Increasing function and decreasing function or Monotonicity of the function

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【人口老龄化引起的消费行为变化】例文

人口老龄化引起的消费行为变化随着全球人口的老龄化趋势加速，人们的消费行为也在发生变化。

老年人作为一支重要的消费群体，其消费行为的变化对整个市场的影响越来越大。

本文将探讨人口老龄化引起的消费行为变化，分析其原因和影响，并提出一些应对策略。

一、人口老龄化带来的消费行为变化1.健康和医疗支出的增加随着年龄的增长，人们需要更多的医疗服务和保健产品。

因此，老年人的医疗支出通常比年轻人更高。

根据美国国家卫生统计中心的数据，65岁及以上的老年人每年的医疗支出是45岁到64岁人群的两倍以上。

在中国，随着医疗保障体系的建设，老年人的医疗支出也在增加。

2.住房需求的变化老年人可能更倾向于住在更小的住房中，因为他们需要更少的空间来生活，并更喜欢便于维护的住房。

此外，老年人可能更希望住在社区或设施中，这些社区或设施提供社交机会和支持服务。

在中国，随着老年人口的增加，一些城市开始建设养老社区，以满足老年人的住房需求。

3.旅游和休闲活动的增加许多老年人拥有更多的自由时间和可支配收入，因此可能更倾向于旅游和参加休闲活动。

在中国，随着老年人口的增加，旅游和休闲活动的市场需求也在增加。

据中国国家旅游局的数据，2019年中国国内老年旅游市场规模达到3500亿元。

4.技术使用的挑战老年人可能对新的技术和数字产品感到陌生，因此可能不会像年轻人那样频繁地使用这些产品。

在中国，一些互联网企业开始针对老年人的特点和需求，开发适合他们使用的产品和服务。

5.家庭和家庭服务的需求老年人可能更倾向于需要家庭和家庭服务，例如家政服务、餐饮服务、家庭保健和安全等服务，以提高他们的生活质量。

在中国，一些企业开始开展针对老年人的家庭服务业务，以满足他们的需求。

二、人口老龄化引起的消费行为变化的原因1.收入水平随着年龄的增长，人们的收入通常会减少，这导致老年人在消费方面更加谨慎。

但是，在一些发达国家，老年人的收入水平相对较高，这也影响了他们的消费行为。

2.健康状况老年人的健康状况通常较差，需要更多的医疗和保健服务。

Calculus--Early Transcendental

1.1.1 Function and Graph . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Representation of Functions . . . . . . . . . . . . . . . . . 8
1.1.3 Piecewise Deﬁned Function . . . . . . . . . . . . . . . . . 9
2.8.4 HIGHER DERIVATIVES . . . . . . . . . . . . . . . . . . 32
3 DIFFERENTIATION RULES
33
3.1 DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNC-
TIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.2 ONE-SIDED LIMITS . . . . . . . . . . . . . . . . . . . . 22
2.2.3 INFINITE LIMITS . . . . . . . . . . . . . . . . . . . . . . 22
2.3 CALCULATING LIMITS USING THE LIMIT LAWS . . . . . . 23
1.2.9 TRANSCENDENTAL FUNCTIONS . . . . . . . . . . . . 13
1.3 NEW FUNCTIONS FROM OLD FUNCTIONS . . . . . . . . . 13
1.3.1 TRANSFORMATIONS OF FUNCTIONS . . . . . . . . . 13

微观经济学考试重点分析解析

微观经济学考试重点一、名词解释(24分）1、消费者剩余（consumer surplus）消费者剩余是一种物品的总效用与其市场价值之间的差额，它衡量的是消费者从某一物品的购买中所得到的超过他们为之支付的那部分额外效用。

2、无差异曲线(indifference curve）无差异曲线是用来表示消费者在一定的偏好、一定的技术和一定资源条件下选择商品时，从两种商品的不同数量组合中，获得相同的满足程度的曲线。

3、预算线（budget line）预算线是指在消费者收入和商品价格既定的情况下，消费者的全部收入所能购买到的两种商品不同数量的各种组合。

4、边际产量（marginal product）一种投入的边际产量是指在其他投入保持不变的情况下,由于增加1单位的该投入而多生产出来的产量或产出。

5、报酬递减规律(law of diminishing returns）报酬递减规律是指在其他投入保持不变的情况下，随着某一投入量的增加,所获得的产出增量越来越少.6、规模报酬不变、递增和递减（constant, increasing and decreasing returns to scale）规模报酬不变、递增和递减是指所有的要素按相同比例增加时，产出相同比例、更大比例或更少比例增加的现象。

7、最低成本原则（least—cost rule）要以最低的成本生产既定的产量，厂商应当购买各种投入（例如劳动力和资本）直到花在每一种投入上的每一美元的边际产量相等.8、需求的价格弹性（price elasticity of demand）需求的价格弹性指的是需求量对价格变动反应程度的指标。

弹性系数等于需求量变动百分比除以价格变动的百分比。

9、生产函数（the production function)生产函数是指在既定的工程、技术和知识条件下,一定量的要素投入与所能生产的最大产量间的函数关系.10、完全竞争市场（perfectly competitive markets）完全竞争市场是指竞争充分而不受任何阻碍和干扰的一种市场结构。

微积分词汇中英对照

微积分中英文对照A：:Absolute convergence :绝对收敛Absolute extreme values :绝对极值Absolute maximum and minimum :绝对极大与极小Absolute value :绝对值Absolute value function :绝对值函数Acceleration :加速度Antiderivative :原函数，反导数Approximate integration :近似积分（法）Approximation :逼近法by differentials :用微分逼近linear ：线性逼近法by Simpson's Rule : Simpson 法则逼近法by the Trapezoidal Rule :梯形法则逼近法Arbitrary constant :任意常数Arc length :弧长Area :面积under a curve :曲线下方之面积between curves :曲线间之面积in polar coordinates :极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution :旋转曲面之面积Asymptote :渐近线horizontal :水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed :平均速率Average velocity :平均速度Axes, coordinate :坐标轴Axes of ellipse :椭圆之对称轴B：Binomial series :二项式级数Binomial theorem：二项式定理C：Calculus :微积分differential ：微分学integral :积分学Cartesian coordinates :笛卡儿坐标一般指直角坐标Cartesian coordinates system : 笛卡儿坐标系Cauch's Mean Value Theorem :柯西中值定理Chain Rule :链式法则Circle :圆Circular cylinder ：圆柱体，圆筒Closed interval ：闭区间Coefficient ：系数Composition of function :复合函数Compound interest :复利Concavity :凹性Conchoid :蚌线Conditionally convergent：条件收敛Cone :圆锥Constant function :常数函数Constant of integration :积分常数Continuity :连续性at a point :在一点处之连续性of a function :函数之连续性on an interval :在区间之连续性from the left : 左连续from the right : 右连续Continuous function :连续函数Convergence :收敛interval of ：收敛区间radius of :收敛半径Convergent sequence :收敛数列series :收敛级数Coordinates：坐标Cartesian :笛卡儿坐标cylindrical :柱面坐标polar ：极坐标rectangular :直角坐标spherical ：球面坐标Coordinate axes :坐标轴Coordinate planes :坐标平面Cosine function :余弦函数Critical point : 临界点Cubic function ：三次函数Curve :曲线Cylinder：圆筒，圆柱体，柱面Cylindrical Coordinates :圆柱坐标D：Decreasing function :递减函数Decreasing sequence :递减数列Definite integral :定积分Degree of a polynomial ：多项式之次数Density :密度Derivative :导数of a composite function :复合函数之导数of a constant function :常数函数之导数directional :方向导数domain of ：导数之定义域of exponential function :指数函数之导数higher ：高阶导数partial ：偏导数of a power function :幂函数之导数of a power series :褰级数之导数of a product ：积之导数of a quotient :商之导数as a rate of change :导数当作变化率right-hand :右导数second :二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant :行列式Differentiable function :可导函数Differential ：微分Differential equation :微分方程partial ：偏微分方程Differentiation :求导法implicit :隐求导法partial ：偏微分法term by term :逐项求导法Directional derivatives :方向导数Discontinuity :不连续性Disk method :圆盘法Distance :距离Divergence :发散Domain :定义域Dot product ：点积Double integral ：二重积分change of variable in :二重积分之变数变换in polar coordinates :极坐标二重积分E：Ellipse :椭圆Ellipsoid :椭圆体Epicycloid :外摆线Equation :方程式Even function :偶函数Expected Valued :期望值Exponential Function :指数函数Exponents , laws of :指数率Extreme value :极值Extreme Value Theorem :极值定理F：Factorial ：阶乘First Derivative Test :一阶导数试验法First octant :第一卦限Focus :焦点Fractions :分式Function :函数Fundamental Theorem of Calculus :微积分基本定理G：Geometric series :几何级数Gradient ：梯度Graph :图形Green Formula :格林公式H：Half-angle formulas :半角公式Harmonic series :调和级数Helix ：螺旋线Higher Derivative :高阶导数Higher mathematics 高等数学Horizontal asymptote :水平渐近线Horizontal line :水平线Hyperbola ：双曲线Hyperboloid ：双曲面I：Implicit differentiation :隐求导法Implicit function :隐函数Improper integral :反常积分，广义积分Increasing，Decreasing Test ：递增或递减试验法Increment :增量Increasing Function :增函数Indefinite integral :不定积分Independent variable :自变量Indeterminate from :不定型Inequality :不等式Infinite point :无穷极限点Infinite series :无穷级数Inflection point :反曲点Instantaneous velocity :瞬时速度Integer ：整数Integral :积分Integrand :被积函数Integration :积分Integration by part :分部积分法Intercepts :截距Intermediate value of Theorem :中值定理Interval ：区间Inverse function :反函数Inverse trigonometric function :反三角函数Iterated integral ：逐次积分L：Laplace transform : Laplace 变换Law of sines：正弦定理Law of Cosines :余弦定理Least upper bound :最小上界Left-hand derivative :左导数Left-hand limit : 左极限Lemniscate ：双钮线Length :长度Level curve :等高线L'Hospital's rule：洛必达法则Limacon ：蚶线Limit :极限Linear approximation：线性近似Linear equation :线性方程式Linear function :线性函数Linearity :线性Linearization :线性化Line in the plane : 平面上之直线Line in space :空间之直线Local extreme :局部极值Local maximum and minimum : 局部极大值与极小值Logarithm :对数Logarithmic function :对数函数M：Maximum and minimum values : 极大与极小值Mean Value Theorem :均值定理Multiple integrals :重积分Multiplier :乘子N：Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line :法线Normal vector :法向量Number :数O：Octant ：卦限Odd function :奇函数One-sided limit ：单边极限Open interval : 开区间Optimization problems :最佳化问题Order ：阶Ordinary differential equation :常微分方程Origin :原点Orthogonal ：正交的P：Parabola :抛物线Parabolic cylinder :抛物柱面Paraboloid :抛物面Parallelepiped :平行六面体Parallel lines :平行线Parameter :参数Partial derivative :偏导数Partial differential equation :偏微分方程Partial fractions :部分分式Partial integration :部分积分Partition :分害UPeriod :周期Periodic function :周期函数Perpendicular lines :垂直线Piecewise defined function :分段定义函数Plane ：平面Point of inflection : 反曲点Polar axis :极轴Polar coordinate :极坐标Polar equation :极方程式Pole ：极点Polynomial ：多项式Positive angle :正角Point-slope form :点斜式Power function : 幂函数Product ：积Q：Quadrant :象限Quotient Law of limit ：极限的商定律Quotient Rule :商定律R：Radius of convergence :收敛半径Range of a function :函数的值域Rate of change :变化率Rational function :有理函数Rationalizing substitution :有理代换法Rational number :有理数Real number :实数Rectangular coordinates :直角坐标Rectangular coordinate system :直角坐标系Relative maximum and minimum : 相对极大值与极小值Revenue function :收入函数Revolution , solid of :旋转体Revolution , surface of :旋转曲面Riemann Sum :黎曼和Right-hand derivative :右导数Right-hand limit :右极限Root ：根S：Saddle point ：鞍点Scalar :纯量、标量Secant line :割线Second derivative :二阶导数Second Derivative Test :二阶导数试验法Second partial derivative :二阶偏导数Sector ：扇形Sequence :数列Series ：级数Set ：集合Shell method :剥壳法Sine function :正弦函数Singularity :奇点Slant, Oblique asymptote :斜渐近线Slope ：斜率Slope-intercept equation of a line :直线的斜截式Smooth curve :平滑曲线Smooth surface :平滑曲面Solid of revolution :旋转体Space :空间Speed :速率Spherical coordinates :球面坐标Squeeze Theorem :夹挤定理Step function :阶梯函数Strictly decreasing :严格递减Strictly increasing :严格递增Substitution rule：替代法则Sum ：和Surface :曲面Surface integral :面积分Surface of revolution :旋转曲面Symmetry :对称T：Tangent function :正切函数Tangent line :切线Tangent plane :切平面Tangent vector :切向量Taylor,s formula :泰勒公式Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分V：Value of function :函数值Variable :变量Vector :向量Velocity :速度Vertical asymptote :垂直渐近线Volume :体积X：X-axis : x 轴X -coordinate : x 坐标X -intercept : x 截距Z：Zero vector :函数的零点Zeros of a polynomial :多项式的零点。

Stationarypoints...

