. Eponential Functions and Their Graphs In this section ou will learn to: evaluate eponential functions graph eponential functions use transformations to graph eponential functions use compound interest formulas An eponential function f with ase is defined f = Note: An transformation of ( ) or =, where > 0,, and is an real numer. = is also an eponential function. Eample : Determine which functions are eponential functions. For those that are not, eplain wh the are not eponential functions. (a) f ( ) = + 7 Yes No () (c) (d) (e) g ( ) = Yes No h h ( ) = Yes No f = ( ) Yes No ( ) = 0 Yes No (f) + f ( ) = + Yes No + (g) g ( ) = ( ) + Yes No (h) h ( ) = Yes No Eample : Graph each of the following and find the domain and range for each function. (a) () f ( ) = domain: range: g( ) = domain: range: Page (Section.) 7 6 7 6 6 7 8 6 7 8
Characteristics of Eponential Functions > 0 < < f ( ) = Domain: Range: Transformations of g() = (c > 0): (Order of transformations is H S R V.) Horizontal: g g + c ( ) = (graph moves c units left) c ( ) = (graph moves c units right) Stretch/Shrink: g ( ) = c (graph stretches if c > ) (Vertical) (graph shrinks if 0 < c < ) Stretch/Shrink: c g ( ) = (graph shrinks if c > ) (Horizontal) (graph stretches if 0 < c < ) Reflection: g g ( ) = (graph reflects over the -ais) ( ) = (graph reflects over the -ais) Vertical: g( ) = + c (graph moves up c units) g( ) = c (graph moves down c units) Page (Section.)
Eample : Use f + ( ) = to otain the graph ( ) = g. Domain of g: Range of g: Equation of an asmptote(s) of g: 7 6 7 6 6 7 8 6 7 8 f = ( ) e is called the natural eponential function, where the irrational numer e (approimatel.788) is called the natural ase. (The numer e is defined as the value that n n + approaches as n gets larger and larger.) Eample : Graph f ( ) = e, on the same set of aes. g ( ) = e, and h( ) = e 7 6 7 6 6 7 8 6 7 8 Page (Section.)
Periodic Interest Formula Continuous Interest Formula nt r A = P + rt n A = Pe A = alance in the account (Amount after t ears) P = principal (eginning amount in the account) r = annual interest rate (as a decimal) n = numer of times interest is compounded per ear t = time (in ears) Eample : Find the accumulated value of a $000 investment which is invested for 8 ears at an interest rate of % compounded: (a) annuall () semi-annuall (c) quarterl (d) monthl (e) continuousl Page (Section.)
. Homework Prolems. Use a calculator to find each value to four decimal places. (a) () π 7 (c). (d) e (e) e (f) 0. e (g) π. Simplif each epression without using a calculator. (Recall: n m n+ m m n mn = and ( ) = ) (a) 6 6 () ( ) (c) ( ) 8 (d) ( ) (e) (f) For Prolems, graph each eponential function. State the domain and range for each along with the equation of an asmptotes. Check our graph using a graphing calculator.. f ( ) =. f ( ) ( ) =. f ( ) = 6. f ( ) = 7. f ( ) = 8. f ( ) = + 9. ( ) = f 0. f ( ) = +. ( ) f = +. f ( ) =. f ( ) = e +. f ( ) = e. $0,000 is invested for ears at an interest rate of.%. Find the accumulated value if the mone is (a) compounded semiannuall; () compounded quarterl; (c) compounded monthl; (d) compounded continuousl. 6. Sam won $0,000 in the Michigan lotter and decides to invest the mone for retirement in 0 ears. Find the accumulated value for Sam s retirement for each of his options: (a) a certificate of deposit paing.% compounded earl () a mone market certificate paing.% compounded semiannuall (c) a ank account paing.% compounded quarterl (d) a ond issue paing.% compounded dail (e) a saving account paing.9% compounded continuousl +. Homework Answers:. (a) 6.; ().8079; (c).0; (d) 7.89; (e).; (f) -.80; (g).8. (a) 6 ; () 9; (c) ; (d) ; (e) ; (f). Domain: (, ) ; Range: ( 0, ) ; = 0. Domain: (, ) ; Range: (, 0) ; = 0. Domain: (, ) ; Range: ( 0, ) ; = 0 6. Domain: (, ) ; Range: ( 0, ) ; = 0 7. Domain: (, ) ; Range: (, ) ; = 8. Domain: (, ) ; Range: ( 0, ) ; = 0 9. Domain: (, ) ; Range: (, ) ; = 0. Domain: (, ) ; Range: (, 0) ; = 0. Domain: (, ) ; Range: (, ) ; =. Domain: (, ) ; Range: (, ) ; =. Domain: (, ) ; Range: (, ) ; =. Domain: (, ) ; Range: (, 0) ; = 0. (a) $,6.; () $,0.67; (c) $,7.0; (d) $,6. 6. (a) $9,0.97; () $,00.96; (c) $,79.9; (d) $,.; (e) $,.6 Page (Section.)
