Exponential Functions

Transcription

1 Exponential Functions n th chapter, a will always be a positive number. For any positive number a>0, re a function f : R! (0, 1) called an exponential function defined as =a x. For example, =3 x an exponential function, g(x) =( 4 17 )x an exponential function. There a big di erence between an exponential function a polynomial. The function p(x) =x 3 a polynomial. Here variable, x, being raed to some constant power. The function =3 x an exponential function; variable exponent. Rules for exponential functions Here are some algebra rules for exponential functions will be explained in class. f n 2 N, n a n product na s. For example, 3 4 = =81 a 0 =1 f n, m 2 N, n a n m = m p a n =( mp a) n a x = 1 a x The rules above were designed so following most important rule exponential functions holds: 178

2 a x a y = a x+y Anor variant important rule above a x a y = ax y And re also following slightly related rule (a x ) y = a xy Examples = 2 p 4= = =7 0 = = = = =15 1 =15 (2 5 ) 2 =2 10 =1024 (3 20 ) 1 10 =3 2 = = 1 (8) 2 3 = 1 ( 3p 8) 2 = = 1 4 * * * * * * * * * * * * * 179

3 The base an exponential function f =a x, n we call a base exponential function. The base must always be positive. Base 1 f anexponentialfunctionwhosebaseequals1 if =1 x nforn, m 2 N we have n f =1 m p n m m = 1 n = mp 1=1 n fact, for any real number x, 1 x =1,s(x) =1 x same function as constant function =1. Forthreason,weusuallydon ttalkmuch about exponential function whose base equals 1. * * * * * * * * * * * * * Graphs exponential functions t s really important you know general shape an exponential function. There are two options: eir base greater than 1, or base less than 1 (but still positive). Base greater than 1. f a greater than 1, n =a x grows taller as it moves to right. To see th, let n 2 Z. We know 1 <a, we know from our rules inequalities we can multiply both sides th inequality by a positive number. The positive number we ll multiply by a n, so we ll have a n (1) <a n a Because a n (1) = a n a n a = a n+1, inequality above same as a n <a n+1 Because last inequality we found true for any n 2 Z, weactuallyhave an entire string inequalities: <a 3 <a 2 <a 1 <a 0 <a 1 <a 2 <a 3 < Keeping in mind a x positive for any number x, a 0 =1, we now have a pretty good idea what =a x looks like if a>1. The y-intercept at 1; when moving to right, grows 180

4 taller taller; when moving to left, becomes shorter shorter, taller taller shrinking taller; taller; towards, when when moving moving but never to to touching, left, left, x-ax. becomes becomes shorter shorter shorter, shorter, shrinking shrinking towards, towards, but but never never touching, touching, x-ax. x-ax. Not only does grow bigger as it moves to right, but it gets bignot in a hurry. doesfor example, grow if we bigger lookas atit moves exponential to right, function but whose it gets base big Not in aonly 2, hurry. does n For example, grow if webigger look at as it moves exponential to right, function butwhose it gets base big in2, a n hurry. For example, if we look at exponential function whose base 2, nf(64) = 2 64 =18, 446, 744, 073, 709, 525, 000 f(64) = 2 64 =18, 446, 744, 073, 709, 525, 000 And 2 n t even f(64) a very= big 2 64 number =18, 446, to744, be 073, using709, for525, a base 000 (any positive number And 2 n t can be even a base, a very bigplenty number numbers to using are much, for a base much(any bigger positive than 2). number And 2 The bigger can n tbe even a base, a very big anplenty number exponential numbers to using function, are much, for a faster much base (any its biggerpositive grows than as 2). number itthe moves bigger can be to a base right. anplenty exponential numbers function, are much, faster much itsbigger than grows 2). asthe Moving it moves bigger to to base left, right. an exponential function, =a x faster it grows. grows small very quickly if a>1. Again Moving Moving if we to to look left, left, at exponential function =a =a x grows x grows whosesmall small basevery very 2, quickly quickly n if if a>1. a>1. Again Againif if we welook lookat at exponential exponential function function f( 10) = 2 10 = 1 whose whose 2 = 1 base base 2, 2, n n f( f( 10) 10) = = = = The bigger base, faster an = = exponential function shrinks as The The it bigger bigger moves to base, base, left. faster faster an anexponential function functionshrinks shrinks as asitwe moves movetoto left. left

5 Base Base less less than than 1 (but (but still still positive). positive). f f a positive positive less less than than 1, 1, n Base we weless we can can than show show 1 (but from fromstill our our positive). rules rules inequalities inequalities f a positive a n+1 n+1 <a <aless n n for for than any any1, nn 2 Z. Z. we That That we means means can show from our rules inequalities a n+1 <a n for any n Z. That means >a >a 3 3 >a >a 2 2 >a >a 1 1 >a >a 0 0 >a >a 1 1 >a >a 2 2 >a >a 3 3 > >a 3 >a So So =a =a x x 2 >a 1 >a 0 >a 1 >a 2 >a 3 > when when base base smaller smaller than than 1 slopes down down as as it itso moves moves to to right, right, but but =a it it x when always always positive. positive. base smaller As As it it moves moves than 1 to toslopes left, left, down as it moves grows grows to tall tall very very right, quickly. quickly. but it always positive. As it moves to left, grows tall very quickly. One-to-one One-to-one onto One-to-one onto Recall Recall an an exponential exponential function function f : : R! (0, (0, 1) ) has has as as its its domain domain set set Recall R has has as as an its its exponential target target function set set (0, (0, 1). ). f : R (0, ) has as its domain set We WeR see see from from has as its target set =a =a x (0, x,, if if ). eir eir a>1or0<a<1, a>1or0<a<1, one-to-one one-to-one We see from onto. onto. Remember Remember =a x, if to to eir check check a>1or0<a<1, if if one-to-one,we one-to-one,we can can one-to-one use use horizontal horizontal onto. line line Remember test test (which (which passes). passes). to check Tocheckwhatrange Tocheckwhatrange if one-to-one,we can use,, we wehorizontal think think compressing compressing line test (which passes). Tocheckwhatrange ontoy-ax. ontoy-ax. f f we we did did,,, we we we would would think see see compressing range range setpositivenumbers, setpositivenumbers, ontoy-ax. f we (0, (0, did 1). )., Since Since we would range range see target target range aresameset, aresameset, setpositivenumbers, onto. onto. (0, ). Since range target 146 aresameset, onto