Mathematics Learning CentreTheﬁrst derivative and stationary pointsJackie Nicholasc 2004University of SydneyThe ﬁrst derivative and stationary pointsThe derivative dydxof a function y =f (x )tell us a lot about the shape of a curve.In this section we will discuss the concepts of stationary points and increasing and decreasing functions.However,we will limit our discussion to functions y =f (x )which are well behaved.Certain functions cause technical diﬃculties so we will concentrate on those that don’t!The ﬁrst derivativeThe derivative,dydx ,is the slope of the tangent to the curve y =f (x )at the point x .If we know about the derivative we can deduce a lot about the curve itself.Increasing functionsIf dy dx>0for all values of x in an interval I ,then we know that the slope of the tangent to the curve is positive for all values of x in I and so the function y =f (x )is increasing on the interval I .For example,let y =x 3+x ,then dydx=3x 2+1>0for all values of x.That is,the slope of the tangent to the curve is positive for all values of x .So,y =x 3+x is an increasing function for all values of x .The graph of y =x 3+x .We know that the function y =x 2is increasing for x >0.We can work this out from the derivative.If y =x 2thendydx=2x >0for all x >0.That is,the slope of the tangent to the curve is positive for all values of x >0.So,y =x 2is increasing for all x >0.The graph of y =x 2.Decreasing functionsIfdydx<0for all values of x in an interval I ,then we know that the slope of the tangent to the curve is negative for all values of x in I and so the function y =f (x )is decreasing on the interval I .For example,let y =−x 3−x ,then dydx=−3x 2−1<0for all values of x.That is,the slope of the tangent to the curve is negative for all values of x .So,y =−x 3−x is a decreasing function for all values of x .The graph of y =−x 3−x .We know that the function y =x 2is decreasing for x <0.We can work this out from the derivative.If y =x 2thendydx=2x <0for all x <0.That is,the slope of the tangent to the curve is negative for all values of x <0.So,y =x 2is decreasing for all x <0.The graph of y =x 2.Stationary pointsWhendydx=0,the slope of the tangent to the curve is zero and thus horizontal.The curve is said to have a stationary point at a point where dydx=0.There are three types of stationary points.They are relative or local maxima,relative or local minima and horizontal points of inﬂection.Relative or local maxima and minima are so called to indicate that they may be maxima or minima only in their locality.They are also called turning points.Points of inﬂection are deﬁned and discussed later.The nature of stationary pointsThe ﬁrst derivative can be used to determine the nature of the stationary points once wehave found the solutions to dydx =0.Relative maximumConsider the function y =−x 2+1.By diﬀerentiating and setting the derivative equal tozero,dydx=−2x =0when x =0,we know there is a stationary point when x =0.We can use the fact that this is the only stationary point (and that we are only dealing with well behaved functions)to divide the real line into two intervals;x <0and x >0.From the derivative we know that sincedydx=−2x >0when x <0the function is increasing for x <0.(Choose a point which is <0,say x =−1,as a test point.)Similarly,sincedydx=−2x <0when x >0the function is decreasing for x >0.(We could use the point x =1as a test point here.)Of course,at the point x =0itself dydx=0.Putting all this together,we can deduce that the stationary point at x =0is a relative maximum.Sincedy dx >0for x <0,dy dx=0for x =0,anddydx<0for x >0,we have a relative maximum at x =0.The graph of y =−x 2+1.We can sum this information up in a table that allows us to see at a glance that the point (0,1)is a maximum.x <00>0y +ve 0−ve y1Relative minimumConsider the function y =x 2−2x +3.By diﬀerentiating and setting the derivative equalto zero,dydx =2x −2=0when x =1,we know there is a stationary point at x =1.Again,we use the fact that this is the only stationary point to divide the real line into two intervals;x <1and x >1.From the derivative we know that sincedydx=2x −2=2(x −1)<0when x <1the function is decreasing for x <1.(Choose a point which is <1,say x =0,as a test point.)Similarly,sincedydx=2(x −1)>0when x >1the function is increasing for x >1.(We could use the point x =2as a test point here.)Of course,at the point x =1itselfdy dx=0.Therefore,we deduce that the stationary point at x =1is a relative minimum.Sincedy dx <0for x <1,dy dx=0for x =1,anddydx>0for x >1,we have a relative minimum at x =1.The graph of y =x 2−2x +3.Again we can summarise this in a table.x <11>1y −ve 0+ve y1We analyse functions with more than one stationary point in the same way. ExampleConsider y=2x3−3x2−12x+4.Then,dydx=6x2−6x−12=6(x2−x−2)=6(x−2)(x+1).So,dydx=0when x=−1or x=2.Therefore the points(−1,11)and(2,−16)are the only stationary points.As before,we use the stationary points to partition the real line into the following intervals:x<−1,−1<x<2,x>2.We can now choose test points in these intervals,say x=−2,x=0and x=3,to determine the sign of the derivative in these intervals.x<−1−1>−1,<22>2y +ve0−ve0+vey 11 −16Therefore the point(−1,11)is a maximum and the point(2,−16)is a minimum. When x=0,y=4so the point(0,4)lies on the graph of the function.We now have enough information to sketch the graph.The graph of y=2x3−3x2−12x+4.。

函数单调性相关英文文献

Data source: Liu Yulian, Fu Peiren, principles of mathematical analysis (I). Third edition. Higher education press, 1992Zhang Zhimin, Zhang Suliang. Function. Science Press, 1985The monotonicity of the functionThe monotonicity of the monotonicity of the function is also called function. The monotonicity of the function on an interval is concerned, it is a local concept. DefinitionIn general, a function f (x) domain of I:If for any belongs to some interval I on the two variables x1, X2, when x1>x2 is f (x1) ≥ f (x2). Then f (x) in this interval is an increasing function of (another argument is monotone non decreasing function). If f (x1) >f (x2), then f (x) in this interval is strictly increasing function (another story is the increasing function).If for any belongs to some interval I on the two variables x1, X2, when x1>x2 is f (x1) ≤ f (x2) f (x). It is in this interval is a decreasing function of (another term for monotone non increasing function). If f (x1) <f (x2), then f (x) in this interval is strictly decreasing function (another argument is a decreasing function).In order to avoid ambiguity, below the monotone non decreasing functions, strictly increasing function, monotone non increasing function, strictly decreasing function such as terminology.NatureIf the function y=f (x) in a certain interval is the increasing function or decreasing function. Then said the function y=f (x) is in the range of (strict) monotonicity, this interval is called y= f (x) monotone interval, image enhancement functions in monotone interval is rising, image reduction function is declining.Be careful.The monotonicity of the monotonicity of the function is also called function;The monotonicity of the function on an interval is concerned, it is a local concept. ExtensionIn mathematics function in the ordered set is monotone (monotone), if they keep the given order. These functions appear first in calculus and later extended to order theory in more abstract structure. Although the concept is generally consistent, two subjects had developed a slightly different terminology. In calculus, we often say that the function is monotone increasing and decreasing monotone, preference terminology monotonic in order theory and the anti monotone or order preserving and order reversal.DefinitionSetF: P → QIs the function between the two with a partially ordered set P and Q ≤. In calculus, which is a function with the usual order subset of real number set, but still maintain the same order theory definition definition is more general.The function f is monotone, if x ≤ y, f (x) ≤ f (Y). So keep the order relation of monotone function.Monotonicity of calculus and real analysisIn calculus, often do not need to resort to order theory Abstract method. As mentioned above, the function is usually a mapping between a subset of real numbers are sorted by the natural order set..Inspired by the monotone function in real on the shape of the graph of the function, also called a monotone increasing (or "non decreasing"). Similarly, the function is called monotone decreasing (or "non incremental"), if the X < y, f (x) ≥ f (y), it reverses the order of.If the order ≥ definition with strict sequence >, are more stringent requirements. Have the function of such nature is strictly increasing. And by reversing the order of symbols, strictly decreasing can get the corresponding. Function of increasing or decreasing strict one one mapping (because < math> a < b< /math> math> a \neq contains < b< /math>).To avoid the term non decreasing and non increasing confusion in strictly increasing and strictly decreasing.In the theory of monotone sequenceIn order theory, not limited to the set of real number, can consider arbitrary partially ordered sets and even pre ordered set. Can also be used in these cases the above definition. But to avoid the term "progressive" and "decline", because once the treatment is not totally ordered sequence is no image motivation attractive. Further, the strict relation of < and > are rarely used in most non order order, so do not intervene in the additional terms of their.Monotone (monotone) function is also called Isotone or order preserving function. Dual concept often called the anti monotone, antitone or sequential inversion. Therefore, the anti monotone function f satisfies the p roperties of X ≤ y contains f (x) ≥ f (Y),For all x and Y domain in its. Easy to see that the compound two monotone function is also monotone.Constant function is monotonic and anti monotonic; conversely, if the F is monotonic and anti monotonic, and if the domain f is a lattice, then f must be a constant function.Monotone function is central to the theory of order. They appear in large numbers in the theme of the article and found in these places in. Monotone function is famous order embedding (x ≤ y if and only if f (x) ≤ f (y) function) and order isomorphism (bijective order embedding).Function of interval editorFeatures(1) the geometric characteristics of the monotonicity of the function: the monotoneinterval function, image enhancement is rising, image subtraction function is declining."When x1 <x2, a f (x1) <f (x2)" is equivalent to y increases with X increasing; "When x1 <x2, a f (x1) >f (x2)" is equivalent to y decreases with the increase of X and.A geometric interpretation: the increment is equivalent to the function of the image from left to right gradually diminishing; equivalent to the function of the image from left to right gradually declined.(2) monotone function is directed at a certain interval, is a local property.Some function is monotonic in the whole domain; part interval in the domain of some function is increasing function, in part on the interval is a decreasing function; some function is a non monotonic function, such as the constant function. Monotonicity of function is a function in a monotone interval on the "whole" in nature, is arbitrary, cannot use the special value instead of.Note: the following properties in the monotonicity.1.f (x) and f (x) +a has the same monotonicity;2.f (x) and a*f (x) have the same monotonicity was in a>0, when a<0, having opposite monotonicity;3 when f (x), G (x) is increasing (decreasing) function, if f (x) *g (x) is a constant greater than zero, are the same as for increasing (decreasing) function; if both constant is less than zero, it is decreasing (increasing) function.Operational properties1 two increase in function and is still increasing function;2 minus the reduction function of increasing function for increasing function;3 two reduction function and is still decreasing function;4 minus function minus increasing function as a decreasing function;In addition to:Function value in the interval number with increasing (decreasing) function, the reciprocal is reducing (increasing) function.Judging method of editingI mage observationThe definition of proofUsing the monotonicity of the function definition that steps:The arbitrary value: let x1, X2 be any in the range of two values, and the x1<x2;The difference of deformation: as f (x2) -f (x1), and factorization, formula, rational methods such as differential to help determine the sign of the difference in thedirection of deformation;The judge set number: F (x2) -f (x1) symbols;The conclusion is: make a conclusion according to the definition (if the difference of >0, is the increasing function; if the difference of <0, is a decreasing function) "Any value -- difference deformation -- judgment -- conclusion no.".Derivative methodThe derivation using derivative formula, and then determine the guiding function and 0 size relations, so as to judge monotonicity, guide function value is greater than 0, that is strictly increasing function, guiding function value is less than 0, that is strictly decreasing function, the premise is the original function must be continuous. Discriminant method monotone pointTheorem: if f (x) n derivative at a certain point, and the first derivative and N-1 derivative which is equal to 0 and the N derivative is not equal to 0 f (x) in the necessary and sufficient conditions for the monotonicity of n is odd, and when the N derivative is greater than 0, f (x) at the point of a strictly increasing and strictly decreasing.Proof: by the extreme second discriminant method of proof and can be launched. Note: this theorem is only f (x) discriminant method in the monotonicity of the point, and can not be monotonic in the point of a field.Discriminant method monotone intervalTheorem two: (necessary and sufficient conditions for monotonicity) if f (x) in the (a, b) can be the guide, then f (x) in the (a, b) is monotone increasing (or monotone decreasing) is necessary in the (a, b) a derivative is greater than or equal to 0 (or the first the derivative is less than or equal to 0).Theorem three: (sufficient condition of strictly monotone) if the interval (a, b) is a derivative of greater than 0 (or a derivative of less than 0), f (x) in the (a, b) is strictly increasing (or strictly decreasing).Note: by theorem three, if f (x) in the (a, b) memory in the continuous derivative, f (x) (a, b) will be in the memory in the monotone interval. In fact, say there exists a point belongs to (a, b), the first derivative is greater than 0 (0 or less), continuous known by the first derivative, there must be a certain field of this point is contained in the (a, b), in this field, the first derivative is greater than 0 (or less than 0), and f (x) in this field, increasing (or decreasing).Theorem four: (necessary and sufficient condition of strictly monotone) if f (x) in the (a, b) can be the guide, then f (x) in the (a, b) is strictly increasing (or strictly decreasing) necessary and sufficient condition is: when the X is (a, b), the two order derivative x greater than or equal to 0 (two order derivative or X is less than or equal to 0) and (a, b) a derivative in any subinterval x not equal to 0.Composite function editorIn the function y=f[g (x) domain] in, let u=g (x), y=f[g (x)] monotonicity by u=g (x) andy=f (U) monotonicity jointly determined, as followsU=g (x) y=f (U) y=f[g (x)]Increasing function of increasing function of increasing functionReducing function of decreasing function of increasing functionIncreasing function of decreasing function of decreasing functionReducing function of increasing function and decreasing functionTherefore, the composite monotonicity of functions available "with increment reduction" to judge, but to consider the domain of some special functions.Note: y=f (x) +g (x) does not belong to the scope of composite function, so this method is not.Note editorIn the use of derivatives are monotone interval function, first to determine the domain of the function, the process of solving the problem only in the domain, the derivative of the symbol to judge the monotone interval function.If a function has more than one monotone interval the same monotonicity, then these monotone interval can not use the "U" connection, and can only use the "comma" or "and" word spaced.Some applications of monotone functionDetermination of concave and convex functionsTheorem five: Let f (x) in the (a, b) can be the guide, then f (x) in the (a, b) on the inside concave (or convex) if and only if x derivative in (a, b) is monotone increasing (or monotone decreasing);The existence of inverse function of decision functionTheorem six: (sufficient condition of inverse function exists) if f (x) in the number set A strictly monotone, then f (x) has inverse function.Theorem seven: (necessary and sufficient conditions of inverse function exists) if f (x) in [a, b] is continuous, necessary and sufficient conditions of inverse function exists is f (x) is strictly monotone function.The proof of inequalityThe proof of the monotonicity inequality is a common and important method. By examples prove.SummaryBy definition, the monotone function promotion, discriminant method and application aspects of the understanding, the monotone function has a deep understanding, a good foundation for the next paper on two monotone function product monotonous writing, also hope to the future foreshadowed in the research field of number.文献来源：刘玉涟，傅沛仁，数学分析讲义(上册).第三版.高等教育出版社，1992章志敏，张素亮.函数.科学出版社，1985函数的单调性函数的单调性也叫函数的增减性。

