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w07finalsol - Mern L191"" Winter-3007 Final Exam...

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Unformatted text preview: Mern L191 "" Winter-3007» Final Exam Sea/Wows 2 1. (9 points) Let R be the region bounded by the curves y = f (33) and y = g(:1:) shown in the graph below. 4_ (b) Set up a definite integral that will give the volume of the solid generated when the region R is revolved about the line y = 4. Slice Papa‘s axis (59 rel/oiufiév‘ é variaE/e is x; At a (:Oémliwa‘le X) ‘flze CroSSISWnilénj“ are washers) W‘H! if”??? Minis [+1520 am? outer Md#&_3‘ ‘1’ 3(1); ‘2. 4"“): Val: K Amman : ‘3 (c) If the base of a solid V is the region R, and if all cross—sections of V that are perpendicular to the 23~axis are squares, set up a definite integral that will give the volume of V. film; are Fer? +0 wram‘s é) Var/nth is x)- A‘f a mom’mé’le )('J Crais‘fiseé’hémf 0” 57"““3 5f“ #9,ij ‘Qfi’j (X) )- 2. (10 points) According to Newton’s Law of Gravity, an object near the earth at a distance of 7" meters from the center of the earth feels a gravitational force of F = k/r2 Newtons, where k is a positive constant that depends on the mass of the object. (a) In a revision of an earlier doomsday prediction, scientists are now forecasting that in the year 2029, asteroid 2004-MN4 will have moved into an orbiting position about the earth, at a distance of 5 x 107 meters from the center of the earth. In terms of the above constant k, compute the work against gravity which is required to push the asteroid from this position to a more desirable distance of 6 X 107 meters. Work In 87(94qu (2 Variable; ”parse. F/r‘) atoms A d’irtanec 1 W- gmg/PMF @407 k k. r: leffl I giant A ,. ... l fins) W: g r‘dr ‘ 7:] “I: ” §“‘:) 5‘40? “Sada? 6M0? 5W0? I) (b) A Hollywood producer suggests that a further effort be made, to send the asteroid from the position of 6 X 107 meters, completely away from the earth. By using a suitable improper integral, compute (again in terms of k) the amount of work against the pull of gravity which is required to send the asteroid “off to infinity.” N rs N 00 Work : g édre “M g éjr =- iim —%] New New : i 7 GXIO7 67W”? r 8 '0 = (IM (.5. 1» it?) = 6 .7 NeMoM—'fwd95. 3. (12 points) Compute the following, showing all work. 1 1 a —-——dx ” ( ) /— 1 332/3 «£55 is an imfvofer ixf'ejm‘ J bamum e'wbjranol is IiSCoVFfmuou‘g af K20. 1 '01, b I 4103 g ”1706?: K1017“:- gr-Lefx = 1"" gJ—vg‘ _, hm "(“ch X 3 x" X26 MO" >53 ciao”f xz’3 ""1 -l O —. Q v b t' V I __l,'m 3X ] + m 3X 3] -1396- tX-éo‘f a V) .