《微积分英文版》课件

Calculus EnglishEdition PPTcourseware目录•Introduction•Fundamentals ofCalculus•Application ofCalculus•Further learningof calculus•Exercises andAnswers01IntroductionCourse IntroductionCourseoverview01This courseintroduces thefundamental conceptsand topicsofcalculus,including limits,continuity,differentiation,integration,and seriesLearningobjectives02Learners willgain adeep understandingof calculusand itsapplicationsin variousfields,such asphysics,engineering,andeconomicsCourse structure03The courseis dividedinto severalmodules,each focusingon aspecifictopic in calculus Learnerscan completethe courseattheir ownpace andin anyorder of the modulesCourseobjectives•Understanding thebasic concepts of calculus:Learners willunderstandthe basicconceptsofcalculus,including limits,continuity,and themeanvalue theory•Mastering the rules ofdifferentiation:Learners willlearn todifferentiate differentfunctions usingtherulesofdifferentiation and understanding the concept of a derivative•Understanding integration:Learners willunderstand theconcept ofintegration,including thefundamentaltheory ofcalculus,and beable tointegrate variousfunctions•Understanding series:Learners willunderstand theconcept ofseries,including convergence and divergence,and beable toevaluate differentseriesThe significanceof learningcalculusApplication inother disciplines:Calculus is thefoundation ofmany disciplines,such asphysics,engineering,and economicsUnderstandingcalculus can help learnersbetter understandthesedisciplinesUnderstand the relationship betweengeometryDevelop analyticalthinking:Learning calculusandalgebra:Through calculus,learners canrequireslearners touse analyticalthinking tounderstand therelationship between geometrysolveproblems Thiscanhelpdevelop learnersandalgebra andhow to apply algebraicthinkingability andproblem-solving skillsmethods to geometricproblems02Fundamentals ofCalculusLimit theory•Limit definition:The limit of a function is the value that the functionapproaches as the input approaches a certain point It isa fundamentalconceptincalculusand is used tostudy the behavior of functions atspecificpoints orin specificregions•Properties of limits:Limits havereceived properties that allowus toregulate andreason aboutthem Propertiessuch as thesum,difference,product,and quota rules forlimits allowus tocalculate limitsof morecomplexfunctions usingsimpler ones•One sidelimits:In additionto thestandard limit,where the input approaches a pointfrom bothsides,there arealsoone sidelimits,where theinput approachesthe pointfrom onlyone directionThese areimportant instudyingdiscontinuities andremovable SingularitiesDerivativeand differential•Derivative definition:The derivative of a function isa measureof therate of change of the function at a givenpoint Itis calculatedbytaking the limitof the ratioof thechange in the functionsoutput tothechange inits input,astheinputapproachesacertain point•Properties of derivatives:Derivatives haveproperties that are similarto thoseoflimitsProperties suchas thesum,difference,product,and quota rules forderivatives allowus tocalculate derivativesof morecomplexfunctions usingsimpler ones•Rules for distinguishing functions:There areseveral rules fordistinguishingfunctions,suchasthe chainrule,theproduct rule,andthequotaruleThese rulescan beapplied tosimplify theprocess offinding derivativesand tounderstand thebehavior offunctionsin variousscenariosIntegrated•Integral definition:The integralof afunction isa measureof theareaunder itscurve Itis calculatedby findingthelimitof thesum ofareasof rectanglesunder thecurve asthe widthof therectanglesapproaches zero•Properties ofintegrals:Integrates havepropertiesthat are similarto thoseofderivativesProperties suchas thesum and subtractionrulesforintegrals allowus tocalculate integralsof morecomplex functionsusing simplerones•Approximation ofintegrals:In practice,we oftenuse numericalmethodstoapproximate integralsTechniquessuch asthe RiemannsumandMonte Carlointegration can be used to estimateintegrals withvarying levelsofaccuracy Thesemethods arespecifically usefulwhen dealingwith functionsthataredifficult orimpossible tointegrateanalytically03Application ofCalculusPhysical applicationsMotionand VelocityCalculusis