Part IV_01 derivative 导数.ppt

PartIVTHEDERIVATIVE TheDerivativeDifferentiabilityandContinuityDifferentiationsofSomeImportantFunctionsBasicRulesofDifferentiation AnIntuitiveExample ConsiderthemaglevexamplefromSection2 4 Thepositionofthemaglevisafunctionoftimegivenbys f t 4t2 0 t 30 wheresismeasuredinfeetandtinseconds Itsgraphis AnIntuitiveExample Thegraphrisesslowlyatfirstbutmorerapidlyovertime Thissuggeststhesteepnessoff t isrelatedtothespeedofthemaglev whichalsoincreasesovertime Ifso wemightbeabletofindthespeedofthemaglevatanygiventimebyfindingthesteepnessoffatthattime Buthowdowefindthesteepnessofapointinacurve SlopesofLinesandofCurves Theslopeofalineisameasureofits steepness foranytwopointsontheline itistheratioofthe rise differenceinthey coordinates tothe run differenceinthex coordinates i e slope y x SlopesofLinesandofCurves Steeplineshavelargeslopes flatterlineshavesmallerslopes Decreasinglines whichgo downhill asxincreasestowardstheright havenegativeslopes Theslopeofalinedoesn tdependonthepairofpointsonthelineusedtocalculateit allpairsofpointsonthesamelinewillgivethesameslope SlopesofLinesandofCurves Forcurvesthataren tlines theideaofasingleoverallslopeisnotveryuseful Intuitively thesteepnessofatypicalcurveisdifferentatdifferentplacesonthecurve soanappropriatedefinitionofslopeforthecurveshouldsomehowreflectthisvariablesteepness SlopesofLinesandofCurves Theslopeatapointofacurveisgivenbytheslopeofthetangenttothecurveatthatpoint SupposewewanttofindtheslopeatpointA ThetangentlinehasthesameslopeasthecurvedoesatpointA A TodefinethetangentlinetoacurveCatapointAonthecurve weshouldconsiderastraightlinethatpassesthroughAandanotherpointPonCdistinctfromA SlopesofLinesandofCurves P A x y C AsthepointPisallowedtomovetowardAalongthecurve thesecantlinerotatesaboutthefixedpointAandapproachesalimitingposition afixedline whichisthetangentlinetothecurveCatthepointA Wecanshowtheprocessmorepreciselyasfollows SlopesofLinesandofCurves Wecanshowtheprocessmorepreciselyasfollows SlopesofLinesandofCurves Wecanshowtheprocessmorepreciselyasfollows SlopesofLinesandofCurves Wecanshowtheprocessmorepreciselyasfollows SlopesofLinesandofCurves Wecanshowtheprocessmorepreciselyasfollows SlopesofLinesandofCurves Wecanshowtheprocessmorepreciselyasfollows SlopesofLinesandofCurves Ingeneral wecanexpresstheslopeofthesecantasfollows SlopesofLinesandofCurves Thus ashapproacheszero theslopeofthesecantapproachestheslopeofthetangenttothecurveatthatpoint SlopesofLinesandofCurves Thus ashapproacheszero theslopeofthesecantapproachestheslopeofthetangenttothecurveatthatpoint Expressedinlimitsnotation TheslopeofthetangentlinetothegraphoffatthepointP x f x isgivenbyifitexists SlopesofLinesandofCurves RatesofChange Wecanseethatmeasuringtheslopeofthetangentlinetoagraphismathematicallyequivalenttofindingtherateofchangeoffatx Definition Theratioofthechangeintheoutputvalueandchangeintheinputvalueofafunctioniscalledasrateofchange Forexamples velocityistherateofchangeindistancewithrespecttotime rateofchangeofvelocityisknownasacceleration slopeofalineorlinearfunctionrepresentstherateofchange AverageRatesofChange Thenumberf x h f x measuresthechangeinythatcorrespondstoachangehinx Thenthedifferencequotientmeasurestheaveragerateofchangeofywithrespecttoxovertheinterval x x h Inthemaglevexample ifymeasuresthepositionthetrainattimex thenthequotientgivetheaveragevelocityofthetrainoverthetimeinterval x x h AverageRatesofChange Theaveragerateofchangeoffovertheinterval x x h orslopeofthesecantlinetothegraphoffthroughthepoints x f x and x h f x h is InstantaneousRatesofChange Bytakingthelimitofthedifferencequotientashgoestozero evaluatingweobtaintherateofchangeoffatx Thisisknownastheinstantaneousrateofchangeoffatx asopposedtotheaveragerateofchange Inthemaglevexample ifymeasuresthepositionofatrainattimex thenthelimitgivesthevelocityofthetrainattimex InstantaneousRatesofChange Theinstantaneousrateofchangeoffatxorslopeofthetangentlinetothegraphoffat x f x isThislimitiscalledthederivativeoffatx TheDerivativeofaFunction Thederivativeofafunctionfwithrespecttoxisthefunctionf read fprime Thedomainoff isthesetofallxwherethelimitexists Thus thederivativeoffunctionfisafunctionf thatgivestheslopeofthetangenttothelinetothegraphoffatanypoint x f x andalsotherateofchangeoffatx TheDerivativeofaFunction FourStepProcessforFindingf x 1 Computef x h 2 Formthedifferencef x h f x 3 Formthequotient4 Compute Example1 Findtheslopeofthetangentlinetothegraphf x 3x 5atanypoint x f x Solution Therequiredslopeisgivenbythederivativeoffatx Tofindthederivative weusethefour stepprocess Step1 f x h 3 x h 5 3x 3h 5 Step2 f x h f x 3x 3h 5 3x 5 3h Step3 Step4 Example2 a Findtheslopeofthetangentlinetothegraphf x x2atanypoint x f x Solution Therequiredslopeisgivenbythederivativeoffatx Tofindthederivative weusethefour stepprocess Step1 f x h x h 2 x2 2xh h2 Step2 f x h f x x2 2xh h2 x2 h 2x h Step3 Step4 Example2 b Findtheslopeofthetangentlinetothegraphf x x2atanypoint x f x Theslopeofthetangentlineisgivenbyf x 2x Now findandinterpretf 2 Solution f 2 2 2 4 Thismeansthat atthepoint 2 4 theslopeofthetangentlinetothegraphis4 Exercises TextbookP14610 12 14 16 18 20 22 DifferentiabilityandContinuity Sometimes oneencounterscontinuousfunctionsthatfailtobedifferentiableatcertainvaluesinthedomainofthefunctionf Forexample considerthecontinuousfunctionfbelow Itfailstobedifferentiableatx a becausethegraphmakesanabruptchange acorner atthatpoint Itisnotclearwhattheslopeisatthatpoint Differentiabilit