数学专业英语课后答案(1)

数学专业英语课后答案2.1数学、方程与比例词组翻译1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place汉译英（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。

数学专业英语课后答案（1）2.1数学、方程与比例词组翻译1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place 汉译英（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。

数学专业英语课后答案1、He kept walking up and down, which was a sure（）that he was very worried. [单选题] *A. sign(正确答案)B. characterC. natureD. end2、20．Jerry is hard-working. It’s not ______ that he can pass the exam easily. [单选题] * A．surpriseB．surprising (正确答案)C．surprisedD．surprises3、--What’s the weather like today?--It’s _______. [单选题] *A. rainB. windy(正确答案)C. sunD. wind4、Many of my classmates are working _______volunteers. [单选题] *A. as(正确答案)B. toC. atD. like5、Mary is interested ______ hiking. [单选题] *A. onB. byC. in(正确答案)D. at6、We need a _______ when we travel around a new place. [单选题] *A. guide(正确答案)B. touristC. painterD. teacher7、While they were in discussion, their manager came in by chance. [单选题] *A. 抓住时机C. 碰巧(正确答案)D. 及时8、Our campus is _____ big that we need a bike to make it. [单选题] *A. veryB. so(正确答案)C. suchD. much9、57．Next week will be Lisa's birthday. I will send her a birthday present ________ post. [单选题] *A．withB．forC．by(正确答案)D．in10、The manager demanded that all employees _____ on time. [单选题] *A. be(正确答案)B. areC. to be11、He has two sisters but I have not _____. [单选题] *A. noneB. someC. onesD. any(正确答案)12、______! It’s not the end of the world. Let’s try it again.（）[单选题] *A. Put upB. Set upC. Cheer up(正确答案)D. Pick up13、My brother usually _______ his room after school. But now he _______ soccer. [单选题] *A. cleans; playsB. cleaning; playingC. cleans; is playing(正确答案)D. cleans; is playing the14、The museum is _______ in the northeast of Changsha. [单选题] *A. sitB. located(正确答案)C. liesD. stand15、______ my great joy, I met an old friend I haven' t seen for years ______ my way ______ town. [单选题] *A. To, in, forB. To, on, to(正确答案)C. With, in, toD. For, in, for16、In the closet（）a pair of trousers his parents bought for his birthday. [单选题] *A. lyingB. lies(正确答案)c. lieD. is lain17、44．—Hi, Lucy. You ________ very beautiful in the new dress today.—Thank you very much. [单选题] *A．look(正确答案)B．watchC．look atD．see18、They all choose me ______ our class monitor.（）[单选题] *A. as(正确答案)B. inC. withD. on19、My mother’s birthday is coming. I want to buy a new shirt ______ her.（）[单选题] *A. atB. for(正确答案)C. toD. with20、--Could you please tell me _______ to get to the nearest supermarket?--Sorry, I am a stranger here. [单选题] *A. whatB. how(正确答案)C. whenD. why21、?I am good at schoolwork. I often help my classmates _______ English. [单选题] *A. atB. toC. inD. with(正确答案)22、He was very excited to read the news _____ Mo Yan had won the Nobel Prize for literature [单选题] *A. whichB. whatC. howD. that(正确答案)23、Becky is having a great time ______ her aunt in Shanghai. （）[单选题] *A. to visitB. visitedC. visitsD. visiting(正确答案)24、In the future, people ______ a new kind of clothes that will be warm when they are cold, and cool when they’re hot.（）[单选题] *A. wearB. woreC. are wearingD. will wear(正确答案)25、This message is _______. We are all _______ at it. [单选题] *A. surprising; surprisingB. surprised; surprisedC. surprising; surprised(正确答案)D. surprised; surprising26、Your homework must_______ tomorrow. [单选题] *A. hand inB. is handed inC. hands inD. be handed in(正确答案)27、Since the war their country has taken many important steps to improve its economic situation. [单选题] *A. 制定B. 提出C. 讨论D. 采取(正确答案)28、How _______ Grace grows! She’s almost as tall as her mother now. [单选题] *A. cuteB. strongC. fast(正确答案)D. clever29、_____ is not known yet. [单选题] *A. Although he is serious about itB. No matter how we will do the taskC. Whether we will go outing or not(正确答案)D. Unless they come to see us30、You cannot see the doctor _____ you have made an appointment with him. [单选题] *A. exceptB.evenC. howeverD.unless(正确答案)。

