46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r)
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- Ethelbert Mills
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1 98 Scion 7.. L w. Thn dw d, so d dw w dw. sin d (sin w)( wdw) w sin w dw L u w dv sin w dw du dw v cos w w sin w dw w cos w + cos w dw w cos w+ sin w+ sin d wsin wdw w cos w+ sin w+ cos + sin +. L w + 9. Thn dw () d, + 9 so d + 9 dw w dw. + 9 w ( ) w dw w dw w dw w L u w dv dw w du dw v w w w w dw w dw w w w w ( w ) + 9 w d w dw ( ) w w ( ) w w dw w w w w w w + 6w 6 + w ( w w + 6w 6) + 7 w d w dw ( 6 6) w w w + w + 6 ( + 6 6) + 6. L ln r. Thn dr, and so r dr r. Using h rsul of Ercis 7, w hav: sin (ln r) dr (sin ) (sin cos ) + ln r [ sin (ln r) cos (ln r) ] + r [ sin (ln r) cos (ln r) ] + 5. L w. Thn dw d. 7 d ( ) d. w w dw Us abular ingraion wih f () w and gw ( ) w. 7. L u n dv cos d n du n d v sin n n n cos d sin (sin )( n d) n n sin n sin d 8. L n u dv sin d n du n d v cos n sin d n n ( )( cos ) ( cos )( n ) d n n cos + n cos d oprigh arson Educaion, Inc. ublishing as rnic Hall.
2 Scion n a 9. L u dv d n a du n d v a n a d n a a n ( ) ( ) a a n d n a n n a, a a d a n 5. L u (ln ) dv d n n(ln ) du d v n n(ln ) n n (ln ) d (ln ) ( ) d n n (ln ) n (ln ) d 5. (a) L 5. f ( ). Thn f(), so d f ( ). Hnc, f ( ) d ( )[ f ( ) ] f ( ) L u dv f ( ) du v f ( ) f ( ) f( ) f( ) f ( )( ) f( ) Hnc, f ( ) d f ( ) f ( ) f( ). L u f ( ) dv d d du f ( ) d d v d f ( ) d f ( ) f ( ) d d 5. (a) Using f ( ) sin and π f() sin, w hav: π, sin d sin sin sin + cos + sin + cos (sin ) + (c) sin 5. (a) Using (c) d d sin sin d d sin d u, du d sin + u du sin + u + sin + + cos (sin ) f ( ) an and π f() an, w hav: π < <, an d an an an ln sc + an + ln cos + an + ln cos (an ) + an d d d an an d an d + u +, du d an u du an ln u + an ln (+ ) + ln cos (an ) ln + ln (+ ) oprigh arson Educaion, Inc. ublishing as rnic Hall.
3 Scion (a) (c) Using f ( ) cos and f() cos, π, w hav: cos d cos cos cos sin + cos sin (cos ) + cos d d cos cos d d cos d u, du d cos u du cos u + cos + sin (cos ) 56. (a) Using f ( ) log and f( ), w hav log d log log + ln log log ln d log d log log d d log d ln d log ln log + ln (c) log 57. L u sc dv sc d du sc an d v an sc d sc an sc an d sc an sc (sc ) d sc an sc d+ sc d sc d sc an sc d+ ln sc + an Add sc d o boh sids. sc d sc an + ln sc + an Mulipl boh sids b. sc d ( sc an + ln sc + an ) L u csc dv csc d du csc co d v co csc d csc co csc co d csc co csc (csc ) d csc co csc d+ csc d csc d csc co csc d ln csc + co Add csc d o boh sids. csc d csc co ln csc + co Mulipl boh sids b. csc d ( csc co + ln csc + co ) + Quic Quiz Scions E oprigh arson Educaion, Inc. ublishing as rnic Hall.
4 Scion 7.. ; sin ; sin ; d sin cos sin π sin / d π / sin sin cos sin π / sin Scion 7. Eponnial Growh and Dca (pp. 5 65) Eploraion hoosing a onvnin Bas. 5 h i 5h 5h h 5, h is h rciprocal of h doubling priod.. A; d dv d v d u du d d +. (a) L and + b in h d diffrnial quaion: ( + b) b b (c) Firs, no ha () () a h d poin (, ). Also, d d ( ), d d d which is a h poin (, ). B h Scond Drivaiv s, g has a local maimum a (, ) log log 5 5log 795. ars. log h h h, h is h rciprocal of h ripling priod. log log log 6. 9 ars. log 5h hi5 h 5 ; h is h rciprocal of h half-lif. 5. log(. ) log 5 5log(. ) 9. 8 ars. log ( ) Quic Rviw 7.. b a oprigh arson Educaion, Inc. ublishing as rnic Hall.
