Chapter 4: Exponential and Logarithmic Functions

Transcription

1 Erciss. 7 EXERCISES. Chatr : Eotial a Logarithmic Fuctios. a. c.. a. c a. 0.5 /.69 c ( ). a ( ) 5 c /.96 5 ( ) ( ) a.. a. o [0, 5] by [0, 0] is highr tha. o [0, 0] by [0, 0] is highr tha 0.0. o [0, 6] by [0, 00] is highr tha for larg valus of. o [0, 000] by [0, 000] 0.0 is highr tha wh is ar Brooks/Col, Cgag Larig.

2 7 Chatr : Eotial a Logarithmic Fuctios c.. o [0, 0] by [0, 0,000] is highr tha for larg valus of. o [0, 5] by [0,,000,000] is highr tha 5 for larg valus of.. will c ay owr of for larg ough valus of.. a. For m (aual comouig), ( r ) t P + simlifis to P(+ r) t. Wh P 000, r 0., a t 8, th valu is ( + 0.) 000(.).59 Th valu is $.59. For quartrly comouig,, P 000, r 0., a t 8. Thus, ( + ) 000( ) 000(.05) 0.76 Th valu is $0.76. c. For cotiuous comouig, P 000, r 0.0, a t 8. Thus, rt P Th valu is $ a. P 00, r 0.0, a, comou wkly, which is 5 tims r yar. This givs a valu of ( + 0.0) 00(.0) 56 00(.0) $96 Th vig is qual to th amout ow aftr thr yars mius th amout loa. This is $96 $00.00 $096.. a. For m (aual comouig), ( r ) t P + simlifis to P(+ r) t. Wh P 000, r 0., a t 8, th valu is ( + 0.) 000(.) Th valu is $ For quartrly comouig,, P 000, r 0., a t 8. Thus, ( 0.0) 000(.0) Th valu is $ c. For cotiuous comouig, P 000, r 0., a t 8. Thus, rt P Th valu is $ For quartrly comouig, r 0.05 a (.05) 56 (.05) $, 6, 9,90 00 Brooks/Col, Cgag Larig.

3 Erciss Th stat rat of 9.5% (comou aily) is th omial rat of itrst. To trmi th ffctiv rat of itrst, us th comou itrst formula, P( + r), with r 9.5% a umbr of ays i a yar. Sic som baks us 65 ays a som us 60 i a yar, w will try both ways. If 65 ays, th r 9.5% Th P( + r) P(.0005).0969P. umbr of ays Subtractig givs , which rss as a rct givs th ffctiv rat of itrst as 9.69%. If 60 ays, th r Th P( + r) P( ).0969P a th ffctiv rat is also 9.69%. Thus, th rror i th avrtismt is 9.85%. Th aual yil shoul b 9.69% (bas o th omial rat of 9.5%). 9. If th amout of moy P ivst at 8% comou quartrly yils $,000,000 i 60 yars, th r a , 000, 000 P( + 0.0), 000, 000 P $869 0 ( + 0.0). For comouig aually, r 0.06 a 5. Prst valu P 000 $ ( + r) (+ 0.06). For 0% comou aually, r 0.0 a. Prst valu P 000 $75. ( + r) (+ 0.0) 8. To trmi th ffctiv rat of itrst, first w trmi r i th cass of 60 ays a 65 ays. For 60 ays, r 0.07 a P( + r) P P 60 Th ffctiv rat is 7.5%. For 65 ays, r 0.07 a P( + r) P P 65 Th ffctiv rat is 7.5%. Th rror i th aual yil shoul b 7.5%. 0. Th yil is $65,000, r , a , 000 P( ) 65, 000 P $, 5 7 ( ). For 6% comou aually, r 0.06 a. Prst valu P 000 $79.09 ( + r) ( ). For % comou aually, r 0. a 0. 0 P( + r).5( + 0.) $6.67 millio 00 Brooks/Col, Cgag Larig.

4 76 Chatr : Eotial a Logarithmic Fuctios 5. To comar two itrst rats that ar comou iffrtly, covrt thm both to aual yils. 0% comou quartrly: P( + r) P(.05) P(.08) Subtractig, Th ffctiv rat of itrst is 0.8%. 9.8% comou cotiuously: r P P P(.00) Subtractig : Th ffctiv rat of itrst is 0.0%. Thus, 0% comou quartrly is bttr tha 9.8% comou cotiuously. 7. Sic th rciatio is 5% r yar, r 0.5. a. P( + r) 5, 50( 0.5) $758 P( + r) 5, 50( 0.5) $, Sic 05 is 0 yars aftr 005, (0) billio % comou quartrly: P( + r) P P(.08) Th ffctiv rat of itrst is 8.%. 7.8% comou cotiuously: r P P P(.08) Th ffctiv rat of itrst is 8.%. Thus 8% comou quartrly (a ffctiv rat of 8.%) is bttr tha 7.8% comou cotiuously (a ffctiv rat of 8.%). 8. For.6% comou aually, r 0.6 a 0. 0 P( + r) 7( + 0.6) $ millio 0. Th oulatio of Chia is giv by.(.006) t billio, whr t is th umbr of yars aftr 005. I 05 (t 0), th oulatio of Chia will b 0.(.006).7 billio. Th oulatio of Iia is giv by.(.05) t billio, whr t is th umbr of yars aftr 005. I 05 (t 0), th oulatio of Iia will b 0.(.05).5 billio. Iia's oulatio will b largr i 05.. a. 00(5) (0.9997) If a mosquito brs 00 mosquitos o avrag, th th 00 chilr will br (00)(00) 00 90, 000 grachilr, a th 90,000 grachilr will br (90,000)(00) 00 7,000,000 grat grachilr. 00(0) (0.9997) Th roortio of light that trats to a th 0. of ft is giv by. a. If th th is ft, 0. 0.() or 6.7% If th th is 0 ft, 0. 0.(0)..% 0.0 or. a. 0.0() f() mg 0.0(8) f(8) 00 5 mg 5. a. f() 0.08(). mg 6. a. If th coloy oubls vry hour, th ish was 50% covr at A.M. 0.08(8) f(8) 0.8 mg If th ish was 50% covr at A.M., it was 5 (50%) covr at 0 A.M. 00 Brooks/Col, Cgag Larig.

