This section reviews the exponential and logarithmic functions. These functions are inverse functions of each other.

Transcription

1 6. Epoetil Logritmic Fuctios ( overview) Tis sectio reviews te epoetil ritmic fuctios. Tese fuctios re iverse fuctios of ec oter. Epoetil ritmic fuctios re ll rou us, mifestig i iverse peome. Epoetil fuctios re ivolve i te rte of growt of bcteri i potto sl, rioctive ecy, memory loss my oter situtios. Logritmic fuctios re preset weever cup of soup cools off (or wrms up), re eve ivolve i every sestio you ve ever eperiece! Epoetil Fuctios Epoetil fuctios re use to moel popultio growt, rioctive ecy, compou iterest, my oter prcticl situtios. Polyomils ivolve terms wit vribles rise to costt powers, like 8. Epoetils ivolve terms wit positive costts rise to vrible powers, like 8. Despite teir similr pperce, tese re fumetlly quite ifferet kis of fuctios. To te rigt is te grp of te epoetil fuctio f() = 2. Notice tt ec time icreses by oe, te fuctio oubles. Epoetils grow fster t y polyomil! Icresig epoetil fuctios of tis type c be use to moel popultio growt compou iterest. Epoetil fuctios rise ytime qutity is irectly proportiol to te rte of cge of tt qutity. Te bove is emple of icresig epoetil fuctio. Net we ll look t ecresig epoetil fuctio. To te rigt is te grp of te epoetil fuctio g() = 2. Notice tt ec time icreses by oe, te fuctio ecreses by lf. y = 2 = = 2 2

2 Te bove ecresig epoetil c be eiter tougt of s umber greter t oe rise to egtive epoet, or s umber less t oe rise to positive epoet. Tis is implie by te properties of epoets. Decresig epoetil fuctios re use to moel rioctive ecy. Oe very prcticl pplictio of epoetil fuctios is te clcultio of compou iterest. Usully iterest is compoue motly. We ll let A(t) eote te mout of moey te ccout s grow to fter t yers, ssumig it s compoue N times per yer t ul iterest rte r. Te Nt () 0 r = A + At Cotiuous compouig of iterest is obtie by tkig te limit s te umber of compouigs per yer, N, pproces ifiity. Lter i tis cpter you ll see tt ow te equtio for cotiuous compouig of iterest, wit iitil mout A 0 ul iterest rte r, is give by: N = A0e rt Ucostrie popultio growt is moele by essetilly te sme equtio A t Pe 0 rt Pt =, were P(t) is te popultio fter t time uits (usully miutes or yers), give iitil popultio. Te costt r is te growt rte per time uit; te lrger r, te fster te popultio grows. Te costt e is specil irrtiol costt, relte to π. Epoetils of tis umber (Euler s costt) re built-i fuctios of ll scietific clcultors, wic mkes suc epoetils esy to clculte. Rioctive ecy is govere by te equtio A( t) = A0e rt,. P 0 were A(t) is te mout of rioctive substce remiig fter t yers, give iitil mout A 0 of te substce. Te costt r epes o te rioctive substce (it s lrger, te fster te substce ecys). Logritmic Fuctios Epoetil fuctios re oe-to-oe, so ve iverse fuctios. Te iverse epoetil fuctios re clle ritmic fuctios. For emple, te iverse of te fuctio is te ritmic fuctio y = 5

3 y =. 5 te to te bse five of Sice te grp of iverse fuctio is but te grp of te fuctio reflecte troug te lie y=, we kow wt te grp of tis ritmic fuctio looks like: Notice from te grp tt ritms ve oly positive umbers i teir omis, just s epoetils oly ve positive umbers i teir rges. Logritmic fuctios ve my prcticl uses, from solvig epoetil equtios to uerstig ow our seses work. Eve te coolig of cup of coffee c be moele usig ritmic fuctios. Sice ritmic fuctios epoetil fuctios re iverse fuctios of ec oter, it follows tt tt =, =. (Remember, fuctios iverse fuctios uo ec oter!) Oe useful wy of tikig bout epoetils ritms is tt oe my write te epoetil C equtio, A = B, i te equivlet ritmic form, B A = C. Logritms re use to get t te epoets we solvig equtios. Some epoetil equtios c be solve by simply writig tem i ritmic form. For emple, to solve te equtio 0 = 7, simply write it i ritmic form, obtiig = 07. Scietific clcultors ve built-i 0 fuctio, usully just writte. Bse 0 s re clle commo ritms. Usig clcultor we c obti umericl pproimtio to our solutio for : =

4 Some ritmic equtios c be solve by writig tem i epoetil form. For emple, to solve 3 te equtio 2 = 3, simply write it i epoetil form, = 2 = 8. Properties of Logritms All te properties of s come from te fct tt tey re iverse epoetil fuctios. My properties c be isttly fou by goig from kow epoetil equtio to its ritmic form. Te followig tree properties re bse o te equivlece: b = c c = b Property of Epoets Property of Logritms 0 = = 0 = ( ) = = = Oter very useful properties of ritms re te followig prouct, quotiet, power rules, s well s te cge of bse formul tt we use to covert from oe ritmic bse to oter: Prouct rule: ( bc) ( b) ( c) = + Quotiet rule: ( b) ( c) b = c Power rule: ( b ) = ( b ) Cge of bse formul: w = Te cge of bse formul is ofte use to clculte rbitrry ritms usig eiter bse 0 ritms (commo s) or bse e ritms (turl s): b b w Sice w w = = lw l b ( = b ), we c rise bot sies to te power to get: = b b ( b = b ), so: Tis is cge of bse formul for epoetil fuctios. I prticulr l = 0 = e.

5 Now A Pic of Clculus Let f =, let s try to compute tis fuctio s erivtive. By efiitio, + f f ( ) lim lim f ' = = lim = lim = lim = = If we ssume tt lim C 0 etils i te et sectios!), te bove becomes: = for some costt C, (o t worry, we ll el wit ll te gory = lim = C = C 0 Net we ll let g = C f ' C f = Te rte of cge of f is irectly proportiol to f. =, try to fi its erivtive. But first, recll te str process for fiig te erivtive of iverse fuctio: f f = f f = f ' f f = [ ci rule ] f ( ) = f ' f Applyig tis wit f = f [ efiitio of iverse fuctio ] [ ifferetite ec sie of te equtio ] = :

6 C = = = C = [ efiitio of iverse fuctio ] [ ifferetite ec sie of te equtio ] [ ci rule, ew ( ) ( C ) = = C erivtive rule for epoetils ] Te rte of cge of y = ( ) is iversely proportiol to [ s epoetils re iverse fuctios ] Emple: Suppose tt f is ivertible ifferetible everywere, tt f =. Suppose furter tt f ' = 7. Te wt is te erivtive of Tus, f f = f f = f ' f f = [ ci rule ] f ( ) = f ' f 2 f, evlute t = 2? [ efiitio of iverse fuctio ] [ ifferetite ec sie of te equtio ] f = 2 = = = f ' 2 ' 7 ( f ) f ()