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Show More drive instruction. The foundation of the program is the Common Core Standards for Mathematics Content and Standards for Mathematical Practice. This series exposes students to highly motivating and relevant problems that offer the depth and rigor needed to prepare them for Calculus and other college-level courses that they will study during their senior year of high school. The Big Ideas Math Algebra 1 book, along with the red Accelerated book, completes the compacted pathway for middle school | 677.169 | 1 |
Geometric Sequences and Series
A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.
In this lesson, we will learn
geometric sequences
how to find the nth term in a geometric sequence
geometric series
how to find the sum of a geometric series
Geometric Sequences
A list of numbers that follows a rule is called a sequence. Sequences whose rule is the multiplication of a constant are called geometric sequences, similar to arithmetic sequences that follow a rule of addition. Homework problems on geometric sequences often ask us to find the nth term of a sequence using a formula. Geometric sequences are important to understanding geometric series.
This video introduces geometric sequences.
Geometric Sequences (Introduction)
A Quick Intro to Geometric Sequences.
This video gives the definition of a geometric sequence and go through 4 examples, determining if each qualifies as a geometric sequence or not.
Geometric Sequences - Find the nth term
Geometric Sequences: A Formula for the' n - th ' Term.
This video derives the formula to find the 'n-th' term of a geometric sequence by considering an example. The formula to find another term of the sequence.
In this video we look at 2 ways to find the general term or nth term of a geometric sequence.
Geometric Series
We can use what we know of geometric sequences to understand geometric series. A geometric series is a series or summation that sums the terms of a geometric sequence. There are methods and formulas we can use to find the value of a geometric series. It can be helpful for understanding geometric series to understand arithmetic series, and both concepts will be used in upper-level Calculus topics.
This video introduces geometric series.
How to find the sum of a geometric series | 677.169 | 1 |
Cooperative Learning & Algebra 1 Secondary Activities Becky Bride (Grades 7–12)
Do you have students who struggle with Algebra? Do they find the problems difficult to understand or of little real-world value? Do they find the repetitive practice boring? The problem may not be Algebra. It may be the way students are learning Algebra. In this book, master teacher, trainer, and celebrated math author Becky Bride will show you step-by-step, activity-by-activity, and lesson-by-lesson how she used cooperative learning structures to help her students succeed with Algebra year after year. When the power of student-to-student interaction is unleashed in Algebra, students enjoy learning more and the abstract algebraic concepts become more concrete and understandable. Chapters cover: working with rational numbers, expressions, equations and inequalities, linear functions and vertical lines, linear systems, polynomials, radicals, and quadratic functions. Transform struggling students into successful mathematicians with motivating teamwork activities. Book includes reproducible transparencies and activities. 464 pages. BBA • $34 Look What's Inside!
Cooperative Learning & Algebra 2 Secondary Activities Becky Bride (Grades 9–12)
Algebra 2 just got engaging! Based on the same successful formula as her other popular high school math books, Becky now offers you Algebra 2 using Kagan's full engagement structures. Your students will have fun, yes fun, as they practice math skills using RallyCoach, Sage-N-Scribe, Quiz-Quiz-Trade, and other interactive structures. More interaction means more learning for everyone. This book is not just a collection of activities. It is a full Algebra 2 curriculum with lessons and activities sequenced to maximize comprehension, complete with blackline activities and projectable pages. Chapters cover: polynomials and polynomial functions, rational expressions and functions, radical expressions and functions, exponential functions, logarithmic functions, piecewise and absolute functions, trigonometry, and sequences and series. 464 pages. BBAT • $34 Look What's Inside!
Cooperative Learning & Pre-Algebra Secondary Activities Becky Bride (Grades 6–10)
Motivate your students to excel with teamwork. You'll be amazed at how well your students will work in teams. They'll pay attention, solve problems together, help each other out when a teammate gets stuck, and discuss Pre-Algebra concepts every day! Master math teacher and celebrated author Becky Bride shares her step-by-step cooperative learning activities and explorations with you. With this terrific resource, Becky takes all the prep work out of teamwork. Chapters cover: whole numbers, integers, decimals, fractions, ratios, proportion, percent, coordinate planes, data analysis, and probability. See why students overwhelmingly prefer working together than working alone. Watch your test scores soar. Book includes reproducible blacklines for transparencies and activities. 312 pages. BBPA • $34 Look What's Inside!