高等数学-微积分第1章(英文讲稿)

C alc u lus (Fifth Edition)高等数学- Calculus微积分（双语讲稿）Chapter 1 Functions and Models1.1 Four ways to represent a function1.1.1 ☆Definition(1-1) function: A function f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B. see Fig.2 and Fig.3Conceptions: domain; range (See fig. 6 p13); independent variable; dependent variable. Four possible ways to represent a function: 1)Verbally语言描述(by a description in words); 2) Numerically数据表述(by a table of values); 3) Visually 视觉图形描述(by a graph);4)Algebraically 代数描述(by an explicit formula).1.1.2 A question about a Curve represent a function and can’t represent a functionThe way ( The vertical line test ) : A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. See Fig.17 p 171.1.3 ☆Piecewise defined functions (分段定义的函数)Example7 (P18)1－x if x ≤1f(x)＝﹛x2if x＞1Evaluate f(0)，f(1)，f(2) and sketch the graph.Solution：1.1.4 About absolute value (分段定义的函数)⑴∣x∣≥0；⑵∣x∣≤0Example8 (P19)Sketch the graph of the absolute value function f(x)＝∣x∣.Solution：1.1.5☆☆Symmetry ，(对称) Even functions and Odd functions (偶函数和奇函数)⑴Symmetry See Fig.23 and Fig.24⑵①Even functions: If a function f satisfies f(－x)＝f(x) for every number x in its domain，then f is call an even function. Example f(x)＝x2 is even function because: f(－x)＝ (－x)2＝x2＝f(x)②Odd functions: If a function f satisfies f(－x)＝－f(x) for every number x in its domain，thenf is call an odd function. Example f(x)＝x3 is even function because: f(－x)＝(－x)3＝－x3＝－f(x)③Neither even nor odd functions:1.1.6☆☆Increasing and decreasing function (增函数和减函数)⑴Definition(1-2) increasing and decreasing function:① A function f is called increasing on an interval I if f(x1)＜f(x2) whenever x1＜x2 in I. ①A function f is called decreasing on an interval I if f(x1)＞f(x2) whenever x1＜x2 in I.See Fig.26. and Fig.27. p211.2 Mathematical models: a catalog of essential functions p251.2.1 A mathematical model p25A mathematical model is a mathematical description of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reduction.1.2.2 Linear models and Linear function P261.2.3 Polynomial P27A function f is called a polynomial ifP(x) ＝a n x n＋a n－1x n－1＋…＋a2x2＋a1x＋a0Where n is a nonnegative integer and the numbers a0，a1，a2，…，a n－1，a n are constants called the coefficients of the polynomial. The domain of any polynomial is R＝(－∞,＋∞).if the leading coefficient a n≠0, then the degree of the polynomial is n. For example, the function P(x) ＝5x6＋2x5－x4＋3x－9⑴Quadratic function example: P(x) ＝5x2＋2x－3 二次函数（方程）⑵Cubic function example: P(x) ＝6x3＋3x2－1 三次函数（方程）1.2.4Power functions幂函数P30A function of the form f(x) ＝x a，Where a is a constant, is called a power function. We consider several cases：⑴a＝n where n is a positive integer ，(n＝1，2，3，…，)⑵a＝1/n where n is a positive integer，(n＝1，2，3，…，) The function f(x) ＝x1/n⑶a＝n－1 the graph of the reciprocal function f(x) ＝x－1 反比函数1.2.5Rational function有理函数P 32A rational function f is a ratio of two polynomials:f(x)＝P(x) /Q(x)1.2.6Algebraic function代数函数P32A function f is called algebraic function if it can be constructed using algebraic operations ( such as addition，subtraction，multiplication，division，and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. Examples: P 321.2.7Trigonometric functions 三角函数P33⑴f(x)＝sin x⑵f(x)＝cos x⑶f(x)＝tan x＝sin x / cos x1.2.8Exponential function 指数函数P34The exponential functions are the functions the form f(x) ＝a x Where the base a is a positive constant.1.2.9Transcendental functions 超越函数P35These are functions that are not a algebraic. The set of transcendental functions includes the trigonometric，inverse trigonometric，exponential，and logarithmic functions，but it also includes a vast number of other functions that have never been named. In Chapter 11 we will study transcendental functions that are defined as sums of infinite series.1.2 Exercises P 35-381.3 New functions from old functions1.3.1 Transformations of functions P38⑴Vertical and Horizontal shifts (See Fig.1 p39)①y＝f(x)＋c，（c＞0）shift the graph of y＝f(x) a distance c units upward.②y＝f(x)－c，（c＞0）shift the graph of y＝f(x) a distance c units downward.③y＝f(x＋c)，（c＞0）shift the graph of y＝f(x) a distance c units to the left.④y＝f(x－c)，（c＞0）shift the graph of y＝f(x) a distance c units to the right.⑵ V ertical and Horizontal Stretching and Reflecting (See Fig.2 p39)①y＝c f(x)，（c＞1）stretch the graph of y＝f(x) vertically by a factor of c②y＝(1/c) f(x)，（c＞1）compress the graph of y＝f(x) vertically by a factor of c③y＝f(x/c)，（c＞1）stretch the graph of y＝f(x) horizontally by a factor of c.④y＝f(c x)，（c＞1）compress the graph of y＝f(x) horizontally by a factor of c.⑤y＝－f(x)，reflect the graph of y＝f(x) about the x-axis⑥y＝f(－x)，reflect the graph of y＝f(x) about the y-axisExamples1: (See Fig.3 p39)y＝f( x) ＝cos x，y＝f( x) ＝2cos x，y＝f( x) ＝（1/2）cos x，y＝f( x) ＝cos（x/2），y＝f( x) ＝cos2xExamples2: (See Fig.4 p40)Given the graph y＝f( x) ＝( x)1/2，use transformations to graph y＝f( x) ＝( x)1/2－2，y＝f( x) ＝(x－2)1/2，y＝f( x) ＝－( x)1/2，y＝f( x) ＝2 ( x)1/2，y＝f( x) ＝(－x)1/21.3.2 Combinations of functions (代数组合函数)P42Algebra of functions: Two functions (or more) f and g through the way such as add, subtract, multiply and divide to combined a new function called Combination of function.☆Definition(1-2) Combination function: Let f and g be functions with domains A and B. The functions f±g，f g and f /g are defined as follows: （特别注意符号(f±g)( x) 定义的含义）①(f±g)( x)＝f(x)±g( x)，domain ＝A∩B②(f g)( x)＝f(x) g( x)，domain ＝A∩ B③(f /g)( x)＝f(x) /g( x)，domain ＝A∩ B and g( x)≠0Example 6 If f( x) ＝( x)1/2，and g( x)＝(4－x2)1/2，find functions y＝f(x)＋g( x)，y＝f(x)－g( x)，y＝f(x)g( x)，and y＝f(x) /g( x)Solution: The domain of f( x) ＝( x)1/2 is [0，＋∞)，The domain of g( x) ＝(4－x2)1/2 is interval [－2，2]，The intersection of the domains of f(x) and g( x) is[0，＋∞)∩[－2，2]＝[0，2]Thus，according to the definitions, we have(f＋g)( x)＝( x)1/2＋(4－x2)1/2，domain [0，2](f－g)( x)＝( x)1/2－(4－x2)1/2，domain [0，2](f g)( x)＝f(x) g( x) ＝( x)1/2(4－x2)1/2＝(4 x－x3)1/2domain [0，2](f /g)( x)＝f(x)/g( x)＝( x)1/2/(4－x2)1/2＝[ x/(4－x2)]1/2 domain [0，2)1.3.3☆☆Composition of functions (复合函数)P45☆Definition(1-3) Composition function: Given two functions f and g the composite function f⊙g (also called the composition of f and g ) is defined by(f⊙g)( x)＝f( g( x)) （特别注意符号(f⊙g)( x) 定义的含义）The domain of f⊙g is the set of all x in the domain of g such that g(x) is in the domain of f . In other words, (f⊙g)(x) is defined whenever both g(x) and f (g (x)) are defined. See Fig.13 p 44 Example7 If f (g)＝( g)1/2 and g(x)＝(4－x3)1/2find composite functions f⊙g and g⊙f Solution We have(f⊙g)(x)＝f (g (x) ) ＝( g)1/2＝((4－x3)1/2)1/2(g⊙f)(x)＝g (f (x) )＝(4－x3)1/2＝[4－((x)1/2)3]1/2＝[4－(x)3/2]1/2Example8 If f (x)＝( x)1/2 and g(x)＝(2－x)1/2find composite function s①f⊙g ②g⊙f ③f⊙f④g⊙gSolution We have①f⊙g＝(f⊙g)(x)＝f (g (x) )＝f((2－x)1/2)＝((2－x)1/2)1/2＝(2－x)1/4The domain of (f⊙g)(x) is 2－x≥0 that is x ≤2 {x ︳x ≤2 }＝(－∞，2]②g⊙f＝(g⊙f)(x)＝g (f (x) )＝g (( x)1/2 )＝(2－( x)1/2)1/2The domain of (g⊙f)(x) is x≥0 and 2－( x)1/2x ≥0 ，that is ( x)1/2≤2 ，or x ≤ 4 ，so the domain of g⊙f is the closed interval[0，4]③f⊙f＝(f⊙f)(x)＝f (f(x) )＝f((x)1/2)＝((x)1/2)1/2＝(x)1/4The domain of (f⊙f)(x) is [0，∞)④g⊙g＝(g⊙g)(x)＝g (g(x) )＝g ((2－x)1/2 )＝(2－(2－x)1/2)1/2The domain of (g⊙g)(x) is x－2≥0 and 2－(2－x)1/2≥0 ，that is x ≤2 and x ≥－2，so the domain of g⊙g is the closed interval[－2，2]Notice: g⊙f⊙h＝f (g(h(x)))Example9Example10 Given F (x)＝cos2( x＋9)，find functions f，g，and h such that F (x)＝f⊙g⊙h Solution Since F (x)＝[cos ( x＋9)] 2，that is h (x)＝x＋9 g(x)＝cos x f (x)＝x2Exercise P 45-481.4 Graphing calculators and computers P481.5 Exponential functions⑴An exponential function is a function of the formf (x)＝a x See Fig.3 P56 and Fig.4Exponential functions increasing and decreasing (单调性讨论)⑵Lows of exponents If a and b are positive numbers and x and y are any real numbers. Then1) a x+y＝a x a y2) a x－y＝a x / a y3) (a x)y＝a xy4) (ab)x+y＝a x b x⑶about the number e f (x)＝e x See Fig. 14，15 P61Some of the formulas of calculus will be greatly simplified if we choose the base a .Exercises P 62-631.6 Inverse functions and logarithms1.6.1 Definition(1-4) one-to-one function: A function f is called a one-to-one function if it never takes on the same value twice；that is，f (x1)≠f (x2)，whenever x1≠x2( 注解：不同的自变量一定有不同的函数值，不同的自变量有相同的函数值则不是一一对应函数) Example: f (x)＝x3is one-to-one function.f (x)＝x2 is not one-to-one function, See Fig.2,3,4☆☆Definition(1-5) Inverse function:Let f be a one-to-one function with domain A and range B. Then its inverse function f－1(y)has domain B and range A and is defined byf－1(y)＝x f (x)＝y for any y in Bdomain of f－1＝range of frange of f－1＝domain of f( 注解：it says : if f maps x into y, then f－1maps y back into x . Caution: If f were not one-to-one function，then f－1 would not be uniquely defined. )Caution: Do not mistake the－1 in f－1for an exponent. Thus f－1(x)＝1/ f(x) ！！！Because the letter x is traditionally used as the independent variable, so when we concentrate on f－1(y) rather than on f－1(y), we usually reverse the roles of x and y in Definition (1-5) and write as f－1(x)＝y f (x)＝yWe get the following cancellation equations:f－1( f(x))＝x for every x in Af (f－1(x))＝x for every x in B See Fig.7 P66Example 4 Find the inverse function of f(x)＝x3＋6Solution We first writef(x)＝y＝x3＋6Then we solve this equation for x:x3＝y－6x＝(y－6)1/3Finally, we interchange x and y：y＝(x－6)1/3That is, the inverse function is f－1(x)＝(x－6)1/3( 注解：The graph of f－1 is obtained by reflecting the graph of f about the line y＝x. ) See Fig.9、8 1.6.2 Logarithmic functionIf a＞0 and a≠1，the exponential function f (x)＝a x is either increasing or decreasing and so it is one-to-one function by the Horizontal Line Test. It therefore has an inverse function f－1，which is called the logarithmic function with base a and is denoted log a，If we use the formulation of an inverse function given by (See Fig.3 P56)f－1(x)＝y f (x)＝yThen we havelogx＝y a y＝xThe logarithmic function log a x＝y has domain (0,∞) and range R.Usefully equations:①log a(a x)＝x for every x∈R②a log ax＝x for every x＞01.6.3 ☆Lows of logarithms :If x and y are positive numbers, then①log a(xy)＝log a x＋log a y②log a(x/y)＝log a x－log a y③log a(x)r＝r log a x where r is any real number1.6.4 Natural logarithmsNatural logarithm isl og e x＝ln x ＝ythat is①ln x ＝y e y＝x② ln(e x)＝x x∈R③e ln x＝x x＞0 ln e＝1Example 8 Solve the equation e5－3x＝10Solution We take natural logarithms of both sides of the equation and use ②、③ln (e5－3x)＝ln10∴5－3x＝ln10x＝(5－ln10)/3Example 9 Express ln a＋(ln b)/2 as a single logarithm.Solution Using laws of logarithms we have:ln a＋(ln b)/2＝ln a＋ln b1/2＝ln(ab1/2)1.6.5 ☆Change of Base formula For any positive number a (a≠1), we havel og a x＝ln x/ ln a1.6.6 Inverse trigonometric functions⑴Inverse sine function or Arcsine functionsin－1x＝y sin y＝x and －π/2≤y≤π / 2，－1≤x≤1 See Fig.18、20 P72Example13 ① sin－1 (1/2) or arcsin(1/2) ② tan(arcsin1/3)Solution①∵sin (π/6)＝1/2，π/6 lies between －π/2 and π / 2，∴sin－1 (1/2)＝π/6② Let θ＝arcsin1/3，so sinθ＝1/3tan(arcsin1/3)＝tanθ＝ s inθ/cosθ＝ (1/3)/(1－s in2θ)1/2＝1/(8)1/2Usefully equations:①sin－1(sin x)＝x for －π/2≤x≤π / 2②sin (sin－1x)＝x for －1≤x≤1⑵Inverse cosine function or Arccosine functioncos－1x＝y cos y＝x and 0 ≤y≤π，－1≤x≤1 See Fig.21、22 P73Usefully equations:①cos－1(cos x)＝x for 0 ≤x≤π②cos (cos－1x)＝x for －1≤x≤1⑶Inverse Tangent function or Arctangent functiontan－1x＝y tan y＝x and －π/2＜y＜π / 2 ，x∈R See Fig.23 P73、Fig.25 P74Example 14 Simplify the expression cos(ta n－1x).Solution 1 Let y＝tan－1 x，Then tan y＝x and －π/2＜y＜π / 2 ，We want find cos y but since tan y is known, it is easier to find sec y first:sec2y＝1 ＋tan2y sec y＝(1 ＋x2 )1/2∴cos(ta n－1x)＝cos y ＝1/ sec y＝(1 ＋x2)－1/2Solution 2∵cos(ta n－1x)＝cos y∴cos(ta n－1x)＝(1 ＋x2)－1/2⑷Other Inverse trigonometric functionscsc－1x＝y∣x∣≥1csc y＝x and y∈(0，π / 2]∪(π，3π / 2]sec－1x＝y∣x∣≥1sec y＝x and y∈[0，π / 2)∪[π，3π / 2]cot－1x＝y x∈R cot y＝x and y∈(0，π)Exercises P 74-85Key words and PhrasesCalculus 微积分学Set 集合Variable 变量Domain 定义域Range 值域Arbitrary number 独立变量Independent variable 自变量Dependent variable 因变量Square root 平方根Curve 曲线Interval 区间Interval notation 区间符号Closed interval 闭区间Opened interval 开区间Absolute 绝对值Absolute value 绝对值Symmetry 对称性Represent of a function 函数的表述（描述）Even function 偶函数Odd function 奇函数Increasing Function 增函数Increasing Function 减函数Empirical model 经验模型Essential Function 基本函数Linear function 线性函数Polynomial function 多项式函数Coefficient 系数Degree 阶Quadratic function 二次函数（方程）Cubic function 三次函数（方程）Power functions 幂函数Reciprocal function 反比函数Rational function 有理函数Algebra 代数Algebraic function 代数函数Integer 整数Root function 根式函数（方程）Trigonometric function 三角函数Exponential function 指数函数Inverse function 反函数Logarithm function 对数函数Inverse trigonometric function 反三角函数Natural logarithm function 自然对数函数Chang of base of formula 换底公式Transcendental function 超越函数Transformations of functions 函数的变换Vertical shifts 垂直平移Horizontal shifts 水平平移Stretch 伸张Reflect 反演Combinations of functions 函数的组合Composition of functions 函数的复合Composition function 复合函数Intersection 交集Quotient 商Arithmetic 算数。