. Applications of Eponential Functions In this section ou will learn to: find eponential equations using graphs solve eponential growth and deca prolems use istic growth models Eample : The graph of g is the transformation of f ( ) =. Find the equation of the graph of g. HINTS:. There are no stretches or shrinks.. Look at the general graph and asmptote to determine an reflections and/or vertical shifts.. Follow the point (0, ) on f through the transformations to help determine an vertical and/or horizontal shifts. 6 6 Eample : The graph of g is the transformation of f ( ) = e. Find the equation of the graph of g. 6 6 Eample : In 969, the world population was approimatel.6 illion, with a growth rate of.7% per 0.07 ear. The function f ( ) =.6e descries the world population, f (), in illions, ears after 969. Use this function to estimate the world population in 969 000 0 Page (Section.)
Eample : The eponential function millions, ears after 986. f ) ( ) = 8.(.0 models the population of Meico, f (), in (a) Without using a calculator, sustitute 0 for and find Meico s population in 986. () Estimate Meico s population, to the nearest million in the ear 000. (c) Estimate Meico s population, to the nearest million, this ear. Eample : College students stud a large volume of information. Unfortunatel, people do not 0. retain information for ver long. The function f ( ) = 80e + 0 descries the percentage of information, f (), that a particular person rememers weeks after learning the information (without repetition). (a) Sustitute 0 for and find the percentage of information rememered at the moment it is first learned. () What percentage of information is retained after week? weeks? ear? Radioactive Deca Formula: The amount A of radioactive material present at time t is given that was present initiall (at t = 0) and h is the material s half-life. t h A = A0 () where A 0 is the amount Eample 6: The half-life of radioactive caron- is 700 ears. How much of an initial sample will remain after 000 ears? Eample 7: The half-life of Arsenic-7 is 7. das. If grams of Arsenic-7 are present in a od initiall, how man grams are presents 90 das later? Page (Section.)
Logistic Growth Models: Logistic growth models situations when there are factors that limit the ailit to grow or spread. From population growth to the spread of disease, nothing on earth can ehiit eponential growth indefinitel. Eventuall this growth levels off and approaches a maimum level (which can e represented a horizontal asmptote). Logistic growth models are used in the stud of conservation io, learning curves, spread of an epidemic or disease, carring capacit, etc. The mathematical model for limited istic growth is given c c : f ( t) = or A = t t, where a,, and c are constants, c > 0 and > 0. + ae + ae t As time increases ( t ), the epression ae and A. Therefore = c is a horizontal asmptote for the graph of the function. Thus c represents the limiting size. 00,000 Eample 8: The function f ( t) = descries the numer of people, f (t), who have 0. 06t + 999e ecome ill with influenza t weeks after its initial outreak in a town with 00,000 inhaitants. (a) How man people ecame ill with the flu when the epidemic egan? () How man people were ill the end of the th week? (c) What is the limiting size of f (t), the population that ecomes ill? (d) What is the horizontal asmptote for this function? Eample 9: The function f (t), after t learning trials. 0.8 = is a model for descriing the proportion of correct responses, + e f ( t) 0. t (a) Find the proportion of correct responses prior to learning trials taking place. () Find the proportion of correct responses after 0 learning trials. (c) What is the limiting size of f (t) as continued trials take place? (d) What is the horizontal asmptote for this function? (e) Sketch a graph of this function. Page (Section.)
