pre-Calculus-Chapter-1.ppt
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pre-Calculus-Chapter-1.ppt
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1.1Lines,IncrementsIfaparticlemovesfromthepoint(x1,y1)tothepoint(x2,y2),theincrementsinitscoordinatesare,1.1Lines,SlopeLetP1=(x1,y1)andP2=(x2,y2)bepointsonanonverticallineL.TheslopeofLis,Q(x2,y1),1.1Lines,Theorem:
Iftwolinesareparallel,thentheyhavethesameslopeandiftheyhavethesameslope,thentheyareparallel.,Proof:
IfL1|L2,then1=2andm1=m2.Conversely,ifm1=m2,then1=2andL1|L2.,1.1Lines,Theorem:
IftwononverticallinesL1andL2areperpendicular,thentheirslopessatisfym1m2=-1andconversely.,Proof:
ADCCDB,m1=tan1=a/hm2=tan2=-h/a,som1m2=(a/h)(-h/a)=-1,1.1Lines,EquationsoflinesPoint-SlopeFormulay=m(xx1)+y1Slope-Interceptformy=mx+bStandardformAx+By=Cy=aHorizontallineslopeofzerox=aVerticallinenoslope,1.1Lines,RegressionAnalysisPlotthedataFindtheregressionequationy=mx+bSuperimposethegraphonthedatapoints.Usetheregressionequationtopredicty-values.,1.1Lines,CoordinateProofsStategivenandprove.Drawapicture.Labelcoordinates,use(0,0)ifpossible.Fillinmissingcoordinates.Usealgebratoproveparallel/perpendicular-slopeequidistant-distanceformulabisect-midpoint,1.1Lines,Provethemidpointofthehypotenuseofarighttriangleisequidistantfromthethreevertices.,Given:
BACisarighttriangleProve:
AM=BM=CM,SinceAM=BM=CM,themidpointofthehypotenuseofarighttriangleisequidistantfromthethreevertices,1.2FunctionsandGraphs,FunctionAfunctionfromasetDtoasetRisarulethatassignsauniqueelementRtoeachelementD.,y=f(x)yisafunctionofx,1.2FunctionsandGraphs,DomainAllpossiblexvaluesRangeAllpossibleyvalues,1.2FunctionsandGraphs,open,closed,halfopened,halfopened,y=mxDomain(-,)Range(-,),1.2FunctionsandGraphs,y=x2Domain(-,)Range0,),1.2FunctionsandGraphs,y=x3Domain(-,)Range(-,),1.2FunctionsandGraphs,y=1/xDomainx0Rangey0,1.2FunctionsandGraphs,Domain0,)Range0,),1.2FunctionsandGraphs,1.2FunctionsandGraphs,FunctionDomainRange,y=x,y=x2,y=|x|,-3,3,0,3,1.2FunctionsandGraphs,DefinitionsEvenFunction,OddFunctionAfunctiony=f(x)isanevenfunctionofxiff(-x)=f(x)oddfunctionofxiff(-x)=-f(x)foreveryxinthefunctionsdomain.,EvenFunctionsymmetricalaboutthey-axis.OddFunction-symmetricalabouttheorigin.,1.2FunctionsandGraphs,OddFunctionsymmetricalabouttheorigin.,EvenFunctionsymmetricalaboutthey-axis.,(x,y),(-x,-y),(-x,y),(x,y),1.2FunctionsandGraphs,Transformationsh(x)=af(x)verticalstretchorshrinkh(x)=f(ax)horizontalstretchorshrinkh(x)=f(x)+kverticalshifth(x)=f(x+h)horizontalshifth(x)=-f(x)reflectioninthex-axish(x)=f(-x)reflectioninthey-axis,1.2FunctionsandGraphs,PieceFunctions,DomainRange,(-,),-3,),1.2FunctionsandGraphs,PieceFunctions,DomainRange,(-,),0,),1.2FunctionsandGraphs,CompositeFunctionsf(g(x),f(x)=x2,g(x)=3x-1,Find:
f(g
(2)g(f(-1)g(f(x)f(g(x),2523x21(3x1)2=9x26x+1,1.3ExponentialFunctions,DefinitionExponentialFunctionLetabeapositiverealnumberotherthan1,thefunctionf(x)=axistheexponentialfunctionwithbasea.,1.3ExponentialFunctions,RulesForExponentsIfa0andb0,thefollowingholdtrueforallrealnumbersxandy.,Usetherulesforexponentstosolveforx.,4x=128