5 Erciss (0) S(0) (0) (0) 0.90 or 90% 9. a. Aftr 5 miuts, t a 60.8(0.5) T(0.5) grs Aftr 0 miuts, t a 60.8(0.5) T(0.5) grs 0. a. (0) % () % + +. If S 00, 0, a r 0.0, th th R- (0.0)(0) Frost mol is I 00( ) 8. Th mol stimats thr will b about 8 wly ifct ol.. Sic 00 is 0 yars aftr 000, (0) 805 $50 thousa. W hav P $8000,, a th valu aftr yars is $0,9.7. 0, ( + r) 0, r ,9.7 r % Th air rssur is ( 0.00) a /00 rct of th origial rssur, whr a is th chag i altitu. 7. a ( 0.00) % So th air rssur cras by about %.. W hav P $9000, 8, a th valu aftr yars is $0, , r + 0,80.65 r % ( 0.05) t rct will still b rst aftr t yars. 50 a. ( 0.05) 0.8 8% 8. a. 00 ( 0.05) % o [0, 00] by [0, 00] o [ 00, 00] by [0, 700] Durig th yar 087 ( 8.08) 60 ft () $0,000 $59, It is ot ossibl to valuat f a with a gativ bas (a < 0) a 0.5, bcaus th fuctio is ot fi for all valus of. For aml, if a, a 0.5, 0.5 f (0.5) ( ), which is ot a ral umbr. 5. lim a lim 0. Th -ais (y 0) is a horizotal asymtot. 5. lim 0 a lim. Th -ais (y 0) is a horizotal asymtot. 00 Brooks/Col, Cgag Larig.

6 78 Chatr : Eotial a Logarithmic Fuctios 5. For vry larg valus of, will b largst bcaus it will hav th largst ot. 5. is a cotiually icrasig fuctio, so if y < y, th <. 55. That its growth is roortioal to its siz. 56. Smiaually, quartrly, aily, cotiuously, bcaus th bfit to th ositor icrass as th itrst is comou mor oft. 57. With 5% mothly, you must wait util th of th moth to rciv itrst, whil for 5% comou cotiuously you bgi rcivig itrst right away, so th itrst bgis arig itrst without ay lay. 59. Drciatio by a fi rctag givs th biggr cras i th first yar, a th straight li givs th biggr cras i th last yar. 58. Daily comouig givs lss rofit tha cotiuous comouig. Th bakr was icorrct. 60. Th cotiuously comou rat is lss tha 0%, bcaus 0% mor at th of th yar accouts for th itrst o th itrst as wll as th itrst o th ricil. EXERCISES.. a. log 5 log 5. a. log 7 log 5 5 log 8 log log 6 log c. log log c. / log log log log. log log 9. / log log. log log f. / log log f. / log log 9. a. 0 l( ) 0. a. 5 l( ) 5 / l l l c. / l l c. / l l. l / l l 5. l(l( )) l bcaus l( ). l l f. ( ) l l f. l(l ) l 0 5. f l(9) l 9 l9 + l l9 l 7. f l( ) l l l l 9. f l + l l l + l l 6. f l + l l l + l l 8. f l l l + l l l 0. f l( 5 ) l 5l l l 00 Brooks/Col, Cgag Larig.

7 Erciss. 79. f l( 5 ) l 5 0. f l( ) + + l l f l f + l( ) 0 5. Th omai of l( ) is th valus of such that > 0 > > or < Th omai is { > or < }. Th rag is. 7. a. W us th formula P( + r) with mothly itrst rat % % 0.0 Sic oubl P ollars is P ollars, w solv P( + 0.0) P.0 l(.0 ) l l.0 l l l Sic is i moths, w ivi by to covrt to yars. A sum at % comou mothly oubls i about.9 yars. To fi how may yars it will tak for th ivstmt to icras by 50%: P( + r).5p ( + r).5 l( + r) l.5 l( + r) l.5 Now, r % % 0.0 thus, l.5 l l( + r) l Sic is i moths, w ivi by to covrt to yars yars A sum at % comou mothly icrass by 50% i about.7 yars. 6. Th omai of l( ) is th valus of such that > 0 > < < < Th omai is { } { y y 0}. < <. Th rag is 8. a. r Sic oubl P ollars is P ollars, w solv P( + 0.0) P.0 l(.0 ) l l. moths l.0 Th moy oubls i..95 yars To fi how may yars it taks to icras by 50%, which is.5p, w solv.5 P P( + 0.0).5.0 l.5 l(.0) l.5.7 moths l.0 Th moy icrass by 50% i.7. yars. 00 Brooks/Col, Cgag Larig.