5 Scion 7.. c ln d.. ln( + ) ln 6 ln ln 85. ln 5. ln 85. ln 5. ln 5. ln ln ln ( + )ln ln ln (ln ln ) ln ln ln 7.. ln. ln ln. ln ln ln. log ln ln ln ln ln ( + ) + + ln ± ± Scion 7. Erciss. d + () ( ) + +, valid for all ral numbrs. d.. + () ( ) + 5 5, valid on h inrval ( 5, 5) d ln ln + + +, valid on h inrval (, ) d ln + + A A A, valid for all ral numbrs 5. ( + ) d + 5 ln / + + / / + / ± A / + A 5 / + 6 5, valid for all ral numbrs. oprigh arson Educaion, Inc. ublishing as rnic Hall.
6 d cos an + an () + an, valid for all ral numbrs. sin cos d sin cos d sin sin + + sin ln ( ), valid for all ral numbrs. d d + ln ( + ), valid for all ral numbrs. d , valid for all ral numbrs. ln d ln u ln du d u du d u + (ln ) + (ln ) + (ln ), valid on h inrval (, ).. () () 5.. () 5. (). () () 5 5 () ln 5. ln Soluion: () 5 (. ln ) or () 5.. () () 6 ( ) 6 ln. ln. ln Soluion: () 6 (. ln ) or () 6 / 5. Doubling im: r A () A ln. 86 ln 86. r.86 Amoun in ars: (. 86)( ) A $, Annual ra: r A () A ()( r 5) 5r ln 5r ln r. 6. 6% 5 Scion 7. oprigh arson Educaion, Inc. ublishing as rnic Hall.
7 Scion 7. Amoun in ars: r A () A [(ln )/ 5]( ) A ln $ 8 7. Iniial dposi: r A () A ( 55. )( ) 898. A 898. A $ Doubling im: r A () A ln. 55 ln. ars Annual ra: r A () A ( r)( ), r. 57 ln r. 57 r ln % Doubling im: r A () A ln. 7 ln 96. ars.7 9. (a) Annuall:. 75 ln ln.75 ln. 9 ars ln.75 Monhl: ln ln + ln. 6 ars ln 75 ( +. ) (c) Quarrl: ln ln.875 ln. 68 ars ln.875 (d) oninuousl:.75 ln. 75 ln. 59 ars 75.. (a) Annuall:. 85 ln ln 85. ln 87. ars ln. 85 Monhl: ln ln + ln 8. ars ln 85 ( +. ) (c) Quarrl: ln ln.65 ln 89. ars ln.65 (d) oninuousl:.85 ln.85 ln 8. ars.85 oprigh arson Educaion, Inc. ublishing as rnic Hall.
8 Scion d 77. d ln 77. ln ( ) 9 ars. 77. d ( ) d ln ln ( ) (a) Sinc hr ar 8 half-hour doubling ims in hours, hr will b 8. 8 bacria. Th bacria rproduc fas nough ha vn if man ar dsrod hr ar sill nough lf o ma h prson sic.. Using, w hav 5., 5 5, and,. Hnc, which givs, or ln. Solving ln,, w hav 5. Thr wr 5 bacria iniiall. W could solv his mor quicl b noicing ha h populaion incrasd b a facor of, i.., doubld wic, in hrs, so h doubling im is hr. Thus in hrs h populaion would hav doubld ims, so h iniial populaion was, ln ln da (a) Half-lif ln ln 8. 6 das 5., ln.5.5 ln das.5 Th sampl will b usful for abou 599 das. 7. Sinc (), w hav: ( )( ) 5 ln 5 ln + ln 5 ln 5. ln.5 Funcion: or (.5 ln.5) Sinc ()., w hav:. ( )( ). ln ln. (ln. ln) Funcion:. (ln. ln ) /. or. 9. A im, h amoun rmaining is ( / ).. This 98 is lss han 5% of h original amoun, which mans ha ovr 95% has dcad alra.. T Ts ( T Ts) ( )( ) 5 65 ( T 65) ( )( ) 5 65 ( T 65) Dividing h firs quaion b h scond, w hav: ln Subsiuing bac ino h firs quaion, w hav: [(ln )/ ]( ) ( T 65) ( T 65) 6 T 65 5 T Th bam s iniial mpraur is 5 F. oprigh arson Educaion, Inc. ublishing as rnic Hall.
9 6 Scion 7.. (a) Firs, w find h valu of. T Ts ( T Ts) ( )( ) 6 ( 9 ) 7 ln 7 Whn h soup cools o 5, w hav: [( / ) ln (/7)] 5 ( 9 ) [( / ) ln (/7)] 5 7 ln ln 7 ln ( ) 7. 5 min ln ( ) 7 I as a oal of abou 7.5 minus, which is an addiional 7.5 minus afr h firs minus. Using h sam valu of as in par (a), w hav: T Ts ( T Ts) [( / ) ln (/7)] 5 ( 5) [ 9 ( 5)] [( / ) ln (/7)] 5 5 ln ln 7 ln. 6 ln 7 I as abou.6 minu. Firs, w find h valu of. Taing righ now as, 6 abov room mpraur manst T s 6. Thus, w hav T Ts ( T Ts) ( )( ) ln 6 (a) T Ts ( T Ts) ( ( / )ln( 76 / ))( 5) I will b abou 5.5 abov room mpraur. T Ts ( T Ts) ( ( / ) ln ( 7/ 6))( ) I will b abou.79 abov room mpraur. (c) s ( ( / )ln(7/6)) T Ts ( T T ) 6 7 ln ln 6 6 ln. 7 min ln ( 6 ) 7 ( ) 6 I will a abou.7 min or.9 hr.. (a) T T 79. 7(. 9) s T (. 9) (c) Solving T and using h ac valus from h rgrssion quaion, w obain 5. 5 sc. (d) Subsiuing ino h quaion w found in par, h mpraur was approimal (a) Nwon s Law of ooling prdics ha h diffrnc bwn h prob mpraur (T ) and h surrounding mpraur ( T s ) is an ponnial funcion of im, bu in his cas T s, so T is an ponnial funcion of im. T [, ] b [, 86] (c) A abou 7 sconds. (d) oprigh arson Educaion, Inc. ublishing as rnic Hall.
