sometimes they even cancel like fractions, but not always! 5. (10 points} Suppose f is a Cl function and z = — Show: Oz + Oz _ n 03: 0}} — ' (jam, 8mm; Nonfitow MW) = flan-em 3% = 3-5 (we—m = ¥’<m~<-+> = f’wm-{m—g) = o 6. (10 points} Suppose f is a Cl function and z Oz Oz 3+3; : f (-r-y )8 how: H’ amt/us I / + ‘ - + =%¢(£K%t—fl) = Hm =£(%3<—’~‘“:;_;§:‘ 9 — * + (-H )H ww = ¥’(%3§3<%) = 181%) ( N L 2, - -9«9 & a I 969: +5 3 - ((%qf\jja)¥l(%4j) 4’ : O m 7. (10 points) A question about. compositions of 2 X 2 linear n1aps: rotations, scales, shears, determinants, inverses (see THQs for examples). 8. (10 points} A question about fill-in-the-box for the differentiation rules. E.g.: (a) Given p(I-} : q(3:}r(3:} and the information below, estimate p(4.1}. q(4} = 8 N4} = —3 5 r’(4} = —9 Tu; LIME/(R gun/Wm; I: Pad») :12“) +P/(A)LL +1.03%) emu, DMP Tuts PM = warm + w) r’wc) AND PH) = 1(4) r01) =(g)(_3) : -17 P W) = (WWW) t 190W”) = (960M190?) = 45-712, = 4? so PWM) = 47+ («mm = aid-w = ~32»; SMALL)DW€ARD WW“) mm + [mm]: AND Filmer RULE: [BMW =([D¢(a)]K)r(a) + maflorwflg AT 3= (1) ) W= (-11) M) = 491(3) = W) - (3’) [pr (“M344 ‘(D “1 iiliii 3] * * < [13} [311]) = <°-°-1'a>[:1+z[3r1:1v] = [2:] we] {12:3 30 mag x [1: + [12:3] - [131:] 9. (10 points) A question about a pathological function. One of: (a) A function for which the limit does not exist. (b) A function for which the partials exist at a point, but it is not continuous there. (c) A function for which the Gateaux derivative exists, but not the Fréchet derivative. 10. (10 points) A question about 5,5 like the ungraded question thq_04_05 part 11. (10 points) A question about. computing Jacobians of differentiable functions. 12. (10 points) A question about linearly approximating function values. 13. (10 points) A question about. quadratic forms (see THQ questions). ...
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