(2)2x=272x=7x=7/2,2x=1/322x=2-5x=-5,1.3ExponentialFunctions,(x3y2/3)1/2x3/2y1/3,27x=9-x+1(33)x=(32)-x+133x=3-2x+23x=-2x+25x=2x=2/5,1.3ExponentialFunctions,49,9,1/9,5,4,5,81,32,2,1,1/8,2,80/9,8,9/4,36,1/25,1/49,4,5,1.3ExponentialFunctions,Domain:
Range:
Increasingfor:
Decreasingfor:
PointSharedOnAllGraphs:
Asymptote:
(-,),(0,),a1,0a1,(0,1),y=0,Propertiesoff(x)=ax,1.3ExponentialFunctions,NaturalExponentialFunctionwhereeisthenaturalbaseande2.718,1.3ExponentialFunctions,(-,),(-,),(-,),(0,),(0,),(0,),Inc.,Dec.,Inc.,(0,1),1.3ExponentialFunctions,Usetranslationoffunctionstographthefollowing.Determinethedomainandrangeofeach.1.f(x)=-5(x+2)32.g(x)=(1/3)(x1)+2,1.3ExponentialFunctions,DefinitionsExponentialGrowth,ExponentialDecay,Thefunctiony=kax,k0isamodelforexponentialgrowthifa1,andamodelforexponentialdecayif0a1.,ynewamountyooriginalamountbbasettimehhalflife,1.3ExponentialFunctions,Anisotopeofsodium,24Na,hasahalf-lifeof15hours.Asampleofthisisotopehasmass2g.Findtheamountremainingafterthours.Findtheamountremainingafter60hours.Estimatetheamountremainingafter4days.Useagraphtoestimatethetimerequiredforthemasstobereducedto0.1g.,1.3ExponentialFunctions,Anisotopeofsodium,Na,hasahalf-lifeof15hours.Asampleofthisisotopehasmass2g.Findtheamountremainingafterthours.Findtheamountremainingafter60hours.,a.y=yobt/hy=2(1/2)(t/15),b.y=yobt/hy=2(1/2)(60/15)y=2(1/2)4y=.125g,1.3ExponentialFunctions,Anisotopeofsodium,24Na,hasahalf-lifeof15hours.Asampleofthisisotopehasmass2g.(c.)Estimatetheamountremainingafter4days.(d.)Useagraphtoestimatethetimerequiredforthemasstobereducedto0.1g.,c.y=yobt/hy=2(1/2)(96/15)y=2(1/2)6.4y=.023g,d.,1.3ExponentialFunctions,Abacteriadoubleeverythreedays.Thereare50bacteriainitiallypresentFindtheamountafter2weeks.Whenwilltherebe3000bacteria?
a.y=yobt/hy=50
(2)(14/3)y=1269bacteria,1.3ExponentialFunctions,Abacteriadoubleeverythreedays.Thereare50bacteriainitiallypresentWhenwilltherebe3000bacteria?
b.y=yobt/h3000=50
(2)(t/3)60=2t/3,Equationswherexandyarefunctionsofathirdvariable,suchast.Thatis,x=f(t)andy=g(t).Thegraphofparametricequationsarecalledparametriccurvesandaredefinedby(x,y)=(f(t),g(t).,1.4ParametricEquations,1.4ParametricEquations,Equationsdefinedintermsofxandy.Thesemayormaynotbefunctions.Someexamplesinclude:
x2+y2=4y=x2+3x+2,1.4ParametricEquations,Sketchthegraphoftheparametricequationfortintheinterval0,3,1.4ParametricEquations,Eliminatetheparametertfromthecurve,Circle:
Ifwelett=theangle,then:
Since:
Wecouldidentifytheparametricequationsasacircle.,1.4ParametricEquations,Ellipse:
Thisistheequationofanellipse.,1.4ParametricEquations,Thepathofaparticleintwo-dimensionalspacecanbemodeledbytheparametricequationsx=2+costandy=3+sint.Sketchagraphofthepathoftheparticlefor0t2.,1.4ParametricEquations,Howistrepresentedonthisgraph?
1.4ParametricEquations,t=0,t=,1.4ParametricEquations,Graphingcalculatorsandothermathematicalsoftwarecanplotparametricequationsmuchmoreefficientlythenwecan.Putyourgraphingcalculatorandplotthefollowingequations.Inwhatdirectionistincreasing?