10 Scion 7. 7 ln 5. Us (s Eampl 5) ln 5. ln.5 57 ln ars ln rar La is abou 6658 ars old. ln 6. Us (s Eampl 5) (a) 7. ln 7. ln.7 57ln. 7, 57 ars ln Th animal did abou,57 ars bfor A.D., in,57 B..E. 8. ln 8. ln.8 57 ln. 8, ars ln Th animal did abou, ars bfor A.D., in, B..E. (c) 6. ln 6. ln 6. 57ln. 6 5, 7 ars ln Th animal did abou 5,7 ars bfor A.D., in,7 B..E. ( ) ln. 5 ln ( ). 5. ( ) ln. 5 ars. 8. r ln( ) r.. ln( ). 6 ars. 9. ( )( ) 8 8. ln 8. A + h: ( ln 8. / ). ln g Abou 585. g will rmain... (a).. ln.. ln r I will a abou 6.9 ars. dp p dh dp dh p dp dh p ln p h + ln p h+ h p h p A Iniial condiion: p p whn h p A A p h Soluion: p p Using h givn aliud-prssur daa, w hav p millibars, so: h p 9 9 ( )( ) 9 ln. m Thus, w hav p. h. oprigh arson Educaion, Inc. ublishing as rnic Hall.
11 8 Scion 7. A 5 m, h prssur is (( / ) ln ( 9/ ) )( 5). 8 millibars. (c) h 9 9 h 9 h ln ln. 977 m ln 9 ( ) 9 ( ) Th prssur is 9 millibars a an aliud of abou.977 m.. B h Law of Eponnial hang, 6.. A hour, h amoun 6. ( ) rmaining will b grams.. (a) B h Law of Eponnial hang, h ( / soluion is ) V V. (/ ). ln. ln. 9. sc I will a abou 9. sconds.. (a) A() A I grows b a facor of ach ar. 5. (a) ln I will a ln. r. (c) In on ar our accoun grows from A o A, so ou can arn A A, or ( ) ims our iniial amoun. This rprsns an incras of abou 7%. ()( r ) 9 ln 9 r ln 9 r. 5 or. 5% ()( r ) ln r ln r. 9 or. 9% r 6. (a) r ln r ln r (c) ln 69., so h doubling im is 69. r which is almos h sam as h ruls. (d) 7 7 ars or. ars 5 5 r () r ln r ln r Sinc ln.99, a suiabl rul is 8 8 or. r i (W choos 8 insad of bcaus 8 has mor facors.) 7. Fals; h corrc soluion is +, which can b wrin (wih a nw ) as. 8. Tru; h diffrnial quaion is solvd b an ponnial quaion ha can b wrin in an bas. No ha ( ) whn ln. r 9. D; A() A 7r ln( ) r ln( ). 99. oprigh arson Educaion, Inc. ublishing as rnic Hall.
12 Scion 7. 9 r / 5. ; A A 99/ r 99 ln (. ) ln(. 5) r 99ln(. 5) r ln(. ) 5. D 5. E; T 68 ( 5 68) ln( 56. ). (. ) ln 57 7 min. 7 dv 5. (a) Sinc acclraion is, d dv Forc m v. d w hav dv dv From m v w g v, d d m which is h diffrnial quaion for ponnial growh modld b ( / m ) v. Sinc v v a, i follows ha v. 5. (a) 55. ( m / ) vm ( m / ) s () v d + Iniial condiion: s () vm + vm vm ( / m) vm s () + vm ( / m) ( ) vm ( / m) vm lim s ( ) lim ( ) vm coasing disanc ( 8. )( 99. ). 998 vm W now ha. and 998 m ( 9. 9). W hav: vm ( / m) s () ( ) /. ( ) 66.. ( ) A graph of h modl is shown suprimposd on a graph of h daa. (c) In ach cas, w would solv ( m / ). If is consan, an incras in m would rquir an incras in. Th objc of largr mass as longr o slow down. Alrnaivl, on can considr h dv quaion v o s ha v changs d m mor slowl for largr valus of m. 56. vm coasing disanc ( 86. )( 8. ) vm ( / m) s () ( ) ( 7. /. 8) s () 97. ( ) s () 97. ( ) A graph of h modl is shown suprimposd on a graph of h daa. oprigh arson Educaion, Inc. ublishing as rnic Hall.
