06exam2sol

06exam2sol - MWLA 4H Aafumn 8606" Exam/Ia Safe/7‘;wa...

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Unformatted text preview: MWLA 4H -' Aafumn 8606 " Exam/Ia Safe/7‘;wa 1. (18 points)'Differentiate1 using the method of your choice. (a) f = ($3I+ 3:62 —.7 5)100 ‘CYXJ a ‘ (X3 +3X'1— (3x2-I- Qx) (b) 9(1):) 2 5293 '+ arcsin(2 — :6) 3m s 5mg; + ’ ' . (_, V l-{Q—xf‘ ) IE . \ Nah: a? you 42” 4 #142 N/e % Jememmy 5“) WSW" g Wm" ’93 o'i-mnefiia'h'on. 7: 52“ I ‘ a? flayfi QK'IME 1975-32; 32,06 g) 3‘55 52*, ($2 _ (C) M33) = W 5"- eagier 4% do - by laj dfimnhaflam 1%» 5)! praJvd/ywh'MF/c/w Mes A (m) = A OMB); m 2. (7 points) (a) Find the linearization of the function flat) = 1n(1 — 33:) at the point a = 0; that is, find the linear function L(a:) that best approximates for values of m near 0. ‘P(X)‘flh,(l’3><) r ‘P(O)=1n(f-o) = O I __ I = ——'—-— . .. = "3 I - ‘5 IF 6‘) Ian (3) a -. 'P {0): {Ti—.0 :: '- g L60 5‘ Law 'le)-{x—a) = 0-3600) = (“3x (1.!(x) Egg-+43%) {at o n'f x=o_ (mm is‘; +1ij {1—33.09 -3x a x near 0.) (b) Use the linearization to estimate 1n(0.97). Is your approximation an overestimate or underestimate of the actual value? Explain fully. WQGM 6F flh(0fl7) as 'P(0.0f)) since flow): In (: 4,-0.0!) = In( 1-0094" (aw). ’Uws’ +0 afpmxrwe flat/aw) is +0 approximm‘e ‘WO 0!) which . ' ) We Can do ()8!le L(0.o:) atom Game 0.0] is Mar 0). so) Amavraaoowuofluhim):. - Wig-Hag.» "Hn's 55 am over or um'eres'hmxfa Jepenofs on -Hoe_ gnaw/y o-P .p Hear O ‘ I/ _3 ,2 We Comrade ‘F (x) =f§(’:§;) -_- 30-530 .63) = —q ([73:51 aim: 5, Thus) ‘pl'S 6! Concave JOWH ‘Fanc‘fiaI/I) Which has f+ always ‘ has 53h” H's (WIVES; 43% _ 77mg) 'Me Verb! L(o-O’) LS an overestima‘te “Par +111 valve. +7090!) I 3. (10 points) Compute the following limits, showing all reasoning. tanmwrc 0 (a) arctanm E5— ) Since ardfan0=o=fan0. l . i - sec-2x..- ' Br LHGPH'a') LIMH' f5 Yew; FL . = x1"; (X:I)(S€czx—l) x‘fl =(0+D(t- I) (b) 3Ln30(2w+=1)““”) °<>° new, SM #5.“; It“ 0 we ,3;“‘J"*"“°°‘ Wax 5 “"4 Lida“) 3'3; Affio Kg" 5% 43m - "K ' . .2. hm / :9— X'H x4“ 2, (3, L’flgoi-ME Me) 4 = hm 31x """ "" Qx—i- I x401: a-t ’4 31-0 .- 4. (8 points) Determine the equations of all asymptotes (horizontal and/ or vertical) of the graph 2 111(932 — 1) y 2:2—x Give complete justification, including relevant limit computations for each asymptote. Fars‘l‘ naf‘e Hat 4119 ohmin a? +h'is 3min}; is restricted: we msf have x"; p0 as well-as xz—X¢O_ Gmsez‘uence 3 domain is (—ooJ—l) 00160). Secondarz Conseguenoe: x=o am]! 3123!" be ‘H19 37””?th “P a warm“, asympv‘oie if He Turk) since 'fhen is no firafk near Level) I Checldn ‘Er horizontal as m A95 1 we Mfg-fa Ldl‘ xi»; mo! y I 2 ' L , oo . . . ', ISM M = ""1 LA 0 Is an -— 4;"... 3., L7 L'llé‘frfnl) rf 5 x6 w x1_x x-aw x(x..’) - 0° 2 l 4/1 _—.... . 2 ' Y O 3"“ ax _ I x—aa QxS“X"-2x+l Xe” a~;—F+;§ 2'0 y F , a] 1 a [mi [MN—l} - 0° . lira x‘rr "13‘ = I’M " : X9 an. 1—1:; Is a!” a" '55 ‘5'”) S“ Hi '5 x-au-fl Rw-l “*ma‘i-izi'i" O. I Coucgvsioh': =0 ls ‘H‘le only Earhaan fiSyMFl'é‘fE . fl.