高等数学双语教材答案Chapter 1: Limits and ContinuitySection 1: Introduction to Limits and ContinuityThe concept of limits and continuity is fundamental in higher mathematics. In this section, we will introduce the basic definitions and properties associated with limits and continuity.1.1 Definitions of LimitsIn order to understand limits, we need to define what it means for a function to approach a particular value. Let f(x) be a function defined on an open interval containing c, except possibly at c. We say that the limit of f(x) as x approaches c is L, denoted bylim (x→c) f(x) = L, if for every ε > 0, there exists a δ > 0 such that |f(x) - L| < ε whenever 0 < |x - c| < δ.1.2 Basic Limit LawsOnce we have a clear understanding of limits, we can explore some basic laws that govern their behavior. These laws include the sum law, constant multiple law, product law, quotient law, and the power law.1.3 ContinuityA function f(x) is said to be continuous at a point c if three conditions are met: (1) f(c) is defined, (2) the limit of f(x) as x approaches c exists, and (3) the limit of f(x) as x approaches c is equal to f(c). We can also discuss continuity on an interval or at infinity.Chapter 2: DifferentiationSection 1: Introduction to DifferentiationDifferentiation is an important concept in calculus that allows us to find the rate at which a function is changing at any given point. In this section, we will introduce the concept of differentiation and its applications.2.1 Derivative DefinitionThe derivative of a function f(x) at a point c is defined as the limit of the difference quotient as h approaches 0. Mathematically, this can be written as f'(c) = lim (h→0) [(f(c + h) - f(c))/h].2.2 Differentiation RulesThere are several rules that allow us to find the derivative of a function quickly. These rules include the constant rule, power rule, sum rule, difference rule, product rule, quotient rule, and chain rule.2.3 Applications of DifferentiationDifferentiation has many applications in various fields, such as physics, economics, and engineering. It can be used to find maximum and minimum values, determine rates of change, and solve optimization problems.Chapter 3: IntegrationSection 1: Introduction to IntegrationIntegration is the reverse process of differentiation. It enables us to find the area under a curve and solve various mathematical problems. In this section, we will introduce the concept of integration and its applications.3.1 Indefinite IntegralsThe indefinite integral of a function f(x) is the collection of all antiderivatives of f(x). It is denoted by ∫ f(x) dx and represents a family of functions rather than a single value.3.2 Integration TechniquesThere are various techniques for finding antiderivatives and evaluating definite integrals. These techniques include basic integration rules, substitution, integration by parts, and trigonometric substitution.3.3 Applications of IntegrationIntegration has numerous applications, such as finding the area between two curves, calculating the length of curves, determining volumes of solids, and solving differential equations.ConclusionIn conclusion, the study of high-level mathematics, particularly limits, continuity, differentiation, and integration, is crucial for a comprehensive understanding of advanced mathematical concepts. This article has provided a brief overview of these topics, highlighting their definitions, properties, and applications. By mastering these concepts, students can develop strong problem-solving skills and apply them in various academic and real-world scenarios.。