微积分英文词汇高数名词中英文对照高等数学术语英语翻

微积分英文词汇,高数名词中英文对照,高等数学术语英语翻译一览V、X、Z:Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分C:Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数。

高等数学英汉对照术语表(打印版)

高等数学英汉对照术语表A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法Approximation by differentials ：用微分逼近Approximation by Simpson’s Rule ：Simpson法则逼近法Approximation by the Trapezoidal Rule ：梯形法则逼近法Approximation linear ：线性逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积Area between curves ：曲线间之面积Area in polar coordinates ：极坐标表示之面积Area of a sector of a circle ：扇形之面积Area of a surface of a revolution ：旋转曲面之面积Area under a curve ：曲线下方之面积Asymptote ：渐近线Asymptote horizontal ：水平渐近线Asymptote slant ：斜渐近线Asymptote vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes of ellipse ：椭圆之轴Axes, coordinate ：坐标轴Binomial series ：二项级数C:Calculus ：微积分Calculus differential ：微分学Calculus integral ：积分学Cartesian ：笛卡儿坐标Cartesian coordinates ：笛卡儿坐标，一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cartesian cylindrical ：柱面坐标Cartesian polar ：极坐标Cartesian rectangular ：直角坐标Cartesian spherical ：球面坐标Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性Continuity at a point ：在一点处之连续性Continuity from the left ：左连续Continuity from the right ：右连续Continuity of a function ：函数之连续性Continuity on an interval ：在区间之连续性Continuous function ：连续函数Convergence ：收敛Convergence interval of ：收敛区间Convergence radius of ：收敛半径Convergent sequence ：收敛数列Convergent sequence series ：收敛级数Coordinate axes ：坐标轴Coordinate planes ：坐标平面Coordinate：s：坐标Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数Derivative as a rate of change ：导数当作变率Derivative as the slope of a tangent ：导数看成切线之斜率Derivative directional ：方向导数Derivative domain of ：导数之定义域Derivative higher ：高阶导数Derivative of a composite function ：复合函数之导数Derivative of a constant function ：常数函数之导数Derivative of a power function ：幂函数之导数Derivative of a power series ：羃级数之导数Derivative of a product ：积之导数Derivative of a quotient ：商之导数Derivative of exponential function ：指数函数之导数Derivative partial ：偏导数Derivative right-hand ：右导数Derivative second ：二阶导数Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程Differential equation partial ：偏微分方程Differentiation ：求导法Differentiation implicit ：隐求导法Differentiation implicit partial ：偏微分法Differentiation implicit term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分Double integral change of variable in ：二重积分之变数变换Double integral in polar coordinates ：极坐标二重积分E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式H:Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Higher mathematics 高等数学/高数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyper boloid ：双曲面Hyperbola ：双曲线I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing Function ：增函数Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Line in space ：空间之直线Line in the plane ：平面上之直线Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallel lines ：并行线Parallelepiped ：平行六面体Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Point-slope form ：点斜式Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rational number ：有理数Rationalizing substitution ：有理代换法Real number ：实数Rectangular coordinate system ：直角坐标系Rectangular coordinates ：直角坐标Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分V、X、Z:Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点。

