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AnnouncementsTopics:
- sections1.3,2.1,1.4,and2.2
*Readthesesectionsandstudysolvedexamplesinyourtextbook!
Homework:
- reviewlecturenotesthoroughly- workonpracticeproblemsfromthetextbookandassignmentsfromthecoursepackasassignedonthecoursewebpage(underthe“SCHEDULE+HOMEWORK”link)
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WorkingWithFunctions
• Reviewaddition,subtraction,multiplication,division,andcompositionoffunctionsonyourown…
• Reviewtransformationsofgraphsandinversefunctions(we’lldoabriefreviewhere)
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InverseFunctions
Thefunctionistheinverseofifand.Eachofandundoestheactionoftheother.Somesimpleexamples:
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f
€
f −1
€
f −1( f (x)) = x
€
f ( f −1(x)) = x
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f −1
€
f
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WhatFunctionsHaveInverses?
Afunctionhasaninverseifandonlyifitisaone-to-onefunction.Afunctionfisone-to-oneifforeveryy-valueintherangeoff,thereisexactlyonex-valueinthedomainoffsuchthaty=f(x).
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HorizontalLineTest
Ifeveryhorizontallineintersectsthegraphofafunctioninatmostonepoint,thenthegraphrepresentsaone-to-onefunction.
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FindingtheInverseofaFunctionAlgorithm:1. Writetheequationy=f(x).2. Solveforxintermsofy.3. Replacexby(x)andybyx.
Note:ThedomainandrangeareinterchangedExample:.
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f −1
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TemperatureConversion
TherelationshipbetweendegreesCelsius(C)anddegreesFahrenheit(F)islinear.Weknowthatcorrespondstoandcorrespondsto.
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32oF
€
0oC
€
100oC
€
212oF
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TemperatureConversion
(a) FindthefunctionthatconvertstoNote:inputisandoutputisDataPoints:Function:
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oC
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oF .
€
slope =change in outputchange in input
=ΔFΔC
=212 − 32100 − 0
=1.8
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(0oC, 32oF) and (0oC, 32oF)€
oC
€
oF
€
F(C) =1.8C + 32
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TemperatureConversion
(b)FindthefunctionthatconvertstoNote:inputisandoutputisOneapproach:FindtheINVERSEofF(c):
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oC.
€
oF
€
oF
€
oC
€
F =1.8C + 32F − 32 =1.8CF − 321.8
= C
∴C(F) =F − 321.8
Note:wedonotinterchangevariablesattheendsinceFandChaveaphysicalmeaning
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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!
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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
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TransformationofanExponentialFunction
GraphRecall:isaspecialirrationalnumberbetween2and3thatiscommonlyusedincalculusApproximation:
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e ≈ 2.718
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e
y
x€
f (x) = e−2x + 3.
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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
€
If f (x) = ax, then f −1(x) = loga x.
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e ln x = x
lnex = x
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GraphsofLogarithmicFunctions
Recall:Forinversefunctions,thedomainandrangeareinterchangedandtheirgraphsarereflectionsinthelineExample:Graph
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f (x) = ln x.€
y = x.
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GraphsofLogarithmicFunctionsy
x
€
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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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!!!
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LawsofLogs
Forx,y>0andpanyrealnumber:
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ln(xy) = ln x + ln yln(x / y) = ln x − ln yln(x p ) = pln x
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SemilogGraphs
Definition:Asemiloggraphplotsthelogarithmoftheoutputagainsttheinput.Thesemiloggraphofafunctionhasareducedrangemakingthekeyfeaturesofcertainfunctionseasiertodistinguish.
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SemilogGraphs
Example:
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SemilogGraphs
Example:Sketchthesemiloggraphof
f (x) =10e−4x.
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Double-LogGraphs
Definition:Adouble-loggraphplotsthelogarithmoftheoutputagainstthelogarithmoftheinput.
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SemilogandDouble-LogGraphs
Example:BloodCirculationTimeinMammalsSketchthesemiloganddouble-loggraphsforthemodel
T (B) =17.73B0.25.
B
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ExponentialModels
Whenthechangeinameasurementisproportionaltoitssize,wecandescribethemeasurementasafunctionoftimebytheformulawhereisthevalueofthemeasurementattimeistheinitialvalueofthemeasurement,andisaparameterwhichdescribestherateatwhichthemeasurementchanges
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S(t) = S(0)eα t
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S(t)
€
S(0)
€
α
€
t
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DoublingTime.
Example:Abacterialculturestartswith100bacteriaandafter3hoursthepopulationis450bacteria.Assumingthattherateofgrowthofthepopulationisproportionaltoitssize,findthetimeittakesforthepopulationtodouble.
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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
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Half-LivesofDrugs
Example:ThinkinginHalf-Lives
**Manydrugsarenoteffectivewhenlessthan5%oftheiroriginallevelremainsinthebody.
#ofhalf-lives amountleftinbody %amountleftinbody
0 M(0) 100
1
2
3
4
5