Cooperative Learning & Geometry High School Activities Becky Bride (Grades 8–12)
Make Geometry come alive using cooperative learning! In this book, you'll receive something you won't find in any textbook—HOW to teach Geometry successfully. You will find over 200 step-by-step activities to enhance Geometry exploration and mastery. In the process of working together through these carefully crafted activities, your students will learn more and enjoy Geometry more than you ever imagined possible! Topics covered: definitions, angles and lines, constructions, triangles, polygons and quadrilaterals, similarity, Pythagorean theorem and special right triangles, area, volume, and circles. Includes reproducible Geometry worksheets and blacklines for transparencies. 440 pages. BBG • $34 Look What's Inside!
Write! Mathematics Multiple Intelligences &
Cooperative Learning Writing Activities Virginia DeBolt (Grades 4–9)
Do you want to move beyond drill and kill, teaching math for understanding? Do you want to sharpen your students' mathematical thinking skills? Do you want your students to help each other learn about mathematics? Do you want to incorporate multiple intelligences in your math instruction? Yes? Here's your road map! Includes 36 cooperative writing activities guaranteed to get students thinking and writing about mathematics. The activities are open-ended, so they'll work with every mathematical topic you teach. 140 pages. BDM • $24 Look What's Inside! Save on the Write! Books Combo!
Mathematics Higher-Level Thinking Questions
The mathematics standards call for moving beyond memorization, rote learning, and application of predetermined procedures. The standards call on teachers to work toward a deeper conceptual understanding and to foster mathematical reasoning. How do we foster such a deep understanding of mathematics concepts? With deep-thinking math questions, of course. You'll find questions for sixteen mathematics topics to promote mathematical thinking and interaction in your class. 160 pages each.
Mathematics Learning Cubes (Grades PreK–3)
Each squishably soft and quiet-to-roll foam cube measures 3 inches. $3 each cube or get a class set of 8 cubes for $16. Save $8! See Entire Series of Learning Cubes
Money Cube
Help students recognize U.S. currency. Add and subtract money. Each side illustrates a different coin or bill: dollar bill, fifty cents, quarter, dime, nickel, and penny. Set of 8 cubes CMLM • $16 Save $8! | 677.169 | 1 |
Critical Thinking Co.'s Understanding Algebra I---(part of the Mathematical Reasoning series)---is a one-year Algebra I course for upper middle school and high school students that teaches basic algebraic concepts and skills. Presenting algebra as generalized arithmetic, this course was designed to help students see the connection between the math they already know and algebra. Over 100 engaging, concept-based activity sheets cover sets and set notation, evaluating expressions and solving inequalities, solving algebraic word problems, polynomials, factoring, linear functions, and more. 362 reproducible pages, softcover. Detailed answer key included in the back of the book. Grades 7-9.
The second volume in the Understanding the Times series, now revised and updatedUnderstand ing the Times Volume 2 compares the tenets of Christianity to five major competing worldviewsFree Delivery Worldwide : Inflation and the Theory of Money : Paperback : Transaction Publishers : 9780202309231 : 0202309231 : 01 Jun 2007 : Assesses the implications of inflationary processes for economic policy. This book synthesizes a general framework within which to illustrate inflationary processes. It reconciles the approaches of demand inflation and cost inflation; and analyzes the determination and behavior of the general price level in an exchange economy.
With his usual flair and reader-friendly style, Ben Witherington III brings us a fresh and distinctive guide to interpreting the Bible. Ideal for courses in Biblical Interpretation, Hermeneutics, and Introduction to the Bible, Reading and Understanding the Bible is unique in that it carefully examines the various genres of literature in the Bible while also explaining how to interpret each within its proper context. Taking a faith-friendly approach to historically based interpretation, it shows students how to read the Bible with a keen awareness of the many and profound differences between the modern world and ancient biblical cultures. It explains how ancient societies worked, how documents were created, who preserved them and why, the patriarchal nature of all ancient cultures, and,...
