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第一章实数集与函数Chapter1SetofRealNumbersandFunctions实数及其性质realnumbeanditsproperties有理数rationalnumber无理数irrationalnumber定义definition命题proposition加plus减minus乘multipliedby/times除over/isto/dividedby绝对值与不等式absolutevalueandinequality三角不等式triangleinequality反三角不等式inversetriangleinequality伯努利不等式Bernoulliinequality确界原理principlesofsupremumandinfimum开区间closedinterval闭区间closedinterval半开区间semi-openinterval半闭区间semi-closedinterval有限区间finiteinterval无限区间infiniteinterval邻域neighborhood去心邻域deletedneighborhood和sum差difference积product商quotient数轴numberaxis/numberline封闭性closeness阿基米德性Archimedeanproperty稠密性density上界与下届upperandlowerbounds有界集boundedset无界集unboundedset存在域existencedomain上确界supremum下确界infimum有序完备集order-completeset实数的完备性completenessofrealnumbers全序域completeorderedfield完备性公理axiomofcompleteness戴德金分割Dedekindcut戴德金性质Dedekindproperty常量与变量constantandvariablequantities函数的定义definitionoffunction定义域domain值域range自变量independentvariable因变量dependentvariable中间变量intermediatevariable单调性monotonicity初等函数elementaryfunction常量函数constantfunction幂函数powerfunction指数函数exponentialfunction对数函数logarithmicfunction三角函数trigonometricfunction反三角函数inversetrigonometricfunction反函数inversefunction复合函数compoundfunction映射mapping逆映射inversemapping像image原像primaryimage分段函数piecewisefunction符号函数signfunction狄利克雷函数Dirichletfunction黎曼函数Riemannfunction有界函数boundedfunction单调函数monotonefunction单调增函数monotoneincreasingfunction严格单调函数strictlymonotonefunction奇（偶）函数odd(even)function最小正周期minimalpositiveperiod绝对值函数absolutevaluefunction恒等函数identityfunction多项式函数polynomialfunction线性函数linearfunction二次函数quadraticfunction有理函数rationalfunction数学分析中的英文单词和短语双曲正弦hyperbolicsine双曲余弦hyperboliccosine三角恒等式trigonometricidentity奇偶恒等式odd-evenidentity余函数恒等式cofunctionidentity毕达哥拉斯恒等式Pythagoreanidentity半角恒等式half-angleidentity积恒等式productidentity和恒等式sumidentity加法恒等式additionidentity倍角恒等式double-angleidentity第二章数列极限Chapter2LimitsofSequences数列极限limitofsequence发散数列divergentsequence无穷小数列infinitesimalsequence收敛数列convergentsequence唯一性定理uniquenesstheorem有界性定理boundednesstheorem保序性inheritingorderproperties保不等式性inheritinginequality子列subsequence严格递增strictlyincreasing单调递增数列monotoneincreasingsequence单调递减数列monotonedecreasingsequence严格递增strictlydecreasing必要条件necessarycondition充分条件sufficientcondition夹逼定理squeezeprinciple柯西收敛准则Cauchyconvergencecritterion第三章函数极限Chapter3LimitsofFunctions函数极限limitoffunction无穷极限infinitelimit单侧极限one-sidedlimit右（左）极限right(left)limit/right(left)handlimit函数极限的性质propertyoflimitoffunction局部有界性localboundedness海涅定理Heinetheorem无穷大量infinity无穷小量的阶orderofinfinitesimal高（低）阶无穷小量infinitesimalofhigher(lower)order同阶无穷小量infinitesimalofsameorder等价无穷小量equivalentinfinitesimalk-阶无穷小量k-orderinfinitesimal垂直渐近线verticalasymptote斜渐近线obliqueasymptote水平渐近线horizontalasymptote第四章函数的连续性Chapter4ContinuityofFunctions自变量的增量incrementofindependentvariable函数的增量incrementoffunction右（左）连续right(left)continuous间断点及其分类discontinuitypointanditsclassification可去间断点removablediscontinuity跳跃间断点jumpdiscontinuity第一类间断点discontinuityofthefirstkind第二类间断点discontinuityofthesecondkind连续函数的局部性质localpropertiesofcontinuousfunction连续函数的复合性质compositionpropertiesofcontinuousfunction闭区间上连续函数的性质propertiesofcontinuousfunctionoverclosedinterval极值定理extremevaluetheorem最大值和最小值定理maximumandminimumvaluetheorem介值性定理intermediatevaluetheorem零点定理zero-pointtheorem一致连续性定理uniformcontinuitytheorem反函数的连续性continuityofinversefunction局部保号性localinheritingorderproperty初等函数的连续性continuityofelementaryfunction第五章导数和微分Chapter5DerivativesandDifferentials有限增量公式finiteincrementformula变化率rateofchange差商differencequotient左（右）导数left(right)derivative导函数derivativefunction可导函数derivablefunction导数的集合意义geometricmeaningofderivative费马定理Fermattheorem达布定理Darbouxtheorem导函数的介值定理intermediatevaluetheoremofderivativefunction导数的四则运算algebraofderivatives和的导数derivativeofsum差的导数derivativeofdifference积的导数derivativeofproduct商的导数derivativeofquotient反函数的导数derivativeofinversefunction最大值maximumvalue最小值minimumvalue复合函数的导数derivativeofcompositefunction对数求导法logarithmicderivative圆的参数方程parametricequationofcircle椭圆的参数方程parametricequationofellipse摆线的参数方程parametricequationofcycloid星形线的参数方程parametricequationofasteroid二阶导数secondderivative三阶导数thirdderivativen阶导数n-thderivative莱布尼茨公式Leibnizformula加速度acceleration物理解释physicalinterpretation微分的概念conceptofdifferential可微函数differentiablefunction线性主部linearprincipalpart自变量的微分differentialofindependentvariable微分的运算法则operationalrulesofdifferential微分形式的不变性invarianceofdifferentialform微分的几何意义geometricmeaningofdifferential高阶微分higher-orderdifferential第六章微分中值定理及其应用Chapter6MeanValueTheoremsofDifferentialsandtheirApplications罗尔中值定理Rollemeanvaluetheorem拉格朗日中值定理Lagrangemeanvaluetheorem柯西中值定理Cauchymeanvaluetheorem泰勒定理Taylortheorem洛必达法则l’Hospittalrule0/0型不定式极限limitofindeterminateformoftype0/0∞/∞型不定式indeterminateformoftype∞/∞其它类型不定式otherindeterminateforms带有皮亚诺型余项的泰勒公式TaylorformulawithPeanoremainder泰勒系数Taylorcoefficient泰勒多项式Taylorpolynomial泰勒公式的余项remainderofTaylorformula带有拉格朗日型余项的泰勒公式TaylorformulawithLagrangeremainder带有拉格朗日型余项的麦克劳林公式MaclaurinformulawithLagrangeremainder函数的极值extremevalueoffunction极值判别testofextremevalue极值的第一充