8 80 Chatr : Eotial a Logarithmic Fuctios 9. a. W us P r with r Sic tril P ollars is P ollars, w solv 0.07 P P l l 5.7 yars 0.07 If P icrass by 5%, th total amout is.5p. W solv 0.07 P.5P l.5 l.5. yars If th rciatio is 0% 0. r yar, th r 0.. W us th itrst formula P( + r) a w solv P( 0.) 0.5P l 0.7 l 0.5 l yars l 0.7. W us th itrst formula P( + r) with r.% 0.0. Sic th icras is 50%, w must fi th umbr of yars to rach.5p. W solv P( + 0.0).5P.0.5 l.5 7. yars l.0 5. W wat to fi th valu of t that roucs (t) t t t l 0. t l ays a. W us P r with r Sic quarul P ollars is P ollars, w solv P P 0.06 l l. yars 0.06 A 75% icras givs.75p, a w solv P P. P( ) P.75 l yars l.09 l l.09 l l. yars l.09. W us P(+ r) with r 0.6, a w solv P( + 0.6) P t 0.0t.6 l.7 yars l t l ays (5) l or 58% 8. 0.t 900, 000, 000, 000( ) 0.t t 0. t l hours Brooks/Col, Cgag Larig.

9 Erciss t st 00( ) 0.t 00( ) 80 0.t t 0. 0.t 0. 0.t l 0. t l 0. wks t t l 0.t l ( ) t 6 hours 0.. To fi th umbr of yars aftr which.% 0.0 of th origial carbo is lft, w solv 0.000t t l 0.0 t l 0.0,00 yars To stimat th ag, w solv 0.000t 0.9 t l yars Th roortio of otassium 0 rmaiig aftr t t millio yars is. If th sklto cotai 99.9% of its origial otassium 0, th t t l l t l t l Thrfor, th stimat of th ag of th sklto of a arly huma acstor iscovr i Kya i 98 is aroimatly.7 millio yars. 5. To fi wh th raioactivity crass to 0.087t 0.00, w solv t l 0.00 t l ays l.7 yars l t t l millio yars Sic rai forsts ar isaarig at a aual rat of.8% 0.08, th r 0.08, a w solv ( 0.08) t t t l yars l 0.98 l(.6 ) l.5 l.6 l.5 l.5.7 yars l.6 00 Brooks/Col, Cgag Larig.

10 8 Chatr : Eotial a Logarithmic Fuctios o [0, 60] by [0, 5] a. 5 yars 55.5 yars. Lt t umbr of yars. Sic th rat is 6% comou quartrly, th r a th amout of moy is t t ( ).05.. o [0, ] by [0, 5] a.. yars.5 yars o [0, 0] by [0, ] a. 9.9 yars. yars. o [0, 6] by [0, ] a..6 yars 6.8 yars. 5. o [0, 0] by [0, ] a. About 9 ays About ays 6. o [0, 5] by [0, 6.] About vry ays o [0, 7000] by [0, ] About 500 yars o [0, 50] by [0.8, ] About 8 millio yars 7. If th amout of raioactiv wast is growig by.% aually sic 000, th th amout of wast yars aftr 000 is 5,000(.). W must fi th valu of such that (.).. l. l l. l l 6.5 l. Th amout will oubl i about 6.5 yars. 8. W must fi th valu of such that (.).. l. l l. l. yars l l. 0. of a yar is aroimatly moths, so th oublig tim for cll ho subscribrs is yars a moths. 00 Brooks/Col, Cgag Larig.

11 Erciss W must fi th valu of such that (.095)..095 l.095 l l.095 l l 7.8 yars l of a yar is aroimatly 0 moths, so th oublig tim for th fift-yar bo is 7 yars a 0 moths l W must fi th valu of such that (.0975) l.0975 l l.0975 l l 7.5 yars l of a yar is aroimatly 6 moths, so th oublig tim for th bo is 7 yars a 6 moths. 5. l( y) l + l y y 5. l ( ) 5. l l l y 55. log 0 y or log y 56. l y l 57. If l( ), th, but is always ositiv. So l( ) is ufi. 59. Cotiuous comouig woul giv th shortst oublig tim. Aual comouig woul giv th logst oublig tim. 58. If l 0, th 0, so woul b, which is ufi. 60. Th rug with a largr absortio costat will hav mor tim btw oss. 6. a. kt kt l kt l l t l k Th half-lif is l k. If k 0.08, half-lif l 9 hours Nt K for t 0, K> 0, b> 0. N(0) N0. Substitut 0 for t. N l N K K K bt a b(0) a 0 a a l K a l N0 l K + l l N l K a a a l K l N l 0 K ( N ) 0 00 Brooks/Col, Cgag Larig.

12 8 Chatr : Eotial a Logarithmic Fuctios 6. f H0 ( ) N Substitut 500 for N. f H 0 H0(0.999) (500) Rucig th frqucy by 6%, H 0.06 H H (0.999) 0.9 H H (0.999) (0.999) l 0.9 l(0.999) l 0.9 l(0.999) l gratios l y a for 0, a >, b> 0. Lt y. a a ( 0) a b b a b b b b l a l l a b l a b y, so y l a. b Th quilibrium oulatio is l a. b 65. a. For r 6%, th oublig tim is aroimatly 7 6 yars. P( ) P.06 l.06 l l.9 yars l If th itrst rat r is comou aually, th formula is P( + r) kp ( + r) k l( + r) l k l k l( + r) 66. a. For r %, th oublig tim is aroimatly 7 7 yars. P(.0) P l(.0) l l 69.7 yars l If th itrst rat r is comou cotiuously, th formula is r P kp r k r l l k l k r EXERCISES.. ( l ) l+ l+. l l l 6 ( ) l. l. l( + ) + 5. l l 6. l (l ) (l ) 00 Brooks/Col, Cgag Larig.

13 Erciss ( + ) l( + ) 6 ( + ) + 8. ( + )( ) l( + ) 8 ( + ) + 9. l( ) 0. l(5 ) 5 5. ( ). ( ) ( + ) (7) 5. ( ) 6. l( ) 7. ( ) ( l ) ()(l ) + l 9. l bcaus l 0. l bcaus l ( + ).. + l( + ) bcaus is a costat 6. 0 bcaus is a costat [l( + ) ] 8. [ l + ( + ) ] + + ( + ) ( + ) ( l + + 5) l + ( ) + ( ) + 0 l+ + (l + ) ( l + 7) ( ) l l. ( l ) l l + + l +. ( l) l ( l ) l l l. ( ) 00 Brooks/Col, Cgag Larig.

14 86 Chatr : Eotial a Logarithmic Fuctios. ( ) ( ) l l t 5. ( t ) ( t ) ( t ) t ( t ) 6. ( t / ) / ( t ) ( t) + + t t t + / / 7. ( t ) ( l t t l t ) ( t ) t+ / t + l t t t t 8. ( t+ lt) t+ lt + t 9. z z z t z z ( z) z z z z z z z 0. ( z) z z z z + z ( z) ( z) z ( z ) ( z ) ( z) z + z + z ( z ) z. ( z ) ( z ) ( z) z + z + ( z ) z. z l 5 5. a. f 6. a. f l 5 5 l f l+ f 5 l 5l l+ 5 0 ( ) 6 f () l 0 f () () l+ () () 7. a. f l( + 8) f + 8 () f () () a. f l f l+ l f () l 9. a. f l( ) f 50. a. 0 f (0) 0 (0) 0 f l( + ) + ( ) f f (0) Brooks/Col, Cgag Larig.

15 Erciss a. 5. a. / f 5l 5. a. f f 5l 5 + 5l+ 5 f ( ) f () 5l + 5 / f () f () 8.66 f ().778 f 5. a. f l( ) () f f f () f () () 9 f ().6 f ().05 / / /5 /5 /5 ( ) 5 5 /5 / /5 /5 5 5 /5 8 / ( ) /6 /6 5 /6 6 6 /6 /6 ( ) /6 ( )( ) /6 0 / ( k k k ) ( k ) k k k k k ( ) ( k ) k ( k) k ( ) ( k ) k ( k) k k k ( ) k k k k k 58. ( k ) k ( k ) k k ( ) k ( k) k ( ) k ( k) k k k ( ) ( ) k k k k k k k o [, ] by [, ] Thr is a rlativ maimum at (0, ); o rlativ miima. Thr ar iflctio oits at about (0.5, 0.6) a ( 0.5, 0.6). o [, ] by [, ] Thr is a rlativ miimum at (0, 0); o rlativ maima. Thr ar iflctio oits at about (, 0.9) a (, 0.9). 00 Brooks/Col, Cgag Larig.

16 88 Chatr : Eotial a Logarithmic Fuctios o [ 5, 5] by [, ] Thr is a rlativ miimum at (0, 0); o rlativ maima. Thr ar iflctio oits at about (, 0.69) a (, 0.69). o [ 5, 5] by [, 9] Thr is a rlativ miimum at (0, ); o rlativ maima. Thr ar o iflctio oits o [, 8] by [, ] Thr is a rlativ maimum at about (, 0.5); rlativ miimum at (0, 0). Thr ar iflctio oits at about (0.59, 0.9) a (., 0.8). o [, 5] by [, ] Thr is a rlativ maimum at about (, 0.7); o rlativ miimum. Thr is a iflctio oit at about (, 0.7) o [, ] by [, ] Thr is a rlativ maimum at about ( 0.7, 0.7); rlativ miimum at about (0.7, 0.7). Thr ar o iflctio oits. (Th fuctio is ot fi at 0.) o [, ] by [, ] Thr ar rlativ miima at about ( 0.6, 0.8) a (0.6, 0.8); thr ar o rlativ maima (Th fuctio is ot fi at 0.) Thr ar iflctio oits at about ( 0., 0.07) a (0., 0.07) 67. y y ( y y ) () yy ' ( y ' + y ) 0 y'( y ) y y y ' y 68. y l y 0 l (0) ( y y) y ' y yy ' (l y + ) 0 y'( y ) l y y l y yl y y ' y y y 69. f 0, f ' 0, 000(0.95) 0.95 f ' () f '() 7800 f '() Aual salary icrass by about $980 r tra yar of calculus t f() t 5 0.t 0.t f '( t) 5(0.) 0.t f '( t). 0.(0) f '(0). f '(0).7866 Th umbr of thousa mgawatts icrass by about. r tra yar. 00 Brooks/Col, Cgag Larig.

