Precalculus MATH 2412 Sectios 3.1, 3.2, 3.3 Epoetial, Logistic ad Logarithmic Fuctios Epoetial fuctios are used i umerous applicatios coverig may fields of study. They are probably the most importat group of fuctios i terms of their applicatio. The mai reaso for this is because their rate of chage is proportioal to their amout at ay time. Form: Let f( ) = ab. Fid ad simplify f( + 1). How is this ew fuctio related to f? b is called the growth factor Domai Rage Itercepts Asymptotes Graphs b > 0, 0< b < 1 Epoetial Growth or Decay. Let a =1. The, f( ) = b
Liear Data vs Epoetial Eample: Determie if the followig data defies a liear or epoetial fuctio model. (a) -3-2 -1 0 1 y 64 32 16 8 4 (b) -2-1 0 1 2 y 3 5.5 8 10.5 13 Applicatio: Compoud Iterest Formula: Derived from the simple iterest formula I = Pr t Eample: Suppose $15,000 is ivested at a iterest rate of 7% compouded aually. Calculate the accout balace after 10 years. Eample: Now suppose the pricipal was compouded mothly. Calculate the accout balace for the same time period.
The Natural Epoetial Fuctio Cotiuous Compoudig Suppose $1 was ivested ito a accout payig 100% aual iterest. Fill i the table below where is the umber of compoudig periods i oe year ad A is the amout i the accout after 1 year. 1+ 1 10 100 1000 10000 100000 1000000 10000000 100000000 This ca be epressed i the followig way: as, 1+ e Or i limit otatio as: lim 1+ = e From this limit, the cotiuous compoudig of iterest formula ca be derived Where A is the amout at ay time t, P is the pricipal or iitial deposit or amout ivested r is the aual rate of iterest ad t is the time i years Eample: Suppose you ivest $6,000 ito a accout that ears 8% iterest compouded cotiuously. How much will you have i the accout thirty years from ow?
Natural Epoetial Growth ad Decay Models: Applicatios: Eample: A job offer at a recet college graduates job fair offers a startig salary of $44,000 with a guarateed raise of 6% each year. Fid a fuctio f that computes the salary durig the th year. Eample: A bacteria culture iitially cotais 10,000 bacteria ad is foud to double i size every 4 hours. Fid a epoetial model of the form f ( ) = Ca that represets the amout of bacterial i the culture at ay time hours. Eample: Bacterial Growth p 263 # 37 Eample: Radioactive Decay p 271 # 33 Eample: Half Life p 272 # 40 Note: the 3 rd problem above ca be modeled with a cotiuous decay model. I order to do this we eed to be able to solve epoetial equatios of the form e = b, where b is a positive costat. We could approimate solutios usig a calculator, however, there is a better way
Logistic Growth Aimal populatios are ot capable of urestricted growth because of limited habitat ad food supplies. Uder such coditios the populatio follows a logistic growth model: c c Pt () = or Pt () = kt 1 + ab 1 + ae kt where c, a, ad k are positive costats. For a certai fish populatio i a small pod c = 1200, k = 11, ad a = 0.2 ad t is measured i years. The fish were itroduced ito the pod at time t = 0. Usig the secod form for logistic growth (a) How may fish were origially put ito the pod? (b) Fid the populatio after 10, 20, ad 30 years. (c) Evaluate Pt () for large values of t. What does the populatio approach as t? (d) Approimate the value of t where the rate of chage of the populatio with respect to time is the greatest. Studets try P 263 # 56 P 272 # 45
Sectio 3.3 The Logarithmic Fuctio A logarithmic fuctio is best thought of as the iverse of a related epoetial fuctio. For eample, sketch a graph of the epoetial fuctio iverse of this fuctio. y = 2. Now, write a equatio for the Defiitio: The logarithmic fuctio with base b Domai *Iverse Properties Commo Logarithm Natural Logarithm Evaluatig Logarithms Eample: Evaluate the followig: (a) log381 (b) log4 64 Graphs ad Iverses (c) 0.67 l e Sketch the followig epoetial fuctios with their iverse o the same set of aes. Write a logarithmic equatio for each iverse. (a) f( ) = 2, (b) 1 f( ) = 2, (c) f( ) = 10, ad (d) f( ) = e.
Solvig Epoetial ad Logarithmic Equatios Eample: Usig the iverse property, covertig to logarithmic form, or equivalet base property, solve each of the followig epoetial equatios (a) 1 2 = (b) 2e = 7 (c) 5 3 6 = 24 16 Eample: Covertig to epoetial form or usig the iverse property, solve each of the followig logarithmic equatios (a) log6 = 3 (b) log3 = 4 (c) l = 6.2 (d) 6log2 3 = 12 Chage of Base Formula: log a log = log b b a