Section 3: Exponential Functions Exponential Functions Definition: The exponential functions are the functions of the form f(x) =a x,wherethe base a is a positive constant with a 6= Properties of the Graphs of f(x) =a x Domain is the set of all real numbers 2 Range is the set of all positive real numbers 3 All graphs pass through the point (0, ) 4 The graph is continuous (no holes or jumps) 5 The x axis is a horizontal asymptote (but only in one direction) 6 If a>, the graph is increasing (exponential growth) 7 If 0 <a<, the graph is decreasing (exponential decay) Example : Graph f(x) =3 x, g(x) =6 x, h(x) = x,andk(x) = 3 x 6 EI **
Rule Properties of Exponents: If a and b are positive numbers and x and y are any real numbers, then a x+y = a x a y 2 a x y = ax a y 3 (a x ) y = a xy 4 (ab) x = a x b x 5 a x = a y if and only if x = y 6 For x 6= 0,a x = b x if and only if a = b Example 2: Solve for x: a) 4 x 2 3x =8 4 b) 25 =54x 5 3x2 change all bases to 24 : (24*24=234 5 2=2 +2 +2 ( z2xixz3 J=zk Used Rule 3?z Iu srrlst 25 4 5 x 2 Example 3: Simplify the expression 5 3 x / te 5x=H change bases to nsn / = J 54 =5 2 4 2=4 3 2 used roles 52=5953 2 used Rate =4k 3 2+2 I} ' 52=54 3 2 Used Rate 0=3 2+4 +2 / xai±etnohy, 52 62 xs *nhpx F ± g 2 6 652 6+4 Distributing 65 6M Role 2 ' 2 Fall 206, Maya Johnson
, * Exponential Function with Base e Exponential functions with base e are the most commonly used Example 4: Simplify ex+5 e 5 x eexjsjn =e t5 = ex +5 =/ej 5+ Rule 2 Distribution Example 5: Solve each equation for x: a) 9 x =3 +x Change bases to 3 ( 39*=34 32 2 =3 '+ Role 3 2 2 = Ltx Rule 5 X +2 +2 X x=3? b) x 2 e x 5xe x =0 ' ( 25 =0 factoring e ( X 5) ( TY 5) =o factoring = 0 ex rs never so we can drop it a 3 Fall 206, Maya Johnson
Applications of Exponential Functions Growth and Decay Applications Functions of the form y = ce kt where c and k are constants and the independent variable t represents time are often used to model population growth and radioactive decay The constant c represents the initial amount The constant k is called the relative growth rate We say that the population is growing continuously at the relative growth rate k Example 6: The population of a particular city grows continuously at a relative growth rate of 54% If 30,000 people currently live in the city, what will be the population in eight years? lee 05454% ) Example 7: The population of an undesirable city is modeled by Pl8azooooeo54lDCa3OoooltojiPltJazooooees4t7462owhent8PPulatoonisy6z@t40y y = ce ( 009t) where t represents the number of years since 950, y represents the population in the t th year, and c is a constant representing the population in 950 a) If the city had a population of 20,000 in 990, what is the value of c? 40=20000 y=ceh 9H zo oo=cef 9l40 ) ) cnetgg#of73l965jt=58yl58)= b) Use the model to predict the population in 2008 73965 Populatronis39T 09 58 3958 4 Fall 206, Maya Johnson
: Finance (NonContinually Compounded Interest) Compound Interest If a principal P (present value) is invested at an annual rate r (expressed as a decimal) compounded m times a year, then the amount F in the account at the end of t years is given by F = P + r mt m Example 8: If $5,000 is invested in an account paying 25% compounded monthly, how much will be in the account at the end of 0 years? 5o P r=ao25( m=l2 tto 25% ) F=50o0( +aq yrx ) Finance (Continuously Compounded Interest) F=$648#P A = Pe rt where P =principal, r=annual interest rate compounded continously (as a decimal), t =Time in years, A =Accumulated amount at the end of t years Example 9: What amount will an account have after five years if $,000 is invested at an annual rate of 325% compounded continuously? 000 A= 000203255 ) F eo325l3i25% ) t=5 /A=$l7645 E ective Yield: The equivalent interest rate if compounding was only done once a year Suppose asumofmoneyisinvestedatanannualrateofr expressed as a decimal and is compounded m times per year The e ective yield is: [ ( +(rm))m is compounded continuously The e ective yield is: E 5 Fall 206, Maya Johnson
Example 0: You have been doing some research and have found that you can either invest your money at an annual interest rate of 355% compounded monthly or 350% compounded continuously Which one would you choose? Fond the etteetove yield for each and see which is higher Let EYM be effective yield when compounded and monthly Eye be foc continuous = 036 EYM =[ t+( 0355/2%2 ( 36 %) I = 0356 ( 356 %) f Yc = e 35 i s wech se3=%compoundedmouthg] Calculator Functions TVM Solver: We can use the TVM Solver on our calculator to solve problems involving compound interest To access the Finance Menu, you need to press APPS > :Finance (Please note that if you have a plain TI83, you need to press 2nd x to access the Finance Menu) Below we define the inputs on the TVM Solver: N =the total number of compounding periods I% =interestrate(asapercentage) PV = present value (principal amount) Entered as a negative number if invested, a positive number if borrowed PMT =paymentamount(0forthisclass) FV =future value (accummulated amount) P/Y = C/Y =the number of compounding periods per year Move the cursor to the value you are solving for and hit ALPHA and then ENTER E ective Yield: We use the C:E ( option on the Finance Menu to compute the e ective rate of interest The inputs are as follows: E (annual interest rate as a percentage, the number of compounding periods per year) 6 Fall 206, Maya Johnson