《大學(xué)文科數(shù)學(xué)》教學(xué)大綱 苗巧云

PAGE《大學(xué)文科數(shù)學(xué)》教學(xué)大綱課程中文名稱：大學(xué)文科數(shù)學(xué)課程英文名稱：CollegeMathematicsforLiberalArts【總學(xué)時(shí)數(shù)】48【周學(xué)時(shí)數(shù)】3【學(xué)分?jǐn)?shù)】3【適用專業(yè)】文科類本科專業(yè)。一、教學(xué)目標(biāo)使學(xué)生通過本課程的學(xué)習(xí)，理解高等數(shù)學(xué)的基本概念，掌握高等數(shù)學(xué)的基本理論和方法。逐步提高學(xué)生的抽象思維能力、邏輯推斷能力、空間想象能力、運(yùn)算能力以及綜合運(yùn)用所學(xué)知識的能力。二、教學(xué)內(nèi)容、要求和學(xué)時(shí)分配（一）函數(shù)、極限、連續(xù)學(xué)時(shí)：8教學(xué)內(nèi)容：（1）緒論（2）函數(shù)（3）極限（4）函數(shù)的連續(xù)性教學(xué)要求：（1）高等數(shù)學(xué)課程的基本介紹，微積分的發(fā)展及思想方法介紹。使學(xué)生初步了解微積分的思想方法，微積分對人類科學(xué)技術(shù)發(fā)展的貢獻(xiàn)，了解高等數(shù)學(xué)課程的基本結(jié)構(gòu)。介紹學(xué)習(xí)高等數(shù)學(xué)的方法和學(xué)習(xí)要求，激發(fā)學(xué)生的學(xué)習(xí)興趣。（2）理解函數(shù)的概念及函數(shù)的四個(gè)基本特性；了解反函數(shù)、復(fù)合函數(shù)及初等函數(shù)的概念；熟悉基本初等函數(shù)的性質(zhì)和圖形。（3）理解極限的概念；了解極限與單側(cè)極限的關(guān)系；掌握極限的性質(zhì)及四則運(yùn)算法則；了解無窮小量、無窮大量以及無窮小的階的概念；會用兩個(gè)重要極限求極限。（4）理解函數(shù)的連續(xù)性的概念；了解初等函數(shù)的連續(xù)性及閉區(qū)間上連續(xù)函數(shù)的性質(zhì)。（二）一元函數(shù)微分學(xué)學(xué)時(shí)：12教學(xué)內(nèi)容：（1）導(dǎo)數(shù)的概念（2）初等函數(shù)的導(dǎo)數(shù)（3）微分（4）導(dǎo)數(shù)的應(yīng)用教學(xué)要求：（1）理解導(dǎo)數(shù)的概念；理解導(dǎo)數(shù)的幾何意義及函數(shù)的可導(dǎo)性與連續(xù)性之間的關(guān)系。（2）掌握導(dǎo)數(shù)的四則運(yùn)算法則、復(fù)合函數(shù)的求導(dǎo)法則及隱函數(shù)的求導(dǎo)法則；熟悉基本初等函數(shù)的求導(dǎo)公式；了解高階導(dǎo)數(shù)的的概念；掌握初等函數(shù)的一階導(dǎo)數(shù)和高階導(dǎo)數(shù)的求法。（3）理解微分的概念；會求函數(shù)的微分。（4）理解拉格朗日定理；掌握用洛必達(dá)法則求極限的方法；掌握導(dǎo)數(shù)的各種應(yīng)用，會利用導(dǎo)數(shù)解決簡單的應(yīng)用問題。（三）一元函數(shù)積分學(xué)學(xué)時(shí)：14教學(xué)內(nèi)容：（1）不定積分（2）定積分（3）定積分的應(yīng)用教學(xué)要求：（1）理解原函數(shù)與不定積分的概念；掌握不定積分的性質(zhì)與基本積分公式；掌握換元積分法和分部積分法。（2）理解定積分的概念與性質(zhì)；熟悉積分上限函數(shù)和牛頓萊布尼茲公式；掌握定積分的計(jì)算法。（3）學(xué)會利用定積分計(jì)算平面圖形的面積、旋轉(zhuǎn)體的體積、函數(shù)的平均值等應(yīng)用方面的問題。注：以下（四）、（五）兩部分內(nèi)容可根據(jù)不同專業(yè)的應(yīng)用方向需求選學(xué)其中之一。（四）微分方程基礎(chǔ)（選學(xué)之一）學(xué)時(shí)：14教學(xué)內(nèi)容：（1）微分方程的基本概念（2）一階微分方程（3）二階微分方程（4）常微分方程的應(yīng)用。教學(xué)要求：（1）了解微分方程的基本概念。（2）掌握可分離變量的微分方程、一階線性微分方程的求解方法。（3）會解三種可降階的高階微分方程、二階常系數(shù)線性齊次微分方程。（4）了解微分方的簡單應(yīng)用，會用微分方程解一些簡單的應(yīng)用問題。（五）線性代數(shù)簡介（選學(xué)之二）學(xué)時(shí)：14教學(xué)內(nèi)容：（1）矩陣及其運(yùn)算（2）矩陣的初等變換與線性方程組的消元法（3）行列式簡介教學(xué)要求：（1）理解矩陣的概念、矩陣的代數(shù)運(yùn)算和轉(zhuǎn)置；了解矩陣的應(yīng)用。（2）掌握矩陣的初等變換及線性方程組的消元法。（3）理解二、三階行列式的定義；掌握行列式的簡單性質(zhì)及克拉默法則。CollegeMathematicsforLiberalArtsCredithours:48Credits：3CourseObjectiveThroughthestudyofthiscourse,liberalartsstudentswillunderstandtheconceptofcollegemathematicsandwillmasterthetheoryandmethodofcollegemathematics.Students’abilityofabstractthinking,logicalinference,spatialimagination,computingandcomprehensiveapplicationwillbegraduallyimproved.CourseOutlines,Goals,AndHoursFunction,LimitsandContinuityhours:8Content:IntroductionFunctionsLimitsContinuityoffunctionsRequirement:Basicintroductionofthecourseofcollegemathematics,andthedevelopmentofcalculusanditsthinkingmethod.Understandthecontributionofcalculustothedevelopmentofscienceandtechnologyinhumanhistory.Understandthelearningmethodsandthecoursestructureofadvancedmathematics.Comprehendtheconceptoffunctionsandthefourbasiccharacteristicsoffunctions;understandtheconceptofinverse,composite,andelementaryfunctions;befamiliarwiththepropertiesandgraphsofelementaryfunctions.Comprehendtheconceptoflimits;understandtherelationoflimitsandone-sidelimits;masterthepropertiesoflimitsandlimitlaws;understandtheconceptofinfinitesimal,infinityandinfinitesimalorder;beabletousetwoimportantlimitsinproblemsolving.Understandtheconceptofcontinuityoffunctions;understandcontinuityofelementaryfunctionsandthepropertiesofcontinuousfunctionsonclosedinterval.DifferentiationofSingleVariableFunctionshours:12Content:DefinitionofderivativesDerivativesofelementaryfunctionsDifferentiationApplicationsofdifferentiationRequirement:Understandtheconceptofderivatives;understandthegeometricmeaningofderivativesandtherelationofdifferentiationandcontinuityofafunction.Masterthelawsofderivatives,chainrule,andimplicitderivatives;befamiliarwithbasicderivativerulesofelementaryfunctions;understandtheconceptofhigher-orderderivatives.Comprehenddifferentiation;solvetheproblemsofdifferentiationoffunctions.ComprehendLagrange’stheorem;applyL’Ho?pital’sRuletoevaluatethelimit;masterallkindsofapplicationsofderivatives;beabletousederivativesinsolvingordinaryproblems.IntegrationofSingleVariableFunctionshours:14Content:IndefiniteintegralDefiniteintegralApplicationsofdefiniteintegralRequirement:Understandtheconceptofantiderivativesandindefiniteintegral;masterthepropertiesofindefiniteintegralandbasicintegrationformulas;mastersolvingtechniquesincludingsubstuitionandintegrationbyparts.Understandtheconceptandpropertiesofdefiniteintegral;befamiliarwiththeupperlimitofintegrationandNewton-Leibniztheorem;masterthemethodofcalculatingdefiniteintegral.Applydefiniteintegraltocomputearea,volumeofrevolution,averageofafunctionandotherproblems.Remark:chooseoneofsection4)and5)belowtostudybasedonmajorrequirements.OrdinaryDifferentialEquationshours:14Content:BasicconceptofdifferentialequationsFirst-orderdifferentialequationssecond-orderdifferentialequationsApplicationsofordinarydifferentialequationsRequirement:Understandthebasicconceptofdifferentialequations.Masterthemethodofsolvingdifferentialequationsofseparablevariablesandfirst-orderlineardifferentialequations.Ablitytosolvethreetypesofhigher-orderdifferentialequations,andsecond-orderlineardifferentialequationswithconstantcoefficients.Understandthesimpleapplicationsofdifferentialequations,beabletoapplydifferentialequationstosomesimpleproblems.IntroductiontoLinearAlgebrahours:14Content:MatricesandtheirsolvingtechinquesElementarytransformationofmatricesandeliminationmethodsoflinearequationsystemIntroductiontodeterminantsRequirement:Understandtheconceptofmatric