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Wall Street Journal Guide To Understanding Money And Investing Morrisipp
The first volume in the Understanding the Times series, now revised and updatedUnderstand ing the Faith Volume 1 is an apologetics handbook that lays the foundation for a Christian worldview by showing how the Bible is both authoritative and trueThe depression of the '80s, the '30s, fiat money, runs on the bank and freezing assets...what does it all have to do with Americans? Written in the same letter format as Richard Maybury's other Uncle Eric books, students will gain a remarkably easy to understand introduction to the causes and effects of inflation and deflation. Using the real-life examples of the assignat and mandat that lead to Napoleon's rise, the freezing of Iranian Assets that led to other Arab investors 'dumping the dollar' and other historical events, Maybury creates a recent economic history of America. 110 pages, softcover.
Daniel Migliore's Faith Seeking Understanding, a standard Christian theology text for more than two decades, explores all of the major Christian doctrines in freshly contemporary ways.This third edition of the book features a number of improvements and additions. Migliore has added a list of books for further reading at the end of each chapter, the glossary is substantially expanded, and a new theological dialogue between Karl Barth and Friedrich Nietzsche joins the other three (at once entertaining and instructive) dialogues in the appendix. New material has been incorporated at various points in the text, including a section on Christians and Muslims.A new generation of students, pastors, and Christian educators, eager to better understand the rich heritage, central themes, and...
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Understanding Business by William Nickels is available now for quick shipment to any US location and this book can be substituted for ISBN 0078023165 or ISBN 9780078023163 or the more current 11th edition. You will save lots of cash by using this used 9th edition which is nearly identical to the newer editions and
Free Delivery Worldwide : Classical Theories of Money, Output and Inflation : Hardback : Palgrave MacMillan : 9780333565629 : 0333565622 : 03 Nov 1992 : Challenges the conventional view that monetarism, or the quantity theory of money, is a necessary part of classical economics and aims to show that the framework upon which classical analysis is based suggests an alternative account of the inflationary process.
This book explains modern macroeconomics. It does not use equations, graphs, diagrams or footnotes. The book is designed to make modern macroeconomics available to those who never had a university course in economics or who had one years ago, now little r
Understanding Society by Margaret L Andersen is available now for quick shipment to any US location and this book can be substituted for ISBN 1305093704 or ISBN 9781305093706 or the more current 5th edition. You will save lots of cash by using this 3rd edition which is nearly identical to the newer editions a
In Understanding Clergy Misconduct in Religious Systems, you'll take an incisive look at why sexual misconduct occurs in religious systems and how to implement proactive strategies for holistic change. Applicable to both Jewish and Christian communities, this illuminating exploration takes a look at the psychology behind scapegoating, why it is perpetuated, and how you can quell the damaging tradition of silence. Understanding Clergy Misconduct in Religious Systems helps you see leaders of religious institutions in a way that the world has been afraid to see them--in a glass clearly. Enriched with metaphoric myths and fairy tales instead of technical jargon, its unique systemic perspective reveals the psychodynamics behind the obsession with family secrets and lets you understand this...
