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. W s his a odd mulipls of π /.. Sinc is h dpndn variabl, I will hav no ffc on h valu of cos. d. Th graph of will loo h sam a all valus of. d. Whn, cos.this can b sn on h graph d nar h origin. A ha poin, h chang in and chang in ar h sam.. Whn π, cos. This can b sn in h d graph a π. A his poin, h chang in is ngaiv of h chang in. 6. This is ru bcaus ach poin on h graph has a ngaiv of islf. Quic Rviw 6.. Ys. d d. Ys. d d d. No. d ( ). Ys. d d d. No. d ( ) 6. Ys. d d ( ) d 7. Ys. sc sc an d 8. No. d d 9. () (). sin cos sin() cos() 7. sc ( ) sc( ). an ln( ) π an () ln(() ) π Scion 6. Erciss. ( sc ) d an. (sc an ) d sc. (sin 8 ) d cos. d ln. d ln an 6. d sin 7. ( cos( )) sin( ) sin 8. cos sin 9. (sc ( )( )) d an. (sin u) cosudu (sin u). sin d cos cos( ), cos. cos d sin sin( ), sin 6 7. du ( 7 ) d 7, 7 u
Scion 6. 6 9. da ( ) d 6 (), A. d c (), ( > ) / 6. d c sc an 7. / 7 an( ) ( ), 7 / an 7 ln an an ( ), an 8. d 6 ln 6 ln() 6(), 7 ln 67 ( > ) 9. dv ( sc an 6) sc sc( ) ( ), V π π sc < <. ds ( ) () (), s d. f d d () a sin( ) sin( ) du d. f d d () a cos u cos d cos. F ( ) f ( ) d a cos F ( ) 9 d () () an d a. G s f Gs () an. Graph (b). (sin ) (sin) > (sin ( )) > 6. Graph (c). (sin ) (sin) > (sin ( )) < 7. Graph (a). (cos ) > (cos) > (cos( ) > 8. Graph (d). (cos) > (cos) > (cos( )) < 9....
66 Scion 6... (, ) Δ Δ d d Δ ( Δ, Δ) (, )... (.,.) (.,.)... (.,.) (.,.)... (.,.66).66.. (, ) d Δ Δ d Δ ( Δ, Δ).. (, )... (.,.) (.,.)... (.,.) (.,.)... (.,.). (, ) d Δ Δ d Δ ( Δ, Δ) (, )... (.,.) (.,.)... (.,.) 6.. (.,.)... (.,.6).6 (, ) d Δ Δ d Δ ( Δ, Δ) 7. (, )... (.9, ) (.9, )... (.8,.99) (.8,.99)... (.7,.97) 8. 9... (, ) Δ Δ d d Δ ( Δ, Δ) (, ).. (., ) (., )... (.,.) (.,.)... (.,.). 6. (, ) d Δ Δ d Δ ( Δ, Δ) (, )... (.9,.) (.9,.).9..9 (.8,.9) (.8,.9).8..8 (.7,.7).7 7. (, ) Δ Δ d d Δ ( Δ, Δ) (, ).. (.9,.) (.9, )... (.8,.) (.8,.)... (.7,.). 8. (, ) d Δ Δ d Δ ( Δ, Δ) (, )... (.9,.) (.9, )... (.8,.) (.8,.)... (.7,.).
Scion 6. 67 9. (a) Graph (b) (b) Th slop is alwas posiiv, so (a) and (c) can b ruld ou. (a) (b) (c) p p. (a) Graph (b) (b) Th soluion should hav posiiv slop whn is ngaiv, zro slop whn is zro and ngaiv slop whn is posiiv sinc slop d. Graphs (a) and (c) don show his slop parn. (, ) (, ) (, ) (a) (b) (c). Thr ar posiiv slops in h scond quadrn of h slop fild. Th graph of has ngaiv slops in h scond quadrn.. Th slop of sin would b a h origin, whil h slop fild shows a slop of zro a vr poin on h -ais... (, ) Δ Δ d d Δ ( Δ, Δ) (, )... (.,.) (.,.)... (.,.6) (.,.6)... (.,.96) (.,.96).6..6 (.,.). Eulr s Mhod givs an sima f(.).. Th soluion o h iniial valu problm is f(), from which w g f(.).6. Th 6.. prcnag rror is hus 9.%. 6. (, ) d Δ Δ d Δ ( Δ, Δ) (, )... (.9,.7) (.9,.7).8..8 (.8,.) (.8,.).6..6 (.7,.6).7,.6)... (.6,.9).9 Eulr s Mhod givs an sima f(.6).9. Th soluion o h iniial valu problm is f(), from which w g f(.6).96. Th prcnag rror is 96. 9. hus %. 96. (. A vr (, ), ( )/ ( )/ )( ), so h slops ar ngaiv rciprocals. Th slop lins ar hrfor prpndicular. 6. Sinc h slops mus b ngaiv rciprocals, g() cos. 7. Th prpndicular slop fild would b producd b sin, so cos for an consan. d 8. Th prpndicular slop fild would b producd b d, so. for an consan. 9. Tru. Th ar all lins of h form. 6. Fals. For ampl, f() is a soluion of d, bu f ( ) is no a soluion of d. 6.. m 6. E. <, >, hrfor d <. 6. B. ( ) 6. A. d.
68 Scion 6. 6. (a) d d d d ( ) Iniial condiion: () Soluion:, > (b) Again,. Iniial condiion: ( ) ( ) ( ) Soluion:, < d (c) For <, d d. d For >, d d. And for, is undfind. d (d) L b h valu from par (b), and l b h valu from par (a). Thus, and. () ( ) ( ) ( ) ( ) 7 Thus, and 7. 66. (a) d (ln ) for > d (b) d d d ln ( ) ( ) ( ) d for < (c) For >, ln ln, which is a soluion o h diffrnial quaion, as w showd in par (a). For <, ln ln ( ), which is a soluion o h diffrnial quaion, as w showd in par (b). Thus, d ln for all cp. d (d) For <, w hav ln ( ), which is a soluion o h difrnial quaion, as w showd in par (a). For >, w hav ln, which is a soluion o h diffrnial quaion, as w showd par (b). Thus, for all cp. d 67. (a) d 6 6 d (b) sin d cos cos d sin (c) d d
Scion 6. 69 68. (a) d 8 8() () 8 d () () () (b) cos sin d sin cos sin cos sin cos d cos sin cos sin cos sin (c) d d 6 6 6 69. (a) d (b) d (c) d d ( ) (d) d d ( ) () d d ( ) / 7. (a) / / d d 6 (b) d d 6 (c) sin sin d cos cos d sin (d) d ( ) d d d ( ) () d ( d sin cos ) cos sin d ( d cos sin )sin cos sin cos Scion 6. Anidiffrniaion b Subsiuion (pp. ) Eploraion Ar fudu ( ) and fud ( ) h Sam Thing?. f ( u) du u du u 6. u ( ) 6. f( u) u ( ) 7 6 d 7. No
7 Scion 6. Quic Rviw 6.. d ( ) ( ) / /. d d ( ) ( ) / / ( ) ( ). d. d. d 6. d 7. d 8. d 8 6 () ( ) ( ) sin () cos ( ) i 8 sin () cos () i sin an cos i cos co sin 9. i (sc an sc ) d sc an. sc an sc sc an sc (an sc ) sc an sc ( csc co csc ) d csc co csc co csc csc co csc (co csc ) csc co csc Scion 6. Erciss. (cos ) d sin. d.. an. ( sc ) d an / 6. ( sc an ) d sc 7. ( co u ) ( csc u) csc u 8. ( csc u ) ( csc uco u) csc uco u 9. ( ). ln ln (ln ). (an u ) u. ( sin u ) u / /. f ( u) du u du u f ( u) d u d d d. f ( u) du u du u f ( u) d u d d u u 7. f ( u) du du u 7 7 f ( u) du d d 7 6. f( u) du sinu du cosu cos f ( u) d sinu d sin d cos 7. u du d du d sin d sin udu cos u hc: cos d d cos ( sin )( ) sin
Scion 6. 7 8. u du d d du cos( ) d cosu du sinu hc: 9. u du d sin( ) d sin( ) cos( )( ) cos( d ) du d sc an d scu anu du scu sc hc: d sc scan sc an d i. u 7 du 7d du d 7 8 7 7 8 7 ( ) d u du u ( ) hc: d ( 7 ) ( 7 ) ( 7) 8( 7 ) d. u du d du d d du 9 9u 9 du 9 u du u an u hc:. u r du r dr du r dr an d an d 9rdr du 9 r u u / / du ( ) u 6 r d hc: r d ( ) 6 6 r 9r r. u cos du sin du sin cos sin u du hc: d d cos cos sin cos sin. u du ( 8) du ( ) du ( ) u cos i 9 ( r )
7 Scion 6.. oninud 8 8 ( ) ( ) u du hc: d ( ) d ( ) ( 8) 8( ) ( ). L u du d d du ( ) u u 6. L u du d sc ( ) d sc u du anu an( ) 7. L u an du sc d / an sc d u du 8. L u θ π du dθ / u / (an ) u ( ) sc θ π an θ π θ sc an d u u du scu sc θ π 9. an( ) d u du d du d anudu ln cos( ) or ln sc( ). (sin ) d u sin du cos d du d cos u d co. L u z du dz du dz cos( z ) dz cos u du sinu. L u co du csc d sin( z ) / co csc d u du. L u ln du d ln 6 6 d u du 7 u 7 7 (ln ) 7 / u (co ) /. L u an du sc d 7 7 an sc d u du 8 i u 8 8 an /. L u s 8 / du s ds du s / ds
Scion 6. 7. oninud / / s cos( s 8) ds u du cos sinu / sin( s 8) d 6. csc d sin L u du d du d csc d csc udu co u co( ) 7. L u cos( ) du sin( )( ) du sin( ) sin( ) cos ( ) u du u cos( ) sc( ) 8. L u sin du cos 6cos ( sin ) d 9. ln u ln 6 u du 6u 6 sin d du du d du ln u ln(ln ) u. an sc d u an du sc d du d sc udu u an d. u du d du d du u ln ln( ) d. u a du d du u an u a a a an 8 an d sin. co cos d L u cos du sind du sind d co u du ln u ln cos ( An quivaln prssion is ln sc.). L u 8 du d du d d u 8 / du / i u 8
7 Scion 6. sc an. sc d sc i sc an d sc sc an d sc an Lu sc an du sc an sc d sc d ln ln sc u du u an csc co 6. csc d csc csc co d csc csc co d csc co Lu csc co du csc co csc d csc du du ln u ln csc co 7. sin d (sin ) sin d ( cos )sin d u cos du sin d ( u ) du u u cos cos 8. sc d (sc )sc d ( an )sc d u an du sc d ( u ) du u u an an 9. sin d ( sin ) d ( cos d ) u du d ( cos udu ) ( u sin u) sin. cos d ( ( cos ) ) d ( cos ) d u du d ( cos u ) du ( sin u u) sin. an an (sc ) d (an sc an ) d u an du sc d ( u udu ) u u an an. (cos sin ) d (cos sin )(cos sin ) d ((cos )) d sin. L u du / u du / u / ( ) ( ) () 8 8 /. L u r du r dr du r dr / r r dr u du i u / ( ) ( )
Scion 6. 7. L u an du sc d an sc d u du π / u ( ) ( ) 6. L u r du r dr du r dr r dr u du ( r ) 7. L u / / du d / du d d () u du / ( ) 8. Lu sin du cos d du cos d π cos d π sin 9. L u u u / du ( ) / ( ) u du du u / / () 7 6. L u cos du sin θ dθ du sin θ dθ 6 cos θ sin θ dθ π / / u du / i u () 7 d 6. u du d 7 du 7 7 9 lnu ln( ) ln u. d 6. u du d du ln u ln ( ) ln( 7). 97 u 6. u du du ln u ln( ) ln u. 69 π / π / π 6. π co d cos d sin u sin du cos d π / du π / π / π lnu ln (sin ) u π / π / d 6. u du d du lnu ln( ) ln( ).8 u d 66. u du d du u ln ln( ). 9 u
76 Scion 6. 67. L u 9, du d. (a) / / d u du u 9 9 9. 8 / (b) d u du 9 / u 9 d 9 9 9. 8 68. L u cos, du sin d. π / (a) ( cos )sind udu u π /6 6 6 6 ( ) ( ) (b) ( cos )sin d udu u 6 6 ( cos ) π / ( cos )sin ( cos ) 6 d π /6 π /6 ( 6 ) 6 () 69. W show ha f ( ) an and f( ), whr f( ) ln cos. cos d f ( ) ln cos d cos d (ln cos ln cos ) d d ln cos d ( sin ) an cos cos f () (ln ) cos 9 π / 7. ln sin 6 sin u sin v sin du cos d d u d ln 6 v d (ln u ln v 6) d du cos co u sin f ( ) co( ) 6 7. Fals. Th inrval of ingraion should chang from [, π / ] o [,], rsuling in a diffrn numrical answr. 7. Tru. Using h subsiuion u f( ), du f ( ) d, 7. D. w hav b a ( b) f ( ) d f du u f f ( a) ln ( ) u ln( f ( b )) ln( f ( a f( b) )) ln f( a). 7. E. d f ( b) f ( a) 7. B. F ( ad ) F( a) F( a) 7 a F ( ) d F( a) F( a) 7 a 76. A. d sin cos d π cos cos( ) π cos 77. (a) L u du d / d u du / u / ( ) d / Alrnaivl, ( ). d (b) B Par of h Fundamnal Thorm of alculus, and, so boh ar d d anidrivaivs of.
