Eponenial Funcions Las updae: February 2008 Secion 1: Percen Growh and Decay Any quaniy ha increases or decreases by a consan percenage is said o change eponenially. Le's look a a few eamples o undersand wha his means. Eample #1.1 Compound Ineres ineres compounded annually An invesmen of $1000 is placed ino a savings accoun wih an annual simple ineres rae of 6%. Simple ineres is ineres ha is compounded (compued and added o he principal) once per year. Le's deermine how he money grows over a 4-year ime period. year = 0: oal amoun equals $1000. year = 1: we keep our iniial $1000 and add on he ineres of 6% of $1000. $1000 + 0.06($1000) = $1000 + $60 = $1060 So he oal amoun afer 1 year is $1060, which is an increase of $60. year = 2: We keep he $1060 and add on he ineres of 6% of $1060. $1060 + 0.06($1060) = $1060 + $63.60 = $1123.60 The oal afer 2 years is $1123.60, which is an increase of $63.60 over he previous year. Now le's coninue his process unil we reach year = 4. The resuls are summarized in he able below. = years A = accoun balance 0 $1000 1 $1000 + 0.06( $1000) = $1060 2 $1060 + 0.06( $1060) = $1123.60 3 $1123.60 + 0.06( $1123.60) = $1191.016 4 $1191.016 + 0.06( $1191.016) $1262.48 There is a simplificaion sep ha we can make o our compuaions above. Again we sar wih $1000. Afer year = 1, we have $1000 + 0.06( $1000). Facoring ou he $1000 resuls in $1000(1 + 0.06) = $1000(1.06) = $1060. Similarly in year = 2, he epression $1060 + 0.06($1060) can be facored as $1060(1+ 0.06) which equals $1060(1.06) = $1123.60. In shor, o find he accoun balance in any year, we simply muliply he previous year's amoun by 1.06. This compuaion is equivalen o aking 106% of he previous year's amoun! We summarize he resuls in he able below. 1
= years A = accoun balance 0 $1000 1 $1000(1.06) = $1060 2 $1060(1.06) = $1123.60 3 $1163.60(1.06) = $1191.016 4 $1191.016(1.06) $1262.48 Noice ha o compue he amoun afer 2 years, we ake he iniial amoun of $1000 and muliply by 1.06 wice. This can be wrien as $1000(1.06) 2 = $1123.60. Likewise he balance afer 3 years is $1000(1.06) 3 = $1191.016. This mehod is summarized in he able below. = years A = accoun balance 0 $1000 1 $1000(1.06) = $1060 2 $1000(1.06) 2 = $1123.60 3 $1000(1.06) 3 = $1191.016 4 $1000(1.06) 4 $1262.48 This mehod o compue he balance afer any number of years is quie powerful. To ge he balance afer 10 25 10 years, compue $1000(1.06) $1790.85. Afer 25 years you'll have $1000(1.06) $4291.87, which means he original invesmen of $1000 will have more han quadrupled! Finally, we can generalize o wrie he accoun balance funcion afer years: A ( ) = $1000(1.06) Here is a graph of he funcion A() over a 25-year domain. (Try o ge a similar picure on your calculaor.) Noe: You can read more abou compound ineres in Eample #1.7 and in Kaseberg s Inermediae Algebra on pp. 455-7. 2
Eample #1.2 Suppose ha he populaion of a baceria culure iniially numbers 200,000,000 and grows seadily a a rae of 3.2% each day. Derive a able showing baceria couns over ime hen wrie a formula. Soluion: Sar by defining o be our independen variable represening elapsed ime in days. Le P be our dependen variable represening millions of baceria. So P (0) = 200. To find he populaion afer 1 day we need o increase he iniial populaion by 3.2%. This is he same as aking 103.2% of he iniial populaion or muliplying 200 million by 1.032. This gives us (200 million baceria)(1.032) = 206.4 million baceria. A able showing baceria couns over he firs 6 days is given below. Afer days he number of baceria (in millions) is given by P ( ) = 200(1.032). The graph of