If y=o Use your clcultor to find the following ( ) 4 Grphing Eponentil functions f / CHAPTER 4 e 07T 0GB 9#= f Section 4 7 e 8cg * ftpzs ste " 4 y y= y= f f f : Grph the following y using trnsltions nd y plotting points f ( ) y y f( ) HA: y= HA: y= y, \ *
'±*, t i I #
EXPONENT RULES p p 9 = p p ( 4 ) 7 = 8 6 4 4 p p 7 = or 0 = p % y z4 p c 4 z 4 = c 4 y ( 7 7z y 4) 4 = ( 7z y 4 7 )4 6 = p p p p 7 p Siplify: = 's ' e =e Mg e ) iv) e (e + ) e (e + ) v) 8 l DeJe!QOe @ i) 4 ii) ( )y iii) ( e+ qqie#ftefeie s**n* ±, 8 q±,, *"
GD Solving equtions y chnging ses If, then = E/ 4 6 E/ 4 E/ 9 ( 4 4 )= z = X= i, z= = =± E/ 4 4 E/ 4*4 # ' Try/ 6 9 7 /=@ =4 Try/ 4 Try/ 4 6 # Try/ 'Ez4 = = #t= } 9=4 = zu =4 # *o# D 7 Try/ 8 4 8* z += ±E # z4 t
Copounded Interest nt A P r n A= Aount lhccount 4 P= Principl r= rte n= pr 's # ofthyit Copounded 6 t= tie ( yers ) pyeur Copounded Continuously rt A Pe How uch will n ccount with $000 copounded onthly with rte of % hve fter yers? p n= r=o os t=s A 0004+0*4*6,4670 A 4000C io#j4lo ) How uch will n ccount with $4000 copounded qurterly with rte of 8% hve fter yers? = 49,8779 How uch will n ccount with $4000 copounded continuously with rte of 8% hve fter yers? 4 How uch will n ccount with $000 copounded continuously with rte of % hve fter yers? A proissory note will py $0,000 t turity yers fro now How uch should you e willing to py for the note now if the note gins vlue t rte of 0% copounded continuously Hint: Find P 4
Continuous Growth nd Decy Section 4 Eponentil growth = initil ount of the sustnce t= tie tht pssed fro tht initil oent A(t)= ount of the sustnce fter tie t k=growth or decy constnt E/ continuous growth of % per yer k=0, t is in yers eple: A(t) = 0 e 00t Initil ount is: Growth constnt is: Grph: 0 : A(t) = 0 e 0t Initil ount is: 0 Decy constnt is: = Grph: 009%0 ss% A(t) = 48e t Initil ount is: Growth constnt is: Grph: A(t) = 0 e 7t Initil ount is: " Decy constnt is: Grph: : Find the eqution for the following nd stte wht t is in ters of ) Initil ount is 00 grs, continuous decy t 4% per yer oo il4t 48 ) Initil popultion is 00, continuous growth t % per week A= Sood ) Initil ount is 0 illigrs, continuous decy of 00% per yer 0 700% >>oo% A=Oe tiskyeuy " 00000T tsn thin weeks yer 6
' ', 0446 4) If the popultion of Meico is round 06 illion people now nd if the popultion decreses continuously t reltive decy rte of 7%, wht will the popultion e in 8 yers? Answer in illions A = " 7T 0680 th in yers A = 06 join "4, 96 illion The tospheric pressure P, in pounds per squre inch, decreses eponentilly with ltitude h, in iles ove se level, s given y P = 47 e 0h ) Wht is the tospheric pressure t se level? 4, > psi ) Wht is the tospheric pressure t 4 iles? P = 4,70 " " = 6 psi c) Is the tospheric pressure incresing or decresing s the height increses? decresing 6 The World helth orgniztion estited tht there were illion deths worldwide fro HIV/AIDS during the yer 004 Assuing tht the nuer continues to increses t reltive growth rte of 4% copounded continuously, estite the nuer of deths fro HIV/AIDS during the yer 00 A A =, @ A ( 6) = } 004 E 0 eo ) = 67 illion dy 6