Scion 6. 77 77. oninud (c) Using NINT o find h valus of and, w hav:.9.797.667 6.787.667.8.869..667.667.667.667.667 (d) d) )...6.6....6.6. d d d d d d 78. (a) F f d [ ( ) ] shouldqual ( ). (b) Th slop fild should hlp ou visualiz h soluion curv F( ). (c) Th graphs of F( ) and f( ) should diffr onl b a vrical shif. (d) A abl of valus for should show ha for an valu of in h appropria domain. () Th graph of f should b h sam as h graph of NDER of F(). (f) Firs, w nd o find F ( ). L u, du d. / d u du u / Thrfor, w ma l F ( ). a) d ( ) ( ) d f( ) b) 79. (a) sin cos d udu u sin (b) sin cos d udu u cos (c) Sinc sin ( cos ), h wo answrs diffr b a consan (accound for in h consan of ingraion). 8. (a) sc an d udu u an (b) sc an d udu u sc (c) Sinc sc an, h wo answrs diffr b a consan (accound for in h consan of ingraion). d cosudu cosudu 8. (a) du. sin u cos u ( No cos u>, so cos u cosu cos u.) d (b) du u sin d udu udu 8. (a) du sc sc an u sc u d (b) du u an d sin / sin i sin cos 8. (a) / sin sin / π/ π sin cos cos sin c) (b) d / π / π / sin ( cos ) [ ( / )sin ] / π ( π )/
78 Scion 6. 8. (a) d an sc udu an an u / π/ π sc udu scu scudu d π (b) udu u u sc ln sc an ln ( ) ln ( ) ln ( ) / π / Scion 6. Anidiffrniaion b Pars (pp. 9) Eploraion hoosing h Righu and dv. u du dv cos v cos d Using for u is nvr a good ida bcaus i placs us bac whr w sard.. u cos du cos sin dv d v d Th slcion of u cos will plac a mor difficul ingral ino vdu.. u cos dusin dv d v d Th slcion of dv d will plac a mor difficul ingral ino vdu.. u and dv cos d ar good choics bcaus h ingral is simplifid. Quic Rviw 6.. ( )(cos )( ) (sin )( ) d. d. d. d cos sin ( ) ln( )( ) ln ( ) ( ) i ( ). an an an 6. cos ( ) cos cos 7. sinπ d cosπ π cosπ cos π π ( ) π π π 8. d d Ingra boh sids. d 9. sin d ( sin ) d Ingra boh sids. ( sin ) d cos ( ) cos. d (sin cos ) d (cos sin ) (sin cos ) cos sin sin cos sin Scion 6. Erciss. sin d dv sin d v sin dcos u du d cos cos d cos sin. d dv d v d u du d d
Scion 6. 79. dv v u du. cos ( ) dv cos sin v cos ( ) u du sin cos( ) sin cos ( ) 9. cos d dv cos d v cos d sin u du d sin sin d dv sin d v sin dcos u du d sin cos cos d sin cos sin 6. d dv d v d u du d d dv v d u du d d 7. d dv d v d u du 6 6 d d dv v d u du d d 8. cos d dv cos d v d cos sin u du d sin sin d dv sin v d sin cos u du d sin 8 cos 8cos d sin 8 cos 6sin 9. ln dv v u ln du ln ln. ln dv v u ln du ln ln 9. ( )sin d ( ) dv sin d v sin dcos u du d ( )cos cos d ( )cos sin ( )cos( ) sin( ) ( )cos sin [, ] b [, ]
8 Scion 6.. d dv v d u du d d ( ) ( ) ( ) [, ] b [, ]. du sc d dv sc d v sc d an w dw d an an d an ln cos an( ) ln cos( ) u an( ) ln cos( ) [.,.] b [, ]. dz ln d dv v d u ln du d d ln ln 6 () () ln ( ) 6 8 6 8 z ln 6 6 [, ] b [, ]. d dv ( ) v ( ) d ( ) u du d / / ( ) d ( ) / / ( ) ( ) / / ( )( ) ( ) ( ) / ( ) / [, ] b [, ] 6. d / / / dv ( ) v ( ) d ( ) u du d / / ( ) d ( ) / 8 / ( ) ( ) / 8 / ( )( ) ( ) 8 / 8 / 8 ( ) ( ) [, ] b [, ] 7. sin d / / / dv d v d u sin du cos d sin cosd dv d v d u cos dusin d sind sin ( cos sin d) sin d (sin cos )
Scion 6. 8 8. cos d dv cos d v cos d sin u du d sin sin d dv sin d v sin dcos u du d cos d sin ( cos cos d) cos d (sin cos ) 9. cos d dv cosd v cosd sin u du d sin sin d dv sin v sin d cos u du d cos d sin ( cos d cos d) cos d ( sin cos ) d ( ). L u dv d du ( ) d v ( ) d ( ) ( ) d L u dv d du d v ( ) ( ) d ( ) ( ) d ( ) ( ) ( 7 7). Us abular ingraion wih f () and g ( ).. sin d dv sind v sindcos u du d cos cos d dv cos d v cosd sin u du d sin d cos ( sin sin d) sin d ( cos sin ). Us abular ingraion wih f () and g ( ). d 8 8. Us abular ingraion wih f () and g ( ) cos. sin cos sin cos 8
8 Scion 6.. Us abular ingraion wih f () and g ( ) sin. π / sin d cos sin cos cos sin π / sin d cos sin π ( ) ( ) π. 7 8 π hc: NINT sin,,,. 7 6. Us abular ingraion wih f () and g ( ) cos. cos d sin cos sin cos 8 sin cos 8 π / cos d sin 8 cos π ( 6 8 ) ( ) 8 π. 6 π hc: NINT cos,,,. π / 7. L u dv cos d du d v sin cos d ( ) sin sin ( d) sin sin d L u dv sin d du d v cos cos d sin ( ) cos cos ( d) ( sin cos ) 9 9 cos d cos d 9 ( sin cos ) 9 cos d ( sin cos ) cos d ( sin cos ) 6 [ ( sin9 cos 9) ( sin( 6) cos ( 6)] 6 [ ( cos9 sin 9) ( cos6 sin 6)] 8. 86 hc: NINT ( cos,,, ) 8. 86 8. L u dv sin d du d v cos sin d ( ) cos ( cos d) cos cos d
Scion 6. 8 8. oninud L u dv cos d du v sin sin d cos ( ) sin sin d ( ) (cos sin ) sin d sin d (cos sin ) sin d (cos sin ) sin d (cos sin ) (cos sin ) 6 [cos( 6) sin( 6)] (cos sin ) 6 (cos 6 sin 6). 8 hc: NINT ( sin,,, ). 8 9. d L u dv d du d v d ( ) ( ) d L u dv d du d v d () 8 8. ln d L u ln dv d du d v d (ln ) ln d ln 9. θ sc θ d θ L u sc dv d du du v No ha w ar old >, so no absolu valu is ndd in h prssion for du. d (sc θ) θ θ θ θ θ θ θ dθ sc θ θ L w θ, dw θ dθ θ sc θ w dw θ sc θ w θ sc θ θ. d L u dv sc an d du d v sc θ scθ sc θ dθ θ scθ ln scθ anθ No : In h las sp, w usd h rsul of Ercis 9 in Scion 6... L u dv sin d du d v cos sin d cos cos d cos sin π (a) sin d sin d π cos sin π( ) () π π