his funcion is displayed below. Noice ha he graph of P() is no linear; lay a sraighedge on i and you'll see ha i curves upwards ever so slighly. General Eponenial Funcions Considering he wo previous eamples, we can sae a generalized formula for a funcion ha grows by a consan rae: y= a b In his formula y is he dependen and is he independen variable. The consan a is he iniial value of y when = 0. Thus, a is he y-inercep. The consan b is called he base. The base b is referred o as he muliplier. (Some es call i he growh facor and ohers he common raio.) In eponenial funcions i is required ha b > 0. 3
Anoher common form of wriing eponenial funcions is y= a (1 + r) In his form of he eponenial funcion, we have replaced b wih 1+ r. The consan r is called he percen rae of change. If our funcion is increasing, r is he growh rae or rae of growh. For decreasing funcions, r is he decay rae or rae of decrease. You can ell from he wo forms of he eponenial funcion ha b = 1+ r In Eample 1.1 we creaed he funcion formula b = 1+ r we can compue ha A ( ) = $1000(1.06). In his formula he base b is 1.06. Using he 1.06 = 1+ r 0.06 = r which means ha he growh rae r is 0.06 or 6%. (We knew his!! ) In Eample #1.2 above, he baceria populaion grows 3.2% daily, so we have a daily growh rae of r = 0.032 and a daily muliplier of b = 1.032. Finding he funcion for an eponenial sequence In our eponenial eamples, we sared wih an iniial y-value of a. Then each subsequen y-value was generaed by muliplying by he base b. The ne eample uses hese ideas in finding a funcion hrough an eponenial sequence of numbers. Eample #1.3 In a hyper-inflaion economy he price of any good increases very fas. Suppose ha he price of a compuer is recorded each day for a week, as displayed below. = days P = price 0 $800.00 1 $808.00 2 $816.08 3 $824.24 4 $832.48 5 $840.81 6 $849.22 7 $857.71 b b We see in he able ha he P-inercep is 800, so a = 800. We can deermine he base b (he 1-day muliplier) by noing ha $800.00b = $808.00. Solving for b yields 808.00 b = = 1.01 800.00 Similarly we can solve he equaion $808.00b= $816.08 o find ha b = 1.01. Thus he eponenial funcion is P ( ) = 800(1.01). 4
We can coninue working down he righ side of he able and find ha b 1.01 beween any wo consecuive y-values. Since b is consan, he price is growing eponenially wih a daily muliplier or base of b = 1.01. Using he formula b= 1+ r, we compue he daily growh rae as follows: b = 1+ r 1.01 = 1+ r 0.01 = r 1% = r We concluded above ha he daily growh rae (inflaion rae) is 1%. Wha is he weekly inflaion rae? Surprisingly i is no 7%! One way o find he weekly inflaion rae is denoe he weekly muliplier by b w. We hen solve $800b w = $857.71 o yield b w = 1.0721. This means ha each week he price is muliplied by 1.0721 or 107.21%, which is idenical o increasing he price by 7.21%. Challenge: use a similar mehod o show ha wih a daily growh rae of 1%, he annual growh rae (annual inflaion rae ) is 3678%. Eample #1.4 Now le's consider a more realisic eample involving compuer prices. Le's say you buy a new compuer priced a $1200 which decreases in value a he rae of 20% yearly. Sudy he wo ables below displaying he compuer s value over ime. You migh wan o compare hese ables wih hose from Eample #1.1. = ime (years) V = value ($) = ime (years) V = value ($) 0 1200 0 1200 1 1200 0.20(1200) = 960 1 1200(1 0.20) = 1200(0.80) = 960 2 960 0.20(960) = 768 2 960(1 0.20) = 960(0.80) = 768 3 768 0.20(768) = 614.4 3 768(1-0.20) = 768(0.80) = 614.4 4 614.4 0.20(614.4) = 491.52 4 614.4(1 0.20) = 614.4(0.80) = 491.52 Noe ha subracing 20% of he previous