Exponential functions
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1 Robero s Noes on Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 1 Eponenial funcions Wha ou need o know alread: Meaning, graph and basic properies of funcions. Power funcions. Wha ou can learn here: The definiion and basic properies of eponenial funcions. You are likel familiar wih he idea of a power funcion, ha is, a funcion of n he form, where he variable is in he base of he power and he eponen consiss of a fied real number. In his secion, we ll sud is a differen pe of funcion where he variable and he consan swich posiion. These funcions are eremel useful and imporan boh in heor and applicaions. Definiion An eponenial funcion is a funcion of he form: a where is he independen variable and a 0 is a posiive consan. Eample: The funcions 2, 10,, as well as and are eponenial funcions. You ma remember ha he similar /3 0.35,,,, are all power funcions funcions. Make sure o disinguish beween hese wo imporan pes of funcions. Wh he demand ha a be posiive? Can we compue powers of negaive numbers? Onl some of hem. For insance, 1/ n n 8 8 eiss if n is odd, bu no if i is even. To preven such complicaions, we resric our aenion o posiive bases. And, b he wa Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 2: Eponenial funcions Page 1
2 Kno on our finger Remember he basic and imporan convenion on he noaion: a a a a Fine, bu hese algebraic definiions don give me a good feel for hese funcions. Wha do heir graphs look like? We can pla wih a graphing calculaor o ge a good saring impression. Here are he graphs of some basic eponenial funcions: ge some more on our own Technical fac For an eponenial funcion a : The domain consiss of all real numbers The range consiss of onl posiive numbers. The graph is above he ais and conains he poin 0,1. The -ais is a horizonal asmpoe. If a 1 he funcion is increasing and he asmpoe is on he righ. If 0a 1 he funcion is decreasing and he asmpoe is on he lef. As an eercise for ou, I will ask ou laer o prove each of hese facs. All ou will need is a basic undersanding of how powers work. And speaking of basic properies of powers, le me remind ou of he following facs Noice ha hese funcions have some common feaures ha can be easil checked b an algebraic argumen and will urn ou o be ver useful. Technical fac For an posiive number a, b and an real numbers, he following are rue: a a a a ; a a ; a a a ab a b Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 2: Eponenial funcions Page 2
3 B he wa, here is a echnical issue relaed o hese combinaions ha ma look pedanic now, bu will become fairl imporan laer, so I encourage ou o clarif i now. Kno on our finger A funcion ha includes oher erms or facors beond a is NOT an eponenial funcion. Eample: The funcions 1 3 or are no eponenial, even hough he involve an eponenial piece. For insance, noice ha some of he common properies of eponenial funcions, such as conaining (0, 1) or having he ais as asmpoe, do no appl here. 1 is indeed an eponenial funcion, no However, he funcion 4 because i has an eponenial piece in i, bu because i can be wrien in he form of an eponenial funcion: I am saring o worr: since he base of an eponenial funcion can be an posiive number, will we have o sud a large number of hem? Rela! I urns ou ha we can alwas change from one base o anoher, and his can be done fairl easil, once we learn how o inver an eponenial funcion hrough a logarihm (coming soon o secion near his one ). Hisoricall, he numbers 2 and 10 have proven o be useful fundamenal bases in man areas of mahemaics, so much so ha he following is a common epression. The funcion eponenial funcion. Definiion 10 is called he common However, for calculus, and in mos scienific applicaions, anoher number urns ou o be much more useful. This number is so special ha i is one of onl wo real numbers ha are commonl known b a leer, he oher number, of course, being. Anoher Greek leer? No, he more common Lain leer e. To undersand wha he number e is, noice ha all eponenial funcions go hrough he poin 0,1, bu wih differen slopes. Now hen Definiions The base of he eponenial funcion whose angen line has a slope of 1 a 0,1 is called e. This number is irraional and is approimae value is The funcion e is called he naural eponenial funcion. Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 2: Eponenial funcions Page 3
4 Here is a graph of he naural eponenial funcion, ogeher wih is angen line, whose slope is 1. e 1 Technical fac If a 1 he quani Q ca increases and follows an eponenial growh process. If a 1 he quani Q ca decreases and follows an eponenial deca process. In order o figure ou wh his fac s rue and under wha specific condiions, ou will have o wai unil ou learn abou differenial equaions. For now, here are wo eamples focused on he mahemaical aspecs of hese processes. Eponenial funcions are eremel useful in boh heor and pracice. You will see heir heoreical uses laer, bu perhaps ou are alread familiar wih he following pracical applicaions, which we will use frequenl. Kno on our finger If a quani Q changes a a rae ha is proporional o is size, hen he size of his quani a an ime is well described b an eponenial model, ha is, b a funcion of he form Q ca Here he consan c represens he iniial amoun, while he consan a represens he proporion of he iniial amoun presen afer one uni of ime. Eample: Mold is known o grow according o an eponenial model. A spo of mold is 2 2 iniiall measured a 12 cm and afer 10 das a 18 cm. Can we esimae he size of he spo afer 90 das, assuming ha he mold is no reaed? If we use he fac ha he eponenial model is of he form ca and ha he iniial amoun is M 12, we know ha he funcion describing he een of he mold is of he form M 12a. B using he informaion abou he size a 10 das we have: a a 1.5 a Therefore our model is S and afer 90 das we can epec a spo of size: Eample: 90 2 cm S A quani epeced o change according o an eponenial model is measured iniiall a 50 unis, while hree hours laer he measuremen is of 48 unis. Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 2: Eponenial funcions Page 4