17 Erciss a. To fi th rat of growth aftr 0 yars, valuat V (0). 0.05t Vt t 0.05t V ( t) 000 (0.05) (0) 0 V (0) Th rat of growth aftr 0 yars is $50 r yar. To fi th rat of growth aftr 0 yars, valuat V (0). 0.05t V () t (0) 0.5 V (0) Th rat of growth aftr 0 yars is $8. r yar. 7. a. To fi th rat of chag at t 0 valuat V (0). 0.5t Vt () 0, t V () t 0,000 ( 0.5) 0.5t (0) V (0) Th valu is crasig by $500 r yar. To fi th rat of chag aftr yars, valuat V (). 0.5() V () Th valu is crasig by $78.05 r yar t Pt t 0.075t P ( t) 6.5 (0.075) t Pt () () 75 t P t ( 0.) 5 I 05, t (0) a. P (0) (0) P (0) It is crasig by 5% r tim uit. I 05, th worl oulatio is growig by millio ol r yar. P () 5 0.() 8. It is crasig by 8.% r tim uit t At (). A ( t). ( 0.05) 0.06 a. 0.05t 0.05t A Th amout rmaiig aftr 0 hours is crasig by 0.06 mg r hour. 0.05() 0. A () Th amout rmaiig aftr hours is crasig by 0.05 mg r hour. 0.05(0) 0 (0) Tt T () t 5.5t.5t a. T (0) 5 Th tmratur is crasig by 5 grs r hour. T.5() () 5 7 Th tmratur is crasig by 7 grs r hour S wkly sals (i thousas) 0. S 900 ( 0.) rat of chag of sals r wk a. 0. S () 90 rat of chag of sals r wk aftr wk 90(0.908) 8. thousa sals r wk S (0) 90 90(0.679) thousa sals r wk aftr 0 wks 78. Nt N ( t) 50, 000 ( 0.) 0, t 50, 000( ) 0.t 0.t a. 0.(0) N (0) 0, 000 0, 000 ol r hour 0.(8) N (8) 0, ol r hour 00 Brooks/Col, Cgag Larig.

18 90 Chatr : Eotial a Logarithmic Fuctios 79. To fi th maimum cosumr itur, solv E 0, whr E D. 0.0 E D E ( 0.0) Sttig E 0, w gt (00 ) 0 $00 W us th sco rivativ tst to show that 00 is a maimum E 5000 ( 0.0) ( 0.0) E (00) < 0 so E is maimiz. 80. To fi th maimum cosumr itur, solv E 0, whr E D() E D E ( 0.05) (0 ) $0 Us th sco rivativ tst to show that 0 is a maimum E 8000 ( 0.05) ( 0.05) E (0) < 0 so E is maimiz ric fuctio (i ollars) a. R rvu fuctio To maimiz R(), iffrtiat R ( 0.0) 00 ( 0. ) R 0 wh 5, which is th oly critical valu. 0. W calculat R () for th sco rivativ tst R 00 ( 0.0)( 0. ) + 00 ( 0.0) ( 0.0)( 0.+ ) ( 0. ) ( 0. ) At th critical valu of 5, R (5) 80 < 0, so R() is maimiz at At 5, 00 (5) Th rvu is maimiz at quatity 5000 a ric $ Brooks/Col, Cgag Larig.

19 Erciss a. R l To maimiz, iffrtiat a solv R () 0. R l l 0 l l Now us th sco rivativ tst. R < 0 for > 0, so R is maimiz. At, l Th rvu is maimiz at 0,086 uits a ric $. 8. a..5t T (0.5) T(0.5) t.5t T 0 (.5) 05.5(0.5) T (0.5) 05.8 Aftr 5 miuts, th tmratur of th br is 57.5 grs a is icrasig at th rat of.8 grs r hour..5t T () T() t.5t T 0 (.5) 05.5() T () 05. Aftr hour, th tmratur of th br is 69. grs a is icrasig at th rat of. grs r hour. 8. a. ( 0.5t N 00,000 ) ( 0.5(0.5) N ) ( t) (0.5) 00,000, t N 00, 000 ( 0.5) 00, (0.5) N (0.5) 00, ,880 Aftr 0 miuts, th umbr of ol who hav har th bullti is about,0 a is icrasig at th rat of 77,880 r hour. ( 0.5t N 00,000 ) ( 0.5() N ) ( t) () 00, , 7 N 00, 000 ( 0.5) 00, 000 N () 00, 000, Aftr hours, th umbr of ol who hav har th bullti is about 55,7 a is icrasig at th rat of, r hour t 0.5() 85. A s of 0 mtrs r sco aftr. scos. 86. o [0.9.9] by [0,0] Aftr 0. scos, Lwis s acclratio was mtrs r sco. Aftr 9. scos, his acclratio was 0.00 mtrs r sco. 87 a. 88. Th fuctio caot b iffrtiat by th owr rul bcaus h bas, is a costat a th variabl, is i th ot. o [5, 80] by [0, 0] f(5).; f (5) 0.8 At ag 5, th fastst ma s tim is hours. miuts a icrasig at about 0.8 miuts r yar. c. f(80).9; f (80) 7.76 At ag 80, th fastst ma s tim is hours 5.9 miuts a icrasig at about 7.76 miuts r yar. 00 Brooks/Col, Cgag Larig.

20 9 Chatr : Eotial a Logarithmic Fuctios a. f '( ) f f 9. c a. 9. l 9. l f f ' f c. l5 5 l5 0 5 l ' Fals: f l f. It os ot ivolv a f atural log. 97. l a 98. l a 99. bt a Nt K for t 0, k> 0, b> 0. bt N a K t bt bt bn( a ) bt ( a t ) bt ( ) a K a b bt a bn(l ) l K bn a bt K bn l K ( N ) If N > K, l K 0 N crasig. If N < K, K oulatio will b icrasig. < a th oulatio will b l > 0 a th N 00. y a for 0, a >, b> 0. b b y a( + ( b)) a ( b) y 0 wh b 0. y 0 at. b Sco rivativ tst b b y a( b) ( b) + a ( b) b b b ab ( + b ) ab ( b ) b b b b ab b y ab ( b ) ab ( ) < 0 ( a >, b> 0) This shows that is i a maimum. b Th oulatio is maimiz wh th artal stock is b. 00 Brooks/Col, Cgag Larig.