year's value is equivalen o aking 80% of he previous year's value. Because we muliply by 0.80 o make he value drop 20% each year, he yearly muliplier is he base b = 0.80. The formula for he value of he compuer as a funcion of ime in years is : V ( ) = $1200(0.80) The formula b= 1+ r sill apples wih decreasing funcions. Solving he equaion 0.80 = 1+ r for r produces r = 0.20. Since r is negaive, we call r he yearly decay rae or rae of decrease. We say ha he value is decreasing by 20% each year. The graph of V() is displayed below. Noice ha he graph ges closer and closer o he horizonal ais as we move o he righ. In his case he horizonal ais is called a horizonal asympoe. 5
Eample #1.5 Suppose ha a high-aliude ho air balloon floas upwards. During he fligh, elevaion and amospheric pressure measuremens are colleced as displayed below. = elevaion P = pressure (miles) (inches of mercury) 0 30 1 27.6 2 25.392 3 23.361 4 21.492 b b For each change in elevaion of 1 mile, we can compue he muliplier b. (Noe: he symbol sands for "implies".) 30b = 27.6 b = 0.92 27.6b = 25.392 b = 0.92 25.392b = 23.361 b = 0.92 23.361b = 21.491 b = 0.92 Because he muliplier b is consan, we conclude ha he pressure is decreasing eponenially as he aliude increases. In his eample he base b is 0.92 which means ha he decay rae is b = 1+ r 0.92 = 1+ r 0.08 = r 8% = r Thus for each gain in elevaion of 1 mile, he pressure decreases by 8%. The funcion relaing pressure o elevaion is given by P ( ) = 30(0.92). For each mile increase we muliply by 0.92. This is equivalen o saying ha ha 92% of he pressure is reained for each increase of 1 mile. The graph is displayed below. Noice ha he curve is almos linear over he domain [0, 10]. 6
Summary: In he formula y = a b, if b > 1 hen y is increasing eponenially wih respec o. if b < 1 hen y is decreasing eponenially wih respec o. if b = 1 hen y = a and y is a consan funcion. In he alernae formula y = a(1 + r) if r > 0 hen y is increasing eponenially wih respec o. if r < 0 hen y is decreasing eponenially wih respec o. if r = 0 hen y = a and y() is a consan funcion. Eample #1.6 Consider he funcion y() = 20(2). When = 0, y(0) = 20. Therefor he graph of y() has a y-inercep of 20. Since b = 2, he variable y is an eponenially increasing funcion of. We ypically use a calculaor o compue oupus for negaive values of. Bu wih his eample we can proceed by hand. 1 1 If = -1, hen y ( 1) = 20(2) = 20 = 10 2 2 1 1 If = -2, hen y ( 2) = 20(2) = 20 = 20 = 5 2 2 4 If = -3, hen y ( 3) = 20(2) 3 1 1 = 20 = 20 = 3 2 8 5 2 We know ha b = 2, so 2 = 1+ r r = 1. This is equivalen o r = 100%. This makes sense considering ha muliplying by b = 2 is equivalen o doubling a quaniy, or increasing a quaniy by 100%. 7
Eample #1.7 Compound Ineres ineres compounded more han once per year In Eample 1.1 an invesmen of $1000 was placed in a savings accoun wih an annual simple ineres rae of 6%. Recall ha simple ineres is ineres ha is compounded once per year. A he end of he 4-year period here was $1262.48 in he accoun. Wha if he ineres were compounded every 6 monhs (semiannually)? How much money would here be afer four years? Banks compue he semi-annual ineres rae by aking he annual ineres rae and dividing by 2. So he semi-annual rae is r = 6% 2 = 3% = 0.03. This means ha he semi-annual muliplier is b = 1.03. If represens he number of 6-monh compounding periods, and A he accoun balance, he accoun balance can be modeled by A ( ) = 1000(1.03) To find he accoun balance afer four years we le = 8 in he funcion (o represen 8 semi-annual compounding periods). 