美国加州数学教材代数04
【美国C++教材PPT】Chapter 04
Testing the Divisor in Program 4-9
Copyright © 2012 Pearson Education, Inc.
Continued…
Testing the Divisor in Program 4-9
Continued…
if Statement in Program 4-2
Copyright © 2012 Pearson Education, Inc.
Flowchart for Program 4-2 Lines 21 and 22
Copyright © 2012 Pearson Education, Inc.
if (expression) statement1; // or block
else if (expression) statement2; // or block
. . // other else ifs . else if (expression)
statementn; // or block
"If it is raining, take an umbrella, else, if it is windy, take a hat, else, take sunglasses”
Copyright © 2012 Pearson Education, Inc.
if/else if Format
12 > 5 is true 7 <= 5 is false
if x is 10, then x == 10 is true, x != 8 is true, and x == 8 is false
zz美国数学本科生、研究生基础课程参考书目
zz美国数学本科生、研究生基础课程参考书目--------------------------------------------------------------------------------转一个港大数学系去哥大的师姐的东西。
那个师姐的最大传奇之处在于，她和他在中科大的男朋友分别是当年从大陆和香港去哥大读数学的唯一的学生……以下正文。
在网上找书的时候恰好看到这个，看着觉得的确是经典书目大全，贴在这里供学弟学妹们参考：）其中所谓第几学年云云，各校要求不同，像我所在的学校，一般学生第一年选三到四门基础课（代数、分析、几何三大类中至少各挑一门），学年末进行qualifying笔试。
第二年开始选自己喜爱方向的高级课程，并通过qualifying口试。
第三年开始做research，并通过第二语言考试（法语或德语或俄语，一般人都选法语，因为代数几何经典大作都是法语的）. 而Princeton就没有基础课，只有seminar类型的课……美国数学研究生基础课程参考书目第一学年秋季学期春季学期几何与拓扑I 几何与拓扑II1、James R. Munkres, Topology较新的拓扑学的教材适用于本科高年级或研究生一年级2、Basic Topology by Armstrong本科生拓扑学教材3、Kelley, General Topology一般拓扑学的经典教材，不过观点较老4、Willard, General Topology一般拓扑学新的经典教材5、Glen Bredon, Topology and geometry研究生一年级的拓扑、几何教材6、Introduction to Topological Manifolds by John M. Lee研究生一年级的拓扑、几何教材，是一本新书7、From calculus to cohomology by Madsen很好的本科生代数拓扑、微分流形教材代数I 代数II1、Abstract Algebra Dummit最好的本科代数学参考书，标准的研究生一年级代数教材2、Algebra Lang标准的研究生一、二年级代数教材，难度很高，适合作参考书3、Algebra Hungerford标准的研究生一年级代数教材，适合作参考书4、Algebra M,Artin标准的本科生代数教材5、Advanced Modern Algebra by Rotman较新的研究生代数教材，很全面6、Algebra：a graduate course by Isaacs较新的研究生代数教材7、Basic algebra V ol I&II by Jacobson经典的代数学全面参考书，适合研究生参考分析基础复分析I 实分析I1、Walter Rudin, Principles of mathematical analysis本科数学分析的标准参考书2、Walter Rudin, Real and complex analysis标准的研究生一年级分析教材3、Lars V. Ahlfors, Complex analysis本科高年级和研究生一年级经典的复分析教材4、Functions of One Complex V ariable I，J.B.Conway研究生级别的单变量复分析经典5、Lang, Complex analysis研究生级别的单变量复分析参考书6、Complex Analysis by Elias M. Stein较新的研究生级别的单变量复分析教材7、Lang, Real and Functional analysis研究生级别的分析参考书8、Royden, Real analysis标准的研究生一年级实分析教材9、Folland, Real analysis标准的研究生一年级实分析教材第二学年秋季学期春季学期代数III 代数IV1、Commutative ring theory, by H. Matsumura较新的研究生交换代数标准教材2、Commutative Algebra I&II by Oscar Zariski , Pierre Samuel 经典的交换代数参考书3、An introduction to Commutative Algebra by Atiyah标准的交换代数入门教材4、An introduction to homological algebra ,by weibel较新的研究生二年级同调代数教材5、A Course in Homological Algebra by P.J.Hilton,U.Stammbach经典全面的同调代数参考书6、Homological Algebra by Cartan经典的同调代数参考书7、Methods of Homological Algebra by Sergei I. Gelfand, Y uri I. Manin高级、经典的同调代数参考书8、Homology by Saunders Mac Lane经典的同调代数系统介绍9、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud 高级的代数几何、交换代数的参考书，最新的交换代数全面参考代数拓扑I 代数拓扑II1、Algebraic Topology, A. Hatcher最新的研究生代数拓扑标准教材2、Spaniers "Algebraic Topology"经典的代数拓扑参考书3、Differential forms in algebraic topology, by Raoul Bott and Loring W. Tu 研究生代数拓扑标准教材4、Massey, A basic course in Algebraic topology经典的研究生代数拓扑教材5、Fulton , Algebraic topology：a first course很好本科生高年级和研究生一年级的代数拓扑参考书6、Glen Bredon, Topology and geometry标准的研究生代数拓扑教材，有相当篇幅讲述光滑流形7、Algebraic Topology Homology and Homotopy高级、经典的代数拓扑参考书8、A Concise Course in Algebraic Topology by J.P.May研究生代数拓扑的入门教材，覆盖范围较广9、Elements of Homotopy Theory by G.W. Whitehead高级、经典的代数拓扑参考书实分析II 泛函分析1、Royden, Real analysis标准研究生分析教材2、Walter Rudin, Real and complex analysis标准研究生分析教材3、Halmos，"Measure Theory"经典的研究生实分析教材，适合作参考书4、Walter Rudin, Functional analysis标准的研究生泛函分析教材5、Conway,A course of Functional analysis标准的研究生泛函分析教材6、Folland, Real analysis标准研究生实分析教材7、Functional Analysis by Lax高级的研究生泛函分析教材8、Functional Analysis by Y oshida高级的研究生泛函分析参考书9、Measure Theory, Donald L. Cohn经典的测度论参考书微分拓扑李群、李代数1、Hirsch, Differential topology标准的研究生微分拓扑教材，有相当难度2、Lang, Differential and Riemannian manifolds研究生微分流形的参考书，难度较高3、Warner,Foundations of Differentiable manifolds and Lie groups标准的研究生微分流形教材，有相当的篇幅讲述李群4、Representation theory: a first course, by W. Fulton and J. Harris李群及其表示论的标准教材5、Lie groups and algebraic groups, by A. L. Onishchik, E. B. V inberg李群的参考书6、Lectures on Lie Groups W.Y.Hsiang李群的参考书7、Introduction to Smooth Manifolds by John M. Lee较新的关于光滑流形的标准教材8、Lie Groups, Lie Algebras, and Their Representation by V.S. V aradarajan最重要的李群、李代数参考书9、Humphreys, Introduction to Lie Algebras and Representation Theory , Springer-V erlag, GTM-9标准的李代数入门教材第三学年秋季学期春季学期微分几何I 微分几何II1、Peter Petersen, Riemannian Geometry标准的黎曼几何教材2、Riemannian Manifolds: An Introduction to Curvature by John M. Lee最新的黎曼几何教材3、doCarmo, Riemannian Geometry.标准的黎曼几何教材4、M. Spivak, A Comprehensive Introduction to Differential Geometry I—V全面的微分几何经典，适合作参考书5、Helgason , Differential Geometry,Lie groups,and symmetric spaces标准的微分几何教材6、Lang, Fundamentals of Differential Geometry最新的微分几何教材，很适合作参考书7、kobayashi/nomizu, Foundations of Differential Geometry经典的微分几何参考书8、Boothby,Introduction to Differentiable manifolds and Riemannian Geometry标准的微分几何入门教材，主要讲述微分流形9、Riemannian Geometry I.Chavel经典的黎曼几何参考书10、Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”V ol 1—3经典的现代几何学参考书代数几何I 代数几何II1、Harris,Algebraic Geometry: a first course代数几何的入门教材2、Algebraic Geometry Robin Hartshorne经典的代数几何教材，难度很高3、Basic Algebraic Geometry 1&2 2nd ed. I.R.Shafarevich.