2023十二月份河南英语四级试卷

2023十二月份河南英语四级试卷December 2023 Henan English CET-4 ExamPart I Listening Comprehension (30 marks)Section A (10 marks)Directions: In this section, you will hear 10 short conversations. At the end of each conversation, a question will be asked about what was said. The conversations and the questions will be spoken only once. After you hear a conversation and the question about it, read the four possible answers on your paper, and decide which one is the best answer to the question you have heard.1. A. Teammates. B. Classmates. C. Colleagues. D. Family members.2. A. Washing dishes. B. Cooking. C. Cleaning the room. D. Doing laundry.3. A. A camera. B. A watch. C. A book. D. A mobile phone.4. A. It’s on the bookshelf. B. It’s in the drawer. C. It’s on the table. D. It’s under the bed.5. A. Sunny. B. Cloudy. C. Rainy. D. Snowy.6. A. Friday. B. Saturday. C. Sunday. D. Thursday.7. A. At a restaurant. B. At a library. C. At a cinema. D. At a park.8. A. Three. B. Four. C. Five. D. Six.9. A. A teacher and a student. B. A boss and an employee. C.A doctor and a patient. D. A salesman and a customer.10. A. By car. B. By subway C. By bus. D. By bike.Section B (10 marks)Directions: In this section, you will hear 4 passages. At the end of each passage, you will hear some questions. Both the passage and the questions will be spoken only once. After you hear a question, read the four possible answers on your paper, and decide which one is the best answer to the question you have heard.Passage oneQuestions:11. A. Traveling around the world. B. Hosting a talk show. C. Teaching. D. Being a lawyer.12. A. She was interested in sports. B. She loved talking to strangers. C. She was good at negotiations. D. Her father was a lawyer.13. A. In a bookstore. B. On TV. C. At school. D. At home.Passage twoQuestions:14. A. Three days. B. Four days. C. Five days. D. Six days.15. A. A sunny day. B. A snowy day. C. A rainy day. D. A windy day.16. A. By car. B. By bus. C. By bicycle. D. On foot.Passage threeQuestions:17. A. They use sign language to communicate. B. They use written language to communicate. C. They use spoken language to communicate. D. They don’t communicate with each other.18. A. At a restaurant. B. At a hotel. C. At a coffee shop. D. Ata theme park.19. A. Two. B. Three. C. Four. D. Five.Passage fourQuestions:20. A. Japan. B. India. C. China. D. Indonesia.21. A. 5 km. B. 10 km. C. 15 km. D. 20 km.22. A. 5 hours. B. 6 hours. C. 7 hours. D. 8 hours.Section C (10 marks)Directions: In this section, you will hear a passage. After you hear a passage, you will have five minutes to take a turn.Part II Reading Comprehension (50 marks)Section A (10 marks)Directions: In this section, you will read a passage with ten questions below. You should answer questions according to the information given in the passage.The Benefits of ReadingReading is a great activity that can help people relax and take their minds off their daily worries. But besides being a great way to unwind, reading also offers a number of other benefits. Research has shown that reading can help improve cognitive function, increase vocabulary, and reduce stress levels.One of the benefits of reading is that it helps improve cognitive function. Reading regularly can help keep your mind sharp and improve your memory. Studies have shown that reading can help improve processing speed, reasoning, and problem-solving skills.Reading also helps increase vocabulary. When you read regularly, you expose yourself to new words and concepts, which can help expand your vocabulary. As you encounter new words in context, you are more likely to remember them and begin using them in your everyday life.Furthermore, reading can help reduce stress levels. Reading can be a great way to escape from the stresses and pressures of everyday life. When you read, you are transported to a different world where you can forget about your problems and relax.Overall, reading is a fantastic activity that offers a wide range of benefits. Whether you enjoy reading novels, magazines, or newspapers, the act of reading can greatly benefit your mental health and well-being.Questions:23. According to the passage, what is one benefit of reading?A. Improving physical health.B. Reducing cognitive function.C. Increasing stress levels.D. Relaxing the mind.24. What does reading regularly help improve, according to the passage?A. Memory.B. Appearance.C. Speed.D. Hearing.25. Why is it beneficial to have an expanded vocabulary?A. To impress others.B. To increase popularity.C. To communicate effectively.D. To become a better athlete.26. How can reading help reduce stress levels?A. By creating more stress.B. By helping people escape reality.C. By increasing cognitive function.D. By raising adrenaline levels.27. What type of world can reading transport people to?A. Reality.B. A magical land.C. The future.D. A dangerous place.28. Which of the following is NOT mentioned as a type of reading material in the passage?A. Novels.B. Magazines.C. Newspapers.D. Comics.29. What can reading help improve?A. Eye color.B. Problem-solving skills.C. Speeding tickets.D. Driving skills.30. According to the passage, how should reading be practiced?A. Once a week.B. Regularly.C. Only during vacations.D. Only in the morning.31. What is one way reading benefits mental health?A. By increasing stress levels.B. By reducing cognitive function.C. By expanding vocabulary.D. By improving reading speed.32. What does research show reading can help improve?A. Calculate.B. Cook.C. Drive.D. Reason.Section B (10 marks)Directions: In this section, you will read a passage with ten questions below. You should answer questions according to the information given in the passage.Personal Branding: Why It’s Important for You r CareerPersonal branding refers to the process of marketing yourself and your career as a brand. Just as big companies usebranding to differentiate themselves in the market, individuals can use personal branding to stand out in their careers.One of the reasons why personal branding is important for your career is that it helps you differentiate yourself from others. In a competitive job market, having a strong personal brand can help you stand out from other candidates and increase your chances of landing your dream job.Personal branding also helps build trust with your audience. By consistently delivering quality work and messages, you can build a reputation for yourself as a reliable and trustworthy person in your field. This can help you attract more opportunities and collaborations in your career.Furthermore, personal branding can help you showcase your unique skills and strengths. By identifying what makes you unique and showcasing it through your personal brand, you can position yourself as an expert in your field and attract the right opportunities to advance your career.Overall, personal branding is an essential tool for your career success. By investing time and effort into building and maintaining your personal brand, you can increase your visibility, build trust with your audience, and attract the right opportunities to take your career to the next level.Questions:33. What does personal branding refer to?A. Marketing products.B. Marketing yourself and your career.C. Marketing companies.D. None of the above.34. How can personal branding help you in a competitive job market?A. By decreasing chances of getting a job.B. By increasing chances of getting a job.C. By changing job preferences.D. By quitting your job.35. What can help you build trust with your audience?A. Consistently delivering quality work and messages.B. Posting random content on social media.C. Ignoring feedback.D. Not engaging with your audience.36. What does personal branding help you showcase?A. Weaknesses.B. Skills and strengths.C. Fears.D. Mistakes.37. How can you position yourself as an expert in your field?A. By not working on your personal brand.B. By being inconsistent.C. By showcasing what makes you unique.D. By pretending you are someone else.38. What can personal branding help you increase?A. Visibility.B. Invisibility.C. Unemployment.D. Failure.39. What can help you attract the right opportunities in your career?A. Not building a personal brand.B. Not showcasing your skills and strengths.C. Not being consistent with your work.D. Showcasing what makes you unique.40. What is an essential tool for your career success?A. Personal branding.B. Personal shopping.C. Personal training.D. Personal therapy.41. What can help you increase your visibility?A. Ignoring feedback.B. Not building a personal brand.C. Consistently delivering quality work.D. Being inconsistent.42. What can personal branding help you attract?A. The wrong opportunities.B. The right opportunities.C. Mishaps in your career.D. Insecurities.Section C (10 marks)Directions: In this section, you will read a passage and answer ten questions. For each question, choose the answer that best completes the sentence.A Memorable Trip to BeijingLast month, I had the opportunity to visit Beijing, and it was an unforgettable experience. One of the highlights of my trip was visiting the Great Wall of China, which was truly breathtaking. The grandeur and history of the Great Wall left me in awe, and I was amazed by the sheer size and scale of it.Another memorable experience was visiting the Forbidden City, which served as the imperial palace for centuries. Walking through the ancient halls and courtyards, I felt like I had stepped back in time to a bygone era. The intricate architecture and stunning decorations of the Forbidden City were a sight to behold.During my trip, I also had the chance to try authentic Chinese cuisine, and it was a culinary delight. From Peking duck to dim sum, each dish was a culinary masterpiece, and I savored every bite. The flavors and textures of the dishes were exquisite, and I found myself craving more with each meal.Overall, my trip to Beijing was an incredible experience that I will never forget. From exploring historical landmarks to indulging in delicious food, each moment was unforgettable, and I hope to visit again in the future.Questions:43. What was one of the highlights of the author’s trip to Beijing?A. Visiting the Great Wall of China.B. Eating dim sum.C. Shopping in ancient markets.D. Playing with pandas.44. How did the author feel when visiting the Great Wall of China?A. Bored.B. Excited.C. Sleepy.D. Frightened.45. Where did the author feel like they had stepped back in time?A. The Great Wall of China.B. The Forbidden City.C. Modern Beijing.D. A fast-food restaurant.46. What was the author amazed by at the Great Wall of China?A. The small size of it.B. The lack of tourists.C. The size and scale of it.D. The terrible weather.47. What served as the imperial palace for centuries?A. The Great Wall of China.B. The Forbidden City.C. Ancient markets.D. Modern restaurants.48. What did the author find to be a culinary delight on their trip?A. Pizza.B. Burgers.C. Tacos.D. Authentic Chinese cuisine.49. What was a culinary masterpiece on the trip?A. Peking duck.B. Hamburger.C. Hot dog.D. Fried chicken.50. What were the flavors and textures of the Chinese dishes like?A. Awful.B. Exquisite.C. Terrible.D. Bland.51. What did the author find themselves craving more of with each meal?A. Chinese food.B. Fast food.C. Soda.D. Dessert.52. What did the author hope to do in the future regarding Beijing?A. Never visit again.B. Visit again.C. Move there.D. Never eat Chinese food again.Part III Writing (60 marks)Section A (20 marks)Directions: For this part, you are allowed 30 minutes to write a short essay. You should write at least 120 words following the outline given below.1. Describe a memorable trip you have taken.2. Explain why this trip was memorable.3. Share any interesting experiences you had during this trip.Section B (20 marks)Directions: Write an English composition in 150-200 words according to the topic given below.Things I Want to Achieve in the Next Five YearsSection C (20 marks)Directions: Write a letter in 120-150 words following the outline given below.1. Describe a memorable trip you have taken.2. Explain why this trip was memorable.3. Share any interesting experiences you had during this trip.You will have one hour and thirty minutes to finish the entire exam.Good luck!。

函数单调性相关英文文献

翻译资料The monotonicity of the functionDefinitionIn general, a function f (x) domain of I:If for any belongs to some interval I on the two variables x1, X2, when x1>x2 is f (x1) ≥ f (x2). Then f (x) in this interval is an increasing function of (another argument is monotone non decreasing function). If f (x1) >f (x2), then f (x) in this interval is strictly increasing function (another story is the increasing function).If for any belongs to some interval I on the two variables x1, X2, when x1>x2 is f (x1) ≤ f (x2) f (x). It is in this interval is a decreasing function of (another term for monotone non increasing function). If f (x1) <f (x2), then f (x) in this interval is strictly decreasing function (another argument is a decreasing function).In order to avoid ambiguity, below the monotone non decreasing functions, strictly increasing function, monotone non increasing function, strictly decreasing function such as terminology.NatureIf the function y=f (x) in a certain interval is the increasing function or decreasing function. Then said the function y=f (x) is in the range of (strict) monotonicity, this interval is called y= f (x) monotone interval, image enhancement functions in monotone interval is rising, image reduction function is declining.Be careful.The monotonicity of the monotonicity of the function is also called function;The monotonicity of the function on an interval is concerned, it is a local concept. ExtensionIn mathematics function in the ordered set is monotone (monotone), if they keep the given order. These functions appear first in calculus and later extended to order theory in more abstract structure. Although the concept is generally consistent, two subjectshad developed a slightly different terminology. In calculus, we often say that the function is monotone increasing and decreasing monotone, preference terminology monotonic in order theory and the anti monotone or order preserving and order reversal.DefinitionSetF: P → QIs the function between the two with a partially ordered set P and Q ≤. In calculus, which is a function with the usual order subset of real number set, but still maintain the same order theory definition definition is more general.The function f is monot one, if x ≤ y, f (x) ≤ f (Y). So keep the order relation of monotone function.Monotonicity of calculus and real analysisIn calculus, often do not need to resort to order theory Abstract method. As mentioned above, the function is usually a mapping between a subset of real numbers are sorted by the natural order set..Inspired by the monotone function in real on the shape of the graph of the function, also called a monotone increasing (or "non decreasing"). Similarly, the function is called monotone decreasing (or "non incremental"), if the X < y, f (x) ≥ f (y), it reverses the order of.If the order ≥ definition with strict sequence >, are more stringent requirements. Have the function of such nature is strictly increasing. And by reversing the order of symbols, strictly decreasing can get the corresponding. Function of increasing or decreasing strict one one mapping (because < math> a < b< /math> math> a \neq contains < b< /math>).To avoid the term non decreasing and non increasing confusion in strictly increasing and strictly decreasing.In the theory of monotone sequenceIn order theory, not limited to the set of real number, can consider arbitrary partially ordered sets and even pre ordered set. Can also be used in these cases the abovedefinition. But to avoid the term "progressive" and "decline", because once the treatment is not totally ordered sequence is no image motivation attractive. Further, the strict relation of < and > are rarely used in most non order order, so do not intervene in the additional terms of their.Monotone (monotone) function is also called Isotone or order preserving function. Dual concept often called the anti monotone, antitone or sequential inversion. Therefore, the anti monotone function f satisfies the properties of X ≤ y contains f (x) ≥ f (Y),For all x and Y domain in its. Easy to see that the compound two monotone function is also monotone.Constant function is monotonic and anti monotonic; conversely, if the F is monotonic and anti monotonic, and if the domain f is a lattice, then f must be a constant function. Monotone function is central to the theory of order. They appear in large numbers in the theme of the article and found in these places in. Monotone function is famous order embedding (x ≤ y if and only if f (x) ≤ f (y) function) and order isomorphism (bijective order embedding).Function of interval editorFeatures(1) the geometric characteristics of the monotonicity of the function: the monotone interval function, image enhancement is rising, image subtraction function is declining."When x1 <x2, a f (x1) <f (x2)" is equivalent to y increases with X increasing; "When x1 <x2, a f (x1) >f (x2)" is equivalent to y decreases with the increase of X and.A geometric interpretation: the increment is equivalent to the function of the image from left to right gradually diminishing; equivalent to the function of the image from left to right gradually declined.(2) monotone function is directed at a certain interval, is a local property.Some function is monotonic in the whole domain; part interval in the domain of some function is increasing function, in part on the interval is a decreasing function; somefunction is a non monotonic function, such as the constant function.Monotonicity of function is a function in a monotone interval on the "whole" in nature, is arbitrary, cannot use the special value instead of.Note: the following properties in the monotonicity.1.f (x) and f (x) +a has the same monotonicity;2.f (x) and a*f (x) have the same monotonicity was in a>0, when a<0, having opposite monotonicity;3 when f (x), G (x) is increasing (decreasing) function, if f (x) *g (x) is a constant greater than zero, are the same as for increasing (decreasing) function; if both constant is less than zero, it is decreasing (increasing) function.Operational properties1 two increase in function and is still increasing function;2 minus the reduction function of increasing function for increasing function;3 two reduction function and is still decreasing function;4 minus function minus increasing function as a decreasing function;In addition to:Function value in the interval number with increasing (decreasing) function, the reciprocal is reducing (increasing) function.Judging method of editingI mage observationThe definition of proofUsing the monotonicity of the function definition that steps:The arbitrary value: let x1, X2 be any in the range of two values, and the x1<x2;The difference of deformation: as f (x2) -f (x1), and factorization, formula, rational methods such as differential to help determine the sign of the difference in the direction of deformation;The judge set number: F (x2) -f (x1) symbols;The conclusion is: make a conclusion according to the definition (if the difference of >0, is the increasing function; if the difference of <0, is a decreasing function) "Any value -- difference deformation -- judgment -- conclusion no.".Derivative methodThe derivation using derivative formula, and then determine the guiding function and 0 size relations, so as to judge monotonicity, guide function value is greater than 0, that is strictly increasing function, guiding function value is less than 0, that is strictly decreasing function, the premise is the original function must be continuous. Discriminant method monotone pointTheorem: if f (x) n derivative at a certain point, and the first derivative and N-1 derivative which is equal to 0 and the N derivative is not equal to 0 f (x) in the necessary and sufficient conditions for the monotonicity of n is odd, and when the N derivative is greater than 0, f (x) at the point of a strictly increasing and strictly decreasing.Proof: by the extreme second discriminant method of proof and can be launched. Note: this theorem is only f (x) discriminant method in the monotonicity of the point, and can not be monotonic in the point of a field.Discriminant method monotone intervalTheorem two: (necessary and sufficient conditions for monotonicity) if f (x) in the (a, b) can be the guide, then f (x) in the (a, b) is monotone increasing (or monotone decreasing) is necessary in the (a, b) a derivative is greater than or equal to 0 (or the first the derivative is less than or equal to 0).Theorem three: (sufficient condition of strictly monotone) if the interval (a, b) is a derivative of greater than 0 (or a derivative of less than 0), f (x) in the (a, b) is strictly increasing (or strictly decreasing).Note: by theorem three, if f (x) in the (a, b) memory in the continuous derivative, f (x) (a, b) will be in the memory in the monotone interval. In fact, say there exists a point belongs to (a, b), the first derivative is greater than 0 (0 or less), continuous known by the first derivative, there must be a certain field of this point is contained in the (a, b),in this field, the first derivative is greater than 0 (or less than 0), and f (x) in this field, increasing (or decreasing).Theorem four: (necessary and sufficient condition of strictly monotone) if f (x) in the (a, b) can be the guide, then f (x) in the (a, b) is strictly increasing (or strictly decreasing) necessary and sufficient condition is: when the X is (a, b), the two order derivative x greater than or equal to 0 (two order derivative or X is less than or equal to 0) and (a, b) a derivative in any subinterval x not equal to 0.Composite function editorIn the function y=f[g (x) domain] in, let u=g (x), y=f[g (x)] monotonicity by u=g (x) and y=f (U) monotonicity jointly determined, as followsU=g (x) y=f (U) y=f[g (x)]Increasing function of increasing function of increasing functionReducing function of decreasing function of increasing functionIncreasing function of decreasing function of decreasing functionReducing function of increasing function and decreasing functionTherefore, the composite monotonicity of functions available "with increment reduction" to judge, but to consider the domain of some special functions.Note: y=f (x) +g (x) does not belong to the scope of composite function, so this method is not.Note editorIn the use of derivatives are monotone interval function, first to determine the domain of the function, the process of solving the problem only in the domain, the derivative of the symbol to judge the monotone interval function.If a function has more than one monotone interval the same monotonicity, then these monotone interval can not use the "U" connection, and can only use the "comma" or "and" word spaced.Some applications of monotone functionDetermination of concave and convex functionsTheorem five: Let f (x) in the (a, b) can be the guide, then f (x) in the (a, b) on the inside concave (or convex) if and only if x derivative in (a, b) is monotone increasing(or monotone decreasing);The existence of inverse function of decision functionTheorem six: (sufficient condition of inverse function exists) if f (x) in the number set A strictly monotone, then f (x) has inverse function.Theorem seven: (necessary and sufficient conditions of inverse function exists) if f (x) in [a, b] is continuous, necessary and sufficient conditions of inverse function exists is f (x) is strictly monotone function.The proof of inequalityThe proof of the monotonicity inequality is a common and important method. By examples prove.SummaryBy definition, the monotone function promotion, discriminant method and application aspects of the understanding, the monotone function has a deep understanding, a good foundation for the next paper on two monotone function product monotonous writing, also hope to the future foreshadowed in the research field of number.函数的单调性定义一般地，设函数f(x）的定义域为I：如果对于属于I内某个区间上的任意两个自变量的值x1、x2，当x1>x2时都有f(x1)≥f(x2).那么就说f(x）在这个区间上是增函数(另一种说法为单调不减函数)。

