6 Appling lulus o furher moelling Conens: A B C D E F Desriing urve properies Curve properies from lulus Invesiging new moels Ienifing moels from grphs Moelling prolems Surge n erminl veloi moels
208 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) In his hper we will ppl he lulus suie in Chper 5 o vrious moelling quesions. We will een he moelling work one in previous hpers, n onsier moels of he following pes: ² liner ² eponenil ² power ² quri ² ui ² logrihmi ² logisi ² erminl veloi ² surge 9 = lre emine in previous work ; ¾ polnomil moels = + ln C =, A>0, >0, C>0 1+Ae = A(1 e ), A>0, >0 = Ae, A>0, >0 A DESCRIBING CURVE PROPERTIES When we esrie he grphil feures of moels, i is imporn o use lnguge h we ll unersn. LOCAL AND GLOBAL MAXIMA AND MINIMA A lol mimum is mimum urning poin. lol mimum A lol minimum is minimum urning poin. lol minimum The glol mimum is he poin wih he mimum vlue of on he omin of he funion. The glol minimum is he poin wih he minimum vlue of on he omin of he funion. lol mimum glol mimum glol mimum (n lol mimum) glol minimum lol minimum glol minimum
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 209 ASYMPTOTES =1 This urve hs horizonl smpoe =1 n veril smpoe =3. As ges ver lrge n posiive, n s ges ver lrge n negive, he grph ges loser n loser o he smpoes. =3 INCREASING AND DECREASING FUNCTIONS Grphs like re inresing for ll. A grph is inresing on some inervl if, s he vlue inreses, he vlue inreses lso. inrese in inrese in As we move from lef o righ we re going up he hill. Grphs like re eresing for ll. A grph is eresing on some inervl if, s he vlue inreses, he vlue ereses. erese in As we move from lef o righ we re going own he hill. inrese in Some grphs onin inervls whih re inresing, s well s inervls whih re eresing. For emple: in r esing 3 e n si re g in e g r sin (3, 2) This funion is eresing for 6 3 n inresing for > 3. (4,-1) This funion is eresing for 0 6 6 4 n inresing for 6 0 n for > 4.
210 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) Emple 1 Fin inervls where he following grphs re inresing or eresing: (4, 3) (8, -2) Inresing for 0 6 6 4. Deresing for 6 0 n for 4 6 6 8. (2, -1) Deresing for 6 2. Inresing for > 2. EXERCISE 6A.1 1 For eh of hese grphs: i give he posiion n nure of n urning poins ii se inervls where he grph is inresing or eresing iii se he equion n nure of n smpoes. (2, 3) e f (0, 2) ( Qw, 6 _ Qr _) (1, -1) g h i (-1, 2) =10 (-4, -3) (2, -2) =-2 2 2 Fin he glol minimum n glol mimum for he following grphs: (3, 17) (4, 5) (-5, 6) (-4, 10) (2, 8) (0, 0) (-1, 1) (-3, -4) (4, -5)
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 211 3 Consier he illusre line. For fie -seps, here re fie -seps, so he slope of he line is onsn. Commen on he slopes of he following urves s he -vlue inreses. e f A B 4 Use quesion 3 o nswer he following: Wh is he slope of urve: i mimum urning poin or minimum urning poin ii on n inervl where he funion is inresing iii on n inervl where he funion is eresing? Wh is hppening o he slopes of he urves : i poin A in 3 ii poin B in 3f? INFLECTIONS AND SHAPE When urve, or pr of urve, hs shpe: M ve looks like: we s h he shpe is onve we s h he shpe is onve. Consier his onve urve: m=1 m=0 m=-1 As inreses, he slope of he ngen ereses. So, he re of hnge is eresing. m=2 =-2 m=-2
212 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) Now onsier his onve urve: =2 m=-2 m=2 As inreses, he slope of he ngen inreses. So, he re of hnge is inresing. POINTS OF INFLECTION m=-1 m=1 m=0 A poin of infleion is poin on urve whih here is hnge of urvure or shpe. A his poin, he urve hnges from onve o onve, or vie vers. or DEMO poin of infleion poin of infleion The ngen he poin of infleion, lso lle he infleing ngen, rosses he urve h poin. EXERCISE 6A.2 1 For he following grphs, ienif whih of he poins A, B, C or D is poin of infleion: B C B C A D A B D C A D 2 For he following funions, fin: i ii iii n poins of infleion inervls where he re of hnge is inresing inervls where he re of hnge is eresing. (-2, 11) =10 (1, 7) (6, 5) (4, 3)
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 213 e f (0, 4) (8, 5) (-1, 3) (1, 3) (5, 2) (2, -1) (2, -1) 3 Wh hppens o he slope of he urve poin of infleion? Hin: Consier Eerise 6A.1 quesion 4gin. B CURVE PROPERTIES FROM CALCULUS So fr we hve esrie feures of urve visul inspeion. However, we re ofen given he equion of funion, n we wn o ienif feures of is grph suh s urning poins, n inervls where he grph is inresing or eresing. Even if we rw he grph of he funion, i is iffiul o preisel loe is feures. In siuions like his, we n use he lulus we lern in he previous hper o eermine ke feures of he grph. Given he slope funion or f 0 (), we n fin he slope of he ngen o he urve n vlue of susiuing h vlue of ino i. For emple: =3-5-2 If = 3 5 2 hen =32 5. Now, when =1, = 3(1)2 5 = 2 So, he poin where =1, he ngen hs slope 2. 1 slope -2 TURNING POINTS A lol mimum The ngens he urning poins re horizonl n herefore hve slope 0: B lol minimum A urning poins, =0. When funion is inresing, he slope of he ngen is posiive. A B When funion is eresing, he slope of he ngen is negive.