数学建模第一章 Modeling Change1.Difference equation （差分方程） Example 1. A Savings CertificateSuppose that you deposited $1,000 into your saving account initially. The interest is paid each month at a rate of 1% per month, then the value of your account will be as follows ：Letn=number of months,n a =the value of the account after n monthsThen we have ,01.01n n n n a a a a =-=∆+ So the difference equation is n n n a a a 01.01+=+ We have the dynamical system(动态系统) model{1000,.......2,1,001.10,1===+a n a a n n*Suppose you withdraw(提取) $50 each month,then the change is5001.01-=-=∆+n n n n a a a aTherefore the model becomes{1000....2,1,0,5001.101==-=+a n a a n n1.2动态方程,1b ra a n n +=+r and b are constants.(1≠r )An equilibrium value(均衡值)（or fixed point(不动点)）of a dynamical system )(1n n a f a =+is a solution a to equation a=f(a),which means that,if a a =0,then all a a n = Thus in this case the equilibrium value is rb a -=1 1st month $1,000 0a 3a 2a 1a 4th month $1,030.30 3rd month $1,020.10 2nd month $1,010Nowbra a b ra a n n +=+=+1Therefore )(1a a r a a n n -=-+Set a a b n n -=, then n n rb b =+1,thus 0b r b n n =,i.e, )(0a a r a a n n -=- hence rbr b a r a n n -+--=1)1(0 In practice, we may write rbc r a n n -+*=1 write C to be determined by 0a 1.3差分方程组【求均衡点 实际意义 说明参数】 Example 1. A Car Rental CompanyA car rental company operates in Orlando and Tampa. A traveler will rent a car in one city and return the car in either of the cities. The company wants to know if there are sufficiently many cars in each city.Let On=number of.cars in Orlando after n days Tn=number of cars in Tampa after n days Then the model is{nn n n n n T O T T O O 7.04.03.06.011+=+=++ To find the equilibrium value:{TO T T O O 7.04.03.06.0+=+=So 4O=3T ，i.e.,if 730=O Total cars and 740=T Total cars,then n O and n T will be unchanged.第二章 The Modeling Process, Proportionality, and Geometric Similarity(几何相似)【写一定的假设（Assumption ）要明确合理】 We already know kx y x y =⇔∝,k is constant.We may also consider .,,ln ,2etc e y x y x y x ∝∝∝Also y=mx+b is a usual assumption,i.e.,x b y ∝-.Geometrically, it is a straight line,which is easy to spot. Example 2. Modeling a Bass （欧洲鲈鱼） Fishing DerbyA fishing club will hold a fishing contest. In order to be environment friendly, the fish will be released immediately after caught. How to determine the weight of a fish?Problem Identification: Determine the weight of a fish in terms of some easily measurable dimensions(度量)Assumption: All fishes are geometrically similar, and the density of a bass is constant.Thus weight W ∝volume V ∝length 3L .that is,3kL W =Model Refinement: We only assume that the cross sectional areas are similar and use another dimension – girth g.Assume effective V ≈ length ⨯average cross sectional areaNow effective length ∝L average cross sectional area ∝2gThus ∝W L 2g ,i.e.,2kLg W =第三章 Least-Squares Criterion:Minimize the sum of the squares of deviations.(最小二乘法)【怎样画散点图 画趋势线 怎样运用最小二乘法公式】 Fitting a Straight LineGiven a collection of data(i i y x ,),i=1,.....m,and a linear model y=ax+b Recall the deviation of the model y=f(x) at (i i y x ,) is )(i i x f y - Thus the least-squares criterion is to minimize 2121)())((∑∑==--=-=mi iimi iib ax y x f y S79LCrossTherefore we need to solve for a and b from0)1()(20)()(211=---=∂∂=---=∂∂∑∑==mi i i i mi i i b ax y b S x b ax y a S That is∑∑∑∑∑=+=+ii i i i i y mb a x y x b x a x )()()(2We get 最小二乘法公式（其中m 为数据个数）)(,)()(,)(22222截距斜率Intercept x x m x y x y x b Slope x x m y x y x m a i iii i i i i i i i i i ∑∑∑∑∑∑∑∑∑∑∑--=--=第四章 Experimental Modeling 【给出一个散点图再给出数据然后怎样变化可让散点图直一点】Thus if the original curve is （1）concave up:（凸）Then usey or lny