微积分中的关键术语

微积分中的关键术语常量：习惯用字母a,b,c,d等表示；变量：习惯用字母x,y,z,u,v,w等表示.函数关系：变量与变量之间的对应关系；极限：变量的变化趋势；导数：变量变化的快慢程度（变化率问题）；微分：函数在某一点处，当自变量有一个微小的改变量时，函数所取得的相应改变量的大小。

dy∆；y≈积分学：已知某个函数F（x）的导函数f(x),求F（x），使()()x'F=xfValue of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线V olume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分C:Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列 series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数。

英文高等数学教材答案

英文高等数学教材答案Chapter 1: Functions and their Graphs1.1 Introduction to Functions1.1.1 Definition of a FunctionA function is a relation that assigns a unique output value to each input value. It can be represented symbolically as f(x) or y = f(x), where x is the input variable and y is the output variable.1.1.2 Notation and TerminologyIn function notation, f(x) represents the output value corresponding to the input value x. The domain of a function is the set of all possible input values, while the range is the set of all possible output values.1.2 Graphs of Functions1.2.1 Cartesian Coordinate SystemThe Cartesian coordinate system consists of two perpendicular number lines, the x-axis and the y-axis. The point of intersection is called the origin and is labeled (0, 0). The x-coordinate represents the horizontal distance from the origin, while the y-coordinate represents the vertical distance.1.2.2 Graphical Representation of FunctionsThe graph of a function is a visual representation that shows the relationship between the input and output values. It consists of all points (x, f(x)) where x is in the domain of the function. The shape of the graph depends on the nature of the function.1.3 Properties of Functions1.3.1 Even and Odd FunctionsAn even function is symmetric with respect to the y-axis, meaning f(-x) = f(x) for all x in the domain. An odd function is symmetric with respect to the origin, meaning f(-x) = -f(x) for all x in the domain.1.3.2 Increasing and Decreasing FunctionsA function is increasing if the output values increase as the input values increase. It is decreasing if the output values decrease as the input values increase. A function can also be constant if the output values remain the same for all inputs.Chapter 2: Limits and Continuity2.1 Introduction to Limits2.1.1 Limit of a FunctionThe limit of a function f(x) as x approaches a particular value c, denoted as lim[x→c] f(x), describes the behavior of the function near that point. It represents the value that the function approaches as x gets arbitrarily close to c.2.1.2 One-Sided LimitsOne-sided limits consider the behavior of the function from only one side of the point. The limit from the left, lim[x→c-] f(x), looks at the behavior as x approaches c from values less than c. The limit from the right, lim[x→c+] f(x), considers the behavior as x approaches c from values greater than c.2.2 Techniques for Evaluating Limits2.2.1 Direct SubstitutionDirect substitution involves substituting the value of the input variable directly into the function to find the limit. This method works when the function is continuous at that point and doesn't result in an indeterminate form (e.g., 0/0 or ∞/∞).2.2.2 Factoring and CancellationFactoring and cancellation can be used to simplify expressions and eliminate common factors before applying direct substitution. This technique is particularly useful when dealing with polynomial functions.2.3 ContinuityA function is continuous at a point if the limit of the function exists at that point, the function is defined at that point, and the left and right limits are equal. A function is called continuous on an interval if it is continuous at every point within that interval.Chapter 3: Derivatives3.1 Introduction to Derivatives3.1.1 Definition of the DerivativeThe derivative of a function f(x) represents the rate at which the function changes with respect to its input variable. It is denoted as f'(x) or dy/dx and is defined as the limit of the difference quotient Δy/Δx as Δx approaches zero.3.1.2 Interpretation of the DerivativeThe derivative represents the slope of the tangent line to the graph of the function at a given point. It provides information about the rate of change, instantaneous velocity, and concavity of the function.3.2 Techniques for Finding Derivatives3.2.1 Power RuleThe power rule states that if f(x) = x^n, where n is a constant, then f'(x) = nx^(n-1). This rule allows us to find the derivative of polynomial functions.3.2.2 Chain RuleThe chain rule is used to find the derivative of composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). This rule is particularly useful when dealing with functions that are composed of multiple functions.3.3 Applications of Derivatives3.3.1 Rate of ChangeThe derivative represents the rate of change of a function. It can be used to determine the instantaneous rate of change at a specific point or the average rate of change over a given interval.3.3.2 OptimizationDerivatives can be used to optimize functions by finding the maximum or minimum values. This involves finding the critical points where the derivative is zero or undefined and analyzing the behavior of the function around those points.Note: This is just a sample outline for the article on the answer key for an English advanced mathematics textbook. The actual content will depend on the specific exercises and problems in the textbook, which cannot be provided without access to the textbook itself.。

2023年3月英语b级考试真题试卷

2023年3月英语b级考试真题试卷2023年3月英语B级考试真题试卷Part I: Listening comprehension (30 points)Section A1. What is the main topic of the conversation?A. Travel plansB. New restaurantC. Sports eventD. Weather forecast2. What does the man suggest the woman do?A. Take a taxiB. Ride a bikeC. WalkD. Drive her car3. Where does the conversation most likely take place?A. At a bookstoreB. At a concertC. At a train stationD. At a restaurantSection B4. What does the woman say about the project?A. It's easyB. It's challengingC. It's interestingD. It's boring5. How does the woman feel about the deadline?A. HappyB. WorriedC. ExcitedD. Relieved6. Why does the man look tired?A. He stayed up lateB. He had a long dayC. He was exercisingD. He was studyingSection C7. Where is the man going next?A. To the airportB. To a meetingC. To a restaurantD. To a hotel8. What does the woman offer to do for the man?A. Pick up his dry cleaningB. Drive him to the airportC. Make a reservationD. Buy him a plane ticket9. What does the woman suggest the man do for his presentation?A. Use more visualsB. Practice moreC. Speak louderD. Be more concisePart II: Reading comprehension (40 points)Section AObsessive-Compulsive Disorder (OCD) is a mental disorder characterized by intrusive thoughts that produce anxiety and by repetitive behaviors aimed at reducing the associated anxiety. Symptoms of OCD can include hand-washing, counting of things, and checking things repeatedly. OCD often centers on themes such as a fear of germs or the need for things to be symmetrical. Individuals who suffer from OCD can sometimes be aware that their obsessions are irrational but still feel compelled to perform the behaviors.10. What is one symptom of OCD?A. Eating too muchB. Exercising regularlyC. Hand-washingD. Reading books11. What do individuals with OCD feel compelled to do?A. EatB. ExerciseC. Perform repetitive behaviorsD. Watch TV12. Why do individuals with OCD perform repetitive behaviors?A. To reduce anxietyB. To socializeC. To make friendsD. To have funSection BRead the following passage about the benefits of reading books:Reading books can have numerous benefits for both physical and mental health. Studies have shown that reading can reduce stress levels, improve brain function, and even increase empathy and compassion. Additionally, reading can improve vocabulary and comprehension skills, enhance creativity, and provide a form of entertainment that is both educational and enjoyable.13. According to the passage, what is one benefit of reading books?A. Reducing stress levelsB. Increasing anxietyC. Improving brain functionD. Decreasing empathy14. How can reading books enhance comprehension skills?A. By reducing vocabularyB. By increasing vocabularyC. By being entertainingD. By being boring15. What is one reason reading books is considered beneficial?A. It increases stress levelsB. It decreases creativityC. It improves brain functionD. It harms physical healthSection CRead the following passage about the importance of exercise:Regular exercise is crucial for maintaining good physical and mental health. Exercise can help reduce the risk of chronic diseases such as heart disease, diabetes, and obesity. Additionally, exercise can improve mood, decrease stress levels, and enhance overall quality of life. It is recommended to engage in at least 150 minutes of moderate-intensity exercise per week for optimal health benefits.16. What is one benefit of regular exercise?A. Increasing the risk of chronic diseasesB. Decreasing moodC. Reducing stress levelsD. Improving overall quality of life17. How much moderate-intensity exercise is recommended per week?A. 50 minutesB. 100 minutesC. 150 minutesD. 200 minutes18. What is one chronic disease that regular exercise can help reduce the risk of?A. DiabetesB. Heart diseaseC. ObesityD. All of the abovePart III: Translation (30 points)Translate the following passage from English to Chinese:Global warming is a pressing issue that requires urgent action. The burning of fossil fuels has led to an increase in greenhouse gases, which trap heat in the atmosphere and contribute to rising temperatures. If we do not take steps to reduce carbon emissions and transition to clean energy sources, the consequences for our planet could be dire. It is essential that we work together to combat climate change and protect the environment for future generations.全球变暖是一个迫切需要采取行动的问题。

微积分词汇中英对照

微积分中英文对照A：:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：原函数，反导数Approximate integration ：近似积分(法) Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之对称轴B：Binomial series ：二项式级数Binomial theorem：二项式定理C：Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西中值定理Chain Rule ：链式法则Circle ：圆Circular cylinder ：圆柱体，圆筒Closed interval ：闭区间Coefficient ：系数Composition of function ：复合函数Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Conditionally convergent：条件收敛Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinates：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆筒, 圆柱体, 柱面Cylindrical Coordinates ：圆柱坐标D：Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变化率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分E：Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理F：Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理G：Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式H：Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Higher mathematics 高等数学Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyperboloid ：双曲面I：Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：反常积分, 广义积分Increasing，Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变量Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限点Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积函数Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分L：Laplace transform ：Laplace 变换Law of sines：正弦定理Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Local extreme ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数M：Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子N：Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数O：Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的P：Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：平行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partition ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Q：Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律R：Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根S：Saddle point ：鞍点Scalar ：纯量、标量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant, Oblique asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Substitution rule ：替代法则Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称T：Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Taylor’s formula ：泰勒公式Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分V：Value of function ：函数值Variable ：变量Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线V olume ：体积X：X-axis ：x轴X -coordinate ：x坐标X -intercept ：x截距Z：Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点。