—— Checkin ‘Far Meal as m sites: Since 7 is q +100 finct'ans CoK‘HhUa'S on 4—inch- domains) We haw y “dB-eh“ is con-finuous an ff: Ogmmt'n. Thus) -Hle graph 5? y at" ml] P‘ELUDI/ [4W6 thfi'.“5yflf+dl‘es find 131»: dkwimmes) «r Hue endpo'm‘ls m" if: domain) x=-I m! x=!. 80+ was-b1! mm cgeck 43¢- ln‘Fm'rfe limHs if Mesa X-vqlves, l‘ 2. _-.- .. .. a “M W -__-_. pm because. x—wF 1‘6: D co 4 J x-> -r' x(x-£) J ’ HM - x64) = a) so fid‘fle 190‘]?er is 0236259! x-a-l r I. anal increases in uésafuf'cmfoc wh‘w a “M 1”“ " r) = _. oo lemme {xi-m 1“6‘"‘)= "°° ml 7 5004' + ' ) HI 36‘ f) ‘13:} x5”): 0) and xa—om 4‘“ x>lJ swim“ ‘H'E 7Wh'3"+ is negat‘l'im “Hal increases in «LSJHE Wine Conclusion: Bo’HA [ya-I in; E are "fihad 5W“! ' (4/9 L’Hé‘pd‘a’f' mfg affirms!) Ver‘fiCaI myMP-fd'l'eS. 5. (10 points) A highway patrol plane-flies above a level, straight road at a steady 120 miles per hour, maintaining an altitude of 3 miles. The pilot sees an oncoming car and uses radar to determine that, at the instant the line—of—sight distance from plane to car is 5 miles, this distance is decreasing at the rate of 160 miles per hour. Find the car’s speed along the highway. (The “line—of—sight distance” between two objects means the length of a line segment drawn directly between the two objects.) ‘ £7: Plane. Lgf- y=+fizlfneops|3H Jr'sl'ance bellman mrgplqnej - an! x=#g__ho'rizau1‘al JI'S‘lunce Mm“ car 3 +119, pain-l an 4m. race! under “Hie plane, ‘4’ o'n ' l ' 1 ’flrena anrfr‘fM-FMQJ f+3=r1) 2x5: = flea“) ‘j—iff)‘ 975E ' so 45% 41,}; means ‘Mq‘f’ ‘Hfle alfrf'MCe x is. cleansing an‘ fie rate 5? 20°“%r bu+ sian 'Hae Plane :3 swag kon‘antilbz m‘ a "a of :20 am.) a as as ’llve Case, as fie mr'is movinj a4 900~I20=!8’0Inf/5rl, Alik: and-her we, +0 mason 'Hais IN" “Fort is +9 1""th 6P 5+ 4: fihfi‘fier Mater! "#65 scum- Snme r” “raw rte mot, M be! an: car’s «wag; (fiyPo-légflcql s‘faanj pain-I3) :3!an uni-M +he distances end. has 4mmng 3.; gr. v Cour i Wane ' W W X -E . ,_ I 0' 5:ch W+X*£—¢°ns+mfl‘, WC I'M “HM 37w * a" " 3!:- ‘fi‘flmfko . ‘flus/ cm. I a? dimer” = -34; ~ 3'; s —(-&Oo)~I20 : 20 “WW. Na'l’e ‘Hmi' §¥=SP€eJchaU mag 55: ’20 "Kt 6. (5 points) Find the absolute maximum and minimum values of the function 9(33) = 2332+833— 10 on the interval [—1, 3]. Show all reasoning. ' Abs. e#MMa occur afier‘ at Bdrm-F: 04‘ +11: 95min or m‘ (TI-H?ch nun/lbw: GP 3’ (Clonal Infieer cri’l‘imf up new! X‘V‘I‘es WA‘“,S'(X) or jig-LIO- i'ms If“? ) which is never wdetineJ J 50+ I ‘1'{¥)=o __ => (-1,“ 3:0 =3 X 3 "(R '. Rheum; 'Hfis is 131'! in +119. $194M! ’flws «firms. M” emir- af “'eanm'fsJ am! we test 'f/rese wakes; 3(4): .a—Fs-ro = —(6 «me- 35%" [3+J‘I“f0‘f' 5a. ’nws’ is Jthe olLSfolu‘k: mamJ amt "I5- is +52 night: min; 7. (6 points) (3.) Give a precise statement of the Mean Value Theorem. 1:? 