8 A (8) = 1000(1.03) $1266.77 We see ha when compounding wice per year for four years we end up wih $1266.77. Tha amoun is $4.29 more han when compounding once per year for four years. The more ofen we compound, he more money we earn (all hings being equal). There is a general formula ha is ofen used o compue he fuure amoun of money in an invesmen. Tha formula is given below. Compound Ineres Formula If n is he number of imes a year ineres is (calculaed) compounded, r S = P 1+ n where S is he fuure value, P is he prese value (also known as he principal or saring amoun), r is he annual rae of ineres (epressed as a decimal), and is he number of years. n To use he compound ineres formula wih he informaion given in Eample 1.7 we would make n = 2, P = 1000, r = 0.06, and = 4 : 0.06 S = 1000 1+ 2 ( ) = 1000 1.03 $1266.77 8 24 You can see ha he compound ineres formula produces he same resul afer as did our funcion ha we creaed in Eample 1.7. I is accepable o use he compound ineres formula raher han firs wriing an eponenial funcion, bu make sure you undersand why i works! 8
Secion 1: Eercises 1. For each funcion idenify he verical inercep a, he base or muliplier b, and he rae of change r. Noe: Soluions o mos odd problems are a he end of hese noes. a) f ( ) = 200(1.075) d) f ( ) = 2050(1.56) b) G ( ) = 30.5(0.78) e) h ( ) = 300(0.45) c) p h( p) = 3 f) y ( ) = 30.52(0.999) 2. Fill ou he able below for he funcion f ( ) = 4(2). Then make a graph. (Try finding values wihou he aid of a calculaor.) -3-2 -1 0 1 2 3 f() 3. Mach he 5 funcions wih heir correc graph. Try o no use your calculaor! equaions a) b) c) d) e) y = 5(1.2) y = 5(1.4) y = 5(0.4) y = 30(0.8) y = 30(0.7) 9
4. The graph of he funcion 2 = 2 + g ( ) is displayed below. a) This graph should be idenical o he one you drew in problem #2 above. Is i? b) Use rules of eponens o prove ha 2 f ( ) = 4(2) and g ( ) = 2 + are idenical funcions. c) Wrie a new funcion h() whose mahemaical epression is differen han ha of f () and g() bu whose graph is idenical. 5. For he following wo ables assume ha he dependen variable grows eponenially wih respec o he independen variable. a) Find he muliplier b for each able. b) Sae he percen increase r. c) Fill in he res of each able d) Wrie a formula for each able. Use funcion noaion. Table A Table B A y 0 20-4 1 25-3 2 31.25-2 3-1 50 4 0 62.5 5 1 6. For he following wo ables assume ha he dependen variable decreases eponenially wih respec o he independen variable. a) Find he muliplier b for each able. b) Sae he percen decrease r. c) Fill in he res of each able d) Wrie a formula for each able. Use funcion noaion. Table A Table B P y 0 120,000-2 1 112,200-1 1000 2 104,907 0 850 3 1 4 2 5 3 10
7. Find he equaion of each graph below. Noe: each graph is eiher linear or eponenial. a) b) c) d) 8. For each graph in he problem above, sae he domain and range. (Assume ha he complee graph is displayed; ha is he graphs do no eend any furher for larger or smaller values of.) 9. Insook deposis $3000 ino a bank fund ha pays 4.6% ineres compounded annually (simple ineres). Wrie he formula ha describes he amoun, A, in he fund as a funcion of, he number of years elapsed since she made he deposi. Then make a able displaying he amoun of money in her fund each year for 10 years. (Assume ha she makes no oher deposis or wihdrawals and he ineres rae remains consan.) 10. Bob invess $40,000 (all of his inheriance money) in a real esae fund. The fund managers buy risky properies and he value of he fund drops 6% each monh. Wrie a formula ha epresses V, he value of Bob's invesmen, as a funcion of m, he number of monhs since he firs invesed. Creae a able on your calculaor displaying he value of Bob's invesmen each monh. On your homework paper, record he value afer 1 year, 2 years, 3 years, ec., up o 6 years. 11
