SM Dae: Secion: Objecive: The average rae of change beween wo poins on a funcion is d. For example, if he funcion ( ) represens he disance in miles ha a car has raveled afer hours, hen finding he slope of he line connecing he poins a = hour and = 4 hours will give he average speed of he car (in miles per hour) during hose hree hours. Slope Formula: The slope beween he poins ( x, y ) and (, ) Examples: Find he slope beween each pair of poins., 4 and, x y is a) (,5 ) and ( 4,9 ) b) ( ) ( ) c) ( 4, 7 ) and (, 5) Examples: For each of he following, draw he line ha connecs he wo poins. Wrie he coordinaes of he wo poins, hen calculae he average rae of change on he specified inerval. 3 a) f ( x) = x + on [ 3,3] b) f ( x) = x 8x 4 on [ 3,0] c) f ( x ) = x +5 on [ 0,3] d) f ( x) = ( x + ) 5 on [ 4,] 4
Example: Find he average rae of change for f ( x) = 3x 5x + 4 on he inerval [ ],3. Sep : he value of he funcion a x = and x = 3. f 3 = f ( ) = ( ) Sep :., 3, ( ) ( ) Sep 3:. Examples: Find he average rae of change for each funcion on he specified inerval. f x x 3 on 5,7 f x = x + 4 on 4, a) ( ) = + [ ] b) ( ) [ ] c) f ( x) = x on [, 4] d) f ( x) = ( x 3) 5 on [ 0,0] Examples: a) Many of he elderly are placed in nursing care faciliies. The cos of hese has risen significanly since 960. Use he able below o find he average rae of change from 000 o 00 and explain wha your resul means. Years Since 960 Nursing Care Cos (billions of dollars) 0 0 4 0 8 30 53 40 96 50 57
b) The graph below shows he heigh, in fee, of an objec launched sraigh up from an iniial heigh of 40 fee. Find he average rae of change from o 4 seconds and explain wha your answer means. Heigh (fee) Time (seconds) c) Suppose 5 flour beeles are lef undisurbed in a warehouse bin. The beele populaion doubles in size every week. The equaion ( ) 5 x P x = can be used o deermine he number of beeles afer x weeks. Complee he able below. Week 0 3 4 5 Beele Populaion Calculae he average growh rae beween weeks and 3. Calculae he average growh rae for he firs five weeks [0,5]. Which average growh rae was higher? Why do you hink i is higher?
SM Dae: Secion: Objecive: Linear Funcions Can be wrien in he form y = mx + b. Graph is a sraigh line. Used o model sequences wih a common firs difference. o Each erm is obained by adding (or subracing) he same number o he previous erm. o e.g.),, 5, 8,, +3 +3 +3 +3 Exponenial Funcions x Can be wrien in he form y = ab. Variable is in he exponen. Used o model sequences wih a common raio. o Each erm is obained by muliplying (or dividing) he previous erm by he same number. o e.g.), 6, 8, 54, 3 3 3 Quadraic Funcions Can be wrien in he form y = ax + bx + c, where a 0. Graph is a parabola. Can be formed by muliplying wo linear funcions. y = x + 5 x 3 o e.g.) ( )( ) Used o model sequences wih a common second difference. o The second difference is he difference beween he numbers in he firs difference. o e.g.), 8, 8, 3, 50, +6 +0 +4 +8 +4 +4 +4 Examples: Deermine wheher he paern would be modeled by a linear funcion, an exponenial funcion, or a quadraic funcion. a) b) c) d) 54, 48, 4, 36, e), 3, 0,, 39, f) 8, 7, 9, 3,
Using a Graphing Calculaor o Find a Regression Equaion Use a graphing calculaor o deermine he quadraic funcion modeled by he given daa: x 3 4 5 6 f ( x ) 9 3 43 69 0 Inpuing Daa Ino a Calculaor Lis Press STAT. Choose Edi If you have daa in your liss, clear hem by highlighing he name of he lis, hen push CLEAR and ENTER. Do no push DEL. f x values ino L. Ener he x values ino L and he ( ) Push ND MODE o ge back o he home screen. Make a Scaer Plo Push ND Y= o bring up he STAT PLOT menu. Selec Plo Turn Plo on by pushing ENTER when On is highlighed. Make sure ha he scaer plo opion is highlighed. If i isn, selec i by pushing ENTER when he scaer plo graphic is highlighed. The Xlis should say L and he Ylis should say L. If i doesn, L can be enered by pushing ND and L by ND. Push GRAPH. If you wan a nice viewing window, push ZOOM, hen selec opion 9, ZoomSa. Creaing a Regression Equaion You do no need o graph a funcion o creae a regression, bu i is a good idea o compare your regression o he daa poins o make sure i is a good model. From he home screen, push STAT. Arrow righ o he CALC menu. Choose opion 5, QuadReg. (If you wan an exponenial regression, choose opion 0, ExpReg.) The home screen should say QuadReg. Type L, L, Y. o Press ND, comma (above he 7), ND, comma, VARS. Arrow righ o Y-VARS, selec opion, Funcion, hen selec Y. Push ENTER. The quadraic regression equaion is f ( x) = 3x x. I has been enered ino Y. Push GRAPH again o compare your regression o he daa.
