lesson1a preview of calculus课程微积分预览.pptx

APreviewofCalculus

WhatDoYouThink?Whatthingscouldbeconsideredthegreatestachievementsofthehumanmind?

ItstheGreatest!ConsiderthatallthesethingsemergedbecauseoftechnologicaladvancesThoseadvancesreliedonCALCULUS!Calculushasmadeitpossibleto:BuildgiantbridgesTraveltothemoonPredictpatternsofpopulationchange

TheGeniusofCalculusisSimpleItreliesononlytwoideasTheDerivativeTheIntegralBothcomefromacommonsenseanalysisofmotionMotionischangeinpositionovertimeAllyouhavetodoisdropyourpenciltoseeithappen

WhatIsCalculusItisthemathematicsofchangeItisthemathematicsoftangentlinesslopesareasvolumesItenablesustomodelreallifesituationsItisdynamicIncontrasttoalgebra/precalcwhichisstatic

WhatIsCalculusOneansweristosayitisalimitmachineInvolvesthreestagesPrecalculus/algebramathematicsprocessBuildingblockstoproducecalculustechniquesLimitprocessThesteppingstonetocalculusCalculusDerivatives,integrals

ContrastingAlgebraCalculusUsef(x)tofindtheheightofthecurveatx=cFindthelimitoff(x)asxapproachesc

ContrastingAlgebraCalculusFindtheaveragerateofchangebetween

t=aandt=bFindtheinstantaneousrateofchangeatt=c

ContrastingAlgebraCalculusAreaofarectangle Areabetweentwocurves

TangentLineProblemApproximateslopeoftangenttoalineStartwithslopeofsecantline

TangentLineProblemNowallowtheΔxtogetsmaller

TheAreaProblemWeseektheareaunderacurve,thegraphf(x)Weapproximate

thatareawith

anumberof

rectanglesSum=31.9Actual=33.33

TheAreaProblemTheapproximationisimprovedbyincreasingthenumberofrectanglesNumberof

rectangles=10Sum=32.92Actual=33.33

TheAreaProblemTheapproximationisimprovedbyincreasingthenumberofrectanglesNumberof

rectangles=25Sum=33.19Actual=33.33