数学专业英语PPT课件

NewWordsitsaltitude,f(x+h)-f(x),representsthedifferenceoftheordinatesofthetwopointsQandP.,考虑斜边为PQ的直角三角形，它的高度f(x+h)-f(x)表示P、Q两点纵坐标的差。,Therefore,thedifferencequotientrepresentsthetrigonometrictangentoftheangleathatPQmakeswiththehorizontal.,因此，差商表示PQ与水平线的夹角a的正切。,TherealnumbertanaiscalledtheslopeofthelinethroughPandQanditprovidesawayofmeasuringthe“steepness”oftheline.,实数tana叫做直线PQ的斜率，它提供了一种度量直线陡峭程度的方法。,Forexample,iffisalinearfunction,sayf(x)=mx+b,thedifferencequotienthasthevaluem,somistheslopeoftheline.,例如，如果f是一个线性函数，那么差商的值为m,那么m就是直线的斜率。,Supposenowthatfhasaderivativeatx.Thismeansthatthedifferencequotientapproachesacertainlimitf(x)ashapproaches0.现在假设f在x有导数。这意味着当h趋于0时，差商趋于某一个极限f(x)。,Whenthisisinterpretedgeometricallyittellsusthat,ashgetsnearerto0,thepointPremainsfixed,QmovesalongthecurvetowardP,andthelinethroughPQchangesitsdirectioninsuchawaythatitsslopeapproachesthenumberf(x)asalimit.,其几何意义为,当h趋于0时,点P保持不动,而点Q沿曲线趋近P,同时,经过PQ的直线以这样的方式改变方向,即其斜率趋于数值f(x),并以它为极限.,ForthisreasonitseemsthatnaturaltodefinetheslopeofthecurveatPtobethenumberf(x).ThelinethroughPhavingthisslopeiscalledthetangentlineatP.正因如此，把曲线在P点的斜率定义成数f(x)看起来是自然的。通过P且具有这个斜率的直线叫做在P点的切线。,本小节重点掌握,1.如果差商以某一个确定的值为极限，则该极限叫做f在x点的导数。2.通过P且具有斜率f(x)的直线叫做在P点的切线。,Ifthedifferencequotientapproachessomedefinitevalueasalimit，thenthislimitiscalledthederivativeoffatx.,Thelinethro