to squeeze the tail downward,OR use 2x or 3x to stretch the tail to theright(2) concave down:(凹)：Then use 2y or 3y to stretch the tail up-ward,OR usex or lnx to squeeze the tail to the left第五章 Simulation Modeling 【给一个随机现象描述模拟过程（random number 随机数） 按概率来分 用公式语言描述结果】Monte Carlo Fair Dice Algorithm Flip of a Fair Coin （抛硬币）： Head Tail0 0.51Let x be a random number in [0,1],define掷骰子第九章. Graphs of Functions as Models （量纲分析） Mass(质量) M Momentum （动量） 1-MTLLength(长度) LWork （功） 22-T ML Velocity （速度）1-LTDensity （密度） 3-ML Acceleration （加速度） 2-LT Viscosity （摩擦系数） 11--T ML Specific weight(重量) 22--T ML Pressure （压力） 21--T ML Force （力） 2-MTL Surface tension(张力) 2-MT Frequency （频率） 1-T Power （功率） 32-T ML Angular velocity （角速度） 1-T Rotational inertia （惯性） 2MLAngular acceleration （角加速度） 2-TTorque （转力距） 22-T ML Angular momentum （角动量） 12-T MLEntropy （能量） 22-T ML Energy （能量）22-T MLHeat22-L MLExample 1. Drag Force on a SubmarineWe are interested in the drag force experienced by a submarine. The main factors are Fluid velocity v,Characteristic dimension r (the length),Fluid density ρ,Fluid viscosity μ.Thus the model is f(D,v,r,ρ,μ)=0We haveD v rρμ2-MTL1-LT L 3-ML11--TMLTo find dimensionless(量纲) products 1)()()()()(11312=-----edcba TMLMLLLTMLTWe haveChoose a and e as free variables,then(1)a=1,e=0:b= -2,d=-1,c= -2,thusρρ221221rvDrDv==∏---(2)a=0,e=1:b= -1,d= -1,c= -1,thusρμμρvrrv==∏---1112Note that21∏is the Reynolds numberHence we have the model )(21∏=∏h,this is )(22ρμρvrhrvD=Suppose we use the model to test the drag force with rrm101=第十章Graphs of Functions as Models【军备竞赛能源危机】军备竞赛 Observations:(1) y is increasing, that is, y'>0. (2) y is concave up, that is, y''≤ 0. (3) If x=my, then y=y0 /sm.We propose the continuous model10,/0<<=s S y y y x ，Similary 10,/0<<=t t x x xy （S,t 为各自的生存率） (1) Change in 0y :If X increases its civil defense, then 0y and y' both increase. Therefore the curve y=f(x) shifts upward and has a larger slope than before.On the other hand, if missiles of Y are more effective, then 0y and y' decrease. Therefore the curve shifts downward and has a smaller slope than before.(2) Change in s:If missiles of Y are well protected, then s increases and y' decreases. Therefore the curve y=f(x) rotates downward and has a smaller slope than before.On the other hand, if the technology and weapon effectiveness of X ’s missiles is improved, then s decreases and y' increases. Therefore the curve rotates upward and has a larger slope than before.(3) Change in exchange ratio e=x/y:If X uses multiple warheads, then e increases. Therefore the curve y=f(x) rotates upward and has a larger slope than before.能源危机（供求曲线）SupposeS(q) = p* + α(q – q*), D(q) = p* – β(q – q*).After a tax of t, the new supply curve is S'(q).The new supply curve isS'(q) = p* + t + α(q – q*).To find the new equilibrium: S'(q) = D(q), that is, p* + t +α(q – q*) = p* – β(q – q*). Thusq1 = q* – t /(α+β) p1 = p* + βt /(α+β). Hence the price increase is p1 – p* = βt /(α+β). Thus,When D(q) is very steep, consumers will pay a larger portion of the tax; When S(q) is very steep, the industry will pay a larger portion of the tax.第十一章 Modeling with a Differential Equation 【画解的曲线（积分曲线）】 Example: Sketch solution curves (integral curves)：)2)(1('-+=y y y Equilibrium: y = – 1, y = 2Equilibrium point y* is stable ify(t) →y* when y0 is close to y*Therefore the equilibrium y* = –1 is stable but y* = 2 is unstable.Example: Sketch solution curves (integral curves)第十二章Modification: If there is no competition, the model is{y k ym dt dy x k xa dt dx )1()1(21-=-=Logistic modelThen the model with competition is{ymnxkymdtdyxabykxadtdx)1()1(21--=--=。