5 To consruc he formula for he funcion ha represens his quani as a funcion of ime, we use he fac ha an eponenial model follows he ca. Since he iniial reading is 50, we know ha 50 formula while since afer hree hours he reading is 48 we have: c, a 0.96 a a Therefore he required funcion is I am confused: didn ou jus sa ha a funcion of he form eponenial? ca is NOT Absoluel, bu noice ha here are called eponenial models, no funcions. This is a small, bu imporan difference. In fac, here are man oher facs and deails abou eponenial funcions ha could be discussed here, bu I will dela hem unil we have a beer mahemaical cone o discuss hem. You will find a firs glimpse of some of hem in some of he Learning Quesions. Summar An eponenial funcion is of he form a, wih 0 a. All eponenial funcions have a one-sided horizonal asmpoe. The mos common eponenial funcion is he one whose angen line a 0,1 has slope 1. Is based is he irraional number denoed b e. Eponenial funcions are used in man applicaions, he mos common being he descripion of growh and deca models. Do no confuse beween eponenial funcions and power funcions Common errors o avoid Do no confuse beween eponenial funcions and funcions involving an eponenial erm or facor. Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 2: Eponenial funcions Page 5
6 Learning quesions for Secion 3.2 Memor quesions: 1. For wha values of a is he eponenial funcion a properl defined? 2. Wha wo feaures of a power makes i an eponenial funcion? 3. If a 0, wha is he domain of he funcion 4. If a 0, wha is he range of he funcion a? a? 5. Which poin is common o all eponenial funcions? 6. Which line is an asmpoe for all eponenial funcions? 7. Which funcion is called he common eponenial funcion? 8. Which funcion is called he naural eponenial funcion? 9. Wha is he value of e rounded o wo decimal places? 10. Wha is he formula for an eponenial model? Compuaion quesions: 1. Which of hese funcions can be considered as power funcions, which ones as eponenial and which ones as neiher? a) 5 3 b) c) e d) 5 e) e f) g) sin h) 3 i) j) Which of he following funcions will have a horizonal asmpoe and on which side? 3 e a) b) 2 c) 5 2 Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 2: Eponenial funcions Page 6
7 3. Given he funcions: 0.5, 0.8, 1, 1.2, 1.5 a) Use our calculaor o obain heir graphs. b) Describe how hese graphs are relaed and how he change as he base ges bigger. 4. Describe he similariies and he difference among he graphs of he funcions 2, 2, 2 and Deermine he eponenial funcion ha goes hrough he poin 3, 9 6. Find he formula for he funcion whose graph is obained from he graph of 2 e b firs reflecing i around he -ais and hen moving i down b 4 unis. Clearl eplain how ou arrive a our conclusion. 7. Consruc he formula for he funcion whose graph is obained b moving he graph of he naural eponenial funcion up 2 unis and o he lef 3 unis. Idenif he horizonal asmpoe of his funcion and he poin a which he slope of he angen line is 1. Theor quesions: 1. If he base of an eponenial funcion is /e, on which side will i have he horizonal asmpoe? 2. If a 0, which quadrans conain he graph of he funcion 3. When is 3 larger han? a? 4. Which eponenial funcion can also be considered as a polnomial? 5. If f is a power funcion and g is an eponenial funcion, is f g a power, an eponenial or neiher? 6. Which poin is common o all funcions of he form 2 a? Proof quesions: 1. In some eponenial model problems, i ma be beer o use a 2 k for growh processes and represens he reciprocal of, respecivel, he doubling ime or half life of he quani. 1 a 2 k for deca processes. Prove ha b doing his, he consan k Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 2: Eponenial funcions Page 7
8 Applicaion quesions: 1. A mahemaical model for he weigh of a cerain bacerial culure is given b he funcion: 2.5 w where w is weigh in grams and >0 is ime in hours. a) Is his an eponenial funcion? b) Obain he graph of his funcion from our calculaor c) Wha is he evenual weigh of he culure? (Here evenual means for large values of ) 2. You wan o predic he deca of a 30g specimen of an isoope whose half-life is 50 ears. a) Which funcion describes his process? b) How much of he specimen will ou have in 10 ears? d) How big was he specimen 20 ears ago? 3. Assume ha he value V() of a car decreases eponeniall, wih a half-life of 5 ears. If he iniial price of ha car is $25,000: a) which formula describes V() as a funcion of ime? b) wha is he value of he car afer 3 ears? c) wha would V(-2) represen? Wha quesions do ou have for our insrucor? Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 2: Eponenial funcions Page 8
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