Objectives for Exponential Functions Activity

Transcription

1 Objecives for Recognize siuaions having a consan percen change as exponenial Creae an exponenial model given wo poins Creae and inerpre an exponenial model in a conex Compound ineres problems Perform exponenial regression 1

2 Exponenial vs Linear Growh 1. How is exponenial growh differen from linear growh? x Explain he difference: Linear: f( x ) Exponenial: gx ( ) Name he following aribues as linear or exponenial, as appropriae. consan percen change consan amoun of change an iniial saring amoun is given muliply he value of y by he same facor for each uni increase in x "he longer i goes, he faser i grows" add (subrac) o he value of y by he same amoun for each uni increase in x 3. Suppose you are hired in a new job wih a saring salary of $30,000. Fill in he able wih your annual salary increases depending on wheher your boss is using a linear model or an exponenial model. Linear Model: Afer each year, you will ge a salary raise of $1500. Tha amoun of increase is consan every year. Exponenial Model: Afer each year, you will always ge a salary raise of 5% from he previous year. Year $ Amoun of Raise Linear Model Salary Linear Model $ Amoun of Raise Exponenial Model Salary Exponenial Model 0 $0.00 $30,000 $0.00 $30, Which model would you raher be paid by? 2

3 4. Read each senence abou a own's populaion growh (decline) and deermine he formula ha maches he saemen. 1. A own sars wih 500 people and grows by 50 people per year. 2. A own sars wih 500 people and grows by 50% per year. 3. A own sars wih 500 people and declines by 50% per year. 4. A own sars wih 500 people and declines by 50 people per year. a. b. c. y (50) 500 d. 500(50) y ) 500 (0.50 e. y 50 y ) ( f. y g. y 500( ) 500 h. y y ) 500 ( 0.50 i. 50 y 500 (1.5 ) 5. Each formula below describes he populaion rend for a ficiious own. Wrie a senence ha describes he own according o each formula, where P represens populaion and represens number of years. a. P 1000 (1.1 ) b. P c. P 3000 (0.75) d. P

4 The Form of an Exponenial Funcion, P = ab 1. The populaions, P, of six owns wih ime in years are given by: i. P = 1000(1.08) ii. P = 600(1.12) iii. P = 2500(0.9) iv. P = 1200(1.185) v. P = 800(0.78) vi. P = 2000(0.99) a. Which owns are growing in size? Which are shrinking? b. Which own is growing he fases? Wha is he annual percen growh rae for ha own? c. Which own is shrinking he fases? Wha is he annual percen decay rae for ha own? d. Which own has he larges iniial populaion (a = 0)? Which own has he smalles? 2. A own has populaion 3000 people a year = 0. Wrie a formula for he populaion, P, in year if he own: a. Grows by 200 people per year. b. Grows by 6% per year. c. Shrinks by 50 people per year. d. Shrinks by 4% per year. 4

5 On Augus 2, 1988, a U.S. Disric Cour judge imposed a fine on he ciy of Yonkers, New York, for defying a federal cour order involving housing desegregaion. The fine sared a $100 for he firs day and was o double daily unil he ciy chose o obey he cour order. 1. Wha was he daily percen growh rae of he fine? 2. Find a formula for he fine as a funcion of, he number of days since Augus 2, If he ciy of Yonkers had waied 30 days before obeying he cour order, wha would he fine have been? 4. In 1988, he annual budge of he ciy was $337 million. If he ciy had chosen o disobey he cour order, a wha poin would he fine have wiped ou he enire annual budge? 5

