11.1 Exponential Functions

Transcription

1 . Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function of the form ny rel number. is clled the bse. An emple of n eponentil function would be f where 0 nd f, nd is. Notice tht the vrible is inside the eponent. This is very different from ny other function we hve delt with. First let us recll some of the properties of eponents.. y y y y. 3. y y 4. Some of these properties will prove to be useful s we proceed through this chpter. Let us now do some bsic evluting of eponentil functions. Emple : Evlute F 3 t F, F 0 nd g t g 3 nd g. By the techniques we lerned in chpter 9 we know lso, F g nd nd F g A bse tht comes in hndy in mny ppliction problems is the following Definition: The nturl bse eponentil function. e is clled the nturl bse. Consequently, f e is clled the nturl The number e is n irrtionl number like which we represent with letter for simplicity. To lern bout the derivtion of the number e tke clculus course.

2 We use the letter e to represent this vlue becuse of the mthemticin Leonrd Euler, who is lrgely responsible for its modern usge. Lets do some problems with the nturl bse. Emple : Let f e. Find f nd 3 f. First, e is such specil vlue it cn be found on your clcultor. It is usully found bove the button tht sys So f e e Now consult your clcultors instruction mnul or spek with your instructor to lern how to properly input this vlue into your clcultor. We get f ln e Similrly, 3 f e e Net we wnt to tlk bout the grphs of eponentil functions. Lets do this by wy of emple. Emple 3: Grph the following on the sme set of is. g 4 h f Assuming we hve no ide wht the grphs of these three functions look like, we hve to go bck to the most fundmentl wy of grphing, by plotting points. f Lets first grph. By plotting severl points we get

3 g 4 Net we grph on the sme is. We get f Notice tht g 4 is rises much fster thn. Also notice tht neither of the grphs will cross the is since the vlues for f nd g could never be negtive. h Lstly we wnt to grph. Intuitively, by wht we lerned in chpter 9, this should be the grph of f reflected bout the y-is. Plotting points will verify this fct. We hve This lst emple tells us tht everything we lerned before bout shifting, reflecting nd dilting works the sme on eponentil functions s it did on the bsic si functions we did before. Summry of Grphs of Eponentil functions y y Domin:,, Rnge: 0, 0, Intercept: 0, 0, All the sme rules for shifting nd reflecting re still vlid. So we will grph eponentil functions in similr fshion to how we grphed bsic function. Emple 4: Grph the following.. f 4 b. g 5. From the previous emple we know tht the shpe of the grph of n eponentil function is slightly different depending on the bse. Therefore, we will hve to plot few points of the bsic function to ccount for this vrition.

4 Here we clerly hve the bsic function y 4 with shift of unit left. So by grphing the bsic function (by plotting 3 points) nd then shifting it we hve b. Similrly we hve the bsic function y 5 with reflection cross the -is, nd shifting units down nd right. Recll we wnt to perform the reflections first. So plotting 3 points nd performing our trnsformtions we hve Finlly, we wnt to introduce one of the mny pplictions inherent to eponentil functions: Compound interest. Essentilly, compounding ones interest mens getting interest on your interest. Tht is, if you get your interest times yer, you first get interest on the mount in the ccount, sy $00 gives you interest of $. Then the second compounding you now get interest on $0. Which would be more thn the interest on $00. Here re the formuls for compounded interest. Compound Interest The mount of money A fter time t, in n ccount with interest rte r, nd principle P is given by the following: For n compoundings per yer: For continuous compounding: A P rt A Pe r n nt Continuous compounding mens tht you get your interest compounded on continuous bsis, i.e. n infinite mount of times per yer. Emple 5: Jon nd Mtt ech hve $0,000 they received from doing some etr work t home. Jon invests his money in n ccount tht pys him 5.3% nd compounds his money dily. Mtt invests his money in n ccount tht pys him 5.% nd compounds his money continuously. Who hs more money t the end of 5 yers?

5 Lets tret ech of them seprtely to determine how much they hve t the end of 5 yers. Jon: First identify which type of interest problem this is. Since it sys compounding dily we will need to use the formul nt r A P. Net identify ech of the vlues involved. n P $0,000, r , n 365 nd t 5 A P r n , , $3, Putting these into the formul we hve Mtt: Similrly we see since this is continuously compounding ccount we need the formul rt A Pe. Here P $0, 000, r nd t 5. Putting these in we get A Pe rt 0,000e nt $, So clerly t the end of the 5 yers, Jon hs more money thn Mtt. So the nswer is Jon Eercises Evlute the given function t the given vlues.. f. g 3 3. h 4. f. g 3. h b. f b. g b. h 0 c. f 0 c. g 0 c. h d. f 4 d. g d. h 4. f 7 5. g 5 6. h 3. f. g. h b. f b. g b. h c. f 0 c. g 0 c. h d. f d. g d. h 4