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Understanding Computers - by Morley is available now for quick shipment to any U.S. location. This edition can easily be substituted for ISBN 1305656318 or ISBN 9781305656314 the 16th edition or 2016 edition or even more recent edition. You will save lots of cash by using this 12th edition which is nearly identi
Understanding And Using English Grammar by Betty Schrampfer Azar is available now for quick shipment to any US location and this book can be substituted for ISBN 0132333317 or ISBN 9780132333313 or the more current 4th edition. You will save lots of cash by using this 3rd edition which is nearly identical to the n
Understanding Global Conflict And Cooperation Joseph S Nye is available now for quick shipment to any U.S. location. This edition can easily be substituted for ISBN 0134403169 or ISBN 9780134403168 the 10th edition or 2016 edition or even more recent edition. You will save lots of cash by using this 8th edition wh
Understanding Weather And Climate by Edward Aguado is available now for quick shipment to any US location and this book can be substituted for ISBN 0321987306 or ISBN 9780321987303 or the more current 7th edition. You will save lots of cash by using this used 4th edition which is nearly identical to the newer edit
Using And Understanding Mathematics by Jeffrey O Bennett is available now for quick shipment to any US location and this book can be substituted for ISBN 0321914627 or ISBN 9780321914620 or the more current 6th edition. You will save lots of cash by using this 4th edition which is nearly identical to the newer edi
Understanding Normal And Clinical Nutrition by Sharon Rady Rolfes is available now for quick shipment to any US location and this book can be substituted for ISBN 1285458761 or ISBN 9781285458762 or the more current 10th edition. You will save lots of cash by using this 7th edition which is nearly identical to the
Understanding Humans by Barry Lewis is available now for quick shipment to any US location and this book can be substituted for ISBN 1111831777 or ISBN 9781111831776 or the more current 11th edition. You will save lots of cash by using this 9th edition which is nearly identical to the newer editions. We have been
Understanding Social Problems Linda A Mooney is available now for quick shipment to any U.S. location. This edition can easily be substituted for ISBN 1305576519 or ISBN 9781305576513 the 10th edition or 2016 edition or even more recent edition. You will save lots of cash by using this 5th edition which is nearly
Grade Inflation by ValenWhile the financial crash and post-recession world has proven young people need to understand economics, Whatever Happened to Penny Candy actually makes it fun, fascinating, and entertaining! Written in Maybury's trademark Uncle Eric letter format, students will learn about recession, money, inflation, boom & bust cycles, government spending, Gresham's Law, and more. Common-sense arguments and down-to-earth explanations create a thorough look at money that's understandable for all. This revised 7th edition now includes updates on runaway inflation, nationals, and legal systems, along with minor updates to reflect current numbers, statistics, dates, data, graphs, charts, and back matter. It also includes a new explanation on how human emotions like fear can affect the economy. 168 indexed...
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Free Delivery Worldwide : Reducing Inflation : Hardback : The University of Chicago Press : 9780226724843 : 0226724840 : 23 Jun 1997 : While there is ample evidence that high inflation is harmful, little is known about how best to reduce it. In this volume, the contributors discuss the consequences of inflation, consider the obstacles facing central bankers in achieving low inflation, and examine how institutions can promote it.
Free Delivery Worldwide : Inflation, Institutions and Information : Hardback : Palgrave MacMillan : 9780333617724 : 033361772X : 01 Nov 1996 : This collection of essays by colleagues and former students pays homage to Axel Leijonhuvfvud. It discusses issues that have featured at the centre of his research for over 30 years: history of thought, philosophy of science and transition dynamics, as well as monetary macroeconomics.
Understanding Food Science And Technology by Peter MuranoUnderstanding Business And Personal Law by Mcgraw-Hillcom We h
Free Delivery Worldwide : Inflation : Hardback : Chapman and Hall : 9780412358708 : 0412358700 : 01 Mar 1993 : Identifies the likely causes of high inflation and assesses the available policy options for preventing or curing it. This book presents an approach that consists of making hypotheses about the economic motivation of individuals, developing a model, and assessing the results.
Free Delivery Worldwide : The Economics of High Inflation : Hardback : Palgrave MacMillan : 9780333563809 : 0333563808 : 13 Nov 1991 : Looks at the theory and practices of inflation as it relates to the financial systems in various countries. This book examines aspects of inflation as they relate to the economy as well as the options available to economists to counter or eliminate the effects of inflation.
Understanding Trademark Law 2005 by Not Avail is available now for quick shipment to any U.S. location. This edition can easily be substituted for ISBN 1422472329 or ISBN 9781422472323 the 2nd edition or 2009 edition or even more recent edition. You will save lots of cash by using this prior edition which is nearl
Understanding Insurance Law by Robert Jerry | 677.169 | 1 |
97801399399active Math for Intermediate Algebra
Elayn Martin-Gay's developmental math textbooks and video resources are motivated by her firm belief that every student can succeed. Martin-Gay's focus on the student shapes her clear, accessible writing, inspires her constant pedagogical innovations. Interactive Math is a new learning system for students that covers the full series of courses in develpmental mathematics in an interactive, multimedia environment. A program complete with instruction, practice, applications, assessment, and a flexible course management system, Interactive Math enables instructors to focus their class time on teaching and working individually with | 677.169 | 1 |
books.google.co.ve - The information-packed introduction to Algebra for Spanish-speaking students is packed with theorems, postulates, and answer keys, and encompasses complete guidelines for all secondary school standards....
