AP微積分課程設(shè)計(jì)

北京中加學(xué)校AP微積分課程的實(shí)施方案正值北京中加學(xué)校建校十五周年之際，為了實(shí)現(xiàn)百年名校的偉大目的，無(wú)論是在管理，還是在教育教學(xué)上都需要不停健全和完善體制機(jī)制。然而，學(xué)科課程的建設(shè)和更新勢(shì)必首當(dāng)其沖，迫在眉睫。暑假在大連舉辦的《北京中加學(xué)校數(shù)學(xué)課程和教學(xué)多元化的探究》教學(xué)研討會(huì)為北京中加學(xué)校的學(xué)科建設(shè)開(kāi)了先河，也奠定了思想基礎(chǔ)。借此良機(jī)，對(duì)于北京中加學(xué)校的特色學(xué)科之一，AP微積分，我們借鑒過(guò)去的教學(xué)經(jīng)驗(yàn)，整合國(guó)內(nèi)外教學(xué)資源，根據(jù)美國(guó)大學(xué)理事會(huì)AP微積分的課程原則，擬定了有關(guān)AP微積分課程的教學(xué)構(gòu)想。一、指導(dǎo)思想本課程是北京中加學(xué)校為學(xué)生開(kāi)設(shè)的一門(mén)國(guó)際數(shù)學(xué)專(zhuān)業(yè)基礎(chǔ)課。開(kāi)設(shè)本課程的目的，在于以美國(guó)大學(xué)理事會(huì)規(guī)定的AP?微積分課程原則為指導(dǎo)，按照理論與實(shí)踐相結(jié)合的原則，通過(guò)對(duì)微積分基本原理及規(guī)律的講授，使學(xué)生系統(tǒng)掌握極限、持續(xù)、導(dǎo)數(shù)和積分等知識(shí)的基本原理、基本內(nèi)容和基本辦法，對(duì)微積分在經(jīng)濟(jì)活動(dòng)中的應(yīng)用有比較清晰的理解，提高學(xué)生專(zhuān)業(yè)詞匯量和閱讀英語(yǔ)原版書(shū)籍的能力，拓寬學(xué)生國(guó)際數(shù)學(xué)視野，使學(xué)生體驗(yàn)到數(shù)學(xué)的價(jià)值和美學(xué)認(rèn)知。課內(nèi)學(xué)時(shí)144，4學(xué)分，從高一第一學(xué)期開(kāi)始開(kāi)設(shè)，高二第二學(xué)期結(jié)束，將近兩個(gè)年授完。二、課程目的AP微積分是在高中學(xué)習(xí)階段有余力、有能力、成績(jī)優(yōu)秀的學(xué)生有機(jī)會(huì)先修的美國(guó)大學(xué)基礎(chǔ)課程以獲得美國(guó)大學(xué)學(xué)分專(zhuān)業(yè)的必修課。規(guī)定學(xué)生在學(xué)完本課程后，掌握本課程的基本原理、基本內(nèi)容、基本辦法及基本知識(shí)，并含有對(duì)所學(xué)的微積分知識(shí)進(jìn)行現(xiàn)實(shí)理解和實(shí)際應(yīng)用的能力，從而順利通過(guò)AP考試。據(jù)此，本課程考核著重于基本知識(shí)的掌握、理解和應(yīng)用分析能力兩個(gè)方面。在各章的考核規(guī)定中，有關(guān)基本概念、基本理論、基本公式、應(yīng)用分析能力的內(nèi)容按“識(shí)記、理解、簡(jiǎn)樸應(yīng)用和綜合應(yīng)用”四個(gè)層次規(guī)定。三、教學(xué)進(jìn)度北京中加學(xué)校AP微積分教學(xué)內(nèi)容及其進(jìn)度計(jì)劃學(xué)期普通班國(guó)際班AP微積分學(xué)時(shí)分派高一第一學(xué)期第一模塊集合（4學(xué)時(shí)）函數(shù)與基本初等函數(shù)（32學(xué)時(shí)）解析幾何（9學(xué)時(shí)）復(fù)合函數(shù)、反函數(shù)以及作圖計(jì)算器的使用高中課程：6學(xué)時(shí)/周，共54學(xué)時(shí)；AP微積分：2學(xué)時(shí)/周，共18學(xué)時(shí)；第二模塊直線與圓的方程（9學(xué)時(shí)）圓錐曲線（8學(xué)時(shí)）三角函數(shù)（16學(xué)時(shí)）極限極限的運(yùn)算法則高中課程：6學(xué)時(shí)/周，共54學(xué)時(shí)；AP微積分：2學(xué)時(shí)/周，共18學(xué)時(shí)；第二學(xué)期第三模塊三角恒等變換反三角函數(shù)導(dǎo)數(shù)導(dǎo)數(shù)的基本公式高中課程：2學(xué)時(shí)/周，共18學(xué)時(shí)；AP微積分：6學(xué)時(shí)/周，共54學(xué)時(shí)；第四模塊立體幾何導(dǎo)數(shù)的運(yùn)算法則高中課程：2學(xué)時(shí)/周，共18學(xué)時(shí)；AP微積分：6學(xué)時(shí)/周，共54學(xué)時(shí)；高二第一學(xué)期第五模塊慣用邏輯用語(yǔ)平面對(duì)量解三角形導(dǎo)數(shù)的應(yīng)用高中課程：2學(xué)時(shí)/周，共18學(xué)時(shí)；AP微積分：6學(xué)時(shí)/周，共54學(xué)時(shí)；第六模塊數(shù)列不等式積分方程微分方程高中課程：2學(xué)時(shí)/周，共18學(xué)時(shí)；AP微積分：6學(xué)時(shí)/周，共54學(xué)時(shí)；第二學(xué)期第七模塊復(fù)數(shù)統(tǒng)計(jì)計(jì)數(shù)原理AP微積分總復(fù)習(xí)AP微積分AB考試高中課程：2學(xué)時(shí)/周，共18學(xué)時(shí)；AP微積分：6學(xué)時(shí)/周，共54學(xué)時(shí)；第八模塊參數(shù)方程極坐標(biāo)AP微積分BC高中課程：4學(xué)時(shí)/周，共36學(xué)時(shí)；AP微積分BC：4學(xué)時(shí)/周，共36學(xué)時(shí)；高三第一學(xué)期總復(fù)習(xí)總復(fù)習(