214 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) A funion is inresing on n inervl if A funion is eresing on n inervl if > 0 for ll on h inervl. 6 0 for ll on h inervl. POINTS OF INFLECTION Consier he grph of = f() longsie, where A is poin of infleion. The grph of is erivive = f 0 () is given elow i. Noie h: ² The urning poins of f() orrespon o he -inereps of f 0 (). This is euse f 0 () =0 he urning poins. ² The infleion poin of = f() orrespons o he urning poin of f 0 (). = f() A = f'() So, o fin he infleion poins of = f(), we grph he erivive funion = f 0 (), n fin he -oorines of n urning poins. We n eermine he -oorine of n infleion poin susiuing he -oorine ino = f(). Emple 2 Consier he funion = 3 3 2 +2. Fin, n solve =0. Use ehnolog o oin grph of he funion. Fin n urning poins. Fin n poins of infleion. e Fin inervls where he funion is: i inresing ii eresing iii onve iv onve. The slope of = f( ) is mimise A, he poin of infleion. = 3 3 2 +2 ) =32 6 So, =0 when 3 2 6 =0 ) 3( 2) = 0 ) =0or 2
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 215 When =0, =0 3 3(0) 2 +2=2 When =2, =2 3 3(2) 2 +2= 2 So, here is lol mimum (0, 2) n lol minimum (2, 2). =32 6 hs he grph given longsie: hs urning poin =1, so he funion hs n infleion poin =1. When =1, =1 3 3(1) 2 +2=0, so here is n infleion poin (1, 0). e i The funion is inresing for 6 0 n > 2. ii The funion is eresing for 0 6 6 2. iii The funion is onve for 6 1. iv The funion is onve for > 1. We n use ehnolog o fin urning poins. Consul he grphis lulor insruions he fron of he ook. EXERCISE 6B 1 For eh of he following funions: i fin n solve =0 ii use ehnolog o oin grph of he funion iii fin he urning poins n poins of infleion of he funion iv fin inervls where he funion is inresing, eresing, onve, n onve. = 3 3 +1 = 2 2 +2 1 =2 3 12 2 +3 = 3 9 2 +24 15 e =5e 2 f =2e g = p h = 8(1 e 2 ) i =2+ln 2 Consier he funion =2e 4 for > 0. Show h = 2(1 4)e 4. Eplin wh =0 onl when = 1 4. Use ehnolog o grph he funion. Fin n urning poins n poins of infleion. e Fin inervls where he funion is: i inresing ii eresing iii onve iv onve.
216 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 20 3 Consier he funion = 1+3e 2 for > 0. Show h = 120 e 2 (1 + 3e 2 ) 2. Eplin wh > 0 for ll vlues of > 0. Use ehnolog o grph he funion. Wh is he signifine of o he grph? C INVESTIGATING NEW MODELS Sine liner, eponenil n power moels hve een emine in n erlier hper, n quri n ui moels hve een emine in Sge 1, we will now invesige he remining four moels. The re he logrihmi, surge, erminl veloi, n logisi moels. INVESTIGATION 1 THE LOGARITHMIC MODEL The logrihmi moel hs he form = + ln where n re onsns. In his invesigion we emine his fmil of urves s he vlues of n hnge. GRAPHING PACKAGE The use of he grphing pkge is reommene. Clik on he ion o open his pkge. A grphis lulor is lso pproprie. I is esirle for suens o eperiene oh ehnologies. Wh o o: 1 Le =2, so = +2ln. Grph he following funions: =5+2ln =3+2ln =1+2ln =0:5+2ln Wrie own our oservions, inluing n urning poins, smpoes, inervls where he funion is inresing or eresing, n inervls where he funion is onve or onve. 2 Le =2, so =2+ln. Grph he following funions: =2+5ln =2+3ln =2+ln =2 ln e =2 3ln f =2 5ln Wrie own our oservions. INVESTIGATION 2 THE SURGE MODEL A surge moel hs he form = Ae where A n re posiive onsns. The inepenen vrile is usull ime, > 0. This moel is use eensivel in he su of meiinl oses where here is n iniil rpi inrese o mimum, n hen slower e o zero. GRAPHING PACKAGE
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 217 Wh o o: 1 Le =4, so = Ae 4. Grph he following funions: =10e 4 =20e 4 =40e 4 Wrie own our oservions. 2 Le A =20, so =20e. Grph he following funions: =20e =20e 2 =20e 3 =20e 5 Wrie own our oservions. INVESTIGATION 3 Wh o o: THE TERMINAL VELOCITY MODEL A erminl veloi moel hs he form = A(1 e ) where A n re posiive onsns. The inepenen vrile is usull ime, > 0. This moel is ofen use when n oje is flling uner he influene of grvi n pprohes limiing veloi. 1 Le =2, so = A(1 e 2 ). Grph he following funions: = 100(1 e 2 ) = 60(1 e 2 ) = 30(1 e 2 ) = 10(1 e 2 ) Wrie own our oservions. 2 Le A = 100, so = 100(1 e ). Grph he following funions: = 100(1 e 5 ) = 100(1 e 2 ) = 100(1 e ) = 100(1 e 0:5 ) Wrie own our oservions. GRAPHING PACKAGE INVESTIGATION 4 C A logisi moel hs he form = 1+Ae C re posiive onsns. The inepenen vrile is usull ime, > 0. THE LOGISTIC MODEL where A, n The logisi moel is useful in limie growh prolems, where growh nno go eon priulr vlue ue o preors or lk of resoures. Wh o o: C 1 Le A = =2, so = 1+2e 2. Grph his funion for: C =30 C =10 C =5 Wrie own our oservions. GRAPHING PACKAGE
