函数英文教育课件

FunctionEnglishPPTcoursewareREPORTING目录FunctionDefinitionFunctiontypeApplicationoffunctionsImageoffunctionTheLimitandContinuityofFunctionsThederivativeanddifferentiationoffunctionsPART01FunctionDefinitionREPORTINGFunctionsareasetofinputandoutputpairsthatspecifytherelationshipbetweentheinputandoutputTheinputofafunctioniscalledthedomain,andtheoutputiscalledtherangeThefunctionisamappingfromthedomaintotherangeDefinitionoffunctions03GraphicalrepresentationDrawacurveonthecoordinateplanetorepresentthefunction01SymbolicrepresentationUsesymbolstorepresentthefunction,suchasf(x)=x^202TabularpresentationListtheinputandoutputpairsinatableThepresentationmethodoffunctionsThepropertiesoffunctionsObjectivefunctionThefunctionisbothonetooneandontoOnfunctionTherangeofthefunctioncoversallpossibleoutputsOnetoonefunctionEachinputresponsetooneandonlyoneoutputObjectivefunctionNotallinputsmaptodifferentoutputsObjectivefunctionNotalloutputsaremappedfromdifferentinputsPART02FunctiontypeREPORTINGSummarywordLinearfunctionisabasictypeoffunction,anditsgraphisastraightline.DetaileddescriptionThegeneralformofalinearfunctionisy=ax+b,whereaandbareconstants,xistheindependentvariable,andyisthedependentvariable.Thecharacteristicoflinearfunctionsisthattheslopeisaandtheinterceptisb.Inmathematicsandphysics,linearfunctionsarewidelyusedtodescribesimplerelationshipsbetweentwovariables.LinearfunctionSummaryAquadraticfunctionreferstoafunctionintheformofy=ax^2+bx+c,wherea,b,andcareconstantsanda≠0.DetaileddescriptionTheimageofaquadraticfunctionisaparabola.Accordingtothepolarityofa,theopeningoftheparabolaisupwardordownward.Aquadraticfunctionhastworootsandcanbesolvedusingtheformulax=[-b±sqrt(b^2-4ac)]/(2a).Quadraticfunctionsarewidelyusedinfieldssuchasmathematics,physics,andengineering.QuadraticfunctionSummaryTriangularfunctionsareatypeofmathematicalfunctionthatstudiestherelationshipbetweentheedgesandanglesofatriangle.DetaileddescriptionCommontrigonometricfunctionsincludesinefunction,cosinefunction,tangentfunction,etc.Thedomainofthesefunctionsisanintervalontherealnumberaxis,witharangeof[-1,1].Trigonometricfunctionsarewidelyusedinthecalculationoftriangles,aswellasintherelationshipbetweentriangleanglesandsidelengths.TrigonometricfunctionSummary:Theinverseproportionalfunctionreferstoafunctionintheformofy=k/x,wherekisaconstantandk≠0.Detaileddescription:Theimageoftheinversescalingfunctionisahyperbola,distributedinthefirstandthirdquadrants.Thecharacteristicoftheinverseproportionalfunctionisthatasxincreases,thevalueofydecreases,andviceversa.Theinverseproportionalfunctionhasawiderangeofapplicationsinphysicsandengineering,suchasdescribingtherelationshipbetweenresistanceandcurrent.ReverseprofessionalfunctionPART03ApplicationoffunctionsREPORTINGTrigonometricFunctionsDescribetherelationshipsbetweensidesandanglesoftriangles,andareessentialinareaslikegeometryandcalculusAnalyticalFunctionsUsedincalculustostudythepropertiesoffunctions,includingcontinuity,differentiation,andintegrityAlgebraicFunctionsUsedtosolvevariousAlgebraicequations,reportrelationshipsbetweenvariables,andsimplifycomplexexpressionsTheApplicationofFunctionsinMathematicsDescribethemotionofobjectsovertime,suchasvelocity,acceleration,andplacementTheyarefundamentalinunderstandingmechanismsMotionFunctionsUsedtodescribewavesinvariousmedia,includingsoundwaves,lightwaves,andelectromagneticwavesWaveFunctionsDescribetheforcefieldgeneratedbyaphysicalsystem,andareusedinfieldslikeelectronicsandgravityPotentialFunctionsTheApplicationofFunctionsinPhysicsUtilityFunctionsUsedtorepresentthepreferencesofindividualsorfirmsandhelpunderstandtheirbehaviorintheeconomyProductionFunctionsDescribehowinputslikelabor,capital,andlandarecombinedtoproduceoutputslikegoodsandservicesMonetaryFunctionsUsedinthestudyoffinanceandmacroeconomicstoreporttheflowofmoneyintheeconomyanditsimpactonpricesandoutputTheApplicationofFunctionsinEconomicsPART04ImageoffunctionREPORTING