— a g _— 3 ¢ 3-8 {81 9b; <3 39 (2-90“ (3- 3a) ‘ fi 95 b / dm . _ l ( ) 0 W 63“ Sim/>64} expmssm hwy/e Mimi 1,), W75? “2)? “fix-Z @ 3P XSOJ‘Mgn 91:6 03$;me f , . ’ => ' X’E 111391 14:} (2: z ”’ X if“, 2 «(I :: , 0 I?” x k ‘ {f 3 05¢ Q “7‘51: O 1"? “Z We [£70wa :1 +11] filos’fr’h’fiow 0'? ‘Qrm g“: ~35m€ d“; .3m36) So 4AM? 671/2312; J. Wk): «((2) 6.0369 ‘5'" 3? A W 49 S i g “m 7 o 6’ «m “’5’ 1‘? a=3sm9é . ‘1 “w 65 3357?“). 9 l. (‘31)? (‘5) I -4 2 . z 49: {(sm ('3‘ ~O)= .. C} 4. (15 points) Match the direction fields below with their difierential equations. (The horizontal variable is t; the vertical is 3;.) Also indicate which two equations do not have matches. / \\\\\\\ \\\\\\\ v V V I 7 I I I I) I 7 I VI, or “none” m \ .u v w v m \ \ “ l//////// _ \\\\\ \ ///////J// _ \\\\ \ lllll /////// _ \\ V 7 \ IIIIII /////// _ \ 3e \ _ lllllll IJA.//////_ W m V Iaan m, m 4 ///////{/////// JIM ”flflfluflfl/flfluflflflfl /////////////// 71V ////_,/ /,_//// /////////////// ////_,/ /,_//// ##/#/// fl////// ////_,///,_//// ,,h,#flh ,hflhhhh 2:21?” m y/Xk1«/1kll?X/V V///?//////7//y a ////_,///,_2/// /////// /////// u ////_,/ /,_//// ,,,,,,, ,,,,,,, m ////_,/H/,_//// _______4_____: ////_,/_/,_//// /////// /////// ////_,/H/,_//// MHMNNMH HHHHHHH ////_,///,_//// ///////4/////// ////_,/%_,//_//// . V 5. (21 points) Each picture below depicts a few possible solution curves to a differential equation chosen from the list at the bottom of the page. (As usual, the t—axis is horizontal, and the y—axis is vertical.) Match each equation to its sketch of solutions; one equation will not have a match. 1, 11, 1117 IV, V, :anop‘fiflm 35+!) safamie vnrs misfits Bmef reason{3€fii ' efm above; hete am w/ffafive Mamas: Only MAE has stapes menvfioem‘o 8e imlefenden‘f' a? 4% Valve 610 = 03/; ’Hfis $24" is infif‘enJa'fjfllj Equation VI, or “none” (at) y’=(t—3)(t—2) If (19 y’ 2 -(y — 3)(y — 2) (a) j}. Simpgfij slap-e i5; fiethive—{Br y<2 0r y>3)flgjd‘l€g ' 31: Mae 5? “6,: none moi/Ia film aims. are mmm‘eni Wit/#3“ :f i . graph 1’) slope is 04% +255 magma; ‘ y; and filopalsneeri’ O 1%» ynmrl‘; Nqulegft. Uni/wk) Am 3; Elm" erIid‘s £03311“ in: slope for fail Z y>31 crthr fag/v6; Memes? MI: fife/3935; £31317, M’tfimgk army has an addi'ffm. \ 1m 6H” 0 seen in audios eE/7mi’mtian.’ E1314 Had/(ts nagnrf‘ive gape ’Fm {>51 {pa} similar {<21 5/3 5 “thee-#3 ”finial“ mph: 1X11?) but +5056; graphs aim have Sicmo ‘chr‘tEa IV. «diam? '.77L~f5 isn‘f‘f alofm'fioné? . j.” fimPLfJ 31(3),?! [30%3362‘1) f‘fflfih’lkf 59y) FILL; Slap? is {>03 tlr‘ba and y>3. Only exp? (4) lug: £111, lamp: ffiamph'flfj slope is O flap-hill) lYfieM/BSS at“); Fla; slope is m: 4%? f>a arriy>3. A/Mozgk @1163) also has #953 pififiW‘es) n‘ 4’5 rams ‘finff slim! 25—9 new :3 find #34275? no?" mi arepkfl {he new! slope “my Uflefi‘neJ near y=3)o 6. (11 points) A new 15-gallon juice dispenser in Branner Hall’s dining room is initially