1.1）function，domain，range，the identity function，the absolute-value function，the real-valued function，real variable2）cube，volume，edge-length，prime，totality3）Hooke's law，stretch，displacement，spring，constant，proportional4）schematic representation，plot，image，output，input5）it is not difficult to imagine，the idea was much too limited2.（1）常用英语字母和希腊字母来表示函数。

Letters of the English and Greek alphabets are often used to denote functions.（2）若 f 是一个给定的函数，x 是定义域里的一个元素，那么记号f(x)用来表示由 f 确定的对应于x 的值。

If f is a given function and if x is an object of its domain, the notation f(x) is used to designate that object in the range which is associated to x by the function f.（3）该射线将两个坐标轴的夹角分成两个相等的角。

The ray makes equal angles with the coordinates axes.（4）可以用许多方式给出函数思想的图解说明。

The function idea may be illustrated schematically in many ways.（5）容易证明，绝对值函数满足三角不等式。

1.1）integer，rational number，irrational number，real number，negative number，the negative，real line，real axis，scale，to the left/right of 2）sum，difference，product，quotient，power，inequality3）axiom，the field axiom，the order axiom4）ordered，entirely complete，Euclidean，appropriate，distinguished，illuminating5）can be deduced formula，formula interchangeably，using a set of formulas，corresponding to an object，proof by induction，the two set to be distinguished2.1）Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”.2）The sum, difference, or product of two integers is an integer, but the quotient of two integers need not be an integer.3）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.4）The geometry often suggests the method of proof of a 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).5）If a set consisting of real numbers satisfies the following conditions we call it an open interval.6）The real number a is the negative number of –a and their absolute values are equal. When a ≠ 0, their notations are different.7）Each real number corresponds to exactly one point on this line and, conversely, each point on the line corresponds to one and only one real number.8）In geometry, the ordering relation among the real numbers can be expressed clearly in real axis.3.1）一个常见的错误是认为x 是一个负数。

高等数学英文教材答案Title: Answer Key for Advanced Mathematics English TextbookI. IntroductionAdvanced Mathematics is a fundamental subject for students pursuing a higher education in STEM fields. To aid in their learning, an English textbook for Advanced Mathematics has been developed. This article serves as an answer key for the textbook, providing students with accurate solutions to exercises and guiding them towards a better understanding of the subject.II. Chapter 1: Functions and Their Graphs1.1 Linear FunctionsExercise 1:Solution: The equation of the line is y = 2x - 3.Exercise 2:Solution: The slope of the line passing through the points (-1, 4) and (3, -2) is -1. The equation of the line is y = -x + 3.1.2 Quadratic FunctionsExercise 1:Solution: The vertex of the parabola y = x^2 - 2x + 1 is (1, 0).Exercise 2:Solution: The equation of the parabola passing through the points (-1, 3), (0, 1), and (2, -1) is y = -x^2 + x + 1.III. Chapter 2: Differentiation2.1 Derivative of FunctionsExercise 1:Solution: The derivative of f(x) = 3x^2 - 2x + 1 is f'(x) = 6x - 2.Exercise 2:Solution: The derivative of g(x) = 5x^3 + 2x^2 - 4x is g'(x) = 15x^2 + 4x - 4.2.2 Rules of DifferentiationExercise 1:Solution: The derivative of h(x) = sin(x) + cos(x) is h'(x) = cos(x) -sin(x).Exercise 2:Solution: The derivative of i(x) = ln(x^2) is i'(x) = 2/x.IV. Chapter 3: Integration3.1 Indefinite IntegralsExercise 1:Solution: The integral of f(x) = 3x^2 - 2x + 1 is F(x) = x^3 - x^2 + x + C, where C is the constant of integration.Exercise 2:Solution: The integral of g(x) = 5x^3 + 2x^2 - 4x is G(x) = x^4/4 +2x^3/3 - 2x^2/2 + C.3.2 Definite IntegralsExercise 1:Solution: The definite integral of h(x) = sin(x) + cos(x) from 0 to π/2 is ∫[0, π/2] (sin(x) + cos(x)) dx = 2.Exercise 2:Solution: The definite integral of i(x) = ln(x) from 1 to e is ∫[1, e] ln(x) dx = 1.V. ConclusionThis answer key provides students using the Advanced Mathematics English textbook with accurate solutions to exercises, enabling them to verify their work and further enhance their understanding of the subject. By following the provided solutions, students can gain confidence in their problem-solving abilities and excel in the field of mathematics.。

