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Understanding Algebra The complete contents of this algebra textbook are available here online. This text is suitable for high-school Algebra I, preparing for the GED, a refresher for college students who need help preparing for college-level mathematics, or for anyone who wants to learn introductory algebra. I am especially pleased to help homeschoolers. Includes a graphing applet, a prime factorization machine, and a prime number list. Brennan uses inserts to answer common questions as the lesson goes on, helping t Author(s): No creator set Exploring Creativity: Japanese students from Jikei Com colleges spend a day at Otis College On October 8 Otis hosted 180 Japanese multimedia students from Jikei COM colleges in Japan, who learned about the American approach to creativity and problem solving. Workshops focused on: connections between brain function and creativity through drawing; expressing emotion through body movement; brainstorming and developing great ideas with ease; and making a habit of creativity. Author(s): No creator setPrincess Presto's Soccer Game Practice spelling the word PASS as Princess Presto and the Super Readers try and help their friends learn how to play soccer! Princess Presto uses the tools of sounding out and writing the letters to make the word P-A-S-S in order for her friends to learn how to play as a team. (2:45) Author(s): No creator setMelanoma Patient on His Cancer Metastasizing By: mdanderson Tim Shiery of Houston, Texas was diagnosed with metastatic melanoma in August of 2007. He had spots on his brain, lungs, liver, skin and bone.License information Related content No related items provided in this feed Sound Carnival: -ou- What's that sound sitting there in the word? Come to the Sound Carnival and find out! The goal for this segment is vowel combination -ou-. (0:45) Author(s): No creator set License information Related content No related items provided in this feed Shock Matrix: -st- Watch Shock master the matrix as he manipulates sounds! The goal for this segment is consonant blend -st-. (1min) Author(s): No creator set
On Math-Atlas, where does "regular" Algebra fit in? On Math-Atlas, where does "regular" Algebra fit in? Where on the Math-Atlas does Algebra I and Algebra II fit? Should I assume "Algebra I and Algebra II" are essentially generalized, introductory courses that cover a subset of branches under the "Abstract Algebra" branch? I'm starting school this Spring and wanted to build a workbook generator to catch myself up to speed. The app will ask you to select a set of math branches* and problem types. I'd therefore like to show the visitor a list of Math branches to choose from. --- * Is a "branch" what different areas of mathematics are even called or is it "field"? I get conflicting results, even within the same articles. Knowing this is important to me, because the web app will be open-sourced and the less mistakes I have on it, the better :P
The course is an introduction to the numerical methods. The main goal consists in explanations of fundamental numerical principles so that students should be able to decide about an appropriate method for problems arising in the other courses or in the technical practice. An important ingredient is the algorithmic implementation of numerical methods and the usage of the standard numerical software. The graduate of this course should know: • to recognize problems suitable for solving by numerical procedures and to find an appropriate numerical method; • to decide whether the computed solution is sufficiently accurate and, in case of need, to assess reasons of inaccuracies; • to propose an algorithmic procedure for solving the problem and to choice a suitable computer environment for its realization. It is necessary to complete Mathematics 1 and Mathematics 2 courses or their equivalents. Recommended Optional Programme Components Common optional components are not offered, students of special interest can participate in departmental activities or can arrange consulting hours with lecturer.
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Course Description Description This rigorous course, drawing on software provided by Thinkwell, delivers a full-year honors curriculum and prepares students for Honors Geometry and Honors Algebra II. Beginning with a brief review of pre-algebra concepts, students move quickly through familiar content. Real-life applications help students to understand the importance of algebra in our world. Topics include: real number system solving equations and inequalities polynomials and exponents factoring and applications rational expressions graphing linear equations solving linear systems roots and radicals quadratic equations Timelines This course offers 9-month, 6-month, and 3-month timelines as guidelines for students to follow in order to finish within their desired time frame.
Highly interactive tutorials and self-test system for individual e-learning, home schooling, college and high school computer learning centers, and distance learning. The product emphasizes on building problem-solving skills. tutorials include the reviews of basic concepts, interactive examples, and standard problems with randomly generated parameters. The self-test system allows selecting topics and length for a test, saving test results, and getting the test review. Topics covered: rectangular coordinate system, functions and graphs, linear equations and inequalities in one variable, systems of linear equations and inequalities, determinants and Cramer s rule, operations with polynomials, factoring polynomials, roots of polynomial equations, rational expressions, exponents and radicals, complex numbers, quadratic functions, conic sections, exponential and logarithmic functions, sequences and series, binomial theorem, counting principles. The demo version contains selected lessons from the full version. The demo is designed to demonstrate the functionality and all features of the product. Reviews Also see ... ...in Home & Education 1) Molecular Genetics 3.1 by F. Triep Multimedia- Software Download A multimedia program on DNA for Biology classes of secundary schools (advanced level), senior high schools and sixth forms colleges. It covers the structure, function and significance of DNA. This demo consist of the introduction, the main menu and 2 of the 6 modules of the complete package, namely:* Module 2: The structure of DNA* Parts of module 6: RNA translationThe full program contains the following modules:* The location of the genetic infomation* The structure of the genetic information* ... 24) NutriGenie Low Carb Diet guess... 5) ECTACO Voice Translator English <-> Spanish ta... 6) NCSPowerMeet A highly configurable program that can be setup to host virtually any meet by changing, loading or creating parameter values as required for a meeting. The program is flexible, easy to use and learn with an intuitive layout that appeals to the eye. * The software can manage meetings with teams and lifter registrations. * Weight and Age coefficient support for Wilks, Reshel, Siff and Sinclair. * Capability to do formula calculations like Siff or Sinclair. * Age and Gender division support. Bio-sh... 7 8) read_write Training of foreign languages on examples. The program user - on his/her own selects any text in a foreign language (at least one sentence) - reads the text into a microphone - then listens to what he/she has read - types (prints on a keyboard) what he/she hears The program is high-effective since it engages all types of human sensors namely - outer hearing, through ears: I hear to type correctly - sight: I see what I write - tactile: I feel the keys through nerve-endings of my fingers - inner h... 1) Adding Three Fractions 1.0 by Polymath Love- Software Download Middle-School (grades 5 through 9) math program written to provide skills in context. Students choose one of two types of problems: ruler fractions (halves, fourths, eighths, sixteenths) or regular fractions. Students must add the given three fractions. Every problem is illustrated with an interactive sketch.... 2) Writing Numbers Part 2).... 3) Talmagi 2.2 by dmagic- Software Download Using keen logic and problem-solving skills, you.ll face multiple math problems that are set up crossword-style. The math equations are set up horizontally, while a set of numbers in glass beakers overlap multiple problems vertically. By moving the numbers in the beakers up or down, try to find the right configuration to solve all of the equations.Talmagi features gorgeous graphics and exciting sound effects to enhance the... 4) 010 Memorizer 010 Memorizer is a powerful program for memorizing numbers. The system works by converting a number into a word or phrase that creates a vivid image in your mind.... 5.... 6) EMSolution Trigonometry Equations short theorem. A translation option offers a way to learn m... 7) Indefinite Integrals Problem Solving Ebook Software Effective indefinite integrals training using problems selected with resolution stepwise of graphical form and with mathematical formThe resolution of indefinite integrals is a primordial factor in any student. The main problems that are presented are not understanding the professor's explanation, not to see the information and the mechanical resolution of the problems that they tell us. To get it it is necessary to have a method of resolution of problems that is effective, simple, elegant and t... 10) Mathomir Basically an equation editor, however not focused over one single equation, but you can write your mathematical artwork over several pages. Illustrate your equations using hand-drawing tools. Use symbolic calculator and function plotter....
Educator - Mathematics: College Calculus Level I with Professor Switkes English \| VP6 650x350 29.970 fps \| MP3 128 Kbps 44.1 KHz \| 8.37 GB Genre: eLearning Dr. Jenny Switkes will help you master the intricacies of Calculus from Limits to Derivatives to Integrals. In Educator's Calculus 1 course, Professor Switkes covers all the important topics with detailed explanations and analysis of common student pitfalls. Calculus can be difficult, but Professor Switkes will show you how to reap the rewards of your hard work, all while showing you the beauty and importance of math. Whether you just need to brush up on your calculus skills or need to cram the night before the final, Professor Switkes has taught mathematics for 10+ years and knows exactly how to help. Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x, theorems on differentiation and antidifferentiation, the chain rule and examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Examples. 1967 edition. Solution guide available upon requestStochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. Calculus is about the very large, the very small, and how things change. The surprise is that something seemingly so abstract ends up explaining the real world. Calculus plays a starring role in the biological, physical, and social sciences. By focusing outside of the classroom, we will see examples of calculus appearing in daily life. Calculus does not have to be difficult. Dr. William Murray knows what it takes to excel in math and will show you everything you need to know about calculus. Dr. Murray demonstrates his extensive teaching experience by clarifying complicated topics with a wide array of examples, helpful tips, and time-saving tricks. Topics range from Advanced Integration Techniques and Applications of Integrals to Sequences/Series. Dr. Murray received his Ph.D from UC Berkeley, B.S. from Georgetown University, and has been teaching in the university setting for 10+ years. Professor Raffi Hovasapian helps students develop their Multivariable Calculus intuition with in-depth explanations of concepts before reinforcing an understanding of the material through varied examples. This course is appropriate for those who have completed single-variable calculus. Topics covered include everything from Vectors to Partial Derivatives, Lagrange Multipliers, Line Integrals, Triple Integrals, and Stokes' Theorem. Professor Hovasapian has degrees in Mathematics, Chemistry, and Classics and over 10 years of teaching experience. For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus. This twelve-lesson series will cover the ins and outs of vector calculus and the geometry of R^2 and R^3. The first 8 video lessons look specifically at vectors and the geometry of R^2 and R^3. This set of videos will cover Coordinate Geometry in Three Dimensional Space, Vectors in R2 and R3, the Dot Product, Orthogonal Projections, the Cross Product, Geometry of the Cross Product, and Equations of Lines and Planes in R3. Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This self-contained textbook gives a complete course on all these topics. It is written by a leading specialist who is also a noted expositor.
Methods of solving polynomial equations lie at the heart of classical algebra. There are two interpretations of the problem of solving an equation, leading to two different approaches to its solution. In most courses, the emphasis is on the structure of the equation and finding a way to express the roots as a formula in terms of the coefficients. The simplest example of such a formula is the quadratic formula, which gives the solution of the equation ax2 + bx + c = 0 as This approach is elegant and leads to some exceedingly profound mathematics. However, for one who actually needs to know a number that satisfies the equation, this approach leaves something to be desired. It works with maximum efficiency in the case of the quadratic equation, but even in that case, if the quantity under the radical is not the square of a rational number, one is forced to resort to approximations in order to get a usable number. For cubic and quartic equations, there are formulas, but they work even less well, since they often involve taking the cube root of a complex number, which is a problem just as complicated as the original equation was, if not more so. Once again, one is forced to resort to numerical approximations. Beyond the fourth degree, the only formulas involve non-algebraic expressions, and are of little practical use. Higher-degree equations are the realm of numerical methods. To understand how numerical methods work, it is useful to begin with the simplest cases and the simplest methods. That is what we are about to do
Take it step-by-step for pre-calculus success! The quickest route to learning a subject is through a solid grounding in the basics. So what you won't find in Easy Pre-calculus Step-by-Step is a lot of…
Algebra For College Students-myworkbook - 7th edition Summary: MyWorkBook provides extra practice exercises for every chapter of the text. MyWorkbook can be packaged with the textbook or with the MyMathLab access kit and includes the following resources for every section: Key vocabulary terms, and vocabulary practice problems Guided Examples with stepped-out solutions and similar Practice Exercises, keyed To The text by Learning Objective References...show more to textbook Examples and Section Lecture Videos for additional help Additional Exercises with ample space for students to show their work, keyed To The text by Learning Objective
This set of Mathcad documents provides an overview of some fundamental calculus operations required of physical chemistry students. There is an introduction to taking derivatives and integrals in Part 1 and Part 2. Part 3 focus on preparing surface and contour plots, which can lead to discussions of partial derivatives. Part 4 provides an introduction to the Runga-Kutta method for solving differential equations. Each document has exercises to enable students to practice the techniques and later use them for preparing homework during the course. These documents require Mathcad 12 or higher. The documents presented here build on concepts learned by working through the Basics collection of documents.
Academics Mathematics 221. Linear Algebra How might you draw a 3D image on a 2D screen and then "rotate" it? What are the basic notions behind Google's original, stupefyingly efficient search engine? After measuring the interacting components of a nation's economy, can one find an equilibrium? Starting with a simple graph of two lines and their equations, we develop a theory for systems of linear equations that answers questions like those posed here. This theory leads to the study of matrices, vectors, linear transformations and geometric properties for all of the above. We learn what "perpendicular" means in high-dimensional spaces and what "stable" means when transforming one linear space into another. Topics also include: matrix algebra, determinants, eigenspaces, orthogonal projections and a theory of vector spaces.
Mathematics: Your skills More in this section Case studies The breadth of mathematics is immense. It is a fundamental subject for much of science/technology, and also for all analytical and model-building activities across a wide range of sectors. Since the spread of topics is so broad in a mathematics degree, some have little in common with others. Statistics is an important mathematics discipline and studying it gives you skills relating particularly to the design and conduct of experimental and observational studies and the analysis of data resulting from them. The analytical approach you practice, trains you to be able to apply theoretical knowledge to problem-solving and to develop and evaluate logical arguments
Emilie Wiesner Research My mathematical research is in representation theory. The idea behind representation theory is to study an algebraic object by looking at how it acts on something else (that is, some "representation" of the original object). For example, the set of transformations that leave a square unchanged form a mathematical object called a group. (One such transformation is rotation by 90° around the center of the square.) We can try to learn more about the abstract group by looking at what it does to the square. More generally I am an algebraist, which means I like to think a lot about ideas from Linear Algebra (Math 231), Abstract Algebra (Math 303), Combinatorics (Math 421) and other ideas from Discrete Mathematics (Math 270, Math 420). I have recently started to learn about applications of discrete mathematics to biology. As a teacher, I am also interested in how students learn mathematics. This has led me to get involved with research in math education. I am particularly interested in the role of mathematics textbooks in undergraduate classes and the ways that students use mathematics textbook. If you would like to learn more about any of my research, feel free to stop by to talk with me. You can find a list of my publications here.
Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences, CourseSmart eTextbook, 13th Edition Description This book is ideal for one- or two-semester or two- or three-quarter courses covering topics in college algebra, finite mathematics, and calculus for students in business, economics, and the life and social sciences. Haeussler, Paul, and Wood establish a strong algebraic foundation that sets this text apart from other applied mathematics texts, paving the way for students to solve real-world problems that use calculus. Emphasis on developing algebraic skills is extended to the exercises–including both drill problems and applications. The authors work through examples and explanations with a blend of rigor and accessibility. In addition, they have refined the flow, transitions, organization, and portioning of the content over many editions to optimize manageability for teachers and learning for students. The table of contents covers a wide range of topics efficiently, enabling instructors to tailor their courses to meet student needs. CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book. Table of Contents Part I. ALGEBRA 0. Review of Algebra 0.1 Sets of Real Numbers 0.2 Some Properties of Real Numbers 0.3 Exponents and Radicals 0.4 Operations with Algebraic Expressions 0.5 Factoring 0.6 Fractions 0.7 Equations, in Particular Linear, Equations 0.8 Quadratic Equations 1. Applications and More Algebra 1.1 Applications of Equations 1.2 Linear Inequalities 1.3 Applications of Inequalities 1.4 Absolute Value 1.5 Summation Notation 1.6 Sequences 2. Functions and Graphs 2.1 Functions 2.2 Special Functions 2.3 Combinations of Functions 2.4 Inverse Functions 2.5 Graphs in Rectangular Coordinates 2.6 Symmetry 2.7 Translations and Reflections 2.8 Functions of Several Variables 3. Lines, Parabolas, and Systems 3.1 Lines 3.2 Applications and Linear Functions 3.3 Quadratic Functions 3.4 Systems of Linear Equations 3.5 Nonlinear Systems 3.6 Applications of Systems of Equations 4. Exponential and Logarithmic Functions 4.1 Exponential Functions 4.2 Logarithmic Functions 4.3 Properties of Logarithms 4.4 Logarithmic and Exponential Equations Part II. FINITE MATHEMATICS 5. Mathematics of Finance 5.1 Compound Interest 5.2 Present Value 5.3 Interest Compounded Continuously 5.4 Annuities 5.5 Amortization of Loans 5.6 Perpetuities 6. Matrix Algebra 6.1 Matrices 6.2 Matrix Addition and Scalar Multiplication 6.3 Matrix Multiplication 6.4 Solving Systems by Reducing Matrices 6.5 Solving Systems by Reducing Matrices (continued) 6.6 Inverses 6.7 Leontief's Input-Output Analysis 7. Linear Programming 7.1 Linear Inequalities in Two Variables 7.2 Linear Programming 7.3 Multiple Optimum Solutions 7.4 The Simplex Method 7.5 Degeneracy, Unbounded Solutions, and Multiple Solutions 7.6 Artificial Variables 7.7 Minimization 7.8 The Dual 8. Introduction to Probability and Statistics 8.1 Basic Counting Principle and Permutations 8.2 Combinations and Other Counting Principles 8.3 Sample Spaces and Events 8.4 Probability 8.5 Conditional Probability and Stochastic Processes 8.6 Independent Events 8.7 Bayes's Formula 9. Additional Topics in Probability 9.1 Discrete Random Variables and Expected Value 9.2 The Binomial Distribution 9.3 Markov Chains Part III. CALCULUS 10. Limits and Continuity 10.1 Limits 10.2 Limits (Continued) 10.3 Continuity 10.4 Continuity Applied to Inequalities 11. Differentiation 11.1 The Derivative 11.2 Rules for Differentiation 11.3 The Derivative as a Rate of Change 11.4 The Product Rule and the Quotient Rule 11.5 The Chain Rule 12. Additional Differentiation Topics 12.1 Derivatives of Logarithmic Functions 12.2 Derivatives of Exponential Functions 12.3 Elasticity of Demand 12.4 Implicit Differentiation 12.5 Logarithmic Differentiation 12.6 Newton's Method 12.7 Higher-Order Derivatives 13. Curve Sketching 13.1 Relative Extrema 13.2 Absolute Extrema on a Closed Interval 13.3 Concavity 13.4 The Second-Derivative Test 13.5 Asymptotes 13.6 Applied Maxima and Minima 14. Integration 14.1 Differentials 14.2 The Indefinite Integral 14.3 Integration with Initial Conditions 14.4 More Integration Formulas 14.5 Techniques of Integration 14.6 The Definite Integral 14.7 The Fundamental Theorem of Integral Calculus 14.8 Approximate Integration 14.9 Area between Curves 14.10 Consumers' and Producers' Surplus 15. Methods and Applications of Integration 15.1 Integration by Parts 15.2 Integration by Partial Fractions 15.3 Integration by Tables 15.4 Average Value of a Function 15.5 Differential Equations 15.6 More Applications of Differential Equations 15.7 Improper Integrals 16. Continuous Random Variables 16.1 Continuous Random Variables 16.2 The Normal Distribution 16.3 The Normal Approximation to the Binomial Distribution 17. Multivariable Calculus 17.1 Partial Derivatives 17.2 Applications of Partial Derivatives 17.3 Implicit Partial Differentiation 17.4 Higher-Order Partial Derivatives 17.5 Chain Rule 17.6 Maxima and Minima for Functions of Two Variables 17.7 Lagrange Multipliers 17.8 Lines of Regression 17.9 Multiple
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Summary: Intended for introductory courses in basic mathematics, this comprehensive text teaches the skills necessary for practical work involving architectural and mechanical drafting, electronics, welding, air conditioning, aviation, and automotive mechanics, and machining and construction. The authors have carefully organized the material to provide flexibility: the text can be used in a lecture class, in a laboratory setting, or for self-paced instruction. Each chapter is...show more divided into frames that present the individual concepts on which the major concepts are based. To ensure student comprehension, each concept is first explained and then illustrated with an example. Questions about the material test students' understanding of the concepts, and the answers are found on the right side of each page. Exercise sets throughout the chapters and self tests at the end of each chapter provide further opportunity for students to master the material. In all, the text presents more than 3,000 problems and exercises. ...show less Former Library book. Shows definite wear, and perhaps considerable marking on inside. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy! $18.78 +$3.99 s/h Acceptable BookSleuth Danville, CA Fast Shipping ! Used books may not include access codes, CDs or other supplements. $76.75 +$3.99 s/h Good LOOK AT A BOOK OH Miamisburg, OH 198505340514293.96 +$3.99 s/h New Bookeyez2 snellville, GA Brand New Title. We're a Power Distributor; Your satisfaction is our guarantee! $295.87 +$3.99 s/h New Extremely_Reliable Richmond, TX Buy with confidence. Excellent Customer Service & Return policy. $340.50 +$3.99 s/h Like
Book Description: Mathematics for Elementary School Teachers, 2/e, provides a unique opportunity for students to develop a clear understanding of mathematical concepts, procedures, and processes, to communicate these ideas to others, and to apply them to the real world.The goal is to achieve the optimum balance between presenting a thorough development of mathematical content and presenting it in a way that is understandable by students. The material has been revised so that it powerfully embodies the new Principles and Standards for School Mathematics of the National Council of Teachers of Mathematics.
credit retrieval course is for students who have already taken this class and not earned credit. Students enrolled in this course will use an online curriculum called Compass Learning Odyssey (CLO). Within the CLO, students will take objective-based assessments which indicate what standards have been met and what skills need to be practiced and demonstrated by the student. Based on the results of the this objective-based assessment, an individual plan (called a CLO learning path) will developed by the teacher and communicated to the student. Successful completion of the CLO learning paths and the final summative assessment earns a passing grade (C) and .5 credit in Algebra 1. The second semester of Algebra 2 covers the following mathematical topics: radical functions, exponential and logarithmic functions, probability and statistics, systems of equations and inequalities, matrices, conic sections, and sequences and series. This course helps meet the state minimum requirements of 2.0 Mathematics credits. Please check with your district for more specific requirements.
