数学与应用数学专业英语期末考试试题

一、英文表示： 1.13.25 thirteen-point-two-five 2.a n b

=log the logarithm of n to the base b equals a 3.5/8 five eighths 4.

3152a -+x the fifth root of a squared plus x to the minus one third 5.0x

5232=--a x to the minus two thirds plus the fifth root of a squared equals zero 6.))((a 2233b ab a b a b +-+=+ a cubed plus b cubed equals a plus b into a squared minus ab plus b squared

二、英语名词定义

1.Equation

An equation is a statement of the equality between two equal number symbols. 2.Function

The modern definition of a function y of x is simply a mapping from a space X to another space Y.A mapping defined when every point x of X has a definite image y, a point of Y. 3.The limit of a sequence {}f(n)(for example L n f =∞

→)(lim n ) : A sequence {}f(n) is said to have a limit L if ,for every positive number ε,there is another positive number N (which may depend on ε)such that

ε<-L n f )( for all n>=N In this case, we say the sequence {}f(n) converges to L and we write

L n f =∞→)(lim n ,or L n f →)( as ∞→n .A sequence which does not converge is called divergent.

4.The derivative of function f(x)

The derivative )x (‘f is defined by the equation h

x f h x f f h )()(lim )x (0-+=→‘ provided the limit exists. The number )x (‘f is also called the rate of change of f at x.

5.Statistical population A statistical population is the set of measurements (or record of some qualitative trait)corresponding to the entire collection of units about which information is sought.

三 几何图形的名称

1.圆 Circle

2.椭圆 ellipse

3.长方形 rectangle

4.正方形 square

5.平行四边形 parallelogram

6.三角形 triangle

7.立方体 cube

8.圆锥 cone

9.曲线 curve 10.双曲线 hyperbola

四．英译汉

1.This device for representing real numbers geometrically is a very worthwhile aid that helps us to discover and understand better certain properties of real numbers. However, the readers should realize that all properties of real numbers that are to be accepted as theorems must be deducible from the axioms without any reference to geometry. This does not mean that one should not make use of geometry in studying properties of real numbers. On the contrary ,the geometry often suggests the method of proof of particular theorem, and sometimes a geometric argument is more illuminating than a purely analytic proof(one depending entirely on the axioms for the real numbers).In this book, geometric arguments are used to large to help motivate or clarify a particular discuss. 译文：几何化地表示实数的方法是一种非常有益的辅助手段，它可以帮助我们发现和更好的了解实数的某些性质。然而，读者应该意识到，那些将要被采用作为定理的所有有关实数的性质时不应该使用几何学。相反地，几何学经常启发特殊定理的证明方法，而且，有时候，几何学方面的论点比纯分析（它完全依赖与实数的公理）的证明更直观。在本书中，几何学的论点会大范围地使用，以帮助人们推动或阐明一些特殊的讨论。

2.Equations are of very great use. We can use equations in many mathematical problems. We may notice that almost every problem gives us one or more statement that something is equal to something; this gives us equations, with which we may work if we need to. To solve an equation means to find the value of the unknown term. To do this, we must change the terms about until the unknown term stand alone on one side of the unknown and the answer to the question. To solve the equation, therefore, means to move and change the terms about without making the equation untrue, until only the unknown quantity is left on one side, on matter which side.

译文：方程的用处很大。我们能将方程用于许多数学问题。我们或许注意到几乎每一个问题都给了我们以一种或多种表示某物和某物相等的说明；这就是给出了方程，如果我们需要的话，我们就可以进行运算、解方程就是找出未知数的值。要做到这点我们必须一项直到使未知项单独处于方程的一边为止，这样一来，就是使得它等于方程另一边的那些项。然后，我们就得到未知数的值也就是问题的答案。因此解方程意味着进行移项，而不是方程失真，直到方程的一边（无论那一边）只留下一个未知数时为止。

3.The study of differential equation is one part of mathematics that, perhaps more than any other, has been directly inspired by mechanics, astronomy, and mathematical physics. Its history began in the 17th century when Newton, Leibniz, and the Bernoullis solved some simple differential equations arising from problems in geometry and mechanics. These early discoveries, beginning about 1690,gradually led to the development of a lot of “s pecial tricks”for solving certain special kings of differential equations, Although these special tricks are applicable in relatively few cases, they do enable us to solve many differential equations that arise in mechanics and geometry. 译文：微分方程的研究是数学的一个部分，它可能比其他部分，更多地直接受到了理学，天文学和数学物理的推动。它的历史开始于17世纪，当时，牛顿，莱布尼茨和伯努利家族解决了一些来源于几何学和力学的简单的微分方程。这些早期发现，大约开始于1690年，逐渐导致了解决一些特定类型的微分方程的大量的“特殊窍门”的发展。尽管这些特殊窍门只适用于相当少的情形，它们确实能使我们解决许多起源与力学和几何学的微分方程。

4. A large variety of scientific problems arise in which one tries to determine something from its rate of change. For example, we could try to compute the position of a moving particle from a knowledge of its velocity or acceleration. Or a radioactive substance may be disintegrating at a known rate and we may be required to determine the amount of material preset after a give time. In example like these, we are trying to determine an unknown function from prescribed information expressed in the form of an equation involving are least one of the derivatives of the unknown function. These equations are called differential equations, and their study forms one of the most challenging branches of mathematics.