11. Suppose ha $2500 is invesed in a cerificae of deposi (CD) ha earns 5% annual ineres. a) If ineres is compounded once per year, wrie a funcion for he amoun of money, A, in he CD as a funcion of he number of years. How much will he CD be worh afer 5 years? b) If ineres is compounded semi-annually, wrie a funcion for he amoun of money, A, in he CD as a funcion of he number of 6-monh periods. How much will he CD be worh afer 5 years? c) If ineres is compounded quarerly (every 3 monhs), wrie a funcion for he amoun of money, A, in he CD as a funcion of he number of quarers. How much will he CD be worh afer 5 years? 12. Suppose ha $50,000 is invesed in a cerificae of deposi (CD) ha earns 6% annual ineres. a) If ineres is compounded once per year, wrie a funcion for he amoun of money, A, in he CD as a funcion of he number of years. How much will he CD be worh afer 5 years? b) If ineres is compounded semi-annually, wrie a funcion for he amoun of money, A, in he CD as a funcion of he number of 6-monh periods. How much will he CD be worh afer 5 years? c) If ineres is compounded weekly, wrie a funcion for he amoun of money, A, in he CD as a funcion of he number of quarers. How much will he CD be worh afer 5 years? 13. Demographics for many of he world's counries can be found on he US Census Inernaional Daabase (IDB) websie (hp://www.census.gov/ipc/www/idbprin.hml ). Below are daa for a few counries: Counry year 2000 populaion (10 6 ) annual growh/decay informaion Algeria 31.2 r = 1.737% Belarus 10.3 r = 0.168% Canada 31.3 r = 0.01015 Hungary 10.1 muliplier = 0.9966 Kenya 30.3 muliplier = 1.01526 a) For each counry find an eponenial funcion P() ha models he saisics given. Assume ha P is measured in millions of people, and ha is he number of years since 2000. b) Which counry is projeced o have more people in 10 years: Algeria, Canada or Kenya? c) Which counry can we projec had more people 10 years ago, Hungary or Belarus? d) Suppose ha Algeria cus is growh rae by 0.2% oday and mainains his new growh rae over he ne 5 years. By wha percen will he populaion increase be reduced? 12
14. Is he variable y an eponenial funcion of he variable in any of he ables below? If so, find he formula y (). If no, is here a non-eponenial formula ha you can find? Table A Table B Table C Table D y y y y 0 5-2 100 0 1 1 1010 1 10-1 50 1 2 2 1020.1 2 15 0 25 2 5 3 1030.30 3 20 1 12.5 3 10 4 1040.60 4 25 2 6.25 4 17 6 1061.52 15. As you swim deeper and deeper in a lake, he amoun of sunligh reaching your locaion decreases. Suppose ha he amoun of sunligh hiing he surface of he lake has a brighness of 800 kilolumens, and he amoun decreases by 8% for each 1 meer increase in deph. a) Idenify variables hen wrie funcion relaing sunligh amoun o deph underwaer. b) Consruc a able of sunligh values for dephs up o 10 meers. Use deph inervals of 1 meer. c) Make a deailed graph displaying he daa from par b). Use graph paper. d) Esimae o he neares enh of a meer he deph where only half he surface sunligh peneraes. 