非常好的代数几何入门教材4、Principles of Algebraic Geometry by giffiths/harris全面、经典的代数几何参考书，偏复代数几何5、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud高级的代数几何、交换代数的参考书，最新的交换代数全面参考6、The Geometry of Schemes by Eisenbud很好的研究生代数几何入门教材7、The Red Book of V arieties and Schemes by Mumford标准的研究生代数几何入门教材8、Algebraic Geometry I : Complex Projective V arieties by David Mumford复代数几何的经典调和分析偏微分方程1、An Introduction to Harmonic Analysis,Third Edition Yitzhak Katznelson调和分析的标准教材，很经典2、Evans, Partial differential equations偏微分方程的经典教材3、Aleksei.A.Dezin，Partial differential equations，Springer-V erlag偏微分方程的参考书4、L. Hormander "Linear Partial Differential Operators, " I&II偏微分方程的经典参考书5、A Course in Abstract Harmonic Analysis by Folland高级的研究生调和分析教材6、Abstract Harmonic Analysis by Ross Hewitt抽象调和分析的经典参考书7、Harmonic Analysis by Elias M. Stein标准的研究生调和分析教材8、Elliptic Partial Differential Equations of Second Order by David Gilbarg 偏微分方程的经典参考书9、Partial Differential Equations ，by Jeffrey Rauch标准的研究生偏微分方程教材复分析II 多复分析导论1、Functions of One Complex V ariable II，J.B.Conway单复变的经典教材，第二卷较深入2、Lectures on Riemann Surfaces O.Forster黎曼曲面的参考书3、Compact riemann surfaces Jost黎曼曲面的参考书4、Compact riemann surfaces Narasimhan黎曼曲面的参考书5、Hormander " An introduction to Complex Analysis in Several V ariables" 多复变的标准入门教材6、Riemann surfaces , Lang黎曼曲面的参考书7、Riemann Surfaces by Hershel M. Farkas标准的研究生黎曼曲面教材8、Function Theory of Several Complex V ariables by Steven G. Krantz高级的研究生多复变参考书9、Complex Analysis: The Geometric V iewpoint by Steven G. Krantz高级的研究生复分析参考书专业方向选修课：1、多复分析2、复几何3、几何分析4、抽象调和分析5、代数几何6、代数数论7、微分几何8、代数群、李代数与量子群9、泛函分析与算子代数10、数学物理11、概率理论12、动力系统与遍历理论13、泛代数*数学基础：1、halmos ,native set theory2、fraenkel ,abstract set theory3、ebbinghaus ,mathematical logic4、enderton ,a mathematical introduction to logic5、landau, foundations of analysis6、maclane ,categories for working mathematican 应该在核心课程学习的过程中穿插选修假设本科应有的水平分析Walter Rudin, Principles of mathematical analysis Apostol , mathematical analysisM.spivak , calculus on manifoldsMunknes ,analysis on manifoldsKolmogorov/fomin , introductory real analysis Arnold ,ordinary differential equations代数：linear algebra by Stephen H. Friedberglinear algebra by hoffmanlinear algebra done right by Axleradvanced linear algebra by Romanalgebra ,artina first course in abstract algebra by rotman几何：do carmo, differential geometry of curves and surfaces Differential topology by PollackHilbert ,foundations of geometryJames R. Munkres, Topology。
美国大学数学教材中文版
美国大学数学教材中文版
1.《数学分析》（Calculus），作者：詹姆斯·萨维奇（James Stewart）
2.《线性代数》（Linear Algebra），作者：弗朗西斯·费舍尔（Francis J. Flanigan）
3.《概率论和统计学》（Probability and Statistics），作者：查
尔斯·贝尔（Charles M. Bell）
4.《微积分》（Calculus），作者：约翰·科恩（John C. Kern）
5.《抽象代数》（Abstract Algebra），作者：约翰·拉德贝克（John B. Ladue）
6.《数学分析：实变函数与复变函数》（Real and Complex Analysis），作者：罗伯特·科尔曼（Robert L. Coleman）
7.《数学分析：积分与微分方程》（Integral and Differential Equations），作者：伊恩·米勒（Ian Miller）
8.《几何学》（Geometry），作者：罗伯特·科尔曼（Robert L. Coleman）
9.《数学分析：空间解析几何》（Analytic Geometry），作者：罗伯特·科尔曼（Robert L. Coleman）
10.《数论》（Number Theory），作者：约翰·科恩（John C. Kern）。
024美国数学教育专业本科生教材《中学数学教学》的评介
第13 卷第3 期数学教育学报Vol.13, No.3美国数学教育专业本科生教材《中学数学教学》的评介李清，李淑文（东北师范大学数学与统计学院，吉林长春130024）关键词：美国；教材；数学教学；探究式教学《中学数学教学》是美国马里兰州马里兰大学数学系数学教育专业本科生使用的教材，与国内高师院校数学教育专业本科生使用的教材《数学教育学》或《中学数学教学论》相比，在编写方式和内容上迥然不同．这本教材没有长篇的理论论述，不是深奥的教育理论加数学例子，而是以一名中学数学教师Casey 的数学教学工作经历为脉络的．该教材详细介绍了Casey 在一年的中学数学教学工作中怎样想、怎样做及其理论依据．该教材还列举了大量其他数学教师正确或错误的教学案例，详细地刻画了教师和学生在数学教学过程中的思维活动，真实地反映了中学数学教学的实际，使学习者能更好地结合具体案例理解数学教育理论．读罢全书，顿觉耳目一新，深感获益匪浅．一名高师数学教育专业的本科生若学习了这本教材，在未来的数学教学实际中，他就会知道应该想些什么、做些什么．另外，该书还详细地论述了美国数学课程标准所提倡的教学理论——探究式教学理论．下面就对这本教材作简要评介，以期对我国数学教育学教材的编写及数学教育的发展有所启示和借鉴．1《中学数学教学》的结构和内容（1）一个数学教师生涯的开始．通过Casey 的亲身经历，收稿日期：2004–03–12基金项目：东北师范大学校内青年基金项目（111494029）介绍了从一名大学毕业生到有一年教龄的新教师所经历的各种各样的数学教学活动及其相应的做法，并结合具体的活动内容讲述了一些发展课程、进行数学教学的思想．（2）在有利于做数学的环境中获得学生的合作．课程实施的成功不仅取决于课程设计的好坏，也取决于教师建立的教学环境．本着这一思想，这一章结合实例讲述了在有利于做数学的环境中，教师获得学生合作的教学策略．这些策略有助于教师创设一个学生能够专心、积极主动地从事学习活动，自由地实验、提问题、暴露思维过程的环境．（3）激发学生从事数学学习活动．这一章内容是第二章内容的延伸．着重介绍了在问题解决教学中培养学生做数学的策略和组织学生从事大组展示、问题讨论会、合作学习、独立工作和家庭作业等学习活动的策略．（4）发展数学课程．介绍了“美国学校数学课程与评价标准（PSSM）”的理念与内容，通过具体案例说明了如何设计和PSSM 理念相一致的数学课，而不是由教材所决定的数学课．由教材所决定的数学课使学生获得的数学是一些无意义的定义、符号、规则和运算法则，而基于PSSM 的数学课则是引导学生做有意义的数学．（5）引导学生建构概念和发现关系．该章结合具体案例说明如何设计和进行探究性教学，如何引导学生建构数学概念、发现数学关系．还介绍了测量、监视、评价学生学习情况和教学目标达成情况的策略．（6）引导学生发展知识和运算技能．讲述了如何运用直接教学策略进行学生获得和记忆数学信息，发展运算技能的教学，也介绍了如何设计小测验识别和更正学生错误的运算执行模式．（7）引导学生用数学进行交流．讲述了如何运用探究式和直接式教学策略，引导学生用数学来组织和交流思想，理解数学语言，以及怎样在课堂上与学生合作交流，包括交谈、讲演、聆听、写作和阅读．（8）引导学生创造性地用数学．如何运用探究式教学引导学生把数学应用到现实生活情境中，培养学生的数学创造力和做数学的愿望．（9）评价和报告学生的数学进展．介绍了一个有效的用72 数学教育学报第13 卷于监控和评价学生发展的系统和如何对学生的数学成就进行地道地终结性评价；一些考试的误用情况和怎样解释标准化和核心课程考试的结果．（10）数学教与学的技术资源．一些学数学和教数学的技术资源（包括在中学各阶段使用的技术手段和课程资源，如实物模型、教科书、期刊、音像材料、网站等）．及其作用．教读者如何评价、选择和运用这些技术资源．（11）数学课程与教学实践案例分析．通过分析若干中学数学教学实例，讲解如何对教学案例进行分析及教师在教学过程中运用的一些策略．从实践层面上总结、复习前10章内容．结合具体实例论述了数学课程与其它学科领域课程进行整合的意义和做法．教材每章都由学习目标、正文、综合学习活动 3 个部分组成．章首是该章应达到的学习目标，章尾是该章的综合学习活动和学习下一章的过渡活动．综合学习活动，一方面可以帮助学习者整合、强化、扩展这一章所学习的内容，另一方面可以检测学生这一章的学习情况，从而针对不足进行复习．2《中学数学教学》的特点分析2.1目的明确且重点突出2.2内容编写有利于教师教学和学习者自学教材在内容编写方面按照学习目标、学习内容（包括理论和案例），应用、复习、检测的顺序设计，符合学生的认知特点，便于教师教学和学习者自学．教材每章的开头为本章的任务、目标．章尾有两部分，一部分是本章的习题，统称为综合实践活动，包括选择题和问答题两种题型，内容主要是该章内容的实验、调查和应用；另一部分为引出下一章内容的过渡活动．这样的设计使得学习者有明确的学习目标，学完一章内容后能结合习题进行复习、检测，并在学习下一章内容前结合相关问题，有针对性地进行思考和实践活动，为下一章内容的学习做好准备．2.3注重内容的整体性和关联性数学教学过程是一个整体，这个过程中的各个环节紧密衔接，任何一个环节出现问题都会影响到教学效果．这本教材重视数学教学过程的整体性和关联性，体现在：（2）各章内容虽相对独立，但又相互关联．例如，该书2.4注重理论与实践的结合该教材没有长篇的纯教育理论的讲解，而是在一小段的理论讲解之前或之后，有相应的、具体的、详细的操作策略、图片或案例．案例很多，共189 个，都是从教学实践中节选出来的，有对教师或学生思维过程的记叙，也有教师和学生的对话，其作用或是引出理论，或是对理论进行解释、说明．如第十一章在介绍了数学学科内容和其它学科内容整合理论后，举例说明了教师如何在教学中进行数学学科内容和其它学科内容的整合的．案例的大意是Casey 先把英语、美术、体育等其它学科教师召集在一起，把自己的教学内容介绍给他们，由他们提建议，指出能够整合的知识．然后Casey参与到其他教师的教学过程中去．在体育课的教学中，Casey 来到了运动场，给学生创设了一个好的问题情境，提出了“如何计算跑道的长度问题”，然后在自己的数学课堂上组织学生分析问题，分组讨论和解决问题．书中除了对该教学案例进行了长篇详细的叙述外，还展示了Casey 与其它学科教师讨论后，总结出的应整合的内容表、不同组学生解决这些问题情况的照片等．这样，教育理论不再抽象，而有了具体的实施方法和步骤，使得理论与实践紧密结合起来．此外，教材每一章的习题也重视理论与实践的结合．如，结合本章所学内容进行调查和实践；学生制定计划参加地区或全国的NCTM 会议；找几本教材进行分析、比较；让学生自己去设计课堂教学，并找两个学生试教，然后把教学的结果在同学之间进行交流反思等．使得学习者能把学到的教育理论运用到实际，提高了学习者的教学能力．2.5重视区别化教学思想教材除了贯彻探究式教学思想外，还在各个章节中讲述了区别化教学思想．由于每个学生都有他们的性格、背景、能力、兴趣、动机、体力、信仰等，因此他们处理事情的方式不同，所以教学应注意学生的差异，不能教条．该书不是纯理论式地讲述区别化教学思想，而是结合具体教学过程进行阐述．如第一章中通过Casey 首次和他所教的班级的学生见面采取考试的做法，说明教师应在对学生的个性特点进行了解后再进行教学，这样才能达到公平．又如第二章中通过举了一个学生违反课堂纪律，教师如何进行管理的例子，说明只有根据学生的个性特点采取相应的管理措施，才能控制好课堂教学，取得良好的教学效果．2.6一位教师在教学工作中会产生或遇到大量的文字、数据第3 期李清等：美国数学教育专业本科生教材《中学数学教学》的评介73《中学数学教学》这本教材现在有 3 个版本，分别出版于1992 年，1996 年，2003 年．上面介绍的是第3 版，版式为16 开，共435 页，这一版在前两版的基础上，在内容、语言上都进行了一定的改动，比较及时、全面地反映了当前美国数学教学的情况，是了解美国数学教育专业本科生数学教育教材情况的有益参考．