Catalog Description: This course is intended to be a one-semester survey of Calculus topics specifically for Biology majors. Topics include limits, derivatives, integration, and their applications, particularly to problems related to the life sciences. The emphasis throughout is more on practical applications and less on theory. Pre-requisite: grade of C in Math 180, or suitable placement score. This course qualifies as a General Education course, G9. Text:Calculus with Applications for the Life Sciences, by Greenwell, Ritchey, and Lial (Addison-Wesley, 2003). General Education Core Skill Objectives 1. Thinking Skills: Students engage in the process of inquiry and problem solving, which involves both critical and creative thinking. (a) The student understands the "big problems" in the development of differential calculus, the tangent problem and the area under the curve problem. (b) The student understands the mathematical concept of Limit. (c) The student explores differentiation and works with differentiation formulas for a variety of functions, including exponential and logarithmic functions, and the applications of these methods, especially to problems from the realm of life sciences (d) The student explores integration and a variety of integration techniques, and applications of these techniques to a variety of problems, especially those related to the life sciences in the computer labs - to solve problems and to be able to communicate solutions and explore options. (c) The student will use the language of mathematics accurately and appropriately. (d) The student will present mathematical content and argument in written form. 3. Life Values: Students analyze, evaluate, and respond to ethical issues from informed personal, professional, and social value systems. (a) The student develops an appreciation for the intellectual honesty of deductive calculus and the role it has played in mathematics and in other disciplines. (b) The student learns to use the language of mathematics - symbolic notation - correctly and appropriately. 5. Technology: (a) The student will demonstrate the basic ability to perform computational and algebraic procedures using a calculator or computer. (b) The student will demonstrate the ability to efficiently and accurately graph functions using a calculator or computer. (c) The student will demonstrate the knowledge of the limitations of technological tools. (d) The student will demonstrate the ability to work effectively with a CAS, such as DERIVE, to do a variety of mathematical work. Course Outline 1. Functions a. Linear functions b. Least squares line c. Properties of functions d. Quadratic functions e. Polynomial and rational functions 2. Exponential and Logarithmic Functions a. Exponential functions b. Logarithmic functions c. Applications: growth and decay 3. The Derivative a. Limits b. Continuity c. Rates of change d. Definition of the derivative e. Derivatives and graphs 4. Calculating the Derivative a. Techniques for finding derivatives b. Derivatives of products and quotients c. The chain rule d. Derivatives of exponential functions e. Derivatives of logarithmic functions 5. Graphs and the Derivatives a. Increasing and decreasing functions b. Relative extrema c. Higher derivatives, concavity, the 2nd derivative test d. Curve sketching 6. Applications of the Derivative a. Absolute extrema b. Applications of extrema c. Implicit differentiation d. Related rates e. Differentials and linear approximation 7. Integration a. Antiderivatives b. Substitution technique c. Area and the definite integral d. The Fundamental Theorem of Calculus e. Area between curves 8. Further Techniques and Applications of Integration a. Numerical integration b. Integration by parts c. Volume and average value Required Course Work Your basic work in this course is to learn the material and develop your problem-solving kills so that you can apply the concepts and methods of the calculus to problems you will encounter along the way. It is important that you attend class regularly and especially that you do the homework. The homework may seem "hidden" to you since it will not be graded, but it precisely in that outside of class practice that you learn the material. There is a rule of thumb that says college courses require roughly 2 hours outside of class for every hour in class and I think this is not at all an overstatement. It takes discipline to leave class on a Wednesday or Thursday, knowing that you have class again the very next day, and still find a couple hours to practice your calculus, but it is exactly that sort of discipline that is required for success in the course. In short: DO YOUR HOMEWORK! Your grade will be based on two primary factors: (1) Exams; and (2) Group work. There will be an exam following chapters 1 and 2, chapters 3 and 4, chapters 5 and 6, and then a final cumulative exam at the end of the course, following chapters 7 and 8. These exams will be done in class subject to the 50 minute time limit, but you can use a set of notes you prepare for the exam and you may use a graphing calculator as well. I see exams as a way to see if you as an individual can actually do what the course asks of you. Secondly, there will be two types of groupassignments: (a) Labs; and (b) practice exams. Every so often, roughly once a chapter, I will take a day and have you work in small groups of 3 or 4 students on a set of problems. I will refer to these problem sets as "labs", in the sense that you will be exploring the concepts and trying to apply them to the problems. I like the idea of asking you to occasionally work in groups – I think the opportunity to discuss the mathematical ideas is extremely valuable. After all, mathematics is a language as well as a way of thinking about things. After each of these labs I will ask you to write up a brief reflection paper about the lab you just did. These will be due the next class period and should comment on the basic idea you were supposed to learn from the lab problems. I will also have you take a "practice exam" during the class period just prior to each individual exam. These practice exams will be done in groups like the labs, and must be turned in by the end of the period, as do the labs. This is an attempt on my part to give you an idea of what to expect on the exam. It is also worth mentioning here that because one of our goals is to develop some skills with technological tools, I will be asking you to use DERIVE on a few problems along the way. DERIVE is a computer program created by Texas Instrument, the calculator people, to perform a very wide variety of mathematical procedures. It is a CSA (Computer Algebra System), which means it can perform algebraic manipulations like solving equations or factoring polynomials, as well as graphing functions in 2 or 3 dimensions. It can also perform calculus procedures and we will have a chance to see some of this power. Instructional Methods My general "lecture" style is more of a give-and-take discussion than simply a rote presentation of material. I like to try to see that the class is following what I am doing and so I want feedback along the way. I often will ask a leading question to see that you are ready for what is about to come. One specific strategy I will use is called Think-Pair-Share. In this case I will ask a question and have each of you think about it for a bit, then pair up with a "neighbor" and finally share the answer with the rest of the class. Not every pair will share with the class each time, but I hope as we do this on a regular basis most of you will have an opportunity to speak. I have already mentioned the "Labs" we will do on occasion. I will randomly assign you to a group and give each of you rotating roles within the process; I'll explain the details when we do that first lab on Wednesday, 7 September. I will try to keep the labs reasonably short so that the group can turn in the work at the end of the period – this is required. I am also requiring each of you to write up a brief reflection paper on each of the labs, due the next class meeting, in which you reflect on what you learned from the lab, or what you think you were supposed to have learned – this will be more or less a "journal". Assessment Strategies The main outcomes I want to see in each of you can basically be listed briefly as: (1) thinking, or problem solving; (2) communicating your mathematical ideas and solutions; and (3) using technology to do some of the work. I will be assessing your ability to do these things throughout the course by grading the labs, the technology assignments, and the exams. I will also be noting your participation in the class room, both in your group work and in general class discussion. Grading System Because this is the first time this course has been taught we are all serving as guinea pigs, and there may very well need to be some adjustments made along the way. But I need to put something down on paper at the start, so here it is. I will then assign letter grades as follows: 90% of possible points for "A", 80% for a "B", 70% for a "C", and 60% for a "D". By the way, I am aware that the biology program requires a grade of at least a "C" in its support courses, so you needn't tell me that if the going gets "close" later on in the course. I wish this wasn't the case, since it is sometimes stressful on me as well as on you, and I don't like "losing" the possibility of giving a "D" grade – sometimes people pass a course but "just barely". Disability Statement If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me and/or Wayne Wojciechowski, the campus ADA coordinator (MC 335Attendance Policy I think that regular attendance in of paramount importance in any course, but perhaps a mathematics course more than most. There may be the occasional exception, a student who is so mathematically talented that she or he can get the material by just reading the text and doing some problems, but for most students it is important to be in class every day. To encourage this I am going to include attendance in the grading scheme in a simple way: 1 point for each day you are there. I'm not going to engage in whether absence is "excused" – too many subtleties and degrees there - just a point a day when you are there. One other note about the labs: because I want to collect them all at the end of class and grade them for the next class meeting, there will be no opportunity to "make them up". You either do it or you don't. But I am making a bit of an allowance here; the total value of the labs is 140, but it is possible to earn 160 points in total – basically I have one lab as a "freebie".
The Everything Guide to Calculus I A step-by-step guide to the basics of calculus—in plain English! By Greg Hill, National Council of Teachers of Mathema Format: SKU# Z9326 Details Calculus is the basis of all advanced science and math. But it can be very intimidating, especially if you're learning it for the first time! If finding derivatives or understanding integrals has you stumped, this book can guide you through it. This indispensable resource offers hundreds of practice exercises and covers all the key concepts of calculus, including: Greg Hill has more than twenty-five years of experience teaching AP Calculus and other advanced math classes. He is a two-time Illinois state finalist for the Presidential Award for Excellence in Mathematics and Science Teaching, and is a member of the Illinois and National Councils of Teachers of Mathematics. Hill has been a College Board consultant and AP Calculus exam grader for the past ten years. He currently teaches at Hinsdale Central High School, and also conducts day- and week-long professional development seminars for AP Calculus teachers. He is the author of CLEP Calculus, a test prep book for the College Board's College Level Entrance Exam. Additional Information SKU Z9326 Author/Speaker/Editor Greg Hill, National Council of Teachers of Mathema File/Trim Size 9 x 9-1/4 Format No ISBN 13 9781440506291 Number Of Pages 320 Retail: $16.95 Your price: $11.53 You save: $5.42
Elementary MathematicsIn secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary schoolstudents, is usually considered college level mathematicsIn the United States, there has been considerable concern about the low level of elementary mathematics skills on the part of many students, as compared to students in other developed countries. The No Child Left Behind program was one attempt to address this deficiency, requiring that all American students be tested in elementary mathematics. "A school is not a factory. Its raison d'être is to provide opportunity for experience." —J.L. (James Lloyd) "Cloud-clown, blue painter, sun as horn, Hill-scholar, man that never is, The bad-bespoken lacker, Ancestor of Narcissus, prince Of the secondary men. There are no rocks And stones, only this imager." —Wallace Stevens (1879–1955) "The longer we live the more we must endure the elementary existence of men and women; and every brave heart must treat society as a child, and never allow it to dictate." —Ralph Waldo Emerson (1803–1882) "He taught me the mathematics of anatomy, but he couldn't teach me the poetry of medicine.... I feel that MacFarland had me on the wrong road, a road that led to knowledge, but not to healing." —Philip MacDonald, and Robert Wise. Fettes (Russell Wade)
TEXTBOOK* Lie Groups: A Problem-Oriented Introduction via Matrix Groups Harriet Pollatsek Can be used as supplementary reading in a linear algebra course, or as a primary text in a bridge course that helps students make the transition to courses that emphasize definition and proofs, as well as for an upper level elective. The work of the Norwegian mathematician Sophus Lie extends ideas of symmetry and leads to many applications in mathematics and physics. Ordinarily, the study of the "objects" in Lie's theory (Lie groups and Lie algebras) requires extensive mathematical prerequisites beyond the reach of the typical undergraduate. By restricting to the special case of matrix Lie groups and relying on ideas from multivariable calculus and linear algebra, this lovely and important material becomes accessible even to college sophomores. Working with Lie's ideas fosters an appreciation of the unity of mathematics and the sometimes surprising ways in which mathematics provides a language to describe and understand the physical world. Lie Groups is an active learning text that can be used by students with a range of backgrounds and interests. The material is developed through 200 carefully chosen problems. This is the only book in the undergraduate curriculum to bring this material to students so early in their mathematical careers. An Instructor's Manual is available to teachers who adopt Lie Groups as a text. Contact our Service Center for details at 1-800-331-1622. A hardcover version of this book is available in our regular store. * As a textbook, Lie Groups does have DRM. Our DRM protected PDFs can be downloaded to three computers.
Exploring Multivariable Calculus is an excellent tool which a student can use to visualize surfaces in 3D. A student can enter up to four equations where some of them can be in another coordinate system such as cylindrical or spherical or even implicit. The graphs are rendered in an easy to visualize manner. The wide variety of menu options allows a student to grab and turn the graphs until they are easy to visualize.
Dublin, GA Geometry...In this subject, the student will learn probability in terms of the basic definition, the binomial distribution, and the normal distribution. Probability is used in inferential statistics. ACT Math basically consists of every branch of Mathematics except for CalculusMost five.
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The study of Mathematics has been a fundamental part of education since ancient times and has lost none of its relevance in modern times. In fact, much the contrary. Mathematics permeates our lives, enables us to communicate across vast distances, shrinks our technology to the palm of our hands, and opens up more and more of the nature of the world to our investigations. We understand full well that many of our students come to use with little of this sense of wonder, rather with a sense of fear and apprehension. We strive constantly to meet our students where they are, bring them into a new relationship with mathematics, and open up the possibilities of mathematics for them and for their careers. Math Curriculum The Mathematics department curriculum has two components, one developmental, the other college-level. The Developmental Mathematics Program For those that are not ready to undertake college-level mathematics, either because they have been away from the subject for a while or because they have yet to fully grasp the fundamentals and put them into practice, the developmental mathematics program provides coursework for students demonstrating deficiencies in mathematical skills. MATH 049, Elementary Algebra, and MATH 149, Intermediate Algebra are consecutive 3-unit courses that prepare students for college-level work in mathematics. We also offer a self-paced version of Intermediate Algebra (MATH 1491) for those who fall somewhere in-between these two traditional courses. Placement in these classes is based on a proficiency examination score. A grade of "C" (2.0) or higher in MATH 049 is required to enroll in MATH 149. A grade of "C" (2.0) or higher in MATH 149 or 1491 is required to enroll in college level mathematics courses. The College-Level Mathematics Program All university students are required to complete at least one college-level mathematics course. Many departments ask for an additional mathematics course, so please check your major department's course requirements. College-level mathematics courses are designated with a number in the 200s, such as Business Math and Statistics, College Algebra and Trigonometry with Descriptive Geometry. In addition, the department may occasionally offer lower- or upper-division topics courses in mathematics. For more information about the Mathematics department curriculum, contact Prof. Marty Tippens, Chair, at marty.tippens@woodbury.edu
Description Bridging the gap between current merchandising math textbooks and retail buying textbooks, The Fundamentals of Retail Buying with Merchandising Math incorporates both buying philosophies and merchandising math. The text's problem-based method of learning incorporates questions and problems that train the reader to think like a buyer and encourage group collaboration and critical thinking. Simulated exercises mimic real-life buying responsibilities. Additionally, the order of the chapters and content within each chapter mimic the training of an assistant buyer in a corporate buying office. Providing a full, broad view of the retail buyer's role, the text also includes the key merchandising math formulae that is the basis of all retail buying analysis. Table of Contents UNIT 1: FORMULAS AND BASIC PRINCIPLES Introduction: Getting Started Chapter One: Sales and Percent Change Chapter Two: Percent to Total Chapter Three: Sell Thru Chapter Four: Stock-to-Sales Analysis Chapter Five: Markup & Markdowns UNIT 2: MANAGING THE BUSINESS Chapter Six: Purchase Orders Chapter Seven: The Six Month Merchandise Plan Chapter Eight: Profit & Loss Statement Chapter Nine: Cooperative Advertising UNIT 3: PRIORITIZING AND NEGOTIATING Chapter Ten: Buying Simulations Chapter Eleven: End-of-Season Negotiations Chapter Twelve: Monday Morning Reports (Sales) Chapter Thirteen: Monday Morning Reports (Markdowns
Larson IS student success. INTERMEDIATE ALGEBRA owes its success to the hallmark features for which the Larson team is known: learning by example, a straightforward and accessible writing style, emphasis on visualization through the use of graphs to reinforce algebraic and numeric solutions and to interpret data, and comprehensive exercise sets. These pedagogical features are carefully coordinated to ensure that students are better able to make connections between mathematical concepts and understand the content. With a bright, appealing design, the new Fifth Edition builds on the Larson tradition of guided learning by incorporating a comprehensive range of student success materials to help develop students' proficiency and conceptual understanding of algebra. The text also continues coverage and integration of geometry in examples and exercises. Additional versions of this text's ISBN numbers Purchase Options List$260208.49 $208.49 Intermediate Algebra Enhanced WebAssign Homework LOE Printed Access Card for One Term Math and Science
Summary: The power and attractiveness of the subject of mathematics is often hidden from students who are in introductory courses. In this new, innovative overview textbook, the authors put special emphasis on the deep ideas of mathematics and present the subject through lively and entertaining examples, anecdotes, challenges and illustrations, all of which are designed to excite the student's interest. The underlying ideas include topics from number theory, infinity, geometr...show morey, topology, probability, and chaos theory. Throughout the text, the authors stress that mathematics is an analytical way of thinking, one that can be brought to bear on problem solving and effective thinking in any field of study. ...show less Surfing the Book Fun and Games: An Introduction to Rigorous Thought Number Theory: The Secret and Hidden Power of Numbers Infinity Geometric Gems Contortions of Space Chaos and Fractals Risky Business Farewell Other Editions of Heart Of Mathematics : An Invitation To Effective Thinking / Text Only:595340796.99 +$3.99 s/h Good Solr Books Skokie, IL No comments from the seller $7.47
Real Analysis Description Modern mathematics and physics rely on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called ``analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Objective To introduce students to the fundamentals of mathematical analysis and to reading and writing mathematical proofs. At the end of this course, students should: understand the axiomatic foundation of the real number system, in particular the notion of completeness and some of its consequences; understand the concepts of limits, continuity, compactness, differentiability, and integrability, rigorously defined; be able to use results and techniques involving these concepts to solve a variety of problems, including types of problems that they have not seen previously; know how completeness, continuity, and other notions are generalised from the real line to metric spaces; and appreciate the Contraction Principle in abstract metric space theory as a powerful tool to solve concrete problems, especially in differential equations. Students should also have attained a basic level of competency in developing their own mathematical arguments and communicating them to others in writing. Content Topics covered are: Basic set theory. The real numbers, least upper bounds, completeness and its consequences. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions, uniform continuity. Differentiation, the Mean Value Theorem. Sequences and series of functions, pointwise and uniform convergence. Power series and Taylor series. Metric spaces: basic notions generalised from the setting of the real numbers. The space of continuous functions on a compact interval. The Contraction Principle. Picard's Theorem on the existence and uniqueness of solutions of ordinary differential equations. Linkage past Linkage present No present linkages have been noted. Linkage future This course is essential for students wishing to study more advanced mathematical analysis, the theory of differential equations, differential geometry, topology, or mathematical physics. The course also provides widely applicable training in constructing and writing rigorous arguments.
Precalculus: Concepts Through Functions - 2nd edition Summary: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry, Second Editionembodies Sullivan/Sullivan's hallmarks-accuracy, precision, depth, strong student support, and abundant exercises-while exposing readers to functions in the first chapter. To ensure that students master basic skills and develop the conceptual understanding they need for the course, this text focuses on the fundamentals:preparingfor class,practicingtheir homework, andreviewingthe concepts. A...show morefter using this book, students will have a solid understanding of algebra and functions so that they are prepared for subsequent courses, such as finite mathematics, business mathematics, and engineering calculus. KEY TOPICS: Foundations: A Prelude to Functions; Functions and Their Graphs; Linear and Quadratic Functions; Polynomial and Rational Functions; Exponential and Logarithmic Functions; Trigonometric Functions; Analytic Trigonometry; Applications of Trigonometric Functions; Polar Coordinates; Vectors; Analytic Geometry; Systems of Equations and Inequalities; Sequences; Induction; the Binomial Theorem; Counting and Probability; A Preview of Calculus: The Limit; Derivative, and Integral of a Function MARKET: For all readers interested in college algebra
Algebra 1 - The Complete Course - Lesson 21: Working with Rational Functions and Equations movie was released Jan 01, 2010 by the TMW Media Group studio. Closed Captioned; Standard Screen; Soundtrack English Departing from traditional methods of teaching math, Dr. Algebra 1 - The Complete Course - Lesson 21: Working with Rational Functions and Equations movie Monica Neagoy uses her extensive knowledge of history and mythology combined with concrete examples and illustrations to introduce algebraic concepts and make learning fun. Algebra 1 - The Complete Course - Lesson 21: Working with Rational Functions and Equations video In this volume, she teaches how to answer questions by functional exploration and how to confirm answers by symbolic manipulation.
An investigation of topics including the history of mathematics, number systems, geometry, logic, probability, and statistics. There is an emphasis throughout on problem solving. Recommended for General Education. For some of you this course might serve to satisfy the math competency requirement, for others this will be just one of the mathematics courses required by your major/minor program. In any case, the main goal of this course is to help you develop and strengthen the foundations of your analytical thinking. Every day, we are faced with numerous questions, such us: Should I run through this yellow light? What should I eat today? Which courses to enroll next semester? . . . We often have to resolve those questions, make appropriate decisions, and then act according to those decisions. The thinking process required for resolving all kinds of questions, puzzles, problems, is known by the name of analytical thinking. We can use mathematics as a convenient tool for working on the analytical thinking skills. In order to achieve this goal of developing and strengthening your analytical thinking skills, via mathematics, our focus will be on the following questions: • What does it mean to do mathematics? • What does it mean to think mathematically? 1 2 MATH 155 WAY OF THINKING SYLLABUS - SPRING 2003 • What does it mean to understand a piece of mathematics? In other words, I expect you to: • Do some mathematics; • Use mathematical reasoning, i.e., ask questions such as: – What does (something) mean? – How did we get from A to B? – Is this (a statement, claim, formula, . . . ) correct? – How do I know that it is correct? • Strive to understand every idea, concept, problem, solution that we encounter in this course. The mathematical content which we will use to achieve these objectives will expose you to a variety of areas of mathematics and thus give you an idea of the importance of mathematics in today's world and a multitude of ways it is being used in practice. We will learn some elements of The General Education aspects of this course. The content and the methods of this course are designed in accordance with general education objectives and the work in this course should help you in developing a number of skills included in the NCTM (National Council of Teachers of Mathematics) 'standards" for mathematics education, and also being among the general education objectives at Viterbo. The main emphasis throughout the course will be on problem solving and developing thinking skills. This includes: (a) writing numbers and performing calculations in various numeration system, (b) solving simple linear equations, (c) exploring the mathematical model of simple and compounded interest rates, and learning how to use those ideas in solving the problems of loan payments, (d) exploring a few major concepts of Euclidean Geometry, focusing especially on the axiomatic-deductive nature of this mathematical system, including a variety of different proofs of the Pythagorean Theorem, (e) develop an ability to use deductive reasoning, in the context of the rules of logic and syllogisms, i.e., learn how to make/recognize a valid argument, (f) some basics of probability and statistics . . . Potential benefits of the course. Mastering this material requires to learn how to reason mathematically, and also how to communicate mathematics. In learning how to do so (on exams, essays, portfolio, and in oral presentations), you will also develop a confidence in your ability to do mathematics. This way you will strengthen your ability to solve problems, analyze arguments, understand abstract concepts. Other benefits of this course include: cultural skills (appreciation of the history of mathematics and its role in today's world, learning how to handle simple loans, etc.), appreciate the beauty and intellectual honesty of deductive reasoning, thereby adding to life value and aesthetic skills. Format: Class sessions will consist of lectures, work in small groups, exams, and individual presentations. I expect students to work out the recommended practice problems and ask for help whenever needed. Resources: Please do not hesitate to contact me for any question you might have; do not let a feeling such as "I am lost . . . " to last. Other resources include: • Internet and the Blackboard software. There is a lot of material on my web page. There will be some quizzes given using the Blackboard. • The Learning Center. • The library. Note that both a video set and a CD set that covers your textbook exist. You can use either of these to hear a lecture again, or just to see/hear another explanation of a particular topic. Grading: The final grade is based on homework, exams, presentations, portfolio, and a (cumulative) final exam. There will be opportunities for a small amount of extra credit. The following grading scale applies to individual exams, and to the overall grade as well: A=90%, AB=87%, B=80%, BC=77%, C=70%, CD=67%, D=60%. The following exceptions to that scale are possible: • An A on the final exam (more than 180/200 points) will raise your grade up, one letter, i.e., a B will turn into an A, a BC will become AB, . . . . • An outstanding presentation, or an outstanding portfolio can raise your grade up a half letter, i.e., a C will turn into a BC, . . . . • If one is failing the course by the end of the semester, but has over 40% average on exams, and earns at least 55% points on the final, he/she can get a D for the final grade. • If one is passing the course by the time of the final exam, but earns less than 30% points (a score less than 60/200), that will result in an F for the final grade. Assignments: • Recommended practice: First 10, middle 5 and the last 5 problems from each Practice Exercises set in each section that we cover; at least one or two of the Application Exercises, at least one of the Writing in Mathematics Exercise, and at least two of the Critical Thinking Exercises. These practice problems will not be graded. However, fell free to ask me for help with any difficulty you might have with those problems. • Four essays, 20 points each: (1) Essay I - Autobiography: Introduce yourself to me in a 1- 2 pages essay. State your name, and where (city/state) you are coming from. The reason you are taking this course, and what mathematics courses you have had before. What was your experience from those courses and what are your expectations, if any, from this course? This assignment is due Friday, January 17. 4 MATH 155 WAY OF THINKING SYLLABUS - SPRING 2003 (2) Essay II - A mathematical story. In order to make the connection of the first assignment (the Autobiography) with the main goals of the course more explicit, I would like you to recall some of your specific experiences of doing mathematics, and tell me a short story (1-2 pages) about it. In particular, I would like you to address the following questions in this story: – Try to recall an experience of you actually doing mathematics. Give an example. Describe, make a story about it. – How about an experience of reasoning mathematically? It would be great if you could give a simple example, and even better if you have had an opportunity to communicate your mathematical reasoning to somebody else. – Did you ever truly understand a piece of mathematics? Give an example. Describe. Explain. If your answer to any of the questions above is negative, i.e., you have not had such an experience, then please try to explain how is that possible. Due: Monday, January 20, 2003. (3) Essay III - World without mathematics: another 1-2 pages 20 points essay. Try to imagine, and describe, a world without mathematics. Due: Friday, January 24. (4) Essay IV - Me, a Mathematician. In this essay, you should answer same questions as in Essay II, but in relation to the material covered in this course. More precisely, the questions are: – Did you ever, during the work in this course, have an experience of actually doing mathematics. Give an example. Describe. Explain. – Did you ever, during the work in this course, have an experience of reasoning mathematically? Give an example. Describe. Explain. – Did you ever truly understand at least one piece of mathematics encountered in this course? Give an example. Describe. Explain. This last essay is due Monday, April 28, 2003 (the last week of class). Homework: At the end of each chapter, there is a Chapter Test. Each one of those tests will be due second class period after the corresponding chapter is covered, and each problem on the "test"is worth 1 point. This rule is a tentative one. Sometimes, I give a different problem set instead of those Chapter reviews. Exams: There will be three in-class exams, worth 100 points each. An exam will typically cover three chapters worth of material. The exams will be closed notes, closed book. However, a calculator and a formula sheet (but not any worked out problem) is allowed. MATH 155 WAY OF THINKING SYLLABUS - SPRING 2003 5 Before each exam, I will give you a take-home practice exam, which will be very much like the actual exam coming. I will grade (25 points) the first one of those, i.e., the "Exam 1 - Practice", but not the others. I will also allow a makeup (up to 50%) of the lost credit for one of the exams. It will be Exam 1 for Section 02, and Exam 2 for Section 01. This makeup will be oral, and will apply to those under 90/100 points on the test, and is to be done within two weeks after the exam. Final Exam: Final exam is a 2-hour, cumulative exam, and is worth 200 points. Portfolio: It should consist of 5 problems, but no two problems should be of the same type (from the same section). Format: You state a problem, write a complete/correct solution to it, and then write a paragraph (or more) explaining why did you choose that particular problem, what did you learn from it, etc.. The portfolio will be worth 50 points. The problems you choose for the portfolio should illustrate the progress in learning mathematics, the change of the perception (if any) of what mathematics is about, the change (if any) in your perception about your abilities to do mathematics. In-class Presentation: The presentation of a proof of the Pythagorean Theorem found on the Internet. Typically, the explanations you will find on the Internet are a bit sketchy. So, part of your job will be to make sure you really understand the proof you are going to present (including filling in the gaps, i.e., the reasons not entirely spelled out in the Internet write-up), and then to clearly explain that proof to your classmates. Sometimes, some people, may find this part quite difficult. Of course, I am here to help you understand and overcome those difficulties, and so please do not hesitate to ask me for help. You should also be prepared for the questions from the audience (myself and/or other students), and it is expected that you listen closely to other presentations and ask any question you might have. The presentation will be worth 35 points. In addition to that, one certain problem for one of the exams, or for the final exam, is going to be: State and prove the Pythagorean Theorem. Last, but not least, the presentations will take about three weeks of class time. I encourage you to use that time to study a lot, catch up, learn some more, and get real ready for the final exam. Also, I will give you several 10-points homework based on some of the questions raised by some presentations. Important University Policies: Those are Viterbo's policies on Attendance, Plagiarism, and Sexual Harassment. You can find the statements at: Disability: Americans with Disability Act: If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me andWayneWojciechowski in Murphy Center Room 320 (796-3085) within ten days to discuss your accommodation needs. 6 MATH 155 WAY OF THINKING SYLLABUS - SPRING 2003 1. Schedule outline Week Section Week Section Jan. 13 12.1 - 12.4 Jan. 20 12.4; 9.1-1.3 Jan. 27 10.1-10.3 Feb. 3 10.4-10.6, Exam 1 Feb. 10 2.1-2.4 Feb. 17 2.5; 3.1-3.3 Feb. 24 3.4-3.7 Mar. 3 4.1-4.4 Mar. 10. Spring break Mar. 17 Exam 2; 5.1-5.2 Mar. 24 6.1 - 6.3; 8.3; No class 03/28 Mar. 31 11.1-11.4 Apr. 7 Exam 3; Presentations Apr. 14 Presentations; Easter vacat. Apr. 21 Presentations; No class 04/25 Apr. 28 Presentations Important dates. Classes begin: January 13. Midterm break: March 10-14. Easter vacation: April 17-21. Easter: April 20. Last day of class: Friday, May 2. : No classes, due to my conferences: • Friday, March 28; • Friday, April 26. Final Exam: From the Final Exam Schedule: The final exam for all Monday 9:00 A.M. classes - on Tuesday, May 6 from 9:50-11:50 a.m.
Teach students how to provide constructed answers to algebra problems with this practical series. The four-page lessons encourage students to show their work, explain how they found each answer, and... Provide instruction and practice for the range of math skills students need to succeed on standardized tests and in everyday life. Each Student Book offers easy-to-follow instructions for basic math...
2012 - 2013 Every Field 8th grader will be taking 8th Grade Algebra during this school year. This is aligned to the MPS Scope and Sequence for Mathematics. In addition, students can elect to take Intermediate Algebra during their BEST class period. Intermediate Algebra is the 9th grade course. Students who sucessfully complete Intermediate Algebra during 8th grade will take Geometry in 9th grade. If you have questions about either 8th Grade Algebra or Intermediate Algebra, please write, call, or visit. Sketchpad Lesson Link -We will be using The Geometer's Sketchpad software to explore algebra concepts. This link connects to my page with activities I've chosen for students. These activities require the Geometer Sketchpad software to be installed on your computer so these are activities we will do in school. A password (given to students) is required.
Course Plan for Beginning Algebra I I. COURSE DESCRIPTION A developmental course covering first year algebra. II. PREQUSITES Successful completion of MTH 2 or appropriate score on the placement test. Students will not be al lowed in the course without the required prequisites. III. INTRODUCTION This course is designed to introduce and develop basic algebraic concepts and skills listed in the out line below . A student must have an 80% average to complete the course successfully and receive a grade of "S" – satisfactory. Otherwise the student will receive a grade of "R" - reenroll, or "U" - unsatisfactory and have to repeat the course. IV. INSTRUCTIONAL MATERIALS The Greatest Common Factor Factoring Trinomials of the Form ax2 + bx + c by the Grouping Method Note: Factoring by Trial and Error is covered in appendices A.1 & A.2 Factoring By Special Products (perfect square trinomials is optional) Solving Quadratic Equations by Factoring Quadratic Equations and Problem Solving Reading Graphs and the Rectangular Coordinate System (stress the Rectangular Coordinate System) Graphing Linear Equations Intercepts Slope andRate of Change The Slope- Intercept Form More Equations of Lines
Personal tools Mathematics Document Actions Mathematics Dixie State College of Utah ABOUT THE DEPARTMENT The Dixie State College Mathematics Department helps students to achieve their academic, career, and life goals, including those related to basic computational skills, mathematical processes, and knowledge that develops real-life applications, modeling and problem solving. The Department's comprehensive and integrated offerings help students master mathematical competencies for career and educational endeavors. The Department offers classes that are useful for skill levels from developmental to selected four-year degree requirements. Copyright 2008, by the Contributing Authors. Cite/attribute Resource. jones. (2007, November 09). Mathematics. Retrieved May 23, 2013, from Dixie State College of Utah Web site: This work is licensed under a Creative Commons License.
Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field - the heat equation, the wave equation, and Laplace's equation. The most common techniques of solving such equations are developed in this book, including Green's functions, the Fourier transform, and the Laplace transform, which all have applications in mathematics and physics far beyond solving the above equations. The book's focus is on both the equations and their methods of solution. Ordinary differential equations and PDEs are solved including Bessel Functions, making the book useful as a graduate level textbook. The book's rigor supports the vital sophistication for someone wanting to continue further in areas of mathematical physics. Examines in depth both the equations and their methods of solution Presents physical concepts in a mathematical framework Contains detailed mathematical derivations and solutions- reinforcing the material through repetition of both the equations and the techniques Includes several examples solved by multiple methods-highlighting the strengths and weaknesses of various techniques and providing additional practice
This handbook is essential for solving numerical problems in mathematics, computer science, and engineering. The methods presented are similar to finite elements but more adept at solving analytic problems with singularities over irregularly shaped yet analytically described regions. The author makes sinc methods accessible to potential users by limiting... more... This book provides a lens through which modern society is shown to depend on complex networks for its stability. One way to achieve this understanding is through the development of a new kind of science, one that is not explicitly dependent on the traditional disciplines of biology, economics, physics, sociology and so on; a science of networks. This... more...... more... Whether you are returning to school, studying for an adult numeracy test, helping your kids with homework, or seeking the confidence that a firm maths foundation provides in everyday encounters, Basic Maths For Dummies, UK Edition, provides the content you need to improve your basic maths skills. Based upon the Adult Numeracy Core Curriculum,... more...
workbook have provided a variety of exercises to help apprenticeship candidates improve their numeracy skills and learn how those skills are used in the trades. The workbook includes sections focusing on measurement and calculation; money math; scheduling, budgeting and accounting; and data analysis. The authors have also included a chart that lists and defines math foundation skills, then provides examples of how each skill might be used in the workplace. For example, they explain that trigonometry is used to determine the size of an unknown side or angle of a triangle and, on a construction site, it might be used to calculate the angles for a circular staircase. In each section, the authors present math exercises based on real-life situations. For instance, in the money math section, the exercises deal with topics like calculating the after-tax cost of an item or the cost of an item after a markup has been included. The section on measurement and calculation includes exercises on calculating how many studs a carpenter will need to frame the exterior walls of a building or how many tiles will be required to cover a specific floor area.
Summary: MATHEMATICS FOR 3D GAME PROGRAMMING AND COMPUTER GRAPHICS, THIRD EDITION, illustrates the mathematical concepts that a game developer needs to develop 3D computer graphics and game engines at the professional level. It starts at a fairly basic level in areas such as vector geometry and linear algebra, and then progresses to more advanced topics in 3D programming such as illumination and visibility determination. Particular attention is given to derivations of key results, ensuring that the reader is not forced to endure gaps in the theory. The book assumes a working knowledge of trigonometry and calculus, but also includes sections that review the important tools used from these disciplines, such as trigonometric identities, differential equations, and Taylor series. Each chapter ends with a summary of important concepts and several exercises. This updated third edition expands upon topics that include projections, shadows, physics, cloth simulation, and numerical methods.
Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum - nothing beyond first courses in linear algebra and multivariable calculus - and the most direct and straightforward approach is used throughout. New features of this revised and expanded second edition include: a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature. Around 200 additional exercises, and a full solutions manual for instructors, available via less
Audio Visual learning aids to learn Math in a fun and effective way. Our Audio Visual learning aids have been designed and developed by education experts. The content is tailored to meet the requirements of each age group. It is interactive, fun and very easy to use. List of chapters in our product Concept of Set Operations on Sets and Venn Diagrams Numbers Integers Fractions Decimals The Number Line Factors and Multiples Powers and Roots Ratio and Proportion Percentage Profit, Loss and Discount Simple Interest Fundamental Concepts of Algebra Operation on Algebraic Expressions Substitution Rel(more)Audio Visual learning aids to learn Math in a fun and effective way. Our Audio Visual learning aids have been designed and developed by education experts. The content is tailored to meet the requirements of each age group. It isShows how corporate responsibility can lead to new markets and solutions to long-standing business problems.Shows how corporate responsibility can lead to new markets and solutions to long-standing business problemsVedic Maths is much more than a magical method of fast calculation.While mastery of its simple sutras - and a little practice - undoubtedly enables one to perform mental computations with lightening speedVedic Maths is much more than a magical method of fast calculation.While mastery of its simple sutras - and a little practice - undoubtedly enables one to perform mental computations with lightening speed(less) variousMagical Book On Quicker Maths is a book that helps its readers tackle the mathematics sections in all competitive exams with minimum effort. Summary Of The Book In Magical Book On Quicker Maths, M. Tyra provides a new and speedier approach to problem solving, one that is simple to understand and eas(more)Magical Book On Quicker Maths is a book that helps its readers tackle the mathematics sections in all competitive exams with minimum effort. Summary Of The Book In Magical Book On Quicker Maths, M. Tyra provides a new and speedier approach to problem solving, one that is simple to understand and easy to apply. It is aimed at aspirants of all competitive exams conducted by banks, the Union Public Service Commission (UPSC), the Staff Selection Commission (SSC), LIC, CPO, UTI, GIC, and similar organizations
Precalculus: Functions Functions Calculus is the mathematical study of change, and real-life things that change are modeled by functions. Precalculus is essentially the study of functions, with a few other related topics that supplement the study of functions and prepare students for calculus. In this text we'll concern ourselves first with learning about general functions, and later with certain types of common functions with special properties, like polynomial, exponential, logarithmic, and trigonometric functions. First, it is of critical importance to understand exactly what a function is. In the following lessons, we'll discuss what makes a function a function, some general properties of functions, and a few basic categories of functions. In this text we'll assume a general knowledge of algebraic principles of solving equations, working with the real numbers, and working with sets. In the last of the upcoming sections, we'll learn how functions behave under operations like addition, subtraction, etc.
Fundamentals of Mathematics, 10th + Student Solutions Manual ISBN10: 1-111-61607-8 ISBN13: 978-1-111-61607-6 AUTHORS: Van Dyke/Rogers/Adams The FUNDAMENTALS OF MATHEMATICS, Tenth Edition, offers a comprehensive and objectives-based review of all basic mathematics concepts. The author helps learners by addressing three important needs: 1) establishing good study habits and overcoming math anxiety, 2) making the connections between mathematics and their modern, day-to-day activities, and 3) being paced and challenged according to their individual level of understanding. The clear exposition and the consistency of presentation make learning arithmetic accessible for all. Key concepts presented in section objectives—and further defined within the context of How and Why—provide a strong foundation for learning and lasting comprehension. With a predominant emphasis on problem-solving skills, concepts, and applications based on "real world" data (with some introductory algebra integrated throughout), this book is suitable for individual study or for a variety of course formats: lab, self-paced, lecture, group, or combined formats. Additional versions of this text's ISBN numbers Purchase Options List$330
2013-2014 University Catalog This course is designed to enhance mathematical literacy and to stimulate interest in and appreciation for mathematics and quantitative reasoning as valuable tools for addressing issues in a constantly changing society. Topics may include, at an introductory level: 1) logical reasoning and problem solving through mathematical games and puzzles; 2) counting and number concepts (number theory and infinity); 3) geometry (Euclidean/non-Euclidean/fractal geometrics, and topology); and 4) probability and statistics. Prerequisite: MATH 1314.
This is a traditional geometry text, requiring the students to prove theorems. It is biblically based throughout and contains one lesson per chapter, relating Geometry and Scripture. Different colors and shading are used to distinguish among postulates, definitions, theorems, and constructions. Exercises are divided into three levels of difficulty. Dominion Thru Math exercises, scattered through each chapter, relate to the chapter openers, and offer the opportunity for students to the use technology in problem solving. Analytic Geometry features, one per chapter, help students to make the algebra-geometry connection. Geometry Around Us features reveal some of Geometry's secret hideouts. Mind over Math brain teasers are included. Geometry Through History introduces students to Mathematicians of the past and their achievements.
Calculus: Calculus teachers recognize Calculus as the leading resource among the "reform" projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. The Sixth Edition uses all strands of the 'Rule of Four' - graphical, numeric, symbolic/algebraic, and verbal/applied presentations - to make concepts easier to understand. The book focuses on exploring fundamental ideas rather than comprehensive coverage of multiple similar cases that are not fundamentally unique. «Show less Calculus: Calculus teachers recognize Calculus as the leading resource among the "reform" projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. The Sixth Edition uses all strands of... Show more» Rent Calculus 6th Edition today, or search our site for other Hughes-Hallett
MiLearn High School Course Complements Curriculum-based high school level math resources in an interactive, "concept, example, application" web design. Courses contain over 500 web pages of step-by step examples and solutions; free demos are available on the site. MiLearn also provides consulting ...more>> mrs. tinashe blanchet - Tinashe Blanchet Courses, collaborations, and other content by Blanchet, who teaches high school math in Marrero, Louisiana. Freely-accessible Moodle classes have included International Baccalaureate (IB) Math Studies and Advanced Math/Pre-Calculus; professional development ...more>> Musing Mathematically - Nat Banting Thoughts from a high school mathematics teacher on the teaching of mathematics, and the learning of mathematics. Posts, which date back to May, 2011, have included "Playing with Mean, Median & Mode," "Attaching a 'Why' to the 'How," "The Mathematics ...more>> Number Bender - Peter Esperanza Freely downloadable algebra, pre-calculus, and AP statistics presentations and worksheets by an Ed.D. candidate who has taught in China, the Philippines, and the U.S.PBS Teachers: Math - PBS Teachers The Public Broadcasting System's Math Service, combining computing and telecommunications technologies to offer interactive data services and interactive video and voice services for education based on the NCTM Standards. The site features: resources, ...more>> Personalized Programming Service Inc. Educational software for teachers and students. Topical software in Math, English, Social Studies, and Science, as well as Quizmaker and Testmaker programs. Download demos and upgrades, browse the products, or order on site. ...more>> Plot Graphs - Philipp Wagner Plot multiple two- or three-dimensional graphs, including of parameterized functions and derivatives, in one and the same graph. Key in equations with the WYSIWYG formula editor. ...more>> Powell's Books Located in Portland, Oregon, USA; the largest independent bookseller in the United States, specializing in technical books and new, used, hard-to-find, and antiquarian titles. An extensive subject and keyword list is available - search database of titles, ...more>> Power Maths - A pre-calculus project - Sidney Schuman A pre-calculus investigation designed to enable students to discover each calculus power rule independently (albeit in simplified form), and hence their inverse relationship. Students are required only to do simple arithmetic and some elementary algebra, ...more>> The Precalculus Algebra TI-83 Tutorial - Mark Turner An online tutorial for using the TI-83 graphing calculator to solve the kinds of problems typically encountered in a college algebra or precalculus algebra course. Step-by-step instructions with full key sequences and animated screen images. Includes ...more>> The Precalculus Page Lesson plans and staff development activities with a problem-solving approach to precalculus, with materials on logarithms, the binomial theorem, and the law of sines, and problems (without answers) on algebra, exponential and logarithmic functions, other ...more>> PreCalculus Problem of the Week - Math Forum Math problems for students who had finished studying topics commonly covered in first-year algebra and high school geometry. From 2002 to 2003, problems involved probability, statistics, discrete math, and trigonometry. The goal was to challenge students ...more>> Precalculus Tutorial - John W. Bales A tutorial for students enrolled in a precalculus, college algebra, or trigonometry course. Bates explains, "It does not, and cannot, replace the textbook or other class resources or assignments. At the instructor's discretion, all or part of this tutorial
Mathematics Home Page Department Chair: Dr. Doug Riley Mission Statement: Our mission is to serve all students at all levels of ability, experience and interest. We foster creative, critical thinking and active problem solving. We seek ways for our students to come to know the beauty and usefulness of mathematics and to appreciate connections to the sciences and arts. Furthermore, we seek to assist our majors in the development of their talents as they prepare for careers involving mathematics. Welcome to the Department of Mathematics. The Department's enthusiastic faculty and well-equipped facilities offer an environment that promotes the growth of students with diverse interests and goals. We offer several courses which, as part of the college's Explorations curriculum, help students in honing their skills in quantitative and logical reasoning, and in learning to apply those skills in their other coursework and future endeavors. But the department also features a rich set of offerings for intermediate and advanced work in these important disciplines. Students will find research, teaching, and internship opportunities, as well as solid preparation for graduate and professional schools and a wide range of exciting careers. Upon completion of the mathematics major, students will be able to write a valid proof of a mathematical statement, write a lucid summary of a scholarly article in mathematics or closely related field, present a coherent explanation of his or her mathematical work in a public setting to a group of peers. The department is housed in the Olin building, where students have access to two computer labs, and all classrooms are outfitted with at least one computer and projector. The building also boasts two recently created computer classrooms where each student has access to both traditional desk space and a new flat-panel PC. The rooms have been thoughtfully laid out in a U-shaped configuration in order to facilitate the kinds of interaction that are so important to effective teaching and learning—between teacher and student, between theory and application, and among peers. When help is needed, students may find it within easy reach in the offices of a talented and welcoming faculty or in the new Quantitative Reasoning Center in Olin 103. For more information, including a course listing, please visit the most recent issue of the Birmingham-Southern College Catalog.