218 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 30 2 Le =2, C =30, so = 1+Ae 2. Grph his funion for: A =1 A =3 A =5 A =10 Wrie own our oservions. 30 3 Le A =2, C =30, so = 1+2e. Grph his funion for: =1 =2 =3 =10 Wrie own our oservions. From hese invesigions, we n mke he following oservions: Logrihmi Moel = + ln, n onsns, > 0. >0 <0 ² The -is is veril smpoe. ² There re no urning poins or infleion poins. ² The funion is eiher lws inresing ( >0) or lws eresing ( <0). Surge Moel = Ae, A n posiive onsns, > 0. ² The -is is horizonl smpoe. ² The funion psses hrough he origin. ² There is mimum urning poin when = 1. ² There is n infleion poin when = 2. Q W Terminl Veloi Moel = A(1 e ), A n posiive onsns, > 0. ² = A is horizonl smpoe. ² The funion psses hrough he origin. ² There re no urning poins or infleion poins. ² The funion is lws inresing, n he re of hnge is lws eresing. Logisi Moel C =, A,, C posiive onsns, > 0. 1+Ae ² = C is horizonl smpoe. ² The funion is lws inresing. C ² There is n infleion poin when = C _wc 2. A =A =C
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 219 Emple 3 A swine flu pien is presrie ourse of n nivirus rug. The moun of rug in he loosrem hours fer he firs ose is ken, is given he surge moel A() = 100 e 2 mg. Skeh he grph of he funion for 0 6 6 4. Show h A 0 () = 100e 2 (1 2). Fin he vlue of for whih A 0 () =0, n eermine he mimum moun of rug in he ssem. Eplin wh is hppening o he moun of rug in he loosrem over ime. e Drw grph of A 0 () = 100e 2 (1 2). f Fin A 0 (0:25) n inerpre our nswer. g Fin he -oorine of he poin of infleion, n inerpre our nswer. A A() = 100e 2 is he prou of 20 u() = 100 n v() =e 2 15 A() = 100e _ -2 ) u 0 () = 100 n v 0 () = 2e 2 10 5 ) A 0 () =u 0 () v()+u() v 0 () fprou ruleg = 100e 2 + 100( 2e 2 ) 1 2 3 4 = 100e 2 (1 2) A 0 () =0 when 1 2 =0 fe 2 6=0g ) = 1 2 Now A( 1 2 ) = 100 1 2 e 2( 1 2 ) ¼ 18:4 So, he mimum moun of rug in he ssem is 18:4 mg, fer hlf n hour. The moun of rug in he ssem rpil inreses for he firs hlf hour unil i rehes mimum of 18:4 mg. I hen ereses quikl firs, hen slowl s i pprohes zero. e 100 80 60 40 20 A'() A'() = 100e _ -2(1-2) 1 2 3 4 f A 0 (0:25) = 100 e 2(0:25) (1 2(0:25)) ¼ 30:3-20 Afer qurer of n hour, he moun of rug in he ssem is inresing re of 30:3 mg per hour. g From e, A 0 () hs urning poin when =1. ) A() hs poin of infleion =1. Afer 1 hour, he re of erese of he rug from he o is mimise.
220 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) EXERCISE 6C 1 When new pin killing injeion is minisere, he effe is moelle E = 750e 1:5 unis, where > 0 is he ime in hours fer he injeion of he rug. Skeh he grph of E gins. Wh pe of moel is his? Wh is he effe of he rug fer: i 30 minues ii 2 hours? Show h E = 375e 1:5 (2 3), n hene eermine when he rug is mos effeive. e In he opering perio, he effeive level of he rug mus e les 100 unis. i When n he operion ommene? ii How long hs he surgeon o omplee he operion if no furher injeion is possile? f Fin he poin of infleion of he grph. Wh is he signifine of his poin? 2 The numer of ns in olon fer monhs is moelle A() = Drw skeh of he funion. Ienif he moel pe. Wh is he iniil n populion? Wh is he n populion fer 3 monhs? e Is here limi o he populion size? If so, wh is i? f A wh ime oes he populion size reh 24 500? 25 000 1+4e. g i ii iii Show h A 0 100 000e () = (1 + 4e ) 2. Fin he ime whih he growh re of he n populion is mimise. Fin he n populion his ime. 3 Ale hs hehe n kes 1000 mg of premol. The onenrion of his rug in her ssem fer hours is moelle he funion P () = 150 0:3 prs per million. Use ehnolog o op n omplee he le of vlues elow: 0 0:5 1 1:5 2 2:5 3 3:5 4 4:5 5 P () 0 41:1 Skeh he grph of = P () for 0 6 6 5. Desrie how he onenrion of premol hnges over he 5 hours. For he rug o e effeive, i is neessr for he onenrion of premol o e ove 15 prs per million. Fin he inervl of ime for whih he rug is effeive. e Fin he ime whih he onenrion of premol is mimum, n eermine he onenrion his ime. f Fin he re of hnge of he premol onenrion fer 1 hour.