Introduction and Remark of the USA Textbook for the Graduates Majoring in Mathematical EducationLI Qing, LI Shu-wen(School of Mathematics and Statistics, Northeast Normal University, Jinlin Changchun 130024, China)Abstract: “Teaching Mathematics in Secondary and Middle School”, a textbook for the graduates majoring in mathematical education in USA, was quite different from “Mathematics Pedagogy” or “Mathematics Instructional Theory in Secondary and Middle School” that were widely used in our country. The USA textbook had some typical characters such as having definite g oal, laid stress on the key points, the writing style being beneficial to and suitable for teachers’ teaching and students’ learni ng by themselves; emphasizing the integrality and relevance of the content, integration of theory and practice, differentiate teaching idea and how teachers managed the teaching materials.Key words: USA; textbook; mathematics teaching; inquiry teaching[责任编校：刘伟娜]数学新课程研究系列（刘兼，黄翔主编）《数学教育的价值》黄翔编著（价待定）《数学课程设计》刘兼，黄翔，张丹编著（19.50 元）《设计合理的数学教学》马复编著（16.10 元）《数学教育评价》马云鹏，张春莉等编著（17.60 元）《数学新课程与数学学习》孔企平，张维忠，黄荣金编著（20.60 元）《数学课程发展的国际视野》孙晓天主编（22.00 元）《数学课题学习的实践与探索》张思明，白永潇编著（19.90 元）地址：北京市西城区德外大街 4 号邮编：100011书讯书讯。
美本高等数学教材目录
美本高等数学教材目录以下是美本高等数学教材的目录：第一章：微积分基础1.1 实数与数轴1.2 函数与映射1.3 极限与连续1.4 导数与微分1.5 中值定理与导数的应用第二章：高等微积分2.1 不定积分与定积分2.2 微积分基本定理与换元积分法2.3 定积分的应用2.4 曲线的长度、曲率与曲边梯形法2.5 多重积分与重心坐标法2.6 广义积分与反常积分第三章：级数与收敛性3.1 数列的极限3.2 数列的收敛性与敛散性判定3.3 黎曼和与积分3.4 级数的收敛性与敛散性判定3.5 幂级数与泰勒级数第四章：微分方程4.1 常系数线性微分方程4.2 变系数线性微分方程4.3 高阶线性齐次微分方程4.4 高阶线性非齐次微分方程4.5 欧拉方程与二阶线性方程4.6 线性方程组与矩阵方程第五章：多元函数与矢量分析5.1 多元函数的极限与连续5.2 偏导数与全微分5.3 多元函数的导数与链式法则5.4 隐函数与隐函数的微分5.5 多元函数的极值与条件极值5.6 多元函数积分与曲线积分5.7 曲面积分与格林公式第六章：多元函数微分学6.1 方向导数与梯度6.2 多元函数泰勒公式6.3 多元函数的最小二乘法6.4 多元函数的泛函极值6.5 多元函数的约束优化问题6.6 多元函数积分的应用第七章：常微分方程7.1 初等方程的解法与初值问题7.2 高阶常微分方程7.3 常系数线性齐次微分方程7.4 常系数线性非齐次微分方程7.5 高阶常微分方程组7.6 线性微分方程与矩阵第八章：变分法与特殊函数8.1 函数的极值与最优化8.2 随机函数与最优随机过程8.3 欧拉方程与变分法8.4 贝塞尔方程与贝塞尔函数8.5 雅各比方程与雅各比函数8.6 数学物理方程与特殊函数第九章：傅里叶级数与变换9.1 傅里叶级数与周期函数9.2 傅里叶级数的收敛性与性质9.3 傅里叶级数与一般函数9.4 波的传播与振动现象9.5 傅里叶变换与拉普拉斯变换9.6 傅里叶变换与偏微分方程第十章：复变函数10.1 复数与复平面10.2 复变函数的极限与连续10.3 复变函数的导数与全纯函数10.4 保解析函数与解析函数10.5 全纯映射与有限变射10.6 复积分与柯西定理以上是美本高等数学教材的目录，本教材全面系统地介绍了微积分、级数与收敛性、微分方程、多元函数与矢量分析、常微分方程、变分法与特殊函数、傅里叶级数与变换、复变函数等数学知识点，旨在帮助学生打好数学基础，提升数学应用能力。
美国algebra1 Chapter04
CHAPTER 4116C HAPTERT ABLE OF C ONTENTS4-1Solving Equations Using More Than One Operation4-2Simplifying Each Side of anEquation4-3Solving Equations That Have the Variable in Both Sides4-4Using Formulas to SolveProblems4-5Solving for a Variable in T erms of Another Variable4-6T ransforming Formulas4-7Properties of Inequalities4-8Finding and Graphing theSolution Set of an Inequality 4-9Using Inequalities to SolveProblemsChapter SummaryVocabularyReview ExercisesCumulative Review FIRST DEGREE EQUA TIONS AND INEQU ALITIES IN ONE V ARIABLE An equation is an important problem-solving tool.A successful business person must make many deci-sions about business practices.Some of these deci-sions involve known facts,but others require the use of information obtained from equations based on expected trends.For example,an equation can be used to represent the following situation.Helga sews hand-made quilts for sale at a local craft shop.She knows that the mate-rials for the last quilt that she made cost $76 and that it required 44 hours of work to complete the quilt.If Helga received $450 for the quilt,how much did she earn for each hour of work,taking into account the cost of the materials?Most of the problem-solving equations for business are complex.Before you can cope with complex equa-tions,you must learn the basic principles involved in solving any equation.Some Terms and DefinitionsAn equation is a sentence that states that two algebraic expressions are equal.For example,x ϩ3 ϭ9 is an equation in which x ϩ3 is called the left side ,or left member ,and 9 is the right side ,or right member .An equation may be a true sentence such as 5 ϩ2 ϭ7,a false sentence such as 6 Ϫ 3 ϭ4,or an open sentence such as x ϩ3 ϭ9.The number that can replace the variable in an open sentence to make the sentence true is called a root ,or a solution ,of the equation.For example,6 is a root of x + 3 ϭ9.As discussed in Chapter 3,the replacement set or domain is the set of pos-sible values that can be used in place of the variable in an open sentence.If no replacement set is given,the replacement set is the set of real numbers.The set consisting of all elements of the replacement set that are solutions of the open sentence is called the soluti on set of the open sentence.For example,if the replacement set is the set of real numbers,the solution set of x ϩ3 ϭ9 is {6}.If no element of the replacement set makes the open sentence true,the solution set is the empty or null set,or {}.If every element of the domain satisfies an equation,the equation is called an identity .Thus,5 ϩx ϭx Ϫ (Ϫ5) is an iden-tity when the domain is the set of real numbers because every element of the domain makes the sentence true.Two equations that have the same solution set are equivalent equations .To solve an equation is to find its solution set.This is usually done by writing sim-pler equivalent equations.If not every element of the domain makes the sentence true,the equation is called a conditional equation ,or simply an equation.Therefore,x ϩ3 ϭ9 is a conditional equation.Properties of EqualityWhen two numerical or algebraic expressions are equal,it is reasonable to assume that if we change each in the same way,the resulting expressions will be equal.For example:5 ϩ7 ϭ12(5 ϩ7) ϩ3 ϭ12 ϩ3(5 ϩ7) Ϫ8 ϭ12 Ϫ8Ϫ2(5 ϩ7) ϭϪ2(12)These examples suggest the following properties of equality:5 1 735123лSolving Equations Using More Than One Operation 117Properties of Equality1.The addition property of equality.If equals are added to equals,the sums are equal.2.The subtraction property of equality.If equals are subtracted from equals,the differences are equal.3.The multiplication property of equality.If equals are multiplied by equals,the products are equal.4.The division property of equality.If equals are divided by nonzero equals,the quotients are equal.5.The substitution principle.In a statement of equality,a quantity may be substituted for its equal.To solve an equation,you need to work backward or “undo”what has been done by using inverse operations.To undo the addition of a number,add its opposite.For example,to solve the equation x ϩ7 ϭ19,use the addition prop-erty of equality.Add the opposite of 7 to both sides.The variable x is now alone on one side and it is easy to read the solution,x ϭ12.To solve an equation in which the variable has been multiplied by a num-ber,either divide by that number or multiply by its reciprocal.