filled with a fruit juice that is 80% orange juice and 20% pineapple juice. Every hour, 10 gallons of juice are consumed. The dispenser is also continuously replenished at this same rate, but due to a supply shortage at Branner, the refilling is being made using an orange—pineapple mixture from neighboring Wilbur Hall that is 40% orange and 60% pineapple. Assume that the dispensed juice is always well—mixed. (a) Write down a differential equation for P(t), the amount of pineapple juice in Branner’s dispenser after t hours. Be sure to state your initial condition, including the units ; involved. pg} 2 ”Mia: apjnzflows op Pineafile juice 0:?th ‘f hauls" 09"; C amwfl,‘ m”? 6.? Pm“fo leGQ in +ank) :14 Sal/hr . SMEQ 3%: m+e (£0092ij (net) :3 {‘04:} M “ rate 00% have its (00)"? 971‘)- 61%)(‘035- 3%)) 5" Ell—:3: 9‘2— 1 in off UM 41+ ) t . E Pufpcv‘biw Cpf‘opoqgtbfT WP”. P63): ‘fi'fs :: 330‘ ‘ w (b) By solving the differential equation, find the amount of pineapple juice in the container after 2 hours. , JP 3MP ’7' @vnrl'g Var'mUeS 3 ""“ 5 [H => 030% - " 3‘2” rug? :3- as P ’ deal. @415 when Ida’s , _ c 146. g- » ' 3 y R++C 3qu girls?!) 50 rm; ,§_.,_% ézfig C/ ,..'- 4c 9 IQ.» Pl: 3 3 $l3“. - [3 I! (a (3 5-) q—P :: i éC/Se/ “(2/3).é 3A e‘H/s (A : $97673) 3) Pa): q~flleflfi€ Since P(0): 3) have 7. (12 points) Solve the following initial value problems. You may use any method or result you like as long as it is fully justified or cited. (a) Z:=22 —22——15, 2(0):— Mef [and l -’ Separwl’iov opvnrleues‘ d2 ‘- 0’; 2 J4 '6 a S J; 21—2245 ' (e-SXM) ) 5° +C ‘ (£-5){z+3) ) and W USC parka! ’Pmdrbm Jewmpasi‘fl’on ; F...“ iii: ,. (E-fiyas)‘. €_5+:2+3 > l A631)+B(E5) {A+3)£1(3A—53)/ So Ar-BJ m! 1:34.55—4—33) 50 g=_§L_—;.A. 17405 F A c ‘V3 p t.‘ 16%” gages) 31172-5 “—0!“ + 822 i ‘ §j“lz‘51'%tjhl%*3l"z%l:§ Solve 4;)!“ 55 in “forms 09 ‘1‘) so 1943‘ = %+C) 7-5 13:45,, ¢€[email protected]*C)-D % ‘ 9. ”5+3 _ , J 12-5....98’é __ 81‘ 314 :21—3‘ 36 4? 1? 2"“3i’9,” e ”4440:? R5 M-SULS‘fka‘ow X: [Ojidfit 97M: Lei M 26 50%;? 5%.»,- 2% ml §f§=g§¢(e—5j[e+3)‘ ’flae. equaliow ificangj can be mwr’rflem i5}: -314 I«—-) , W/liCl/o {5 mmgnizaléle as 0x, basil/c flit/«flan will; k=~z mo? K=2 We can cHSg ‘flw %m:2(a pwvsaleol a; file all um J so +fo {me = m?) = 8’ lfAiz‘é D {H} . 8{ Q ' HAG: M - . ., -, 8 ‘3 NOW X-‘l Whm‘lz~C’2) so ’I~ Egg—3 =>A :3 171413 [2“): “391% -3] A wlu'irk $3 eyvivnle/rl' ’79: {kg above ‘ dy 2?; — my (b) d— = a: :1: - my your answer as an implicit curve in ac and y.) , y(3) = 2. (Do not attempt to solve for y as a function of 11:; leave Séfam‘l'mw o‘p vaw‘qLies wi/l work 5‘9 MAS—Cam "actor RHS :1er a» x-Farf Mal A -' midi." 7’0 %: 62x42 3 [(5% ) ’"7‘7 X(i~y) ) $0 MJQQOQ «tidy: §:2(~JK X . Mad SWNS?” ’;> “”9054 %*’)°‘x 8. (16 points) A certain population of animals is affected by seasonal variations. The rate at which the population grows is proportional to both the current population size P and to cosz(%t); i.e., it is proportional to their product. (Here If is the time measured in months.) Suppose the initial relative growth rate (i.e., 1 —P’ when t - 0) is2 — Oper month. (a) Write a differential equation which models the growth of this population. I Pa) 3 ‘HMMk a‘l‘ wlu‘QL popula‘h‘am 3”” 3 so I I , V E (f) 5 k P msé(%-E) ’fiar Sam k‘ (caWSi’an‘i' (SpiOfOfGr’l'icMa’IJy) . P', I Since $7 , 2—; when {=0 we ["th (“3,510 = i Se 2,!— J Thus) ’an o/Fggnm‘lml €?Ua+p'om is (b) Suppose the initial population is 400. Use Euler’s method with h = 3 to estimate the population after 9 months. 542? O C (’60) P0) :: (0) H00) ‘ We we”? %53 5+€f3$ "re IPAQEQ, +119) efim‘lQQ S’lep l 43’: {'th = Z 0M0? ) l\ 1% g+kaifimsaFg€J 5 HOO+ 3° 2:; [1‘00 C03 80 400+3~ao = #60. 523,031: 1931335: 6 We J P PM“ 2“ P 33(33): q€O+3glo°Lléo 3.22; = 4601*3-2-2 two 0 : 450 ‘b \\ [\x-D + #:4110316va 2 Lf€0 4° 3ng e 960-09325?) : #60 + 3 33.; = [Mo—teflwfl (c) Solve the differential equation (again using an initial population of 400) to find an exact expression for the population after t months. \ ‘ olP - WNf'Q ,‘5 J. Pam?— 4’14 fink is a: separaL/e, Pygmééw. 2 C 8 o a i' a j P: #5 weltww 2 A9} fie w (an) Wk”: ‘ HUF’JCD) we «swam qoogAeq%(b+§s3Me} “4562/4 ' 1‘ +3;- ' W Mcfl'vame )5 Phi) : wa 9010? ”5”}fl‘) 9. (13 points) Two species, A and B, live in a closed ecosystem where they are allowed to interact. Their populations as functions of time, A(t) and B (t) (in numbers of beings; here t is in months), are modeled by the equations engine dt _ 2 6000 E=4B_A_B dt 50 (a) Describe the nature of the relationship between the two species: is it one of competition, cooperation, or predator and prey, and how can you tell? (If the relationship is predator and prey, don’t forget to explain how to tell which species is which.) in “absence 0‘0 8 {Le-350) I ‘l‘W‘ 5714 = :4 50 pop A can't Saw/mg 7-— ) Cr ’fltc the o ' ¢ ’ fl (3 S “K F A {LLA O), ‘flun jogislflg) Sn For B grows ex‘qum 224/ér_ 613. l’ , , ’lez 4 6000 ’l’erm m $01? rho/lattes ”firm"— +1.9, armfii rah can A is POSHM‘é/ M+£m+€§ 107 in‘l'frndfloms wrf'lx 5985195 8 1‘ Meanwhile] {[1 “9,150 4am M g ) r . JO {malicvrlias “Hm—l BS jmwth M41 is ”again/J7 a‘hégi‘gyi 57 mamasm/ lWA‘MCF/oms‘ WW“ A ’ 41:95, +er is a NMimshlp 45" prw/aizorwo/ prey) lefll E Mfief’fl)’ “”1 Amrmel’ (b) Explain in words the meaning of the concept “equilibrium solution” for this system, and then compute all these equilibrium solutions. H/ auélférium 50/”le mean; at Pair (AJB) 0‘? +wa moo/Miran $1.265 450+ age, ’ilk? equating; 501’ do Wt" chat/#3:: in size over 47”“) flat ,5) 51233“ m0 Jrim. "two popoim‘bh; ’fot cam sustain ‘fémsaiwg Ly [94%de 'flze mtg—m! gmwfi or (fem? a? ‘flvzir Sizes ageing!