College Algebra (3rd Edition) 9780321466075 ISBN: 0321466071 Edition: 3 Pub Date: 2007 Publisher: Addison Wesley Summary: These authors have created a book to really help students visualize mathematics for better comprehension. By creating algebraic visual side-by-sides to solve various problems in the examples, the authors show students the relationship of the algebraic solution with the visual, often graphical, solution. In addition, the authors have added a variety of new tools to help students better use the book for maximum effecti...veness to not only pass the course, but truly understand the materialDowningtown, PAShipping:Standard, ExpeditedComments:This book has some WRINKLED pages but can still be used without any problem or hassle. Ships with... [more]This book has some WRINKLED pages but can still be used without any problem or hassle. Ships within 24 hrs of your order. Open Mon - Fri. May have some notes/highlighting, slightly worn covers, general wear/tear. [
IMS is the only institute Island-wide, which offers students the opportunity to select numerous combinations from the broad spectrum of Unitised Mathematics offered by examining boards of the United Kingdom. More specifically we offer teaching for the following units: The above units can be used in a variety of combinations so that students can obtain up to three A levels. Six of the above units are required for an A level award and three for an AS level award. In order for two combinations of six units each to be considered as two separate A levels, they must not have units in common. Leading the way internationally we have managed to create a programme to suit the individual needs of students, within the framework of the syllabus, consisting of the various units which make up up to three A levels in Mathematics. Typically: For students requiring only one mathematical A level: It would be appropriate to pursue A Level Pure Mathematics (Core Maths units 1, 2, 3 and 4, Further Pure 1 and 2). This combination consists of core and advanced pure mathematics and is particularly valuable in all of the science fields as well as in all of the social sciences, Law and Economics. As a rule, this two-year course is taken in the fourth and fifth year of secondary education. For students requiring two mathematical A levels, we offer the following two choices: A Level Pure Mathematics (Core Maths units 1, 2, 3 and 4, Further Pure 1 and 2) and A Level Statistics (units 1 to 6). A Level Statistics covers a wide spectrum of subjects in diverse fields of study such as psychology, biology, physics, medicine etc. For students who have successfully passed the IGCSE Mathematics while in their third year of secondary school, we offer a two-year course for their fourth and fifth year and a one-year Statistics course in the fifth or sixth year. For students who have successfully passed the IGCSE Mathematics in the second year of secondary school, the programme will be completed a year earlier. A Level Mathematics (Core Maths units 1 to 4, Mechanics 1 and 2) and A Level Further Maths (Further Pure 1 to 3, Mechanics 3 to 5). This combination of units is invaluable to those whose aim is to continue in the fields of study of mathematics, engineering, physics computer science and information technology. For students who have successfully passed the IGCSE Mathematics in the third year of secondary school, we offer a one-year Mathematics course for their fourth year and a one-year Further Mathematics course in their fifth year. For students who have successfully passed the IGCSE Mathematics in their second year of secondary school, the programme will be completed a year earlier. For students who want to receive three A level awards in mathematical subjects, their only option is as follows: Mathematics (Core Maths units 1 to 4, Mechanics 1 and 2), Further Mathematics (Further Pure 1 to 3, Mechanics 3 to 5) and Statistics (units 1 to 6). These courses, are extremely helpful in gaining a place at a top university in the United Kingdom to study mathematics, engineering, physics, information technology, medicine, architecture and computer science. A typical route for a student would be to take the IGCSE Mathematics while in third year of secondary school and thereafter the following courses: Mathematics (Core Maths units 1 to 4, Mechanics 1 and 2) in one year while in fourth year of secondary school. Further Mathematics (Further Pure 1 to 3, Mechanics 3 to 5) in one year while in fifth year of secondary school. Statistics (units 1 to 6) in one year while in fifth or sixth year of secondary school. For student taking the IGCSE Mathematics while in second year of secondary school the following courses are recommended: Mathematics (Core Maths units 1 to 4, Mechanics 1 and 2) in one year while in third year of secondary school. Further Mathematics (Further Pure 1 to 3, Mechanics 3 to 5) in one year while in fourth year of secondary school. Statistics (units 1 to 6) in one year while in fifth year of secondary school.
Algebra I is a mathematics class in which students will begin to learn more complex algebraic concepts. Topics in algebra I include solving equations, inequalities, and proportions, creating and assessing graphs, functions, and linear equations, and polynomials and factoring, among other concepts. Algebra II Algebra II is a mathematics class in which students build on concepts previously learned in algebra I. Topics in algebra II include matrices, quadratic equations and functions, polynomials and polynomial functions, basic trigonometric concepts, sequences and series, logarithmic functions, and probability and statistics. Calculus Calculus is a branch of mathematics that focuses on limits, functions, integrals, and derivatives. Calculus has applications in science, engineering, higher mathematics, and more. Differential Equations A course in differential equations involves the study of equations that represent real-world systems requiring multiple variables changing at different rates. Discrete Math Discrete math is the study of mathematical structures that are fundamentally discrete as opposed to continuous. It has been characterized as the branch of mathematics dealing with countable sets. Topics in discrete math include game theory, set theory, logic, probability, and number theory, among others. Geometry Geometry is the branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Topics in geometry include finding area, volume, and perimeter of a variety of shapes, properties of triangles, the Pythagorean Theorem, angles, and symmetry, among others. Pre-Algebra Pre-Algebra, as the name suggests, prepares students for algebra. Topics in pre-algebra include properties of operations, operations with integers, fractions, decimals, and negative numbers, solving basic algebraic expressions, roots, exponents, area, perimeter, and volume. Pre-Calculus Pre-Calculus, as the name suggests, prepares students for success in calculus. Topics in pre-calculus include exponential, rational, polynomial, and logarithmic functions and their properties, real and complex numbers, and limits, among others. Statistics Statistics is the science of the collection, organization, and interpretation of data. Statistics applies to a variety of fields including math, science, technology, and business, just to name a few. Statistics involves reading and interpreting tables and graphs, learning about qualifiers of data like mean, median, range, and mode, as well as more advanced topics and skills. Trigonometry Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and angles. Topics in trigonometry include the Pythagorean Theorem, the law of sines and cosines, trigonometric functions, right triangles, angles, graphs of trigonometric functions, and circles.
Thorough, well-written, and encyclopedic in its coverage, this text offers a lucid presentation of all the topics essential to graduate study in analysis. While maintaining the strictest standards of rigor, Professor Gelbaum's approach is designed to appeal to intuition whenever possible. Modern Real and Complex Analysis provides up-to-date treatment of such subjects as the Daniell integration, differentiation, functional analysis and Banach algebras, conformal mapping and Bergman's kernels, defective functions, Riemann surfaces and uniformization, and the role of convexity in analysis. The text supplies an abundance of exercises and illustrative examples to reinforce learning, and extensive notes and remarks to help clarify important points. Read Online Now at Wiley Online Library An online version of this product is available through our subscription-based content service. Read Online
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education. Chapter 2Notation. Recall that F(R3 ) denotes the set of all dierentiable real-valued functions f : R3 R and V(R3 ) denotes the set of all dierentiable vector elds on R3 . 2.1 7. This is what you are supposed to show here: given u Tp (Rn ) with \|u AME 21241 Laboratory 3: Beam Bending and Strain Transformation IntroductionBeams are structural elements used to carry loads transverse to their length. They carry loads by bending. They are used in many engineering designs such as buildings, cars, EXERCISES1220. During the construction period, the expenditures will be charged as follows:Land Building Land ImprovementsPurchase. $390,000 Land survey. 5,200 Fees for search of title for land. 600 Building permit. 3,500 Temporary quarters for c Chapter 4 Cost-Volume-Profit Analysis SolutionsEXERCISESE1. With high fixed costs, a business is quite risky. This follows because if sales fail to meet expectations, the company may have a large loss. Paul may be able to make rent a variable expeKarush-Kuhn-Tucker Theorem We seek first order conditions for local optimality for the following formulation of nonlinear programming:(1)Minimize f ( x) Subject to h( x) = 0 g ( x) 0where x is a vector in En, h maps into Ek, and g maps into Em Syllabus topic: Friends, recreation and pastimesSample task 4 This is a Part B task. (10 marks) A new shopping centre has just opened near your house. You have spent the day there with a friend. Write a diary entry about your day. Write approximat Syllabus topic: Family life, home and neighbourhoodSample task 4 This is a Part B task. (10 marks) You are home staying in Japan. Write an email to your Japanese teacher in Australia about your life in Japan. Write approximately 200-250ji in Japan Math 60440 Midterm Exam March 20, 2009 The Honor Code is in effect for this examination. All work is to be your own. You may use any material in the handouts on the web site. You may consult any books or internet material you can find. The only 1 B. A Linear Combination of Atomic Orbitals: H2 Mathematically, atomic orbitals are combined through something called a linear combination. I promised wed do this without math, so lets try and get a feel for what this is conceptually. Consider the s Natural Law, Natural Rhetoric, and Rhetorical PerversionsJeffrey J. MaciejewskiAbstract. Observers, including the Catholic Church, have consistently demonstrated a keen ability to identify instances of rhetoric, such as advertising, that are dista Justice as a Nexus of Natural Law and RhetoricJeffrey J. MaciejewskiIntroduction Only recently have we begun to build a bridge connecting natural law philosophy and rhetoric.1 At the outset one might ask: Why do they need to be connected in the rs CS2204 DIGITAL LOGIC & STATE MACHINE DESIGNDIGITAL SYSTEM DESIGN EXAMPLESPRING 2009A Vending Machine ControllerWe will design a vending machine controller as a digital system by using the finite state machine (FSM) technique. The design is done Goal of Research To derive conclusions from a body of data and discover what was previously unknown tools are needed to facilitate this goal there are many kinds of tools because of the varied nature of the types of research, but there is a genera Databases IlluminatedChapter 13 Databases and the InternetDatabases and the WWW WWW is a loosely organized information resource Many websites use static linked HTML files can become inconsistent and outdated Many organizations provide dynamic MeteoritesThis meteorite, a basalt lava rock nearly indistinguishable from many Earth rocks, provided the first strong proof that meteorites could come from Mars. Originally weighing nearly 8 kilograms (17.6 pounds), it was collected in 1979 in the James Thurston HIST 405 Dr. Stephan December 4, 2006 CRITICAL REVIEW: LARSON'S INTERNAL IMPROVEMENT Although much has been written about American internal improvements of the early nineteenth century, historian John Lauritz Larson's Internal Improvem C. The Probability of Observing a State: Pj Let us return to the problem at hand. We are trying to find the probability of observing a given quantum state (say, Ej) in a system selected from a canonical ensemble. To state the same goal another way, w CHM 446 Statistical Mechanics. Exam II. Thursday, Sep. 30, 2004. There are five questions on the exam, each worth 20 points. Choose FOUR and complete them for a total of 80 points. (If you do more than four, I will choose one at random to not grade.) VI. The Relation to the Combined Law of Thermodynamics To bring nonmechanical variables like temperature and entropy into the discussion, we combine the above considerations with thermodynamics. By virtue of the first postulate, we can associate our CHM 446: Statistical Mechanics. Exam IV. Tuesday, Nov. 23, 2004. There are five questions on the exam, each worth 20 points. Choose FOUR and complete them for a total of 80 points. (If you do more than four, I will choose one at random to not grade.) CHM 446: Statistical Mechanics. Exam V. Thursday, Dec. 16, 2004. There are five questions on the exam, each worth 20 points. Choose FOUR and complete them for a total of 80 points. (If you do more than four, I will choose one at random to not grade.) EXPLOITATION interactionsbiophagy - feeding on living organisms predation - prey is captured and consumed parasitism - "host" is fed upon, but usually not killed parasitoid - insect parasite that lays eggs on host; larvae feed on and eventually kill Echinacea angustifolia DC - purple cone flowerThe coneflowers are in the aster family (Asteraceae), having the same large central disk with numerous tiny flowers arrayed in a pleasing symmetrical pattern. The big difference is that the disk swells Deterministic push-down automataWhen we talk about deterministic push-down automata (DPDA), our definition will be different from that of finite automata, but the idea remains the same: at any point of execution we shouldn't have more than one choic Fundamentals8/31/07Why parallel algorithms? There are many computationally intensive problems we want to solve in a reasonable amount of time Current sequential computing is limited by hardware and often insufficientBasics The running time of Polytechnic UniversityMA 2132 MIDTERMPrint Name: Signature: ID #: Instructor/Section:April 7, 2006Directions: You have 90 minutes to answer the following questions. You must show all your work as neatly and clearly as possible and indicate the Polytechnic UniversityMA 2132 MIDTERMPrint Name: Signature: ID #: Instructor/Section: Andy TsangJanuary 9th, 2007Directions: You have 90 minutes to answer the following questions. You must show all your work as neatly and clearly as possible an ENCM 369 Winter 2006 Mid-Session Test Problem 3 Answers Part a. To encode a jal instruction, the assembler would need the address of the target instruction, but that address wont be known until the linker combines all the text segments from the objec
Calculus for the 21st Century Philosophy of the Course Traditionally, calculus has been presented from an analytical point of view often devoid of meaningful applications. Calculators and computers, with their powerful numeric, graphic, and symbolic tools, provide new opportunities for taking a multiple representation approach to the study of calculus. In particular, greater use of visualization, approximation, and prediction can be made in calculus instruction. A modern calculus course should foster in students an appreciation and skill that allows them to apply their mathematical knowledge in a variety of practical situations. Ideally, successful completion of the course would provide students with the ability to pick up a newspaper and recognize the calculus that surrounds them. We hope students will apply their knowledge to situations that they face in their everyday lives. Calculus students should look back on their learning experience with favor and in such a way that they desire to continue their pursuit of mathematics. Through technology, students may now take an active role in their learning. We are now able to create an environment which is rich with technology, nurtures curiosity, and promotes action. Mathematics is an experimental science and should be treated as such. Therefore, employing lesson plans that include laboratory activities, discovery exercises, individual projects, applied problems, writing exercises, and open-ended questions should be an integral part of the course. A Sampling of the Course It is not our purpose to detail a first year calculus course. However, the following illustrates an approach to calculus that utilizes hands-on experience and technology. This approach makes learning functions, limits, continuity, derivatives, integrals, approximation, and their applications a more enriching experience for the students and teachers. LIMITS AND CONTINUITY Limits are critical to the study of calculus. While the development of a rigorous definition is necessary, formal proofs may be de-emphasized in a first course. Limits should be approached numerically, graphically, and analytically. Graphing calculators are wonderful tools to help develop a clear-sighted concept of limits. An intuitive understanding of the e and d definition can be explored using the idea of local linearity. Looking at problems like (1 + )x both numerically and graphically greatly enhances understanding. Introducing L'Hopital's Rule early in the course is desirable. Students need to examine, graphically and analytically, the relationship between left and right hand limits, continuity, and local linearity. DERIVATIVES AND THEIR APPLICATIONS Because calculus is the study of change, the derivative and anti-derivative continue to be the focal point of this study. The definition of derivative and the relationship between differentiation and continuity must be emphasized as well as important theorems like The Mean Value Theorem. The derivatives of polynomial, rational, trigonometric, exponential, logarithmic and piece-wise functions must be studied. In addition, students need to have a working knowledge of implicit differentiation, logarithmic differentiation, the chain rule, product rule and quotient rule. Numerical estimates of the derivative should also be emphasized. Applications of the derivative should include related rates, maximum/minimum, and motion problems. Questions in these areas should be realistic and focus on applications. The derivatives and their relationship to slopes, concavity, and the linearization of a curve continue to be important components of calculus. Newton's Method and similar iterative techniques should be part of the curriculum. INTEGRALS AND THEIR APPLICATIONS Relating motion and the anti-derivative to area and the Fundamental Theorem of Integral Calculus is a primary goal. Given a rate of change, a student should be able to construct the function. The integral as the infinite sum should be explored by several methods including rectangles, mid-point, trapezoid, and Simpson's Rule. With technology, these methods can be explored without tedious computations. DIFFERENTIAL EQUATIONS Differential equations are a common theme throughout a first year calculus course. They provide a wonderful opportunity for students to model real life situations. Numerical methods to solve differential equations, along with associated error analysis, help the student to understand the real world of applied mathematics. It is not necessary to solve differential equations solely by analytical methods when other approaches are just as enriching. SEQUENCES AND SERIES Students should have a thorough understanding of geometric series and the concept of estimating functions with infinite series. Graphical relationship, the ratio test, the comparison test, intervals of convergence, and error analysis should be addressed, especially with new technology. Course Books and Resources Several major calculus reform projects are currently in progress throughout the country. In selecting a calculus textbook, much consideration should be given to both the use of technology and the curriculum content. Major projects have been undertaken at Duke, Harvard, NCSSM, Ohio State, Oregon State, Smith, and St. Olaf's College among others. Associated with many of these are training institutes which are funded by the NSF. Other institutes and inservice opportunities exist. These programs are exciting opportunities for teachers to learn how to incorporate technology into the teaching of calculus and select curriculum materials based on current thinking in the field. Also, close attention should be paid to periodicals, newsletters and journals as a means for staying current with new ideas, technology, and trends in calculus reform. Focus on the Future The power of current and future technology can no longer be ignored in classroom instruction. We are faced with technology that is changing at an exponential pace. Teachers must look at how and what they teach in this environment of change. The availability of technology has caused changes in the curriculum. Some content will receive less emphasis, some content will receive more emphasis, and solutions to problems that were previously inaccessible are now possible. Open-ended problems and mathematical modeling need to be an integral part of the calculus curriculum. Writing and group activities are important to constructing and applying knowledge. Teachers must assume the role of a life-long learner and must convey this role to their students. Modes of assessment, including the Advanced Placement examination will, of necessity, change to reflect the use of technology in instruction.
to see why it comes out as it does. If they see that this is all perfectly logical, and they know how to do it, the day is a big success—and in fact, this sort of justifies the whole unit on matrices. As a final point, mention what happens if the equations were unsolvable: matrix A will have a 0 determinant, and will therefore have no inverse, so the equation won't work. (You get an error on the calculator.)