used to analyzethe motionof objectsandunderstandtheir velocity,speed,and directionIt allowsus tocalculate theexact positionof anobject atanygiven timeGravityand PotentialInphysics,calculus is used tostudy theforce ofgravity andpotential energyIthelps in understanding therelationshipbetweendistance,force,and energyAccelerationand KineticEnergyCalculus is used toanalyze acceleration,which is the rateof changein velocityItalso helps inunderstandingtheconceptof kineticenergy,which isthe energyofmotionEconomic applicationsSupplyand DemandCurvesIn economics,calculations areused toanalyze supplyand demandcurves Ithelpsinunderstanding therelationshipbetween priceand quantitydemand orsupplyMarginal AnalysisCalculusisusedin marginalanalysis to understandtheeffects ofsmall changesin inputsor outputsonthe overallcost orrevenueOptimization ProblemsCalculusisusedto solveoptimization problemsin economics,so asto find the optimallevel ofproductionthat maximizesprofit orsocial welfareScientificcomputing applicationsNumericalAnalysis ComputerSignal ProcessingGraphicsCalculusisusedin numericalIn computergraphics,calculus isIn signalprocessing,calculus isanalysisto solvemathematical usedtounderstandthe conceptsusedtoanalyze signals,such asproblemsthataredifficult orof differentialgeometry andaudio orvideo signals,andimpossible tosolve analyticallytopology,which areessential forextract meaningfulinformationTechniques likeinterpolation,creating realistic3D modelsand fromthem Techniqueslike Fourierdifferentiation,and integrationanimations analysisand wavelettransformsare usedextensively arebased oncalculus04Further learningof calculusMultivariateCalculusMultivariate functionsDefinition:A functionthat assignsa valueto eachpoint ina spaceof twoormore dimensionsMultivariateCalculus•Properties:Continuity,differentiation,integrity,etcMultivariate CalculusLimitsand continuityDefinition:A limitisthevaluethatafunctionapproaches astheinputapproachesacertainpointContinuitymeans that thefunctiondoesnt haveany breaksor jumpsat anypointMultivariate Calculus•Properties:One sidelimits,absolutecontinuity,uniform continuity,etcMultivariate CalculusDifferentiationDefinition:The derivativeofafunction ata point istheslope of the tangentline tothe graph ofthefunctionat that pointIt canbe usedtofindtherateofchangeof afunctionProperties:Derivative ofcomposite functions,chain rules,product rules,quotarules,etcDifferential equationDefinitionTypes MethodsofsolutionA differentialequation isan Ordinaldifferential equationsSeparation ofvariables,equation thatcontains oneor ODEsand partialdifferential integrationby parts,Laplacemore unknownfunctions andequations PDEsODEs describetransforms,Fourier series,etctheir derivativesItcanbeusedto thebehaviorofafunctionof onemodelreal worldphenolmena variableover time,while PDEssuchas populationgrowth,describe thebehavior ofprojectilemotion,etc functionsof multiplevariablesover spaceand timeReal analysisDefinition01Real analysisisthestudy ofproperties andbehaviorsof realnumbers Itincludes topicssuch assequences,series,continuity,differentiation,integrity,etcProperties02Arithmetic operationson sequencesand series,convergenceanddivergence ofsequences andseries,properties ofcontinuous functions,etcApplications03Realanalysishas applicationsin fieldssuch ascalculus,probability theory,statistics,numericalanalysis,etc.It providesa solidfoundation forfurtherstudies inmathematics andother disciplines05Exercises andAnswersExercise sectionExercise1Exercise3Calculate thederivativeofDetermine whicha functionthegiven functionatapoint is increasing,decreasing,orconsistent onan intervalExercise2Exercise4Find thelocal maximumSketchthegraphof aandminimum valuesofafunctionand identifyanyfunctionsymbols ordiscontinuitiesAnswer andAnalysis•Answer1:The derivativeofthegiven functionatthepointisX•Answer2:The localmaximum andminimum valuesofthefunction areY andZ,respectively•Answer3:The function isincreasingon theinterval fromA toB•Answer4:The graphofthefunctionisas follows:It startsat pointC,increases toa maximumatpointD,then decreasestoaminimum atpoint Ebefore increasingagainto pointF,where itlevels offand approachesasymptote G.There areno discontinuitiesinthefunctionTHANK YOU感谢各位观看。