(Remember multiplying by the reciprocal is the same as dividing by the number.) To solve 6x ϭ24,divide both sides by 6 or multiply both sides by .6x ϭ246x ϭ24orx ϭ4x ϭ4To solve ,multiply each side by the reciprocal of which is 3.x ϭ15In the equation 2x ϩ3 ϭ15,there are two operations in the left side:mul-tiplication and addition.In forming the left side of the equation,x was first mul-tiplied by 2,and then 3 was added to the product.To solve this equation,we must undo these operations by using the inverse elements in the reverse order.Since the last operation was to add 3,the first step in solving the equation is to add its opposite,Ϫ3,to both sides of the equation or subtract 3 from both sides(3)x 35(3)5x35513x 35516(6x )516(24)6x6524616x 1 7 5 1927 27x512118First Degree Equations and Inequalities in One Variableof the equation.Here we are using either the addition or the subtraction prop-erty of equality.orNow we have a simpler equation that has the same solution set as the original and includes only multiplication by 2.To solve this simpler equation,we multi-ply both sides of the equation by ,the reciprocal of 2,or divide both sides of the equation by 2.Here we can use either the multiplication or the division property of equality.orAfter an equation has been solved,we check the equation,that is,we verify that the solution does in fact make the given equation true by replacing the vari-able with the solution and performing any computations.Check:2x ϩ3 ϭ152(6) ϩ3 ϭ1512 ϩ3 ϭ1515 ϭ 15 ✔To find the solution of the equation 2x ϩ3 ϭ15,we used several properties of the four basic operations and of equality.The solution below shows the math-ematical principle that we used in each step.2x ϩ3 ϭ15Given(2x ϩ3) ϩ(Ϫ3)ϭ15 ϩ(Ϫ3)Addition property of equality 2x ϩ[3 ϩ(Ϫ3)]ϭ15 ϩ(Ϫ3)Associative property of addition 2x ϩ0ϭ12Additive inverse property 2x ϭ12Additive identity property ϭMultiplication property of equality ϭAssociative property of multiplication 1x ϭ6Multiplicative inverse property x ϭ6Multiplicative identity propertyThese steps and properties are necessary to justify the solution of an equationof this form.However,when solving an equation,we do not need to write each of the steps,as shown in the examples that follow.12(12)C 12(2)D x12(12)12(2x )2x 5122x 25122x 562x 51212(2x )512(12)x 56122x 1 323 2x551523122x 1 35152x 1 3 1 (23)515 1 (23)2x 512Solving Equations Using More Than One Operation 119EXAMPLE 1Solve and check:7x ϩ15 ϭ71Solution How to Proceed(1)Write the equation:(2)Add Ϫ15,the opposite of ϩ15 toeach side:(3)Since multiplication and division areϭinverse operations,divide each x ϭ8side by 7:(4)Check the solution.Write the solution7x ϩ15 ϭ71in place of x and perform the computations:71 ϭ71 ✔Answer x ϭ8Note:The check is based on the substitution principle.EXAMPLE 2Find the solution set and check:ϭ Ϫ18SolutionAnswer The solution set is {Ϫ20}.EXAMPLE 3Solve and check:7 Ϫx ϭ9Solution METHOD 1.Think of 7 Ϫ x as 7 ϩ(Ϫ1x ).35x 2 656 1 15 5717(8) 1 15 5715677x 7120First Degree Equations and Inequalities in One VariableϭϪ12ϭx ϭϪ2053(212)53A 35x B 35x 16 1635x 2 65 218Addition property of equality Multiplication property of equalityCheckϪ6ϭϪ18Ϫ18 ϭ Ϫ18 ✔212 2 6 521835(220) 2 6 5 21835x7x ϩ15ϭ71Ϫ15Ϫ157x ϭ56CheckAddition property of equality Division property of equalityx ϭϪ2METHOD 2.Add x to both sides of the equation so that the variable has apositive coefficient.How to Proceed(1)Write the equation:7 Ϫ x ϭ9(2)Add x to each side of the equation:7 Ϫ x ϩx ϭ9 ϩx7 ϭ9 ϩx(3) Add Ϫ9 to each side of the equation:Ϫ9 ϩ7 ϭϪ9 ϩ9 ϩxϪ2 ϭxThe check is the same as for Method 1.Answer {Ϫ2} or x ϭϪ2Writing About Mathematics1.Is it possible for the equation 2x ϩ5 ϭ0 to have a solution in the set of positive real num-bers? Explain your answer.2.Max wants to solve the equation 7x ϩ15 ϭ71.He begins by multiplying both sides of the equation by ,the reciprocal of the coefficient of x .a.Is it possible for Max to solve the equation if he begins in this way? If so,what would be the result of multiplying by and what would be his next step?b.In this section you learned to solve the equation 7x ϩ15 ϭ71 by first adding the opposite of 15,Ϫ15,to both sides of the equation.Which method do you think is better? Explain your answer.Developing SkillsIn 3 and 4,write a complete solution for each equation,listing the property used in each step.3.3x ϩ5 ϭ354.ϭ1512x 211717EXERCISES21x 21522171(2x ) 5 927272x 52Solving Equations Using More Than One Operation 1217 Ϫ x ϭ99 ϭ9✔7 1 2 597 2 (؊2) 59In 5–32,solve and check each equation.5.55 ϭ6a ϩ76.17 ϭ8c Ϫ 77.9 Ϫ 1x ϭ78.11 ϭ15t ϩ169.15 Ϫ a ϭ310.11 ϭϪ6d Ϫ 111.8 Ϫ y = 112.ϭ1213.ϭϪ814.12 ϭ15.16.ϭ3017.7.2 ϭ18.ϭ519.Ϫ2 ϭ20.21.4a ϩ0.2 ϭ522.4 ϭ3t Ϫ 0.223.ϭ524.13 ϭ25.ϭ4726.0.04c ϩ1.6 ϭ027.15x ϩ14 ϭ1928.8 ϭ18c Ϫ 129.30.0.8r ϩ19 ϭ2031.32.842 Ϫ162m ϭϪ616Applying Skills33.The formula F ϭgives the relationship between the Fahrenheit temperature F andthe Celsius temperature C .Solve the equation 59 ϭto find the temperature in degrees Celsius when the Fahrenheit temperature is 59°.34.When Kurt orders from a catalog,he pays $3.50 for shipping and handling in addition to thecost of the goods that he purchases.Kurt paid $33.20 when he ordered six pairs of socks.Solve the equation 6x ϩ3.50 ϭ33.20 to find x ,the price of one pair of socks.35.When Mattie rents a car for one day,the cost is $29.00 plus $0.20 a mile.On her last trip,Mattie paid $66.40 for the car for one day.Find the number of miles,m ,that Mattie drove by solving the equation 29 ϩ0.20x ϭ66.40.36.On his last trip to the post office,Hal paid $4.30 to mail a package and bought some 39-centstamps.He paid a total