21 Erciss To maimiz R(r), solv R (r) 0. R () r a b 0 r a br 0 a b r Usig th sco rivativ tst, R () r a < 0 r so R(r) is maimiz. 0 a. y log a for a > 0 a > 0 y a y l l a l yl a l y l a l loga l a 0. To fi th maimum coctratio, solv A (t) 0, whr 0. 0.t 0.6t At () ( ) t 0.6t 0.t 0.6t ( 0.) ( 0.6) 0.6t 0.t A () t t 0.6t t 0.6t l(0. ) l(0. ) l(0.) + ( 0. t) l 0. + ( 0.6 t) 0.t l 0. l 0. t.0 l0. l0. 0. Now us th sco rivativ tst. 0.6t 0.t A ( t) 0. ( 0.6) 0. ( 0.) 0.t 0.6t (.0) 0.6(.0) A (.0) so A(t) is maimiz. Th maimum coctratio occurs at about.0 hours. y l l a yl a l y l a l y a loga y log l a l a 0. ( l a) a (l a) sic l 05.. l. (l.) l0.6 (l0.6) a. 0 (l 0)0 (l ) (l ) c. (l ) () (l ). 5 (l 5)5 (6 ) 6(l 5) 5. (l ) ( ) ( l ) Brooks/Col, Cgag Larig.

22 9 Chatr : Eotial a Logarithmic Fuctios 08. a. 5 (l 5)5 (l ) (l ) c. (l ) () ( l ) (l 9)9 (0 ) 0(l 9) 9. 0 (l0)0 ( ) ( l0)0 09. a. log (l ) log 0( ) (l 0)( ) c. log ( ) (l )( ) 0. a. log (l ) log ( + ) (l )( + ) c. 0 log ( ) (l 0)( ) EXERCISES.. a. l f ( t) l t l t l f( t) l t t t t For t, th rlativ rat of chag is. For t 0, th rlativ rat of chag is a. l f ( t) l t l t l f( t) l t t t t For t, th rlativ rat of chag is. For t 0, th rlativ rat of chag is a. 0.t l f( t) l00 l00 + l l t l f( t) 0. t 0.t. a. 0.5t l f( t) l00 l00 + l l00 0.5t l f( t) 0.5 For t 5, th rlativ rat of chag is 0.. For t, th rlativ rat of chag is 0.5. t 0.5t 5. a. t t l f ( t) l t l f( t) t For t 0, th rlativ rat of chag is (0) 0. t 7. a. l f ( t) l t l f( t) t t For t 0, th rlativ rat of chag is (0) a. 8. a. t t l f ( t) l t l f( t) t For t 5, th rlativ rat of chag is (5) 75. t t l f ( t) l t l f( t) t For t 5, th rlativ rat of chag is (5) Brooks/Col, Cgag Larig.

23 Erciss a. l f( t) l 5 t / l 5( t ) l 5 + l( t ) l f( t) 0 + t ( t ) For t 6, th rlativ rat of chag is (6 ) a. l f( t) l00 t+ l00 + l( t + ) l f( t) 0 + t ( t+ ) For t 8, th rlativ rat of chag is. (0) 0. A l A l(. ) l. + l l l A A. (0.5) l A 0.5 A 0.5. For t 0, th rlativ rat of chag is l A (0) 0.5 or 50%. Th stock is arciatig by 50% r yar.. B l B l( ) l + l l + 0. l B B (0.) l B 0. B 0. For t, th rlativ rat of chag is l B () 0. or %. Th stock is arciatig by % r yar.. 0.0t l Nt l( ) 0.0t. (0.0) l Nt t 0.0t For t 0, th rlativ rat of chag is 0.0(0). (0.0) 0.0(0) or 0.7%.. 0.0t l Nt l(0. +. ) 0.0t. (0.0) l Nt t 0.0t 0.+. For t 0, th rlativ rat of chag is 0.. (0.0) or 0.77%. 5. a. 0.0t l Pt l( +. ) 0.0t. (0.0) 0.0t l Pt t t 0.0t For t 8, th rlativ rat of chag is 0.0(8) or.%. 0.0(8) +. If th rlativ rat of chag is.5% 0.005, th 0.0t t t 0.0t t t t l 0.06 ( 0.05 ) t l ( 0.05) 5. Th rlativ rat of chag will rach.5% i about 5. yars. 6. a. 0.05t l Pt l(6 +.7 ) 0.05t.7 (0.05) 0.05t l Pt t 0.05t 0.05t For t 8, th rlativ rat of chag is (8) 0.05 or.5% (8) Th rlativ rat of chag will rach.5% i about 8. yars. 00 Brooks/Col, Cgag Larig.

24 96 Chatr : Eotial a Logarithmic Fuctios 7. D 00 5, 0 8. a. a. Elasticity of ma is D ( 5) 5 E D Evaluatig at 0, E (0) Th lasticity is lss tha, a so th ma is ilastic at 0. ( 8) E (5) 0 5 (5) 5 E(5) Sic E(5) >, th ma is lastic. 9. a. ( ) E 0. a. (0) (0) 00 E(0) Sic E (0), th ma is uitary lastic. ( ) E (5) (5) 75 E(5) Sic E (5) <, th ma is ilastic.. a. ( 00) E. a. ( 500) E Sic E (), th ma is uitary lastic. Sic E (), th ma is uitary lastic.. a. E / [ (75 ) ]( ) / (75 ) (75 ). a. / [ (00 ) ]( ) E / (00 ) 00 (50) 50 [75 (50)] 50 E(50) Sic E (50) >, th ma is lastic E(0) Sic E (0) <, th ma is ilastic. 5. a E 6. a E Sic E (0) >, th ma is lastic. Sic E (5) >, th ma is lastic. 0.0 (000 )( 0.0) 0.05 (6000 )( 0.05) 7. a. E a. E E (00) 0.0(00) E (00) 5 Sic E (00) >, th ma is lastic. Sic E (00) >, th ma is lastic. 00 Brooks/Col, Cgag Larig.