8 Scion 6.. oninud π π (b) sin d sin d π π π π cos sin π() π( ) π π π π π (c) sin d sin d sin d π π π. W bgin b valuaing ( ) d. π L u dv d du ( ) d v ( ) d ( ) ( ) d L u dv d du d v ( ) d ( ) ( ) d ( ) ( ) ( ) Th graph shows ha h wo curvs inrsc a, whr.. Th ara w s is ( ) d ( ) (. 888 ) (.86 ).76. Firs, w valua cos. L u dv cos du v sin cos sin sin L u dv sin du v cos cos sin cos cos cos (sin cos ) cos (sin cos ) Now w find h avrag valu of cos for π. π Avrag valu cos π π cos π π (sin cos ) π π ( ) ( ) π π.9 π 6. Tru. Us pars, ling u, dv g()d, and v f(). 7. Tru. Us pars, ling u, dv g()d, and v f(). 8. B. cos d sin cos sin S problm. sin d cos sin S problm. h ( ) sin 9. B. sin ( ) d dv sin( ) d v sin( ) d cos u du d cos( ) cos( ) d cos sin( ).. csc d dv csc d v csc dco u du d co co d co ln sin.. ln d dv d v d u ln du d ln d ln. (a) L u dv d du d v d d ( ) (b) Using h rsul from par (a): L u dv d du d v d d ( ) ( )
Scion 6. 8. oninud (c) Using h rsul from par (b): L u dv d du d v d d ( ) ( 6 6) (d) n d d d d d d n ( ) n n n n n or n n n n n( n) n n ( ) ( n)! ( ) ( n!) () Us mahmaical inducion or argu basd on abula ingraion. Alrnal, show ha h drivaiv of h answr o par (d) is n : d n n n d ( n n n ( ) n n ( ) ( n!) ( ) n! ) n n n [ n n( n) n n ( ) (!) n ( ) n!] d ( ) d n n n n n n n n n ( ) ( n!) ( ) n!] n n n [ n n( n) n n ( ) ( n!) ( ) n! ] n n n n( n ) n nn ( )( n) n ( ) n! ] n d. L w. Thn dw, so d dw w dw. sin d (sin w)( wdw) wsin wdw L u w dv sin wdw du dw v cos w w sin wdww cos w cos wdw w cos w sin w sin d w sin wdw w cos w sin w cos sin. L w 9. Thn dw () d, so 9 d 9 dw w dw. 9 w dw ( ) wdw w 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 ( 9 ) 9. L w. Thn dw d. 7 w d ( ) d w dw. Us abular ingraion wih f () w and gw ( ) w. w w w w w w dw w w 6w 6 w ( w w 6w6) 7 w d w dw ( w w 6 w 6 w ) 6 ( 6 6) 6. L ln r. Thn dr,andsodr r. r Using h rsul of Ercis, w hav: sin (ln r) dr (sin ) (sin cos ) ln r [ sin (ln r) cos (ln r) ] r [in s (ln r) cos (ln r) ] n 7. L u dv cos d n du n d v sin n n n cos d sin (sin )( n d) n n sin n sin d
86 Scion 6. n 8. L u dv sin d n du n d v cos n n n sin d ( )( cos ) ( cos )( n ) d n n cos n cos d n 9. L u dv d a n a du n d v a n d ( a a n ( n ) d) n a n d, a a a n a n a a a. L u (ln ) dv d n n n(ln ) du d v n n (ln ) d (ln ) ( ) n n (ln ) d n n (ln ) n (ln ) d. (a) L f ( ). Thn f( ), so d f ( ). Hnc, f ( ) d ( ) f ( ) f ( ) (b) 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 d f ( ) d v d f d f d f ( ) ( ) ( ) d. (a) Using f ( ) sin and f( ) sin, π π, w hav: sin d sin sin sin cos sin cos (sin ) d (b) sin d sin sin d, sin d u du d sin u du sin u (c) cos (sin ) sin d. (a) Using f ( ) an and f( ) an, π π < <, w hav: d an an an an ln sc ( Scion 6., Eampl ) an ln cos an ln cos (an ) d (b) an d an an d d an d u, du d an u du an ln u an ln ( ) (c) ln cos (an ) ln ln ( ). (a) Using f ( ) cos and f( ) cos, π, w hav: cos d cos cos cos sin cos sin (cos ) d (b) cos d cos cos d cos u, du d cos u du cos u cos d d (c) sin (cos )
Scion 6. 87 6. (a) Using f ( ) log and f( ), w hav logd log log ln log log ln d (b) logd log log d d log (c) log d ln d log ln log ln Quic Quiz Scion 6. 6.. E.... A. d dv d v d u du d d. (a) (b) L and b in h diffrnial quaion: d ( b) b b (c) Firs, no ha d () () a h poin (, ). Also, d d ( ), which is a h d d d poin (, ). B h Scond Drvaiv s, g has a local maimum a (, ). Scion 6. Eponnial Growh and Dca (pp. 6) Eploraion hoosing a onvnin Bas. h. h is h rciprocal o h doubling priod. h. log h log log 7. 9 ars. log. h. h is h rciprocal o h ripling priod. h. log h log log 6. 9 ars. log. h. h is h rciprocal o h half lif. h 6.. log(. ) h log log(. ) 9. 8 ars. log Quic Rviw 6.. a b. c ln d. ln( ). 6 6 ln6 ln6. 8.. ln 8. ln. ln 8. ln. ln.. 68 l n 8. 6. ln ln ( )ln ln ln (lnln) ln ln ln. 7
88 Scion 6. 7.. ln. ln ln. ln ln. ln. l og 9. 8. ln ln ln ln 9. ln( ). ln ± ± Scion 6. Erciss. d () (), valid for all ral numbrs. d.. () ( ), valid on h inrval (, ) d ln ln, valid on h inrval (, ) d ln ± A,valid for all ral numbrs. d ( ) 6. ln ( ) / c / A / 6 /, valid for all ral numbrs 7. d cos d an an () an, valid for all ral numbrs. cos sin sin cos d sin sin sin ln ( ), valid for all ral numbrs. 8. d d ln ( ), valid for all ral numbrs. 9. d., valid for all ral numbrs.. ln d ln d u ln du d u du u (ln ) (ln ) (ln ), valid on h inrval (, ).