高等数学教材答案下册英语Unit 1: Functions and Their GraphsChapter 1: Linear Functions1.1 Functions and Their Representations1.2 Domain and Range1.3 Linear Functions and EquationsChapter 2: Quadratic Functions2.1 Graphs of Quadratic Functions2.2 Solving Quadratic Equations2.3 Quadratic Functions and Their Transformations Chapter 3: Exponential and Logarithmic Functions3.1 Exponential Functions and Their Graphs3.2 Logarithmic Functions and Their Graphs3.3 Exponential and Logarithmic EquationsUnit 2: Limits and ContinuityChapter 4: Limits and Continuity4.1 Limits and Their Properties4.2 Continuity and Its Properties4.3 Computing LimitsChapter 5: Derivatives5.1 The Derivative and its Interpretation5.2 Differentiation Techniques5.3 Applications of DerivativesChapter 6: Higher-Order Derivatives6.1 Higher-Order Derivatives and Their Interpretations 6.2 The Chain Rule6.3 Implicit DifferentiationUnit 3: IntegrationChapter 7: Antiderivatives and Indefinite Integrals 7.1 Antiderivatives and Their Properties7.2 Indefinite Integrals7.3 Substitution MethodChapter 8: Definite Integrals and Their Applications 8.1 Definite Integrals and Their Properties8.2 Applications of Definite Integrals8.3 Numerical IntegrationChapter 9: Techniques of Integration9.1 Integration by Parts9.2 Trigonometric Integrals9.3 Trigonometric SubstitutionUnit 4: Differential Equations and Applications Chapter 10: First-Order Differential Equations 10.1 Separable Differential Equations10.2 Linear Differential Equations10.3 Applications of Differential Equations Chapter 11: Applications of Differential Calculus 11.1 Optimization11.2 Related Rates11.3 Newton's MethodChapter 12: Sequences and Series12.1 Sequences and Their Limits12.2 Infinite Series12.3 Convergence TestsUnit 5: Multivariable CalculusChapter 13: Functions of Several Variables 13.1 Functions of Two or More Variables13.2 Partial Derivatives13.3 Optimization of Functions of Two VariablesChapter 14: Multiple Integrals14.1 Double Integrals14.2 Triple Integrals14.3 Applications of Multiple IntegralsChapter 15: Vector Calculus15.1 Vector Fields15.2 Line Integrals15.3 Green's TheoremChapter 16: Differential Calculus of Vector Fields16.1 Gradient Fields and Potential Functions16.2 Divergence and Curl16.3 Stokes' TheoremI hope the above chapters and sections provide a comprehensive overview of the answers to the exercises and problems in the textbook. Remember to utilize this answer key as a useful tool to check your understanding and progress in studying advanced mathematics.。