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 221 4 The spee of esen of skiver seons fer jumping from plne, is given v() = 180(1 e 0:2 ) ms 1. Skeh he grph of = v(). Wh pe of moel is his? Desrie he veloi of he skiver over ime. Fin he veloi fer: i 1 seon ii 3 seons. e Fin he verge re of hnge of he veloi over he ime inervl =1 o =3. f Fin v 0 (). g Fin v 0 (3) n inerpre our nswer. h Fin he limiing veloi of he skiver. 5 During sorm, he volume of wer ollee in nk fer hours is moelle V () = 12 + 10 ln lires, > 1. Drw grph showing he volume ollee over 20 hours. Fin V 0 (). A he grph of he erivive o our grph in. Fin V (6) n V 0 (6), n inerpre our nswers. e Desrie wh is hppening o oh he volume n re of hnge of volume over ime. 6 The numer of usomers in resurn hours fer noon is moelle N() =2 3 15 2 +24 +25, 0 6 6 6. e Fin N 0 (). Fin when N 0 () =0, n inerpre our nswer. Wh ws he les numer of usomers in he resurn? During wh ime inervl ws he numer of usomers eresing? Skeh he grph of N 0 (), n eplin he signifine of he poin where N 0 () is minimum. 7 Suppose f() =Ae, where A n re posiive onsns. Show h f 0 () =Ae (1 ). Show h f() hs urning poin when = 1. D IDENTIFYING MODELS FROM GRAPHS The spee of moorike is mesure one seon inervls s i eleres from sop sign. The following is oine, n isple on he ser plo: Time (s) 1 2 3 4 5 6 Spee (km h 1 ) 10 14 17 20 22 24 Bse on he shpe forme he ser plo, logrihmi or power moel pper mos pproprie for his. 25 20 15 10 5 spee (kmh _ -1) 0 0 2 4 6 ime (s)
222 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) The following summr m prove useful in ienifing pproprie moels: ² liner = m + m<0 m>0 ² power = m m>1 0<m<1 m<0 ² eponenil = e >0 <0 ² quri = 2 + + vere >0 <0 vere ² ui = 3 + 2 + + >0 urning poins eis >0 no urning poins eis <0 <0 urning poins eis no urning poins eis ² logrihmi = + ln >0 <0 ² erminl veloi = A(1 e ) =A A>0,>0
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 223 ² surge = Ae A>0,>0 C ² logisi = =C 1+Ae C>0,A>0,>0 EXERCISE 6D 1 From he lis given longsie, eermine he possile moels for wih he following ser plos: Mss Power ² liner ² quri ² ui ² eponenil ² power ² logrihmi ² logisi ² erminl veloi ² surge Gin e f Weigh Demn Fore g h i Kilomeres Tonnge Q j k l S N
224 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 2 For he following ser plos, eplin wh he suggese moel is no eple: P A power moel of he form = m, >0, m>1 T A logrihmi moel of he form P = + ln, >0 D e A logisi moel of he form C T =, A, n C>0 1+Ae V f A ui moel of he form D = 3 + 2 + + N A erminl veloi moel of he form V = A(1 e ), A n >0 3 For he following ses: i rw ser plo of he ii sugges he mos pproprie moel for he. An eponenil moel of he form N = e, >0, <0 0 1 2 3 4 5 6 7 8 0 7:9 12:6 15:5 17:3 18:4 19:0 19:4 19:6 1 3 4 6 8 10 12 14 16 17 20 0:7 1:7 2:8 6:7 14:1 23:9 32:1 36:7 38:7 39:1 39:8 0 3 5 8 10 13 15 30 17:7 12:6 9:2 10:1 15:7 22:5 0 3 7 10 15 20 30 40 60 80 0 67 105 110 100 81 45 22 5 1
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 225 E MODELLING PROBLEMS In his seion we will emp o fi n pproprie moel o se of. We hve previousl fie liner, eponenil, n power moels o. In his seion we will emine whih m e: ² eponenil ² power ² quri ² ui ² logrihmi ² logisi In Chper 4 we use nurl logrihms o rnsform ino liner, hen use liner regression o fin he equion onneing he vriles. Now we will use ehnolog o fin he moel of es fi, in he sme w we fin liner moels. Consul he grphis lulor insruions he fron of he ook if ou require ssisne. Emple 4 Beri re ulive on gr ples. The weigh W of eri presen is reore wo hour inervls. Time ( hours) 2 4 6 8 10 12 14 Weigh (W grms) 0:09 0:36 1:02 1:82 2:27 2:45 2:46 e Drw ser plo of he. Eplin wh he logisi moel is mos pproprie for he. Fin he logisi moel of es fi. Esime he weigh of he eri fer 7:5 hours. A wh re is he eri weigh inresing fer 6 hours? W 2 1 or 2 4 6 8 10 12 14 The poins inrese slowl firs, hen fser, hen more slowl gin s he pproh limiing vlue. This ehviour is onsisen wih logisi moel. Using ehnolog, he logisi moel of es fi is W ¼ 2:484 1+94:69e 0:6960.
226 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 2:484 When =7:5, W ¼ ¼ 1:64 1+94:69e 0:6960(7:5) Afer 7:5 hours, he weigh of he eri is 1:64 g. W e Using ehnolog, when =6, ¼ 0:417 Afer 6 hours, he weigh of he eri is inresing 0:417 grms per hour. If here is more hn one possile moel whih ppers suile for he, we n use he r 2 vlues ssoie wih eh moel (eep for he logisi moel) or else he one of he prolem, o eermine he mos pproprie moel. Emple 5 A ole of wer hs een lef ou in he sun, n is now ple insie refrigeror. The wer emperure is mesure ever 15 minues. The emperures re: Time ( mins) 15 30 45 60 75 90 105 120 Temperure (T o C) 31 19 12 7 4 3 2 1 Drw ser plo of he. The ppers o follow eiher logrihmi or n eponenil urve. i Fin he moel of es fi for eh of hese moel pes. Inlue he r 2 vlues. ii Whih moel is mos pproprie? Eplin our nswer. Use he moel o esime he wer emperure fer 20 minues in he refrigeror. Fin he re whih he wer is ooling when =40. 40 30 20 T or 10 0 0 30 60 90 120 150 i Logrihmi The logrihmi moel of es fi is T ¼ 69:1 14:7ln r 2 ¼ 0:977