'P is oliPfiuu‘fr'ala‘e-on [atqu ‘Hren {Lye axis/f5 c Marceau. duo! £3 Suck "that We) 5 «FM-1%; t-« (14"; et'so mr‘f‘a‘lnk +0 5dr “1"? ’F 3'5 AFFEWHHE. .0"- (qJ‘I’) M con-hm: 0" (b) Let f be a. difierentieble function such that f (0) = 0 and f’ g 1 for all cc. Use the Mean Value Theorem to Show that f'(2) 79 3. Consider ‘Hm in’f'erwf [0’51] Gm which ‘F is Certain? atll'q‘EPevrh'aUe) flan Mm. ‘rs sum 1: Men O and a? sack #m' ‘P = 'Ffa)"F(°) am J Le.) SucL 7 Hal) = Q‘PTC) +‘F{O) = (EH-“63). (“lists by MVT.) Bufs‘mcexrprfif always; we Emu. in parfi’cofar ‘flnrf‘ flfcflflé’r'filflhfatéf c) 529ch s 2. 11m; *«°(:<)= we) -< a, w So «96w; 8. (15 points) Consider the function Mar) = 3:234/3 — 3321/3. (a) Find all points where the graph of h crosses the cc—axis. Mar Mar kw 3x204). Thu graph up la crosseexénxu': Whtn 560:0) Le. when %xfi(x-q)=o' ‘nwsJ £141.“ x50 .0, mqu and 4M zero-crossrhfls ' 04'! 1112 PM“ (0,0) a”! ((410) v (b) On what interval(s) is h increasing? decreasing? Explain completely. Fm! kw mt ‘lr‘zc‘f'crt 166:) = x"3- {VS -_— {#0- , ) , ngas'l olefin-Ml»: in‘bnals when Nr his paséh‘rga- ner’rmJ Sc we shn‘ 15w»: all paid: wlcn: I" could New; change sign: when: effher' W60 undefiueea Or 30. (’fln‘s SPIHS flit X-axls info regions of “13'5"”! Siam.) Sin-Ce “'64) = 51- hr 3: unafrfineo! Men x2704 anal x"? " ‘1’ is 22m when x=l, in’lewa‘ Sign o‘P x-l Sir cf“ 5:6 3530: (VP ‘1, ’1 jflcmluj/R'Ecreqsruj ,9 X< o Q Q h Jmfnj O< X< f I 6 6 (—3) I1 dacreasinj x>t (-9 (-5 6 A Fatwasfnj 405’ la is Masha on (—oo’O) 0(0) (wag chPf (-95,) a: we”); and a'ncmsinj on (L ab) _ (c) On what interva1(s) is h concave up? concave down? Expiain completein Fina, 51%) am! Swarm} : Lina)" é; "’éx'fi 1“ éxq$$6<+ 2) _ Warned dfierMI-ne Ham's m W62) is 9.3mm” “gawk”; so we mm; by 'Qan‘aj ag Pow: when 5!: and! tiny-3' 5,3“ : “be”. ewer h'figmlceneol or “7‘32 5'0 . Since. “‘6‘ * , “elsewhere; nil-en sun-OJ on! k“ 33 28“ Wk Xawa ‘ MWA! 330: GP X-I'Z a“? it"s Sb" 5? in” 5 mmtu x<-;2 ® 6) (-13 '1 W op “QC X4 0 Q 6 69 h concave Jam .................................. ........................ .. 6 . ('9 ‘* WW- v9 “05) l" ;5 CmcamuP on @mama.) U (03%)) m"! concave. down on (“3220). (C?) Using the information yon’ve found, sketch the graph 3; m Mm). Label and provide the (:12, y) coordinates of all local extrema and inflection points. mww 9%! number if wo "6’" W”! 0'? a Wfica‘ 43:25.” (has m)! m m mwm ‘flwsg in a“ 4km are, We inficcfien points {when In wages cmwfih W one, local min 6vka Wfifles deer. 4b 9:01)} M No ’9Cg{ mx‘ 9. (10 points) A farmer Wishes to build an Eshaped fence (see picture) along a straight river bank so as to create two identical rectangular pastures. With 300 feet of fencing material, What is the largest total area he can enclose? Justify completely. I , I x ' x ,p ’ / ence Lei K5 {enj'Hn 0'? one. Future sink parallel +3 we‘ler‘) y: JP out Silk Pei-P 'f‘a Whaler", ‘ ' - Wm‘e ’fiqen {41¢ 'Fm}nj limit-Minn is ‘H-qf 3y—rax =300, Note-Had state YZOJWE 5w; x559) and aka since )rzoJ “than YleO. f ‘ [N 1‘0 I ‘ ' = __ _ % ywn‘f‘f); we WIS mamas IS Q 52K),__ Rx(3o§;zx)) So we cm look ‘Hm q'lesa’lu‘fe MAY Value 6F +31 ‘FUHc‘hag == g‘xa'OO-ax) on 'H‘te r'A‘Fer‘Va’ [0,50]. AUMbETS GP Q1 ‘firsf :.- j';(amxn§x‘) = 200-— gm - ’flm 87x)l5”€VCFWJ€£hJ 541‘ Q7): ’1 r 2 J x 0 WA?” ZOO".