of $13.66.Find s ,the number of stamps that he bought,by solving the equation 0.39s ϩ4.30 ϭ13.66.An equation is often written in such a way that one or both sides are not in sim-plest form.Before starting to solve the equation by using additive and multi-plicative inverses,you should simplify each side by removing parentheses if necessary and adding like terms.Recall that an algebraic expression that is a number,a variable,or a prod-uct or quotient of numbers and variables is called a term .First-degree equations in one variable contain two kinds of terms,terms that are constants and terms that contain the variable to the first power only.95C 13295C 13213w 1 652217514 2 x45t 1 7 5 2 23y14x 1 119d 2 1251712y5 1 3a 4 1 94m 5235m 5t 4545234y 23x 3a 8122First Degree Equations and Inequalities in One VariableLike and Unlike TermsTwo or more terms that contain the same variable or variables,with corre-sponding variables having the same exponents,are called like terms or similar terms .For example,the following pairs are like terms.6k and k5x 2and Ϫ7x 29ab and 0.4aband Two terms are unli ke terms when they contain different variables,or thesame variable or variables with different exponents.For example,the following pairs are unlike terms.3x and 4y5x 2and 5x 39ab and 0.4aand To add like terms,we use the distributive property of multiplication overaddition.9x ϩ2x ϭ(9 ϩ2)x ϭ11x Ϫ16d ϩ3d ϭ(–16 ϩ3)d ϭϪ13dNote that in the above examples,when like terms are added:1.The sum has the same variable factor as the original terms.2.The numerical coefficient of the sum is the sum of the numerical coeffi-cients of the terms that were added.The sum of like terms can be expressed as a single term.The sum of unlike terms cannot be expressed as a single term.For example,the sum of 2x and 3cannot be written as a single term but is written 2x ϩ3.EXAMPLE 1Solve and check:2x ϩ3x ϩ4 ϭϪ6Solution How to Proceed Check(1)Write the equation:2x ϩ3x ϩ4 ϭϪ6(2)Simplify the left side bycombining like terms:(3)Add Ϫ4,the additiveinverse of ϩ4,to Ϫ6 ϭϪ6✔each side:(4)Multiply by ,themultiplicative inverse of 5:(5)Simplify each side.x ϭϪ2Answer Ϫ215(5x )515(210)1524 2 6 1 4 5262(22) 1 3(22) 1 4 52647x 2y383x 3y22113x 2y392x 2y3Simplifying Each Side of an Equation 1232x ϩ3x ϩ4ϭϪ65x ϩ4ϭϪ6Ϫ4Ϫ45xϭϪ10Note:When solving equations,remember to check the answer in the original equation and not in the simplified one.The algebraic expression that is on one side of an equation may contain e the distributive property to remove the parentheses solving the equation.The following examples illustrate how the distributive and asso-ciative properties are used to do this.EXAMPLE 2Solve and check:27x Ϫ3(x Ϫ6) ϭ6Solution Since Ϫ3(x Ϫ6) means that (x Ϫ6) is to be multiplied by Ϫ3,we will use thedistributive property to remove parentheses and then combine like terms.Note that for this solution,in the first three steps the left side is being simplified.These steps apply only to the left side and only change the form but not the numerical value.The next two steps undo the operations of addition and multi-plication that make up the expression 24x ϩ18.Since adding Ϫ18 and dividing by 24 will change the value of the left side,the right side must be changed in the same way to retain the equality.How to Proceed(1)Write the equation:27x Ϫ3(x Ϫ6) ϭ6(2)Use the distributive property:27x Ϫ3x ϩ18ϭ6(3)Combine like terms:(4)Use the addition property of equality.Add Ϫ18,the additive inverse of ϩ18,to each side:(5)Use the division property ofequality.Divide each side by 24:(6)Simplify each side:x ϭCheck(1)Write the equation:27x Ϫ3(x Ϫ6) ϭ6(2)Replace x by (3)Perform the indicated computation:6 ϭ6 ✔Answer x ϭ212122 562272 1 392 562272 1 1832 5627A 212B 2 3A 2612B 5 627A 212B 2 3A 212 2 6B 5 621221224x24521224124First Degree Equations and Inequalities in One Variable24x ϩ18ϭ6Ϫ18Ϫ1824x ϭϪ12Simplifying Each Side of an Equation125 Representing Two Numbers withthe Same VariableProblems often involve finding two or more different numbers.It is useful toexpress these numbers in terms of the same variable.For example,if you knowthe sum of two numbers,you can express the second in terms of the sum and thefirst number.•If the sum of two numbers is 12 and one of the numbers is 5,then theother number is 12 Ϫ5 or 7.•If the sum of two numbers is 12 and one of the numbers is 9,then theother number is 12 Ϫ9 or 3.•If the sum of two numbers is 12 and one of the numbers is x,then theother number is 12 Ϫx.A problem can often be solved algebraically in more than one way by writ-ing and solving different equations,as shown in the example that follows.Themethods used to obtain the solution are different,but both use the facts statedin the problem and arrive at the same solution.EXAMPLE 3The sum of two numbers is 43.The larger number minus the smaller number is5.Find the numbers.Solution This problem states two facts:The sum of the numbers is 43.The larger number minus the smaller number is 5.In other words,the larger number is 5 more than the smaller.(1)Represent each number in terms of the same variable using Fact 1:the sum of the numbers is 43.Let xϭthe larger number.Then,43 Ϫxϭthe smaller number.(2)Write an equation using Fact 2:The larger number minus the smaller number is 5.___________________________________↓↓↓↓↓xϪ(43 Ϫx)ϭ5(3)Solve the equation.(a) Write the equation:x Ϫ(43 Ϫx ) ϭ5(b) To subtract (43 Ϫx ),add its opposite:x ϩ(Ϫ43 ϩx )ϭ5(c)Combine like terms:(d) Add the opposite of Ϫ43 to each side:(e) Divide each side by 2:x ϭ24(4)Find the numbers.The larger number ϭx ϭ24.The smaller number ϭ43 Ϫx ϭ43 Ϫ24 ϭ19.Check A word problem is checked by comparing the proposed solution with the factsstated in the original wording of the problem.Substituting numbers in theequation is not sufficient since the equation formed may not be correct.The sum of the numbers is 43:24 ϩ19 ϭ43.The larger number minus the smaller number is 5:24 Ϫ19 ϭ5.Reverse the way in which the facts are used.(1)Represent each number in terms of the same variable using Fact 2:the larger number is 5 more than the smaller.Let x ϭthe smaller number.Then,x ϩ5 ϭthe larger number.