高等数学的英文版教材答案I. Introduction to Higher MathematicsHigher Mathematics is a fundamental subject that plays a crucial role in various fields of study, such as engineering, physics, economics, and computer science. To fully comprehend the concepts and principles of Higher Mathematics, it is essential to have access to accurate and comprehensive textbooks. This article aims to provide an overview of the English version of a Higher Mathematics textbook, focusing on its content, structure, and the importance of having reliable answer keys.II. Content of the English Version of Higher Mathematics TextbookThe English version of the Higher Mathematics textbook covers a wide range of mathematical topics, including calculus, linear algebra, differential equations, probability theory, and mathematical modeling. Each chapter delves into specific concepts and provides detailed explanations, accompanied by illustrative examples to enhance understanding.1. Calculus:The section on calculus is divided into differential calculus and integral calculus. It introduces key principles such as limits, derivatives, and integrals. Various calculus techniques, such as the chain rule, product rule, and fundamental theorem of calculus, are thoroughly explained.2. Linear Algebra:The linear algebra section encompasses topics like vector spaces, matrices, determinants, and eigenvalues. It illustrates the fundamentaltheories of linear algebra and introduces essential concepts, including linear transformations, spanning sets, and eigenvectors.3. Differential Equations:In the differential equations chapter, students learn about different types of differential equations, such as first-order, second-order, and higher-order equations. The solution methods for both homogeneous and non-homogeneous equations are outlined, along with applications in various fields.4. Probability Theory:The section on probability theory introduces students to basic concepts, such as random variables, probability distributions, and expected values. It covers topics like binomial, Poisson, and normal distributions, enabling students to apply probability theory in real-life situations.5. Mathematical Modeling:The mathematical modeling section explores the process of formulating a mathematical model to describe real-world phenomena. It highlights the importance of applying mathematical techniques to analyze and solve practical problems.III. Structure of the English Version of Higher Mathematics TextbookThe English version of the Higher Mathematics textbook follows a well-organized structure, with each chapter building upon the previous one. It presents complex mathematical content in a systematic manner, ensuring a progressive learning experience for students.1. Chapter Organization:Each chapter begins with an introduction that outlines the key concepts and learning objectives. It is followed by concise explanations and illustrative examples that aid in conceptual understanding. The chapters conclude with a summary of the main points covered, allowing for a quick review.2. Exercises and Problems:Throughout the textbook, numerous exercises and problems are provided to reinforce the learned concepts. Students can practice solving mathematical problems to enhance their problem-solving skills and apply the newly acquired knowledge. The textbook also offers answers to the exercises for self-assessment.IV. Importance of Reliable Answer KeysHaving reliable answer keys is crucial for students studying Higher Mathematics. These answer keys serve as a valuable resource to verify the accuracy of their solutions and improve their problem-solving abilities. Furthermore, they provide immediate feedback, allowing students to identify and rectify any mistakes made during the learning process.V. ConclusionThe English version of the Higher Mathematics textbook serves as a comprehensive guide for students studying Higher Mathematics. With its well-structured content, illustrative examples, and reliable answer keys, the textbook enhances students' understanding of mathematical concepts and facilitates their learning experience. It equips them with essential skillsrequired for various fields, enabling them to excel in their academic and professional pursuits.。

Mathematica入门教程第1篇第1章MATHEMATICA概述 (3)1.1 M ATHEMATICA的启动与运行 (3)1.2 表达式的输入 (4)1.3 M ATHEMATICA的联机帮助系统 (6)第2章MATHEMATICA的基本量 (8)2.1 数据类型和常数 (8)2.2 变量 (10)2.3 函数 (11)2.4 表 (14)2.5 表达式 (17)2.6 常用的符号 (19)2.7 练习题 (19)第2篇第3章微积分的基本操作 (20)3.1 极限 (20)3.2 微分 (20)3.3 计算积分 (22)3.4 无穷级数 (24)3.5 练习题 (24)第4章微分方程的求解 (26)4.1 微分方程解 (26)4.2 微分方程的数值解 (26)4.3 练习题 (27)第3篇第5章MATHEMATICA的基本运算 (28)5.1 多项式的表示形式 (28)5.2 方程及其根的表示 (29)5.3 求和与求积 (32)5.4 练习题 (33)第6章函数作图 (35)6.1 基本的二维图形 (35)6.2 二维图形元素 (40)6.3 基本三维图形 (42)6.4 练习题 (46)第4篇第7章MATHEMATICA函数大全 (48)7.1 运算符和一些特殊符号，系统常数 (48)7.2 代数计算 (49)7.3 解方程 (50)7.4 微积分 (50)7.5 多项式函数 (51)7.6 随机函数 (52)7.7 数值函数 (52)7.8 表相关函数 (53)7.9 绘图函数 (54)7.10 流程控制 (57)第8章MATHEMATICA程序设计 (59)8.1 模块和块中的变量 (59)8.2 条件结构 (61)8.3 循环结构 (63)8.4 流程控制 (65)8.5 练习题 (67)--------------习题与答案在68页-------------------第1章Mathematica概述1.1 Mathematica的启动与运行Mathematica是美国Wolfram研究公司生产的一种数学分析型的软件，以符号计算见长，也具有高精度的数值计算功能和强大的图形功能。

