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1、MITOpenCourseWarehttp://ocw.mit.edu18.01SingleVariableCalculusFall2006ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.Lecture118.01Fall2006Unit1:DerivativesA.Whatisaderivative?•Geometricinterpretation•Physicalinterpretation•Importantforany
2、measurement(economics,politicalscience,finance,physics,etc.)B.Howtodifferentiateanyfunctionyouknow.d�xarctanx�•Forexample:e.Wewilldiscusswhataderivativeistoday.Figuringouthowtodxdifferentiateanyfunctionisthesubjectofthefirsttwoweeksofthiscourse.Lecture1:Derivatives,Slope,Velocity,
3、andRateofChangeGeometricViewpointonDerivativesyQSecantlineTangentlinePf(x)xx+∆x00Figure1:AfunctionwithsecantandtangentlinesThederivativeistheslopeofthelinetangenttothegraphoff(x).Butwhatisatangentline,exactly?1Lecture118.01Fall2006•ItisNOTjustalinethatmeetsthegraphatonepoint.•
4、Itisthelimitofthesecantline(alinedrawnbetweentwopointsonthegraph)asthedistancebetweenthetwopointsgoestozero.Geometricdefinitionofthederivative:LimitofslopesofsecantlinesPQasQ→P(Pfixed).TheslopeofPQ:Q(x+∆x,f(x+∆x))00SecantLine∆f(x,f(x))00P∆xFigure2:Geometricdefinitionofthederivati
5、veΔff(x0+Δx)−f(x0)�lim=lim=f(x0)Δx→0ΔxΔx→0Δx��������“differencequotient”“derivativeoffatx0”1Example1.f(x)=xOnethingtokeepinmindwhenworkingwithderivatives:itmaybetemptingtopluginΔx=0Δf0rightaway.Ifyoudothis,however,youwillalwaysendupwith=.YouwillalwaysneedtoΔx0dosomecancellation
6、togetattheanswer.11����Δfx0+Δx−x01x0−(x0+Δx)1−Δx−1====ΔxΔxΔx(x0+Δx)x0Δx(x0+Δx)x0(x0+Δx)x0TakingthelimitasΔx→0,−1−1lim=2Δx→0(x0+Δx)x0x02Lecture118.01Fall2006yxx0Figure3:Graphof1xHence,�−1f(x0)=2x0Noticethatf�(x)isnegative—asistheslopeofthetangentlineonthegraphabove.0Findingthet
7、angentline.Writetheequationforthetangentlineatthepoint(x0,y0)usingtheequationforaline,whichyoualllearnedinhighschoolalgebra:y−y=f�(x)(x−x)0001−1Pluginy=f(x)=andf�(x)=toget:00x0x2001−1y−=2(x−x0)x0x03Lecture118.01Fall2006yxx0Figure4:Graphof1xJustforfun,let’scomputetheareaofthetr
8、ianglethatthetangentlineformswiththex-andy-axes(seetheshadedr