(2)Write an equation using the first fact.(3)Solve the equation.(a) Write the equation:x ϩ(x ϩ5) ϭ43(b) Combine like terms:(c) Add the opposite of 5 to each side:(d) Divide each side by 2:x ϭ19(4)Find the numbers.The smaller number ϭx ϭ19.The larger number ϭx ϩ5 ϭ19 ϩ5 ϭ24.(5)Check.(See the first solution.)Answer The numbers are 24 and 19.2x 25382The sum of the numbers is 43.______________________↓↓↓x ϩ(x ϩ5)ϭ43Alternate Solution 2x 25482126First Degree Equations and Inequalities in One Variable2x Ϫ43ϭ5ϩ43ϩ432x ϭ482x ϩ5ϭ43Ϫ5Ϫ52x ϭ38Writing About Mathematics1.Two students are each solving a problem that states that the difference between two num-bers is 12.Irene represents one number by x and the other number by x ϩ12.Henry repre-sents one number by x and the other number by x Ϫ12.Explain why both students arecorrect.2.A problem states that the sum of two numbers is 27.The numbers can be represented by x and 27 Ϫx .Is it possible to determine which is the larger number and which is the smaller number? Explain your answer.Developing SkillsIn 3–28,solve and check each equation.3.x ϩ(x Ϫ6) ϭ204.x Ϫ(12 Ϫx ) ϭ385.(15x ϩ7) Ϫ12 ϭ46.(14 Ϫ3c )ϩ7c ϭ947.x ϩ(4x ϩ32) ϭ128.7x Ϫ(4x Ϫ39) ϭ09.5(x ϩ2) ϭ2010.3(y Ϫ9) ϭ3011.8(2c Ϫ1) ϭ5612.6(3c Ϫ1) ϭϪ4213.30 ϭ2(10 Ϫy )14.4(c ϩ1) ϭ3215.25 Ϫ 2(t Ϫ5) ϭ1916.18 ϭϪ6x ϩ4(2x ϩ3)17.55 ϭ4 ϩ3(m ϩ2)18.5(x Ϫ3) Ϫ30 ϭ1019.3(2b ϩ1) Ϫ7 ϭ5020.5(3c Ϫ2) ϩ8 ϭ4321.7r Ϫ(6r Ϫ5) ϭ722.8b Ϫ4(b Ϫ2) ϭ2423.5m Ϫ2(m Ϫ5) ϭ1724.28y Ϫ6(3y Ϫ5) ϭ4025.3(a Ϫ5) Ϫ2(2a ϩ1) ϭ026.0.04(2r ϩ1) Ϫ0.03(2r Ϫ5) ϭ0.2927.0.3a ϩ(0.2a Ϫ0.5) ϩ0.2(a ϩ2) ϭ1.328.Applying SkillsIn 29–33,write and solve an equation for each problem.Follow these steps:a.List two facts in the problem.b.Choose a variable to represent one of the numbers to be determined.e one of the facts to write any other unknown numbers in terms of the chosen variable.e the second fact to write an equation.e.Solve the equation.34(8 1 4x ) 2 13(6x 1 3) 5 9EXERCISESSimplifying Each Side of an Equation 127f.Answer the question.g.Check your answer using the words of the problem.29.Sandi bought 6 yards of material.She wants to cut it into two pieces so that the differencebetween the lengths of the two pieces will be 1.5 yards.What should be the length of each piece?30.The Tigers won eight games more than they lost,and there were no ties.If the Tigers played78 games,how many games did they lose?31.This month Erica saved $20 more than last month.For the two months,she saved a total of$70.How much did she save each month?32.On a bus tour,there are 100 passengers on three buses.Two of the buses each carry fourfewer passengers than the third bus.How many passengers are on each bus?33.For a football game,of the seats in the stadium were filled.There were 31,000 empty seatsat the game.What is the stadium’s seating capacity?A variable represents a number.As you know,any number may be added toboth sides of an equation without changing the solution set.Therefore,the samevariable (or the same multiple of the same variable) may be added to or sub-tracted from both sides of an equation without changing the solution set.For instance,to solve 8x ϭ30 ϩ5x ,write an equivalent equation that hasonly a constant in the right side.To do this,eliminate 5x from the right side byadding its opposite,Ϫ5x ,to each side of the equation.METHOD 1METHOD 2Check8x ϭ30 ϩ5x 8x ϭ30 ϩ5x 8x ϩ(Ϫ5x ) ϭ30 ϩ5x ϩ(Ϫ5x )8(10) 3x ϭ3080 x ϭ1080 ϭ80✔Answer:x ϭ10To solve an equation that has the variable in both sides,transform it into anequivalent equation in which the variable appears in only one side.Then,solvethe equation.5 30 1 505 30 1 5(10)45128First Degree Equations and Inequalities in One Variable8x ϭ30 ϩ5x Ϫ5x Ϫ5x 3x ϭ30x ϭ10EXAMPLE 1Solve and check:7x ϭ63 Ϫ2xSolution How to ProceedCheck (1)Write the equation:7x ϭ63 Ϫ2x (2)Add 2x to each side of7(7) the equation:49 49 ϭ49✔(3)Divide each side of theequation by 9:(4)Simplify each side:x ϭ7Answer x ϭ7To solve an equation that has both a variable and a constant in both sides,first write an equivalent equation with only a variable term on one side.Thensolve the simplified equation.The following example shows how this can bedone.EXAMPLE 2Solve and check:3y ϩ7 ϭ5y Ϫ3Solution METHOD 1METHOD 2Check3y ϩ7 ϭ5y Ϫ33(5) ϩ7 15 ϩ7 22 ϭ22 ✔ϭy ϭ5y ϭ5Answer y ϭ5A graphing calculator can be used to check an equation.The calculator candetermine whether a given statement of equality or inequality is true or false.Ifthe statement is true,the calculator will display 1;if the statement is false,the calculator will display 0.The symbols for equality and inequality are found in the menu.TEST 10252y22102222y225 25 2 35 5(5) 2 39x 956395 63 2 145 63 2 2(7)Solving Equations That Have the Variable in Both Sides 1297x ϭ63 Ϫ2x ϩ2x ϩ 2x 9x ϭ633y ϩ7ϭ5y Ϫ3Ϫ5y Ϫ5y Ϫ2y ϩ7ϭϪ3Ϫ7Ϫ7Ϫ2y ϭϪ103y ϩ7ϭ5y Ϫ3Ϫ3y Ϫ3y 7ϭ2y Ϫ3ϩ3ϩ310ϭ2yTo check that y ϭ5 is the solution to the equation 3y ϩ7 ϭ5y Ϫ3,firststore 5 as the value of y .then enter the equation to be checked.ENTER :5 3 75 3 DISPLAY :EXAMPLE 3The larger of two numbers is 4 times the smaller.If the larger number exceedsthe smaller number by 15,find the numbers.Note:When s represents the smaller number and 4s represents the largernumber,“the larger number exceeds the smaller by 15”has the followinge any one of them.1.The larger equals 15 more than the smaller,written as 4s = 15 ϩs .2.The larger decreased by 15 equals the smaller,written as 4s Ϫ15 ϭs .3.The larger decreased by the smaller is 15,written as 4s Ϫs ϭ15.Solution Let s = the smaller number.Then 4s = the larger number.Check The larger number,20,is 4 times the smaller number,5.The larger number,20,exceeds the smaller number,5,by 15.Answer The larger number is 20;the smaller number is 5.4s ϭ15 ϩs Ϫs Ϫs 3s ϭ15s ϭ54s ϭ4(5) ϭ20The larger is 15 more than the smaller._____________________________↓↓↓↓↓4s ϭ15ϩs4s ϭ15 ϩsENTER ؊ALPHA ENTER TEST 2nd ؉ALPHA ENTERALPHA STO Ł130First Degree Equations and Inequalities in One VariableThe calculator displays 1 which indi-cates that the statement of equality is true for the value that has been stored for y .EXAMPLE 4In his will,Uncle Clarence left $5,000 to his two nieces.Emma’s share is to be$500 more than Clara’s.How much should each niece receive?Solution (1)Use the fact that the sum of the two shares is $5,000 to express each sharein terms of a variable.Let x ϭClara’s share.Then 5,000 Ϫx ϭEmma’s share.