Examples:. Use a graphing calculaor o find a quadraic funcion ha models he daa. x 0 3 4 5 f ( x ) 3 3 7 3. From 97 o 998 he U.S. Fish and Wildlife Service has kep a lis of endangered species in he Unied Saes. The able below shows he number of endangered species. Find an appropriae exponenial equaion o model he daa. Le x equal he number of years since 97. Year 97 975 978 98 984 987 990 993 996 Number of Species 0 58 07 73 359 473 63 8 080 3. The pesicide DDT was widely used in he Unied Saes unil is ban in 97. DDT is oxic o a wide range of animals and aquaic life, and is suspeced o cause cancer in humans. The half-life of DDT can be 5 or more years. Half-life is he amoun of ime i akes for half of he amoun of a subsance o decay. Scieniss and environmenaliss worry abou such subsances because hese hazardous maerials coninue o be dangerous for many years afer heir disposal. Wrie an equaion o model he daa below. Le x equal he number of years since 97. Year 97 98 99 00 Amoun of DDT (grams) 50 9.8.9 0.4 4. The able shows he average movie icke price in dollars for various years from 983 o 003. Find a model for he daa. Years Since 983, 0 4 8 6 0 Movie Ticke Price, m 4.75 4.07 3.65 4.0 5.08 6.03
SM Dae: Secion: Objecive: Exponenial Growh: When a quaniy increases by a fixed percen each year, or oher period of ime, he amoun of ha quaniy, A, afer ime periods is given by he following formula: ( ) A = A + r 0 A 0 = iniial (saring) amoun r = growh rae, as a decimal = # of ime periods A = final amoun Exponenial Decay: When a quaniy decreases by a fixed percen each year, or oher period of ime, he amoun of ha quaniy, A, afer ime periods is given by he following formula: ( ) A = A r 0 A 0 = iniial (saring) amoun r = growh rae, as a decimal = # of ime periods A = final amoun Examples: Deermine wheher he equaion shows growh or decay and deermine he growh or decay rae. A = 400(.7) Growh or Decay? Rae: A =,000( 0.83) Growh or Decay? Rae: A = 987(.005) Growh or Decay? Rae: A = 347( 0.9975) Growh or Decay? Rae: Tips: If a sory problem gives you a saring amoun and an amoun afer ime period, you can find he Amoun of Change growh rae by using: Rae = Saring Amoun To solve an equaion like 500 = 300(.) graphically: x o Graph y = 500 and y = ( ) 300.. o Make he graphs fi in your window by pushing ZOOM. Choose 0:ZoomFi. If he graphs sill don fi, push WINDOW and change he x-coordinaes (make Xmax be a bigger number). o Find he inersecion of he wo graphs. Push ND TRACE. Choose 5:inersec. Move he cursor along one graph unil you are abou where he graphs cross. Hi ENTER. Repea for he oher graph. Hi ENTER again o mark your guess. o The x-coordinae on he boom of he screen is your answer. If a problem asks you wheher an equaion represens growh or decay and you can ell, graph he equaion and see if he graph goes up from lef o righ (growh) or down from lef o righ (decay). If a problem asks abou Effecive Ineres Rae, he answer will always be he choice ha is higher han he ineres rae he problem gives you.
Examples:. Ted wans o sar a new exercise program. He walks 0.5 miles on he firs day, and increases he disance he walks by 0% each day. a. Wrie an exponenial growh model ha describes he siuaion. b. How far will Ted walk on he 5 h day? c. Afer how many days will Ted be walking 3 miles? (Solve graphically).. The price of a scienific calculaor, which currenly sells for $89, decreases by 6% each year. a. Wrie an exponenial decay model ha describes he siuaion. b. How much will he calculaor cos afer 5 years? c. Afer how many years will he price of he calculaor be $50? (Solve graphically). 3. A blue whale calf weighed 5,000 pounds a birh (or a ime = 0). One week laer he baby blue whale weighed 6,750 pounds. a. Find he growh rae. b. Wrie an equaion o model he weigh of he blue whale, in pounds, a any ime. c. If i coninues growing a is curren rae, wha will he blue whale calf weigh in 8 weeks? d. When will he calf weigh 00,000 pounds? (Solve graphically). 4. Hugo wans o sell his car. The value of his car has been decreasing by 0% per year. Is curren value is $000. If he car is en years old, how much was i worh when he bough i? 5. A school has room for 500 sudens. The suden populaion is growing a a rae of 3.6% per year. If he school will be full in 6 years, how many sudens are here now?