13 Scion 7. basd on an insananous ra of chang which is a limi of avrag ras as h incrmn in im approachs. [, ] b [, ] 57. (a) ( + ) ,.78, r ( + ) , 7.876, r ,.687, ( + 5. ) (c) As w compound mor ims, h incrmn of im bwn compounding approachs. oninuous compounding is 58. (a) To simplif calculaions somwha, w ma wri: a a a mg ( ) v () a a a ( + ) a mg a + a mg ( + ) a + mg a + Th lf sid of h diffrnial quaion is: dv mg a a m m ( )( + ) ( a ) d mg a a ma ( + ) ( ) g mg a a m ( + ) ( ) m a mg a ( + ) Th righ sid of h diffrnial quaion is: mg v mg mg a + mg a + mg + a a ( ) + + a ( + ) mg a ( + ) a mg a ( + ) Sinc h lf and righ sids ar qual, h diffrnial quaion is saisfid. mg And () v, so h + iniial condiion is also saisfid. oprigh arson Educaion, Inc. ublishing as rnic Hall.
14 Scion 7.5 mg a a a lim v ( ) lim i a a a + a mg lim + a mg + mg (c) mg Th limiing vloci is. mg 6 79 f/sc 5. Th limiing vloci is abou 79 f/sc, or abou mi/hr. Scion 7.5 Logisic Growh (pp ) Eploraion Eponnial Growh Rvisid.. ( ) 9, 6 ( ) 88 ( ). 97. No; his numbr is much largr han h simad numbr of aoms.. 5, ( ) log5. 9 hours log 5. 5, Eploraion Larning From h Diffrnial Equaion. will b clos o zro whn is clos o d and whn is clos o M.. is half h valu of M a is vr. M. Whn, ( M d ) is a is maimum.. Whn h iniial populaion is lss han M, h iniial growh ra is posiiv. 5. Whn h iniial populaion is mor han M, h iniial growh ra is ngaiv. 6. Whn h iniial populaion is qual o M, h growh ra is. 7. lim ( ) M, rgardlss of h iniial populaion. Th limi dpnds onl on M. Quic Rviw (, ) oprigh arson Educaion, Inc. ublishing as rnic Hall.
15 Scion lim () 7. As,., and 8. 6 lim ( ) + 5 (. ( ))., so 9. From problms 6 and 7, h wo horizonal asmpos ar and Scion 7.5 Erciss. A ( ) + B ( ), B 8 B, A( ) + ( )( ) A. A ( ) + B ( + ) + 6, B( + ) ( ) + 6 5B B, A( ) ( ) + 6 5A A. A( + 5) + B( ) 6 5, B( 5 ) 6 ( 5) 7B B, A( + 5) 6 7A A. A ( + ) + B ( ), B( ) 6B B, A( + ) 6A A 5. S problm. d + d ln ln + ln + ( ) 6. S problm. + 6 d d + ln + + ln + ( ) ln + ( + ) d u du d du + + ln u + u + ln( ) A B + d d A( ) + B( + ) oprigh arson Educaion, Inc. ublishing as rnic Hall.
16 Scion , B( + ) B, A( ) A + + d + ln d an + + d an d A B ( + )( ) A ( ) + B( + ) 7, B( ( ) + ) 7 B, A 7 A ln d d 5 A B + ( )( ) A ( ) + B( ), B( () ) () B, A A A + d ln ln A B ( + )( ) A( ) + B( + ) 8 7, 8 B B 5 B, A( ) 8 7 A( ) 8 7 5A 5 A + d + ln + + ln + ( ) ln d + 7 A B ( + 7) A ( + 7) + B 5+ 7, 7B 5( 7) + 7B B, A( + 7) 5( ) + 7A A + d ln + ln ln d A B 6 + ( ) A ( ) + B 6, B ( ) 6 B B ( ) ln + ( ) oprigh arson Educaion, Inc. ublishing as rnic Hall.
17 Scion , A( ) ( ) 6 A 6 A + d ln ln + ln + du d A B + + ( + )( ) A ( ) + B ( + ), B( + ) B B, A( ) A A u + d + u ln + + ln + u ln + + F ( ) d d A B ( + )( ) A ( + )( ) + B ( ) + ( + ),, B B, A A + + d + F ( ) ln + ln + + ln + F ( ) ln G () d d + d + d A B + + ( )( + ) A ( + ) + B ( ), B( ) B B, A( + ) A A G () + + d ( ) ( + ) + ln ln ln + d u du d du ln u+ ln + u d + u + du ( ) d du ln u ln u oprigh arson Educaion, Inc. ublishing as rnic Hall.