6 Disance Traveled Suppose you ravel a disance of m miles in h hours. The able gives cerain values for m and h. ime elapsed in hours, h disance in miles, m Average rae of change (miles per hour) Wha kind of funcion is his? linear or exponenial? Wha do you noice abou he average rae of change? Is i consan? Give a formula for disance raveled as a funcion of ime. In he conex of his problem, inerpre he slope in your formula. Wha does i mean? Mockingbird Populaion Suppose you coun he number of mockingbirds in your backyard and find ha here are 10. The nex day you coun again, and find ha he number is now 20. On he hird day, you coun again, and find 40 mockingbirds! Assuming ha his paern of growh coninues, fill in he able. Day, d Number of birds, N Average rae of change (birds per day) Wha kind of funcion is his? linear or exponenial? Wha do you noice abou he average rae of change? Is i consan? Give a formula for he number of mockingbirds as a funcion of ime. Wha growh facor did you use in your formula? In he conex of his problem, wha does i mean? 6

7 Exponenial Growh & Decay 1. Given he percen change, find he growh (or decay) facor: a. Growh of 10% per year b. Growh of 1% per year c. Growh of 90% per year d. Growh of 25% per year e. Growh of 100% per year f. Decline of 10% per year g. Decline of 1% per year h. Decline of 90% per year i. Decline of 25% per year 2. Given he growh (or decay) facor, find he percen change. Wrie your answer as a growh or decay rae. a b c d e f. 4 7

8 1. A colony of baceria sars wih 300 organisms and doubles every week. Weeks Baceria a. Wrie an exponenial equaion o represen he daa in he able. b. How many baceria will here be afer 8 weeks? 2. The populaion of a small own is 7,000, and is growing a a rae of 12% per year. a. Wrie an exponenial equaion o represen he populaion growh. b. Wha will he populaion of he own be in 15 years? grams of a radioacive elemen decays a rae of 9.5% per day. a. Wrie an exponenial equaion o represen his scenario. b. How much radioacive maerial will be lef in 2 weeks? 4. Y = 15000(1.17) x describes he growh of Happy College. Wha informaion does his formula give us abou he college? 5. There are 950 sudens enrolled in Mah 150 a he beginning of he semeser. If sudens drop a he rae of 1% per week, how many sudens will be enrolled during he 15 h week of he semeser? 6. Imagine ha 2000 people cach a cold, all a he same ime. Half of hose who are sick ge well each day. a. Wrie an equaion o represen he number of people who are sick on any given day. b. How many people will be well on day 7? 8

9 Finding Exponenial Formulas For each of hese exercises, use he mehod of common raios o find each exponenial formula: Common Raio Mehod ab ab x2 x1 y y Find a formula for he exponenial funcion ha passes hrough he wo poins (0, 1) and (2, 100). 2. Find a formula for he exponenial funcion ha passes hrough he wo poins (0, 1) and (4, 1/16). 3. Find a formula for he exponenial funcion ha passes hrough he poins (7, 3.21) and (13, 3.75). 9

10 Compound Ineres n r A formula used for compounding is B P 1, where B is he balance, P is he saring amoun, r n is he annual growh rae, n is he number of compounding periods in one year, and is he number of years. 1. Wha does he fracion r n represen? 2. Wha does he produc n represen? 3. Suppose you inves $1000 in a fund pays 5% compounded daily. Afer 10 years, wha is he balance in your accoun? Nominal versus Effecive Raes Example: An accoun pays ineres a he rae of 5% per year compounded monhly. Nominal Rae = 5% (he adverised rae; does no accoun for compounding) Effecive Rae = 5.12% (wha you acually earn because of compounding): = Anoher invesmen earns 4.5% compounded daily. Wha is is effecive rae? 10

11 Coninuous Compounding A formula used for coninuous compounding is B Pe r, where B is he balance, P is he saring amoun, r is he annual ineres rae and is he number of years. The irraional number e is called he naural number and is used for he base in his formula. Recall ha e = For coninuous growh, how many compounding periods per year are here? 2. Suppose an accoun pays ineres a he rae of 5% compounded coninuously. a. If you inves $1000 in his accoun, wha will your balance be afer one year? b. In his case, wha is your effecive growh rae? 3. If you pu $1000 in an accoun ha pays 6% compounded daily, isn he balance growing coninuously? Why no use B Pe r? Explain. 11