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 227 Eponenil The eponenil moel of es fi is T ¼ 49:0e 0:0318 r 2 ¼ 0:995 ii The eponenil moel hs higher vlue of r 2. Also, he logrihmi moel will he owrs 1 over ime, wheres he eponenil moel will pproh zero from ove, whih is more onsisen wih how he emperure will ehve. So, he eponenil moel is mos pproprie. Using he eponenil moel, when =20, T ¼ 49:0e 0:0318(20) ¼ 26:0. So, fer 20 minues, he emperure of he wer is 26:0 o C. T ¼ 49:0e 0:0318 ) T ¼ 49:0( 0:0318)e 0:0318 ¼ 1:56e 0:0318 T When =40, ¼ 1:56e 0:0318(40) ¼ 0:437 So, when =40, he wer is ooling 0:437 o C per minue. NOTE ON EXPONENTIAL MODELS Eponenil moels in his hper re wrien in he form = e, s oppose o = in Chper 4. This llows us o ifferenie he moel, n mkes he moel more onsisen wih oher moels isusse in his hper whih lso involve e. When using ehnolog o fin n eponenil moel, he Csio gives he moel in he form = e, s shown in he ove emple. However, he Tes Insrumens lulors give he moel in he form =, s shown longsie. We n use he rule = e ln o onver his o he form = e : T ¼ 49:0(0:969) ) T ¼ 49:0(e ln 0:969 ) ) T ¼ 49:0(e 0:0318 ) ) T ¼ 49:0e 0:0318 EXERCISE 6E 1 A hemispheril owl is fille wih wer. The volume of wer require o reh ifferen wer levels is given in he le elow. 40 m h m
228 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) h (m) 0 3 6 9 12 15 18 V (ml) 0 550 2050 4350 7250 10 600 14 250 e Oin ser plo of he. A ui moel is he es fi for he. Fin he ui moel for his. Use he moel o fin he volume when he owl is full. The volume of sphere is given V = 4 3 ¼r3. Use his formul o hek our nswer o. Use he moel o esime he volume of wer when he heigh is 10 m. 2 Wer is kep onsn emperure s i flows hrough hik wlle pipe. The imeer of he pipe is 20 m n he wlls of he pipe re 10 m hik. The emperure of he pipe wll vries from insie o ousie s he he espes hrough he wlls. The emperure mesuremens re: Disne ( m) 11:2 13:8 15:7 16:9 17:4 19:1 Temperure (T o C) 81 66 55 51 48 42 10 m 10 m ross-seion of he pipe Drw ser plo of he. Fin he logrihmi moel of es fi. Fin he emperure he poin hlf-w hrough he wll. Esime he emperure of: i he inner wll ii he ouer wll. e Fin T, n hene eermine he re whih he emperure is flling he poin hlf-w hrough he wll. 3 Air is pumpe ino spheril lloon onsn re, n he rius of he lloon is mesure vrious imes: Volume (v lires) 18:2 27:8 39:6 58:2 73:1 96:8 Rius (R m) 16 19 21 24 26 28 Drw ser plo of he. Eplin wh power moel ppers suile. Fin he power moel of es fi. Wh is he rius when here is 80 lires of ir in he lloon? A wh re is he rius inresing he insn when v =80? Give our nswer in m L 1. 4 The mss of rioive susne is mesure eh er for si ers: Time ( ers) 0 1 2 3 4 5 6 Mss (M grms) 5:7 5:3 4:9 4:6 4:3 4:0 3:7 Drw ser plo of he. For he ser plo, i ppers h eiher n eponenil or liner moel is mos pproprie. i Fin he moel of es fi for eh of hese moel pes. Inlue he r 2 vlues. ii Whih moel is he mos pproprie? Eplin our nswer. Use our moel o esime he remining mss of he susne fer 10 ers. The hlf-life of rioive susne is he ime ken for i o reue o hlf is originl mss. Wh is he hlf-life of his susne? m
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 229 5 In smll ommuni, everone evenull hers rumour. A smll group of people inverenl sr rumour, n he proporion of people who hve her he rumour is reore on n hourl sis uring he. Hours (h) 0 1 2 3 4 5 6 7 8 9 10 Proporion (P ) 0:02 0:04 0:09 0:18 0:33 0:54 0:72 0:86 0:91 0:96 0:98 Drw ser plo of he. Wh is he mos likel moel pe for he? Fin he moel of es fi. Esime he proporion of people who hve her he rumour fer 5:5 hours. Fin he re whih he rumour is spreing fer: i 2 hours ii 5 hours. 6 A onror igs wells o mimum eph of 100 m. He wns formul o fin he os of igging wells of vrious ephs. Using from some of his previous jos, he onsrus le of oss: Deph ( m) 15 30 45 58 68 84 8000 14 000 21 000 32 000 46 000 84 000 The onror suspes h eiher quri or ui moel is mos pproprie. i Fin he moel of es fi for eh of hese moel pes. Inlue he r 2 vlues. ii Whih moel is he mos pproprie? Fin he os of igging well whih is: i 50 m eep ii of mimum eph. Fin C when he eph is 50 m. Wh is he mening of his resul? e Cos ($C) Drw ser plo of he. A wh eph oes he re of hnge of os C 7 Consier gin he for he moorike on pge 221: sr inresing? Time ( seons) 1 2 3 4 5 6 Spee (S km h 1 ) 10 14 17 20 22 24 The ppers o follow eiher logrihmi or power urve. i Fin he logrihmi moel of es fi. ii Fin he power moel of es fi. iii Whih moel o ou hink is mos pproprie? Eplin our nswer. Using he moel of es fi, esime: i he spee of he moorike fer 4:5 seons ii he re whih he moorike s spee is inresing fer 3 seons. 8 The profi when mking n selling spee os per monh is given in he following le: Numer of os () 5 7 10 14 19 24 Profi ($P 000) 50 136 233 300 290 172 Oin ser plo of he. Whih moel pe is mos likel o fi he?
230 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) e f g h i Fin he moel of es fi. Use he moel o esime he profi for he monh if: i 0 os re sol ii 8 os re sol. Eplin our nswer o i. How mn os shoul e sol o mimise he profi? Chek our nswer o f iffereniing he moel. Give possile reson wh he urve ereses for lrger prouion. For wh prouion level oes loss our? 9 Dimons re egorise oring o heir quli. The vlues of 7 high quli gre-f imons of ifferen weighs re given in he le elow: Weigh ( rs) 0:32 0:41 0:67 0:81 1:03 1:27 2:11 Vlue ($D) 430 690 1750 2500 4000 5900 15 500 Drw ser plo of he. Fin he: i power moel ii eponenil moel of es fi. Whih moel es fis he? Eplin our nswer. Use he moel from o fin he vlue of 1:5 r gre-f imon. e f Wh is he vlue of Wh is D D when =1:5? Inerpre his resul. posiive for ll vlues of? 10 A wine glss hs proli ross-seion. Wine is e o he emp glss in 20 ml los, n eh ime he heigh of he wine ove he vere is mesure. v (ml) 20 40 60 80 100 H (m) 3:6 5:0 6:2 7:2 8:0 Oin ser plo of he. 9 m H m Fin he moel whih es fis he. Esime he pi of he glss. vere Fin he heigh of he wine when he glss is hlf pi. 11 The imes ken for populion of roens o reh erin levels re shown in he le elow: Populion (p) 48 150 290 560 2150 Time (w weeks) 0 5 8 11 17 From ser plo of he, whih moel pe seems o e possile fi for he? Fin he moel of es fi. Use he moel o esime he numer of weeks neee for he populion o reh: i 400 ii 3000: Sugges resons wh his moel m e inpproprie for lrger populion sizes.