. Homework Prolems. Find the equation of each eponential function, g (), whose graph is shown. Each graph involves one or more transformation of the graph of f ( ) =. (a) () (c) 6 6 6 6 6 6. Find the equation of each eponential function, g (), whose graph is shown. Each graph involves one or more transformation of the graph of f ( ) = e. (a) () (c) 6 6 6 6 6 6. In 970, the U. S. population was approimatel 0. million, with a growth rate of.% per ear. 0.0 The function f ( ) = 0.e descries the U. S. population, f (), in millions, ears after 970. Use this function to estimate the U. S. population in the ear 0.. A common acterium with an initial population of 000 triples ever da. This is modeled the t formula P ( t) = 000(), where P(t) is the total population after t das. Find the total population after (a) hours () da (c) ½ das (d) das. Assuming the rate of inflation is % per ear, the predicted price of an item can e modeled the t function P t) = P (.0), where P 0 represents the initial price of the item and t is in ears. ( 0 (a) Based on this information, what will the price of a new car e in the ear 0, if it cost $0,000 in the ear 000? () Estimate the price of a gallon of milk e in the ear 0, if it cost $.9 in the ear 000? Round our estimate to the nearest cent. Page (Section.)
6. The 986 eplosion at the Chernol nuclear power plant in the former Soviet Union sent aout 000 0 kirams of radioactive cesium-7 into the atmosphere. The function f ( ) = 000(0.) descries the amount, f (), in kirams, of cesium-7 remaining in Chernol ears after 986. If even 00 kirams of cesium-7 remain in Chernol s atmosphere, the area is considered unsafe for human haitation. Find f (60) and determine if Chernol will e safe for human haitation 06. 00,000 7. The istic growth function f ( t) = descries the numer of people, f (t), who have t + 000e ecome ill with influenza t weeks after its initial outreak in a particular communit. (a) How man people ecame ill with the flu when the epidemic egan? () How man people were ill the end of the fifth week? (c) What is the limiting size of the population that ecomes ill? 90 8. The istic growth function P( ) = models the percentage, P () of Americans who 0. + 7e are ears old with some coronar heart disease. (a) What percentage of 0-ear-olds have some coronar heart disease? () What percentage of 80-ear-olds have some coronar heart disease?. Homework Answers:. (a) g ( ) = + + + ; () g ( ) = ; (c) g ( ) = +. (a) g ( ) = e ; () g ( ) = e ; (c) g ( ) = e. aout.7 million. (a) aout 7; () 000; (c) aout 96; (d) 9000. (a) aout $,97.; () $.0 6. 0; no 7. (a) aout 0 people; () aout 88 people; (c) 00,000 people 8. (a) aout.7%; () aout 88.6% Page (Section.)
. Logarithmic Functions and Their Graphs In this section ou will learn to: change arithmic form eponential form evaluate natural and common arithms use asic arithmic properties graph arithmic functions use transformations to graph arithmic functions The arithmic function with ase is the function f ( ) =. For > 0 and > 0,, = is equivalent to =. Eample : Complete the tale elow: Logarithmic Form Eponential Form Answer 0 00 = = 7 = 8 = a 0 0 e e = 7 6 = 6 0 = = e = = 6 Page (Section.)
Eample : Evaluate each of the following arithms mentall without using a calculator: 0 00 = 0 000 = = = = = = 0.0 = = 8 = = = 7 7 = = 7 = = Basic Logarithmic Properties Involving One Inverse Properties = ecause = 0 ecause = ecause = ecause Eample : Evaluate each of the following without a calculator. 0 0 = e 9 e = 0 0 = e e = n = = n = + = 0 = Page (Section.)
f = ( ) and f ( ) = If ( ) =, then f = f are inverse functions of each other. If If ( ) =, then f = f 0 ( ) e, then f = f = Eample : Graph ( ) = and g ) f ( =. 7 6 7 6 6 7 8 6 7 8 = Characteristics of Inverse Functions: = Domain: Range: Domain: Range: Page (Section.)
Eample : Use f ( ) = to otain the graph g ( ) = ( + ) +. Also find the domain, range, and the equation of an asmptotes of g. Domain: Range: Asmptote(s): 7 6 7 6 6 7 8 6 7 8 Eample 6: Use f ( ) = to otain the graph g( ) = ( ). Also find the domain, range, and the equation of an asmptotes. Domain: Range: Asmptote(s): 9 8 7 6 9 8 7 6 6 7 8 9 0 6 7 8 9 0 Page (Section.)