' 004 Douling An ccount doules in vlue every yers with n initil ccount lnce of $4000 Fill in the tle to show the lnce every yers kffk I Nuer of yers 0 6 9 8 Account lnce $4,000 k 64k 8k 6k tch The douling forul is A(t) = A 0 () t tie to doule Find the forul for the following E/ Initil ount is 00 grs, tie to doule is 4 yers A loocgt Try/ Initil popultion is 4 illion, tie to doule is 0 yers 4(z)e Crete forul nd use it to nswer the following The use of insecticide DDT is no longer llowed in ny countries ecuse of its longter dverse effects If frer uses pound of ctive DDT, ssuing its hlflife is yers, how uch will still e ctive fter yers? = In 96, Gordon Moore, founder of Intel, predicted tht the nuer of trnsistors tht could e plced on coputer chip would doule every yers This hs coe to e known s Moore's lw IN 970,,00 trnsistors could e plce on chip Use Moore's lw * to predict the nuer of trnsistors in 0 7 A : 004 = 0 00 0 At )( " =6 0 "
Hlflife I counted 000 old spores in y throo rug If I pply lech every dy, then the hlflife of the popultion of old spores is dys Fill in the tle to show the nuer of old spores every dys Nuer of dys 0 6 9 8 Nuer of old spores,000 Soo 0 6 6 Te iooc' The hlflife forul is A(t) = A 0 ( ) hlf life t Find the forul for the following E/ Initil ount is 00 grs, hlflife is 4 yers Try/ Initil popultion is 4 illion, hlflife is 0 yers A 4Ctt Crete forul nd use it to nswer the following A rdioctive isotope hs hlflife of 46 hours If we strt with 00 illigrs, how ny illigrs would e left fter dy? ooc 's Aczy ) rocks ' = 699g The use of insecticide DDT is no longer llowed in ny countries ecuse of its longter dverse effects If frer uses pound sof ctive DDT, ssuing its hlflife is yers, how uch will still e ctive fter yers? A=zs(± ACS )=s( I ) = 87l 8
Foruls Logrtiths Section 4 riths y y Eqution type Eqution type Eqution type y y y y r 4 c r c, ln e 6 7 0, 8 or 0 error, >0 9 ln ln Use forul Solve for 8 7 0 000 0 00 00 0 4 4 4 4 4 8 4 8 or 4 8 gives n error when checked, so 4 is the only nswer X=4 Coine s This ens the insides re = gives n error essge when checked, so No Solution 8 4 Need to ring the vrile down using forul 4 ln 6 4966 6 4 ( )ln 6 ln 4 ( )7979 86944 7979 778 86944 778 4046 ln 4 7 e 006 q 0 006 ln e ln 0 Using forul 006 ln0 006 ln0 006 006 66866 y y Rewrite s rith 4 8 0%84 4 6 09 :S y 0=0 'E=4 0 9 Rewrite s n eponentil eqution 9 8 4 9 =9 4=8 = zef 9
Solving 6 9 = oe E±t# 9= =/ I Siplifying 9 7 wht 's the eponent? Log se 0 nd e 09,0=00 %=069 = ln = Solve 00 0 0 µ 000 io =Ko lot lototo ki= 0 0 =z oo= = } nx±4&lne4= l%e g luo=, % /= " > To ee " 9ee4 ' e =e4 g=e =4 7=4 " 0 's