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 231 12 Dmien hrows rike ll verill upwrs. The isne ove he groun vrious imes is eermine from igil vieo eviene. The following le of vlues gives he isne ove he groun seons fer he ll ws relese. Time ( seons) 1:2 1:8 2:4 2:9 3:6 4:2 4:7 5:2 Disne (D m) 30:1 39:2 45:0 47:1 45:8 40:7 33:9 24:9 Using ser plo eviene, wh moel seems o fi he? Fin he moel of es fi. Use he moel o esime he heigh of he ll 2 seons fer relese. How fr ove he groun ws he ll when relese? e Fin he grees heigh, n he ime fer relese when his heigh ws rehe. f Fin he ol ime he ll ws in he ir. g Fin he veloi D of he ll: i when i ws relese ii fer 3 seons iii when i hi he groun. h Show h he elerion of he ll ws onsn hroughou is fligh. Hin: Aelerion is he re of hnge of veloi. 13 Severl ers go he Willing Golf Course greens were urn off ue o wering wih highl sline wer from is ms. This ourre le in summer when he ms were ver low in wer. As he wer level lowere ue o use n evporion, he sl onenrion inrese. During he following summer, he he greenkeeper kep reors of he volume of wer in he m n he sl onenrion. The resuls were: e Volume (V ML) 45 32 24 14 8 4 Sl onenrion (C%) 0:012 0:023 0:036 0:068 0:125 0:258 From ser plo, sugges he mos likel moel pe for he. Fin he moel of es fi. Wh ws he sl onenrion when he volume of wer in he m ws 20 ML? A wh re ws he sl onenrion inresing when he volume of wer in he m ws 20 ML? Wer from he m shoul no e use when he onenrion eees 0:2%. Wh is he volume of wer in he m when his ours? 14 A smll isln off Cpe York hs n unne of roens whih re he mjor foo soure of hreene speies of rown snke. A few pirs of rown snkes were inroue o he isln whih ws previousl snke free. Gulls n oher irs on he isln keep he snke populion in hek. The populion of snkes over mn ers hs een reore. Time ( ers) 1 2 3 4 5 6 7 8 Snke populion 47 108 215 384 560 675 753 780 Drw ser plo of he n sugges possile moel. Fin he moel of es fi. Esime he populion fer 4:5 ers. Is here limi o he populion size? If so, wh is i? e A wh re ws he populion inresing fer: i 2 ers ii 4 ers iii 6 ers?
232 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 15 The imes ken for rioive susne o reh erin msses re given in he following le: Mss (m grms) 3:0 2:5 2:0 1:5 1 0:5 Time ( hours) 4:0 7:9 12:6 18:7 27:4 42:0 e f Oin ser plo of he n inie whih moel pes es fi he. Fin he moel of es fi. Use he moel o fin he iniil mss of he susne. Use he moel o esime he ime require for he susne o e o: i 1:8 grms ii 0:2 grms. Whih of he wo esimes in woul e more relile? Eplin our nswer. Ke proue 5:37 grm smple of he susne n oserve is e. A le of vlues ws foun. Wh woul e he moel for m() in his se? F SURGE AND TERMINAL VELOCITY MODELS If ou emine our grphis lulor menu for possile moels, ou will noie h i oes no irel hnle surge or erminl veloi siuions. However, if we perform simple lgeri rnsformions on whih we suspe is fie one of hese moels, we n hen use grphis lulor o omplee he sk. SURGE MODELS Noie h if = Ae hen = Ae, whih is n eponenil moel. So, if we suspe he grph of gins is fie surge moel, we n plo he grph of gins, n hek wheher n eponenil moel is goo fi for hese poins. Emple 6 The effe of pin-killing injeion fer hours is shown in he following le: Time ( hours) 0:00 0:10 0:25 0:50 0:75 1:00 1:25 1:50 1:75 2:00 Effe (E unis) 0 56 84 84 58 42 22 10 5 3 Oin ser plo of he n sugges suile moel for i. Plo vlues of E gins using ehnolog, n fi n eponenil moel o his. Rewrie he moel in he form E = Ae. When is he hemil mos effeive? e An operion n onl ke ple when he effe is les 60 unis. i A wh ime n he operion ommene? ii How long hs he surgeon o omplee he operion?
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 233 A surge moel ppers mos suile. E E The eponenil moel is E gins E ¼ 804:5e 3:147 We remove he poin (0, 0) s Pp is unefine. The surge moel is E ¼ 804:5e 3:147 E = 804:5e 3:147 + 804:5( 3:147e 3:147 ) fprou ruleg = 804:5e 3:147 (1 3:147) Now, he hemil is mos effeive when E =0 ) = 1 ¼ 0:3178 hours ¼ 19 minues 3:147 e i When E =60 for he firs ime, ¼ 0:103 hours ¼ 6 min 11 s. So, he operion n sr 6 minues n 60 11 seons fer he injeion is given. ii When E =60 for he seon ime, ¼ 0:721 hours ¼ 43 min 16 s. So, he surgeon hs 43 6 = 37 minues o omplee he operion. TERMINAL VELOCITY MODELS If = A(1 e )=A Ae, hen A = Ae, whih is n eponenil moel. So, if we suspe n follow erminl veloi moel, we n plo A gins (where A is he limiing vlue of he ), n hek wheher n eponenil moel is goo fi. Emple 7 A mel sphere of mss 1 kg is roppe ino eep reservoir. Is spee is mesure vrious imes s follows: (s) 0 0:2 0:4 0:6 0:8 1 1:5 2 S (m s 1 ) 0 1:2864 2:0212 2:4409 2:6807 2:8178 2:9551 2:9890
234 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) Drw ser plo of he n sugges suile moel for i. Is here vlue whih S is pprohing for lrger vlues of? Le his vlue e A. Drw ser plo of_ A S _ gins n fi n eponenil moel o he. Rewrie he moel in he form S = A(1 e ). e Use he moel o fin he spee of he sphere fer 0:5 seons. f The re of hnge of spee wih respe o ime is elerion. Wh is he elerion fer 1 seon? Yes, S is pprohing 3, so A =3. A erminl veloi moel ppers suile. A-S gins S A-S The eponenil moel is A S ¼ 3e 2:804 Sine A =3, he moel is 3 S ¼ 3e 2:804 ) S ¼ 3 3e 2:804 ) S ¼ 3(1 e 2:804 ) e When =0:5, S ¼ 3(1 e 2:804(0:5) ) ¼ 2:26. So, fer 0:5 seons, he spee of he sphere is 2:26 ms 1. f Using ehnolog, when =1, S ¼ 0:510. So, fer 1 seon, he elerion is 0:510 ms 2. EXERCISE 6F 1 The following le gives he verge pin relief effeiveness (APRE) of 500 mg spirin le over 12-hour perio. Time ( hours) 0 0:5 1 2 3 4 6 9 12 APRE (P unis) 0 2:9 3:6 5:0 5:5 5:2 3:9 2:5 1:1 Oin ser plo of he n se he moel pe whih es fis he.