25 Erciss Th ma fuctio is D ( ) To trmi whthr th alr s to rais or lowr th ric to icras rvu, w to trmi th lasticity of ma. D E D ( )( 0.00) ( ) Wh th cars sll at a ric of $,000, E 0.00(, 000) (,000) (, 000) 5 Sic E 8>, to icras rvus th alr shoul lowr rics.. To trmi th lasticity of ma, w cosir / { [50,000(.75 ) ( )] } E / 50, 000(.75 ) (.75 ) Wh th far is 75 cts, E (.5).5 (.75.5).5.5 Sic ma is lastic, raisig th far will ot succ.. D (0 + ) 0 0+ D E D 0+ E (6) Sic E <, icrasig rics shoul icras 8 rvus. Ys, th commissio shoul grat th rqust. 0. To trmi th lasticity of ma, w cosir ( ) E Wh th ric is $5, E (5) 5 5 Sic ma is lastic, th iscouts will succ.. To trmi th lasticity of ma, w cosir / [0,000(75 ) ( )] E / 80,000(75 ) (75 ) Wh th ric is 50 cts, E(50) 50 (75 50) Sic E(50), raisig rics will ot succ.. To trmi th lasticity of ma, w cosir (9.5 )( 0.0) E Wh th ric is $0 r barrl, E (0) 0.0(0).8 Sic ma is lastic, it shoul lowr rics. 5. To trmi th lasticity of ma, w cosir 0.06 (.5 )( 0.06) E Wh th ric is $0 r barrl, E (0) 0.06(0) 7. Sic ma is lastic, it shoul lowr rics. 6. E.859 (.509)( 0.859) Brooks/Col, Cgag Larig.

26 98 Chatr : Eotial a Logarithmic Fuctios 7.. (7.88)( 0. ) E a. E 0.75 Sic ma is ilastic, raisig rics will rais rvu. c. $, ( ) 9. a. E a. E Sic ma is ilastic, raisig rics will E(0) rais rvu. c. $0,0 c. E(89.999) , E( ) , 999, As aroachs 90, E aroachs ifiity.. Th mioit is 5. E(5) For D a b, ( b) b a b a b E b(0) (b) For 0, E (0) a b(0) 0. (c) Wh th valu aroachs b, E aroachs ifiity. a a b, b b a a a a ( b ) a b a a a a a ( ) a a b b () For th mioit, E.. 5. rct chag i ma E rct chag i ric % % or 5% f. l f l... f l f l. f l f l f. l f l l 6. ( ) E ( )( ) E E 00 Brooks/Col, Cgag Larig.

27 Rviw Erciss for Chatr ( ) E ( )( ) E E 8. Dma will icras wkly. 9. Dma will icras strogly. 50. Elastic for hatig oil a ilastic for oliv oil. (It is commoly us.) 5. Ilastic for cigartts (thy ar habit-formig) a lastic for jwlry. c c 5. a. + E c c If E( ), th w kow from art (a) that D c c. 5. c a ( c) E c 5. c If S a, th c a c S ( a )( c) Es c S c a 55. If S a, th S ( a ) Es S a REVIEW EXERCISES FOR CHAPTER. a.. a. For quartrly comouig, r a 8. P( + r) 0, 000( + 0.0) 0, 000(.0) $8,85. For comouig cotiuously, r 0.08 a 8. r P (0.08)(8) 0, 000 $8, 96.8 If th rciatio is 0% r yar, r 0% 0. a t. Th formula for th valu is t t V 800,000( 0.) 800,000(0.8) Aftr yars, its valu is V 800, 000(0.8) $7, 680. For 6% comou quartrly, r a. P( + r) ( ).06 For 5.98% comou cotiuously, r a. r P % comou cotiuously has a highr yil. 0.t. For Drug A, Ct (). Aftr hours, 0.8 C() t For Drug B, Ct (). Aftr hours, C().0 Drug B has a highr coctratio. 00 Brooks/Col, Cgag Larig.

28 00 Chatr : Eotial a Logarithmic Fuctios Th valu t corrsos to / 8 0 a C () 65, 56 mgabits. São Paulo will ovrtak Tokyo i about 60 yars, or urig th yar r 0% 5% 0.05 a. P( ) P.05 l.05 l l.05 l l l Sic is i half yars, w ivi by to covrt to yars. About 7. yars. P(.05).5P.05.5 l.05 l.5 l.05 l.5 l.5 l half yars 8.. yars 8. a. For comouig cotiuously, r 7% P P 0.07 l l 9.9 yars 0.07 To icras by 50%, P P.5 l yars Th roortio of otassium 0 rmaiig aftr t t millio yars is. Sic 97.% 0.97 rmais, t t l 0.97 t l millio yars ol t. W wat to solv Nt,000,000( ) with N(t) 500, t, 000, 000( ) 500, t t t t l 0.5 t l hours 0. Sic 99.9% rmais, t t l millio yars ol Sic th rat of icras is % 0.0 r yar, r 0.0. To fi wh ma will icras by 50% 0.5,.5 P P( + 0.0).5 (.0) l.5.7 yars l.0 00 Brooks/Col, Cgag Larig.