Scion 6. 89. (). (). (). (). () () () ln. ln (. ln ). Soluion : () or () i. () () 6 ( ) 6 ln. ln. ln Soluion: () 6 (. ln ) or () 6 i /. Doubling im: A () A r. 86. 86 ln. 86 ln 86. r.86 Amoun in ars: 86 A (. )( ) $, 97. 6. Annual ra: r A () A ( r)( ) r ln r ln r. 6. 6% Amoun in ars: r A () A [( ln )/ ]( ) A ln i $ 8 7. Iniial dposi: r A () A (. )( ) 898. A 898. A $ 6.. 7 Doubling im: A () A r. 6. ln. ln. ars. 8. Annual ra: A () A r ( r)( ),. 7. 7 r. 7 ln r. 7 r ln. 7 7. % Doubling im: A () A r. 7. 7 ln. 7 ln 96. ars.7 9. (a) Annuall:. 7 ln ln.7 ln. 9 ars ln.7 (b) Monhl:. 7. 7 ln ln ln. ar. ln 7 6 s (c) Quarrl:. 7 ln ln.87 ln ln.87 (d) oninuousl:. 7. 68 ln. 7 ln. 9 ars. 7 ars
9 Scion 6.. (a) Annuall:. 8 ln ln. 8 ln 8. 7 ars ln. 8 (b) Monhl: (c) Quarrl:. 8. 8 ln ln ln. ars. ln 8 8. 8 ln ln.6 ln ln.6 (d) oninuousl:. 8 89. ln. 8 ln 8. ars. 8.. 77 77. ln. 77 ln ( / ) 9 ars. 77 ars. ln ln ( / ) 6. 67. (a) Sinc hr ar 8 half-hour doubling ims in hours, hr will b 8 8. bacria. (b) Th bacria rproduc fas nough ha vn if man ar dsrod hr ar sill nough lf o ma h prson sic.. Using, w hav, and,,. Hnc, which givs, ln, or ln. Solving,, w hav. Thr wr 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,. 8.. 9. ln 9. 8. ln 9.. 8 das 8. ln ln 6. (a) Half-lif. das. 8 6. (b). ln.. ln. 99. das. Th sampl will b usful for abou 99 das. 7. Sinc ( ),w hav: ( )( ) ln ln ln ln. ln. (. ln.) Funcion: or 8. Sinc ( ).,w hav:. ( )( ). ln ln. (ln. ln ). 8 (ln. ln ) / Funcion:. or. 9. A im, h amoun rmaining is. ( / ). 99. This is lss han % of h original amoun, which mans ha ovr 9% has dcad alra.. T Ts ( T Ts) ( )( ) 6 ( T 6) ( )( ) 6 ( T 6) Dividing h firs quaion b h scond, w hav: ln Subsiuing bac ino h firs quaion, w hav: [(ln )/ ]( ) ( T 6) ( T 6) 6 T 6 T Th bam s iniial mpraur is F.
Scion 6. 9. (a) Firs, w find h valu of. T Ts ( T Ts) ( )( ) 6 ( 9 ) 7 ln 7 Whn h soup cools o, w hav: [( / ) ln (/7)] ( 9 ) [( / )ln (/7)] 7 ln ln 7 ln 7. min ln 7 I as a oal of abou 7. minus, which is an addiional 7. minus afr h firs minus. (b) Using h sam valu of as in par (a), w hav: T Ts ( T Ts) ( ) [ ( )] [( 9 / ) [( / ) ln (/7)] ln ln 7 ln. 6 ln 7 ln (/7)] I as abou.6 minus. Firs, w find h valu of. Taing righ now as, 6 abov room mpraur mans T T s 6. Thus, w hav T Ts ( T Ts) ( )( ) 7 6 7 6 ln 7 6 (a) T T ( T T ) 6. s s ( ( / ) ln ( 76 / ))( ) I will b abou. abov room mpraur. (b) T T ( T T ) 6. 79 s s ( ( / ) ln ( 7/ 6))( ) I will b abou.79 abov room mpraur. (c) T T ( T T ) s s ( ( / )ln (7/6)) 6 7 ln 6 ln 6 ln (/6). 7 min ln ( 7/ 6) I will a abou.7 min or.9 hr.. (a) T T s 79. 7(. 9) (b) T 79. 7(. 9) (c) Solving T and using h ac valus from h rgrssion quaion, w obain. sc. (d) Subsiuing ino h quaion w found in par (b), h mpraur was approimal 89.7.. (a) Nwon s Law of ooling prdics ha h diffrnc bwn h prob mpraur (T ) and h surrounding mpraur (Ts) is an ponnial funcion of im, bu in his cas Ts, so T is an ponnial funcion of im. (b) T 79. 96. 97 [, ] b [, 86] (c) A abou 7 sconds. (d) 76.96. Us ln (s Eampl ). 7. ln. ln. 7 ln. 668 ars ln rar La is abou 668 ars old. 6. Us ln (s Eampl ). 7 (a) 7. ln 7. ln.7 7 ln. 7, 7 ars ln Th animal did abou,7 ars bfor A.D., in,7 B.. (b) 8. ln 8. ln.8 7 ln. 8, ars ln Th animal did abou, ars bfor A.D., in, B..
9 Scion 6. 6. oninud (c) 6. ln 6. ln. 6 7 ln. 6, 7 ars ln 7. Th animal did abou,7 ars bfor A.D., in,7 B.. ln( / ). ln( / ) ars.. 8. r ln ( ) r. r ln( ). 6 ars. 9. ( )( ) 8 8. ln 8. A h: ( ln. 8/ ). ln. 8 8. g Abou 8. g will rmain.... ln.. ln. 6. 9 r I will a abou 6.9 ars.. (a) dp p dh dp dh p dp dh p ln p h ln p h h p p A h Iniial condiion: p p whn h p A A p Soluion: p p h Using h givn aliud-prssur daa, w hav p millibars, so: h p ( )( ) 9 9 9 ln. m Thus, w hav p. (b) A m, h prssur is 9 (( / ) ln ( / ))( ). 8 millibars. (c) 9 h 9 h 9 ln ( 9 / ) h ln 9.77 m ln ( 9 / ) Th prssur is 9 millibars a an aliud of abou.977 m.. B h Law of Eponnial hang, 6.. A hour, h amoun rmaining will b 6. ( ). 88 grams.. (a) B h Law of Eponnial hang, h soluion is (b). V V (/ ). (/ ) ln. ln. 9. sc I will a abou 9. sconds.. (a) A () A I grows b a facor of ach ar. (b) 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%.. (a) 9 ( r)( ) ln9 r ln9 r. or. % (b) ( r)( ) ln r ln r. 9 or. 9% h