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数学专业英语试题及参考答案数学专业英语试题一、词汇及短语1. For a long period of the history of mathematics, the centric place of mathematical methods was occupied by the logical deductions “在数学史的很长的时期内，是逻辑推理一直占据数学方法的中心地位”2. An equation is a statement of the equality between two equal numbers or number symbols.equation ：“方程”“等式”等式是关于两个数或数的符号相等的'一种陈述3. In such an equation either the two members are alike, or become alike on performance of the indicated operation. 这种等式的两端要么一样，要么经过执行指定的运算后变成一样。

注“two members”表等号的两端alike 相同的一样的On the performance of …中的“on”引导一个介词短语做状语Either…or…4. is true “成立”5. to more and change the terms移次和变形without making the equation untrue 保持方程同解数学专业英语试题二、句型及典型翻译1. change the terms about 变形2. full of ：有许多的充满的例 The streets are full of people as on a holiday(像假日一样，街上行人川流不息)3. in groups of ten…4. match something against sb. “匹配”例 Long ago ,when people had to count many things ,they matched them against their fingers. 古时候，当人们必须数东西时，在那些东西和自己的手指之间配对。

2.3集合论的基本概念单词、词组1.1集set，子集subset，真子集proper subset，全集universal subset，空集void/ empty set，基地集the underlying set 1.2正数positive number，偶数even integer，图形diagram，文氏图V enn diagram，哑标dummy index，大括号brace 1.3可以被整除的be divisible by，两两不同的distinct from each other，确定的definite，无关紧要的irrelevant/inessential 1.4一样的结论the same conclusion，等同的效果equivalent effect，用大括号表示集sets are designated by braces，把这个图形记作A：this diagram is designated by letter A，区别对象to distinguish between objects，证明定理to prove theorems，把结论可视化to visualize conclusions/consequences汉译英2.1由于小于10且能被3整除的正整数组成的集是整数集的子集。

The set consisting of those positive integers less than 10 which are divisible by 3 is a subset of the set of all integers.2.2如果方便，我们通过在括号中列举元素的办法来表示集。

When convenient，we shall designate sets by displaying the elements in braces.2.3用符号￠表示集的包含关系，也就是说，式子A￠B表示A包含于B。

数学专业英语（1）、Given ε>0,there exists a number N >0 such that ε<-a a n for all N n ≥ 译文：对给定的ε>0，存在一个数N >0使得不等式ε<-a a n 对所有的N n ≥都成立。

（2）、The function )(x f approaches infinity as x tends to zero.译文：当x 趋于0时，函数)(x f 趋于∞。

（3）、Suppose D is an open set with its closure in G .译文：假定D 是一个开区间，且其闭包在G 中。

（4）、Suppose )(x f is a function on domain D such that M x f <)( for all x ∈D ,where M is a constant .译文：假定)(x f 是区域D 上的一个函数，使得对所有x ∈D ，不等式M x f <)(成立，其中M 是一个常数。

（可用“satisfying ”代替上述“such that ”。

）（5）、表示推理的根据常用“by 短语”，有时也用“according to ”。

By Lemma 2 we have x y ≥.译文：根据引理2可推出x y ≥。

（6）、有时用现在分词表示“经过……而得到……”（推论）。

Integrating the above inequality twice,we see that . log )(0t t c t y ≥译文：将上一不等式两次积分得到. log )(0t t c t y ≥。

（7）、表示充分必要条件The sufficient and necessary condition for the equality isα>0 and ≥p 3. 译文：该等式成立的充分必要条件是α>0 且≥p 3。