The Natural Logarithmic Function: Differentiation. The Natural Logarithmic Function

Transcription

1 60_00.q //0 :0 PM Pag CHAPTER Logarihmic, Eponnial, an Ohr Transcnnal Funcions Scion. Th Naural Logarihmic Funcion: Diffrniaion Dvlop an us propris of h naural logarihmic funcion. Unrsan h finiion of h numbr. Fin rivaivs of funcions involving h naural logarihmic funcion. Th Naural Logarihmic Funcion Rcall ha h Gnral Powr Rul Th Grangr Collcion JOHN NAPIER (0 67) Logarihms wr invn b h Scoish mahmaician John Napir. Alhough h i no inrouc h naural logarihmic funcion, i is somims call h Napirian logarihm. n n C, n n Gnral Powr Rul has an imporan isclaimr i osn appl whn n. Consqunl, ou hav no foun an anirivaiv for h funcion f. In his scion, ou will us h Scon Funamnal Thorm of Calculus o fin such a funcion. This anirivaiv is a funcion ha ou hav no ncounr prviousl in h. I is nihr algbraic nor rigonomric, bu falls ino a nw class of funcions call logarihmic funcions. This paricular funcion is h naural logarihmic funcion. Dfiniion of h Naural Logarihmic Funcion Th naural logarihmic funcion is fin b ln > 0., Th omain of h naural logarihmic funcion is h s of all posiiv ral numbrs. From h finiion, ou can s ha ln is posiiv for > an ngaiv for 0 < <, as shown in Figur.. Morovr, ln 0, bcaus h uppr an lowr limis of ingraion ar qual whn. = If >, > 0. = If <, < 0. If >, hn ln > 0. If 0 < <, hn ln < 0. Figur. EXPLORATION Graphing h Naural Logarihmic Funcion Using onl h finiion of h naural logarihmic funcion, skch a graph of h funcion. Eplain our rasoning.

2 60_00.q //0 :0 PM Pag SECTION. Th Naural Logarihmic Funcion: Diffrniaion = ln (, 0) Each small lin sgmn has a slop of Figur.. NOTE Slop fils can b hlpful in ging a visual prspciv of h ircions of h soluions of a iffrnial quaion. To skch h graph of ln, ou can hink of h naural logarihmic funcion as an anirivaiv givn b h iffrnial quaion. Figur. is a compur-gnra graph, call a slop (or ircion) fil, showing small lin sgmns of slop. Th graph of ln is h soluion ha passs hrough h poin, 0. You will su slop fils in Scion 6.. Th following horm liss som basic propris of h naural logarihmic funcion. THEOREM. Propris of h Naural Logarihmic Funcion Th naural logarihmic funcion has h following propris.. Th omain is 0, an h rang is,.. Th funcion is coninuous, incrasing, an on-o-on.. Th graph is concav ownwar. = = = = = = = = = = = ln = = = = Th naural logarihmic funcion is incrasing, an is graph is concav ownwar. Figur. Proof Th omain of f ln is 0, b finiion. Morovr, h funcion is coninuous bcaus i is iffrniabl. I is incrasing bcaus is rivaiv f Firs rivaiv is posiiv for > 0, as shown in Figur.. I is concav ownwar bcaus f Scon rivaiv is ngaiv for > 0. Th proof ha f is on-o-on is lf as an rcis (s Ercis ). Th following limis impl ha is rang is h nir ral lin. lim ln an lim ln 0 Vrificaion of hs wo limis is givn in Appni A. Using h finiion of h naural logarihmic funcion, ou can prov svral imporan propris involving opraions wih naural logarihms. If ou ar alra familiar wih logarihms, ou will rcogniz ha hs propris ar characrisic of all logarihms. LOGARITHMS Napir coin h rm logarihm, from h wo Grk wors logos (or raio) an arihmos (or numbr), o scrib h hor ha h spn 0 ars vloping an ha firs appar in h book Mirifici Logarihmorum canonis scripio (A Dscripion of h Marvlous Rul of Logarihms). THEOREM. Logarihmic Propris If a an b ar posiiv numbrs an n is raional, hn h following propris ar ru.. ln 0. lnab ln a ln b. lna n n ln a. ln a ln a ln b b