16. In an aricle on caffeine consumpion, columnis Vicky Lowery wries, A Red Bull spokeswoman said 1.5 billion cans of he drink were consumed worldwide in 2003, a 10% increase from he previous year. New York Times, May 11, 2004. I s Fizzy and he Can is Nice, Bu Coffee May be Cheaper. a) Assume ha Bed Bull sales coninues o grow by he same percenage each year. Deermine he yearly growh rae and yearly muliplier for sales. b) Idenify variables and wrie an eponenial funcion epressing Red Bull sales over ime. c) Esimae he sales of Red Bull in he year 2001. d) Esimae when sales would reach 2 billion cans. 17. Suppose ha you ren a car in King Couny. In addiion o renal coss, you pay a special a of approimaely 10% (for ballparks, ransporaion, ec.) as well as he sandard sales a of 8.8%. Assume ha hese aes are compounded; ha is, afer one a is compued, he oher a is imposed on oal of he renal fee and he firs a. a) Does i maer which a is compued firs? Answer yes or no, hen jusify wih mahemaics. b) Wha is he oal amoun of a? (Epress as a percen.) 18. Your sock in Fasmoney.com jumps 10% on Tuesday. On Wednesday he company repors poor earnings and he sock drops 10%. Your friend says, "No worries. The wo price changes even ou." Is your friend correc? Answer yes or no, hen jusify wih mahemaics. 13
19. Try he following eperimen: On a shee of paper, draw a square wih sides of lengh 1 inch. Then subdivide he square ino 4 equal squares. Now ake each of hose squares and subdivide hem. See below. sage 0 sage 1 sage 2 Coninue his process several more imes, each ime subdividing squares by four addiional squares. Then complee he following able. lengh of side of individual area per individual sage # squares oal area square square 0 1 1 in. 1 in. 2 1 in. 2 1 4 0.5 in. 0.25 in. 2 1 in. 2 2 3 4 5 6 *** *** *** *** *** n 14
: Soluions 1. a) verical inercep = 200, muliplier = 1.075, growh rae = 0.075 or 7.5% c) verical inercep = 1, muliplier = 3, growh rae = 2 or 200% e) verical inercep = 300, muliplier = 0.45, decay rae = -0.55 or -55% 3. Maching 5. Table A: a) b = 1.25 b) 25% increase c) A ( 3) = 39. 063, A ( 4) = 48. 828, A ( 5) = 61. 035 d) A ( ) = 20(1.25). Table B: a) b = 1.25 b) 25% increase c) y ( 4) = 25. 6, y ( 3) = 32, y ( 2) = 40, y ( 1) = 78. 125 d) y ( ) = 62.5(1.25). 7. a) y ( ) = 0.5 + 6 b) c) y ( ) = 20 + 200 d) y ( ) = 6(2) y ( ) = 100(0.25) 9. The funcion is A ( ) = 3000(1.046). A() A() 0 $3000.00 6 $3929.27 1 $3138.00 7 $4110.01 2 $3282.35 8 $4299.07 3 $3433.34 9 $4496.83 4 $3591.27 10 $4703.68 5 $3756.47 11. a) A ( ) = 2500(1.05). In 5 years he CD is worh b) A ( ) = 2500(1.025). In 5 years he CD is worh c) A ( ) = 2500(1.0125). In 5 years he CD is worh 5 A (5) = 2500(1.05) = $3190.70 10 A ( ) = 2500(1.025) = $3200.21 20 A ( ) = 2500(1.0125) $3205.09 13. a) Algeria: P A( ) = 31.2(1.01737), Belarus: P B ( ) = 10.3(0.99832), Canada: P C ( ) = 31.3(1.01015), Hungary: P H ( ) = 10.1(0.9966), Kenya: P K ( ) = 30.3(1.01526) b) Algeria, wih PA( 10) 37. 1 million people. 15
c) Belarus, wih P ( 10) 10. 47 million people. B d) wihou cu: P A( 5) 34. 01. wih cu: P A( 5) 33. 67. Change of -0.34 million people. Relaive change of 0.34 / 34.01 0. 01 or a drop of 1%. 15. a) Le S = brighness in kilo-lumens, le d = deph in meers. b) S ( d) = 800(0.92). d d S d S 0 800 6 485 1 736 7 446 2 677 8 411 3 623 9 378 4 573 10 348 5 527 c) see below d) a lile over 8 meers deep 17. a) No, he order of compounding doesn' maer. Toal cos can be compued as (renal cos)(1.10)(1.088) or (renal cos)(1.088)(1.10). b) The overall muliplier is (1.10)(1.088) = 1.1968, which means he overall a is 19.68%. 19. lengh of side of individual area per individual sage # squares oal area square square 0 1 1 in. 1 in. 2 1 in. 2 1 4 0.5 in. 0.25 in. 2 1 in. 2 2 16 0.25 in 1/16 in. 2 1 in. 2 3 64 0.125 in 1/64 in. 2 1 in. 2 4 256 0.0625 in 1/256 in. 2 1 in. 2 5 1024 0.03125 in 1/1024 in. 2 1 in. 2 6 4096 0.015625 in 1/4096 in. 2 1 in. 2 *** *** *** *** *** n 4 n 0.5 n n n inches 1/4 = 4 in. 2 1 in. 2 16