“ 1‘48 effing a"? pmcya'lmn/Fveyiuj, «£004 ail/Ma We factor ‘[email protected] QXPYESSibm ’91P Wfl‘ mi‘r 0M , A ( , a? 2 3” a J,“ 630 B‘éooo) , AB A 5 , CL? A {S constam’f/ ’l’hem £520) 5,3 420 0,. 8530005 :1?- §;(400~A) . 80+ ‘4‘ 24:0 and 8I'5Qeng’l'qrd/ 41‘1“ O: 35:3:(300‘O) 53 8:0 ‘ / =5 O a ' . I - ’18 '7 f? B 900 ha! BIS (mafia/5+) “”10: O~ F:%”[email protected]_A)J $5 ASZOOi 47105 m eeuia/Iiriq owe (14,8) :[OJ 0)! am? [(74, 8) : (200.: 3000) i Quick reference: E 2 ‘3 + 6000 'V 6000 (8'36”) dB AB _: __ , 3 dt 43 50 ” 53(300’A) (c) Suppose that at time t = 0 months, we have A(0) 2 210 and 3(0) : 4000 beings. Use the diflerential equations to predict the two populations in one month’s time (i.e., at t = 1); be as mathematically precise as possible, and Show all reasoning. _ ’, ' I AA» A f) a 210 9 0 A4 {50) 4/26 exit/Mien; Awful? ‘f/ra‘i a? - 6m (Promo) ~ F ' WO:%~=35J J8 - s an 2? ' gafm-AF 3934-00) = “800 if. species A is 3%? ”l A ”lie 0‘? 35 ie’lj‘flm‘fl‘ Mae 8 is s/émklrrnj of? 800 Lell’jy J "M [m ‘ We can filima‘l'e {17“ l'II’fleoy-IQFPF‘OXH’WFPBAJorby(m E/gh—ffle Wilma!) 4245* in 0M 3 9 e FPO 0‘ law W, be {0 (3701 “'4 Size 0% POPUIGL 50M 8 VJ“ $1 limo g " i 325 G 56 "i5; ' I m ’ - (0“? WW? OI’CUV‘DL’AE 31,455“ MY, Peggié/fi) Q? by ww‘lfle (JUESSQS ,‘M Smaller infirm/5 04 15W} ”9 29/ Qs’hilmrl‘m {403V 0M [Mal 4‘8 meelves‘ (flap, 7 00+ "”16 fishy? mam/w i5- 5 #2,?”ngin 3 a“? a“? 3f 1 (d) Will the first month’s trend (that you identified in (0)) continue indefinitely? Explain fully how you are able to tell. ilk; Wewi” 061‘ mm 0mm seemj 8 drop me A rise. 4'3 see #00) Mia 0%.“? «is $59!»! as B pig/0’33 below 3600 m 5,926 {Mz’cé M73415 occur as Gary as fine imam" mom—{A} anagram? +5 102% (cjlf esfimfle) / a a}? a? “71%; will fire news) ,+ Jaws on #12 Mar 8-3000, flfi; will muse species/ils PaPuioxi’vr‘MA 40. start drappinj (31 Sam“ (OHM; EQLQViorj) $00.10. £5 ’fi‘f, BWJL/q! mflfiVblA/H’l of" Pnpvllcimh B é: 50,,“ 5;; A b“: 430530 40 Mew 206)) and 9PM pe’llt’m‘fla/ gzréosil'c‘ [-‘fi‘HéMs‘ in 1‘42 fO/Jvle‘i’im’, cm)“ 9W“ Ea PNCIIVC‘FGOQ wffll ’E'r‘fhfl‘ iflWSfI3a+IOHJ 10. (12 points) Determine whether each of the series below converges or diverges. Indicate clearly which tests you use and how you apply them. (a) Té 12:16 1 n1 hm 911'- : h“ (M I, - fin Raffle +€S+ 91—9 0‘“ may“, Erna)" n! 2 - hm (moi e" {M M! _ (529) ) ”60g ”4 eDZ-IQIH-i ”4m 82"?! 0Q N“ 50 ”Hue series (Emerges r (b) :00: (30 — 7—35) We “m (304» 2 30-02 30¢0) ram—3 m A?“ 59 5/ ‘MA 12.31!“ ’93!“ Diver mace/J 1%! $01K div/938,3 . 