exponential functions - mcmaster universityclemene/1ls3lectureoutlines/1ls3_2.2.pdf · exponential...
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ExponentialFunctions
Anexponentialfunctionisafunctionoftheformwhereisapositiverealnumbercalledthebaseandisavariablecalledtheexponent.Domain:Range:
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f (x) = ax
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a
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x
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x∈ R
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y > 0*Note:PleasereviewEXPONENTLAWSonyourown!
GraphsofExponentialFunctions
y
x
y
x
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f (x) = 3x
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f (x) = 12( )x
Whena>1,thefunctionisincreasing. Whena<1,thefunctionisdecreasing.
y=0isahorizontalasymptote
TransformationofanExponentialFunction
GraphRecall:isaspecialirrationalnumberbetween2and3thatiscommonlyusedincalculusApproximation:
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e ≈ 2.718
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e
y
x€
f (x) = e−2x + 3.
LogarithmicFunctions
Theinverseofanexponentialfunctionisalogarithmicfunction,i.e.Cancellationequations:Ingeneral: Forexponentials&logarithms:
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aloga x = x
loga ax = x
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f ( f −1(x)) = x
f −1( f (x)) = x
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If f (x) = ax, then f −1(x) = loga x.
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e ln x = x
lnex = x
GraphsofLogarithmicFunctions
Recall:Forinversefunctions,thedomainandrangeareinterchangedandtheirgraphsarereflectionsinthelineExample:Graph
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f (x) = ln x.€
y = x.
GraphsofLogarithmicFunctions
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f −1(x) = ex
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(−1,e−1)
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(0,1)€
(1,e)€
e ≈ 2.7 y
x
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y = x
GraphsofLogarithmicFunctions
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f −1(x) = ex
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(−1,e−1)
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(0,1)€
(1,e)€
e ≈ 2.7
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y = xy
x
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(e,1)
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(1,0)
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(e−1,−1)€
f (x) = ln x
Memorize!!!
LawsofLogs
Forx,y>0andpanyrealnumber:
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ln(xy) = ln x + ln yln(x / y) = ln x − ln yln(x p ) = pln x
SemilogGraphs
Definition:Asemiloggraphplotsthelogarithmoftheoutputagainsttheinput.Thesemiloggraphofafunctionhasareducedrangemakingthekeyfeaturesofcertainfunctionseasiertodistinguish.
SemilogGraphs
Example:
SemilogGraphs
Example:Sketchthesemiloggraphof
f (x) =10e−4x.
Double-LogGraphs
Definition:Adouble-loggraphplotsthelogarithmoftheoutputagainstthelogarithmoftheinput.
SemilogandDouble-LogGraphs
Example:BloodCirculationTimeinMammalsSketchthesemiloganddouble-loggraphsforthemodel
T (B) =17.73B0.25.
B
ExponentialModels
Whenthechangeinameasurementisproportionaltoitssize,wecandescribethemeasurementasafunctionoftimebytheformulawhereisthevalueofthemeasurementattimeistheinitialvalueofthemeasurement,andisaparameterwhichdescribestherateatwhichthemeasurementchanges
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S(t) = S(0)eα t
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S(t)
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S(0)
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α
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t
DoublingTime.
Example:Abacterialculturestartswith100bacteriaandafter3hoursthepopulationis450bacteria.Assumingthattherateofgrowthofthepopulationisproportionaltoitssize,findthetimeittakesforthepopulationtodouble.
Half-LivesofDrugs Half-life
Tetrahydrocannabinol … Marijuana (infrequent users)
1.3-3 days
Marijuana (frequent users) 1-10 days
Marijuana (if taken orally as pills) 25-36 hours
Marijuana (smoking/inhaling) 1.6-59 hours
LSD (Lysergic acid diethylamide) 3-5 hours
MDMA … ecstasy Methylenedioxymethamphetamine
6-10 hours
Caffeine adults 4-5 hours
Caffeine infants 10-20 hours
Caffeine with oral contraceptives 5-10 hours
Caffeine (if pregnant) 9-11 hours
Caffeine (liver disease) several days
Codeine (Tylenol 3) 3-6 hours
Demerol (pain killer) 3-5 hours
Morphine (pain killer) 2-3 hours
Heroin (IV or inhaled) 3-5 minutes
Cocaine (benzoylmethylecgonine) 1 hour
Psilocin … magic mushrooms, shrooms
2-3 hours
Phencyclidine … rocket fuel, killer weed, angel dust
7-46 hours
Half-LivesofDrugs
Example:ThinkinginHalf-Lives
**Manydrugsarenoteffectivewhenlessthan5%oftheiroriginallevelremainsinthebody.
#ofhalf-lives amountleftinbody %amountleftinbody
0 M(0) 100
1 0.5M(0) 50
2 0.52M(0) 25
3 0.53M(0) 12.5
4 0.54M(0) 6.25
5 0.55M(0) 3.125
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