3 60_00.q //0 :0 PM Pag CHAPTER Logarihmic, Eponnial, an Ohr Transcnnal Funcions Proof Th firs propr has alra bn iscuss. Th proof of h scon propr follows from h fac ha wo anirivaivs of h sam funcion iffr a mos b a consan. From h Scon Funamnal Thorm of Calculus an h finiion of h naural logarihmic funcion, ou know ha ln. So, consir h wo rivaivs an lna a a ln a ln 0. Bcaus lna an ln a ln ar boh anirivaivs of, h mus iffr a mos b a consan. lna ln a ln C B ling, ou can s ha C 0. Th hir propr can b prov similarl b comparing h rivaivs of ln n an n ln. Finall, using h scon an hir propris, ou can prov h fourh propr. ln a b lnab ln a lnb ln a ln b Eampl shows how logarihmic propris can b us o pan logarihmic prssions. EXAMPLE Epaning Logarihmic Eprssions f() = ln a. ln 0 ln 0 ln 9 Propr 9 b. Rwri wih raional ponn. ln ln g() = ln ln Propr c. ln 6 ln6 ln Propr ln 6 ln ln Propr. ln ln ln ln ln ln ln ln ln ln ln ln Figur. Whn using h propris of logarihms o rwri logarihmic funcions, ou mus chck o s whhr h omain of h rwrin funcion is h sam as h omain of h original. For insanc, h omain of f ln is all ral numbrs cp 0, an h omain of g ln is all posiiv ral numbrs. (S Figur..)

4 60_00.q //0 :0 PM Pag SECTION. Th Naural Logarihmic Funcion: Diffrniaion = Ara = =.7 is h bas for h naural logarihm bcaus ln. Figur. Th Numbr I is likl ha ou hav sui logarihms in an algbra cours. Thr, wihou h bnfi of calculus, logarihms woul hav bn fin in rms of a bas numbr. For ampl, common logarihms hav a bas of 0 an hrfor log 0 0. (You will larn mor abou his in Scion..) Th bas for h naural logarihm is fin using h fac ha h naural logarihmic funcion is coninuous, is on-o-on, an has a rang of,. So, hr mus b a uniqu ral numbr such ha ln, as shown in Figur.. This numbr is no b h lr. I can b shown ha is irraional an has h following cimal approimaion Dfiniion of Th lr nos h posiiv ral numbr such ha ln. = ln (, ) (, ) ( 0, 0) (, ) FOR FURTHER INFORMATION To larn mor abou h numbr, s h aricl Unpc Occurrncs of h Numbr b Harris S. Shulz an Bill Lonar in Mahmaics Magazin. To viw his aricl, go o h wbsi Onc ou know ha ln, ou can us logarihmic propris o valua h naural logarihms of svral ohr numbrs. For ampl, b using h propr ln n n ln n n ou can valua ln n for various valus of n, as shown in h abl an in Figur.6. (, ) (, ) If n, hn ln n. Figur.6 ln Th logarihms shown in h abl abov ar convnin bcaus h -valus ar ingr powrs of. Mos logarihmic prssions ar, howvr, bs valua wih a calculaor. EXAMPLE Evaluaing Naural Logarihmic Eprssions a. ln 0.69 b. ln.66 c. ln 0..0