18 Scion d + + d u du ( ) d du + ln u+ + ln + u d + d u du d du + + ln u+ + ln + u. (a) individuals individuals (c) ( ). 6( )( ) d 6 individuals pr ar.. (a) 7 individuals 6. (a) 5 individuals 5 individuals (c) ( 5) 5 ( 5)( 5 5) d 6. 5 individuals pr ar d ( ) ( ). 6 d A B + ( ) A( ) + B, B B 5., A( ) A A ln ln ( ). + ln.. c. c +. ( ) + 8 c. + c 5 individuals (c) ( 5). 8( 5)( 7 5) d 98 individuals pr ar. [, 7] b [, ] 5. (a) individuals 6 individuals (c) ( 6). ( 6)( 6) d 7 individuals pr ar. oprigh arson Educaion, Inc. ublishing as rnic Hall.
![d A B + ( ) A + B, B B c, A( ) A A +. 7 +. ln ln. + + 59 ( ) ln.. c. c +. ( ) + c 59. [, ] b [, ] d 5.](/docs-images/81/84353185/images/19-1.jpg "( 5 ) ( 5 ) 5 d A B + 5 ( 5 ) A( 5 ) + B 5, 5B B., A( 5 ) 5A A... 5 + 5 + 5. 5 c oprigh arson Educaion, Inc.")
19 6 Scion ( 7 ) d 8. d ( 7 ) 9. A B + 7 ( 7 ) A( 7 ) + B 7, 7B B 7, A( 7 ) A ln ln( 7 ) ln c c ( ) + c [, 5] b [, 7] ( ) ( ). ( ) d. d A B + ( ) A + B, B B c, A( ) A A ln ln ( ) ln.. c. c +. ( ) + c 59. [, ] b [, ] d 5. ( 5 ) ( 5 ) 5 d A B + 5 ( 5 ) A( 5 ) + B 5, 5B B., A( 5 ) 5A A c oprigh arson Educaion, Inc. ublishing as rnic Hall.
![. (a). (a) ( ) ln ln 5. 5 + 5 ln 5. 5 5. c 5 5. c + 5 5. ( ) c + 5 c 99 5 5. + 99 [, ] b [, 5] () 8 7 +.. 8. 7. + M M + A This is a logisic growh modl wih 7. M and. 7. ( ) 8 8.](/docs-images/81/84353185/images/20-0.jpg "+ Iniiall hr ar 8 rabbis. () 5 +. 5. + M M + A This is a logisic growh modl wih M and 5.. ( ) 5. + Iniiall sudn has h masls.. (a). 5( 5 ) d ( M ) Thus,.5 and M 5. M 5 M 5.")
20 . (a). (a) ( ) ln ln ln c 5 5. c ( ) c + 5 c [, ] b [, 5] () M M + A This is a logisic growh modl wih 7. M and. 7. ( ) Iniiall hr ar 8 rabbis. () M M + A This is a logisic growh modl wih M and 5.. ( ) 5. + Iniiall sudn has h masls.. (a). 5( 5 ) d ( M ) Thus,.5 and M 5. M 5 M 5. + A + A Iniial condiion: () A + A 5 A 5 Formula:. 5 + Scion ln 8 ln 8 7. ws ln ln I will a abou 7. ws o rach guppis, and abou.8 ws o rach 5 guppis.. (a). ( 5 ) ( M ) d Thus,. and M 5. M 5 M. + A + A Iniial condiion: () 8, whr rprsns h ar 97. oprigh arson Educaion, Inc. ublishing as rnic Hall.
21 8 Scion A 8( + A) 5 5 A Formula: 5 (), or approimal +. / 5 () Th populaion () will round o 5 whn () / ( 9. 5)( ) , 89. ln 55, 89 (ln 55, 89 ln ) 8. 8 I will a abou 8 ars. 5. ( M ) d d M ( ) Q R + M ( M ) QM ( ) + R, MQ Q M M, MR R M M M + + ( M ) + M + ( M ) M ln M M M M M + A M M + A 6. (a) M ( ) d d M ln( M ) + c M c L A hn M A. lim ( ) M A M (c) Whn. (d) This curv has no inflcion poin. If h iniial populaion is grar han M, h curv is alwas concav up and approachs M asmpoicall from abov. If h iniial populaion is smallr han M, h curv is alwas concav down and approachs M asmpoicall from blow. 7. (a) Th rgrssion quaion is [ 5, 7] b [, 6], 79. 9, lim , 7 popl. oprigh arson Educaion, Inc. ublishing as rnic Hall.
22 Scion (c) (d), , , , or in.. ( M ) d 7 (. 5 ) (, ). 8. (a) Th rgrssion quaion is , , lim. ( ) , 79. Tru; h graph will b a logisic curv wih lim ( ) and lim ( ).. D; 6.. B; M., 9 so a mos 9% of h populaion will b infcd. Th rmaining % will no b infcd.. D; ( )( + ) d A B + + ( )( + ) A ( + ) + B ( ), B( ) B B, A( + ) A A 8 d ln + ln (c) (d) 58, , , or in.. ( M ) d 7 ( 66. ) ( ). B 5. (a) No ha > and M >, so h sign of is h sam as h sign of d ( M )( m). For m < < M, boh M and m ar posiiv, so h produc is posiiv. For < mor > M, h prssions M and m hav opposi signs, so h produc is ngaiv. 9. Fals; i dos loo ponnial, bu i rsmbls h soluion o ( ) ( 9 ). d oprigh arson Educaion, Inc. ublishing as rnic Hall.