%K=O ® 75. Thus K=75 is ‘Hcg owl; crHicnl number 6‘0 Q. Clasmldlrfcrvni we +¢sf (3(0)) (9(3)) @050) a»! dam m “(at ' 62(0): @050) :0) 50+ 62(75): 51-75- 3931'?) 5- ria-7550: 7300'. znws 75005uawl‘ect Is 1% maximum amen. Angm—h‘m +9 CIQSPJ En‘ferw, MLQA: Since Qwugx We. Mlle 'H'Wf' ) Ql{x)<o Wl‘i’" 7" 75 am! 'Hm'l' )0 W“ 'x< 75. Thus) lay ‘flce first Derivth 1st" 42V ABSo/ufe 5rth Values Q has #5 abs max ml K=7$J W4 as lac-Fth 'sz win is QK7SJ=7500_ (Coufdjurt a: £25”)! Mafch signch 0” is nejfi‘hifi: ‘Er ally) So “Hm-f 6? is concave Jaw“ ewrrfiem) and 44w: ham Max art 'n‘s [one cri-Hml numherA 10. (5 points) A certain television network produces a new episode of the hit show Hysterical Husbands each week; the amount of profit the show brings into the network is ultimately a function of the amount of money 71 the network spends to produce it. (In show business, 1) is known as the production value.) Specifically, the profit P depends on production value 1), each measured in thousands of dollars, according to the formula I I 18000 p m . __ — . (U) 500 (30 v + 600) v, v > 0 What is the maximum profit the network can make, and what production value u should the network choose in order to achieve this profit? Show all reasoning. We ward («bx-lute) WXFMM Value 0P P61) 431* V in (0J 0°) ‘ Critical #3- 0-? F". F;,,,( P7,), -m_w0mq(4).(v+€w)zfl I = 5-!800000-(maof1 I 5 9000000 r z I . (Va-600) PYV) is new undefind) since W600>OI and a: 61+;ij 35 “My: deg": P’(v)=o when m,, 50 (W600) =9 (wacnf -~ “7000000 -=(3ooc:v)Z 7‘7 v+5go = $3000 '57 V: ~€003330CLO (by! reject v=—600—3000/ since VDO) _ ‘9 v: 2400. 17,;5 ;5 4% only crib'ca/ mmLer, fl W9. can characterize ’H‘fi MGVM‘ 6P P gem- {+5 lane Office! novel)” [2); aim/fin 4113- §iay 6F either P’ or F“? I -' Of‘l'ion ' " 057313 Pf: Nate PIA/J: 71.] . wile 3000 . H 0<V<QLI00) 'H1-en %>') So PYV)>OJ luff when V>SWOO} ‘Hren o<i§§ <1) So P7v)<O_ T1105, v=R400 is a pic 0F «lame/0+9 maximum , . a " .n "3 fl " .- OP‘H ma US! 3 P Nah P [‘0‘ “afiOOOW/YV'PGDO) 3<O ’Fdr' a” V>O which man: FHuI- P hau- qn alDSoIW‘E MON d‘f #3 [We ) cr'rl'lcqf writer. Wily!i M598 V=Q¢I00459ummf {of/4,3 6:8.é’RHMI‘l/r‘on) i: ‘l'lve PmJucf‘ion ' ~H1a‘l" Maximrzes profit) am! Fatwa) 1:: 500(3onzlfi'290) —;qoo =500.gq..aq00 =- ‘fllovsamd Jot/2’3 {ll-C. #7-gnfl'l/I'Oh) is flu, Mammy». pY'd'fi‘f. 11. (6 points) The value 1 is a pretty lousy approximation to Using Newton’s method, find a rational number (Whole number or fraction) which is closer, and then another one which is still closer. Simplify your answers to a reasonable form, and Show all of your steps. We. wan/1+ +he 28'!) GP +52. ‘EMCHOVI -P(x).—: )?_3\ J and We,” 'NeV/I'OH’S Xl=f 4b 36f XL “"09 x3 ) ‘H‘i "93+ “(We OPPNlea‘Hong. . r 7... Wm: Xy-Hq Kn”? and ) SO X 7" X’ = .9..— 1 3) L 3 = :1 W344i? 3 36$ 33 = .9... 93-52-33 3 92-3?- 2 i- eff-5‘1 == ifl-[Q- 3 Ic-q 3 l-‘i ,fi,5 .. 15-5.; " 3 $31 _ 751 7a ...
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06exam2sol - MWLA 4H Aafumn 8606" Exam/Ia Safe/7‘;wa...

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