Scion 6. 9 6. (a) r r ln r ln r (b) (c) ln 69., so h doubling im is 69. which is almos r h sam as h ruls. (d) 7 7 ars or. ars 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. Th corrc soluion is, which can b wrin (wih a nw ) as. 8. Tru. Th diffrnial quaion is solvd b an ponnial quaion ha can b wrin in an bas. No ha ( ) whn /(ln ). 9. D. A() A r 7 ln( ) r. 99 7 ln( ).. 99.. A A. D. r / 99/r 99 ln(. ) ln(. ) 99 ln(. ) r ln(. ). E. T 68 ( 68) 9 68 7 7. 6 7. 6. (. ) 68 7 68 ln 7 7min. 7. (a) Sinc acclraion is dv, w hav Forc m dv v. (b) From m dv v w g dv m v, which is h diffrnial quaion for ponnial growh modld b ( v / m ). Sinc v v a, i follows ha v. (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 quaion dv m v o s ha v changs mor slowl for largr valus of m. ( m / ) vm. (a) s v ( m / ) () Iniial condiion: s( ) vm vm vm s ( m / ) vm () vm ( ) ( m / ) vm ( m / ) vm (b) lim s ( ) lim ( ) vm. coasing disanc ( 8. )( 99. ). 998 W now ha vm 998. and. m ( 9. 9) Using Equaion, w hav: vm ( m / ) s () ( ) /. ( ) 66.. ( )
9 Scion 6.. oninud A graph of h modl is shown suprimposd on a graph of h daa. Graphical suppor:, 6. vm coasing disanc ( 86. )( 8. ) 97. 7. vm ( s () ( / m ) ) ( 7. /. 8) s () 97. ( ). 8866 s (). 97( ) A graph of h modl is shown suprimposd on a graph of h daa. r...689.667.68,.687,.687.. 687 Graphical suppor:.,. [, ] b [, ] 7. (a).97.78.769,.78,.78. 78 Graphical suppor:, (b) r 6.97 7.6 7.7, 7.876, 7.889 7. 89 (c) As w compound mor ims, h incrmn of im bwn compounding approachs. oninuous compounding is basd on an insananous ra of chang which is a limi of avrag ras as h incrmn in im approachs. 8. (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: m dv m mg a a ( )( ) ( a ) ma mg a a ( ) ( ) m g mg a a ( ) ( ) m a mg a ( )
Scion 6. 9 8. oninud 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. a a a mg (b) lim v ( ) lim i a a a mg lim mg Th limiing vloci is a a mg mg. mg (c) 6 79 f /sc. Th limiing vloci is abou 79 f/sc, or abou mi/hr. Scion 6. Logisic Growh (pp. 6 76) Eploraion Eponnial Growh Rvisid. ( ) 96 (.. ( ) ). 97 9. No. This numbr is much largr han h simad numbr of aoms.., ( ) log. 9 hours log Eploraion Larning From h Diffrnial Equaion. dp will b clos o zro whn P is clos o M.. P is half h valu of M a is vr.. Th graph bgins a downward rnd a half h carring capaci, causing a dclin in growh ras.. Whn h iniial populaion is lss han M, h iniial growh ra is posiiv.. 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 a a maimum. 7. lim P ( ) M. lim P ( ) dpnds onl on M. Quic Rviw 6.. ). ). ). ). (, ) 6 6. lim f ( ) 6. ( ) 6 7. lim f ( ) (. ( )) 6 8. ( ) (. ( )) 9. From problms 6 and 7, h wo horizonal asmpos ar and 6.
96 Scion 6.. 6 Scion 6. Erciss. A ( ) B ( ), B8 B, A( ) ( )( ) A. A ( ) B ( ) 6, B( ) ( ) 6 B B, A( ) ( ) 6 A A. A ( ) B ( ) 6, B( ) 6( ) 7B B, A( ) 6 7A A. A ( ) B ( ), B( ) 6B B /, A( ) 6A A /. S problm. d d ln ln ( ) ln ( ) 6. S problm. 6 d d 6 ln ( ) ln ( ) ( ) ln ( ) 7. ) 8 8 8 d u du d du ln u u ln( ) 8. 9) 6 9 d 9 A B d A ( ) B ( ), B( ) B /, A( ) A / / d / ln d 9. an d. 9 an 7. d A B 7 A ( ) B( ) 7, B( ( ) ) 7 B /, A( / ) 7 A d ln
Scion 6. 97... d A B A ( ) B( ), B( ( ) ) ( ) B, A A A d ln( ) ln( ) ln ( ) 8 7 A B 8 7 A( ) B( ) 87, B 8 7 B B, A( ) 8 7 A ( ) 87 A A d ln ln ( ) ln 7 d A B 7 A ( 7) B 7, 7B ( 7) 7B B, A( 7) ( ) 7A A d 7 ln ln ( 7) ( ) ln 7 6. d A B 6 A ( ) B 6, B ( ) 6 B B, A( ) ( ) 6 A 6 A d ln ln ln 6. du d A B A ( ) B ( ), B( ) B B, A( ) A A u d u ln ln u ln 7. F ( ) d d A B A ( )( ) B ( ) ( ),, B B, A A d F ( ) ln ln ln F ( ) ln
98 Scion 6. 8. G ( ) ) A ( ) B ( ), B( ) B B, A ( ) A A G () ( ) ( ) ln ln ln 9. d u du d du ln u ln u. d u du ( ) d du ln u u ln. d ) d u du ( ) d du ln u u ln. d ) d u du d du u ln u ln. (a) individuals. (b) individuals. (c) dp ( ). 6( )( ) 6 individuals pr ar.. (a) 7 individuals. (b) individuals. (c) dp ( ). 8( )( 7 ) 98 individuals pr ar.. (a) individuals. (b) 6 individuals. (c) dp ( 6). ( 6)( 6) 7 individuals pr ar. 6. (a) individuals. (b) individuals. (c) dp ( ) ( )( ) 6. individuals pr ar. 7. dp. 6 P( P) dp. 6 P( P) A B P P A( P) BP P, B B. P, A( ) A A.... 6 P P dp P P dp. ln Pln ( P).
Scion 6. 99 7. oninud P ln. P. c P. c P. ( ) 8 c P. c [, 7] b [, ] dp 8.. 8 P( 7 P) dp P 7 P. 8 ( ) A B P 7 P A( 7 P) BP P 7, 7 B B. P, A( 7 ) A... P 7 P dp. 8 P P dp 6. 7 ln P ln ( 7 P). 6 7 P ln 6. P 7 6. c P 7 6. c P 7 6. ( ) c c 69 7 P 6. 69 [, ] b [, 7] 9.. dp. P( P) dp P( P). A B P P A( P) BP P, B B. 8 P, A( ) A A. 8. 8. 8 P 7 P dp. P P dp. ln P ln ( P). P ln. P. P P c 9 P. 9 dp [, ] b [, ] c. ( ) P P. ( ) c c dp P( P) A B P P A( P) BP P, B B. P, A( ) A A... P P dp P P dp.