微积分专有名词中英文对照

微积分专有名词中英文对照absolutely convergent 绝对收敛absolute value 绝对值algebraic function 代数函数analytic geometry 解析几何antiderivative 不定积分approximate integration 近似积分approximation 近似法、逼近法arbitrary constant 任意常数arithmetic series/progression （AP）算数级数asymptotes （vertical and horizontal)(垂直/水平）渐近线average rate of change 平均变化率base 基数binomial theorem 二项式定理，二项展开式Cartesian coordinates 笛卡儿坐标（一般指直角坐标）Cartesian coordinates system 笛卡儿坐标系Cauch's Mean Value Theorem 柯西均值定理chain rule 链式求导法则calculus 微积分学closed interval integral 闭区间积分coefficient 系数composite function 复合函数conchoid 蚌线continuity （函数的）连续性concavity (函数的）凹凸性conditionally convergent 有条件收敛continuity 连续性critical point 临界点cubic function 三次函数cylindrical coordinates 圆柱坐标decreasing function 递减函数decreasing sequence 递减数列definite integral 定积分derivative 导数determinant 行列式differential coefficient 微分系数differential equation 微分方程directional derivative 方向导数discontinuity 不连续性discriminant （二次函数）判别式disk method 圆盘法divergence 散度divergent 发散的domain 定义域dot product 点积double integral 二重积分ellipse 椭圆ellipsoid 椭圆体epicycloid 外摆线Euler's method （BC）欧拉法expected valued 期望值exponential function 指数函数extreme value heorem 极值定理factorial 阶乘finite series 有限级数fundamental theorem of calculus 微积分基本定理geometric series/progression (GP)几何级数gradient 梯度Green formula 格林公式half—angle formulas 半角公式harmonic series 调和级数helix 螺旋线higher derivative 高阶导数horizontal asymptote 水平渐近线horizontal line 水平线hyperbola 双曲线hyper boloid 双曲面implicit differentiation 隐函数求导implicit function 隐函数improper integral 广义积分、瑕积分increment 增量increasing function 增函数indefinite integral 不定积分independent variable 自变数inequality 不等式ndeterminate form 不定型infinite point 无穷极限infinite series 无穷级数infinite series 无限级数inflection point (POI) 拐点initial condition 初始条件instantaneous rate of change 瞬时变化率integrable 可积的integral 积分integrand 被积分式integration 积分integration by part 分部积分法intercept 截距intermediate value of Theorem :中间值定理inverse function 反函数irrational function 无理函数iterated integral 逐次积分Laplace transform 拉普拉斯变换law of cosines 余弦定理least upper bound 最小上界left—hand derivative 左导数left-hand limit 左极限L’Hospital's rule 洛必达法则limacon 蚶线linear approximation 线性近似法linear equation 线性方程式linear function 线性函数linearity 线性linearization 线性化local maximum 极大值local minimum 极小值logarithmic function 对数函数MacLaurin series 麦克劳林级数maximum 最大值mean value theorem (MVT)中值定理minimum 最小值method of lagrange multipliers 拉格朗日乘数法modulus 绝对值multiple integral 多重积分multiple 倍数multiplier 乘子octant 卦限open interval integral 开区间积分optimization 优化法，极值法origin 原点orthogonal 正交parametric equation (BC）参数方程partial derivative 偏导数partial differential equation 偏微分方程partial fractions 部分分式piece—wise function 分段函数parabola 抛物线parabolic cylinder 抛物柱面paraboloid ：抛物面parallelepiped 平行六面体parallel lines 并行线parameter ：参数partial integration 部分积分partiton ：分割period ：周期periodic function 周期函数perpendicular lines 垂直线piecewise defined function 分段定义函数plane 平面point of inflection 反曲点point—slope form 点斜式polar axis 极轴polar coordinates 极坐标polar equation 极坐标方程pole 极点polynomial 多项式power series 幂级数product rule 积的求导法则quadrant 象限quadratic functions 二次函数quotient rule 商的求导法则radical 根式radius of convergence 收敛半径range 值域（related) rate of change with time （时间)变化率rational function 有理函数reciprocal 倒数remainder theorem 余数定理Riemann sum 黎曼和Riemannian geometry 黎曼几何right-hand limit 右极限Rolle’s theorem 罗尔(中值）定理root 根rotation 旋转secant line 割线second derivative 二阶导数second derivative test 二阶导数试验法second partial derivative 二阶偏导数series 级数shell method （积分)圆筒法sine function 正弦函数singularity 奇点slant 母线slant asymptote 斜渐近线slope 斜率slope-intercept equation of a line 直线的斜截式smooth curve 平滑曲线smooth surface 平滑曲面solid of revolution 旋转体symmetry 对称性substitution 代入法、变量代换tangent function 正切函数tangent line 切线tangent plane 切（平）面tangent vector 切矢量taylor’s series 泰勒级数three—dimensional analytic geometry 空间解析几何total differentiation 全微分trapezoid rule 梯形（积分）法则。

分数阶粘弹性固体模型的滞后和预调

Hysteresis and Precondition ofFractional-order Viscoelastic SolidModels1Li Yan Xu MingyuInstitute of Applied Math, School of Mathematics and System Sciences,Shandong University,Jinan 250100,P.R.Chinalandc@ xumingyu@AbstractIn this paper the hysteresis and precondition of fractional-order viscoelastic solidmodels were studied under the condition of loading and unloading of saw-tooth wave.It contains the monotonic and limit property of hysteresis and precondition. It wasshown that the hysteresis loop's width decreases with increase of period and is greateror equal to zero. At the same time it was shown that under the condition of quasi-lineartheory the above conclusions also hold.Keywords: viscoelastic solids, fractional-order Kelvin model, saw-tooth wave,precondition, hysteresis, quasi-linear theory.1IntroductionThe mechanical characteristics of viscoelastic solids are mainly creep, relaxation, hysteresis and precondition[1~4]. In order to simulate the experimental results with fidelity a series of models are quoted. In the 1940's, Scott Blair[5] and Gerasimov[6] independently proposed a fractional-order Newton model. Then Gerasimov[6] was the first to consider an Abel kernel for the relaxation modulus in Boltzmann's general theory of viscoelasticity. Bagley and Torvik[7] had given several meaningful conclusions and special cases. Gemant[8] was the first to propose a fractional-order viscoelastic model. Caputo[9] introduced a fractional-order Voigt solid model. A more appropriate representation of solid behavior is the fractional-order Kelvin model[10]. Y.C.Fung proposed quasi-linear theory[1]. Carew, et al.[11] combine fractional-order Kelvin model with quasi-linear theory.In this paper the hysteresis and precondition of fractional-order viscoelastic solid models were studied under the condition of loading and unloading of saw-tooth wave. It contains the monotonic and limit property of hysteresis and precondition. It was shown that the hysteresis loop's width decreases with increase of period and is greater or equal to zero. At the same time it was shown that under the condition of quasi-linear theory the above conclusions also hold.1 The work was supported by the National Natural Science Foundation of China (Grant No.10272067) and by the Doctoral Foundation of the Education Ministry of P.R.China (Grant No.20030422046).2 Fractional-order Kelvin model and saw-tooth wave inputThe constitutive equation of fractional-order Kelvin model is [3,4]:)()()()(00t D E t E t D t tR R t ετεστσαασααε+=+, (1) where t is time, )(t σand )(t εare stress and strain respectively. And ετ,στand are characteristicrelaxation time, characteristic retardation time and rubbery modulus respectively, in which,,R E 111−=µητεεστµητ+=−1010µ=R E . The initial condition is . The creep modulus and relaxation modulus are respectively[3,4]:010σττεασαε−=）（R E )()]((1[1)(t H t E E t C R ασαασαεασττττ−−−=, (2) )()]((1[)(t H t E E t K R αεααεασαεττττ−−−=, (3) where ∑∞=+Γ−=−0)1(])([)((n nn t tE αττασασαis Mittag-Leffler function and is unit step function. It is obvious that the relaxation modulus given by (3) is a monotone decreasing function with increase of time t [12].)(t H )(t K For convenience, let the saw-tooth strain )(t ε,as shown in Fig.1, be input of the model. It has the following expression:∑=−−−+=][1)()()1(2)()(t k k k t H k t t tH t ε. (4)Obviously, )(t ε has the following characters:)()2(t t εε=+,)),0[(+∞∈t 0)2(=n ε,1)12(=+n ε,（N n ∈） )22()2(a n a n −+=+εε,（]1,0[∈a ）⎪⎩⎪⎨⎧++∈+∈].22,12[decreasing ]12,2[increasing )(n n t function n n t function is t εFig.1: )(t εas a function of time .t3 Hysteresis and preconditon3.1 HysteresisIn the experiments of viscoelastic solids in which the periodic loading and unloading are inputted, if two curves of loading and unloading are not coincident with each other, this phenomenon is called hysteresis[1].Because equation (1) is linear, it is easy to prove that the Boltzmann superposition principle holds good for it with the form:∫−=td t K t 0)()()(ττετσo(5)or∫−+=t d t K t K t 0)()()()0()(ττετεσo. (6)Where is the fractional-order relaxation modulus given by equation (3), and denote derivatives with respect to )(t K )(τo K )(τεoτ.To study the hysteresis property of fractional-order Kelvin model let us consider the curves of periodic loading and unloading. For this end, we focus our attention on the sign and monotonicity of)22()2(a n a n −+−+σσ, (7)where ,. Equation (7) stands for the difference between loading and unloading curves in the period and at the point N n ∈]1,0[∈a th n )1(+)(a ε(or )2(a −ε). Substituting and a n +2a n −+22 into (6) yields:∫+−+++=+a n d a n K a n K a n 20)2()()2()0()2(ττετεσo, (8)∫−+−−++−+=−+a n d a n K a n K a n 220)22()()22()0()22(ττετεσo. (9)Substituting (8) and (9) into (7) and after straightforward calculation, we have:)22()2(a n a n −+−+σσ,)22()]([)]2()22()][([)]2()22()][([2222220∫∫∫−+++−−+−+−+−−−+−+−+−−−+−=an an an n nd a n K d a n a n K d a n a n K ττετττετετττετετooo（10）where the function )2()22(τετε−+−−−+a n a n is shown in Fig.2. Its piecewise analytic expression is)2()22(τετε−+−−−+a n a n.210)]},22())2(2()[22(2))]2(2())1(2([2))]1(2())1(2()[12(2))]1(2()2([2)]2()2([2{]2[0≤≤−−−−−−−−+−−−−+−−−+−−−−−−−−−−−−−−−+−−−−=∑=a for n H a n H n a n H a n H a a n H a n H n a n H a n H a a n H n H n ττττττττττττττor.121)]},22())1(2()[22(2))]1(2())2(2()[1(2))]2(2()2()[12(2)]2())1(2()[1(2))]1(2()2([2{]2[0≤≤−−−+−−−−++−−−−−−−−−−−−−−−−−−−−−−−−+−−−−−=∑=a forn H a n H n a n H a n H a a n H a n H n a n H a n H a a n H n H n ττττττττττττττ Two bases of the trapezoid shown in Fig.2 are )210(21≤≤−a a (or 12−a )121(≤≤a ) and 1,and the base angle is .2arctanFig.2：)2()22(τετε−+−−−+a n a n as a function of τIt is obvious that when =τa k ±−12))1,0[},,,3,2,1{(∈∈a n k L ,we have0)]12(2[)]12(22[≥−−−+−−−−−+a k a n a k a n εε, 0)]12(2[)]12(22[≤+−−+−+−−−+a k a n a k a n εε, and )]12(2[)]12(22[a k a n a k a n −−−+−−−−−+εε)]}12(2[)]12(22[{a k a n a k a n +−−+−+−−−+−=εε.Because is a positive monotone decreasing function, we have the following inequality:)(t K o−,))]12(2())12(22()[12())]12(2())12(22()[12(a k a n a k a n a k K a k a n a k a n a k K +−−+−+−−−++−−≥−−−+−−−−−+−−−εεεεoo(11)namely,0)]2()22()][([20≥−+−−−+−∫nd a n a n K ττετετo. (12)Because and )(t K o−)(t ε are all nonnegative function on ),0[+∞and]22[τε−−+a n 0]2[≥−+−τεa n ,])2,2[(a n n +∈τwe arrive at0)]2()22()][([22≥−+−−−+−∫+an nd a n a n K ττετετo, (13)and0)22()]([222≥−−+−∫−++an an d a n K ττετo. (14)The three signs of equality in (12)~(14)do not hold simultaneously, provided .1≠aWhen ,we have 1=a0)122()12(=−+−+n n σσ.In sum, equation (10) is greater or equal to zero, that is to say,0)22()2(≥−+−+a n a n σσ, (15)and the sign of equality holds, provided and only provided 1=a . As for the monotonicity and varingof hysteresis loop’s width with time, we will discuss later. 3.2 PreconditionIn the experiments of viscoelastic solids in which the periodic loading and unloading are inputted, the differences among periods are decreasing gradually. This phenomenon is called precondition. In order to get repeated data, every kind of materials needs precondition[1].In order to study precondition property of fractional-order Kelvin model, we need just proving that )22()2(++−+b n b n σσ for )2,0[∈∀b is greater than zero and decreasing to vanishing along with increasing to infinite. Using formula (6), we can obtainn )22()2(++−+b n b n σσ])22()()22([])2()()2([22020∫∫+++−+++++−−+++=b n b n d b n K b n d b n K b n ττετεττετεoo.0)22()]([222>−++−=∫+++bn bn d b n K ττετo(16)In course of deriving (16) the periodicity of )(t ε and the positive monotonicity of were used. Further because the periodicity of )(t K o−)2(τε−+b n as a function of τ, equation (16) is a monotonic decreasing function of .Taking the limit of (16) we have the following result:n )22()2(lim ++−++∞→b n b n n σσ∫++++∞→−++−=b n bn n d b n K 222)22()(lim ττετo∫++++∞→−<bn bn n d K 222)]([limττo0=. (17)Therefore, using (16) and (17) we have0)22()2(lim =++−+∞→b n b n n σσ. (18)In sum, we can draw the conclusion that )2,0[∈∀b the difference )22()2(++−+b n b n σσ is greater than zero and decreasing to vanishing along with increasing to infinite. That is why there is precondition in fractional Kelvin model of viscoelastic solids.n 4 Further discussion about hysteresis and preconditionUsing (6) and the property of )(t ε we obtain∫+−+−=+−2)2()]([)2()(t td t K t t ττετσσo. (19)Differentiating both sides of (19) with respect to t and using the monotonicity of yield)(t K o−)2()]([))]2(2[()]([)2()]([)]2()([2t t K t t K d dt t d K dtt t d t t−+−−+−+−+−+−=+−∫+ετετττετσσo o o ∫+−+−=2)2()]([t t d dtt d K ττετo. (20)0)]([)]([211<−+−−=∫∫+++t t t td K d K ττττooTherefore, )2()(+−t t σσis a monotonic decreasing function of t .In arriving at (20) we used the following piecewise expression of⎩⎨⎧+−−−=−+)]2([)2(t tt τττε ]2,1[]1,[++∈+∈t t t t ττas a function of τ with ]2,[+∈t t τ as shown in Fig.3. Therefore⎩⎨⎧−=−+11)2(dt t d τε).2,1()1,(++∈+∈t t t t ττFig.3：)2(τε−+t as a function of τ, with ]2,[+∈t t τMoreover in view of the fact that )2(22a n a n −+≤+ and using (20) we have:0)]222()22([)]22()2([≥+−+−−+−++−+a n a n a n a n σσσσ,namely,0)]2)1(2())1(2([)]22()2([≥−++−++−−+−+a n a n a n a n σσσσ. (21) The obtained result (20) means that )22()2(a n a n −+−+σσis a decreasing function of ,that is to say, the width of hysteresis loops is monotone decreasing along with the increasing of , the number of period .Moreover, we have n n 1+n 0)22()2(≥−+−+a n a n σσ, using (21) we obtain that)]22()2([lim a n a n n −+−+∞→σσis a positive number dependent on )1,0(∈a . Letting)]22()2([lim )(a n a n a A n −+−+=∞→σσ,then we have0)]022()02([lim )0(=−+−+=∞→n n A n σσ, (0=a )which is a special case of precondition, and0)]122()12([lim )1(=−+−+=∞→n n A n σσ. (1=a )Further because )2()22(τετε−+−−−+a n a n and)22(τε−−+a nare bounded, and is decreasing to vanishing along with the increasing of )]([τoK −τ, we have)]22()2([lim )(a n a n a A n −+−+=∞→σσ∫∫∫−+++∞→++∞→+∞→−−+−+−+−−−+−+−+−−−+−=an an n an n n nn d a n K d a n a n K d a n a n K 2222220)22()]([lim)]2()22()][([lim )]2()22()][([lim ττετττετετττετετooo∫−+−−−+−=+∞→nn d a n a n K 20)]2()22()][([lim ττετετo.5 Discussion of hysteresis and precondition with quasi-linear theoryUsing the quasi-linear theory proposed by Y.C.Fung[1] we quote the definition of generalizedcreep and relaxation modulus as follows:)0()()(+=C t C t J and )0()()(+=K t K t G .If the strain is step function, the developing process of stress is a function of time t and strain ε.Using ),(t K ε to express this process, we have)()(),()(εεe T t G t K =,1)0(=+G , (22)where is a function of strain and is called elastic response . If strain )()(εe T )(t ε is a continuous function of time under the condition that the superposition principle holds, the stress reads:t ∫+−=te d d dT t G t 0)()]([)()(τττετσ. (23)If t t T e ∂∂)]([)(εand tt G ∂∂)( are continuous on ),0(+∞, using integration by parts and commutativeproperty of convolution in equation (23) we have∫+−+=+te e d t T d dG t T G t 0)()()]([)()]([)0()(ττεττεσ. (24) Because , (24) can be rewritten as1)0(=+G ∫+−+=te e d t T d dG t T t 0)()()]([)()]([)(ττεττεσ. (25) Using the definition of elastic response[1] we have)]22([)]2([)()(a n T a n T e e −+=+εε, (26))]2([)]([)()(+=t T t T e e εε, (27)0)]2([)(=n T e ε, (28)and⎪⎩⎪⎨⎧++∈+∈].22,12[decreasing ]12,2[increasing )]([)(n n t function n n t functionis t T e ε (29)Where ,, provided is an increasing function of L ,2,1,0=n ]1,0[∈a )()(εe T ε and .Then substituting and into 0)0()(=e T )]([)(t T e ε)(t G )(t εand we have the same conclusions for hysteresisand precondition.)(t K Next, we cite an example which is applied to heart valve tissues to verify the above mentioned theory.The elastic response is represented by[11]⎪⎩⎪⎨⎧>+−+−+−≤+−=.)()()()1()(23)(T T T T Tb e g f e dc e a Tεεεεεεεεεεεεε(30)where a,b,c,d and e are fitted parameters , T εis a specified transition strain, and are the corresponding tension and slope. Authors of [11] have demonstrated that quasi-linear fractional-order viscoelastic theory can be used to model the stress response of porcine aortic valves with good fidelity.Obviously, (30) is an increasing function of f g ε on ),0(+∞∈ε, and . Therefore (30) satisfies (26)~(29). Using (30) we can easily get the same conclusions for hysteresis and precondition as we get in sections 3 and 4.0)0()(=e T6 Concluding remarksFractional-order viscoelastic models exhibit high fidelity to experiment data, so, we choose fractional-order Kelvin model and quasi-linear theory to discuss viscoelastic materials’ hysteresis and precondition. We explain hysteresis and precondition of viscoelastic models to a certain extent. It should be noted that besides Boltzmann superposition principle and the characters of inputted strain (i.e. symmetry, periodicity and initial value with null) in this paper, we used only the monotone decreasing of generalized relaxation modulus and its derivative . Therefore, as for other models of viscoelastic solid, all conclusions obtained in this paper hold good, provided is monotone decreasing and is monotone decreasing to vanishing. )(t K )]([t K o−)(t K )]([t K o−References[1] Fung.Y.C.Biomechanics:Mechanical(1993), Properties of Living Tessues.2nd ed. New York: Springer. [2] Roderic S. Lakes(1999),Viscoelastic Solids,CRC Press, Boca Raton London New York Washington, D.C. [3] XU Mingyu , TAN Wenchang(2003),Representation of the constitutive equation of viscoelasticmaterials by the generalized fractional element networks and its generalized solutions.Sciense in China Series G.Vol.46.No.2.145-157.[4] Alan Freed , Kai Diethelm,et al.(2002), Fractional-Order Viscoelasticity(FOV):Constitutive development using the fractional calculus:First annual report.NASA/TM-2002-211914. [5] G.W.Scott Blair(1944,1986),Analytical and integrative aspects of the stress-strain-timeproblem,Journal of Scientific Instruments21(1944),80-84. On the fractional calculus model of viscoelastic behavior,Journal of Rhelogy 30(1986),133-155.[6] A.N.Gerasimov(1948),A generalization of linear laws of deformation and its application to inner friction problems. [7] R.L.Bagley and P.J.Torvik(1983),A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology 27,201-210. [8] A.Gemant(1936).A method of analyzing experimental results obtained from elsto-viscous bodies,Physics 7,311-317. [9] M.Caputo(1967),Linear models of dissipation whose Q is almost frequency independent-II. The Geophysical Journal of the Royal Astronomical Society 13,529-539. [10] M.Caputo , F.Mainardi(1971),Linear models of dissipation in anelastic solids,Rivista del Nuovo Cimento 1,161-198. [11] Evelytn O. Carew,Todd C. Doehring, et al .(2000), Role of Precondtioning and Recovery Time inRepeated Testing of Aortic Valve Tissues:Validation Through Quasilinear Viscoelastic Theory. Annals of Biomedical Engineering.Vol28,pp1093-1100. [12]Kenneth ler,Stefan G. Samko(2001).Completely monotonic functions.Integr.Transf.and Spec. Funct.2001,Vol.12,No4,389-402.。