23 Scion 7.5 ( M )( m) d M ( )( d ) ( )( ) d ( )( ) d ( ) + ( ) ( )( ) d + d + d ln + ln + ln + / ± / A / / A A / / ( + A ) A + / A + / + A (c) A + + A ( + A) A+ A A 9A A 9 / ( 9 ) + () / + ( 9 ) / ( ) + ( 9) () / 9+ / ( 8 + ) () / 9+ oprigh arson Educaion, Inc. ublishing as rnic Hall.
24 Scion 7.5 (d) [, 75] b [, 5] No ha h slop fild is givn b. ( )( ). d () ( M )( m ) d M M ( M )( m) d M M m M m ( M )( m) d ( m) + ( M ) M m ( M )( m) d M M m + M m d M M m + d M m M M m ln M + ln m + M m M m ln + M M m ( M m) / M ± M m A ( M m) / M M ( M ) m ( M ) A ( M m) / M ( M m) / M ( M m) / M mm / ( + A ) AM + m AM + A AM + m AM + m () + A + A ()( + A) AM + m A ( ( ) M) m ( ) + m ( M m) / M m () () m A () M M () Thrfor, h soluion o h diffrnial quaion is ( M m) / M + () whr ( M m) / M A M AM m m. + A () oprigh arson Educaion, Inc. ublishing as rnic Hall.
25 Scion (a) L u ; hn au, d a du a d a du a + a + a u a du a + u an ( u) + a an + a a L u ; hn au, d a du a d a du a a a u a du a u du a u du a ( u+ )( u ) A B + u+ u ( u+ )( u ) Au ( ) + Bu ( + ) u, A( ) A u, B( ) B du a + u+ u du a u+ u u + ln + a u + ln a + a a + a ln + a a (c) L u a +, du d d du ( a ) + u u + + a + 7. (a) 5 d ( + ) A B ( + ) ( + ) A ( + ) + B 5, B 5, A( + ) 5 5 A() 5 A d ( ) ln d ( + ) A B ( + ) ( + ) ( + ) ( + ) A ( + ) + B ( + ) + 5, 5 A ( + ) + B ( + ) 5 5, 9A + B 5, B 5 + A, 6A + B 5 5, B 5 + A 5 + A A A B 5 + () d ( ) ( ) ( + ) 8. (a) This is ru sinc A B + + ( ) ( ) A( ) + B( ) + ( ) A ( ) + B ( ) , A B+ 5, 9, A+ B+ 5 Thn A B A+ B 6 A A B 5 oprigh arson Educaion, Inc. ublishing as rnic Hall.
26 (c) d ( ) ( ) 5 9 ln + ( ) Quic Quiz Scions 7. and 7.5. ;, 5 5, 5 ln ( ) ln( / ). 87. ;. A; 8, ln( / ) F( ) cos( a ) d F() a F( ) cos( ) d 5 F() 5 cos( ) d. 9 [Us NINT (cos( ),, 5, ) o valua h ingral.] d ( )( + ) A B + + ( )( + ) A ( + ) + B ( ), B B, A A d ln d 5 d ( ) 5 A B + ( ) A( ) + B, B B., A A ln 5 5 / + () + A A. (a) 5 / ( ) Scion 7.5 lim ( ) 5 / ( ). + + A 5 / ( ) ( ) A 5. lim ( ) 5 / ( ). 5 + (c) Spara h variabls. dy d Y 5 lny + 5 /5 / Y whr /5 / Y (d) lim /5 /5 lim lim ( ) / /5 /5 / oprigh arson Educaion, Inc. ublishing as rnic Hall.
27 hapr 7 Rviw hapr 7 Rviw Erciss (pp. 77 8).. π / π / sc θ dθ anθ π an an d + ( ) +. L u +, du d, du d. 6 d 8 ( + ) 8 u du u L u, du d, du d sin( ) d sin udu 5. L u sin, du cos d π / / 5sin cos d / 5u du 5 / 5 u 5 ( ) 6. + / / d ( + ) d ( ) + / ( 6) + ( ) L u an, du sc d π / sc d u du u 8. L u ln r, du dr r 9. ln r dr u du / r / u ( ) d ( + )( + ) A B ( + )( + ) A ( + ) + B ( + ), B( + ) B oprigh arson Educaion, Inc. ublishing as rnic Hall.