(2)Use the fact that Emma’s share is $500 more than Clara’s share to writean equation.(3)Solve the equation to find Clara’s share.2,250 ϭxClara’s share is x ϭ$2,250.(4)Find Emma’s share:5,000 Ϫx ϭ5,000 Ϫ2,250 ϭ$2,750.(1) Use the fact that Emma’s share is $500 more than Clara’s share to expresseach share in terms of a variable.Let x ϭClara’s share.Then x ϩ500 ϭEmma’s share.(2)Use the fact that the sum of the two shares is $5,000 to write an equation.(3)Solve the equation to find Clara’s sharex ϩ(x ϩ500) ϭ5,000Clara’s share is x ϭ$2,250.2x ϩ500ϭ5,000Ϫ500Ϫ5002x ϭ4,500x ϭ2,250Clara’s share plus Emma’s share is $5,000.__________________________↓↓↓↓↓x ϩ(x ϩ 500)ϭ5,000Alternate Solution 5,000Ϫx ϭ500 ϩx ϩx ϩx 5,000ϭ500 ϩ2x Ϫ500Ϫ5004,500ϭ2xEmma’s share is $500 more than Clara’s share.___________________________________↓↓↓↓↓5,000 Ϫx ϭ500ϩxSolving Equations That Have the Variable in Both Sides 131(4)Find Emma’s share:x ϩ500 ϭ2250 ϩ500 ϭ$2,750.Check $2,750 is $500 more than $2,250,and $2,750 ϩ$2,250 ϭ$5,000.Answer Clara’s share is $2,250,and Emma’s share is $2,750.Writing About Mathematicsus said that he finds it easier to work with integers than with fractions.Therefore,in order to solve the equation ,he began by multiplying both sides of the equation by 4.3a Ϫ28 ϭ2a ϩ12Do you agree with Milus that this is a correct way of obtaining the solution? If so,what mathematical principle is Milus using?2.Katie said that Example 3 could be solved by letting equal the smaller number and x equal the larger number.Is Katie correct? If so,what equation would she write to solve the problem?Developing SkillsIn 3–36,solve and check each equation.3.7x ϭ10 ϩ2x4.9x ϭ44 Ϫ2x5.5c ϭ28 ϩc6.y ϭ4y ϩ307.2d ϭ36 ϩ5d 8.9.0.8m ϭ0.2m ϩ2410.8y ϭ90 Ϫ2y 11.2.3x ϩ36 ϭ0.3x 12.13.5a Ϫ40 ϭ3a 14.5c ϭ2c Ϫ8115.x ϭ9x Ϫ7216.0.5m Ϫ30 ϭ1.1m 17.18.7r ϩ10 ϭ3r ϩ5019.4y ϩ20 ϭ5y ϩ920.7x ϩ8 ϭ6x ϩ121.x ϩ4 ϭ9x ϩ422.9x Ϫ3 ϭ2x ϩ4623.y ϩ30 ϭ12y Ϫ1424.c ϩ20 ϭ55 Ϫ4c25.2d ϩ36 ϭϪ3d Ϫ5426.7y Ϫ5 ϭ9y ϩ2927.3m Ϫ(m ϩ1) ϭ6m ϩ128.x Ϫ3(1 Ϫx ) ϭ47 Ϫx 29.3b Ϫ8 ϭ10 ϩ(4 Ϫ8b )30.31.18 Ϫ4n ϭ8 Ϫ2(1 ϩ8n )32.8c ϩ1 ϭ7c Ϫ2(7 ϩc )33.8a Ϫ3(5 ϩ2a ) ϭ85 Ϫ3a34.4(3x Ϫ5) ϭ5x ϩ2( x ϩ15)35.3m Ϫ5m Ϫ12 ϭ7m Ϫ88 Ϫ536.5 Ϫ3(a ϩ6) ϭa Ϫ1 ϩ8a 23t 2 1154(16 2 t ) 2 13t 414c 5934c 1 44234x 1 2453x214y 5114y 2 8x 44A 34a 2 7B 54A 12a 1 3B 34a 2 7512a 1 3EXERCISES132First Degree Equations and Inequalities in One VariableSolving Equations That Have the Variable in Both Sides133In 37–42,a.write an equation to represent each problem,and b.solve the equation to find each number.37.Eight times a number equals 35 more than the number.Find the number.38.Six times a number equals 3 times the number,increased by 24.Find the number.39.If 3 times a number is increased by 22,the result is 14 less than 7 times the number.Find thenumber.40.The greater of two numbers is 1 more than twice the smaller.Three times the greaterexceeds 5 times the smaller by 10.Find the numbers.41.The second of three numbers is 6 more than the first.The third number is twice the first.The sum of the three numbers is 26.Find the three numbers.42.The second of three numbers is 1 less than the first.The third number is 5 less than the sec-ond.If the first number is twice as large as the third,find the three numbers.Applying SkillsIn 43–50,use an algebraic solution to solve each problem.43.It took the Gibbons family 2 days to travel 925 miles to their vacation home.They traveled75 miles more on the first day than on the second.How many miles did they travel eachday?44.During the first 6 month of last year,the interest on an investment was $130 less than dur-ing the second 6 months.The total interest for the year was $1,450.What was the interest for each 6-month period?45.Gemma has 7 more five-dollar bills than ten-dollar bills.The value of the five-dollar billsequals the value of the ten-dollar bills.How many five-dollar bills and ten-dollar bills does she have?46.Leonard wants to save $100 in the next 2 months.He knows that in the second month he willbe able to save $20 more than during the first month.How much should he save each month?47.The ABC Company charges $75 a day plus $0.05 a mile to rent a car.How many miles didMrs.Kiley drive if she paid $92.40 to rent a car for one day?48.Kesha drove from Buffalo to Syracuse at an average rate of 48 miles per hour.On thereturn trip along the same road she was able to travel at an average rate of 60 miles perhour.The trip from Buffalo to Syracuse took one-half hour longer than the return trip.How long did the return trip take?49.Carrie and Crystal live at equal distances from school.Carries walks to school at an averagerate of 3 miles per hour and Crystal rides her bicycle at an average rate of 9 miles per hour.It takes Carrie 20 minutes longer than Crystal to get to school.How far from school doCrystal and Carrie live?50.Emmanuel and Anthony contributed equal amounts to the purchase of a gift for a friend.Emmanuel contributed his share in five-dollar bills and Anthony gave his share in one-dollar bills.Anthony needed 12 more bills than Emmanuel.How much did each contribute toward the gift?。
美国数学参考书目
美国数学本科生、研究生基础课程参考书目在网上找书的时候恰好看到这个，看着觉得的确是经典书目大全，贴在这里供学弟学妹们参考：）其中所谓第几学年云云，各校要求不同，像我所在的学校，一般学生第一年选三到四门基础课（代数、分析、几何三大类中至少各挑一门），学年末进行qualifying笔试。
第二年开始选自己喜爱方向的高级课程，并通过qualifying口试。
第三年开始做research，并通过第二语言考试（法语或德语或俄语，一般人都选法语，因为代数几何经典大作都是法语的）. 而Princeton 就没有基础课，只有seminar类型的课。
第一学年几何与拓扑：1、James R. Munkres, Topology：较新的拓扑学的教材适用于本科高年级或研究生一级；2、Basic Topology by Armstrong：本科生拓扑学教材；3、Kelley, General Topology：一般拓扑学的经典教材，不过观点较老；4、Willard, General Topology：一般拓扑学新的经典教材；5、Glen Bredon, Topology and geometry：研究生一年级的拓扑、几何教材；6、Introduction to Topological Manifolds by John M. Lee：研究生一年级的拓扑、几何教材，是一本新书；7、from calculus to cohomology by Madsen：很好的本科生代数拓扑、微分流形教材。