5 60_00.q //0 :0 PM Pag 6 6 CHAPTER Logarihmic, Eponnial, an Ohr Transcnnal Funcions Th Drivaiv of h Naural Logarihmic Funcion Th rivaiv of h naural logarihmic funcion is givn in Thorm.. Th firs par of h horm follows from h finiion of h naural logarihmic funcion as an anirivaiv. Th scon par of h horm is simpl h Chain Rul vrsion of h firs par. THEOREM. Drivaiv of h Naural Logarihmic Funcion L u b a iffrniabl funcion of... u > 0 ln u u u ln, > 0 u u, EXAMPLE Diffrniaion of Logarihmic Funcions EXPLORATION Us a graphing uili o graph an ln in h sam viwing winow, in which 0. an 8. Eplain wh h graphs appar o b inical. a. b. u ln u ln u u c. Prouc Rul ln ln ln ln ln. Chain Rul ln ln ln ln u u Napir us logarihmic propris o simplif calculaions involving proucs, quoins, an powrs. Of cours, givn h availabili of calculaors, hr is now lil n for his paricular applicaion of logarihms. Howvr, hr is gra valu in using logarihmic propris o simplif iffrniaion involving proucs, quoins, an powrs. EXAMPLE Diffrnia Logarihmic Propris as Ais o Diffrniaion f ln. Soluion Bcaus f ln ln ln ou can wri f. Rwri bfor iffrniaing. Diffrnia. inicas ha in h HM mahspac CD-ROM an h onlin Euspac ssm for his, ou will fin an Opn Eploraion, which furhr plors his ampl using h compur algbra ssms Mapl, Mahca, Mahmaica, an Driv.

6 60_00.q //0 :0 PM Pag 7 SECTION. Th Naural Logarihmic Funcion: Diffrniaion 7 EXAMPLE Diffrnia Logarihmic Propris as Ais o Diffrniaion f ln. Soluion f ln ln ln ln f 6 Wri original funcion. Rwri bfor iffrniaing. Diffrnia. Simplif. NOTE In Eampls an, b sur ou s h bnfi of appling logarihmic propris bfor iffrniaing. Consir, for insanc, h ifficul of irc iffrniaion of h funcion givn in Eampl. On occasion, i is convnin o us logarihms as ais in iffrniaing nonlogarihmic funcions. This procur is call logarihmic iffrniaion. EXAMPLE 6 Logarihmic Diffrniaion Fin h rivaiv of,. Soluion No ha > 0 for all. So, ln is fin. Bgin b aking h naural logarihm of ach si of h quaion. Thn appl logarihmic propris an iffrnia implicil. Finall, solv for. ln ln ln ln ln, Wri original quaion. Tak naural log of ach si. Logarihmic propris Diffrnia. Simplif. Solv for. Subsiu for. Simplif.

7 60_00.q //0 :0 PM Pag 8 8 CHAPTER Logarihmic, Eponnial, an Ohr Transcnnal Funcions Bcaus h naural logarihm is unfin for ngaiv numbrs, ou will ofn ncounr prssions of h form ln u. Th following horm sas ha ou can iffrnia funcions of h form ln u as if h absolu valu sign wr no prsn. THEOREM. Drivaiv Involving Absolu Valu If u is a iffrniabl funcion of such ha u 0, hn. ln u u u Proof If hn an h rsul follows from Thorm.. If hn u u > 0, u, u < 0, u, an ou hav ln u lnu u u u u. u EXAMPLE 7 Drivaiv Involving Absolu Valu Fin h rivaiv of f ln cos. = ln ( + + ) (, ln ) Rlaiv minimum Th rivaiv of changs from ngaiv o posiiv a. Figur.7 Soluion Using Thorm., l u cos an wri ln cos u u EXAMPLE 8 sin cos an. Loca h rlaiv rma of ln. Simplif. Fining Rlaiv Erma Soluion Diffrniaing, ou obain. ln u u u u cos Bcaus 0 whn, ou can appl h Firs Drivaiv Ts an conclu ha h poin, ln is a rlaiv minimum. Bcaus hr ar no ohr criical poins, i follows ha his is h onl rlaiv rmum (s Figur.7).