30 Mathematics Lessons Using the TI-15 Helps younger learners grasp mathematical concepts and skills with lessons that integrate calculator use. This book provides step-by-step mathematics lessons that incorporate the use of the TI-15 calculator throughout the learning process. The lessons present mathematics in a real-world context and cover each of the five strands, including numbers and operations, geometry, algebra, measurement, and data analysis and probability. Teacher Resource CD Included
Synopses & Reviews Publisher Comments: This edition of Swokowski's text is truly as its name implies: a classic. Groundbreaking in every way when first published, this book is a simple, straightforward, direct calculus text. It's popularity is directly due to its broad use of applications, the easy-to-understand writing style, and the wealth of examples and exercises which reinforce conceptualization of the subject matter. The author wrote this text with three objectives in mind. The first was to make the book more student-oriented by expanding discussions and providing more examples and figures to help clarify concepts. To further aid students, guidelines for solving problems were added in many sections of the text. The second objective was to stress the usefulness of calculus by means of modern applications of derivatives and integrals. The third objective, to make the text as accurate and error-free as possible, was accomplished by a careful examination of the exposition, combined with a thorough checking of each example and exercise
Chapter 1: Numbers and Points The word number typically refers to a quantity, which may be definite as in ``the number of people in line'', indefinite as in ``a number of students took the class''; the dictionary definition typically lists over a dozen common meanings. In this course, a variety of types of numbers will be studied, including the numbers in each of the following sets: The positive integers or natural numbers : 1, 2, 3, .... The integers which includes the positive integers, zero, and the negative integers: -1, -2, -3, .... The rational numbers is the set of numbers which can be written as m/n where m and n are integers with n non-zero. Rational numbers can be expressed as finite or repeating decimals. The real numbers include the rational numbers as well as irrational numbers such as and . Every real number has a possibly infinite decimal expansion. The complex numbers which include the real numbers as well as the . Every complex number can be expressed in the form where and are real numbers. On each of these sets, is defined the binary operations addition and multiplication. For each ordered pair of numbers from one of these sets, there are well defined numbers and called the sum and the product of and in the same set. One expresses this property by saying that the sets are closed under addition and multiplication. These operations have simple geometric interpretations. For example, if two line segments are of length x and y respectively, then connecting the two together end-to-end yields a line segment of length x + y. Similarly, the area of a rectangle with length x and width y is precisely xy. By repeatedly combining numbers via addition and multiplication, one can make complicated expressions such as . By substituting various values of x and y into this expression, one gets numerical values for the expression. One finds that regardless of which values of x and y you use, the numerical value is the same as that obtained by the expression . The process of verifying this is one of the skills that you have already mastered in earlier algebra courses. The idea is that you can use a number of properties of numbers to successively simplify the first expression until you get to the second one. Amongst these properties are: Addition and Multiplication are commutative: a + b = b + a and ab = ba. These properties are referred to as the commutative, associative, and distributive laws. Before going on, we should clear up some ambiguity. We said that was obtained by a succession of additions and multiplications. This is true, but there are many different ways of doing this, e.g. after one gets the value of , , and , one could add the third to the sum of the first two or add the first to the sum of the last two. Of course, the result would be the same, and it is the associative law for addition that guarantees this. For this reason, one typically does not even bother to specify the order by adding in parentheses. Similarly, one didn't put parentheses to indicate the order of evaluation of the product of the three factors in . Another ambiguity occurs in the expression . Does this mean add the product of 4 and x to y or does it mean multiply 4 times the sum of x and y? This is more serious because the two ways of evaluating the expression give different answers. This is resolved by the rules for order of evaluation of expressions. In this case, the rules say that you evaluate multiplications before additions. So, the meaning of the expression is and not . The rules for order of evaluation will be stated in detail at the end of the next section. Exercise 1.1: Using the geometric interpretation of addition and multiplication given above for positive real numbers, give geometric interpretations of the commutative, associative, and distributive laws. For example, the commutative law for addition might be illustrated by saying that if you join a line segment of length x end-to-end with a line segment of length y, then the length of the result is the same whether you measure it from one end or from the other. Here are two more important properties of the rational numbers , the real numbers , and the complex numbers : There is a number 0 such that a + 0 = a for all numbers a and there is a number 1 not equal to 0 such that for all numbers a. Any such numbers are called additive or multiplicative identities. For any number , there is a number such that . Such a number is called an additive inverse of . Similarly, if is number other than 0, then there is a number called a multiplicative inverse or reciprocal of such that . The numbers 0 and 1 are unique: Proposition 1.1: There is at most one additive identity and at most one multiplicative identity. Proof: Suppose and are two additive identities. Letting play the role of in the definition of an additive identity , we have . Similarly, one has by letting play the role of in the definition of being an additive identity. Because of the commutative law, one has The same argument shows that multiplicative identities are also unique. We can define subtraction by and division by if b is non-zero. Alternative ways of denoting division are: . Exercise 1.2: Which of the five sets of numbers , , , , and are closed under subtraction? Which are closed under division? Are subtraction and/or division commutative and/or associative? Does addition distribute over multiplication? Does division distribute over addition? Do the natural numbers and/or the integers have additive and/or multiplicative identities? What about inverses? Any set closed under addition and multiplication which satisfies the commutative, associative, and distributive laws as well as has identities and inverses is referred to as a field. Here is the formal definition: Definition 1.1: A field is a set F together with with two binary operations '+' and '·' called addition and multiplication which satisfy the following conditions: F is closed under addition and multiplication. Addition and multiplication are commutative. This means that and for all a and b in F. Addition and multiplication are associative. This means that and for all a, b, and c in F. Multiplication distributes over addition. This means that for all a, b, and c in F. (Identities) There is an element 0 in F such that a + 0 = a for all . There is an element 1 in F different from 0 such that for all . (Inverses) For every in F, there is an element denoted in F such that . For every in F other than 0, there is an element denoted in F such that . Remark: If all the conditions except for the requirement that multiplicative inverse exist are true, then F is called a commutative ring with identity. Many of the common rules of algebra follow from the fact that one is working in a field. One very important example is: Proposition 1.2: Let F be a field. If is in F, then . If and in F satisfy , then either or are zero. Proof: Let be an arbitrary element of F. Then . Let b be the additive inverse of . Then applying it to the last equation, we get Suppose ab = 0. If a is zero, there is nothing more to prove. On the other hand, if , then a has a multiplicative inverse c and so . Corollary 1.1: If a is an element of a field F, then -a = (-1)a. Proof: . Corollary 1.2: 0 has no multiplicative inverse. Proof: If this were false, then one would have But, Combining these, we conclude that . But this contradicts the property that 0 and 1 are not equal. Because the field properties imply most of the rules of algebra, they help us understand why these rules are true. For example, you know that the product of two negative numbers is always positive. But why should this be true? Why not make a new rule, e.g. that the product of two negative numbers should be negative. The answer is that you could do this, but then one of the field properties would no longer be true. Exercise 1.3: Show that in a field, one has (-a)(-b) = ab. Hint: Fill in reasons for each of the following steps: Here are some more common rules of algebra: Proposition 1.3: Let F be a field containing a, b, c, and d where b and d are non-zero. Then if and only if If c is also non-zero, then Proof: (i) By the definition of multiplicative inverse, one must show that Starting from the left hand side, one can simplify the expression as shown here. Each of the above steps involves one of the fundamental properties of fields; make sure that you can justify each step with the appropriate property. (ii) This like any "if and only if" statement is really two assertions: If then If then To prove assertion (a), assume that To see that , multiply both sides of by to get . Now simplify each side: and Combining results, one gets (As before, make sure that you can justify each step.) Now, let's show assertion (b). Assuming that and that both and are non-zero, we know that both and have multiplicative inverses. So, we need only multiply both sides of by and simplify. Here are the details: and (Label each line with the property that justifies the step.) (iii) Simplify starting from the left hand side: Did you justify each step? (iv) One has: (v) This is the usual rule for adding fractions. Notice that property (iv) allows one to convert the fractions so that they have a common denominator . Here are the detailed steps: (vi) This is the rule for simplifying fractions of fractions. It is the same as Using property (ii), this is the same as showing But, we can see this by simplifying each side: and This completes the proof of Proposition 1.3. Exercise 1.3: (i) Redefine addition on the real numbers: With the new definition of addition and the usual multiplication, do the real numbers still form a field? Which of the field properties are true and which are not. We defined multiplication by making be the area of a rectangle with sides of length m and n respectively. Suppose that we based things on the area of a triangle with given base and height instead. With the new definition of multiplication (and the usual definition of addition), do the real numbers form a field? What is the multiplicative identity? 1.2.1 Mathematical Induction The positive integers are the numbers 1, 2, 3, .... Every positive integer n has a successor n' = n + 1. Every positive integer n except 1 is the successor of a unique positive integer, viz. n - 1. The most important property of the set of positive integers is Principle of Mathematical Induction: If is any property of the positive number such that is true. If is true for a positive integer , then is true for the successor of Then is true for all positive integers This principle can be used for both definitions as well as for theorems. For example, one can use it to define the usual operations on the positive integers using only the successor operation: Let be the property of the positive integer n that m + n is defined. Then (i) assures us that is true and (ii) assures us that is true of n' if it is true of n. So, is true for all positive integers n. Similarly, one sees that we can define multiplication by: Definition 2.2: The product of two positive integers and is defined by: for all positive integers If is defined, then One can also use the principle of mathematical induction to prove theorems. For example, we can verify that our addition operation is associative: Proposition 2.1: Addition is associative. Proof: Let be the property of the positive integer n that (a+ b) + n = a + (b + n) for all positive integers a and b. In the case where n = 1, is true because (Why is each step true?) Assuming that is true for n, let's show it for n'. One has So, must be true for all positive integers Proposition 2.2: Addition is commutative. Proof: Let be the property of the positive integer n that a + n = n + a for all positive integers a. We will show that is true by using the principle of mathematical induction. Let be the property of the positive integer a that a + 1 = 1 + a. We know that is true because 1 + 1 = 1 + 1. If is true for some positive integer a, then and so is true for all positive integers and so is true. Now suppose that is true for n. Then and so is true for all positive integers n. (Did you justify each step?) Proposition 2.3Multiplication distributes over addition. Proof: Let be the property of the positive integer n that a(b + n) = ab + an for all positive integers a and b. Then is true. because Suppose that is true for n. Then it is also true for n' because and so is true for all positive integers n. Exercise 2.1:Use induction to prove the following results: Multiplication is associative. (a + b)c = ac + bc for all positive integers a, b, and c. Multiplication is commutative. Exercise 2.2: (Laws of Exponents) Use induction to define where a is number (real or complex) and n is a positive integer. Use induction to prove that where a is a number and m and n are positive integers. Use induction to prove that where a is a number and m and n are positive integers. Assuming that you want these properties to hold for all integers m and n, what are the only possible definitions for ? What about where n is a negative integer? Exercise 2.3: (Sums of Powers) Show by induction that the sum of the first n positive integers is precisely n(n+1)/2. Show by induction that the sum of the squares of the first n positive integers is precisely n(n + 1)(2n + 1)/6. 1.2.2 Binomial Theorem It will often be useful to expand powers of a binomial a + b. For small positive integer value of n, you can do this using the distributive law. For example, Exercise 2.4: Show that It should be clear that for any given positive integer n, we can obtain a formula for ; but also that it is becoming more and more tedious as n increases. The goal is to find a general formula. Here is a first approximation: Proposition 2.4: (Binomial Theorem) For each positive integer n, one can write where the coefficients are all positive integers. In fact, one has for k = 1, ..., n-1 and Proof: Define by induction using the formulas in the statement of the proposition. Let be the property of the positive integer n which is true provided that the can be expressed as in the statement of the proposition. Then is clearly true because Assume that is true for n so that So is true for all positive integers n. The equations defining the coefficients appear quite complicated, however, if we write them down in the form of a triangle with the row containing for k = 0, 1, ..., n from left to right: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 From the diagram, we see that the formulas for the simply says that each row begins and ends with a 1 and every other entry is the sum of the two numbers in the row above it immediately to its left and to its right. This triangle is called Pascal's triangle and gives a simple way of computing the coefficients. Exercise 2.5: Define n! (pronounced n factorial) by 0! = 1! = 1 and (n+1)! = (n+1)n!. So, n! is the product of all positive integers less than or equal to n. Another interpretation of the number is the number of combinations of objects taken at a time which is a formal way of saying the number of subsets of elements which can be formed from a given set of elements. Exercise 2.6: In Pascal's triangle, note that each row reads the same from left to right as from right to left, i.e. is palindromic. Express this relation as an equation involving the . Explain why the relation is true. 1.2.3 Order of Operation The rules for order of operations are used to determine the precise order in which operations are to be carried out when evaluating an expression. Using them allows one to write expressions with far fewer parentheses making them both more readable and less error prone. On the other hand, the rather large number of operators makes for a rather complicated set of rules. The goal is to make your expressions readable and correct; sometimes it is better and clearer to add a set of parentheses even though the rules indicate that they are not really necessary. Here are the rules. You should read through the rules now and start using them. Because they are complicated, you will need to refer back to them a few times before they become clear. I am purposely not specifying exactly which numbers we are talking about because we want the material to apply to any of several types of algebraic quantities. Variables like x or y are letters that stand for numbers. By parentheses we mean any of the usual types of parentheses seen in algebraic expressions; they include rounded parentheses (), square brackets [], and curly brackets {}; in the rules below, we will use the rounded parentheses, but any other kind can be used as well. Definition 2.4: An expression is a particular kind of string of numbers, variables, operators, and parentheses. The following rules can be used to determine if such a string is an expression: Every number and every variable is an expression. If E is an expression, then so is ( E ). If E is an expression, then so is -E. If E1 and E2 are expressions and op is any of the binary relations '+', '-', '·', '÷', and '^'then E1 op E2 is also an expression. The only strings which are expressions are those which can be shown to be expressions by applying the above rules a certain number of times. Some fine points: Sometimes we will omit the multiplication operator ·. You can handle these expressions either by adding in the operator or by defining a new operator which is simply a space character. One can add the alternative slash '/' character as an alternate for the division symbol '÷'. Division is also often indicated with a horizontal bar. To handle this, simply replace the expression with the numerator between parentheses followed by ÷ and the denominator between parentheses. Sometimes, one does not assume that the exponentation operator is evaluated from left to right. In such cases, you need to be sure to put in additional parentheses so that the order of evaluation is explicit. Any expression has a well defined value. To guarantee this property, one needs to agree on the order in which an expression is evaluated. For example, if one evaluates the addition in 4 + 5 · 3 before the multiplication, the value is 27; but evaluating the multiplication first gives a value of 19. In order to avoid this kind of ambiguity, expressions are always evaluated using the so-called order of operations. The rule is: Evaluate parenthesized subexpressions starting from the first innermost parenthesized expression. If E has value v, then the value of (E) is also v. Use the remaining rules in order to evaluate a subexpression with no parentheses. The value of a number is the number itself; the value of a variable is the number which it represents. Powers should be evaluated first, from left to right. Unary minus operations should be evaluated next starting from the innermost. Multiplications and divisions should be done next; do this left to right. Additions and subtractions should be done next; again do this left to right. Lemma 3.1: Let r be a positive integer. There is no positive integer m such that Proof: We prove this by induction using the property which is true of m provided that We know that is true because 1 is not the successor of any positive integer. Now suppose that is true for If were not true for its successor then But then and so because each positive integer is the successor of at most one positive integer. This contradiction shows that must be true for and so the lemma is true. We can use the lemma to show that no two of the conditions , , and can be simultaneously true. There are three cases: If and , then If and , then If and , then In all three cases, we get a contradiction with the assertion of the lemma and so no two of the conditions can be simultaneously true. It remains to show that at least one of the three conditions must always be true. Let m be a fixed positive integer and be the property of a positive integer n that is true provided that at least one of and is true. is true because either or else for some positive integer p; in the second case, and so If is true for the positive integer n, then there are three cases: If then there is an r with But then and so If then and so . If there is a positive integer with If then On the other hand, if is not 1, then it is the successor of some positive integer, say and so One has and so In all three cases, we see that the condition is true and so holds for all positive integers and so trichotomy is true. Exercise 3.1: Prove that if then Definition 3.2: If then the difference is defined to be 1.3.2 Descent Descent is a variant on the principle of mathematical induction: Descent Principle: Let be a property which may or may not be true of the positive integer n. If is false for at least one positive integer then there is a smallest integer for which is false, i.e. is false for and true for all with . Proof: Suppose the descent principle is false for some property . Let be the property of the positive integer which is true provided that is true for all positive integers less than or equal to must be true; otherwise, would be false and clearly n = 1 would be the smallest number for which would be false because 1 is smaller than every positive integer except 1. Now suppose that is true for If were not true, then would be false but would be true for all numbers k less than . But then would be the smallest positive integer for which was false which contradicts the assumption that the descent principle is false for . 1.3.3 Unique Factorization Definition 3.2:Let and be positive integers. is a factor of (or is a multiple of ) if there is a positive integer such that A factor of is a proper factor if is prime if it is not equal to 1 and has no proper factors. Examples of primes are 2, 3, 5, 7, 11,.... Exercise 3.2: Replace the order relation in the statement of Proposition 3.1 with the property that be a factor of Which of the conclusions of the Proposition are still true? Which are false? Exercise 3.3: Show that if is a factor of then Proposition 3.2 Every positive integer can be factored into a product of primes. In other words, if is a positive integer, then there are prime positive integers ..., (where is a non-negative integer) such that Proof: This is a proof by descent. If the Proposition is false, then there is a smallest positive integer for which the assertion is false. cannot be equal to 1 or prime and so it can be written as product of proper factors: Since each of the factors is smaller than , one has and where the and are primes. But then Proposition 3.3 (Euclid) There are infinitely many primes. Proof: If not, then there are only finitely many. Let ..., be the complete list of them. Consider the positive integer Let be any prime factor of we know that for some = 1, 2, ..., Then But then which means that is a prime factor of 1. By Exercise 3.3, it follows that is less than or equal to 1 and so must be equal to 1 contrary to the assumption that is prime. Clearly, positive integers might be factored in more than one way as product of primes, e.g. As it turns out, the factorization is unique except for the order of the factors (and the way it is parenthesized): Proposition 3.4 (Division Theorem) If and are positive integers, then there are non-negative integers and such that and Furthermore, the numbers and are unique. Proof: (i) Existence: Let and be positive integers for which there are no such numbers and For the fixed value choose to be the smallest such that and do not exist. We cannot have because that would allow for Also, one cannot have as this would allow for By trichotomy, it follows that and so for some positive integer But then and so there must be non-negative numbers and such that and and But then contrary to the assumption that no appropriate and exist. (ii) Uniqueness: Suppose where and are non-negative integers with and If then and so Otherwise, assume that (swapping the roles of q, r and u, v if necessary. Re-arranging, we get which is a contradiction because the right hand side is smaller than n and the left hand side is greater than equal to n. Theorem 3.1 (Fundamental Theorem of Arithmetic) Every positive integer can be factored as a product of primes and this factorization is unique up to order of the factors. Proof: By Proposition 3.2, we need only show the uniqueness assertion. Let us begin with Lemma 3.2 If is a prime factor of the product then is a factor of either or (or both). Proof: If this is not true in general, then let be the smallest prime for which there is a counter-example. Of all such counter-examples choose one with the smallest possible value for , and of all these choose one with the smallest possible value for n. Using the division theorem, we know that one can write and where and are smaller than Since , we see that is a factor of and cannot be a factor of either or of without being a factor of or respectively. So, we can assume by the choice of and that they are both smaller than Since is a factor of there is a with . Further, and so But then every prime factor of must be smaller than . We have a factor of and so by the minimality of it must be that is either a factor of or of In either case, we could divide both sides of by and get a contradiction with the minimality of the choice of either or So, it must be that has no prime factors and so But then contradicting the fact that is a prime. This proves the lemma. Corollary 3.1: If is a prime factor of then is a prime factor of at least one of the factors ..., Proof: Suppose not. Choose a counter-example with the smallest possible value of . Then divides and so Lemma 3.2 tells us that is either a factor of or of . But, it can't be a factor of the second quantity because this would give us a counter-example with a smaller . So, must be a factor of contrary to assumption. If factorization can be non-unique, then let be the smallest positive integer which has at least two factorizations into products of primes differing other than in the order of the factors. No prime can occur in both factorizations; otherwise, would be a smaller positive integer with at least two factorizations. If is any prime factor of the first factorization, then is a factor of some prime of the second factorization by Corollary 3.1. So for some positive integer Since is a prime, it must be that and so contrary to assumption. A numeral is a symbol used to represent a number. The standard numeration system is the Arabic numeral system. A base 10 Arabic numeral is a sequence of digits 0, 1, 2, 3, ..., 9. The numeral where each is a digit represents the number For example, 3147 means . Decimal numerals are similar, e.g. . Sometimes it is more convenient to use bases other than 10. A base b Arabic numeral (where b is a positive integer bigger than 1) uses digits 0, 1, ..., b-1 and the numeral where each is a digit represents the number The commonly used bases other than 10 are 2 and 16. Base 2 numerals are called binary numbers and base 16 numerals are called hexadecimal numbers. Hexadecimal numbers use the letters A through F (or a through f) to denote the digits 10 through 15 respectively. For example, the decimal number 3147 is equal to the Number C4B in hexadecimal because 11 + 16(4) + 162(12) = 3147. You can check that in binary, the same number is 110001001011. One can also work with decimals in other bases as well; for lack of a better term, base 2 decimals will be referred to as binary decimals. The usual algorithms for doing operations with base 10 numbers work fine in other bases. However, most people find it easier to convert the numbers to base 10, do the computation in base 10, and convert the answer back to the desired base. Doing base conversions is easy. We have already seen how to convert to base 10. If you wanted to convert to say base 16, then take the number and divide by 16. The remainder is the lowest digit. Dividing the quotient by 16 gives a remainder which is the next digit, etc. Example 4.1: To convert the base 10 number 3145 to hexadecimal, divide 3145 by 16 to get So the lowest digit is eleven which is denoted B. Next divide 196 by 16 to get and so the next digit is 4. Finally, and so the high order digit is twelve or C. The number 3147 in base 10 is equal to the number C4B in base 16. Example 4.2:Calculate First, let's calculate it using the same algorithm as one uses in school. Here is the calculation: C4B C4B --- 8739 312C 9384 ------ 971DF9 To understand this, recall that and so by the distributive law: The three terms on the right are written above in the three lines between the horizontal bars. Instead of writing multiplying the lines by 10 (hex) and (hex), the numbers are simply written one and two columns to the left of the where they would normally be written. To see how one calculates , write this as Verify that this is precisely what you do when you do multiplication in base 10; then check the other two subproducts and the final addition. Now check your work: We know that C4B is 3145 in base 10. So its square in base 10 is in base 10 which converted to base 16 is 971DF9. Check the computation! Of course, if the two answers do not match, there must be an error in the computation. Repeating decimals are possibly infinite decimals whose digits eventually start to repeat themselves, numbers like 3.14141414... Such numbers are actually rational numbers. To see this, we will need: Proposition 4.1: (Geometric Series) One has In particular, if x is a number less than 1 in absolute value, then Proof: Use the distributive law to expand out the left side of the first equation. All but two of the terms subtract out leaving the two terms on the right hand side. To see the second assertion, divide both sides of the first equation by . As grows larger and larger, the term approaches 0 because . We can use the Proposition to evaluate our infinite decimal: A similar argument shows that any repeating decimal represents a rational number. Conversely, if you have a rational number p/q, then you can expand it out into a decimal by using the standard long division algorithm. At each stage in the computation, the remainder is a number between 0 and q - 1. As soon as the remainder repeats, the sequence of digits also repeat; so, the decimal expansion of a rational number is repeating. The formal result is: Proposition 4.2: Every repeating decimal represents a rational number and every rational number has a repeating decimal expansion. Although we argued in base 10 numerals, everything works anlaogously regardless of which base we work in. Corollary 4.1: There exist real numbers which are not rational. Proof: Any decimal expansion which is not repeating represents a real number which is not rational. Remark 4.1: The square root of 2 is irrational. For otherwise, we could write where and are integers with non-zero. Squaring both sides and multiplying through by , one gets . Now express and as products of primes and substitute these into this last equation. Because integers factor uniquely, we have a contradiction, because the left side of the equation would have an odd number of factors of 2 and the right side would have an even number of factors of 2. Definition 5.1: An ordered field F is a field (i.e. a set with addition and multiplication satisfying the conditions of Definition 2) with a binary relation < which satisfies: (Trichotomy) For every pair of elements a and b in F, exactly one of the following is true: a < b, a = b, and b < a. (Transitivity) Let a, b, c be arbitrary elements of F. If a < b and b < c, then a < c. If a, b, and c in F satisfy a < b, then a + c < b + c. If a, b, and c in F satisfy a < b and 0 < c, then ac < bc. Fact: If F is an ordered field, then 0 < 1. Proof: By Definition 2, and so by trichotomy, if the the fact were wrong, then we would have a field F with 1 < 0. By property iii, we would have 1 + (-1) < 0 + (-1) and so 0 < -1. But then using property iv, we would have . By Proposition 2, the left side is 0 and so . This contradicts trichotomy and so the assertion must be true. If F is an ordered field, an element a in F is called positive if 0 < a. Proposition 5.1: The set P of positive elements in an ordered field F satisfy: (Trichotomy) For every a in F, exactly one of the following conditions holds: a is in P, a = 0, and -a is in P. (Closure) If a and b are in P, then so are a + b and ab. Proof: (i) By property i of the Definition 3, exactly one of a < 0, a = 0, and 0 < a must be true. If a < 0, then by property iii of Definition 3, we have a + (-a) < 0 + (-a) and so 0 < -a. Conversely, if 0 < -a, adding a to both sides gives a < 0. So the three conditions are the same as -a is in P, a = 0, and a is in P. (ii) Suppose a and b are in P. Then 0 < a and by property iii of Definition 3, we have 0 + b < a + b and . Since 0 < b and b < a + b, transitivity implies that 0 < a + b. Since by Proposition 2, we have 0 < ab. Remarks: i. In one of the exercises, you will show that, if a field has a set P of elements which satisfy the conditions of Proposition 8, then the field is an ordered field assuming that one defines a < b if and only if b - a is in P. ii. An element a of an