thq_05-key - Exam 165 MATH 2451 thq_05 T-C Use Course...

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Unformatted text preview: Exam : 165 MATH 2451 thq_05 T-C- Use Course : Multivariable Calculus Due Date : 2099-02-04 (Wed) Instructor : McCary KEZ (First Name) (Last Name) 2016-02-12 16:21 (10 points} Find all values of a and b which make the function differentiable at. all points. You must. use left. and right. 1. limits, the limit. definition of continuity at. a point, and the limit. definition of derivative at. a point. a332+bx+4 3:32 J60”): (133+?) I>2 1H5; Fuumonl :5 9L; (lair: Even/WHERE EXCEPT foééwb/ 96:3, 5/0 PIECES A212 POLY§. F012 SE Ar 99:52,) NEED: (m)= (M an fiéx‘tmey’) = EQEGME) = flaw we 7la+ll+¢ = «CZ/Mb Qa+b+tf : o Foe {Iii AT ¢6=.2) a! 1%) = xx»:- Jfg”; (QMHLJ) : {66:54 (0‘) fa+b = a 3a+b = O Cox/191mm» mesa mthrronc, 2. (10 points} {a} Compute using the limit definition of partial derivative. (b) Compute using the limit definition of partial derivative. {c} Show f is not continuous at 3: _ I my (.3 0 otherwise _ [430 ‘77— ., mm (s) 4:) ‘ “Lg? Lb ll 0 @ jzmx PAW F0 (g) .3 LIMI’r 02m” 0N P/‘r’rH ‘ LIMIT ,. DN-E NM g2! The purpose of this exercise is to illustrate )ne of the weaknesses of partial derivatives: even if all partials exist at a point, the function may still be discontinuous! This is in contrast to scalar calculus where the existence of 9’01} implies continuity at a. See the comment about symbolic calculators at the end of the next exercise, which also holds for this exercise. 3. (10 points} Find 82f and 82f using the limit definition of partial derivative. a a?! a 593N933 (93:69}; 2 .2 . _. '3 PARTI/HJ I _ lava-$2341?) 7&0 > Foe THE 9/ MIXED I _ n I t: 93(0 =u<3—i((o)+m)-g§(%) . O .- erw l‘ne 9x 0) hr)" kl __.— AT (9‘); 5’ -1» saw-o j 0K) - l’WO k '0 \~ 0 u Jam ¥((°ll(ad ’ " > - < -kgi_o 143:0 l!» - M90 h J» L 3’0) M L. = w v» (7) -— M r) 4 L1 . o Noflm J» awn-aw) 9 M M 99693 ‘7 Pm k i M l’ ,u sumo 952(6) = o W k ) ' k“ In. W 9:? ‘ 9L +4941 9'- ’9'” W 2 (a 2i : (ml-4x1 9-1 K 9” (maul : 1 40 (um: 9. m 9: 6o 0—19 _ —”;§—#) 9 09 axill- WM gjgxm 7; ago) 0 orMERwICIE ) WH)’ DOES ms WT WNW/(Um cum/1W3 macaw? (with; 9._ or i _ i-fizér) a 7M ' (K m j 0 otMEtheg The purpose of this exercise is to show that mixed partials in different orders are not always equal. Don’t get lost in the limit formalism try plotting f and its first and second partials on W'olfram Alpha (or, even better, download SAGE Math and start using it, MATH majors}. Can you see why the mixed partials are not equal? Feel free to use a symbolic calculator to compute the partials of f away from 6, but remember that you must use the limit definition to compute the partials at [-5 because [-5 is not in the domain of that rational expression. 4. (10 points} Compute the Jacobian. You must. Show your work and not. use a symbolic calculator. (a) I cos(3:} f = 332?} + :92 ‘ 5111032 — -M (04) 0 [if = M5 06””‘5 (MW-9) m m4 (“’45) (b) I /I2 + 92 9(a) = ' (5111(I39D2 % ’21—- .24— [TW] = W 'x. 3 w (W)? (MW) 9 WW “W” °‘ 5. (10 points} Let = Show, using limits, that for any number m, lim (f([}+ h} — m} —m h) = n (1) 31—30 but lim 100a} + h} — fan — m a) = f} (93 31—30 is never true: there is no m such that m h is a good approximation to the increment Af at f}. Fm m NR (90 J/W‘;<Iul~ioI-W‘ ‘>0 AM“ MDT? A>o - M(iflm\=1—wx "” 1)” Imo“ V”- Am: Jl-m =4-m —> 1 =1 IS Fifi/‘35 \I m a" ‘LIMTT (9.} D.N-E. WV» Tug; exeamg 6mm; WW THERE «:5 No LINEAR FuNLTIo'J or 1m; maewsm wurcn MppoxtMMES lad AT 0) Em: a mow." tr Is US NEW. m,uo WI (75 MW)\i WI.) N0 DOB; LIMIT (:0 weak? “0) 3/0 THEQE 139 ND WI- who” 246.22% V/ LE MEIR 0 M» 1/55 ruwe ME 0mm MOT£OM5 as Danni/“IVE WHERE N“) Is DIN—,7 4V”: ,4; A WEN Dearuj-TIUE, BUT [46) :5 N07— Vwa‘W FR’ELHET DIFF- Does LIMIT (4) weak? fig 7 Vin) THE/I MIQBEAT 0. The purpose of this exercise is to wean you off of the scalar calculus definition of derivative and to get you accustomed to the Fréchet definition. ...
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