ThedrawingmethodoffunctiongraphDirectmethodDrawingthegraphoffunctiondirectlyonthecoordinatedpaperaccordingtothegivenfunctionexpressionInterpolationmethodUsingknownpointstointerpolatethegraphoffunction,andthendrawthegraphApproximationmethodUsingknownpointstoapproximatethegraphoffunction,andthendrawthegraphThegraphoffunctioniscontinuousatallpointswithinthedomainContinuityDiscontinuitySymmetryThegraphoffunctionmayhavejumpdiscontinuitiesorremovablediscontinuitiesSomefunctionshavesymmetryproperties,suchasevenfunctionsandoddfunctions030201CharacteristicsoffunctiongraphsTranslationScalingRotationReflectionTransformationoffunctionimages01020304ShiftingthegraphoffunctionhorizontallyorverticallyEnlargingorreducingthegraphoffunctioninbothhorizontalandverticaldirectionsRotatingthegraphoffunctionaroundafixedpointReflectingthegraphoffunctionacrossthexaxisoryaxisPART05TheLimitandContinuityofFunctionsREPORTINGDefinition:ThelimitofafunctionfatapointaisthevaluethatfapproachesastheinputapproachesaLimitoffunctionTypesLeftLimit:ThelimitoffasxapproachesafromtheleftsideRightLimit:ThelimitoffasxapproachesafromtherightsideLimitoffunctionTwoSideLimit:ThelimitoffasxapproachesafrombothsidesLimitoffunctionPropertiesTheSqueezeTheorem:Ifa≤g(x)≤b,andlim(f(x)=L)asxapproachesc,thenlim(g(x)=L)asxapproachescContinuityandLimits:Ifafunctioniscontinuousatapoint,thenithasthesamevalueatthatpointasitslimitsfrombothsidesLimitoffunctionDefinition:Afunctionfiscontinuousatapointaifthelimitoffasxapproachesaexistsandisequaltof(a)Continuityoffunctions03TheoriesofContinuity:Iffandgarebothcontinuousatapoint,thensoaretheirsum,difference,product,andquota(providedg(x)≠0)01Properties02IntermediateValueTheorem:Iffiscontinuouson[a,b]andf(a)<0<f(b),thenthereexistatleastonecin(a,b)sothatf(c)=0ContinuityoffunctionsAfunctioniscontinuousatapointifandonlyifithasthesamevalueatthatpointasitslimitsfrombothsidesLimitandContinuityDiscontinuitiescanberemovable,jump,orinfiniteAremovablecontinuityisapointwherethefunctionhasahole,andthevalueonbothsidesoftheholeisthesameAjumpdiscontinuityisapointwherethefunctionhasahiddenjumpinvalueAninfinitediscontinuityisapointwherethefunctionbenefitsunboundDiscontinuitiesTherelationshipbetweenlimitandcontinuityPART06ThederivativeanddifferentiationoffunctionsREPORTINGSummaryUnderstandingthedefinitionandpropertiesofderivativesiskeytolearningfunctions.DetaileddescriptionThederivativedescribestherateatwhichafunctionvaluechangeswiththeindependentvariableandisanimportantindicatorofthelocalvariationofthefunction.Ithassomebasicproperties,suchaslinearity,additivity,multiplicability,andchainrule,whicharewidelyusedinthestudyofmonotonicity,extremum,andtangentlinesoffunctions.DefinitionandpropertiesofderivativesSummary:Masteringthecalculationmethodofderivativesisanimportantstepinlearningfunctions.Detaileddescription:Thecalculationmethodsforderivativesincludethederivativesofpolynomialfunctions,powerandexponentialfunctions,trigonometricfunctions,andcompositefunctions.Forcompositefunctions,weneedtousethechainruletocalculatederivatives.Inaddition,therearederivativesofsomespecialfunctions,suchasnaturallogarithmicfunctions,sinefunctions,andcos