ordered field F is said to be negative if and only if a < 0. iii. It is convenient to use the other standard order relations. They can all be defined in terms of <. For example, we define a > b to mean b < a. Also, we define to mean either a < b or a = b and similarly for . iv. The absolute value function is defined in the usual way: Proposition 5.2: Let a and b be elements of an ordered field F. \|-a\| = \|a\| (i.e. and ) (Triangle Inequality) Proof: i. By Trichotomy, we can treat three cases: a > 0, a = 0, and a < 0. If a > 0, then -a < 0 and so \|a\| = a and so \|-a\| = -(-a) = a. If a = 0, then -a = 0 and so \|a\| = 0 = \|-a\|. If a < 0, then -a > 0 and so \|a\| = -a and \|-a\| = -a. In all three cases, we have \|a\| = \|-a\|. ii. Again, we can treat three cases: If a > 0 or a = 0, then \|a\| = a and so . If a < 0, then adding -a to both sides gives 0 < -a and so a < -a by transitivity. In this case we have \|a\| = -a and so a < \|a\|. We could argue the other inequality the same way, but notice that we could also use our result replacing a with -a. (Since it holds for all a in F, it holds for -a.) The result says , where we have used assertion i. Adding a - \|a\| to both sides of the inequality gives the desired inequality. iii. Once again, do this by considering cases: If , then \|a + b\| = a + b. Since and , we can add b to both sides of the first inequality and \|a\| to both sides of the second one to get and . Using transitivity, we get as desired. Now suppose that . Then adding -a - b to both sides of the inequality gives -a + (-b) > 0. Applying the result of the last paragraph, we get . But a + b < 0 means that \|a + b\| = -(a + b) and so where we have used assertion i for the last step. This completes the proof. As noted earlier, one can write numerals in any integer base b > 1. The choice b = 10 has the advantage of being most familiar; but choosing b = 2 often makes the proofs a bit simpler - basically, each additional digit cuts an interval in two equal pieces which is easier to handle than 10 equal pieces. For this section, we will use base 10; but after reading it, you should go back and verify that everything works regardless of the choice of the base. In some later sections, we will use base 2 in order to have simpler proofs. We start by formalizing our notion of infinite decimal. Definition 6.1: i. An infinite decimal is an expression of the type , where is an integer, and is an infinite sequence of decimal digits (i.e. integers between 0 and 9). ii. Every such infinite decimal defines a second sequence of finite decimals where . iii. One says that the infinite decimal represents the number r (or has limit r) if can be made arbitrarily close to zero simply by taking k sufficiently large. Definition 6.2: i. An ordered field F is said to be Archimedean if, for every positive a in F, there is a natural number N with a < N. ii. An Archimedean ordered field F is called the field of real numbers if every infinite decimal has a limit in F. Given any element a in F, we can form an infinite decimal for a. First, we can assume that a is positive, since the case where a = 0 is trivial, and if a < 0, then we can replace a with -a. Next, we see why we needed to add the Archimedean property to the above definition. Without it, we would not know how to get the integer part of a: Since F is Archimedean, the set of natural numbers N with a < N is non-empty and so there it has a smallest element b. Let . Then if . Choose to be the decimal digit such that and let , so that again . Assuming that we have already defined for some natural number k, the quantities and with , define by induction the digit so that and let , so that . The infinite decimal was defined so that with . So this infinite decimal has limit a. We say that this is the infinite decimal expansion of the element a in F. Proposition 6.1:i. Every element a in F is the limit of the infinite decimal expansion of a. ii. The decimal expansion of every rational number is a repeating decimal, i.e. except for an initial segment of the decimal, the decimal consists of repetitions of a single string of digits. iii. Every repeating decimal has limit a rational number. Proof: The first assertion has already been proved. For the second assertion, note that the definition of the sequence of digits is completely determined by the value of . If a = r/s is rational with r and s integers, then is a rational number with denominator (a factor of ) s. Furthermore, since , if is rational with denominator s, then so is . By induction, it follows is rational with denominator s for every k. Since lies between 0 and 1 and is rational with denominator s, it follows that there are at most s possible values for . The following principle is called the pigeonhole principle: If s + 1 objects are assigned values from a set of at most s possible values, then at least two of the objects must be assigned the same value. By the pigeonhole principle, there are subscripts i and j with such that . As indicated at the beginning of the proof, it follows that the sequence of digits starting from must be the same as the sequence of digits starting from and so the decimal repeats over and over again the cycle of values . The third assertion is easy to prove -- it is essentially the same as our calculation of the limit of the infinite decimal expansions of 1/3 and 1/7. The formalities are left as an exercise. Example 2: The field of real numbers contains many numbers which are not rational. All we need to do is choose a non-repeating decimal and it will have as its limit an irrational number. For example, you might take where at each step one adds another zero. Proposition 6.2: Every a > 0 in the field of real numbers has a positive -root for every natural number n, i.e. there is a real number b with . Proof:It is easy to show by induction that, if , then for every natural number n. So the function is an increasing function. By the Archimedean property, we know that there is a natural number M > a. Again by induction, it is easy to see that . Now consider the set S of all numbers m with . Clearly 0 is in this set. If every successor of an element of S lies in S, then the principle of mathematical induction would imply that S would be the set of all non-negative numbers contrary to the fact that we have already identified a number M not in S. So, let be an element of S such that is not in S. We know that the -root of a must lie between and . Next evaluate for integers j from 0 to 10. The values start from a number no smaller than a and increase to a number larger than a. Let be the largest value of j for which the quantity is at most a. Repeating the process, one can define by induction an infinite decimal such that the -power of the finite decimal differs from a by no more than . Let b be the limit of the infinite decimal, and be the values of the corresponding finite decimals. Then we have and and so it is reasonable to expect that . This is in fact true. Using the identity for geometric series, we see that: . But then the triangle inequality gives where C is a positive constant which does not depend on k. Since this holds for all positive integers k, it follows that . If a is a positive element of any ordered field, we know that because the set of positive numbers is closed under multiplication. Since we also have , it follows by trichotomy that the square of any element in an ordered field is always non-negative. In particular, such a field cannot contain a solution of . We would like to have a field where all polynomial equations have a root. We will define a field called the field of complex numbers which contains the field of rational numbers and which also has a root, denoted i, of the equation . In a later chapter, it will be shown that, in fact, contains a root of any polynomial with coefficients in . This result is called the Fundamental Theorem of Algebra. Let us first define the field of complex numbers. Since it is a field which contains both the field of real numbers and the element i, it must also contain expressions of the form z = a + bi where a and b are real numbers. Furthermore, there is no choice about how we would add and multiply such quantities if we wanted the field axioms to be satisfied. The operations can only be: and where we have used the assumption that . It is straightforward, but a bit tedious to show that these operations satisfy all the field axioms. Most of the verification is left to the exercises. But let us at least indicate how we would show that there are multiplicative inverses. Let us proceed heuristically -- we would expect the inverse of a + bi to be expressed as but this does not appear to be of the desired form because there is an i in the denominator. But our formula from geometric series shows how to rewrite it: We have . This is just what we need: Of course, we have proven nothing. But we now have a good guess that the multiplicative inverse might be . It is now an easy matter to check that this does indeed work as a multiplicative inverse. Proposition 7.1: The set of all expressions a + bi, where a and b are real and i behaves like , is a field if we define operations as shown above. We have already seen that the field cannot be ordered. Nevertheless, we can define an absolute value function by . The conjugate of a complex number is a + bi is defined to be a - bi and is denoted . Proposition 7.2: Let w and z be complex numbers. Then \|w\| = \|-w\| \|wz\| = \|w\|\|z\| \|z\| = 0 if and only if z = 0. (Triangle Inequality) . . If r is a real number, its absolute value is the same as a real number as it is if it is considered to be the complex number . Proof: These are all left as exercises except for the triangle inequality. To prove the triangle inequality, let and where and are real numbers. To show the triangle inequality, it is enough to show that Substituting in the values for and one sees that this will hold provided that This simplifies to the equivalent inequality This in turn would be true if the square of the left side were less than or equal to the square of the right side, i.e. which is equivalent to But subtracting from both sides and factoring gives which is obviously true. So, the triangle inequality is also true. Exercise 7.1 (i) Prove the rest of Proposition 7.2 Prove that in the triangle inequality, one has equality if and only if one of and is a non-negative real multiple of the other. Example 8.1: To solve the general linear equation ax + b = c for x where a, b, and c are constants with a non-zero, one can assume that x is a solution so that ax + b = c. Add -b to both sides of the equation to get (ax + b) + (-b) = c + (-b). This simplifies to ax = c - b. Multiplying both sides by a-1 and simplifying gives x = a-1(c - b). We have shown that this is the only possible solution. Substituting it into the original equation and simplifying verifies that this value is indeed a solution. So, the equation has the unique solution x = a-1(c - b). Example 8.2: To solve the general quadratic equation ax2 + bx + c = 0, we can restrict ourselves to the case where a is non-zero. (Otherwise, the equation is linear and Example 3 applies.) If x is a solution of the equation, then one can divide both sides by a and simplify to get x2 + (b/a)x + c/a = 0. If 2 is not zero, then one can complete the square to get (x + b/(2a))2 = (b/(2a))2 - c/a. If the right side was square, then we could solve for x to get Furthermore, one can check that this actually is a solution of the original equation. This last equation is known as the quadratic formula. In particular, if we are working in the field of real numbers, then we have completely solved the quadratic; there are two, one, or no solutions when b2 - 4ac is positive, zero, or negative respectively. In the case of the field of complex numbers, things are even simpler as we will show that all complex numbers have square roots. Another example is the general system of 2 linear equations in two unknowns x and y: ax + by = e, cx + dy = f. What makes the general system of 2 linear equations appear difficult is that both equations involve both variables. There are two approaches: One could solve the first equation for one of the variables in terms of the other. Then substitute this into the second equation giving an equation in only the second variable. Then proceed as in the easy case. One could subtract an appropriately chosen multiple of one equation from the other in order to obtain an equation involving only one variable. As before, one needs to check that the possible solutions one obtains do indeed satisfy the original equations. Example 8.3: Consider the system: 2x + 3y = 5, 4x - 7y = -3. Assume that x and y are satisfy both equations. One can proceed using either method: Solving for x using the first equation, gives x = (5/2) - (3/2)y. Substituting this into the second equation gives 4((5/2) - (3/2)y) - 7y = -3, which can be used to find y = 1. Substituting this back into our expression for x yields x = 1. One then checks that the values x = 1, y = 1 do indeed satisfy the original equations. If one subtracts twice the first equation from the second equation, one gets (4x - 7y) - 2(2x + 3y) = -3 - 2(5) or -13y = -13. This gives y = 1 and substituting this value back into the first equation gives 2x + 3 = 5 or x = 1. As before, we need to check that x = 1, y = 1 does indeed satisfy the original two equations. Example 8.4: Find all the solutions of the system of equations: , . Assume that one has a solution x, y. Solving the second equation for y, one gets a value y = 1 -x which when substituted into the firs equation gives: or . Collecting terms, we get a quadratic . Factoring and using the second part of Proposition 1, gives x = 0 or x = 1. Substituting these values into our expression for y, gives two possible solutions (x,y) = (0, 1) and (x,y) = (1, 0). Substituting each of these into the original equations, verifies that both of these pairs are solutions of the original system of equations. If we have more than two variables and more equations, we can apply the same basic strategies. For example, if you have three linear equations in three unknowns, you can use one of them to solve for one variable in terms of the other two. Substituting this expression into the two remaining equations gives two equations in two unknowns. This system can be solved by the method we just described. Then the solutions can be substituted back into the expression for the first variable to find all possible solutions. When you have these, substitute each triple of numbers into the original equations to see which of the possibilities are really solutions. One can also use the second approach as is illustrated by the next example. Example 8.5: Solve the system of equations: x + y + z = 0, x + 2y + 2z = 2, x - 2y + 2z = 4. Assume that (x, y, z) is a solution. Subtracting the first equation from each of the other two equations gives y + z = 2, -3y + z = 4. Now subtracting the first of these from the second gives -4y = 2 or y = -1/2. Substituting this into the y + z = 2 gives z = 5/2. Finally, substituting these into the first of the original equations gives x = -2. So the only possible solution is (x, y, z) = (-2, -1/2, 5/2). Substituting these values into the original equations shows that this possible solution is, in fact, a solution of the original system. This section is an informal review of some elementary notions of analytic geometry. Let's start with the number line. Choose a line and mark off two points 0 and 1 on the line. By marking off segments of length equal to that between 0 and 1, one can define points 2, 3, etc. Moving in the other direction, one gets -1, -2, etc. To each rational number, one can associate a point on the line; e.g. 1/2 is the point midway between 0 and 1; 7/4 is the point a quarter of the way between 0 and 7, etc. Real numbers can be represented as infinite decimals; we can associate them with the points obtained as limits of the finite decimals obtained by throwing away the tail end of the decimal. For example, 1.2121212... is the point which is the limit of the points 1, 1.2, 1.21, 1.212, etc. At least intuitively, it appears that we have defined a one-to-one correspondence between the real numbers and the set of points on the line. To name the points in the plane, simply use the cartesian product of the real numbers with itself. Geometrically, this corresponds to taking two perpendicular number lines intersecting at 0. These are called the x-axis and y-axis respectively. Each point on the plane can be projected perpendicularly onto each of the two axes. The name the point is the ordered pair (x, y) where x is the number associated with the projection of the point on the x-axis and y is the number associated with the projection of the point on the y-axis. The diagram below shows the point (3, 2): Recall that the complex numbers can be written in the form a + b i where and a and b are real numbers. So, a complex number is essentially the same thing as an ordered pair (a, b) of real numbers. This allows one to think of the plane as being the set of real numbers. The number a is called the real part and the number b is called the imaginary part of the complex number a + b i. In the above diagram, the point (3, 2) corresponds to the complex number 3 + 2i. Proposition 9.1: (Distance Formula) If and are two points in the plane, then the distance between them is . Proof: Let . Then is a right triangle with right angle at . The legs have length and respectively. The distance formula now follows by the Pythagorean Theorem. (This theorem will be proved in a next chapter.) If (a, b) and (c, d) are two points in the plane, then one can define their sum to be (a + c, b + d). Note that this is precisely the point corresponding to the sum z + w of the complex numbers z = a + bi and w = c + di corresponding to the two points. The addition is the so-called parallelogram law Let L be the line segment from the origin (0, 0) to (a, b) and M be the line segment from the origin to (c, d). If we move M parallel to itself so that it starts at (a, b), then its other end-point will be at (a + c, b + d). So the line segments R from (a, b) to (a + c, b + d) and S from (c, d) to (a + c, b + d) combine with L and M to make a parallelogram whose diagonal starts from the origin and ends at the sum of z = (a, b) and w = (c, d). You can also think of this as a triangle rule: To add z and w, start with a line segment from the origin to (a, b), then move the line segment from the origin to (c, d) parallel to itself until it is starting from (a, b); the final end-point is the sum z + w. Let and be any two complex numbers with We can think of as defining a line segment M from the origin to If is a positive real number, then corresponds to a line segment obtained by stretching M by a factor of t. If is a negative real number, then the segment is streched by the a factor of but goes in the oppositive direction as does M. When you add to it, the complex numbers in the set make up a line through and parallel to M. Definition 9.1 A line is any set of points of the form where and are complex numbers with The corresponding set of points in the plane are also referred to as a line. Proposition 9.2: (Midpoint Formula) If and are two points in the plane, then the point . is on a line through and and the distance between and either end-point is equal to half the distance between the end-points. Proof: The line is the one defined by and Letting t = 1/2 gives the point M. The assertions about the distance are easy to verify using the distance formula. Intuitively speaking, a function is a rule which associates with each element of a set an element of a set . The set is called the domain of the function and the set is called the codomain of the function. A function with domain and codomain is often denoted and the number in associated by the rule to the element in is denoted The range of is the set of all elements of that are associated to at least one element of the domain For example, function which squares each real number is a function with domain and codomain both equal to the field of real numbers. For each real number one has and the range of the function is the set of non-negative real numbers. On the other hand, the square root function has domain the non-negative real numbers and The codomain might be the set of non-negative real numbers or any set containing this set. For any function one can form the set of all ordered pairs where is in the domain of This set is called the graph of the function. In the special case where the function has domain and codomain contained in the set of real numbers, the graph of f can be thought of as a subset of the number plane. Remark 1.10.1 One should think of the graph of a function as a visual representation of the function. In the special case in which the domain and codomain are subsets of the real numbers, one has: The domain is the set of x-coordinates of points in the graph. The range is the set of y-coordinates of points in the graph. Every vertical line intersects the graph in at most one point. The last property is often referred to as the vertical line test Example 1.10.1 Not every subset of the number plane is the graph of a function. For example, the circle with center at the origin and radius 1 is not the graph of a function. This is because there are vertical lines which intersect the graph in more than one point. For example, the y-axis intersects the circle at which means that there cannot be a rule which associates to 0 a single number and still have both of these points in the graph. On the other hand, the part of the circle which has y-coordinate non-negative is the graph of the function which assigns to every in the closed interval [-1, 1] the value Many times the function will be ambiguously specified by simply giving the rule for associating elements of the domain with elements of the codomain, without specifying a domain and codomain. In this case, one normally assumes one is dealing with the largest domain for which the rule makes sense, and this domain is referred to as the natural domain. For example, the natural domain of the squaring function is the set of all real numbers, even though there are other functions with domains any specific subset of the real numbers. In some cases, one needs to intuit the meaning of what one means by the largest domain for which the rule makes sense; for example, the squaring function is also defined on the field of complex numbers. Remark 1.10.2 Because it is difficult to formalize what one means by a rule, a formal definition of function usually is a definition of the graph of the function. For example, one could say that a function with domain and codomain is any subset of the set of all ordered pairs where and such that for every there is exactly one ordered pair in first coordinate equal to One then writes to indicate that 1.10.1 Operations on Functions Let and be functions with domains some set codomains some field Then one can combine and to form new functions: The sum of and is defined by The difference of and is defined by The product of and is defined by The quotient of and is defined by Note that the domain of the first three functions is but the domain of the fourth function is the set of for which Finally, if and are two functions, then the composition of the two functions is defined by Example 1.10.2 If is the squaring function and is the cubing function, then is the function Also, is the function with The function is the reciprocal function, which is only defined for non-zero real numbers. Finally, the composition is the function which raises numbers to their sixth power. Exercise 1.10.1 Show that the operations sum and product are commutative but that the other three operations are not. Show that the operations sum, product, and composition are associative, but the other two are not.
1. Use basic matrix operations and the algebra of matrices in practical problems. Possible applications may be drawn from areas such as Kirchoff's laws, Leontieff model of an interacting economy, Markov chains, method of least squares, singular value decomposition and fourier coefficients of a function. 2. Understand the concepts of vector spaces, subspaces, basis, independence and dependence, dimension, coordinates, rank of a matrix, inner product. 3. Use the dependency relationship algorithm and the Gram-Schmidt orthogonizational process. 4. Understand linear transformations, range and null space of a linear transformation, the correspondence principle and similarity. 5. Understand properties of the determinant function and the cofactor expansion of determinants. 6. Understand the concepts of eigenvalues and eigenvectors. 7. Understand the concepts of quadratic formsMethods of presentation can include lectures, discussion, experimentation, audio-visual aids, small-group work and regularly assigned homework. Calculators/computers will be used when appropriate. Mathematica, Derive and TI-92 calculators are available for use at the College at no charge. Course may be taught as face-to-face, media-based, hybrid or online course. VIII. Course Practices Required (To be completed by instructor) IX. Instructional Materials Textbook information for each course and section is available on Oakton's Schedule of Classes. Within the Schedule of Classes, textbooks can be found by clicking on an individual course section and looking for the words "View Book Information". Textbooks can also be found at our Mathematics Textbooks page. A computer algebra system is required. X. Methods of Evaluating Student Progress (To be determined and announced by the instructor) Evaluation methods can include grading homework, chapter or major tests, quizzes, individual or group projects, calculator/computer projects and a final examination
Book Description: This new textbook in Signals and Systems provides a pedagogically-rich approach to what can oftentimes be a mathematically 'dry' subject. Chaparro introduces both continuous and discrete time systems, then covers each separately in depth. Careful explanations of each concept are paired with a large number of step by step worked examples. With features like historical notes, highlighted 'common mistakes,' and applications in controls, communications, and signal processing, Chaparro helps students appreciate the usefulness of the techniques described in the book. Each chapter contains a section with Matlab applications. * pedagogically rich introduction to signals and systems using historical notes, pointing out 'common mistakes,' and relating concepts to realistic examples throughout to motivate learning the materialintroduces both continuous and discrete systems early, then studies each (separately) in more depth laterextensive set of worked examples and homework assignments, with applications to controls, communications, and signal processing throughoutprovides review of all the background math necessary to study the subjectMatlab applications in every chapter
11th Grade Math Embed or link this publication Description Differential Calculus :- Differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus. The primary objects of Popular Pages p. 1 11th grade math 11th grade math differential calculus differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change it is one of the two traditional divisions of calculus the other being integral calculus the primary objects of study in differential calculus are the derivative of a function related notions such as the differential and their applications the derivative of a function at a chosen input value describes the rate of change of the function near that input value the process of finding a derivative is called differentiation geometrically the derivative at a point equals the slope of the tangent line to the graph of the function at that point for a real-valued function of a single real variable the derivative of a function at a point generally determines the best linear approximation to the function at that point differential calculus and integral calculus are connected by the fundamental theorem of calculus which states that differentiation is the reverse process to integration know more about how to factor cubic polynomials tutorcircle.com page no 1/4 p. 2 differential equations a differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders differential equations play a prominent role in engineering physics economics and other disciplines differential equations arise in many areas of science and technology specifically whenever a deterministic relation involving some continuously varying quantities modeled by functions and their rates of change in space and/or time expressed as derivatives is known or postulated this is illustrated in classical mechanics where the motion of a body is described by its position and velocity as the time value varies newton s laws allow one given the position velocity acceleration and various forces acting on the body to express these variables dynamically as a differential equation for the unknown position of the body as a function of time in some cases this differential equation called an equation of motion may be solved explicitly an example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air considering only gravity and air resistance the ball s acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance gravity is considered constant and air resistance may be modeled as proportional to the ball s velocity this means that the ball s acceleration which is a derivative of its velocity depends on the velocity finding the velocity as a function of time involves solving a differential equation.differential equations are mathematically studied from several different perspectives mostly concerned with their solutions the set of functions that satisfy the equation learn more how to draw a heptagon tutorcircle.com page no 2/4 p. 3 random variable random variable or stochastic variable is a variable whose value is subject to variations due to chance i.e randomness in a mathematical sense as opposed to other mathematical variables a random variable conceptually does not have a single fixed value even if unknown rather it can take on a set of possible different values each with an associated probability a random variable s possible values might represent the possible outcomes of a yet-to-be-performed experiment or an event that has not happened yet or the potential values of a past experiment or event whose already-existing value is uncertain e.g as a result of incomplete information or imprecise measurements they may also conceptually represent either the results of an objectively random process e.g rolling a die or the subjective randomness that results from incomplete knowledge of a quantity the meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself but instead related to philosophical arguments over the interpretation of probability the mathematics works the same regardless of the particular interpretation in use random variables can be classified as either discrete i.e it may assume any of a specified list of exact values or as continuous i.e it may assume any numerical value in an interval or collection of intervals the mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution the realizations of a random variable i.e the results of randomly choosing values according to the variable s probability distribution are called random variates tutorcircle.com page no 3/4 page no 2/3
Intermediate Algebra - 4th edition Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief.Intermediate Algebra, Fourth Editionwas written to provide students with a solid foundation in algebra and to help them transition to their next mathematics course. The new edition offers new resources like theStudent Organizerand now includesStudent Resourcesin the back of the book to help...show more students on their quest for success. ...show less All text is legible, may contain markings, cover wear, loose/torn pages or staining and much writing.Missing components. SKU:9780321726377-5-1 $36.68 +$3.99 s/h Acceptable Extremely_Reliable Richmond, TX Buy with confidence. Excellent Customer Service & Return policy. $37.78 +$3.99 s/h Acceptable BookSleuth Danville, CA Fast Shipping ! Used books may not include access codes, CDs or other supplements.40.38$49.97 +$3.99 s/h LikeNew Nivea Books Lynnwood, WA PAPERBACK Fine 0321726375$50.27 +$3.99 s/h New Textbook Charlie Nashville, TN Brand new instructor's edition. May contain answers and/or notes
Visitors Why Mathematics at Guilford? While many people associate mathematics with calculations and arithmetic, there is much more to math than simply crunching numbers. In its most general form, math is sometimes described as the science of patterns. Some of the patterns mathematicians explore include algorithms, sets, sequences, graphs, networks, functional relations, statistical data, and geometric and topological structures. Since the analysis and understanding of patterns is important in virtually every discipline, the ideas and methods of mathematics can be applied in almost any field. Sometimes mathematical analysis allows for the prediction of certain patterns (or at least of their likelihood). Other times, just as importantly, mathematics reveals that making a prediction with reasonable certainty is impossible. Students who are well versed in math will be better prepared for employment and for graduate work in any field that deals with data analysis, quantitative reasoning, or logical deduction. Mathematics students will also be better able to understand recent advances in subjects where mathematical methods are routinely applied. Even in fields such as law and philosophy, where computational issues may not be emphasized, the use of logical thinking as required by mathematical proofs is a valuable skill. Many majors at Guilford, including Business Management, Biology, Chemistry, and Physics, already require mathematics courses. However, the increasing use of mathematical methods and terminology in many fields, scientific and otherwise, is a great reason to study more than just the bare minimum of mathematics. Questions about infinity, higher dimensions, the limitations of computing, and the prediction of future events are just some of the topics up for grabs. If you are a current or prospective student who wants to know more about the different math courses Guilford has to offer, please contact any of the mathematics faculty, and we'd be happy to tell you more.