Scion 6.. oninud ln Pln ( P). P ln. P. c P. c P. ( ) c c 99 P. 99 [, ] b [, ]. (a) P () 8. 7. 8. 7. M A This is a logisic growh modl wih.7 and M. (b) P( ) 8 8. Iniiall hr ar 8 rabbis.. (a) P ().. M A This is a logisic growh modl wih and M. (b) P( ). Iniiall sudn has h masls.. (a) dp. P( P). P( P) ( ) M PM P Thus,. and M. M P A. A Iniial condiion: P( ) 6 6 A A A Formula: P. (b).... 8. ln 8 ln 8. 7. ws.. 6... ln ln. 8. I will a abou 7. ws o rach guppis, and abou.8 ws o rach guppis.. (a) dp. P( P). P( P) ( ) M PM P Thus,. and M. M P A. A Iniial condiion: P( ) 8, whr rprsns h ar 97. 8 A 8( A) A 7. 986 8 Formula: P (), or approimal. / P (). 7. 986
Scion 6.. oninud (b) Th populaion P() will round o whn P () 9.. 9.. / 9... ( 9. )( ).., 89. ln, 89 (ln,89 ln ) 8. 8 I will a abou 8 ars.. dp P( M P) dp PM ( P) Q R P M P RP Q( M P) P, MQ Q M P M, MR R M M M dp P M P ( ) P ( M P) dp M M P ln M P M M P M M A P M P M A 6. (a) dp M ( P) dp M P ln( M P) c M P c A P M A (b) lim P ( ) 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 79. 9 P 8.. [, 7] b [, 6] (b) P(. ) 79 9,. ( ). 7 popl. 8 79. 9 (c) P 8... 79. 9. 8,.. 6.. 6 or. (d) dp 7 P( M P) (. ) P( 79. 9 P). 8. (a) Th rgrssion quaion is 879. 8 P 8 77.. (b) P(. ) 879 8,. ( ). 8 79 8 77 879. 8 (c) P 8 77... 879. 8 8. 77,.. 687. 6 or.. (d) dp 7 P( M P) (. 66 ) P( 879. 8 P) 9. Fals. I dos loo ponnial, bu i rsmbls h soluion o dp P( ) ( 9) P.
Scion 6.. Tru. Th graph will b a logisic curv wih lim P ( ) and lim P ( ).. D. 6.. B. ( ) 9. 9.. ( ) (..) 9 %. D.. B. d ( )( ) A B A ( ) B ( ), B( ) B B, A( ) A A d 8 ln ln. (a) No ha > and M >, so h sign of dp is h samas h sign of ( MP)( Pm). For m < P < M, boh MP and Pmar posiiv, so h produc is posiiv. For P< mor P> M, h prssions M P and P m hav opposi signs, so h produc is ngaiv. (b) dp M ( MP)( Pm) dp P P ( )( ) dp ( P)( P) dp ( P)( P ) ( P ) ( P) dp ( P)( P) dp P P dp P P ln P ln P P ln P P / ± P P P A / / / P A AP / / P( A ) A / A P / A A (c) A ( A) A AA 9A A 9 / ( / 9) P () / ( / 9) / ( ) ( 9) P () / 9 / ( 8 ) P () / 9 (d) [, 7] b [, ] No ha h slop fild is givn b dp. ( P)( P). dp () M ( MP)( Pm) M dp ( MP)( Pm) M M m dp M m ( MP)( Pm) ( P m) ( MP) dp M m ( MP)( Pm) M dp M m M P P m M M P P m dp M m M ln M P ln P m M m M P m M m ln M P M P m ( Mm) / M ± M P P m M P A ( Mm) / M ( Mm) / M P m ( MP) A ( Mm) / M ( Mm) / M P( A ) AM m P A ( Mm) / M M m ( Mm) / M A AM m AM m P( ) A A
hapr 6 Rviw. oninud () P( )( A) AM m AP ( ( ) M) mp( ) A m P ( ) P( ) m P( ) M M P( ) Thrfor, h soluion o h diffrnial quaion is P AM ( Mm) / M m P( ) m whr A ( Mm) / M A M P( ). 6. (a) an a a (b) a ln a a (c) a 7. (a) ln (b) ( ) 8. (a) This is ru sinc A B ( ) ( ) (b) A ( ) B ( ), A B, 9, A B A B 9 B 9 (c) ln ( ) Quic Quiz.... d. A. ( )( ) A B A ( ) B ( ), B B /, A A / / / d ln A ( ) B ( ) ( ) dp P P. dp P( P) A B P P A( P) BP P, B B. P, A A... P P dp P ln / P P / P( ) / ( ) A A. lim P ( ) / ( ). (b) P( ) / ( ) A A 66. lim P ( ) / ( ). 66 (c) Spara h variabls. dy Y lny Y / / whr / / Y (d) lim / / / lim lim / / / ( ) hapr 6 Rviw Erciss π / π / π. sc θ dθ anθ an an. d ( )
hapr 6 Rviw. L u du d du d 6 d 8 ( ) u du 8 u 9 9 8 9 9 8. L u du d du d sin( ) d sinudu. L u sin 6. du cos d π / / / sin cos d u du / / 7. L u an i u / ( ) d d ( ) ( ) du sc d π / an / 6 ( ) ( ) 8 8 7 8 8 sc d du u u 8. L u lnr du r dr lnr / dr u du r / u ( ) 9. d 6 ( )( ) A B A ( ) B ( ), B( ) B, A( ) A A d 6 ln( ) ln( ) ln 6. d 6 ( ) A B 6 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 sin d u du ln u ln sin
hapr 6 Rviw. L u du d du d d u du / i u / ( ) /. L u du du u du u ln ln( ) ln. L u θ du dθ θ sc an d θ θ θ θ sc u an udu scu sc θ. L u ln du an(ln ) anudu sinu cosu du L w cosu dw sinudu w dw ln w ln cosu ln cos(ln ) 6. L u du d sc( ) d sc udu ln sc u an u ln sc( ) an( ) 7. L u ln du d d ln u du ln u ln ln 8. / / / 9. Us abular ingraion wih f ( ) and g ( ) cos. cos d sin cos 6sin 6cos. L u ln dv d du d v ln d ln d ln d ln
6 hapr 6 Rviw. Lu 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 sin d cos sin sin d [ cos sin ] sin cos. L u dv d du d v d d L u dv d du d v d 9 9 9 7 9 7. d ( )( ) A B A ( ) B ( ), B( ) B B /, A( ) A A / d / / d ln. d ( )( ) A B A ( ) B( ), B( ( ) ) ( ) B B, A S 9 A A d ln ( ) ( ). d d d 6 ( ) 6 Graphical suppor:
hapr 6 Rviw 7 6. d d d d () Graphical suppor: 7. ln ( ) ln( ) ln( ) Graphical Suppor: cscθ π cscθ 9. ( ) d ( ) d ( ) d () ( ) d ln () ln Graphical suppor: L f( ) ln. Wfirsshowhgraphof f ( ), >, along wih hslopfildfor f ( ). 8. cscθco θ dθ cscθco θdθ cscθco θdθ
8 hapr 6 Rviw 9. oninud W now show h graph of f ( ) along wih h slop fild for f ( ).. dr ( ) cos dr ( ) cos dr ( ) cos r sin r ( ) r sin dr ( ) ( sin ) r cos r ( ) r cos dr (cos ) r sin r( ) r sin Graphical suppor: W firs show h graph of r sin along wih h slop fild for r cos.. d d d ln () Graphical suppor:. ( )( ) d d ( ) d ( ) ln ( ) Graphical suppor: N, w show h graph of r cos along wih h slop fild for r sin. Finall w show h graphof r sin along wih h slop fild for r cos.. ( ) ( ) A B A( ) B, B, A
hapr 6 Rviw 9. oninud. ln ln ln c c A ( ).. A A 9 9 -. d. ( ) d. ( ) A B. A( ) B(. ), B(. ) B, A A...... ln ln. ln.. c. A ( ).( ) A A 9 9.. sin 6. 7. 8. 9. Graph (b).. Graph (d).. Graph (c).. Graph (a).. (,) Δ Δ Δ ( Δ, Δ ) d d (,)... (.,.) (.,.)... (.,.) (.,.)... (.,.6).6. (,) Δ Δ Δ ( Δ, Δ ) d d (,)... (.9,.) (.9,.)... (.8,.) (.8,.)... (.7,.6).6. W s h graph of a funcion whos drivaiv is sin. Graph (b) is incrasing on [ π, π ], 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 (,,, ).
