Modeling With Exp & Power Functions

8/10/2019 Modeling With Exp & Power Functions

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Modeling withExponential & Power

Functions


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Definition of the Exponential

FunctionThe exponential functionfwith base bis defined by

f(x) = bx or y= bx

Where bis a positive constant other than and xis any real number.

The exponential functionfwith base bis defined by

f(x) = bx or y= bx

Where bis a positive constant other than and xis any real number.

/

Here are some examples of exponential functions.

f (x) = 2x g(x) = !x h(x) = "x#

$ase is 2. $ase is !. $ase is ".


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Text ExampleThe exponential functionf (x) = ".%&(!.&')x describes the

number of *+rin,s expected to fail- when the temperature is x.

*n the mornin, the 0hallen,er was launched- the temperature was

"- colder than any previous experience. ind the number of *+

rin,s expected to fail at this temperature.

Solution $ecause the temperature was "- substitute " for xandevaluate the function at ".

f (x) = ".%&(!.&')x This is the ,iven function.

f (") = ".%&(!.&')31 1ubstitute " for x.

f (") = ".%&(!.&')31 =".


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Relationships of

Exponential (y = x! & "ogarithmic

(y = logbx! Functions

# y = logbx is thein$erse of y = x

y = x


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%on$erting etween

Exponents & "ogarithms# 'EEXPONENT= P)*ER

# +2= ,-

# + is the ase. / is the exponent.

,- is the power.

# 's a logarithm0

logBASEP)*ER=E1P)2E2T

# log 4,- = /


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The exponential function fwith ase ais

defined y

f(x! = ax

where a5 60 a,0 andxis any real

numer.

For instance0

f(x! = 7x

and g(x! = 6.8x

are exponential functions.


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The ;raph of f(x! = ax0 a5 ,

y

x

(60

,!

Domain< (90 !

Range< (60

!

ori>ontal'symptote y= 6


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The ;raph of f(x! = ax0 6 ? a?,

y

x

(60

,!

Domain< (90 !

Range< (60

!ori>ontal

'symptote y= 6


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Experimental Data

3f data from an experiment is analysed- sayxandy- and plotted-

and it is found to form an exponential ,rowth curve thenxandy

are related by the formula4

y= a b x

(a and b are constants)


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When you are ,iven more than 2 points

of data- you can decide whether an

exponential model fits the points by

plottin, the natural logarithmsof the

y values against the x values.

3f the new points (x, ln y or log y)fit a

linear pattern- then the ori,inal points(x-y) fit an exponential pattern.


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(@/0 A! (@,0 B! (60 ,! (,0 /!

(x0 lny!

(@/0 @,.74! (@,0 @.-:! (606! (,0 .-:!


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Finding an exponential model

# %ell phone suscriers ,:44@,::3

# t= C years since ,:43

t , / 7 + 8 - 3 4 : ,6

y ,.- /.3 +.+ -.+ 4.: ,7., ,:.7 /4./ 74./ +4.3

lny 6.+3 6.:: ,.+4 ,.4- /.,: /.8: /.:- 7.7+ 7.-+ 7.4:


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# se / points to write the linear euation.

# (/0 .::! & (:0 7.-+!

# m= 7.-+ @ .:: = /.-8 = .73:: 9 / 3

# (y @ .::! = .73: (x 9 /!

# y @ .:: = .73:x @ .384# y = .73:x ./77 LINEAR MODEL FOR (t,ln!

# The y $alues were lnGs & xGs were t so

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Modeling With Exp & Power Functions