Course Number and Title Number of Credits Minimum Number of Instructional Minutes Per Semester Prerequisites Math Placement Test score of 7 or MATH103 (C or better) Corequisites None Other Pertinent Information A comprehensive departmental final examination is required for this course. Catalog Course Description This course provides a preparation in mathematics for students interested in elementary education. Topics include elementary logic, sets, relations, functions, numeration systems, whole numbers, integers, and number theory. Required Course Content and Direction Learning Goals: define the whole numbers using equivalence classes, define the operations for whole numbers using sets, order whole numbers using sets, calculate multiples and powers of whole numbers, use mental math and estimation; work with numeration systems including Hindu-Arabic, Roman numerals, and other ancient systems; work with positional numeration systems; perform calculations involving positional numeration systems and other bases; and develop an understanding of the importance of place value and groupings in decimal system and other base systems; complete prime factorizations and use this to calculate the greatest common divisor and greatest common multiple of two or more numbers; use divisibility rules to test for divisibility; and define the integers from the whole numbers and develop an understanding of the properties of the integers using the definition of the integers.; define the operations for the integers using equivalence classes; order the integers and simplify expressions involving absolute value and negative exponents. use methods, concepts and theories in new situations(Application Skills). demonstrate an understanding of solving problems by: recognizing the problem reviewing information about the problem developing plausible solutions evaluating the results Planned Sequence of Topics and/or Learning Activities: elementary logic inductive reasoning deductive reasoning patterns and pattern recognition algorithms sets, relations, and functions set operations artitions as equivalence relations and equivalence classes unctions whole numbers definition of whole numbers properties of whole numbers operations for whole numbers using sets ordering whole numbers using sets multiples and powers of whole numbers mental math and estimation numeration systems Hindu-Arabic, Roman numerals and other ancient systems positional numeration systems and other bases operations using the decimal system and other bases number theory factors, factorizations, and prime numbers divisibility rules greatest common divisor and Euclidean Algorithm least common multiple integers define integers from whole numbers properties of integers operations for integers and their algorithms ordering integers absolute value negative exponents Assessment Methods for Core Learning Goals: Course Students will apply mathematical concepts and principles to identify and solve problems through informal assessment (oral communication among students and between teacher and students) and formal assessment (may include homework, quizzes, exams, projects, and comprehensive final). Core (if applicable) Math or Science: Assigned problems require the student to translate a descriptive problem into mathematical statement and solve. Critical Thinking and Problem Solving: Critical thinking and problem solving skills are required when creating Venn Diagrams and evaluating the information in the diagram to answer questions posed about the problem. They are also assessed when solving Logic problems. Other Evaluative Tools: Exams, quizzes, class participation, and projects as specified in the individual instructor's course format are utilized. Reference, Resource, or Learning Materials to be used by Students: A departmentally selected textbook will be used. Details will be provided by the instructor of each course section. See course format. Teaching Methods Employed Section VIII is not being used in new and revised syllabi as of 12/10/08.
New Textbook Related Products Error Patterns in Computation: Using Error Patterns to Improve Instruction Summary As your students learn about mathematical operations and methods of computation, they may adopt erroneous procedures and misconceptions, despite your best efforts. This engaging book was written to model how you, the teacher, can make thoughtful analyses of your student's work, and in doing so, discover patterns in the errors they make. The text considers reasons why students may have learned erroneous procedures and presents strategies for helping those students. You will come away from the reading with a clear vision of how you can use student error patterns to gain more specific knowledge of their strengths on which to base your future instruction. Book jacket. Table of Contents PART ONE DIAGNOSIS AND INSTRUCTION 1 (90) CHAPTER One Computing with Paper and Pencil in an Age of Calculators and Computers 3 (6) Paper-and-Pencil Computation Procedures Today 3 (3) Importance of Conceptual Understanding 6 (1) Error Patterns in Computation 7 (2) CHAPTER Two Diagnosing Error Patterns in Computation 9 (35) Learning Error Patterns 10 (5) Overgeneralizing and Overspecializing 15 (2) Encouraging Self-Assessment 17 (3) Using Graphic Organizers for Diagnosis 20 (2) Using Tests as a Part of Diagnosis 22 (5) Using Computers for Diagnosis 27 (1) Interviewing: Observing, Recording, and Reflecting 27 (13) Getting at a Student's Thinking 29 (3) Observing Student Behavior 32 (1) Recording Student Behavior 33 (1) Watching Language: Yours and Theirs 34 (1) Probing for Key Understandings 35 (3) Designing Questions and Tasks 38 (2) Guiding Diagnosis of Written Computation 40 (4) CHAPTER Three Providing Needed Instruction in Computation 44 (49) Understanding Concepts and Principles 46 (4) Numeration 47 (1) Equals 48 (1) Other Concepts and Principles 49 (1) Acquiring Specialized Vocabulary 50 (1) Using Models and Manipulatives 51 (3) Recalling the Basic Facts 54 (4) Stressing Estimation 58 (2) Teaching Students to Compute 60 (6) Talking and Writing Mathematics 66 (3) Using Graphic Organizers for Instruction 69 (5) Using Calculators and Computers 74 (2) Using Alternative Algorithms 76 (2) Subtraction of Whole Numbers: The Equal Additions Method 77 (1) Subtraction of Rational Numbers: The Equal Additions Method 77 (1) Using Cooperative Groups 78 (2) Monitoring Progress with Portfolios 80 (2) Guiding Instruction 82 (9) Focus on the Student 82 (1) Teach Concept and Skills 83 (1) Provide Instruction 83 (1) Use Concrete Materials 84 (1) Provide Practice 85 (6) PART TWO IDENTIFYING, ANALYZING, AND HELPING STUDENTS CORRECT SPECIFIC ERROR PATTERNS
Batten is great for the Life Con and Survival Models stuff if you already have some knowledge of the material. He just provides a lot of problems and some useful shortcuts that are not taught in the Bowers text. So if you have no prior knowledge of the material than I would not use Batten.
*** is supposed to provide a comprehensive education that prepares students for the professional environment where *y may encounter the need for knowledge in many areas in addition to their particular field of academic focus. Mathematics skills are required in * common daily applications, such as financial negotiations, personal finances, complying with U.S. tax codes, mortgages, and myriad other specific circumstances where ignorance of basic mathematics is detrimental * one's interests. Likewise, we live in an age where science permeates our lives in practically every respect. Modern motor vehicles employ sophisticated internal mechanisms; we take regular medications to maintain * health * address diseases, some of which require an understanding of potential adverse interactions that can have deadly consequences; and our kitchen appliances include microwave ovens emitting microwave radiation to cook * food. Perhaps the most ubiquitous aspects * * inclusion of science into our daily lives is the * computer * many of us use for essential communications, research, and even on-line shopping and bill-paying. Maintaining personal computer hardware and implementing regular software updates requires a degree of scientific understanding that one learns in general * classes in college. One of * most important aspects of scientific appreciation is the nature * * scientific method, * applies just as well to non-* matters. Mandatory college science c*ses ****** * to the principle of scientific inquiry, which may be even more * than much of the subject matter actually covered in the class. More than ever ******e, * is a part of American life and, therefore, a person who is ignorant of * pr*ciples is h*icapped by virtue of that ignorance. The world * rapidly becoming smaller in the sense that increasing globalization now enables domestic business entities to branch out * foreign markets and reach potential customer * client bases that greatly increase revenue * over * strictly domestic scope * modern business. In that sense, the usefulness of multiple language fluency is much greater * it was for previous generations of college students. Whereas in the recent past American * studied the so-called "classic" languages * as French, Greek, and Latin, those * represented purely academic achievements rather than functional necessity. However, today, the complexion of American society h* also c*anged significantly in the last few decades and bilingual capacity, especially in Spanish, is becoming a virtual necessity in modern American bus*ess as well as in ordinary, every day social life. Further*, the modern * economy is so unstable that choices of career made in college may no longer * a permanent narrow vocational application. Unlike previous generations of American students, contemporary adults may undergo more * changes through* a ****** than *ir parents, * of whom reta*ed the same position, (or at least * same vocational field) from the time of their college graduation until their retirement. Consequently, ***ing one's educational focus to one specific area of instruction
Edgenuity Geometry is a two-semester, hands-on and lecture-based course featuring an introduction to geometry, including reasoning and proof and basic constructions. Triangle relationships (similarity and congruency) and... Edgenuity Algebra is a two-semester course that provides in-depth coverage of writing, solving and graphing a variety of equations and inequalities, as well as linear systems. Functions are a central theme of the course... Algebra II begins with a review of algebraic properties and equation and inequality solving. Students will study relations and functions, including linear, quadratic, and radical functions, and be able to graph these... In this course students will use their prior knowledge to learn and apply Algebra II skills. This course includes topics such as functions, radical functions, rational functions, exponential and logarithmic functions,... The Chemistry course takes students into the composition, structure, and reactions of matter. This course encourages students to ask questions about things that occur in nature and determine the underlying chemical... The purpose of this year-long Physics course is to help students to see physics as a way to understand their world. To this end, we ask them to think critically, while sharpening their observation skills, analyze and... This year-long course is designed to prepare students for the rigor of college and post-secondary studies relative to our fast-paced economic growth and their role in how they will enter a future where self-study and... Economics will provide students with an understanding of the structure and workings of the economic system of the United States in a global setting. This course requires students to master fundamental economic concepts....
Stock Status:In Stock Availability: Usually Ships in 5 to 10 Business Days Product Code:9781877085703 About this book Author Information The Connections Maths 10 Stage 5.3 / 5.2 / 5.1 Teaching and Assessment Bookincludes many resources that makes using the Connections series the most effective and user-friendly series available. The resources in this book include: - A teaching program referenced to the student book - Syllabus notes - Detailed guidance on teaching each topic - Outcomes clearly stated and cross-referenced to the student books - Assessment ad reporting strategies - Overview and summary of every chapter and exercise in the student book - Relevant internet sites and further research questions - All this material is also provided on CD-ROM to allow for printing and customising. A. Kalra and J. Stamell Connections Maths is a comprehensive, full colour, 6-book series that meets all the requirements of the new Years 7–10 course. The series will engage, motivate and support students of all abilities. The page design, the vibrant use of colour, and the range of photos and cartoons make each page an interactive learning experience. Each textbook is accompanied by a student CD-ROM.
College Algebra: A Concise Approach COLLEGE ALGEBRA: A CONCISE APPROACH, 1/E textbook Software Only: The software enables students to have unlimited practice and homework problems. It also has access to definitions, objectives and examples like a textbook. Download the software ISBN 10 Digit ISBN 13 Digit ISBN Software Only 0-918091-59-4 978-0-918091-59-8 1st Edition Hard Cover Textbook & Software Bundle 1-935782-04-5 978-1-935782-04-9 1st Edition Hard Cover Textbook 1-935782-02-9 978-1-935782-02-5 Software Table of Contents Chapter 1: Number Systems and Fundamental Concepts of Algebra 1.1 The Real Number System 1.2 The Arithmetic of Algebraic Expressions 1.3 Properties of Exponents 1.4 Properties of Radicals 1.5 Polynomials and Factoring 1.6 The Complex Number System Chapter 2: Equations and Inequalities of One Variable 2.1 Linear Equations in One Variable 2.2 Linear Inequalities in One Variable 2.3 Quadratic Equations in One Variable 2.4 Higher Degree Polynomial Equations 2.5 Rational Expressions and Equations 2.6 Radical Equations Chapter 3: Linear Equations and Inequalities of Two Variables 3.1 The Cartesian Coordinate System 3.2 Linear Equations in Two Variables 3.3 Forms of Linear Equations 3.4 Parallel and Perpendicular Lines 3.5 Linear Inequalities in Two Variables 3.6 Introduction to Circles Chapter 4: Relations, Functions, and their Graphs 4.1 Relations and Functions 4.2 Linear and Quadratic Functions 4.3 Other Common Functions 4.4 Transformations of Functions 4.5 Combining Functions 4.6 Inverses of Functions Chapter 5: Polynomial Functions 5.1 Introduction to Polynomial Equations and Graphs 5.2 Polynomial Division and the Division Algorithm 5.3 Locating Real Zeros of Polynomials 5.4 The Fundamental Theorem of Algebra Chapter 6: Rational Functions and Conic Sections 6.1 Rational Functions and Rational Inequalities 6.2 The Ellipse 6.3 The Parabola 6.4 The Hyperbola Chapter 7: Exponential and Logarithmic Functions 7.1 Exponential Functions and their Graphs 7.2 Applications of Exponential Functions 7.3 Logarithmic Functions and their Graphs 7.4 Properties and Applications of Logarithms 7.5 Exponential and Logarithmic Equations Chapter 8: Systems of Equations 8.1 Solving Systems by Substitution and Elimination 8.2 Matrix Notation and Gaussian Elimination 8.3 Determinants and Cramer's Rule 8.4 The Algebra of Matrices 8.5 Inverses of Matrices 8.6 Linear Programming 8.7 Nonlinear Systems of Equations Chapter 9: An Introduction to Sequences, Series, Combinatorics, and Probability
Instructor Class Description The Concept of Number Explores the concept of number from an historical perspective and the modern mathematical perspective. Stresses the new properties of "'number"', starting with counting numbers and progressing to the concept of a field. Class description The concept of number will be studied from a historical perspective and the modern mathematical perspective. Starting with counting numbers and progressing to the concept of a field, the new properties of "number" will be stressed. As a capstone, modern error correcting codes of the digital highway will be constructed using the "arithmetic" and properties of finite fields. The course will have two goals. First, the students will be exposed to a historical view of number to show how the modern concept of number came about. The second goal will be to develop the basic mathematical thinking of the students by stressing the properties of numbers and how these properties allow us to do our everyday "arithmetic". Student learning goals General method of instruction Recommended preparation Prerequisites are an honest intellectual curiosity and the willingness to work hard. Class assignments and grading The only way that students can learn to do mathematics is to try to do it. Therefore, there will be problems assigned along with the historical readings. The personal computer will be used in the course. Besides the homework problems, there will be three quizzes and a final Debra Teresa Salas-Haynes
Thats the goddamn problem with math. All the the teachers i have had, just do the examples that are shown so goddamn easily, and then set us on to work. The first two tasks are fine, but then everything fucking explodes and we have to calculate why the vikings couldnt fly on jet-skis. Yeah, when you major in mathematics, you don't finish your "basics" until you're done with calc 3 I have a buddy who hasn't done anything as simple as trigonometry in so long, he does a lot of it in his head. Most of the sine, cosine, secant, and their inverses he has memorized and when he tutors me he will finish a problem before I have time to say, "wat.' ANY college level course? bro i did pre calc junior year in high school, took senior year off, and the class they placed me in my college is like algebra 2, we learned about graphs, so its not any college level course, because there are some dumbshit courses out there
Lynn Foshee Reed Lynn Foshee Reed has been a mathematics instructor at Maggie L. Walker Governor's School (MLWGS) in Richmond, Virginia, since 1998. Reed primarily taught calculus – AP Calculus AB as well as dual enrollment Calculus I, Calculus II, and Multivariable Calculus – in conjunction with Virginia Commonwealth University. She has also taught Algebra II, Trigonometry and Mathematical Analysis, and History of Mathematics at MLWGS. Previously, Reed taught mathematics at Virginia Tech for eight years. Reed originates from southern Illinois and she received her Bachelor of Arts in Mathematics and minored in chemistry at the University of Evansville. After two years of teaching high school, she enrolled at Ohio State University where she received both her Master of Arts and Master of Science in Mathematics. Reed earned her Gifted Education endorsement in 2002 and achieved National Board Certification in Adolescent and Young Adult Mathematics in 2009. Reed is Secondary Mathematics Representative on the board of the Virginia Council of Teachers of Mathematics (VCTM), and she was VCTM's representative to the Virginia Educational Technology Advisory Committee. Reed is a past president of the Greater Richmond Council of Teachers of Mathematics (GRCTM). She is one of the coordinators of the Calculus Network of Richmond, and she regularly presents at GRCTM and VCTM conferences. Reed is a more than 20-year member of both the National Council of Teachers of Mathematics and the Mathematical Association of America (MAA). Reed has experience as an AP Reader of AP Calculus exams, and she is a frequent calculus instructor for the Virginia Advanced Study Strategies review sessions. Lynn welcomes opportunities to stretch her understanding of the world in general and mathematics in particular, and tries to demonstrate a delight in mathematics and joy in learning new things to her students and colleagues. She participated in the Institute for the History of Mathematics and Its Use in Teaching and the MAA's summer 2005 Professional Enhancement Program workshop on the "Mathematics of Asia's Past." She has participated in MAA mini-courses, most recently "Geometry and Art" and "Dance and Mathematics" at the 2012 Joint Mathematics Meetings. Lynn brought university-affiliated guest speakers to her school to expose students to mathematics in unexpected contexts like Sudoku, origami, sports, and history. "I have always been a proponent of using multidisciplinary explorations in teaching mathematics, and I emphasize to my students how mathematics opens doors to many careers and fields of interest. The mission of my school emphasizes government and international studies, and I strive to instill the viewpoint that mathematics permeates and supports nearly all other disciplines, and that as future leaders, they will benefit from studying as much mathematics as possible." Reed is serving her Fellowship at the National Science Foundation, Arctic and Antarctic Sciences Division, Office of Polar Programs.
Stumbled across a book at Amazon - the Financial Calculations Workbook. It has all the equations I've ever used in finance, from simple operating margins to the Black Schole Model. Each equation has a worked example and there are test questions with solutions. Check it out. I will be do the teacher interview next week, pls advice me about interview questions, tips and interview process. You can ref some interview questions as follows: Tell me about yourself? What are your biggest strengths for Teacher? Why did you leave your last job? What are your career goals for Teacher? What kind of salary would you require to accept the position: Teacher? Why should we hire you over the other candidate? What is your system for evaluating student work? What would be the ideal philosophy of a school for you? What have been your most positive teaching experiences? Could a student of low academic ability receive a high grade in your classes? You can ref more 170 teacher interview questions & answers at: azjobebooks.info/170-teacher-interview-questions or 103 common interview questions and answers
I am going to take Calculus 2 next semester... And I would not say I am a math genius or anything... :/ So do you know of anything helpful to understand Calculus 2 very well? I want to study some over the break to have a head-start. Thank you!!! :) Be distrustful of your grade from Calculus 1. Restudy as much as you can from Calculus 1 before starting Calculus 2, since you might have earned a good grade, but you also probably learned a few things less well than you think, and you could find those things to be necessary for your ability to learn Calculus 2. Make a stack of flashcards of derivatives/integrals/trig identities and study it until you have it burned into your brain. You should be able to blurt out in your sleep any of the basics with zero hesitation. You will need to able to do u-subs in your head, so if you falter with any of the basic derivatives/integrals, you're setting yourself up for pain. I'm pretty much in the same position as the OP. Just finished Calc I (which refers to an introduction to limits, derivatives, and integrals) and will be starting Calc II in January. In my opinion, I think it depends on what you learn from best. Myself, I've found that (generally) the textbooks of today are too bloated with silly pictures and whatnot. I enjoy so much more reading from a book written in the 50s-80s. I'm an engineering student, so as tempting as it is to try something like Spivak or Apostol, they are a bit too rigorous for me, and I've found a book entitled 'Modern Calculus with Analytic Geometry' by A.W. Goodman (1967). There are a few proofs, but not as deep as other books. My point being, I think if you find something you enjoy, wether it be old books, new books, paul's online math notes, videos, anything, find what that is and learn from it. Older books are written in a way that appeals to me, and it makes it natural for me to keep reading from them, making the learning process more enjoyable.
Algebra Interactive is an HTML-based book for undergraduate students in all fields of science, e.g. mathematics, computer science, physics and (electrical) engineering. The main theme is algebraic structures, ranging from the elementary structures of the integers, polynomial rings in one variable and symmetric groups to the abstract notions of groups, rings and fields. It focuses on an algorithmic approach to these algebraic structures. The novel features of Algebra Interactive are the interactive examples and tools for computing in the various structures. Many of these tools and examples use the computer algebra package GAP, which comes with Algebra Interactive, as a back engine. Moreover, abstract notions are enlived by Java applets, which go beyond the classical examples given in traditional textbooks. The book also presents numerous exercises. Apart from the classical type of exercise (with hints and solutions), the book also presents (interactive) multiple-choice questions on each page and at the end of each chapter. On this web-site you can find the following information. A demo of the first chapter of Algebra Interactive. (Notice that the gapplets do not work!)
Can some student help me? I have a an algebra exam coming up Wednesday and I am completely lost. I require a fair amount of assistance especially with some math questions in math homework answers for free that are very confusing. I don't want to wade through any self-help website and I will greatly prize any direction in this domain. Regards! Could you give several details regarding the question? I can help you if you clarify what exactly you are searching for. Recently I discovered software product that helps in understanding math questions easily. You may get guidance on any topic related to math homework answers for free plus more, thus I recommend examining it. It is great to know that you want to improve your mathematics skills as well as being making efforts to do so. I think you could run Algebra Buster. This is not precisely some tutoring application program but it provides answers to math questions in a genuinely \| an extraordinarily step-by-step manner. The best feature of this tool is it is very easy to use. There are umpteen examples provided under different fields which are quite helpful to learn more about a specific content. Exercise it. Wish you good luck with math. Algebra Buster is the software application that I have exploited through lots of algebra classes of instruction - College Algebra, Pre Algebra plus Algebra 1. It's an extraordinary piece of math software. I recall going over questions with exponent rules, equation properties plus scientific notation. I could simply type in a problem from the workbook, click on execute I'd have a stepwise result for my math courses. I regularly recommend this application.