8 60_00.q //0 :0 PM Pag 9 SECTION. Th Naural Logarihmic Funcion: Diffrniaion 9. Compl h abl blow. Us a graphing uili an Simpson s Rul wih n 0 o approima h ingral.. (a) Plo h poins gnra in Ercis an connc hm wih a smooh curv. Compar h rsul wih h graph of ln. (b) Us a graphing uili o graph for 0.. Compar h rsul wih h graph of ln. In Erciss 6, us a graphing uili o valua h logarihm b (a) using h naural logarihm k an (b) using h ingraion capabiliis o valua h ingral /.. ln. ln 8.. ln ln 0.6 In Erciss 7 0, mach h funcion wih is graph. [Th graphs ar labl (a), (b), (c), an ().] (a) (c) Erciss for Scion. / In Erciss 6, skch h graph of h funcion an sa is omain. (b) () 7. f ln 8. f ln 9. f ln 0. f ln. f ln. f ln. f ln. f ln. f ln 6. g ln In Erciss 7 an 8, us h propris of logarihms o approima h inica logarihms, givn ha ln 0.69 an ln (a) ln 6 (b) ln (c) ln 8 () ln 8. (a) ln 0. (b) ln (c) ln () In Erciss 9 8, us h propris of logarihms o pan h logarihmic prssion. 9. ln 0. ln. ln z. lnz. ln a. lna. ln 6. ln 7. ln zz 8. ln In Erciss 9, wri h prssion as a logarihm of a singl quani. 9. ln ln 0. ln ln ln z. ln ln ln. ln ln ln. ln ln. ln ln ln In Erciss an 6, (a) vrif ha f g b using a graphing uili o graph f an g in h sam viwing winow. (b) Thn vrif ha f g algbraicall.. 6. f ln, f ln, In Erciss 7 0, fin h limi. 7. lim ln 8. lim ln lim ln 0. lim ln In Erciss, fin an quaion of h angn lin o h graph of h logarihmic funcion a h poin, 0.. ln. ln S for work-ou soluions o o-numbr rciss. (, 0) > 0, 6 g ln ln g ln ln ln 7 (, 0) 6

9 60_00.q //0 :0 PM Pag 0 0 CHAPTER Logarihmic, Eponnial, an Ohr Transcnnal Funcions. ln. ln In Erciss 70, fin h rivaiv of h funcion.. g ln 6. h ln 7. ln 8. ln 9. ln 0. ln. f ln.. g ln.. lnln 6. lnln 7. ln 8. ln 9. f ln 60. f ln ln 6. lnsin 6. lncsc cos lncos ln sin ln cos ln sc an sin ln ln 69. f 70. g In Erciss 7 76, (a) fin an quaion of h angn lin o h graph of f a h givn poin, (b) us a graphing uili o graph h funcion an is angn lin a h poin, an (c) us h rivaiv faur of a graphing uili o confirm our rsuls. 7. f ln, f sin ln,, 0 7. f ln,, f ln, f ln sin, f ln, In Erciss 77 an 78, us implici iffrniaion o fin /. 77. ln ln 0,, 0 In Erciss 79 an 80, us implici iffrniaion o fin an quaion of h angn lin o h graph a h givn poin. 79. ln, 80. ln,, ln, ln, 0 h ln 0, f ln In Erciss 8 an 8, show ha h funcion is a soluion of h iffrnial quaion. 8. ln 8. ln In Erciss 8 88, loca an rlaiv rma an inflcion poins. Us a graphing uili o confirm our rsuls. 8. ln 8. ln 8. ln ln ln Linar an Quaraic Approimaions In Erciss 89 an 90, us a graphing uili o graph h funcion. Thn graph P f f an Funcion Diffrnial Equaion 0 0 ln P f f f in h sam viwing winow. Compar h valus of f, P, an P an hir firs rivaivs a. 89. f ln 90. f ln In Erciss 9 an 9, us Nwon s Mho o approima, o hr cimal placs, h -coorina of h poin of inrscion of h graphs of h wo quaions. Us a graphing uili o vrif our rsul. 9. ln, 9. ln, In Erciss 9 98, us logarihmic iffrniaion o fin / Wriing Abou Concps 99. In our own wors, sa h propris of h naural logarihmic funcion. 00. Dfin h bas for h naural logarihmic funcion. 0. L f b a funcion ha is posiiv an iffrniabl on h nir ral lin. L g ln f. (a) If g is incrasing, mus f b incrasing? Eplain. (b) If h graph of f is concav upwar, mus h graph of g b concav upwar? Eplain.

