derivatives - oocities：衍生物oocities

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1、DerivativesOverridingprincipalistheconceptofchange(speed,weather,populations,..)I.AveragespeedAnobjectisdroppedfroma500ftbuilding.Fromphysicsweknowthatthedistanceabovethegroundis.Sketchagraphofdistanceversustime.Findtimeuntilobjecthitstheground.ComputeaveragespeedontheintervalusingGeometricallythea

2、veragespeedistheslopeofthesecantline.II.InstantaneousvelocityGreeksrealizedthatatanygiveninstantevenamovingobjectisfrozen.Newtonandhiscontemporariesgaveuponthenotionofspeedataninstantandreplaceditwithaveragespeedsovershorterandshorterintervals.Computeinstantaneousspeedatthetimeusing.Geometricallyth

3、einstantaneousspeedistheslopeofthetangentline.III.ThederivativeatapointCanapplythesameprocesstoanyfunction.TheinstantaneousrateofchangeofatthepointisdefinedbyorlettingIV.ThederivativeasafunctionWecanapplytheprocessoffindingthederivativetomanydifferentvaluesandinfactforeveryvaluethereisacorrespondin

4、gvalueforthederivative.ForanyfunctionwedefinethederivativefunctionThereareseveralequivalentnotationsforthederivativefunctionincludingI.InterpretationsofthederivativeAsaslopeandasarateofchange.Ex.SupposeCiscostandtistime.Thenistherateofchangeofcost.II.GraphicalderivativesFunctionDerivativeIncreasing

5、PositiveDecreasingNegativeConstantZeroConcaveupIncreasingConcavedownDecreasingInflectionpointMax/MinSmoothmax/minZeroHA(x=a)HA(x=a)VA(y=a)VA(y=0)SharpcornerundefinedIII.AlgebraicshortcutsforcomputingderivativesTableofknownderivatives·Powerrule·TrigonometricfunctionsCombinations·Sums,Differences,Pro

6、ducts,Quotients·Chainrule·ImplicitdifferentiationI.FocusontheoryLimitsIntuitivedefinitionmeansthatasgetscloseto,getscloseto.Existenceoflimits(limitscanfailtoexistiftherearesmalljumps,unboundedbehaviororexcessiveoscillations)Rigorousdefinitionmeansthatforeverythereexistsasuchthatwhenever.ContinuityI

7、ntuitivelyafunctioniscontinuousifyoucansketchitsgraphwithoutliftingyourpencil.Fromarigorousperspectiveafunctioniscontinuousatif(i)isdefined(ii)exists(iii)DifferentiabilityAfunctionisdifferentiabilityifitpos