Annotation A DIFFERENT WAY TO LEARN DIFFERENTIAL EQUATIONS Now anyone with an interest in stepping up to higher math can do so—without formal training, unlimited time, or a genius IQ. In "Differential Equations Demystified," award-winning math professor Steven Krantz provides an effective, anxiety-free method to get past common obstacles on the road to success in higher math and science. With "Differential Equations Demystified," you master the subject one step at a time—at your own speed. This self-teaching guide offers unique "Math Notes" and "You Try It" exercises, problems at the end of each chapter to pinpoint weaknesses, and a 100-question final exam to reinforce the great information in the entire book. If you want to master differential equations fearlessly, here's a fast and effective self-teaching course to help you do just that. Get ready to — Solve both ordinary and partial differential equations— Work detailed examples and stimulating do-it-yourself exercises Discover multiple ways to solve equations, moving seamlessly into higher dimension math concepts Apply the analytical power of differential equations in engineering and science, Score better on science, math, and standardized tests A fast, effective self-teaching course, "Differential Equations Demystified" is the perfect shortcut to confidence and skill with crucial mathematical skills. Publisher Description Here's the perfect self-teaching guide to help anyone master differential equations— Equations and Boundary Value Problems, Numerical Techniques, and more. Review From review by M. Henle, Oberlin College Differential equations is an important subject that lies at the heart of the calculus. Here one sees how the calculus applies to real-world problems. Differential Equations DeMystified, (to use the spelling on the cover) is ...a serious, straightforward work. In style and substance this book is like standard differential equations books...The emphasis is consistently...on the computations needed to find...solutions to specific equations... Choice 20050201 Krantz asserts that if calculus is the heart of modern science, differential equations are the guts. Writing for those who already have a basic grasp of calculus, Krantz provides explanations, models, and examples that lead from differential equations to higher math concepts in a self-paced format. He includes chapters on first-order and second-order equations, power series solutions and spatial functions, Fourier series, Laplace transforms, numerical methods, partial equations and boundary value problems. His models come from engineering, physics and other fields in math. He includes solutions to the exercises and a final exam. Sci-Tech Book News 20041201 Author Biography Steven Krantz, Ph.D., is Chairman of the Mathematics Department at Washington University in St. Louis. An award-winning teacher and author, Dr. Krantz has written more than 45 books on mathematics, including Calculus Demystified, another popular title in this series. He lives in St. Louis, Missouri
Reference Electronic books (ebooks) are used more and more in education and mathematics is an important part of this. Design Science is working with the publishing community, ebook standards organizations, and reading system vendors to ensure that mathematical notation is handled optimally. more> Design Science has several products that allow you to publish web pages containing mathematics. Read this section to learn more about the kinds of web pages you can create, strategies for creating them, and which products and technologies are required. more> White papers are short, somewhat technical, documents that express Design Science's position on a technology. In addition, we publish Math on the Web: A Status Report to give you an objective report about available Math on the Web software and what to shop for while investigating long-term software solutions. more> The MathType SDK is available to developers who want to write software that makes use of MathType's capabilities. Applications include customizing the special commands that MathType installs into Microsoft Word, modifying MathType's translators, and extending MathType's knowledge of fonts and characters. more> Here you'll find our own version of the of the official W3C MathML 2.0 Test Suite with additional tests that exercise features of MathPlayer, our free MathML diaplay engine for the Microsoft Internet Explorer browser. more> To address the needs of people with visual impairments, Design Science is working to make technical and medical documents more accessible. The project is funded in part by a National Science Foundation grant awarded to Design Science through the Small Business Innovation Research (SBIR) Program. more> Design Science has partnered with ETS on a three year Institute of Education Sciences development grant to significantly improve accessibility of mathematical content in the classroom. The three main areas of focus are improved speech, the ability to navigate larger math equations, and to allow math to be read in Word documents. To learn more about these efforts and the progress we've made, see the grant status pages.
The picture on the front of this book is an illustration for Totakahini: The tale of the parrot, by Rabindranath Tagore, in which he satirized education as a magnificent golden cage. Opening the cage addresses mathematics education as a complex socio-political phenomenon, exploring the vast terrain that spans critique and politics. Opening the cage... more... This book consists of eighteen articles in the area of `Combinatorial Matrix Theory' and `Generalized Inverses of Matrices'. Original research and expository articles presented in this publication are written by leading Mathematicians and Statisticians working in these areas. The articles contained herein are on the following general topics: Exploring the ways in which maths skills can be learned through cross-curricular projects on arts and music, this book presents maths as a meaningful and exciting subject which holds no fear for children. more... concept of understanding in mathematics with regard to mathematics education is considered in this volume, the main problem for mathematics teachers being how to facilitate their students' understanding of the mathematics being taught. more...
CLEP College Mathematics You are about to start our College Mathematics CLEP course. This quiz practice will prove your prowess in Algebra, Geometry, Functions, Sets, Logic, Real Number Systems, and Statistics, and help you successfully pass the exam. We will help you by providing an equally rigorous practice exam course that simulates what you will experience on exam day. First, lets break down what you will find on the exam. CLEP College Mathematics Study Guides CLEP College Mathematics Practice Welcome to the CLEP Math Quiz. These tests are grouped into several sections to make your study easier. The sections include the following: Algebra and Geometry Part one of the College Math test. Concepts include: Complex numbers, Logarithms and exponents, Applications from algebra and geometry, Perimeter and area of plane figures, Properties of triangles, circles, and rectangles, The Pythagorean theorem, Parallel and perpendicular lines, Algebraic equations and inequalities. Real Number System Part five of the College Math test. Concepts include: Prime and composite numbers, Odd and even numbers, Factors and divisibility, Rational and irrational numbers, Absolute value and order, Open and closed intervals. Probability and Statistics Part six of the College Math test. Concepts include: Counting problems, including permutations and combinations, Computation of probabilities of simple and compound events, Simple conditional probability, Mean, median, mode, and range. Concept of standard deviation. Simulated CLEP Mathematics Practice Test This is a full length simulated CLEP exam providing a similar experience to the the actual College Board CLEP exam. This timed CLEP exam is geared to prepare you for the pace and format that you'll need to succeed. Use it to test your knowledge while getting as close as you can to the real-life conditions of the exam.
Welcome to Integral Math Solutions. We strive to be mathematical superheroes, and since all superheroes need a good origin story, we want to explain why we exist. As a kid, I liked math because I was good at it. As a student, I liked math because I thought it was fascinating. As an adult and a professional mathematician, I like math because these days, the world hinges on it. Like it or not, the world is a mathematical place, and just like everyone needs to know how to read, everyone needs to know how to do math. I feel strongly that it should be no more acceptable to say "I am just no good at math," than to throw one's hands in the air and say "I just can't read." Our society does not tolerate the latter, but does largely tolerate the former. I am on a mission to change this. The title of my course gives an extreme view on the importance and relevance of mathematics. Still, whether you are a student, a teacher, or a professional, you are living in a mathematical world. I started this company in order to help you through it.
Design Calculations COURSE Code: DMAT 100 4 Credits Course Description This course covers the fundamentals of solid and analytical geometry, ratio, proportion, trigonometry and elementary statistics. The emphasis is on understanding applied concepts and estimates. This is a bridging course for the degree program in Industrial Design and must be successfully completed to continue in the program.
Revise GCSE Maths The Course Our maths course can teach you any topic on the GCSE curriculum. Whether you're having trouble with your trigonometry homework, or still can't understand quadratic equations the night before the exam, we've got a video that can help! The course is taught by one of Britain's best teachers and is broken down into hundreds of mini-lessons. Each topic includes: Detailed explanations of the core concepts Worked examples to reinforce what you've learned Practice questions for you to try yourself before watching an in-depth solution All this adds up to one of the internet's most comprehensive GCSE Maths resources. In fact, if you were to watch all of our videos back to back, it would take more than 25 hours! However, our easy to use menu system allows you to find exactly what you need in a matter of seconds. This means that whether you're looking for a detailed explanation as part of a long revision session, or just trying to quickly brush up on a basic concept, RevisionBox can help. Teacher A graduate of the University of Reading, Simon's clear, methodical style has won his videos rave reviews. Simon believes strongly in using real world situations to explain Maths problems, and his innovative approach has ensured that he has a loyal following at RevisionBox. Sample Videos You can watch all videos on this course absolutely FREE, using the "Watch Videos" menu at the top left, or you can start with these sample links.
The College's developmental math classes are taught in the lab which is located inside the College's Alford L. Sweatt, Jr. Science and Technology Building. In addition, students may use the equipment for individual tutorial assistance, computer access, instructional resources and monitored testing and assignments. "We wanted a modern, open and inviting space," NCC Mathematics and Sciences Department Chair Ginny Stokes told the group. "We wanted to remove barriers to student learning. Students come in here and know that they can focus on math. They don't have to wait on instruction. They can move at their own pace." Math instructors demonstrated how to graph equations and use the calculator and explained many interactive features. Special features facilitate quick responses to student work allowing them to correct their work immediately or continue working through problems. For more information, contact Ginny Stokes at 252-451-8273 or This email address is being protected from spam bots, you need Javascript enabled to view it
People who don't learn or understand this material probably won't use it, but people who do may be surprised to find where it is useful. This applies not just to the content of the course, but to its association with careful, creative thinking. It will probably be up to you to find places where you can use this mathematics. But depending on your career, you may find that things that are now obvious to you are not known to others; or on the other hand, you may find it taken for granted that you know this material and much more. But most likely, you may actually use the subject of this course and the skills you've gained, without even realizing it. In reality, the questions and complaints mentioned above are all too frequently tacit, and it may be that much more difficult to bring these issues to a point of real discussion. Sometimes these complaints only show up on teachers' end-of-term evaluations. There are certainly more useful responses for individual students in individual situations than those offered here. The key point, however, is for the teacher to be able to listen to these kinds of questions and implicit challenges as having serious substance in them, that strike to the root of the problems of teaching and learning mathematics. ACKNOWLEDGMENTS The authors are grateful to the editor for his very useful suggestions. BIOGRAPHICAL SKETCHES Sandra Keith is a professor of mathematics at St. Cloud State University MN. Just as Einstein allegedly wanted to ride on a ``beam of light'', she has been interested in getting into the minds of students to understand how they think! She has worked with exploratory writing assignments and other interactive teaching methods. She served as director and edited Proceedings for the National Conference of Women in Mathematics and Sciences and was assistant editor of Winning Women (MAA). Her interests include better public relations for mathematics, improving the mathematical environment for women and minorities, better advising, and mathematical networking. Jan Cimperman is an assistant professor at the same school. Her interests include mathematics education, particularly, teaching elementary teachers. She frequently gives workshops on the MCTM Standards and the use of manipulatives to explore mathematical concepts at the K-6 level. She is interested in the variety of ways in which students learn.
Success in your calculus course starts here! James Stewart's CALCULUS texts are world-wide best-sellers for a reason: they are clear, accurate, and filled with relevant, real-world examples. With
GMAT Math is easy! Well that is what I thought when I was preparing for the exam. But I knew better when I missed a perfect 800 score in the GMAT because of some mistakes on the math section. And from my personal experience, I can say that a lot of students with Math/science background feel the same way. On the other hand, there are students who shudder at the very thought of facing a math problem. Well, neither of these approaches are conducive to a high math score. What is required is a thorough study of the fundamentals, a basic grasp of the concepts, and developing an ability to apply these concepts to the gmat type problems. Then comes the ability to solve a gmat problem in multiple ways, the ability to use shortcuts when stumped, and the ability to guess intelligently. Whether you are a novice or a math expert, you do need to brush up /build up your fundamentals, and then go on to the tougher problems. And this is Exactly what the 'Winners' Guide to GMAT Math does. It helps you to develop a solid understanding of the underlying concepts, builds upon this understanding by providing various different types of examples, exposes you to alternative ways of looking at a particular problem, and finally shows you how to use shortcuts. Unique Features of the Winner's Guide to GMAT Math - Part I Comprehensive coverage of Algebra, Arithmetic, Sets Geometry & Coordinate Geometry for the GMAT. All Theory & Questions based around the Actual GMAT questions that have appeared on these topics in the recent past. All topics & subtopics covered extensively over 210+ pages. Over 180+ fully solved problems to ensure in depth understanding of All concepts. No Superfluous Material. You study ONLY what is required for the GMAT. No learning difficult concepts or theories that will never get tested on the GMAT. Instant Delivery: Since this is an eBook, you will be able to download it instantaneously after you have made the payment. - Definition - Importance of Base for Calculations - Concept of Percentage Change - Difference between the Percentage Point Change and the Percentage Change - Calculating Percentage Values through additions - Percentage Change Graphic - Effect of a Change in Both Numerator and Denominator on the Ratio - Practice Problems (After completing your Transaction, click on the 'Back to Merchant' button. You will then be taken to a page where you can instantaneously download the eBook) Having trouble paying? Try our alternative payment method. Worried about security? Rest assured, our payment processors use the latest 128 bit SSl technology and are Verisign Certified. Click here for more details. Check out our Math Special Offer & get a 30% Discount click here for Details. Money Back Guarantee Try this eBook for a FULL TWO MONTHS at OUR COST If you are not happy with this guide for whatever reason, you can ask for a full refund, NO QUESTIONS ASKED at anytime. Just Quote the Paypal Receipt Number and the money will be refunded to you. No questions asked!!
Euclideanspace.com - Mathematics and Computing - Martin BakerOverview of Euclideanspace.com Euclideanspace.com is ranked 1,165,271 in the world (among the 30 million domains). This site is estimated worth $5073USD. This site has a good Pagerank(4/10). It has 244 backlinks. It's good for seo website. EuclideanspaceArticles on math education --curriculum and pedagogy-- by Henri Picciotto, a math teacher, curriculum developer, and consultant. Many sample activities, and some complete books, with an emph...Articles on math education --curriculum and pedagogy-- by Henri Picciotto, a math teacher, curriculum developer, and consultant. Many sample activities, and some complete books, with an emphasis on manipulative and technological tools.more CPM Educational Program strives to make middle school and high school mathematics accessible to all students. It does so by collaborating with classroom teachers to create problem-based text...CPM Educational Program strives to make middle school and high school mathematics accessible to all students. It does so by collaborating with classroom teachers to create problem-based textbooks and to provide the professional development support necessamore Used by over 70,000 teachers & 1 million students at home and school. Studyladder is an online english literacy & mathematics learning tool. Kids activity games, worksheets and lesso...Used by over 70,000 teachers & 1 million students at home and school. Studyladder is an online english literacy & mathematics learning tool. Kids activity games, worksheets and lesson plans for Primary and Junior High School students in United Stamore Get free math help by watching free math videos online from algebra and geometry to calculus and college math. Understand your high school math homework by watching free math videos online f...Get free math help by watching free math videos online from algebra and geometry to calculus and college math. Understand your high school math homework by watching free math videos online from your own free math help tutor.more Summary statistics for euclideanspace.com Title: Mathematics and Computing - Martin Baker Alexa Rank: 1,165,271 Pagerank: Take this button and put in your website Website Worth: $5073 USD SEO score: 34% Date Registered: Site Age: IP-address: Description:Whois Lookup For euclideans
There are a number of free algebra 1 worksheets for you to download, print, or solve online. The worksheets cover a variety algebra levels. Begin by selecting the free algebra worksheet you would like to have. This will take you to the web page of the algebra word problems worksheet. You then have several options. You can print the worksheet, download the corresponding PDF file, or complete the free algebra worksheet online. The online feature works as long as you are using a modern web browser, your iPad or other tablet device. Now you are all ready to start solving algebra equations.
This course develops students' ability to recognize, represent, and solve problems involving relations among quantitative variables. Key functions studied are linear, exponential, power and periodic functions using graphic, numeric and symbolic representations. Students will also develop the ability to analyze data, to recognize and measure variation and to understand the patterns that underlie probabilistic situations This course prepares students to take the Advanced Placement Calculus Exam. It covers limits, differential calculus, and integral calculus. Terminology, theory, notation, and in-depth problems guide students through a rigorous study of calculus. Extensive use of the graphing calculator will be involved throughout the year. Students will review and extend their knowledge of linear, quadratic, exponenetial, logarithmic, polynomial, trigonometric, step, and absolute value functions. Students will apply elements of probability, theory, and concepts of statistical design to interpret statistical findings. Other skills emphasized include imporving arithmetic skills, algebraic maniputlation, solving equations without calculators, and solving systems of equations. This course is designed to prepare students with tools necessary to be successful when taking the MCA II. Students will learn test-taking skills and be exposed to the MCA II testing format in addition to learning how to use the TI-83 graphing calculator. Students will work on practice tests and study key components of the MCA II exam.
Theory and Problems of Elementary Algebra Buy PDF List price: $16.95 Our price: $12.99 You save: $3.96 (23%) "This third edition of the perennial bestseller defines the recent changes in how the discipline is taught and introduces a new perspective on the discipline. New material in this third edition includes: A modernized section on trigonometry An introduction to mathematical modeling Instruction in use of the graphing calculator 2,000 solved problems 3,000 supplementary practice problems and more "
Media Math Calculator This new CAB tool will calculate any media formula promptly; aiding in the planning process. Click on any term to get started. Media Math This PowerPoint provides a comprehensive overview of media math with specific examples for each formula; synopsis of Nielsen national and local measurement systems; demographic derivations and some basic terminology.
EXPECTED STUDENT COMPETENCIES TO BE ACQUIRED: The successful student at the end of the course will be able produce well-written correct solutions for problems similar to those assigned for homework in this course. ASSIGNMENTS: Homework will be assigned daily and will occasionally be collected as a check on how you are keeping up. Although most of the homework assignments will not be collected, that doesn't mean you don't have to do it! A major part of learning mathematics involves DOING mathematics! Also, homework is useful in preparing for the type of questions, which may appear on quizzes or exams.Many homework problems will be given on quizzes and some on tests. Evaluations:There will be given two tests and one final exam during this term.There will also be given quizzes twice a week, except on weeks when we have a test, in which case only one quiz will be given.. Tentative Test Dates: Test 1 June 8, 2010 Test 2 June 17, 2010 Final Exam Date June 29, 2010 at 8:00 AM GRADING: Your success in meeting the course objectives will be measured by your scores on homework, quizzes, lab activities, three one-hour exams, and a cumulative final exam. The weights of the various components of your grade in determining your final course grade are shown below, along with the grade scale for the course. WEIGHTS: GRADE SCALE 1. Two exams (100 points each) 90-100 A 70-74 C 2. Quizzes, homework (150 points) 85-89 B+ 65-69 D+ 3. Cumulative Final Exam (150 points) 80-84 B 60-64 D Final average score 75-79 C+ 0-59 F NOTES: One quiz/homework grade will be dropped before determining your final quiz average. There will be no makeup quizzes.There will be no makeup tests, except under exceptional (documented) circumstances.In the case you cannot take an exam at the scheduled time, contact the instructor before the test (or as soon as possible after), to arrange a make up.In no case, will any student be allowed to have more than one make up exam during the term. If you leave right after taking a quiz, your quiz might not be graded NOTES: Please silence your cell phones during class time.No use of cell phones or any electronic device during class times or exam times. SPECIAL NOTES: If you have a physical, psychological, and/or learning disability which might affect your performance in this class, please contact the Office of Disability Services, 126A B&E, (803) 641-3609, and/or see me, as soon as possible. The Disability Services Office will determine appropriate accommodations based on medical documentation. ATTENDANCE POLICY: I may occasionally take attendance. It is highly recommended that the student not miss any class. However, the Attendance Policy established by the Department of Mathematical Sciences states that the maximum number of unexcused absences allowed in this class before a penalty is imposed is four for a regular semester. ACADEMIC CODE OF HONESTY: Please read and review the Academic Code of Conduct relating to Academic Honesty located in the Student Handbook. If you are found to be in violation of this Code of Honesty, a grade of F(0) will be given for the work. Additionally, a grade of F may be assigned for the course and/or further sanctions may
Mathematics at Hills Road Maths students enjoy the subject and find it interesting - nearly half the students at Hills Road take one of the maths courses as part of their overall programme. Maths complements other subjects, supporting and enhancing understanding in the physical or social sciences or providing breadth and balance to an arts- or languages-based programme. Maths is a good training for the mind, helping to develop logical thinking and problem-solving skills – the kind of analytical processes that have helped solve problems of all kinds for thousands of years. Students with maths qualifications are numerate and highly employable in a variety of areas as diverse as computing, engineering, finance, data analysis and mathematical modelling. Entry with: GCSE grade A in Mathematics. • Opportunities to use graphical calculators and computers to help you develop your mathematical understanding • Extra-curricular programme including trips, competitions and talks • No coursework • Range of courses allowing some choice in applied modules: Decision maths develops a variety of approaches to problem-solving and decision-making in industry and commerce . Statistics analyses situations where outcomes are uncertain and takes probability as its starting point. Mechanics investigates the way in which objects move and how they are affected by forces. It also appears in physics courses and would be particularly useful for potential engineers. • The possibility of following a Further Maths AS course as an option in Year 13. Awarding body:AQA AS Level Units Unit 1: Core 1 Unit 2: Core 2 These two units form the "tools of the trade" – the information and set of techniques used to solve problems. They cover topics such as algebra and trigonometry, which appear in GCSE courses, and develop them further. They also cover some new topics, most notably calculus. Unit 3: Applied Unit 1 (either Mechanics, Statistics or Decision depending on the course option chosen) • A purpose-built Mathematics Centre with teaching rooms grouped around a resource area containing 12 networked computers with access to the Internet and College Intranet and also a library area. • Stocks of text books and other materials • A large teaching team of dedicated and enthusiastic subject specialists. • Additional individual tuition in the lunchtime workshop, as well as other support schemes. • Advice and preparation for further study in mathematics and related disciplines. "The topics we have covered are very varied, so the lessons are always interesting. You do not repeat the same material for weeks on end! The teaching styles are also varied which helps people to grasp the new concepts. The different learning styles and classroom activities help us to remember the techniques taught. If you have a problem with anything there is always a member of staff around to help, as well as specific lunchtime surgeries. A highlight of the course was encountering new areas of maths and discovering topics I really enjoy." Pippa Cadd "I enjoy maths because its challenging, the simplicity of either right or wrong, and the buzz you get when you get a hard question right. It's not just textbook work and this also makes it more interesting and enjoyable." George Proctor
Mathematics Home Page Department Chair: Dr. Doug Riley Mission Statement: Our mission is to serve all students at all levels of ability, experience and interest. We foster creative, critical thinking and active problem solving. We seek ways for our students to come to know the beauty and usefulness of mathematics and to appreciate connections to the sciences and arts. Furthermore, we seek to assist our majors in the development of their talents as they prepare for careers involving mathematics. Welcome to the Department of Mathematics. The Department's enthusiastic faculty and well-equipped facilities offer an environment that promotes the growth of students with diverse interests and goals. We offer several courses which, as part of the college's Explorations curriculum, help students in honing their skills in quantitative and logical reasoning, and in learning to apply those skills in their other coursework and future endeavors. But the department also features a rich set of offerings for intermediate and advanced work in these important disciplines. Students will find research, teaching, and internship opportunities, as well as solid preparation for graduate and professional schools and a wide range of exciting careers. Upon completion of the mathematics major, students will be able to write a valid proof of a mathematical statement, write a lucid summary of a scholarly article in mathematics or closely related field, present a coherent explanation of his or her mathematical work in a public setting to a group of peers. The department is housed in the Olin building, where students have access to two computer labs, and all classrooms are outfitted with at least one computer and projector. The building also boasts two recently created computer classrooms where each student has access to both traditional desk space and a new flat-panel PC. The rooms have been thoughtfully laid out in a U-shaped configuration in order to facilitate the kinds of interaction that are so important to effective teaching and learning—between teacher and student, between theory and application, and among peers. When help is needed, students may find it within easy reach in the offices of a talented and welcoming faculty or in the new Quantitative Reasoning Center in Olin 103. For more information, including a course listing, please visit the most recent issue of the Birmingham-Southern College Catalog.
Calculus Essentials For Dummies (For Dummies (Math & Science)) sticks to the point with content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical two-semester high school calculus class or a college level Calculus I course, from limits and differentiation to integration and infinite series. This guide is also a perfect reference for parents who need to review critical calculus concepts as they help high schoolIn an artful interweaving of mathematics and literature, this series of lively mathematics activities jumps off from one of the well-known "Frog and Toad" stories, "The Lost Button." The story leads ... For students competing for the decreasing pool of college scholarships, writing a stellar entrance essay can make all the difference. With discrete explanations of vital grammar rules, common usage ... Learning a new language is a fun and challenging feat for students at every level. Perfect for those just starting out or returning to Spanish after time away, Spanish Essentials For Dummies focuses ...

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