Common Logs Natural Logs Logarithmic Properties General Logarithm (ase = ) Common Logarithm (ase = 0) Natural Logarithm (ase = e) Eample 7: Evaluate each of the following without a calculator:. 0 a = 0 = ln ln( + ) e = e = ( 8) = ( ( 8) ) = ln( 0) = Eample 8: Solve each of the equations changing to eponential form. ( ) = ( ) = = Page (Section.)
+ Eample 9: Let f ( ) = 6 (a) Find the domain and range of f. () Find the equation of the asmptote for the graph of f. (c) Evaluate f ( ). (d) Find the and -intercepts of f. (e) Find an equation for the inverse of f. (f) Find the domain and range of the inverse. Eample 0: Let f ( ) = ( + 9) (a) Find the domain and range of f. () Find the equation of the asmptote for the graph of f. (c) Evaluate f (8). (d) Find the and -intercepts of f. (e) Find an equation for the inverse of f. (f) Find the domain and range of the inverse. Page 6 (Section.)
. Homework Prolems Write each equation in arithmic form.. =. = 9 Write each equation in eponential form.. 9 = 7. m n = p. = 7 6. 6 = 7. 0 000 = 8. π π = 9. = 0 0. = 8 Find each value of without using a calculator.. 8 6 =. 6 8 =. 8 =. 8 =. = 8 6. 8 = 7. 9 = 8. = 9. 7 = 0. 8 = 0. π =. π = 7. =. 9 =. 8 9 = 6. ln e = 7. ln = 8. ln = 0 9. ln( ) = 0 0. ln =. Graph f ( ) = + and g ( ) = ( ) on the same graph. Find the domain and range of each and then determine whether f and g are inverse functions. For prolems -, use the graph of f ( ) = and transformations of f to find the domain, range, and asmptotes of g.. g ( ) = ( + ). g( ) = +. g( ) = ( ). g ( ) = ( ) For prolems 6-9, use the graph of asmptotes of g. f ( ) = ln and transformations of f to find the domain, range, and 6. g( ) = ln 7. g( ) = ln 8. g ( ) = + ln( + ) 9. g( ) = ln( ) Use a calculator to find each value to four decimal places. 0. 0 7... e. ln 6. ( )(ln ) Evaluate or simplif each epression without using a calculator.. 000 6. 0. ln. ln e. 7. 0 8. 000 7 ln e. 7 7 0 9. ln. e. ( 8) 6. ( ( 7)) 7. ln( ( 6)) 8. (ln e ) ln e Page 7 (Section.)
Solve each of the arithmic equations first changing the equation into eponential form. 9. ( ) = 60. = 6. ( + ) = 6. ( ) = Find the inverse function, f ( ), for each function. 6. f ( ) = ln 6. f ( ) = ( ) 6. f ( ) = e 66. = =g() f ( ) = + For each of the functions elow, find (a) the domain and range, () the equation of the asmptote of the graph, (c) the - and -intercepts, (d) the equation for the inverse function, and (e) the domain and range of the inverse function. + 67. f ( ) = 68. f ( ) = ( + 9). Homework Answers:. =. = 9. 7 = 6. = 6 7. 0 = 000 8. π = π... 9 7 =. m p = n = 8 9. 0 = 0.. 6. - 7. 8 8. 9. 7 0... 7. 6. = 7. = e 8. = 9. = 0. = =f() e. Domain of f: (, ); Range of f: (, ) ; Domain of g: (, ) ; Range of g: (, ); f and g are inverse functions. Domain: (, ) ; Range: (, ) ; Asmptote:... -. 8. Domain: ( 0, ) ; Range: (, ) ; Asmptote: = 0. Domain: (,0) ; Range: (, ) ; Asmptote: = 0. Domain: (, ) ; Range: (, ) ; Asmptote: = 6. Domain: ( 0, ) ; Range: (, ) ; Asmptote: = 0 7. Domain: ( 0, ) ; Range: (, ) ; Asmptote: = 0 8. Domain: (, ) ; Range: (, ) ; Asmptote: = 9. Domain: (,) ; Range: (, ) ; Asmptote: = 0..0....609....087. 6. - 7. ½ 8. 7 9. 0 0. 0.. 7. -7.. 6. 0 7. 0 8. 0 9. {0} 60. {9/} 6. {-¾} 6. {-, } 6. f ( ) = e 6. f ( ) = 0 + 6. f ( ) = ln 66. f ( ) = 67. (a) Domain: (, ); Range: (, ) ; () = ; (c) -int: -; -int: ; (d) f ) = ( + ) ; (e) Domain: (, ); ( Range: (, ) 68. (a) Domain: ( 9, ); Range: (, ) ; () = 9; (c) -int: 7; -int: -; (d) f ( ) = + 9 ; (e) Domain: (, ); Range: ( 9, ) Page 8 (Section.)
. Applications of Logarithmic Functions In this section ou will learn to: use arithms to solve geo prolems use arithms to solve charging atter prolems use arithms to solve population growth prolems Richter Scale Charging Batteries Population Douling Time If R is the intensit of an earthquake, A is the amplitude (measured in micrometers), and P is the period of time (the time of one oscillation of the Earth s surface, measured in seconds), then A R = P If M is the theoretical maimum charge that a atter can hold and k is a positive constant that depends on the atter and the charger, the length of time t (in minutes) required to charge the atter to a given level C is given C t = ln k M If r is the annual growth rate and t is the time (in ears) required for a population to doule, then ln t = r Eample : Find the intensit of an earthquake with amplitude of 000 micrometers and a period of 0.07 second. Eample : An earthquake has a period of ¼ second and an amplitude of 6 cm. Find its measure on the Richter scale. (Hint: cm = 0,000 micrometers.) Eample : How long will it take to ring a full discharged atter to 80% of full charge? Assume that k = 0. 0 and that time is measured in minutes. Eample : The population of the Earth is growing at the approimate rate of.7% per ear. If this rate continues, how long will it take the population to doule? Page (Section.)
. Properties of Logarithms In this section ou will learn to: use the product, quotient, and power rules epand and condense arithmic epressions use the change-of-ase propert Properties of Eponents n = n m n = m n 0 = n m n+ m m n mn = ( ) n n n n n a a ( a ) = a = n = Logarithmic Properties Involving One = = 0 Inverse Properties = = Product Rule ( MN) = M + N Quotient Rule M N = M N Power Rule p M = p M M, N, and are positive real numers with. Eample : Use the product rule to epand the arithmic epressions. ( MN) = M + N (a) () 000 (c) 000 = (d) ln ( + ) (e) ln e (f) ln (z + ) Page (Section.)
Eample : Use the quotient rule to epand the arithmic epressions. M N = M N (a) e () ln (c) ln A BC (d) (e) e ln (f) 8 Eample : Use the power rule to epand the arithmic epressions. p M = p M (a) () (c) ln (d) ln (e) Beware of these frequentl occurring errors!!! Eample : Use properties of arithms to epand each arithmic epression as much as possile. Simplif whenever possile. (a) ( ) () a (c) e + ln Page (Section.)
Eample : Let = A, = B, and = C. Write each epression elow in terms of A, B, and C. (a) 6 () 9 (c) 0 8 Condensing Logarithmic Epressions (Write as a single arithm with a coefficient of.) Product Rule Quotient Rule Power Rule M M + N ( MN) M N = p N p M = M = Eample 6: Use properties of arithms to condense each arithmic epression as much as possile. Write the epression as a single arithm with a coefficient of. Simplif whenever possile. (a) + () + 8 (c) ln + ln ln (d) ( + ) (e) ln + ln( + ) (f) 6ln + ln ln ln Page (Section.)
(g) ( ln a lnc) + ln( d + e) (h) ( ) + 0 + ( ) Change of Base Propert: a M M M = = = a ln M ln Eample 7: Evaluate each of the following using our calculator. Round to decimal places: (a) 8 = () = (c) = (d) π 00 = Eample 8: Write each of the following as a single term that does not contain an arithm: (a) e ln0 ln () (c) 0 + Page (Section.)
. Homework Prolems Determine whether each statement is true or false.. a = a +. = a. 0 = a. =. = 6. ( + ) = + ln 7. ln + ln = ln 8. ln = 9. ln(8 ) = ln() 0. 0 =. A B A =. ln( ) + ln = ln() B Use properties of arithms to epand each epression. Simplif whenever possile.. 9. 00. 6 6. ln e 7. 8. 9. 6 0. z. ln e. a. c d e ln. e 6 6 Let = A, = B, and = C. Write each arithmic epression in terms of A, B, and C.. 0 6. 7. 8. 6 7 0 Use properties of arithms to condense each arithmic epression as much as possile. Write the epression as a single arithm with a coefficient of. Simplif whenever possile. 9. + 9 0. 8 +. 96. ln + ln ln9. ln e ln+ ln. +. 6. ln( ) ln( + ) Evaluate each of the following using our calculator. Round answers to four decimal places. 7. 8. 9. π 0. π.... Write each of the following as a single term without arithms.. e ln +ln. 0. e ln 6 ln 6. 7. 6 + 0. Homework Answers:. True. True. False. False. True 6. False 7. False 8. True 9. True 0. True. False. True. 9 +.. 6. ln Page (Section.)
7. + 8. 9. + 0. z. + ln. d a + c. ln. 6 6. A + C 6. C A 7. C A B 8. B A C 9. 0... ln 9. + ln.. 6. ln( ) 7..80 8..609 9..7 0..060. -.609. -0.... 6. 7. 7 Page 6 (Section.)
.6 Eponential and Logarithmic Equations (Part I) In this section ou will learn to: solve eponential equations using like ases solve eponential equations using arithms solve arithmic equations using the definition of a arithm solve arithmic equations using -to- properties of arithms appl arithmic and eponential equations to real-world prolems convert = a to an eponential equation using ase e Definition of a Logarithm = is equivalent to = Inverse Properties = = Log Properties Involving One = = 0 Product Rule ( MN) = M + N Quotient Rule M N = M N Power Rule p M = p M One-to-One Properties If If If M N = then M = N. M = N then M = N. M = N then M = N. Eample : Solve each equation epressing each side as a power of the same ase. (a) + = () 9 = (c) e e e = e 6 Page (Section.6)
Eample : Solve = 0 using the one-to-one propert for (a) common arithms and () natural arithms. Eample : Solve e = 60 Steps for solving eponential equations:.... Eample : Solve 0 + = 8 Eample : Solve = 0 Page (Section.6)
Eample 6: Solve + = Eample 7: Use factoring to solve each of the following equations. (Hint: Use sustitution or short-cut method learned in Section.6.) (a) e e = 0 () = 0 Eample 8: Solve ( + ) = Steps for solving arithmic equations:... Page (Section.6)
Eample 9: Solve + ( + 7) = Eample 0: Solve ln = Eample : Solve ( + ) ( ) = Eample : ( + 7) = (7 + ) Steps for solving equations using -to- properties:... Eample : 7 = Eample : ln( ) = ln(7 ) ln( + ) Page (Section.6)
Periodic Interest Formula Continuous Interest Formula nt r A = P + rt n A = Pe Eample : How long will it take $,000 Eample 6: How long will it take $,000 to grow to $00,000 at 9% interest to grow to $00,000 at 9% interest compounded continuousl? compounded quarterl? Eample 7: What interest rate is needed for $,000 to doule after 8 ears if compounded continuousl? (Round rate to nearest hundredth of a percent.) Part II: Eponential Growth and Deca Deca Model A = rt Pe f ( t) = A0e A = A0e kt kt Growth Model A 0 = A = k = t = Page (Section.6)
Eample 8: In 00 the world population was approimatel 6. illion. If the annual growth rate averaged aout.% per ear, write an eponential equation that models this situation. Use our model to estimate the population for this ear. Eample 9: The minimum wage in 970 was $.60. In 000 it was $.. (a) Find a growth model for this situation. Steps for finding growth/deca model: kt. Use A = A0e to find k. (Estimate k to 6 decimal places.). Sustitute k into growth/deca model: kt = A e A 0 () Estimate the minimum wage for this ear. (c) Estimate the minimum wage 0 ears from now. (d) Based on this model, when will the minimum wage e $0.00? Page 6 (Section.6)
Eample 0: If ou have an account with interest rate k, how long will it take our mone to doule if compounded continuousl? How long will it take it to triple? Eample : Caron- testing is used to determine the age of fossils, artifacts and paintings. Caron- has a half-life of 7 ears. (a) Find an eponential deca model for Caron-. () A painting was discovered containing 96% of its original Caron-. Estimate the age of the painting. (c) An art collector plans to purchase a painting Leonardo DaVinci for a considerale amount of mone. (DaVinci lived from -9). Could the painting in part () possil e one of DaVinci s works? Eample : Arsenic-7 has a half-life of 7. das. (a) Find an eponential deca model for this situation. Page 7 (Section.6)
() How long will it take for % of the original amount of arsenic-7 to remain in a lood sstem? (Round to nearest da.) (c) What is the deca rate (per da) to the nearest tenth of a percent. Eample : In 000 the population of Israel was approimatel 6.0 million and 00 it was projected to grow to 0 million. (a) Find an eponential growth model for this situation. () Estimate the average annual growth rate to the nearest tenth of a percent. Epressing an Eponential Model in Base e: = ae ln = a is equivalent to (ln ) = ae or Eample : Rewrite each equation in terms of ase e. (a) 68() = () 000(.) = (c).7(0.) = (d) = (.) Page 8 (Section.6)
.6 Homework Prolems Solve each equation epressing each side as a power of the same ase.. + = 9. = 7 e. e e =. e Solve the eponential equations. Round answers to two decimal places. + = 9. = 7 6. e = 0 7. 0 + 6 = 0 8. = 7 9. e 7 =, 0. + =. e + = 0. 0 0. 6 e = 7 Solve factoring. Write the answer using eact values.. e e + = 0. e e + = 0. e e = 6. e + e = Solve each arithmic equation. 7. ( + ) = 8. ln( ) = 0 9. + ( ) = 0. ( ) + ( + ) =. ( + ) =. ( + ) = ( + ) +. ( + ) = ( + ). ln( ) + ln( + ) = ln( 8). How long will it take $000 to grow to $0,000 at % if compounded (a) continuousl and () compounded quarterl. 6. What interest rate is needed for $000 to doule after 0 ears if compounded continuousl? 0.08t 7. The formula A =.9e models the population of Teas, A, in millions, t ears after 00. (a) What was the population in Teas in 00? () Estimate the population in 0? (c) When will the population reach 7 million? 8. You have $00 to invest. What interest rate is needed for the investment to grow to $000 in two ears if the investment is compounded quarterl? 9. What interest is needed for an investment to triple in four ears if it is compounded (a) semiannuall and () continuousl? 0. How much mone would ou need to invest now in order to have $,000 saved in two ears if the principal is invested at an interest rate of 6.% compounded (a) quarterl and () continuousl?. A certain tpe of radioactive iodine has a half-life of 8 das. kt (a) Find an eponential deca model, A = A0e, for this tpe of iodine. Round the k value in our formula to si decimal places. () Use our model from part (a) to determine how long it will take for a sample of this tpe of radioactive iodine to deca to 0% of its original amount. Round our final answer to the nearest whole da. Page 9 (Section.6)
. The population in Tanzania in 987 was aout. million, with an annual growth rate of.%. If the population is assumed to change continuousl at this rate. (a) estimate the population in 008. () in how man ears after 987 will the population e 0,000,000?. The half-life of aspirin in our loodstream is hours. How long will it take for the aspirin to deca to 70% of the original dosage?. The growth model A (a) What is Meico s growth rate? 0.0t = 07.e descries Meico s population, A, in millions t ears after 006. () How long will it take Meico to doule its population?.6 Homework Answers:. 7... 0. {.0} 6. {.90} 7. {.06} 8. {6.06} 6 9. {.8} 0. {-.80}. {.8}. {.99}. {0, ln }. {0, ln }. {ln } 6. ln 7. 6 8. e 9. 0. {6}.. {}. 9. φ. (a).9 ears; ().9 ears 6. 6.9% 7. (a).9 million; () 6.0 million; (c) 0 8..% 9. (a) 9.%; () 7.7% 0. (a) $,6.; () $,0.70. (a) A = A e 0 0.0866t () 7 das. (a) 0.68 million; () 6.0 ears. 6. hours. (a).%; () aout 8 ears Page 0 (Section.6)