Pre-Calculus - Chapter 3 Sections Notes

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1 Pre-Clculus - Chpter 3 Sectios Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = = 1/( ) c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied ( ), where > 0 1 the epoet is ( rel umer). Chrcteristics o Epoetil Fuctios the -itercept is (0,1). the domi is ll rel umers. the rge is >0. the - is is the horizotl smptote. there re o verticl smptotes. Agi, ote tht the vrile is i the epoet s opposed to the se, d the se is restricted to eig positive umer other th 1. Also ecuse >0 d 1, the we c rech the coclusio tht either 0<<1 or >1. Epoetil uctio whe >1: 2 ( ) The vlues o this uctio strt smll ver smll, so smll tht the're prcticll idistiguishle rom " = 0", which is the - is d the, oce the strt growig, the grow ster d ster, so st tht the shoot right up through the top o our grph. This is icresig uctio d descries epoetil growth (See 3.5). E: popultio growth or moe i svigs ccout).

2 You m hve herd o the term "epoetil growth".(sectio 3.5). Our uctio () ove douled ech time we icremeted. Tht is, whe ws icresed 1, icresed to twice wht it hd ee. This is the deiitio o epoetil growth: tht there is cosistet ied period durig which the uctio will doule (or triple, or qudruple, etc). So i ou her someod climig tht the world popultio is doulig ever thirt ers, ou kow he is climig epoetil growth. t r Compoud Iterest: A P 1 rt Cotiuous Compoudig: A Pe where A= Fil Amout, P= Pricipl, r=rte, = umer o times iterest is pid or compouded ech er d t=umer o ers. Epoetil uctio whe 0<<1: 1 ( ) 3 The vlues o this uctio strt ver lrge, gettig smller d smller hl (or third, or...) t ech step, so smll tht the're prcticll idistiguishle rom " = 0", which is the -is. This is decresig uctio d descries epoetil dec (See 3.5). E: whe qutit loses vlue epoetill over time). B lookig t epoetil equtio or its grph, ou will e le to correctl ideti which tpe o chge it represets, growth or dec. (See sectio 3.5) Trsormtios o Epoetil Fuctios: (See Pge 392) As with other tpes o grphs, represets pret grph. The sme techiques used to trsorm the grphs o other uctios ou hve studied c e pplied to grphs o epoetil uctios. Emples: c ( ) 3 mes the pret grph hs ee shited c uits to the let. c ( ) 3 mes the pret grph hs ee shited c uits to the right. ( ) 5 c mes the pret grph hs ee shited c uits up. ( ) 5 c mes the pret grph hs ee shited c uits dow. ( ) 3 mes the pret grph hs ee shited uits to the right d uits dow. ( ) mes the pret grph hs ee relected over the -is. ( ) mes the pret grph hs ee relected over the -is. Whe grphs re trslted, the -itercept d the smptote move with the grph. Rememer: to id the - itercept o the grph o uctio or reltio, ou must set =0 d solve or.

3 Sectio 3.1 Pg. 393 The umer e is irrtiol umer which ppers s the se o m epoetil uctios. 1 The umer e is deied s the vlue tht 1 pproches s gets lrger d lrger. As the pproimte vlue o e to 4 deciml plces is e Sectio 3.2 Pg. 400 Logrithms: Logrithmic uctios re the iverse o epoetil uctios. For emple i (4, 16) is poit o the grph o epoetil uctio, the (16, 4) would e the correspodig poit o the grph o the iverse rithmic uctio. Logrithmic uctios: or, where > 0 1 Chrcteristics o Logrithmic uctios: the -itercept is (1,0). the domi is >0. the rge is ll rel umers. the - is is the verticl smptote. there re o horizotl smptotes. Need to kow how to: Compre the chrcteristics o rithmic uctios to the chrcteristics o epoetil uctios. Evlute uctios. Grph uctios. Fid the domi o uctios. Chge rom uctio to epoetil uctio. Chge rom epoetil uctio to uctio.

4 Commo Logrithms d Nturl Logrithms. The most commo ses re the se 10 d the se e. Logrithms with se 10 re clled commo rithms, d rithms with se e re turl rithms. The techiques to simpli epressios d solve equtios with rithms re the sme, regrdless o the se. O our clcultor, the se 10 rithm is oted, d the se e rithm is oted l. ( l e ). Properties o commo d turl s. See pges 407 & 409. SECTION 3.3 Properties o Logrithms. EXPANDING AND COMPRESSING LOG EXPRESSCIONS m d re positive umers, is positive umer other th 1, d p is rel umer. Propert Product Quotiet Power m m p m p Deiitio m m Equlit I m, the m m CHANGING THE BASE OF A LOGARITHM There re iiite umer o ses d ol ew uttos o our clcultor. You c covert rithm with se tht is ot 10 or e to equivlet rithm with se 10 or e. Let,, d e positive rel umers such tht 1d 1 (rememer must e greter th 0). The c e coverted to the se the ormul

5 Emple o CHANGE OF BASES: Fid 3 7 to ccurc o si decimls. Note tht the swer will e etwee 1 d 2 ecuse d, d 7 is etwee 3 d Accordig to the chge o rithm rule, 3 7 c e writte s. Whe the se is 10, we c leve o the 10 i the ottio. Thereore c e writte. Use our clcultor to id the swer. (As: ). Let's check the swer. I , our swer is correct.. Close eough. Wh is't it 7 ectl? Note: The Chge o Bses Propert works with se, s log s the ses re the sme. For clcultor purposes, we c ol use se 10 or e. ADD YOUR OWN NOTES FOR 3.4 SOLVING EXP. AND LOG FUNCTIONS AND 3.5 EXPONENTIAL MODELING: GROWTH AND DECAY. Miscelleous Notes: I ule to solve equtio or simpli epressio s is, tr oe o the ollowig: 1) Chge rom epoetil orm to or l orm. 2) Chge rom or l orm to epoetil orm. 3) Drop or l rom oth sides o equtio i there is sigle or l o ech side d the ses re the sme. 4) Isert or l o oth sides o equtio. 5) Perorm chge o ses. 6) Epd the epressio usig the properties o s. 7) Compress the epressio (write s sigle ) usig the properties o s. 8) Kow how our clcultor works usig, l, d e. 9) Whe solvig equtios, check the solutios (i cse o them do ot work). 10) d so o l e l e 1 l e l e d so o

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