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 235 Plo P gins, n fin he eponenil moel for his. Rewrie he moel in he form P = Ae. Wh is he APRE ime i =5 ii =10? e A wh ime is here mimum pin relief? f A wh ime oes he re of hnge of APRE sr o inrese? g If he les proue eple pin relief when he APRE level is 2 or more unis, eween wh imes oes he le proue eple relief? 2 A phrmeuil ompn evelops new pin-killing hemil whih is suppose o e n improvemen on he one illusre in Emple 6. The effe of he new hemil over ime is given in he le elow: Time ( hours) 0:00 0:10 0:25 0:50 0:75 1:00 1:25 1:50 1:75 2:00 Effe (E unis) 0 59 90 93 64 46 22 8 4 1 Oin ser plo of he n sugges moel whih fis i. Plo E gins using ehnolog n fi n eponenil moel o his. Rewrie he moel in he form E = Ae. Fin E, n hene eermine when he hemil is mos effeive. e The operion n onl ke ple when he effe is les 60 unis. i A wh ime n he operion ommene? ii How long hs he surgeon o rr ou he operion? f Commen on he ifferenes eween his rug n h of Emple 6. 3 The owner of ompuer sofwre shop verises new informion gme lle D FACTO. Ineres in he new prou from elephone lls n shop enquiries is reore on weekl sis, long wih he umulive sles of he prou. The on ineres n umulive sles ws ule n is shown elow. Time ( weeks) 0 1 2 3 4 5 6 7 8 Ineres (I people) 0 42 67 83 58 32 15 7 4 Time ( weeks) 1 2 3 4 5 6 7 8 Cumulive sles (s iems) 8 34 67 92 101 108 111 112 For he ineres : i rw ser plo, n sugges moel whih losel fis he ii fin he moel of es fi iii use he moel o esime he of highes ineres iv esime he ineres in week 9: For he umulive sles : i rw ser plo, n sugges he mos likel moel pe ii fin he moel of es fi iii eermine he week in whih sles were me he grees re.
236 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 4 A somh virus spres hroughou high shool. The numer of suens N ffee fer s is shown on he ser plo longsie. µ N When ln is ploe gins, he is liner, n he equion of he line of es fi µ N is ln = 0:351 + 3:92. 60 N 50 40 30 20 10 0 0 5 10 15 20 Fin he equion onneing N n, in he form N = Ae. Hene, fin he surge moel onneing N n. How mn suens were ffee fer one week? Fin N, n hene eermine he re whih N is inresing when =2. 5 A skiver jumps from smll eroplne n her spee is reore erin imes. Time ( seons) 0 2 4 7 10 15 20 30 40 50 Spee (S km h 1 ) 0 42:7 76:2 113:7 139:8 166:9 181:9 194:5 198:4 199:5 Drw ser plo of he n sugges moel whih es fis i. Is here vlue whih S is pprohing s ges lrger? Cll i A. Drw ser plo of A S gins, n fi n eponenil moel o he. Rewrie he moel in he form S = A(1 e ). e Use he moel o fin he spee of he skiver 6 seons fer she jumpe from he eroplne. f Wh is her elerion fer 6 seons of free-fll? 6 The urren in irui is shown in he following le: Time ( milliseons) 0 40 120 360 740 1240 2500 3250 3800 Curren (I mps) 0 6:15 17:07 41:06 61:79 73:30 79:46 79:88 79:96 Drw ser plo of he n sugges suile moel. Is here vlue whih I is pprohing s ges lrger? Cll i A. Drw ser plo of A I gins, n fi n eponenil moel o he. Rewrie he moel in he form I = A(1 e ). e Use he moel o fin he urren in he irui when: i = 500 milliseons ii = 8000 milliseons.
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 237 REVIEW SET 6A 1 For eh of he following grphs, fin: i he posiion n nure of n urning poins ii he inervls where he grph is inresing or eresing iii he posiion of n poins of infleion iv he inervls where he grph is onve or onve v he equions of n smpoes. (1, 4 Qw ) (-1, 1) 4 (-2, 3) (- Qw, 3 ) Qw (2, 0) (-4, 2) (-2, -Qw_) 2 The sopping isne of r is he isne rvelle in he ime eween he river ppling he rke, n he r oming o omplee hl. A es ws rrie ou o fin he sopping isnes of r vrious spees. The resuls were: Veloi (v km h 1 ) 40 60 70 80 90 110 120 Sopping isne (s meres) 29 55 67 96 112 168 223 Drw ser plo for he. Fin he equion of he ui moel of es fi. Use he moel o esime he sopping isne : i 100 km h 1 ii 65 km h 1. Fin s v for v > 0. Hene fin he inervls where he urve is: i inresing or eresing ii onve or onve. 3 Dimons n e snheill proue in speill equippe lorories. Unforunel, wih urren ehnolog, i is slow n epensive proess. The oss of prouing imons of vrious hiknesses re given elow. (-4, 1) 1Qw =-1 =-3 4 =2 Thikness (T oms) 100 200 300 400 500 1000 2000 Cos (C ollrs) 27:9 84:5 162 256 366 1109 3363 Drw ser plo for his n sugges possile moel. Fin he moel whih es fis he. Use his moel o prei he os of mking imon: i 250 oms hik ii 1500 oms hik iii 5000 oms hik. Fin C, n eermine he re whih he os is inresing for hiknesses of: T i 200 oms ii 2200 oms. 4 The roon hnnel of P-TV nework regulrl onus surves of is susriers o fin ou whih roons re populr n whih re no. I hen uses he resuls o eie whih roons o show ne seson. The populri of he roon Poker Mn,
238 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) roon ou poke size monsers who oul e ugh n rine o pl poker, re s follows: De of surve Jn 2008 June 2008 Jn 2009 April 2009 June 2009 Viewers (V ) 0 17 524 21 930 19 464 17 976 De of surve Sep 2009 Jn 2010 June 2010 De 2010 Viewers (V ) 15 026 11 890 7921 4929 B leing e he numer of monhs fer Jnur 2008 ( =0 for Jnur 2008, =5 for June 2008, n so on) onsru ser plo of V gins. Sugges possile moel pe for his. Fin he moel of es fi. Use he moel o prei he numer of viewers: i when =8 ii in Sepemer 2010. Fin V, n hene fin he monh in whih Poker Mn ws mos populr. e Fin V when =8, n inerpre our nswer. f If mngemen eies o srp ll shows whose populri rops elow 1000 viewers, when will Poker Mn e nelle? 5 Vivin runs smll grening sore, speilising in gren gnomes. She purhses ll of her gnomes from Muhlk n Dngerfiel Gnomes of Disinion, who sell o her isoun re whih vries oring o he size of her orer. The os of n orer of g gnomes is given C = 609 ln g 672 ollrs. Fin he os of orering 25 gnomes. Fin he numer of gnomes Vivin mus orer for he ol os o e $1500. Fin C, n show h he os is lws inresing. Wh n e si ou he g re of inrese? REVIEW SET 6B 1 For eh of he following grphs, fin: i he posiion n nure of n urning poins ii he inervls where he grph is inresing or eresing iii he posiion of n poins of infleion iv he inervls where he grph is onve or onve v he equions of n smpoes. =5-1 1 =2 (-2, Qw ) (-1, - Qw ) (2, 4) (3, 3) ( Qw, 2) 2 =1
APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 239 2 A group of onservioniss, primril onerne wih he welfre of n enngere ree of lemming, lerne of n isln off he Souh Ausrlin os wih n unn foo suppl for lemmings n ver few liffs. The inroue few pirs of lemmings o he isln n hen reore he populion erl inervls. The populion of lemmings ers fer he progrm egn is moelle 2570 P () = 1+84e 0:9. Esime how mn pirs of lemmings were originll inroue o he isln. Show h he re of hnge in he lemming populion is given P = 194 292e 0:9 (1 + 84e 0:9 ) 2. Fin he lemming populion n he re whih i ws hnging fer 30 monhs. Will he lemming populion inrese inefiniel, or will i pproh some limiing vlue? e Fin he vlue of whih he re of growh of he populion is mimum, n fin his mimum re of growh. 3 Tehnolog hs mn ppliions in mhemis. One of hese ppliions is he simulion of projeile moion. For emple, n ire uhor hrew her mlfunioning ompuer off he op of ll uiling n hen reore is veloi vrious imes on is w o he groun. The veloi of he ompuer seons fer i ws hrown is given V = 50(1 e 0:35 ) ms 1. Fin he veloi of he ompuer: i s i psse 10h sore winow, 6:6 seons fer eing hrown ii jus efore i hi he groun, fer 8:3 seons. Fin V V. Evlue when =4. Inerpre our nswer. Does he veloi pproh mimum vlue? If so, wh is i? 4 Kh n Krin run he ompuer sore where he uhor from 3 ough her replemen ompuer. Kh is he slesperson n Krin onsrus n elivers he ompuers. Beween hem, he re le o sell n isph up o 150 ompuers per week. Their profi figures for he ls eigh weeks re: Compuers sol () 43 115 0 101 82 19 144 62 132 Profi ($P ) 2790 3250 700 2820 2710 1260 5180 3090 4230 Drw ser plo for he. Whih moel pe will es fi he? Fin he moel of es fi. Use he moel o prei he profi me selling 90 ompuers. How mn ompuers nee o e sol o mke profi of $5000 per week? e Fin P. f Sugges how mn ompuers Kh n Krin shoul sell per week o mimise heir profi. Wh is his mimum weekl profi? Hin: Rememer h he n onl proue 150 ompuers per week.
240 APPLYING CALCULUS TO FURTHER MODELLING (Chper 6) 5 The populion of zers in her fer ers is given Z() = 80 + 35 ln(2 +1), > 0. Skeh he grph of Z() gins. Fin: i he iniil populion ii he populion fer 5 ers. i Fin Z 0 (). ii Hene, eermine he re whih he zer populion is inresing fer 2 ers. 6 A smple of rioive meril ws suie over 20 er perio. The mss M of he rioive meril remining ws reore over h ime: Time ( ers) 0:5 1 1:5 2 5 10 15 20 Mss (M grms) 48:0 46:1 44:1 42:4 32:7 21:2 14:7 9:7 Consru ser plo for his. The fis eiher logrihmi, power, or n eponenil moel. i Fin he moel of es fi for eh of hese moel pes. Inlue he r 2 vlues. ii Whih moel is mos pproprie? Eplin our nswer. Use our moel o lule he mss of rioive meril: i when he smple ws ken ii fer 7 ers. The hlf-life of rioive susne is he ime ken for he mss of he susne o hlve. Fin he hlf-life for his meril. e Fin he re of e fer: i 1 er ii 10 ers. f Is he re of e inresing or eresing?