代数：1、Abstract Algebra Dummit：最好的本科代数学参考书，标准的研究生一年级代数材；2、Algebra Lang：标准的研究生一、二年级代数教材，难度很高，适合作参考书；3、Algebra Hungerford：标准的研究生一年级代数教材，适合作参考书；4、Algebra M,Artin：标准的本科生代数教材；5、Advanced Modern Algebra by Rotman：较新的研究生代数教材，很全面；6、Algebra：a graduate course by Isaacs：较新的研究生代数教材；7、Basic algebra Vol I&II by Jacobson：经典的代数学全面参考书，适合研究生参考。
加州初中数学教材
加州初中数学教材篇一：中美初中数学教材中美初中数学教材“函数”内容比较-------以人教版初中数学教材和美国Prentice Hall 教材为例程丽摘要纵观近、现代数学的发展可知，函数是描述运动、变化的基本概念。
数学中许多概念或由函数派生，或可归之为函数观点研究。
可以说函数概念的产生，本身就标志着数学思想方法的重大转折--由常量数学到变量数学。
函数的应用，更使得数学的面貌，从对象到理论、方法、结构，发生了根本的变化。
基于这些原因，就中学数学而言，函数的重要性是不容置疑的，它己经成为中学数学中的纽带，但同时它又是学生最难理解的内容之一。
本文对中美初中数学教材中“函数”模块的内容进行了比较研究。
关键词中美；初中数学；教材比较；函数一、研究的问题目前，人们对教科书的认识已由“教”的材料向“学”的材料转变，这也就意味着，教科书的编写不仅要考虑到教师的“教”，更主要的还要考虑到学生的“学”[1]。
近年来学术界对教科书的研究也正处在逐步的深入当中，其中有纵向上的对我国历届教科书的比较研究，也有与国外发达国家教科书进行的横向比较分析。
从横向比较来说，虽然已有一些与美国教育比较的研究成果，但是这些比较研究大都是针对中美教育思想与制度差别的整体宏观比较，而针对中学阶段的中美两国的教科书的比较研究还相对较少[2]。
因此，在我国的国际教科书比较领域内，本研究具有一定的理论价值。
本研究立足于中美初中数学教育发展的历史渊源与现状，主要选取在中美两国具有代表性的初中数学教科书“人教版”和“Prentice Hall”中“函数”的课程内容进行比较研究，对中美初中数学教科书的编写进行了系统而全面的分析。
本文采取比较法和内容分析法。
深入到微观层面对中美教科书编写方式进行对比分析，精心选择初中数学“函数”作为典型案例式，对两个版本教材的共性和差异性进行了详细讨论。
二、中美初中数学“函数”课程容量的比较课程容量是一个比较宽泛的概念。
美国六年级数学知识点总结
美国六年级数学知识点总结第一章：整数
1.了解自然数、整数和负数的含义
2.掌握整数的加减法规则
3.掌握整数的乘法和除法
4.掌握整数的比较大小
5.掌握整数的运算规则：先乘除后加减
第二章：分数
1.了解分数的含义和基本属性
2.掌握分数的加减法规则
3.掌握分数的乘法和除法
4.掌握分数的大小比较
5.掌握分数的化简与约分
6.了解分数与整数的关系
第三章：小数
1.了解小数的含义和基本属性
2.掌握小数的加减法规则
3.掌握小数的乘法和除法
4.掌握小数的大小比较
5.掌握小数的化简与约分
6.了解小数与分数的关系
第四章：代数
1.了解代数的含义和基本符号
2.了解代数式的含义和形式
3.掌握代数式的加减法规则
4.能够根据代数式计算
5.了解代数式与数学问题的关系
第五章：几何
1.了解平面图形的基本形状和性质
2.了解立体图形的基本形状和性质
3.掌握图形的面积和周长的计算
4.了解角的概念和性质
5.了解相似和全等图形的概念
6.了解平行线、垂直线和交错线的性质
第六章：数据分析
1.了解统计学的基本含义和方法
2.掌握数据的收集和整理方法
3.掌握数据的图表表示方法
4.了解平均数、中位数和众数的计算方法
5.了解概率的概念和计算方法
以上就是美国六年级数学知识点的总结，希朇对大家有所帮助。
美国的数学教材高等数学
美国的数学教材高等数学高等数学作为数学学科中的重要分支，对于美国的数学教育具有重要的地位。
美国一直以来注重培养学生的数学素养和创造力，因此在高等数学教材的编写和选择上相当慎重。
本文将介绍美国的数学教材高等数学的特点和影响。
一、教材体系概述美国的高等教育系统与中国有所不同，大学本科专业教育主要围绕选修课程展开。
高等数学作为全球范围内普遍开设的课程之一，涉及面广，教材的编选也相对多样化。
1.1 教材类型多样美国的高等数学教材种类繁多，分为教材和参考书两大类。
教材主要针对本科生编写，内容涵盖了高等数学的各个分支，包括微积分、线性代数、概率统计等。
参考书则侧重于对特定数学领域的深入探究和研究。
1.2 教材编写理念美国的高等数学教材编写侧重于培养学生的数学思维和解决问题的能力。
教材往往注重理论与实践相结合，鼓励学生通过实际问题的应用来理解和掌握数学知识。
此外，美国的教材注重培养学生的创新意识和团队合作能力，教材中常包含一些启发式的问题和讨论活动，鼓励学生积极参与。
1.3 教材内容覆盖全面美国的高等数学教材注重内容的全面性和连贯性。
教材往往从基础概念出发，逐步推进，形成完整的知识体系。
此外，教材还会涉及一些前沿的数学研究和应用成果，以培养学生的兴趣和好奇心。
二、教材特点分析2.1 注重学生思维培养与中国的高等数学教材相比，美国的教材更注重培养学生的思维方式和解决问题的能力。
教材中常包含一些开放性问题和案例研究，鼓励学生灵活运用数学知识进行分析和推理，培养学生的逻辑思维和创新能力。
2.2 强调数学与实际应用的结合美国的高等数学教材强调数学在实际生活和工作中的应用价值。
教材中常通过实例和案例引入数学知识的应用场景，让学生意识到数学与实际问题的密切关系，提高学习的兴趣和动力。
2.3 尊重学生个体差异美国的教材编写注重尊重学生的个体差异。
教材中往往包含不同学习层次和兴趣爱好的内容，鼓励学生选择适合自己的学习路径。
此外，教材还提供了丰富的练习题和习题答案，供学生自主练习和验证。
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functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. (Key, CAHSEE)
Chapter 4 Get Ready For Chapter 4 185
EXPLORE
4-1
Algebra Lab
Steepness of a Line
In mathematics, you can measure the steepness of a line using a ratio.
Reinforcement of Standard 7AF3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. (Key, CAHSEE)
4 Label each of the tabs with a lesson number. The last tab is for the vocabulary.
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184 Chapter 4 Analyzing Linear Equations
GET READY for Chapter 4
Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2
Option 1
Take the Online Readiness Quiz at .
rise and the run of the ramp. Then calculate the
RISE
RUN
RISE RUN
ratio _ rriusne. Record the data
in a table like the one at
the right.
Step 2
Keeping the rise the same, move the books to make the ramp steeper. Measure the rise and run, and
3. MAKE A PREDICTION Suppose you want to construct a skateboard ramp
that is not as steep as the one shown at the left. List three different
sets
of
_ rise
Key Vocabulary
point-slope form (p. 220) rate of change (p. 187) slope (p. 189) slope-intercept form (p. 204)
Real-World Link
Space Exploration Linear equations are used to model a variety of real-world situations, including the cost of the U.S. space program.
• Stack three books on your desk.
• Lean a ruler on the books, creating a ramp.
• Tape the ruler to the desk.
RISE RUN
ACTIVITY
Step 1 Measure and record the
run
measurements
that
will
your predictions by calculating the
result in a less steep ramp.
ratio
_ rise
run
of
each
ra graph and draw a line
Analyzing Linear Equations Make this Foldable to help you organize information about writing linear equations. Begin with four sheets of grid paper.
1 Fold each sheet of grid paper in half along the width. Then cut along the crease.
through
the
origin
with
a
_ rise
run
ratio
greater
than the original line. Then draw a line
through the origin with a ratio less than the
original line. Explain your reasoning using
2 Staple the eight half-sheets together to form a booklet.
3 Cut seven lines from the bottom of the top sheet, six lines from the second sheet, and so on.
Take the Quick Check below. Refer to the Quick Review for help.
Simplify. (Prerequisite Skill)
1.
_2
10
2.
_8
12
4. _ -4
8
5.
_ -5
-15
7.
_9
3
8.
_18
12
3.
_2
-8
6.
_ -7
-28
9. -_26
calculate the ratio _ rriusne.
Repeat three times and record the data.
Step 3 Start with the last measurements from Step 2. Keeping the run the same, add a book to increase the rise of the ramp. Measure and record the rise and run, and calculate the ratio. Repeat one time, adding another book, and record the data.
=
_ -1
7
or
-_17
Simplify. Since the signs are different, the quotient is negative.
EXAMPLE 2
_ Evaluate
a-b c-d
if
a
= 2,
b
=
5,
c
=
-3,
d
=
-12.
_ a - b
c-d
Original expression
Standard 8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. (CAHSEE)
y (2, 4)
(1, 2)
rise run
O
x
the words rise and run.
186 Chapter 4 Analyzing Linear Equations
4-1 Rate of Change
and Slope
Main Ideas
• Use rate of change to solve problems.
= _2 -_ 5 Substitute 2 for a, 5 for b, -3 for c,
(-3) - (-12) and -12 for d.
=
_ -3
9
=
_ -3 ÷ 3
9÷3
=
_ -1
3
or
-_13
Simplify.
Divide -3 and 9 by their GCF, 3. Simplify. The signs are different, so the quotient is negative.
EXAMPLE 3
Write the ordered pair for A.