10 60_00.q //0 :0 PM Pag SECTION. Th Naural Logarihmic Funcion: Diffrniaion Wriing Abou Concps (coninu) 0. Consir h funcion f ln on,. (a) Eplain wh Roll s Thorm (Scion.) os no appl. (b) Do ou hink h conclusion of Roll s Thorm is ru for f? Eplain. Tru or Fals? In Erciss 0 an 0, rmin whhr h samn is ru or fals. If i is fals, plain wh or giv an ampl ha shows i is fals. 0. ln ln ln 0. If ln, hn 0. Hom Morgag Th rm (in ars) of a $0,000 hom morgag a 0% inrs can b approima b whr is h monhl pamn in ollars. (a) Us a graphing uili o graph h mol. (b) Us h mol o approima h rm of a hom morgag for which h monhl pamn is $67.. Wha is h oal amoun pai? (c) Us h mol o approima h rm of a hom morgag for which h monhl pamn is $068.. Wha is h oal amoun pai? () Fin h insananous ra of chang of wih rspc o whn 67. an () Wri a shor paragraph scribing h bnfi of h highr monhl pamn. 06. Soun Innsi Th rlaionship bwn h numbr of cibls an h innsi of a soun I in was pr cnimr squar is 0 log 0 I 0 6. > 000 Us h propris of logarihms o wri h formula in simplr form, an rmin h numbr of cibls of a soun wih an innsi of 0 0 was pr squar cnimr. 07. Moling Daa Th abl shows h mpraur T ( F) a which war boils a slc prssurs p (pouns pr squar inch). (Sourc: Sanar Hanbook of Mchanical Enginrs) p T p T ln, ( am) A mol ha approimas h aa is T ln p 7.9p (a) Us a graphing uili o plo h aa an graph h mol. (b) Fin h ra of chang of T wih rspc o p whn p 0 an p 70. (c) Us a graphing uili o graph T. Fin lim Tp an p inrpr h rsul in h con of h problm. 08. Moling Daa Th amosphric prssur crass wih incrasing aliu. A sa lvl, h avrag air prssur is on amosphr (.07 kilograms pr squar cnimr). Th abl shows h prssurs p (in amosphrs) a slc alius h (in kilomrs). h p (a) Us a graphing uili o fin a mol of h form p a b ln h for h aa. Eplain wh h rsul is an rror mssag. (b) Us a graphing uili o fin h logarihmic mol h a b ln p for h aa. (c) Us a graphing uili o plo h aa an graph h mol. () Us h mol o sima h aliu whn p 0.7. () Us h mol o sima h prssur whn h. (f) Us h mol o fin h ra of chang of prssur whn h an h 0. Inrpr h rsuls. 09. Tracri A prson walking along a ock rags a boa b a 0-mr rop. Th boa ravls along a pah known as a racri (s figur). Th quaion of his pah is 0 ln (a) Us a graphing uili o graph h funcion. (b) Wha is h slop of his pah whn an 9? (c) Wha os h slop of h pah approach as 0? Tracri 0 0. Conjcur Us a graphing uili o graph f an g in h sam viwing winow an rmin which is incrasing a h grar ra for larg valus of. Wha can ou conclu abou h ra of growh of h naural logarihmic funcion? (a) f ln, g (b) f ln, g. Prov ha h naural logarihmic funcion is on-o-on.. (a) Us a graphing uili o graph ln. (b) Us h graph o inif an rlaiv minima an inflcion poins. (c) Us calculus o vrif our answr o par (b).