hapr 6 Rviw 7. (iv) Th givn graph loos h graph of, which saisfis and (). d 8. Ys, is a soluion. 9. (a) dv 6 dv ( 6) v Iniial condiion: v whn v. (b) v() ( ) 6 6 Th paricl movs 6 m. [, ] b [, ]. (a) Half-lif ln ln. 6 ln. 69. 6 (b) Man lif. T T ( T T ) s. 89 ars s T ( ) Us h fac ha T 8 and o find. ( )( ) 8 ( ) 8 9 7 9 ln 7 (( / ) ln ( 9/ 7)) T ( ) (( / ) ln ( 9/ 7)) 7 ( ) (( / ) ln ( 9/ 7)) 8 6 9 ln 7 ln6 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.. T T ( T T ) s s W hav h ssm: 9 Ts ( 6 Ts) Ts ( 6 Ts) 9 Ts Ts Thus, and 6 Ts 6 Ts Sinc ( ), his mans: 9 T s T 6 Ts s 6 Ts ( 9 Ts) ( Ts)( 6Ts) 78Ts Ts 8 79Ts Ts Ts Th rfrigraor mpraur was.. Us h mhod of Eampl in Scion 6... 99 ln. 99 7 ln. 99 ln. 99. ln Th paining is abou. ars old.. Us h mhod of Eampl in Scion 6.. Sinc 9% of h carbon- has dcad, % rmains.. ln. 7 ln. ln. 8, 9 ln Th charcoal sampl is abou 8.9 ars old. 6. Us 988 9 6 ars. r 7 r r ln ln ln r. 6 Th ra of apprciaion is abou., or.%. 7. Using h Law of Eponnial hang in Scion 6. wih appropria changs of variabls, h soluion o h diffrnial quaion is L( ) L, whr L L( ) is h surfac innsi. W now. 8, so ln. and our quaion bcoms 8 / 8 L ( ) L (ln. )( / 8 ) L. W now find h dph whr h innsi is on-nh of h surfac valu.
hapr 6 Rviw 7. oninud / 8. ln. ln 8 8 ln. 9. 8 f ln. You can wor wihou arificial ligh o a dph of abou 9.8 f. A 8. (a) V ( c ) A c V A ln c V A ln c V ( A/ V ) c ( A/ V ) c ± ( A/ V ) c± c D ( A/ V ) Iniial condiion whn c D c D ( Soluion: c c A / ( ) V ) (b) lim ( ) lim[ c ( c ) ] c ( A/ V ) 9. (a) p ().. M. This is p whr M, A, and. A Thrfor, i is a soluion of h logisic diffrnial quaion. dp M PM P dp ( ), or P( P). Th carring capaci is. (b) P( ). Iniiall hr wr infcd sudns. (c). 6.. ln.. ln 9. das. I oo abou 6 das. 6. Us h Fundamnal Thorm of alculus. d d d sin d ( ) (sin ) ( ) d (sin ) d (cos )( ) 6 cos( ) 6 Thus, h diffrnial quaion is saisfid. Vrif h iniial condiions: ( ) (sin ) ( ) ( ) sin( ) 6. dp P. P 8 dp. P 8 P P dp. ( 8 ) ( 8 P) P dp. P( 8 P) p dp 8 P. ln P ln 8 P. P ln. 8 P 8 P ln. P 8 P. P 8 P. ± P 8. A P 8 P. A Iniial condiion: P( ) 8 A A 6 A 8 Soluion: P. 8 8 P 6. Mhod ompar graph of ln wih ln NDER 9. Th graphs should b h sam. Mhod ompar graph of NINT( ln ) ln wih. Th graphs should b h sam or 9 diffr onl b a vrical ranslaion.
hapr 6 Rviw 6. (a),, (. 6). 6 ln ln. 6 ln. ln. 6 I will a abou. ars. (b).,, 6. 6 ln. 6 ln.. 6 I will a abou. ars. d 6. (a) f ( ) u ( ) u ( ) d d g ( ) u ( ) u ( ) d (b) f( ) g( ) u () u () u() u() u () 786. 6. (a) Th rgrssion quaion is.. 69. 9 Th graph is shown blow. (b) (. ) 786,. ( ). 7 86 popl. 9 69 (c) dp 7 7. 69 P( 786. P) (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 79. 96(. 97) [, ] b [, 9] (b) 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( ) 7976.. 67. (a) of h own has hard h rumor whn i is sprading h fass. (b). ( ) A B. A( ). B, B. 8, A 8.. ln.... A ( ) A A 9 9.. ( ). (c) 9. Solv for o obain ln 8. das. 68. (a) dp ( 6 P). Spara h variabls o obain dp 6 P dp P 6 ln P 6 P 6 6 P 6 P () 6 (b) 6 / ln. 69 i. 69 (c) lim( 6 ) 6
Scion 7. 69. (a) Spara h variabls o obain dv v 7 ln v 7 v 7 7 7 v 7 v 7 (b) lim( 7) 7 f pr scond (c) 7 ln. sconds hapr 7 Applicaions of Dfini Ingrals Scion 7. Ingral as N hang (pp. 78 89) Eploraion Rvisiing Eampl 8 8. s () ( ) 8 s( ) 9 8 Thus, s (). 8. s( ). 6 This is h sam as h answr w found in Eampl a. 8. s( ). This is h sam answr w found in Eampl b. Quic Rviw 7.. On h inrval, sin whn π, π, or. Ts on π poin on ach subinrval: for,sin ;for π π,sin ; for,sin ; and for π,sin. Th funcion changs sign a π, π, and. Th graph is f(). ( )( ) whn or. Ts on poin on ach subinrval: for, ; for, ; and for,. Th funcion changs sign a and. Th graph is f(). has no ral soluions, sinc b ac ( ) ()() 8 <. Th funcion is alwas posiiv. Th graph is f(). ( ) ( ) whn Ts on poin on ach subinrval: for, ; for, ; and or.,. Th funcion changs sign a. Th graph is f(). On h inrval, cos whn, π, π, π or. Ts on poin on ach subinrval: for π 8, cos, for, cos for π, 6 cos π; and for, cos 8.. Th funcion changs sign a π π,, and π. Th graph is f() 6. whn. On h rs of h inrval, is alwas posiiv. 7.. Ts on poin on ach subinrval: for, ; for,. Th funcion changs sign a. Th graph is f()