The Exponential Function

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Abel Gregory
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1 The Epoetil Fuctio Defiitio: A epoetil fuctio with bse is defied s P for some costt P where 0 d. The most frequetly used bse for epoetil fuctio is the fmous umber e E.: It hs bee foud tht oyge cosumptio of oe yer-old slmo icreses epoetilly with the speed of swimmig ccordig to e where is the speed i feet per secod. ) Wht is the oyge cosumptio whe the fish re still? b) Wht is the oyge cosumptio t speed of feet per secod? E.: Slmoell bcteri grow rpidly i wrm plces. Suppose the umber of bcteri preset i potto sld (eposed from choppig bord previously used to cut up chicke!) is give by 500 where is the umber of hours fter eposure. ) If the potto sld is left out o the tble, how my bcteri re preset oe hour lter? b) How my were preset iitilly? c) How quickly will the umber of bcteri icrese to,000? d) How ofte do the bcteri double? E.: The umber of Iteret users ws estimted to be 6 millio i 995 d 05 millio i 999. The growth c be modelled with epoetil fuctio. ) Fid the fuctio f0 tht models the umber of Iteret users, where 0 (i yers) correspods to 995, d f () gives the umber of users i millios. b) Sketch f (). c) Wht will the umber of Iteret users be i 05? The Logrithmic Fuctio I our previous Iteret questio we hd the fuctio describig user growth. How would I clculte whe the Iteret will rech billio users? Defiitio: For 0,, d 0, y log mes y. Now e is very populr umber so isted of writig clled the turl logrithm. log e hve specil symbol l,

2 Defiitio: The turl logrithm of, l c mes e c. l, is the power of e eeded to get. i.e. Properties of logrithms: Let d y be y positive rel umbers d r be y rel umber. Let be positive rel umber,. The log y log log y log y log r log r log log log y log 0 (s 0 ) log r r logb log log (b positive rel umber, b ) b E.: Solve 6 for. E.: Solve e 9 for. E.: Solve E.4: Solve 5 l( ) for. 4 for. E.5: Solve for. E.6: Solve for. E.7: Suppose for certi food product the umber of bcteri preset is give by 0.t f ( t) 500e where t is time i dys fter pckig of the product d the vlue of f (t) is i millios. ) If the product cot be sfely ete fter the bcteri cout reches 000 millio, how log will this tke? b) If t 0 correspods to J, wht dte should be plced o the product? E.8: The umber of yers N(r) sice two idepedetly evolvig lguges split off from commo cestrl lguge is pproimted by N( r) 5000lr where r is the proportio of the words from the cestrl lguge tht re commo to both lguges ow. 4

3 ) Fid N (0.9) d N (0.). b) How my yers hve elpsed sice the split if 70% of the words of the cestrl lguge re commo to both lguges tody? c) If two lguges split off from commo cestrl lguge bout 000 yers go, fid r. Trigoometric Fuctios Mesurig Agles: degree is rdis d to chge from degrees to rdis. rdi is degrees d to chge from rdis to degrees. Eg rdis d rdis is Trigoometric Fuctios: Mke sure you kow if your clcultor is i degrees or rdis! A fuctio y f () is periodic if there eists positive rel umber such tht f ( ) for ll i the domi of the fuctio. The smllest positive vlue of is clled the period of the fuctio. Note: si( ) si d cos( ) cos. d re periodic. Their period is. Note: the sie d cosie grphs re the sme shpe, oly shifted horizotlly. Ad cos si( ) d si cos( ). Note: d hve mplitude d period. hs period d hve mplitude d period. They re d shifted horizotlly by.d verticlly by, respectively. 5

4 E.: Fid ll E.: Fid ll E.: Fid ll E.4: Fid ll E.5: Solve cos for E.6: Fid solutio to si( ) 4 where. E.: Solve for. E.8: The Trsylvi Hypothesis clims the full moo hs effect o helthrelted behviour. A study ivestigtig this effect foud sigifict reltioship betwee the phse of the moo d the umber of geerl prctice cosulttios tiowide, give by: ( 6) y 00.8cos, 4.77 where y is the umber of cosulttios s percetge of the dily me d is the dys sice the lst full moo. ) Wht is the period of this fuctio? b) There ws full moo o Aug 4, 00. O wht dy i Aug 00 does this formul predict the mimum umber of cosulttios? Wht percet icrese would be predicted for tht dy? c) Wht does the formul predict for Aug 7, 00? 6

5 Etr Trigoometry Idetities d Properties csc( ), si( ) si( ) t( ), cos( ) sec( ), cos( ) cos( ) cot( ). si( ) cot( ), t( ) si ( ) cos ( ), sec ( ) t ( ). si( ) si( ), cos( ) cos( ), si( 90 ) cos( ), cos( 90 ) si( ), si( 80 ) si( ), cos( 80 ) cos( ). si( A B) si AcosB cosasib, cos( A B) cos AcosB si Asi B. i rdis i degrees si( ) 0 cos( ) t( ) 0 Specil Vlues for Trigoometric Fuctios

6 Polyomil Fuctios You kow bout lier fuctios b, d qudrtic fuctios b c, well more geerlly we hve polyomil fuctios:... 0 where,,,, 0, re costts clled coefficiets, is oegtive iteger clled the degree of the polyomil, d the ledig coefficiet is where 0. For : 0 A lier fuctio is polyomil of degree. For : 0 A qudrtic fuctio is polyomil of degree. For 0 hve 0 A costt fuctio is polyomil of degree 0. We re lredy quite fmilir with the specil cse This is clled power fuctio. Notes: A polyomil of degree c hve t most turig poits. I the grph of polyomil fuctio of eve degree, both eds go up or both eds go dow. For polyomil fuctio of odd degree, oe ed goes up d oe ed goes dow. If the grph goes up s becomes lrge, the ledig coefficiet must be positive. If the grph goes dow s becomes lrge, the ledig coefficiet is egtive. Note: Though polyomil fuctios c grow fst, epoetil growth is lwys fster. Ad ote tht both evetully domite logrithmic growth. 8

7 Rtiol Fuctios p( ) A rtiol fuctio is defied by, where p () d q () re polyomil fuctios q( ) d q ( ) 0. (Note tht sice y vlues of givig q ( ) 0 re ecluded from the domi, rtiol fuctios ofte hve grphs with oe or more breks.) Notes: If the vlues of f () pproch umber L s gets lrger d lrger ( ), y L is clled horizotl symptote. If umber k mkes the deomitor 0 i rtiol fuctio but does ot mke the umertor 0, the the lie k is verticl symptote. 9

Content: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.

Content: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B. Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe