NAME DATE PERIOD MATH ANALYSIS HONORS UNIT 6 EXPONENTIAL FUNCTIONS DATE TOPIC ASSIGNMENT 10 0 10/19 10/ 10/ 10/4 10/5 10/6 10/9 10/0 10/1 11/1 11/ TOTAL
Mah Analysis Honors Workshee 1 Eponenial Funcions For each problem, 1-8, - Graph each funcion. - Idenify domain and range. - Describe he end behavior of he funcion. - Idenify he horizonal asympoe. 1 f g h 1 4 4 j 5 6 m p 1 1 7 r 100 1.5 8 1 s 1.5 Mah Analysis Honors Workshee Eponenial Growh and Decay Tell wheher he funcion is an eponenial growh or eponenial decay funcion, and find he consan percenage rae of growh or decay. 1 P.5 1.09 f 5606 0.9968 g 47 Deermine he eponenial funcion ha saisfies he given condiions. 4 Iniial value = 5, increasing a a rae of 17% per year 5 Iniial populaion = 50,000, decreasing a a rae of.6% per year 6 Iniial heigh = 18 cm, growing a a rae of 5.% per week 7 Iniial populaion = 8,900, increasing a a rae of 1.7% per year 8 The 000 populaion of Jacksonville, Florida was 76,000 and was increasing a he rae of 1.49% each year. A ha rae, when will he populaion be 1 million? 9 The populaion of Smallville in he year 1890 was 650. Assume he populaion increased a a rae of.75% per year. a) Esimae he populaion in 1915 and 1940. b) Predic when he populaion reached 50,000. 10 The populaion of Tusin in he year 010 was 78,890. Assume he populaion increases a a rae of 1.8% per year. a) Esimae he populaion in 015 and also 05 if he populaion coninues o increase a he given rae. b) Predic when he populaion will reach 100,000 people. c)
Mah Analysis Honors Workshee Logarihmic Funcions Evaluae he logarihmic epression wihou using a calculaor 1) log 4 4 ) log 6 1 ) log 5 5 1 4) log 6 5 6 5) log10 4 6) log 4 16 7) lne 8) ln1 9) ln0 4 10) ln e 11) ln 1 1) 7 log 7 e 7 1) e ln6 9 14) log 4 15) ln e 16) log 4 Use a calculaor o evaluae each epression. Round your answer o hree decimal places. 17) ln 5 18) ln 5 ln 10 19) 0.04 0) ln 0.1 1) Find a so ha he graph of f log a conains he poin, Solve each equaion ) log ) 6) 5 log 1 1 4) log 8 log 4 7) log 8 8) e 10 Hin: change o log form 5) log 4 1 9) e 5 8 Mah Analysis Honors Workshee 4 Inverse Funcions and Graphs of Logarihmic Funcions Find he inverse of each funcion. Sae he domain of f. Label he inverse as g. 1) f 4 ) f 4 ) f 4 4) f 1 5) Use f and g from # and evaluae f g and g f. You should ge he same answer for boh. Evaluae wihou using a calculaor 6) log 9 7) ln e 8) ln 1 e 9) log 4 log 8 Graph each funcion. i. Sae he domain. ii. Idenify any asympoes iii. Describe he end behavior iv. Verify your graph wih a graphing calculaor. 10) f ln 11) f e 1 14) f e 15) f e 1) f ln 16) f log 5 1) g ln 17) f log
Mah Analysis Honors Workshee 5 Eponenial and Logarihmic Funcions Review 1. When a person akes a dosage of D milligrams of ibuprofen, he amoun A (in milligrams) of medicaion remaining in he person s bloodsream afer hours can be modeled by he equaion A D0.71. Find he amoun of ibuprofen remaining in a person s bloodsream for a dosage of 500 mg afer 4 hours.. Each March from 008 o 01, a websie recorded he number of referrals i received from Inerne search engines. The resuls can be modeled by y 500 1.50 where y is he number of referrals and is he number of years since 008. a. Idenify he iniial amoun, he growh facor, and he annual percen increase. b. Graph he funcion and sae he domain and range. Esimae he number of referrals he websie received from Inerne search engines in March 01. c. Use your graph o esimae he year in which here were abou 5500 referrals.. Populaion From 1990 o 000, he populaion of California can be models by P 9,816,519 1.018 where is he number of years since 1990. a. Wha was he populaion in 1990? b. Wha is he growh facor and annual percen increase? c. Esimae he populaion in 007. 4. Graph each funcion. Sae he domain of each. a. f 1 b. f 4 ln 4 1 5. Find he value of each logarihm wihou using a calculaor. a. ln e b. ln 1 e c. log 1 d. log 7 4 100 6. Solve each equaion. log 4 a. b. ln 1 c. log 41 1 7. Find he inverse of each funcion. Label he inverse f a. f 4 1 b. f e.
Mah Analysis Honors Workshee 6 Wrie he epression as a sum or difference of logarihms. 1) ln 8 ) log 5) ln y 6) log 4 y Properies of Logarihms ) log y 5 4) log y 4 7) log 100 8) log y Wrie he epression as a single logarihm. 9) log 5 u 4 log 5 v 10) log u log v 11) log log 1) 4log 4log y log yz 1 1 1) ln ln ln 1 14) ln ln y 15) log 1 log log 1 16) log 5 log a 1 a 17) log log 1 Change of Base Formula: log a M log M loga ln M ln A This formula can be used in order o solve logarihms of oher bases besides 10 and e. I is useful for he calculaors ha don have he capabiliies o solve log b M. You can change o eiher base 10 or base e. 18) log 1 19) log 5 8 0) log e 1) log 7 ) Solve for y: ln y ln lnc ) Solve for y: ln y 4 5 lnc
Mah Analysis Honors Workshee 7 Eponenial and Logarihmic Equaions Solve each equaion log log 8 1) 4 ) log 4 4 log 4 9 ) ln ln 4 4) ln 1 ln 5) log 4 6) log log 1 4 4 7) 10 8) 14 9) e 6 4 96 11) 10 1 1 0 1) 5 10) In 1-16, approimae he logarihm using he properies of logarihms, given ha logb 0.56, logb 0.5646, and logb5 0.871. Round your resul o four decimal places. 1) logb 5 14) logb 0 15) logb 5 16) logb 9 Use a graphing calculaor o solve each equaion. 17) e 18) ln 1 19) e ln 4 1 For #0 and #1, use A A h 0 of he given subsance. for radioacive decay, where A0 represens iniial amoun and h represens he half-life 0) The half-life of radium is 1690 years. If 10 grams are presen now, how much will be presen in 50 years? When will here be.5 grams remaining? 1) The half-life of radioacive poassium is 1. billion years. If 0 grams are presen now, how much will be presen in 1000 years? When will here be 15 grams remaining? 0.00 ) The size P of a cerain insec populaion a ime (in days) obeys he funcion P 500e a) Deermine he number of insecs a 0 days. b) Wha is he populaion afer 10 days? c) When will he insec populaion reach 800?. ) The populaion of he Unied Saes in 000 was 8 million people and was growing a a rae of 1.1% per year. Assuming ha his growh rae coninues, when will he populaion he Unied Saes be 0 million people? When will he populaion be 55 million people? 4) The value of a cerain auomobile can be modeled by V 14,51(0.8). a) Is he value of he car decreasing or increasing? Wha is he % growh/decay rae? b) According o he model, when will he car be worh $9000? c) According o he model, when will he car be worh $000? 5) For seleced years from 1985 o 004, he average salary y (in housands of dollars) for public school eachers for he year can be modeled by he equaion y 1.56 14.584ln, where 5 represens 1985. During which year did he average salary for public school eachers reach $44,000?
Answers: 1) 6 6) 4 11) ) 1 7) ln10 ln 1) ln5 ln ln50 ) 1 1 e 4 4) 8) ln14 ln 1 e 1 9) ln 5) 5 10) 5
Mah Analysis Honors Workshee 8 Mahemaics of Finance Know he following formula: Ineres compounded k imes per year: A P 0 1 r k k 1 Find he amoun A accumulaed afer invesing a principal amoun of $1500 for 5 years a an ineres rae of 7% compounded quarerly. Find he amoun A accumulaed afer invesing a principal amoun of $500 for 10 years a an ineres rae of 5% compounded quarerly. Find he amoun A accumulaed afer invesing a principal amoun of $40,500 for 0 years a an ineres rae of.8% compounded monhly. 4 Find he amoun A accumulaed afer invesing a principal amoun of $5,00 for 5 years a an ineres rae of 4.5% compounded monhly. 5 If John invess $00 in a savings accoun wih a 9% ineres rae compounded quarerly, how long will i ake unil John s accoun has a balance of $4150? 6 If Joan invess $8000 ino a reiremen accoun wih 9% ineres rae compounded monhly, how long will i ake for your iniial invesmen o double in value? 7 Willis is he financial officer of privae universiy wih a responsibiliy for managing a large donaion. If he invess $1.5 million a an ineres rae of 8% compounded quarerly, how long will i be unil he accoun eceeds $.75 million? 8 Wha ineres rae compounded monhly is required for an $8500 invesmen o riple in 5 years? 9 Deermine how much ime is required for an invesmen o riple in value if ineres is earned a he rae of 6.5% compounded monhly. 10 Which invesmen is more aracive, 5% compounded monhly or 5.1% compounded only quarerly? Answers: 1) $1.17 ) $575.67 ) $86, 496.6 4) $77,765.69 5) 6.6 years 6) 7.70 years 7) 11.567 years 8) r =.17% 9) 17.64 years 10)
Mah Analysis Honors Workshee 9 Mahemaics of Finance 1 Dylan invess $1000 ino an accoun wih 8% ineres compounded monhly. Wha is he amoun in he accoun afer 7 years? Dylan invess $1000 ino an accoun wih 7.% ineres compounded coninuously. Wha is he amoun in he accoun afer 7 years? Erica invess $000 ino an accoun wih.8% ineres compounded coninuously. How long will i ake for he accoun o reach $10,000? 4 Hannah invess $500 ino an accoun wih 4.78% ineres compounded coninuously. How long will i ake for he accoun o reach $15,000? 5 Jerome will be buying a used car for $15,000 in years. How much money should he ask his parens for now so ha, if he invess i a 5% compounded coninuously, he will have enough o buy? 6 John will require $000 in 6 monhs o pay off a loan. How much money does he need o inves in an accoun paying % compounded monhly so ha in 6 monhs he will have eacly $000? 7 On January 1, Kim places $1000 in a cerificae of deposi (accoun) ha pays 6.8% compounded coninuously and maures in monh. Then Kim places he $1000 and he ineres earned in a savings accoun ha pays 5.5% compounded monhly. How much does Kim have in he accoun on July 1? 8 Suppose Lehua has access o invesmens ha will pay 9% ineres compounded semiannually or 8.5% compounded coninuously. If she is o inves $1000, which is he beer opion? Give eamples o jusify your reasoning. 9 Wha will a $450,000 house cos 5 years from now if he inflaion rae over ha period averages % compounded annually? 10 Wha rae is required o increase he value of a $000 invesmen o $0,000 in 5 years if ineres is compounded coninuously? Answers 1) $1747.4 ) $1666.96 ) 4.564 years 4) 0.445 years 5) $1,910.6 6) $955.9 7) $105.11 8) 9) $51,67. 10) 7.7%
Mah Analysis Honors Workshee 40 Eponenial and Logarihmic Funcions Review For 1-5, solve each equaion. 1 e 1 5 8 6 4 log 4 4 ln 4 ln 5 5 log log 4 1 6 6 Wrie each epression as a single logarihm. 6 1 log log 7 log 9 log 7 1 8 1 log log log log 9 The half-life of radioacive cobal is 5.7 years. There are iniially 100 grams of radioacive cobal presen. a) Epress he amoun of cobal P remaining as a funcion of ime. b) How much will be presen in 0 years? c) When will here be 15 grams remaining? 10 Susan invess $4800 ino an accoun ha pays 6.% ineres compounded quarerly. a) Epress he amoun of money A in he accoun as a funcion of ime. b) How much will be in he accoun in 10 years? c) When will here be $1,000 in he accoun? 11 The populaion of Merociy is 1,000 and is decreasing by.4% each year. a) Epress he populaion P of Merociy as a funcion of ime. b) How many people will be in Merociy in 15 years? c) When will here be 50,000 people in Merociy? 1 Wha ineres rae compounded coninuously is required for a $,000 invesmen o grow o $6,500 in 5 years? 1 How long would i ake for an iniial invesmen of $150 o grow o $1800 if i is placed ino an accoun ha pays 5.4% ineres compounded coninuously? 14 For f 4 9 0 8, use a graphing calculaor o find: a) he real zeros of f. b) The local maima and local minima Answers 1) -0.609 ) -0.89 ) 1.140 4) 0.014 5) -1 7 6) log 7) log 4 8) log 4 6 9) a) P 100 1 10) 5.7 A 48001.0155 b) 7.04 g $8880.5 c) 14.44 years 16.194 years 11) 1) 5, 80,.07 1) 10.1% 14) 6.75 years P 1000 0.976 15) 4,, 1 4 85,48 Ma (-.5, 60.75) 7.055 years Min (1, -5) 4