11. (6 points) Suppose that the power series i an(9: + 2)” n=0 converges if a: = —7 and diverges if x = 7. Decide which of the following series must converge, must diverge, or may either converge or diverge (inconclusive). Circle your answer. You do not need to justify your answers. (a) If a: : —8, the power series Converges Diverges (b) If :E = 1, the power series Diverges Inconclusive (c) If x = 3, the power series Converges Diverges (d) If a: = —11, the power series . Converges Diverges @ (e) If at = 5, the power series Converges Diverges @ (f) If x = —5, the power series @ Diverges Inconclusive file CCVerfE’V‘ flit-“‘2 ‘ LE,“ R 33¢ +539, (UM/(flown) ffiihilbs 0‘? COMWVje‘y’Ce. we knew R? E’V—(CZEE : 5. Shade. x257 Ieadls 42; convergence J 55mm x=7 leads’fo divergence) we knew E5 [74 (—1)! :: Cf Of) 31mph +145 §le+75m 0% Wm X’QXiS‘. Afié—Ffl—‘a <z—4——_—4»—————4~—>,> »7' *2 7 x QW- Ced. Div. ”“103, any X WWI/A iX’(”Z)‘ >(l 3m aim? 8140:)” any X WHlx leG‘Z) i <' 5 3/915 camwwaence; but since we Emow Axe/“4&3 MGM ”$001! R) if [)0- (—2)! lies Earl/88m 5X 5] incl-(min) We 931i 1‘" cane/ode any/“A03. 12. (13 points) (a) Find, showing all your steps, the degree—4 Taylor polynomial T4(x) with center 0 for the function f($) = sina: + cos 1:. 3’) ‘Pw ") V‘”[o) C“ W i O SMX/rcasx . l a!“ l l msxfisimc l ,L: l I’. Z ”Sim—Cost “I 7-!- s ’1— ’ 2; . .. w if i “C 4‘5 ‘5’ "'f 3 95x mx I 3! g . i d / Sir +Cos>< l "J ,- L! ‘i ("I 2L, ‘1 ‘ i" as m) = £ch n=0 (b) Use T4 to obtain an estimate for sin(1—10) + cos(%). (You do not need to simplify your answer.) We use/'46,” 40 931‘qu ‘FCflrméflmi: $405) was 1.6%) = (0) Compute the 5th derivative f (5) (3:) of f (x) and explain Why if‘5)|32 fowl” Em Pant (“l ) “(1)60: 50.4% +603 X (2413‘) I) ) ) “Ml 3° l“; w) = Com-56M , ill/t {gale ‘fkd‘l H‘mfiz)’ : lcoxx ’5‘th I S ‘m3x)+/«~Sim( I 2 [cosx ) + [3w ‘ ) owl since (cosxf (m1 {5M4 an: bfil‘ 04M)“ 9+ "M“ l we have J we), f Icosxl+lflldls H’l ‘9 Z (d) Use the fact from part (0) (even if you were unable to verify it) to draw a conclusion (in sentence form) about the accuracy of your estimate from part (b); be as mathematically precise as you can, and cite all of your reasoning. B)! fiybrs inciua/rlx/J «gr any X I” 7"“ 1”?”le [”20 M] we 1’9”? ‘fl’o‘f mm: imam); 5 3M? M5 * )6 Jo ) “gem M 1550911443 [fan/5M on .1 l] B (C) 5'5 a) (l)- 'x 7 ) rice fr: «Pd H‘sléz ’l’orr’fl/ X, mm? M?“ M50?) 55 {flail gerrorl 2 l‘DM‘TMlé 3K ”(’45 42! X 6” El (1. 5f 1"“ in par‘l'i'wlar) ’e' KC’I‘ ,- WB Lillie “”le [0 1M : WM; é)! S“ 3w"? 5! i0“ 1 “ll”: means; ‘Hm’l the €9ltm:z,‘l{ USIMJ 114/) oli‘mrs gm “the ’frve Valve 0'? $?n(/fé")+CO-€(;L) 2 . 1% 105.5! UNS’ 034““??? 5° {0‘ U lay no mm 41?!!!” 'flus l3 flcellen‘f,) ...
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