28 hapr 7 Rviw 5, A( + ) A A + d ln( + ) ln( + ) ln + 6. d + 6 ( ) A B ( ) A ( ) + B + 6, B () + 6 B, A( ) ( ) + 6 A 6 A + d ln + ln( ) 6ln. L u sin, du cos d, du cos d cos d du sin u ln u + ln sin +. L u +, du d du d d / u du + / u + / ( + ) +.. L u + 5, du d du d d 5 du + u ln u + ln ln( + 5) + L u, du dθ θ θ sc an dθ sc u an u du θ θ θ scu+ sc + θ 5. L u ln, du an (ln ) an u du sin u du cosu L w cos u dw sin u du dw w ln w + ln cosu + ln cos(ln ) + 6. L u, du d sc( ) d sc u du ln sc u+ an u + ln sc( ) + an ( ) + 7. L u ln, du d d du ln u ln u + ln ln + oprigh arson Educaion, Inc. ublishing as rnic Hall.
29 6 hapr 7 Rviw 8. d d / / d / + + sin d cos + sin + sin d [ cos+ sin ] + sin cos + 9. Us abular ingraion wih g() cos... cos d f() and sin + cos 6sin 6cos + L u ln dv d 5 du d v ln d ln 5 5 d 5 ln d ln L u dv sin d du d v cos sin d cos + cos d Ingra b pars again L u dv cos d du 9 d v sin sin d cos + sin 9 sin d.. L u dv d du d v d + d L u dv d du d v + + d + d d 5 d 5 ( + 5)( 5) A B ( + 5)( 5) A ( 5) + B ( + 5) 5 5, B( 5+ 5) 5 B 5 5 B 5, A( 5 5) 5 A 5 5 A ln + d oprigh arson Educaion, Inc. ublishing as rnic Hall.
30 d d + ( )( + ) A B ( )( + ) A ( + ) + B( ) 5+, B( ( ) ) 5( ) + B B, A A A + d + ln + ln + ln ( ) ( + ) d + + d + + d () d + d + d + + d + + () d + d + d + ln + + ( ) ln( ) + ln ( + ) + 8. csc θ co θ dθ csc θ co θ dθ csc θco θ dθ csc θ + π + csc θ + hapr 7 Rviw 7 oprigh arson Educaion, Inc. ublishing as rnic Hall.
31 8 hapr 7 Rviw 9.. d( ) d d( ) d d( ) d + + () + + ( + ) d + ln + + ( + ) d + ln + () ln + dr ( ) cos d dr ( ) cosd dr ( ) cosd r sin + r () r sin dr ( ) ( sin ) d r cos + r () + r cos dr (cos ) d r sin + r() r sin. + d d + d + ln () ( + )( + ) d ( + ) d + ( + ) d + ln ( ) +. ( ) d d ( ) A B + ( ) A( ) + B, B, A + d ln ln + ln c c + A oprigh arson Educaion, Inc. ublishing as rnic Hall.
32 hapr 7 Rviw 9 ( ). + ( ) A A ( ) d d. ( ) A B +.. ( ) A( ) + B(. ), B(. ) B, A A ln ln. + ln.. c. + A ( ) 5 +.() A A sin d+ 5 + d+ oprigh arson Educaion, Inc. ublishing as rnic Hall.
33 hapr 7 Rviw Graph. Slop lins ar vrical for poins on h lin.. Graph (d). Slop lins ar vrical for poins on h lin.. Graph (c). Slop lins ar horizonal for poins on h - and -as. Slops ar posiiv in Quadrans I and III. Slops ar ngaiv in Quadrans II and IV.. Graph (a). Slop lins ar horizonal for poins on h - and -as. Slops ar posiiv in Quadrans II and IV; ngaiv in Quadrans I and III.. (, ) + d d ( +, + ) (, )... (.,.) (.,.)... (.,.) (.,.)... (.,.6.6. (, ) d d ( +, + ) (, )... (.9,.) (.9,.)... (.8,.) (.8,.)... (.7, W s h graph of a funcion whos drivaiv is sin. Graph is incrasing on [ ππ, ], whr sin is posiiv, and oscillas slighl ousid of his inrval. This is h corrc choic, and his can b vrifid b graphing NINT sin,,,. 6. W s h graph of a funcion whos drivaiv is. Sinc > for all, h dsird graph is incrasing for all. Thus, h onl possibili is graph (d), and w ma vrif ha his is corrc b graphing NINT (,,, ). 7. (iv) Th givn graph loos h graph of, which saisfis and (). d oprigh arson Educaion, Inc. ublishing as rnic Hall.
34 hapr 7 Rviw d 8. Ys,, so. d d Sinc (),. d Thn d +. Sinc (),. is a soluion. 9. (a) (a) dv + 6 d dv ( + 6) d v + + Iniial condiion: v whn + v + + v () d ( + + ) d Th paricl movs 6 m. [, ] b [, ] ln Half-lif ln 65. ln Man lif. 859 ars 5. T Ts ( T Ts) T ( ) Us h fac ha T 8 and 5 o find. ( )( 5) 8 ( ) ln 5 7 (( 5 / ) ln ( 9/ 7)) (( 5 / ) ln ( 9/ 7)) T ( ) 7 ( ) (( 5 / ) ln ( 9/ 7)) ln ln ln 6 7 min ln ( 9/ 7) I oo a oal of abou 7 minus o cool from F o 7 F. Thrfor, h im o cool from 8 F o 7 F was abou 9 minus. 5. T Ts ( T Ts) W hav h ssm: 9 Ts ( 6 Ts) Ts ( 6 Ts) 9 Ts Ts Thus, and 6 Ts 6 Ts Sinc ( ), his mans: 9 Ts T 6 T 6 T s s ( 9 Ts) ( Ts)( 6 Ts) 5 78Ts + Ts 58 79Ts + Ts Ts Th rfrigraor mpraur was. 5. S Eampls and 5 in Scion 7.. Us h fac ha h half-lif of - is 57 ars o find : ( 57) ln 57 ln ( ) ln Th paining conains 99.5% of is original arbon-. ln ln ln( 995. ) ln( 995. ). ln Th paining is abou. ars old. s oprigh arson Educaion, Inc. ublishing as rnic Hall.