译文：大量的科学问题人们根据事物的变化率去确定改事物（的量）。例如，我们可能试图根据速度或加速度的知识计算一个移动微粒的位置；又如，某种放射性物质可能正在以已知的速度进行衰变，需要我们确定在给定的时间后遗留物质的总量。在类似的例子中，我们力求通过以方程的形式表述的指定来确定未知函数，而这种方程至少包含了未知函数的一个导数。这些方程被称为微分方程，而且它们的研究形成了数学上最具有挑战性的分支之一。

5. In discussing any branch of mathematics, be it analysis, algebra, or geometry, it is helpful to use the notation and terminology of set theory. This subject, which was developed by Boole and Cantor in the latter part of the 19th century, has had a profound influence on the development of mathematics in the in the 20th century. It has unified many seemingly disconnected ideas and has helped to reduce many mathematical concepts to their logical foundations in an elegant and systematic way. In mathematics, the word “set” is used to represent a collection of objects viewed as a single entity. The individual objects in the collection are called elements or members of the set, and they are said to belong to or to be contained in the set. The set is said to contain or be composed of its elements. In many applications it is convenient to deal with abstract sets. Abstract set theory has been developed to deal with such collections of arbitrary objects, and from this generality the theory derives its power.

译文：

在讨论数学的任何一个分支时，也许是分析学，代数学，几何学，使用集合论的记号和术语是非常有益的。这个学科是由布尔和康尔在19世纪后期发展起来的，它对于20世纪数学的发展具有深远的影响。它统一了许多看起来似乎不相关的概念，而且以简练的、系统化的方式帮助人们把许多数学概念归并到它们的逻辑基础。在数学中，这个词“集合”被用来表示看成一个单个整体的一些物体的集合。集合中单个的物体称为集合的元素和成员，集合被称为包含或由元素组成。在许多应用中，

处理抽象集是比较方便的。抽象集合论已经被发展来处理任意物体所做成的集合，而且集合论从这种广泛性中获得了力量。

6. In discussions involving probability, one often sees phrases from everyday language such as “two events are equally likely,” “an event is impossible,” or “an event is certain to occur.” Expressions of this sort have intuitive appeal and it is both pleasant and helpful to be able to employ such colorful language in mathematical discussions. Before we can do so, however, it is necessary to explain the meaning of this language in terms of the fundamental concepts of our theory. Because of the way probability is used in practice, it is convenient to imagine that each probability space (S, B, P) is associated with a real or conceptual experiment. The universal set S can then be thought of as the collection of all conceivable outcomes of the experiment, as in the example of coin tossing discussed in the foregoing section. Each element of S is called an outcome or a sample and the subsets of S that occur in the Boolean algebra B are called event. The reasons for this terminology will become more apparent when we treat some examples.

译文：

在涉及概率的讨论中，人们常会看到来自日常用语的词汇，例如：“两个事件是等可能的”，“一个时间是不可能的”，或者“一个事件肯定会发生”。这样的表达具有直观的吸引力，而且它既会让人感到愉快，同时能够在数学讨论中使用如此绘声绘色的语言也是有益的。然而，在我们能够这样做之前，我们有必要解释根据理论的基本概念解释这种语言的意义。根据实践中使用概率的方式，假设每个概率空间(S,B,P)与一个真实的或者概念上的试验相联系是方便的。将全集S看成这个试验所有可以想象到的结果的集合，就像前面一节中掷硬币试验的那个例子一样。S中的每一个元素叫做一个结果或者样本，而且出现在布尔代数B中的S的子集称为事件。使用这个术语的原因当我们讨论某些例子的时候会越来越清楚。

五、请将下面一段话翻译成英语。

（1）数学来源于人类的社会实践，例如，工农业生产，商业活动，军事行动及科技研究。而且反过来，数学服务于实践且在所有的领域中扮演了重要角色。如果没有数学的应用，没有哪个现代科学技术分支能够有规律的向前发展。

译文：

Mathematics comes from man’s social practice, for example, ind ustrial and agricultural production, commercial activities, military operations and scientific and technological researches. And in turn, mathematics serves the practice and plays a great role in all fields. No modern scientific and technological branches could be regularly developed without the application of mathematics.

（2）十七世纪工业的迅猛发展促进了经济和技术的发展，并且要求处理变量。从常量到变量的飞跃产生了两个新的数学分支——解析几何与微积分，它们都属于高等数学。现在，高等数学中有许多分支，当中有数学分析，高等代数，微分方程，函数论等等。

译文：

The rapid development of in 17th century promoted the progress of economics and technology and required dealing with variable quantities. The leap from constants to variable quantities brought about two new branches of mathematics –analytic geometry and calculus, which belong to the higher mathematics. Now there are many branches in higher algebra, differential equations, function theory and so on.