xí)畢業(yè)會(huì)考AP微積分BC高中課程：4學(xué)時(shí)/周，共36學(xué)時(shí)；AP微積分BC：4學(xué)時(shí)/周，共36學(xué)時(shí)；第二學(xué)期微積分其它大學(xué)預(yù)修課程AP微積分BCAP微積分BC考試高中課程：4學(xué)時(shí)/周，共36學(xué)時(shí)；AP微積分BC：4學(xué)時(shí)/周，共36學(xué)時(shí)；四、課程內(nèi)容Chapter2LimitsandDerivatives第二章極限和導(dǎo)數(shù)TeachingContent教學(xué)內(nèi)容TeachingRequirementsandObjectives教學(xué)規(guī)定和目的Time學(xué)時(shí)TheTangentandVelocityProblems切線和速率問(wèn)題Thestudentwillapplythederivativetosolveproblems,includingtangentandnormallinestoacurve,curvesketching,velocity,acceleration.TheLimitofaFunction函數(shù)的極限Thestudentwilldefineandapplythepropertiesoflimitsoffunctions.Thiswillincludelimitsofaconstant,sum,product,quotient,one-sidedlimits,limitsatinfinity,infinitelimits,andnonexistentlimits.CalculatingLimitsUsingtheLimitLaws運(yùn)用極限法則計(jì)算極限Continuity持續(xù)性Thestudentwillstatethedefinitionofcontinuityanddeterminewhereafunctioniscontinuousordiscontinuous.Thiswillincludecontinuityatapoint;continuityoveraclosedinterval;andgraphicalinterpretationofcontinuityanddiscontinuity.LimitsatInfinity;HorizontalAsymptotes無(wú)窮遠(yuǎn)處極限和水平漸近線Thestudentwilldefineandapplythepropertiesofelementaryfunctions,includingalgebraic,trigonometric,exponential,andcompositefunctionsandtheirinverses,andgraphthesefunctionsusingagraphingcalculator.Propertiesoffunctionswillincludedomains,ranges,combinations,odd,even,periodicity,symmetry,asymptotes,zeros,upperandlowerbounds,andintervalswherethefunctionisincreasingordecreasing.Thestudentwillalsodefineandapplythepropertiesoflimitsoffunctions.Thiswillincludelimitsofaconstant,sum,product,quotient,one-sidedlimits,limitsatinfinity,infinitelimits,andnonexistentlimits.Tangents,Velocities,andOtherRatesofChange切線、速度和其它的變化率Derivatives導(dǎo)數(shù)Thestudentwillfindthederivativeofanalgebraicfunctionbyusingthedefinitionofaderivative.Thiswillincludeinvestigatinganddescribingtherelationshipbetweendifferentiabilityandcontinuity.TheDerivativeasaFunction導(dǎo)函數(shù)Review復(fù)習(xí)Chapter3DifferentiationRules第三章導(dǎo)數(shù)法則TeachingContent教學(xué)內(nèi)容TeachingRequirementsandObjectives教學(xué)規(guī)定和目的Time學(xué)時(shí)DerivativesofPolynomialsandExponentialFunctions多項(xiàng)式函數(shù)和指數(shù)函數(shù)的導(dǎo)數(shù)Thestudentwillapplyformulastofindthederivativeofalgebraic,trigonometric,exponential,andlogarithmicfunctionsandtheirinverses.TheProductandQuotientRules導(dǎo)數(shù)的乘法和除法運(yùn)算法則Thestudentwillapplyformulastofindthederivativeofthesumofelementaryfunctions.RatesofChangeintheNaturalandSocialSciences自然科學(xué)和社會(huì)科學(xué)中的變化率Studentswillbeabletounderstandthemathematicalmodelingprocessofderivatives(ratesofchanges)intherealworldofTrigonometricFunctions三角函數(shù)的導(dǎo)數(shù)StudentswillbeabletousethedifferentiationrulesoftrigonometricfunctionsTheChainRule鏈?zhǔn)椒▌tThestudentwillapplyformulastofindthederivativeofthesum,product,quotient,inverseandcomposite(chainrule)ofelementaryfunctions.ImplicitDifferentiation隱函數(shù)求導(dǎo)Thestudentwillfindthederivativeofanimplicitlydefinedfunction.HigherDerivatives高階導(dǎo)數(shù)Thestudentwillfindthehigherorderderivativesofalgebraic,trigonometric,exponential,andlogarithmicfunctions.ofLogarithmicFunctions對(duì)數(shù)函數(shù)的導(dǎo)數(shù)Thestudentwilluselogarithmicdifferentiationasatechniquetodifferentiatenon-logarithmicfunctions. HyperbolicFunctions雙曲函數(shù)Thestudentwillbeabletounderstandthedefinitionofhyperbolicfunctions,andsolveforitsderivatives.LinearApproximationsandDifferentials線性逼近和微分Thestudentwillapplythederivativetosolveproblems,includingtangentandnormallinestoacurve,curvesketching,velocity,acceleration,relatedratesofchange,Newton'smethod,differentialsandlinearapproximations,andoptimizationproblems.Review復(fù)習(xí)Chapter4ApplicationsofDifferentiation第四章導(dǎo)數(shù)的應(yīng)用TeachingContent教學(xué)內(nèi)容TeachingRequirementsandObjectives教學(xué)規(guī)定和目的Time學(xué)時(shí)MaximumandMinimumValues極大值和極小值Thestudentwillbeabletounderstandextremevaluesofafunction,findcriticalvaluesofafunctionandfindextremevaluesofafunction.TheMeanValueTheorem中值定理Thestudentwillstate(withoutproof)theMeanValueTheoremforderivativesandapplyitbothalgebraicallyandgraphically.HowDerivativeAffecttheShapeofaGraph導(dǎo)數(shù)是如何變化圖像的形狀Thestudentwillgraphthesefunctionsusingagraphingcalculator,includingunderstandingandusingtheFirstDerivativeTestandtheSecondDerivativeTesttodeterminemin’sandmax’s.IndeterminateFormsandL’Hospital’sRules不定式和洛必達(dá)法則Thestudentwillusel'Hopital'sruletofindthelimitoffunctionswhoselimitsyieldtheindeterminateforms:0/0andinfinity/infinityOptimizationProblems最優(yōu)化問(wèn)題 ThestudentwillbeabletousederivativestosolveoptimizationproblemsNewton’sMethod牛頓法則ThestudentwillbeabletouseNewton’smethodtoapproximaterootsofanequation.原函數(shù)（反導(dǎo)數(shù)）Thestudentwillbeabletounderstandtheconceptofanantiderivative,thegeometryoftheantiderivativeandthatofslopefieldsandalsoworkrectilinearmotionproblemswithantiderivativesReview復(fù)習(xí)Chapter5Integrals第五章積分TeachingContent教學(xué)內(nèi)容TeachingRequirementsandObjectives教學(xué)規(guī)定和目的Time學(xué)時(shí)AreasandDistances面積和距離Thestudentwillidentifythepropertiesofthedefiniteintegral.ThiswillincludetheFundamentalTheoremofCalculusandthedefiniteintegralasanareaandasalimitofasumaswellasthefundamentaltheorem.TheDefiniteIntegral不定積分Thestudentwillcomputeanapproximatevalueforadefiniteintegral.ThiswillincludenumericalcalculationsusingRiemannSumsandtheTrapezoidalRule.TheFundamentalTheoremofCalculus微積分基本定理Thestudentwillidentifythepropertiesofthedefiniteintegral.ThiswillincludetheFundamentalTheoremofCalculusandthedefiniteintegralasanareaandasalimitofasumaswellasthefundamentaltheorem.Theintegralfromatoxoff(t)d(t)dt/dx=f(x)IndefiniteIntegralsandtheNetChangeTheorem不定積分和原函數(shù)定理Thestudentwillfindtheindefiniteintegralofalgebraic,exponential,logarithmic,andtrigonometricfunctions.TheSubstitutionRule換元積分法Thestudentwillfindtheindefiniteintegralofalgebraic,exponential,logarithmic,andtrigonometricfunctions.Thespecialintegrationtechniquesofsubstitution(changeofvariables)andintegrationbypartswillbeincluded.Review復(fù)習(xí)Chapter6ApplicationsofIntegration第六章積分的應(yīng)用TeachingContent教學(xué)內(nèi)容TeachingRequirementsandObjectives教學(xué)規(guī)定和目的Time學(xué)時(shí)AreasBetweenCurves曲邊面積Thestudentwillapplythedefiniteintegraltosolveproblems.Theseproblemswillincludefindingdistancetraveledonalineandvelocityfromaccelerationwithinitialconditions,growthanddecayproblems,solutionsofseparabledifferentialequations,theaveragevalueofafunction,areabetweencurves,volumesofsolidsofrevolutionabouttheaxesorlinesparalleltotheaxesusingdisc/washerandshellmethods,andvolumesofsolidswithknowncross-sectionalareas.Volumes體積Thestudentwillapplythedefiniteintegraltosolveproblems.Theseproblemswillincludeareabetweencurves,volumesofsolidsofrevolutionabouttheaxesorlinesparalleltotheaxesusingdisc/washerandshellmethods,andvolumesofsolidswithknowncross-sectionalareas.VolumesbyCylindricalShells圓柱體體積Thestudentwillapplythedefiniteintegraltosolveproblems.Theseproblemswillinclude…areabetweencurves,volumesofsolidsofrevolutionabouttheaxesorlinesparalleltotheaxesusingdisc/washerandshellmethods,andvolumesofsolidswithknowncross-sectionalareas.Work物體功Thestudentwillapplythedefiniteintegraltosolveproblems.Theseproblemswillincludefindingdistancetraveledonalineandvelocityfromaccelerationwithinitialconditions,growthanddecayproblems,workdone.AverageValueofaFunction實(shí)函數(shù)均值Thestudentwillapplythedefiniteintegraltosolveproblems.Theseproblemswillincludetheaveragevalueofafunction.DensityFunction密度函數(shù)Thestudentwillapplythedefiniteintegraltosolveproblems.Theseproblemswillincludefindingdistancetraveledonalineandvelocityfromaccelerationwithinitialconditions,growthanddecayproblems.Review復(fù)習(xí)Chapter7TechniquesofIntegration第七章積分技巧TeachingContent教學(xué)內(nèi)容TeachingRequirementsandObjectives教學(xué)規(guī)定和目的Time學(xué)時(shí)IntegrationbyParts分部積分法Thestudentwillfindthedefiniteandindefiniteintegralofalgebraic,exponential,logarithmic,andtrigonometricfunctions.Thespecialintegrationtechniquesofsubstitution(changeofvariables)andintegrationbypartswillbeincluded.TrigonometricIntegrals三角函數(shù)積分TrigonometricSubstitution三角函數(shù)替代In