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(1) calculate the guide function(求出函数的导函数)
(2) for solving inequality of F '(x) >0, we obtain the solution set,According to the solution set to write a monotone
Judging method
Basic method for judging the monotonicity of the function is 1 The definition of law(定义法) 2 The image method（图想法） 3 Complex function method（复合函数法）
Increasing function and decreasing function or Monotonicity of the function
Rise
y y x 1
o
x
Drop
y
y x 1
o
x
局部上升或下降
Localyrise or fall
y x2
o
x
能函用数图的象这上种动性点质P称（为x，Moyn）o的ton横ic、ity纵o坐f th标e
∴∵∵fxx(11x＞＜1)－0x，2f，(xx∴22＞)x＞20－0，，x∴1即＞x1f0x(，x2＞1)＞∴0 f(xx2x2)1x2x1 ＞0
∴f(x)= 1 在(0，+∞)上是减函数. x
Solution two：(用导数方法证)
∵f′(x)=(
1)′=(－1)·x－2=－ x
1，x＞0， x2
关f在u系n某c一来tio区说n（间明内函上，数升的或单下调降性趋）势吗？
When the x value increases, the function value y increases -- image gradually increased in the range;
When the value of x increases, y decreases the function value -- image gradually decreased in the range.
函数单调性体现出了函数值y随自变量x的变化而变化的情况,而导数也正是研究自变量的增加量与函数值的增加量之间的关系，于是我们设想
一下能否利用导数来研究单调性呢？
1. 函数的导数与函数的单调性的关系：
曲线y=f(x)的切线的斜率就是函数y=f(x)的导
数.从函数 y x 2 4x 3 的图像可以看到：y
一、函数单调性定义
1．增函数
In general, a function y=f (x) domain of definition for I, if for any a range of D domain of I within the two variables x1, X2, when x1<x2, there are f (x1) <f (x2), then we say that f (x) in the interval D is increasing function.
结论
定号A fixed number
Conclusio返n 回
Monotone function reflects the function value of Y changes with the independent variable x and the change of the situation and the derivative is also to study the relationship between the independent variable and the value of the function to increase the amount of increase in the amount of,so we can imagine to study the monotonicity of using derivative?
function of the interval.
新疆王新敞
奎屯
设函数y=f(x)在某个区间内有导数，如果在
这个区间内y ′ >0，那么y=f(x)为这个区间内的
增函数；如果在这个区间内y ′ <0，那么y=f(x)
为这个区间内的减函数.
y ′ >0
增函数
y ′ <0
减函数
Derivative method
结论：
A function y=f (x) is the derivative in a certain range, if in this interval of Y '>0, then y=f (x) is increasing function of the interval; if in this interval of Y' <0, then y=f (x) is a decreasing
11
Conclusion: judge the monotonicity of the function with the derivative
Use derivative method to determine the monotonicity of the function of the step is:（用导数法确定函数的单调性时的步骤)
y=f(x) =x2－4x+3
切线的
f′(x)
斜率
fx = x2-4x+3
(2，+∞)
增函数
正
＞0
B
(－∞,2) O 函数；在（
，2）内，切
线内，切线的，函数 y = f(x)的值值随x的增大而减增大而
减函数
负
0时，函数y = f(x) 在区间区
＜0
，2）内为内为1减.
2
3
A
x
在区间（2，）内，切线的斜率为正，函数y=f(x)的值随着x的增大而增大，即 f ′(X) >0时函数y=f(x) 在区间( ,2)内为增函数；在区间（，2）内，切线的斜率为负，函数y=f(x)的值随着x的增大而减小，即f ′(X) < 0时，函数y=f(x) 在区间（，2）内为减函数.
increasing interval(求解不等式f ′(x)>0，求
得其解集，再根据解集写出单调递增区间)
(3) for solving inequality of F '(x) <0, we obtain the solution set,According to the solution set to write a monotonically
定义法
The main steps主要步骤 1.Value 任取x1，x2∈D，且x1<x2； 2. Make a difference作差f(x1)－f(x2)； 3. Deformation
变形（通常是因式分解和配方）；
4. A fixed number定号（即判断差f(x1) －f(x2)的正负）； 5. Conclusion下结论
In general, for a given interval of the function f(x), if f ′(x)>0 ,then we say that f(x) in this interval is increased; If the f ′(x) <0 then say f(x)in this interval is a decreasing function.
证明：在区间 1, 上任取两个值 x1, x2 且 x1 x2 取值
则
f
( x1 )

f
(x2 )

( x1

1) x1

( x2

1 x2
)
11 (x1 x2 ) ( x1 x2 )
Value
作差Make a difference

(
x1
ห้องสมุดไป่ตู้
x2
)

(
x2 x1
x1 x2
一般地，设函数y=f(x)的定义域为I，如果对于定义域I内的某个区间D内的任意两个自变量x1， x2，当x1<x2时，都有f(x1)<f(x2)，那么就说f(x)在区间D上是增函数．
2．减函数
In general, a function y=f (x) domain of definition for I, if for any a range of D domain of I within the two variables x1, X2, when x1<x2, there are f (x1) >f (x2), then we say that f (x) in the interval D is decrease function. 一般地，设函数y=f(x)的定义域为I，如果对于定义域I内的某个区间D内的任意两个自变量x1，x2，当x1<x2时，都有f(x1)>f(x2)，那么就说f(x)在区间 D上是减函数．
1 ∴x2＞0，∴－ x 2 ＜0. ∴f′(x)＜0，
∴f(x)= 1 在(0，+∞)上是减函数.
x
点评：比较一下两种方法，用求导证明是不是更简捷一些.如果是更复杂一些的函数，用导数的符号判别函数的增减性更能显示出它的优越性.
)
变形 Deformati

(
x1