(a)x=t2,y=t3(b)(c)x=sec,y=tan;-/2/2,1.4ParametricEquations,ParametricequationscaneasilybeconvertedtoCartesianequationsbysolvingoneoftheequationsfortandsubstitutingtheresultintotheotherequation.,1.4ParametricEquations,(a)x=t2,y=t3,1.4ParametricEquations,(b),(c)x=sect,y=tantwhere-/2t/2,Hint:
sec2tan2=1,1.4ParametricEquations,1.4ParametricEquations,Findaparametrizationforthelinesegmentwithendpoints(2,1)and(-4,5).,x=2+aty=1+bt,whent=1,a=-6whent=1,b=4,x=26tandy=1+4t,CartesianEquationm=(51)/(-42)=-2/3,y=mx+b1=(-2/3)
(2)+bb=7/3,y=(-2/3)x+7/3,1.5FunctionsandLogarithms,Afunctionisone-to-oneiftwodomainvaluesdonothavethesamerangevalue.,Algebraically,afunctionisone-to-oneiff(x1)f(x2)forallx1x2.,Graphically,afunctionisone-to-oneifitsgraphpassesthehorizontallinetest.Thatis,ifanyhorizontallinedrawnthroughthegraphofafunctioncrossesmorethanonce,itisnotone-to-one.,Tobeone-to-one,afunctionmustpassthehorizontallinetestaswellastheverticallinetest.,one-to-one,notone-to-one,notafunction,(alsonotone-to-one),1.5FunctionsandLogarithms,1.5FunctionsandLogarithms,Determineifthefollowingfunctionsareone-to-one.(a)f(x)=1+3x2x4(b)g(x)=cosx+3x2(c)(d),1.5FunctionsandLogarithms,Theinverseofaone-to-onefunctionisobtainedbyexchangingthedomainandrangeofthefunction.Theinverseofaone-to-onefunctionfisdenotedwithf-1.,Domainoff=Rangeoff-1Rangeoff=Domainoff-1,f1(x)=yf(y)=x,Toprovefunctionsareinversesshowthatf(f-1(x)=f-1(f(x)=x,1.5FunctionsandLogarithms,Toobtaintheformulafortheinverseofafunction,dothefollowing:
Letf(x)=y.Exchangeyandx.Solvefory.Lety=f1(x).,Inversefunctions:
Givenanxvalue,wecanfindayvalue.,Switchxandy:
Inversefunctionsarereflectionsabouty=x.,Solvefory:
1.5FunctionsandLogarithms,1.5FunctionsandLogarithms,Provef(x)andf-1(x)areinverses.,1.5FunctionsandLogarithms,Determinetheformulafortheinverseofthefollowingone-to-onefunctions.(a)(b)(c),1.5FunctionsandLogarithms,Youcanobtainthegraphoftheinverseofaone-to-onefunctionbyreflectingthegraphoftheoriginalfunctionthroughtheliney=x.,1.5FunctionsandLogarithms,1.5FunctionsandLogarithms,1.5FunctionsandLogarithms,Sketchagraphoff(x)=2xandsketchagraphofitsinverse.Whatisthedomainandrangeoftheinverseoff.,Domain:
(0,)Range:
(-,),1.5FunctionsandLogarithms,Theinverseofanexponentialfunctioniscalledalogarithmicfunction.,Definition:
x=ayifandonlyify=logax,1.5FunctionsandLogarithms,Thefunctionf(x)=logaxiscalledalogarithmicfunction.,Domain:
(0,)Range:
(-,)Asymptote:
x=0Increasingfora1Decreasingfor0a1CommonPoint:
(1,0),Findtheinverseofg(x)=3x.,Definition:
x=ayifandonlyify=logax,1.5FunctionsandLogarithms,loga(ax)=xforallxalogax=xforallx0loga(xy)=logax+logayloga(x/y)=logaxlogaylogaxn=nlogax,CommonLogarithm:
log10x=logxNaturalLogarithm:
logex=lnxAlltheabovepropertieshold.,1.5FunctionsandLogarithms,Thenaturalandcommonlogarithmscanbefoundonyourcalculator.Logarithmsofotherbasesarenot.Youneedthechangeofbaseformula.,wherebisanyotherappropriatebase.,1.5FunctionsandLogarithms,$
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