35 hapr 7 Rviw 55. Sinc 9% of h arbon- has dcad, % rmains. W showd in roblm 5 ha, for ln arbon-,. 57 ln 57. ln ln(. ) ln(. ) 895, ln Th sampl is abou 8,95 ars old. 56. Us ars. r r 6r ln ln r. 5 6 Th ra of apprciaion is abou.5, or 5.%. 57. L L whr rprsns h dph in f and L is h surfac innsi. Whn 8 f, L L, so 8 L L 8 ln 8 ln ( ) ln 8 8 W wan o now h dph a which L L L ln 8 L ln 8. ln ln(. ) 8 8 ln(. ) ln You can wor wihou arificial ligh o a dph of abou 59.8 f. 58. (a) 59. (a) A ( c ) d V A d c V A ln c + V A ln c V ( A/ V ) c ( A/ V ) c ± ( A/ V ) c± ( A/ V ) c+ D Iniial condiion whn c+ D c D Soluion: ( ) ( A + / V ) c c ( A/ V ) lim ( ) lim [ c+ ( c) ] c (c) 5 5 () M This is whr M 5, M + A. A, and 5. Thrfor, i is a soluion of h logisic diffrnial quaion. ( M ), or d 5 5 ( d ). Th carring capaci is 5. 5 (). + Iniiall hr wr infcd sudns ln ln das. I oo abou 6 das. oprigh arson Educaion, Inc. ublishing as rnic Hall.
36 hapr 7 Rviw 6. Us h Fundamnal Thorm of alculus. d sin + ( + + ) d d d d (sin ) + ( + ) d (sin + + ) d (cos )( ) + 6 cos( ) + 6 Thus, h diffrnial quaion is saisfid. Vrif h iniial condiions: () (sin ) + () + () sin( ) d d 8 8. d 8 8. d ( 8 ) A B ( 8 ) A( 8 ) + B 8, A 8, B +. 8 d ln ln 8. + ln ln ± 8. A 8. + A Iniial condiion: () A + A 6 A 5 8 Soluion: Mhod ompar graph of ln wih ln NDER. Th graphs 9 should b h sam. Mhod ompar graph ln of NINT( ln ) wih. 9 Th graphs should b h sam or diffr onl b a vrical ranslaion. 6. (a),, (. 6). 6 ln ln. 6 ln. 5 ln 6. I will a abou. ars. 6. (a) 6.,, 6. ln. 6 ln. 6. I will a abou. ars. f ( ) d u( ) d u( ) d g ( ) d u( ) d u( ) d f( ) g( ) u () d () u d u () d+ () u d u () d 65. (a) Th rgrssion quaion is Th graph is shown blow. 7, 86. lim , , 86 popl. oprigh arson Educaion, Inc. ublishing as rnic Hall.
37 hapr 7 Rviw (c) ( 786. ) d (d) Th carring capaci drops o 67,.6, which is blow h acual populaion. Th logisic rgrssion is srongl affcd b poins a h rms of h daa, spciall whn hr ar so fw daa poins bing usd. Whil h fi ma b mor dramaic for a small daa s, h quaion is no as rliabl. 66. (a) T (. 97) (c) ( ) + A A Solv for o obain 5ln 8. das. [, ] b [ 5, 9] Solving T () graphicall, w obain 9. sc. Th mpraur will rach afr abou 9. sconds. (c) Whn h prob was rmovd, h mpraur was abou T () (a) of h own has hard h rumor whn i is sprading h fass.. ( ) d A B + ( ) A( ) + B, A, B +. d ln. + c. c.. A. + A 68. (a) ( 6 ). Spara h variabls d o obain d 6 d 6 ln () 6 (c) 5 6 ln lim ( 6 ) (a) Spara h variabls o obain dv d v + 7 ln v+ 7 + v v+ 7 v 7 (c) lim ( 7) 7 f pr scond 7 ln. 5 sconds oprigh arson Educaion, Inc. ublishing as rnic Hall.
Chapter 6 Differential Equations and Mathematical Modeling
6 Scion 6. hapr 6 Diffrnial Equaions and Mahmaical Modling Scion 6. Slop Filds and Eulr s Mhod (pp. ) Eploraion Sing h Slops. Sinc rprsns a lin wih a slop of, w should d pc o s inrvals wih no chang in.
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