六、名词翻译

Addition 加法 higher algebra 高等代数 concept 概念 constant 常数 definition 定义 division 除法Equality等式equation方程 equation of condition 条件等式 differential equation微分方程 linear equation二次方程

Figure插图formula 公式function函数geometry 几何学identity恒等式mathematical analysis 数学分析mathematics数学Elementary mathematics初等数学 multiplication 乘法 root根 set集合 subtraction 减法 term项theorem 定理 variable变量expression 表达式 fraction分数 parenthesis 圆括号 ratio比例 angle 角 arc弧 major arc 优弧minor arc劣弧 center 圆心 chord 弦 circle 圆周 circumference 周长 cone 圆锥 cube 立方体 curve 曲线cylinder 柱体 diameter 直径 line 直线 line segment 直线段 broken line折线 straight line 直线 plane几何pyramid棱锥radius 半径 ray 射线 semicircle 半圆 side 边 solid 立体 sphere 球面 surface 曲面 triangle 三角形acute angle 锐角adjacent side 邻边 complementary 互余的 hypotenuse 斜边 initial side of an angle 角的始边obtuse angle钝角 right angle直角 right triangle 直角三角形 straight angle 平角 supplementary 互补的 the terminal 终边 vertex 顶点 venn diagram 文氏图 even integer偶数 if and only if当且仅当 positive number 正数roster notation枚举法 subset 子集universal set 全集 void set 空集 composite 合数 odd number 奇数 prime 素数

reduce to 化简为

whole number [非负]整数 difference 差 field 域 geometric interpretation 几何意义 induction （数学）归纳法 proof by induction 归纳法证明 inequality 不等式 integer 整数 irrational 无理的 irrational number 无理数

negative number 负数 product 积 quotient 商sum 和 decimal 小数 distributive property 分配律 exponent 指数fraction 分式 index 指数 minus 减号 multiplication distributive over addition 乘法对加法的分配 numerator 分子 plural 复数 plus 加号、正号 power 乘方 abscissa 横坐标 analytic geometry 解析几何area 面积 circular region 圆域 coordinate 坐标 coordinate axis 坐标轴hypotenuse 斜边 integral 积分 ordinate 纵坐标 the origin 坐标原点 parabolic 抛物弓形 quadrant 象限 region 区域 triangle 三角形 right triangle 直角三角形parabola 抛物线 the absolute-value function 绝对值函数 cube 立方体edge 边、棱 function idea 函数思想 the identity function 恒等函数 limit 极限 prime 质数 range 值域 real variable 实变量the real-valued function 实值函数 the triangle inequality 三角不等式 volume 体积 characteristic function 特征函数 inverse function 反函数 mapping 映射 simple function 简单函数 complex-valued sequence 复值序列 converge 收敛 diverge 发散imaginary part 虚部 infinite sequence 无穷序列 real part 实部 real valued sequence 实值序列 recursion formula 递推公式sequence 数列approach zero 趋于0 average rate 平均变化率 average value 平均值 derivative 导数 the first derivative 一阶导数 the second derivative 二阶导数 difference quotient 差商 fraction 分数 open interval 开区间linear function 线性函数 numerator 分子 slope 斜率 tangent 正切 directed line 有向直线

directional derivative 方向导数the derivative on the left 左导数the derivative on the right 右导数 evolution 开方 involution 乘方 logarithm 对数the naturalsystem 自然对数系 partial derivative 偏导数 approximate evaluation 近似估计 differential equation 微分方程exponential function 指数函数initial-valued problem 初值问题 integrate 对…积分integration 积分 logarithm function 对数函数 ordinary differential equation 常微分方程 partial differential equation 偏微分方程trigonometric function 三角函数 inverse trigonometric function 反三角函数 definite integral 不定积分 rational function 有理函数inproper integral 反常积分 total differential 全微分 integrand function 被积函数 approximate 约等于 arbitrary constant 任意常数 complex number 复数 evaluate 求值 factor 因式imaginary number 虚数 step function 阶梯函数 strict inequality 严格不等式

plane analytic geometry 平面解析几何 dependent variable 因变量 impossible event 不可能事件 matrix 矩阵 quadrant 象限 closed interval 闭区间 probability theory 概率论 tangent line 切线 operation research 运筹学 combinatorial mathematics 组合数学 linear space 线性空间 the Differential and integral calculus 微积分

ordinary differential equation 常微分方程 algebra 代数学 Trigonometry 三角学 dimension 维数 proper set 真子集

rational number 有理数 denominator 分母 coordinate system 坐标系 vector 向量 domain 定义域 series 级数 initial condition 初始条件 polynomial 多项式 primitive function 原函数

Find the area under the parabola 222,y x x =-+above the x axis , and between 0x =and 1x =

Calculate the derivative of the following function x

y x =