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Unformatted text preview: l I OKCC VDSS 'Elmz gm How does the change in the base value affect the shape of the general look of an
exponential graph? ‘0” As the We Mommas ‘ Mai/Cal W is
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Mal—axisfmn graph 61mm
Properties of Graphs of Exponential Functions: Let f(x) = 19" , b > 0, b #1. Then th. be . graph of ﬁx): 1) Is cow for all real numbers 7C
2] Has no sharp corners ' ' 3] Passes through the point Q76? (0 1 I)
4) Lies above the gazjs which is a horizontal as tote eit 7 . x ~—> ea or x ——> —oo but not both
5) Increase as x increases if b>.1, decreases . . '  e   ' e e 3 I,
6] ﬁx) is one to one, that is intersects any horizontal line at most once; Shaw {was}: pass—PS ill/a
Tegtl‘ll— is m +0 W ' ' 1 ._
Ex: Solve for t: (3)23 = 9M , 3) If x is rational, b’c can be expressed as a radical Solve the following exponential equations: .. it 2 40
(3'): (alt 7*! Q of  l
—2.t 7' 26k, (9) x 7 3C
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Ex: Solveforx:272'“l;—='94”:_"’_:Z_ e) ax —; g
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@ (5&5 x 5 3 'r :9 3_[2x+q)¥_: 20m) I2 23$
9M ‘1'3235 __ " Properties ofreal number exponents for 19‘ , b > 0, b #1 u—.__. 1) If x is real, 19" is a unigue real number example: 3"2 '——v 2) Ifxis an integer, bx is known I example: 32 '_‘__;———_—_=5 example: 335:: #37 ____.——r 4) If x is irrational, approximate 19" with br where example 3‘5 z 31‘2 Remember: All the properties related to exponents will hold true for 1—4
8 .5;
b .L
5' ——
.II' l} “base 6:: wax?wa Note: The base e is called the natural base and for Our purpos e is an irrational number which {iasf‘appifoximatelyiequal to 23182818284 ...... .. g——————_._—. ‘ The base e is sometimes easier to use than the other base ‘ yinte‘rcept: ( 0 3 l) xintercept: N 0 we
Asymptote: H1)?! 21) n‘hl A33th
‘7’ 1: O How do we define e to be 2.718E8 ..... .. 1 I '
Let’s investigate the values of function f(x) = [1 + (—)] . as x gets larger and larger
___.  ' x ‘.'.__..—‘.._.—_‘:
l x
f(x) =[1+()]
x 51'6013792‘115
240%1 3 89—
524119012551?)  31"] 1 3? usesz, Observe: As x gets bigger and bigger the value of f(X] approaches Ihsllnnwm ‘
 27 vapa W 05, W \n‘ Theorem : Let a be any real number. The instantaneous rate of cha e of the 5 w“
function f(x) = eI at x m a is e__a._ So the instantaneous rate of change of the function Sf ._
f(x) =‘ex at the point (a,e“) is ea. This property is unique to the exponential ag— ‘e @ W‘Q —.__., function (x) x e" or constant multiples of the exponential function such as r em
f(x) = ke" Where k is a constant. ’ You will have to remember and use this theorem fre uentl 1 when ou take Calculus. ‘1  Ex: In a biology experiment, let N t b  the number ofbacteria in a colony after t
hours, where = 0 corres ends to the ime the ex eriment be ins. Suppose that during the period from t = 4 hours to 7 = 8 hours the number of bacteria is modeled 4e:+[1l=‘l by the experimental function N(t_) : I' et:
'trfﬁr , NM: 4‘9
be , were? L'. b] Find the WORM population over the time period
55:57 '
Xi: {=15 Kits? m: NOS) = 48 Y2: NEW=49 a I‘_ A__ 1’ 6
éstmhvwﬁg‘ we or we.  49 4 0" ... 7 5 .797—“Xi 7";
" M 1.26185) f 5 c) Use your knowledge about the exponential function to compute  t. Instantaneous Rate of Change of the population at the instanrg ours Tm‘lmnlawm W ‘3? t?"
Wsz r=> =4 46"? g
l M £57 1,154: —= 467”
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“so (steels, Yes Sign? we ‘5 Recall the general graph of the exponential function y = 2". This function passes the horizontal line test. Thus it is one to one and has an inverse.  r
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L091 «t 5; Tm i 'l X l = 9 33115 2': 3/
Definition: The logarithm base of geno ent to I which b must be raised to yield x. __ ' 7 ' I Another words, ' ‘A '— Y r E > b :1 r or , .7 .. A WP  New
Ll  “we‘ll/02w Leta: owl exponent llorm This relationship allows one to convert logarithmic equation to exponential
equation and Vice versa. ~_
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—l _ 300 '2 x 3 3‘3 ‘ 1034K K60 : By K (X) ’ U93 5K
3 MOE . \n; Ex: Now let’s sketch the graphs of y = 1053);, y = logx and y = lnx on the same set of axes. You can figure out the logarithmic graphs from their correSponding
exponential graphs by treatinﬁt :4  . __ verses K . 8A0 3 Ex: Sketch where b is any positive constant by specifying the domain, range, asymptote and inter ept : (‘10 '. ‘R‘V 'I § ' :Domajn 120/00
EOWWM! (~00;°°) ' ; C_oo}oo> NEW? (ON) :0 ‘Xi‘ni‘: (he) \ ‘ .
Y‘ I n+ Note. The domain any 1 ; function is (0, ('3
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i
.’ ;<
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>2
H
C) Ex: Find__the'domain"ofy =ln[x ]
' )6 +53 x=q
. 1 (3+4 5*4) =C? )[—0.= —7 (wry.
. y remembering shifﬁhting and reﬂecting. Specify the = '"
domain, range, 1n ercepts and asjrmp ﬁes. ‘ " +0) [3 h—H I
_ ,r_..., ._.._._._., Ex: Decide which quantity is larger? log23 or log32 Remember the conversion equation between log and exponential equation. Evaluate: 1) 10g10 In general 10gb b. :1 '5 _______________ ,. M ____________ ,__ ' .  1: ...
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 Winter '08
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 Calculus

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