Epnding riths r y y y 4 c r c y Write s the su nd difference of riths y y y 09 +%4 oqgtogyogsmtogsy@dueyeaoyiitiyzogzoqzl09ztogztogzxogx z Writing s single rith y *:) igtgy i%c y y loss 'C i%e D i%e*j Specil Proles,, e ln 6 7 0 8 or 0 error, >0 4 ln e 4 0 0 0 DNE 9 ln ln Yhht lon 'YYIT=E 0 YOST 4 7 8 0 YFFIM Yhttse
/ Find the doin of (+) Find the doin of () Find the doin of : Xt 4>0 > X +4+4 +4=0 Hr y 4*0 +4>0 470 7 E N#h =± 4 =+7 Dik (9) vcz,d) Grphing Logrithic functions " lis µ!! zx== tz >o " * Wht is the Doin of ech: > >o =o Wht is the reltion of the doin nd the verticl syptote? Find the doin of the function Find the VA (Verticl syptote) lel it = Trnslte Grph the following: Doin: Doin: > 0 VA: = VA: =
Eponentil nd Logrithic Equtions Section 4 } Toledo los " EC#D=So= }S== y Tz Ydq= ' c%z= oeri4se?k ## %(*9)=0% 0 Es 0 += *o ) ) ln e c) 4 ( 4) kotyqoscnhsioscthonlgtt4ytzfoiooqtlosscftdt0@ost4oe0cqiyxctjroitlsifotrfxqtiy q d) 4 e) ( 7) ( 9) f) ln 4 ln ln 0 9+4=+4 't Egf 0=68+ ty @#
}= Try: g) ( ) h) ( 7) ( 7) i) 4 hint: fctor (ogp=4oqx u=los * n e E y% =ioo=l =v' too Solve the following to three decil plces z ++ ne0i0 6 = no * l k=x ' 8*=8 0 oo6=h' ± 6= + 0 Foos 0%=4 6687 ± f 0% ( ) 7 ) Try: ) 8 6 ) 09,,s=, 4 647 c) ) e '' the' X 0 4 e d) 8 e 4
Applictions How long will it tke nk How long will it tke nk An eponentil decy ccount with $000 copounded ccount with $4000 copounded is given y the forul onthly t % to rech 40,000? continuously t 8% tke to doule? t A 4e Find the hlf life A tgs#t=88yers866yiyt PCltrDntAPertz4etPsoooTf4A49oooP400oroosA80odlnztInetnizr0o84YEysqoCHFIttEf4ytgeoio8tLnEfy8Clt9tDltng@0o8tZn8nCltoEYtnoo8t046D4YYoIkoty 0 'Y ' to # Creting n eqution 4 According to US Census Bureu estites, The popultion of the United Sttes ws 7 illion on July, 980, nd 49 illion on July, 990 Let t e the nuer of yers since 980 A) Find the eponentil growth odel ssuing the continuous growth rte Wht is the growth rte? 980,7, ( ) t (990,49%) ( ( t A ), 0,7, ) (74%) Attoekt #Tco#9 A= zzekt A=e0 49=7 ek "D growth rte 098=0 " 09%0 In,098 =/ne t k= 0009 *098=,y B) Estite the popultion of the US tody Go online nd copre A= illions 7,*9%=7, k 0,
Try: Cron 4 dting A onkey fro refuse deposit ner the Strit of Mgelln hd 40 grs of the cron4 00004t If the forul for ount for cron 4 left fter t yers is given y A( t) A0e, then wht is the hlflife of cron 4? A 0 Initil ount (ount of cron 4 when the orgnis died) Cron Dting 6 The hlflife of cron 4 is ctully 70 yers The sic forul for decy odel for cron 4 is 00004t A( t) A0e where t is in yers ) If rndo oject hs 6 % of its cron4 left in 004, then find when the oject died or de fro ded stuff c) After 000 yers how uch cron 4 will rein? 6
04t 7 The nuer of grs of rdioctive sustnce present fter t hours is given y A( t) 00e ) How ny grs of the sustnce re there initilly? ) How ny grs will there e fter hours? c) When will there e 00 grs reining? d) Wht is the hlflife of the sustnce (in hours)? 7