6 * * * * * * * * * * * * * Where exponential functions appear Exponential functions are closely related to geometric sequences. They appear whenever you are multiplying by same number over over over again. The most common example in population growth. f a population a group increases by say 5% every year, n every year total population multiplied by 105%. That, after one year population 1.05 times what it originally was. After second year, population will be (1.05) 2 times what it originally was. After 100 years, population will be (1.05) 100 times what it originally was. After x years, population will be (1.05) x times what it originally was. nterest rates on credit cards measure a population growth sorts. f your credit card charges you 20% interest every year, n after 5 years not making payments, you will owe (1.20) 5 = times what you originally charged on your credit card. After x years not making payments, you will owe (1.20) x times what you originally charged. Sometimes a quantity decreases exponentially over time. Th process called exponential decay. f a tree dies to become wood, amount carbon in it decreases by % every year. Scientts measure how much carbon in something died, use exponential function =( ) x to figure out when it must have died. (The number base th exponential function because = ) Th technique called carbon dating it can tell us about htory. For example, if scientts dcover wood used to build a fort came from trees died 600 years ago, n fort was probably built 600 years ago. * * * * * * * * * * * * * e Some numbers are so important in math y get ir own name. One such number e. t a real number, but it not a rational number. t s very near to but not equal to rational number 27 =2.7. The

7 importance number e becomes more apparent after studying calculus, but we can say something about it here. Let s say you just bought a new car. You re driving it o lot, odometer says it s been driven exactly 1 mile. You are pulling out lot slowly at 1 mile per hour, for fun you decide to keep odometer speedometer so y always read same number. After something like an hour, you ve driven one mile, odometer says 2, so you accelerate to 2 miles per hour. After driving for something like a half hour, odometer says 3, so you speed up to 3 miles an hour. And you continue in th fashion. After some amount time, you ve driven 100 miles, so you are moving at aspeed100milesperhour.theodometerwillsay101afteralittlewhile, n you ll have to speed up. After you ve driven 1000 miles ( here s where story starts to slide away from reality) you ll have to speed up to 1000 miles per hour. Now it will be just around 3 seconds before you have to speed up to 1001 miles per hour. You re traveling faster faster, as you travel faster, it makes you travel faster, which makes you travel faster still, things get out h very quickly, even though you started out driving at a very reasonable speed 1 mile per hour. f x number hours you had been driving for, was dtance car had travelled at time x, n exponentialfunction with base e. n symbols, =e x. alculus studies relationship between a function slope function. n previous example, function was dtance travelled, slope dtance travelled speed car moving at. The exponential function =e x has at every number x same slope as value. That makes it a very important function for calculus. For example, at x =0,slope =e x f(0) = e 0 =1. That means when you first drove o lot (x =0)odometerread1mile, your speed was 1 mile per hour. After 10 hours driving, car will have travelled e 10 miles, you will be moving at a speed e 10 miles per hour. (By way, e 10 about 22, 003.) * * * * * * * * * * * * * 184

8 Exerces For #1-11, write each number in simplest form without using a calculator, as was done in Examples in th chapter. (On exams you will be asked to simplify problems like se without a calculator.) 1.) ) ( )3 ( ) 3 3.) ) ) ) (3 200 ) ) ) ) ) ) (5 4 7)

9 For #12-20, decide which only number x satfies given equation. 12.) 4 x =16 13.) 2 x =8 14.) 10 x =10, ) 3 x =9 16.) 5 x = ) ( 1 2 )x =16 18.) ( 1 4 )x =64 19.) 8 x = ) 27 x = ) Suppose you accidentally open a canter plutonium in your living room 160 units radiation leaks out. f every year, re half as much radiation as re was year before, will your living room ever be free radiation? How many units radiation will re be after 4 years? 22.) Your uncle has an investment scheme. He guarantees if you invest in stock h company, n you ll earn 10% on your money every year. f you invest $100, you uncle right, how much money will you have after 20 years? (Note when you earn 10%, you ll have 110% what you had before.) 186

10 For #23-31, match numbered functions with ir lettered s. 23.) 2 x 24.) ( 1 3 )x 25.) 2 x ) ( 1 3 )x 1 27.) ( 1 3 )(x 1) 28.) 2 x 29.) 5(2 x ) 30.) ( 1 3 )x 31.) 2 (x+1) A.) B.).) D.) E.) F.) G.) H.).) 187

11 For #32-40, match numbered functions with ir lettered s. 32.) e x 33.) ( 1 2 )x 34.) e x 1 35.) ( 1 2 )x ) ( 1 2 )(x+1) 37.) e x 38.) 2(e x ) 39.) ( 1 2 )x (x 1) 40.) e A.) B.).) GO GO D.) E.) F.) 00 GO 00 GO G.) H.).) 188