4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS

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1 Funcions Modeling Change: A Preparaion for Calculus, 4h Ediion, 2011, Connally 4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS

2 Growing a a Consan Percen Rae Example 2 During he 2000 s, he populaion of Mexico increased a a consan annual percen rae of 1.2%. (no amoun Since he populaion grew by he same percen each year, i can be modeled by an exponenial funcion. Ex: Calculae he populaion of Mexico for he years afer In 2000, he populaion was 100 million. Funcions Modeling Change: A Preparaion for Calculus, 4h Ediion, 2011, Connally

3 The populaion grew by 1.2%, so Pop. in 2001 = Pop. in % of Pop. in 2000 Pop. in 2001 = (100 Pop. in 2001 = 100 ( facored 100 Pop. in 2001 = 100 (1.012 Pop. in 2001 = million STOP HERE

4 Year One (2001 Wihou he final muliplicaion ( ( (1.012

5 Year Two The populaion grew by 1.2% again, so Pop. in 2002 = Pop. in % of Pop. in 2001 = [ (100] [ (100] = [100 (1.012] [100 (1.012] simplify = [100 (1.012] { } facored [100 (1.012] = [100 (1.012(1.012] = 100 (

6 Year Two( ( ( ( and in general...for year 100 ( 1012.

7 Funcions Modeling Change: A Preparaion for Calculus, 4h Ediion, 2011, Connally Growing a a Consan Percen Rae Example 2 coninued Populaion of Mexico The populaion of Mexico increased by slighly more each year han i did he year before, because each year he increase is 1.2% of a larger number. Year ΔP, % increase in populaion P, populaion (millions The projeced populaion of Mexico, assuming 1.2% annual growh P, populaion (millions year

8 Growh Facor vs. Percen Growh Rae The growh facor is equal o ( 1 + growh rae The Growh Facor of an Increasing Exponenial Funcion: In Example 2, he populaion grew by 1.2%, so, New Populaion = Old Populaion + 1.2% of Old Populaion = ( * Old Populaion = * Old Populaion We call he growh facor. Funcions Modeling Change: A Preparaion for Calculus, 4h Ediion, 2011, Connally

9 Example 3 Carbon-14 is used o esimae he age of organic compounds. Over ime, radioacive carbon-14 decays ino a sable form. The decay rae is 11.4% every 1000 years. For example, if we begin wih a 200- microgram (μg sample of carbon-14 hen

10 Growh Facors vs. Percen Growh Raes The growh facor is equal o ( 1 + growh rae The Growh Facor of a Decreasing Exponenial Funcion: In Ex 3, he carbon-14 changes by 11.4% every 1000 yrs. New Amoun = Old Amoun 11.4% of Old Amoun = (1.114 * Old Amoun = * Old Amoun Alhough represens a decay facor, we use he erm growh facor o describe boh increasing and decreasing quaniies. Funcions Modeling Change: A Preparaion for Calculus, 4h Ediion, 2011, Connally

11 Funcions Modeling Change: A Preparaion for Calculus, 4h Ediion, 2011, Connally A General Formula for he Family of Exponenial Funcions An exponenial funcion Q = f ( has he formula form f ( = a b, a 0, b > 0, where a is he iniial value of Q (a = 0 and b, he base, is he growh facor. The growh facor is given by b = 1 + r where r is he decimal represenaion of he percen rae of change. If here is exponenial growh, hen r > 0 and b > 1. If here is exponenial decay, hen r < 0 and 0 < b < 1.

12 Funcions Modeling Change: A Preparaion for Calculus, 4h Ediion, 2011, Connally Applying he General Formula for he Family of Exponenial Funcions Example 6 Using Example 2, find a formula for P, he populaion of Mexico (in millions, in year where = 0 represens he year Soluion In 2000, he populaion of Mexico was 100 million, and i was growing a a consan 1.2% annual rae. The growh facor is b = = 1.012, and a = 100, so P = 100( Because he growh facor may change evenually, his formula may no give accurae resuls for large values of.

13 EXAMPLES If you sar wih 500 iems. (Show How many do you have afer he following change? 100% increase 42% decrease 42% decrease followed by a 42% increase

14 The populaions, P, of six owns wih ime in years are given by: (i P 1000( 108. (ii P 2500( 0. 9 (iii P 800( (iv P 600( 112. (v P 1200( (vi P 2000( 0. 99

15 (a Which owns are growing in size? Which are shrinking? (i P 1000( 108. (ii P 2500( 0. 9 (iii P 800( (iv P 600( 112. (v P 1200( (vi P 2000( 0. 99

16 (b Which own is growing he fases? Wha is he annual percen growh rae for ha own? (i P 1000( 108. (ii P 2500( 0. 9 (iii P 800( (iv P 600( 112. (v P 1200( (vi P 2000( %

17 (c Which own is shrinking he fases? Wha is he annual percen "decay" rae for ha own? (i P 1000( 108. (ii P 2500( 0. 9 (iii P 800( (iv P 600( 112. (v P 1200( % (vi P 2000( 0. 99

18 (d Which own has he larges iniial populaion (a = 0? Which own has he smalles? (i P 1000( 108. (ii P (iii P (iv P 600( 112. (v P 2500( ( ( larges smalles (vi P 2000( 0. 99

19 The amoun (in milligrams of a drug in he body hours afer aking a pill is given by: A( 25( (a Wha is he iniial dose given? (b Wha percen of he drug leaves he body each hour? (c Wha is he amoun of drug lef afer 10 hours? (d Esimae afer how many hours is here less han 1 milligram lef in he body?

20 Radioacive gallium-67 decays by 1.48% every hour; here are 100 milligrams iniially. (a Find a formula for he amoun of gallium-67 remaining afer hours. (b How many milligrams are lef afer 24 hours? Afer 1 week?

21 A one-page leer is folded ino hirds o go ino an envelope. If i were possible o repea his kind of ri-fold 20 imes, how hick would he leer be? (A sack of 150 pieces of saionery is one inch hick

22 Pollued waer is passed hrough a series of filers. Each filer removes 85% of he remaining impuriies. Iniially, he unreaed waer conains impuriies a a level of 420 pars per million (ppm. Find a formula for L, he remaining level of impuriies, afer he waer has been passed hrough a series of n filers.

23 1. Which is greaer, ao or a1? a b 0 ( 0 a 1 ( b 1 0

24 2. Which is greaer, bo or b1? a b 0 ( 0 a 1 ( b 1 0

25 3. Wha happens o o if ao is increased while he oher quaniies remain fixed? a b 0 ( 0 a 1 ( b 1 0

26 4. Wha happens o o if b1 is decreased while he oher quaniies remain fixed? a b 0 ( 0 a 1 ( b 1 0

27 Show work o prove his represens an Exponenial x f (x

28 Generae a possible exponenial formula for he skech.

29 45 2, , 3

30 Figure 3.12 shows he balance, P, in a bank accoun. (a Find a possible formula for P = f( assuming he balance grows exponenially. (b Wha was he iniial balance? (c Wha annual ineres rae does he accoun pay?

31 P dollars P = f( 8,5000 3,2000 years

32 Le p(x 2 x q(x 2 x Esimae he values of x such ha p(x < q(x.

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