6 Evlute the given function t the given vlues. Round your nswers to three deciml plces. 7. f e 8. g e 9. h e. f. g 0. h 0 b. f b. g b. h 3 c. f 0 c. g c. h d. f d. g d. h 4 0. f e. g e. h 6 4. f. g 0. h 0. b. f b. g b. h 3 c. f c. g c. h.8 d. f 0 d. g d. h f 0 4. g 5. f. g 0.5 b. f. b. g 0.75 c. f 0.3 c. g.3 d. f 3.5 d. g.68 Let f, g 5 nd h 7. Find the following. 5. f t 6. g 7. f 8. f 9. g 0. f h. g h. h g 3. h f 4. f g 5. g f 6. f f Grph the following. 7. f 5 8. g 3 9. h f 3. h 4 3. g g h f g t3 g t f f g t 7 t 4. h 3 4. h f f t4 g t f 3 47 f f e 49. g e 50. h e 5. Jennifer invests $500 into college fund for her son. The bnk gives her n nnul interest rte of 5.% nd compounds it monthly. How much money will Jennifer hve for her son when he is redy for college in 5 yers? 5. John invests $7,000 in bnk ccount tht compounds his money dily t n interest rte of 4.9%. Wht mount of money will John hve t the end of 5 yers? 3 5

7 53. After winning the lottery, Ptrick wnts to invest his money in some money mrket ccounts. The bnk offers Ptrick one ccount, which compounds qurterly t 5.8%, nd one tht compounds dily t 5.6%. Which ccount should Ptrick invest in if he strts with $7,000,000? 54. While reding the Sundy pper, Chrles notices two different bnks offering different types of sving ccounts. Bnk of the Sequois dvertises n ccount with 6.% interest rte compounded dily nd Bnk of the Redwoods dvertises n ccount with 6.% interest rte compounded continuously. Into which bnk should Chrles invest his life svings of $30,000 if he is going to withdrw the money in 0 yers for retirement? 55. Jordn nd Ethn re the best of friends. They re lso fierce competitors. With this in mind they ech wnt to invest $500. Jordn is going to invest his money into n ccount which pys him.5% interest nd compounds it seminnully nd Ethn is going to invest in n ccount which pys him.% nd compounds in continuously. At the end of yer, who hs brgging rights? 56. Trcy just received $6,000 bonus for mking gret sle. She wnts to invest her money wisely. The bnk offers Trcy one ccount, which compounds dily t 4.8%, nd one tht compounds continuously t 4.7%. Which ccount should Trcy invest in? 57. In problem 55 we met Jordn nd Ethn. If they worked together nd combined their money into one ccount when they strted, they would hve received more totl interest. Which ccount would produce the most mount of interest nd how much more would it be? 58. Eric wnts to be millionire. Insted of going on gme show or going to Vegs, he decides to invest in Certificte of Deposits t his locl bnk. So if Eric hs $75,000 to invest in n ccount tht compounds nnully. Wht rte would Eric hve to get in order to hve one million dollrs t the end of yers? 59. Json needs money for college. He figures tht he cn mke it though public school with $30,000. Json s locl bnk offers him svings ccount tht compounds nnully. If Json strts with $0,000, wht rte would be needed for Json to hve his college money in yers? 60. Eric still wnts to be millionire. But relizing the impossibility of his sitution in problem 58, he hs come up with new pln. He decides to invest in mutul fund tht gurntees him n interest rte of 7.% compounded bimonthly. How much would Eric hve to invest to hve one million dollrs t the end of 0 yers? 6. Jred wnts to sve some money for riny dy. He figures tht if he should lose his py for months $8,000 should be enough to mke it through. If Jred invests his money for 5 yers into n ccount tht compounds dily t n interest rte of 7.3%, how much money would Jred hve to invest? 0.36t 6. A certin type of bcteri increses ccording to the model time in hours. How much bcteri will be present in 4 hours? 0 hours? P t 00e where t is the 600e where t is t 63. A certin type of fungus increses ccording to the model the time in dys. How much bcteri will be present in 7 dys? 30 dys? P t

8 000e where t 64. The popultion of smll town increses ccording to the model t is the time in yers. How mny people re in the town initilly? How mny people will be in the town fter 0 yers? 50 yers? 65. The popultion of spiders in n bndon house increses ccording to the model 0.05t Pt 0e where t is the time in weeks. How mny spiders were in the house initilly? How mny spiders will be in the house fter 3 weeks? yer? t 66. A rdioctive mteril decys ccording to the model / 5730 P t Q t 30 where t is in yers nd Q is in grms. How much mteril is present initilly? How much will be left fter 500 yers? t 67. A rdioctive mteril decys ccording to the model / 60 Q t 55 where t is in yers nd Q is in grms. How much mteril is present initilly? How much will be left fter 0,000 yers?