29 Rviw Erciss for Chatr 0. a. If th itrst rat is 6.5% comou quartrly, r ( ) (000) +.5 l.5 l ( ) 5. quartrs 6. yars If th itrst rat is 6.5% comou cotiuously, r (000) l.5 6. yars l l 6.. a. To rach 0% 0. of th ol, (t) t t t l 0.7 t l0.7 ays 0.0 To rach 0% 0. of th ol, t () t t 0.6 t l ays 0.0 ( ) l( ) ( ) 7. l ( ) / + ( ) 8. l + / ( + ) + / / l l l /. l ( l ). ( l ) l + l. ( ). 5. l ( ) + 6. l / l / l l l / 7. (5 + l + ) 0+ l + 0+ l + 8. ( + l ) 6 + l l 00 Brooks/Col, Cgag Larig.

30 0 Chatr : Eotial a Logarithmic Fuctios 9. ( ) 6 () ( ) ().. Rlativ miimum at (0,.) Rlativ maimum at (0, 6).. a. 0. S S 500 ( 0.) 50 0.() S () 50 6 Sals ar icrasig by 6,000 aftr wk. 0.(0) S (0) Sals ar icrasig by 55,000 aftr 0 wks.. a. 0.08t At () t 0.08t A ( t).5 ( 0.08) (0) A (0) Th amout of th rug i th bloostram immiatly aftr th ijctio is crasig by 0. mg r hour. 0.08(5) A (5) Th amout of th rug i th bloostram aftr 5 hours is crasig by 0.08 mg r hour. 5. Pt () 00 00l( t+ ) P ( t) 00 rat of chag t + P (5) rat of chag aftr 5 scos Th rat of chag aftr 5 scos is crasig by % r sco. 6. a. 0.t Tt () t 0.t T 5 ( 0.).5 0 T (0).5.5 grs r hour 7. a. 0.t Nt 0, 000( ) 0.t 0.t N ( t) 0,000 ( 0.) () N () Aftr hour, th rat of chag i th umbr of iform ol is icrasig by 6667 r hour. 0.5 T (5).5. grs r hour 0.(8) N (8) Aftr 8 hours, th rat of chag i th umbr of iform ol is icrasig by 86 r hour. 00 Brooks/Col, Cgag Larig.

31 Rviw Erciss for Chatr t V() t 50t 0.08t 0.08t V t t t 0.08t 0.08t ( 0.08) 00t t W st V (t) 0 to maimiz V(t). 0.08t 0.08t 0 00t t 0.08t 0 t (5 t) A critical valu is t 5. Now w us th sco rivativ tst. 0.08t 0.08t 0.08t 0.08t V ( t) t ( 0.08) 8t t ( 0.08) 0.08t 0.08t 0.08t 0.08t 00 8t 8t + 0.t 0.08t 0.08t 0.08t 00 6t + 0.t V (5) < 0 so V is maimiz. Th rst valu is maimiz i 5 yars a. 0.5 R 00 R ( 0.5) ( 0.5 ) R 0 wh, which is th oly critical valu R 00 ( 0.5)( 0.5 ) + 00 ( 0.5) ( 0.5)( 0.5+ ) ( 0.5 ) R () 50 < 0, so R is maimiz at. Thus, quatity 000 a ric () 00 $7.58 maimiz rvu. 0. a. R (5 l ) 5 l R 5 l l R 0 wh, which is th oly critical valu. R < 0 wh so R is maimiz. 5 l( ) Thus, th quatity a th ric $ maimiz rvu. 00 Brooks/Col, Cgag Larig.

32 0 Chatr : Eotial a Logarithmic Fuctios. To fi th maimum cosumr itur, solv E 0, whr E D. 0.0 E (5,000 ) E 5, , 000 ( 0.0) , E 0 wh 50, which is th oly critical valu. W us th sco rivativ tst to show that 50 is a maimum E 5, 000 ( 0.0) ( 0.0) (50) 0.0(50) E (50) (50) < 0 so E is maimiz. Th ric $50 maimizs cosumr itur... o [ 5, 5] by [ 5, 5] Th fuctio has a rlativ maimum at about (,.69) a a rlativ miimum at (0, 0). Thr ar iflctio oits at about (,.7) a (6,.). o [ 5, 5] by [ 5, 5] Th fuctio has a rlativ maimum at about ( 0.7, 0.) a a rlativ miimum at about (0.7, 0.). Thr ar iflctio oits at about ( 0., 0.07) a (0., 0.07).. To fi th maimum cosumr itur, solv E 0, whr E D E (00 ) E ( 0.00) E 0 wh th ric is about 769, which is th oly critical valu E ( 0.00) (769) 0.00(769) E (769) (769) 0.00(769) 0.6 < 0 so E is maimiz. Th ric $769 maimizs cosumr itur R (0 ) R ( ) R () 0 wh th quatity is about 0. R 0 ( ) R (0) (0) (0) 0.5 < 0 so R is maimiz. Th quatity 0 maimizs rvu (0) (0) (0) 00 Brooks/Col, Cgag Larig.

33 Rviw Erciss for Chatr t Gt () t 0.0t (0.0) l Gt 0.0 t t 0.0t If t 0, th th rlativ rat of chag is % G () t 0.0(0) 0. For t 0, % Gt () 0.0(0) D ( ) E D 6 6 E() Sic E() <, ma is ilastic. Raisig rics will icras rvu. / 00(600 ) D ( ) 00 E D 00(600 ) / 00(600 ) (600 ) E (50) (600 50) 500 Sic E <, ma is ilastic. Raisig rics will icras rvu.. D ( 0.58 ) E D 0... Dma is ilastic. 5. Rlativ rat of chag of P l P. l P (9) At 9, % (9) (9) 5. a. Sic ma is lastic, th alr shoul lowr th ric. c. E(8.7) ; thus at $8700 lasticity quals. o [0, 7] by [0, 5] D E D ( ) Th ma is lastic at $0,000 bcaus E(0) Brooks/Col, Cgag Larig.