ppt__my space54个百分点我的空间ppt课件

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1、Integrals5TheFundamentalTheoremofCalculus5.4TheFundamentalTheoremofCalculusTheFundamentalTheoremofCalculusisappropriatelynamedbecauseitestablishesaconnectionbetweenthetwobranchesofcalculus:differentialcalculusandintegralcalculus.Itgivesthepreciseinve

2、rserelationshipbetweenthederivativeandtheintegral.TheFundamentalTheoremofCalculusThefirstpartoftheFundamentalTheoremdealswithfunctionsdefinedbyanequationoftheformwherefisacontinuousfunctionon[a,b]andxvariesbetweenaandb.Observethatgdependsonlyonx,whic

3、happearsasthevariableupperlimitintheintegral.Ifxisafixednumber,thentheintegralisadefinitenumber.Ifwethenletxvary,thenumberalsovariesanddefinesafunctionofxdenotedbyg(x).TheFundamentalTheoremofCalculusIffhappenstobeapositivefunction,theng(x)canbeinterp

4、retedastheareaunderthegraphofffromatox,wherexcanvaryfromatob.(Thinkofgasthe“areasofar”function;seeFigure1.)Figure1Example1–AFunctionDefinedasanIntegralIffisthefunctionwhosegraphisshowninFigure2andfindthevaluesofg(0),g(1),g(2),g(3),g(4),andg(5).Thens

5、ketcharoughgraphofg.Figure2Example1–SolutionFirstwenoticethat.FromFigure3(i)weseethatg(1)istheareaofatriangle:Tofindg(2)weagainrefertoFigure3(ii)andaddtog(1)theareaofarectangle:Figure3(i)Figure3(ii)=(12)=1=1+(12)=3Example1–SolutionWeestimatethatt

6、heareaunderffrom2to3isabout1.3,soFort3,f(t)isnegativeandsowestartsubtractingareas:cont’dFigure3(iii)Figure3(iv)3+1.3=4.34.3+(–1.3)=3.0Example1–SolutionWeusethesevaluestosketchthegraphofginFigure4.Noticethat,becausef(t)ispositivefort3,wekeepaddi

7、ngareafort3andsogisincreasinguptox=3,whereitattainsamaximumvalue.Forx3,gdecreasesbecausef(t)isnegative.cont’dFigure3(v)3+(–1.3)=1.7Figure4TheFundamentalTheoremofCalculusForthefunction,wherea=1andf(t)=t2,noticethatg(x)=x2,thatis,g=f.Inotherwords

8、,ifgisdefinedastheintegraloffbyEquation1,thengturnsouttobeanantiderivativeoff,atleastinthiscase.AndifwesketchthederivativeofthefunctiongshowninFigure4byestimatingslopesoftangents,wegetagraphlikethatoffinFigure2.Sowesuspectthatg=finExample1too.Figur