Algebra: con gráficos y 6.523 ejercicios y problemas con respuestas
The information-packed introduction to Algebra for Spanish-speaking students is packed with theorems, postulates, and answer keys, and encompasses complete guidelines for all secondary school standards. | 677.169 | 1 |
Essentials of Integration Theory for Analysis is, as the title states, a textbook on measure theory for analysis. It does not, as the author writes, embed measure theory in topics in which measure theory plays a central role, such as probability or Fourier analysis. However, there are examples from both in the book.
Essentials is a significantly expanded version of the author's A Concise Introduction to the Theory of Integration from 1990. I see that according to the Springer web site Concise was updated in 1998 to 262 pages (not so concise anymore), but I don't have a copy of that edition handy. Compared to the 1990 edition that I used when I first learned measure theory, Essentials has about 90 more pages than Concise, with new sections on the rate of convergence of Riemann approximations, Steiner symmetrization, Fourier series and transforms, and Daniell integration. There are also more exercises.
The target audience is students who have completed a good analysis course (baby Rudin, say).
Stroock writes beautifully and is full of insight and advice. For example he states that "the essence of any theory of integration is a divide and conquer strategy" and then goes on to describe what would be needed in "a reasonable notion of measure." It is my belief that these messages to the student learning the material are very beneficial, and help place the material in context.
On the negative side, some notation is used without definition. For example, the notation for the interior of a set is used on page 1 without being explained.
The difference between the two books is exactly what you would expect from their titles — one is a very concise introduction, the other a more detailed, fleshed out text. Each is extremely well suited to its stated purpose.
Peter Rabinovitch is a Systems Architect at Research in Motion, and a PhD student in probability. He s currently writing about applications of Mallows permutations. | 677.169 | 1 |
Product Description
Boost your students understanding of Saxon Math with DIVE's easy-to-understand lectures! Each lesson concept in Saxon's textbook is taught step-by-step on a digital whiteboard, averaging about 10-15 minutes in length; and because each lesson is stored separately, you can easily move about from lesson-to-lesson as well as maneuver within the lesson you're watching. Taught from a Christian worldview, Dr. David Shormann also provides a weekly syllabus to help students stay on track with the lessons. DIVE teaches the same concepts as Saxon, but does not use the problems given in the text; it cannot be used as a solutions guide.
System Requirements:
Mac OS 10.3.9-10.4.x
Windows 98, 2000, ME, XP, Vista, 8, 10
Quicktime Download Required
Please Note! The current edition of Saxon Math 7/6 is the 4th Edition. This 3rd edition is offered for families using older versions of Saxon | 677.169 | 1 |
In-depth treatment of concepts underlying common topics in the middle and high school mathematics curriculum. Topics include number systems, polynomial and transcendental functions, analytic geometry, theory of equations, and measurement. Prerequisite: MATH 150 and 166 and 215. Open only to mathematics teaching majors. | 677.169 | 1 |
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I am currently a mathematics teacher. One of my favorite memories is bonding with the "Math Square and the Arbitrary Point". Other favorite memories include the Math Department picnics and developing a relationship with my professors that some students will never have an opportunity to do.
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Mathematics Courses
MATH 095 Fundamentals of Mathematics (3)
Topics include exponents, radicals, factoring, linear and quadratic equations, graphing of linear and polynomial functions, area, volume, systems of equations, and problem solving. This course does not satisfy the core math requirements.
MATH 120 Mathematics in the Liberal Arts (3)
Designed to implement NCTM curriculum standards with emphasis on problem-solving, patterns and relationships, functions, estimation, and mathematical connections to other disciplines. Topics are chosen from this list: functions, mathematical modeling, basic trigonometry, geometry, astronomy, music, elementary statistics, voting methods, and logic. No prior knowledge of any of these topics is assumed. There are no prerequisites for MATH 120 - Math Placement exams are not required for this course.
MATH 130 Mathematics for Contemporary Society (3) (Formerly MATH 110)
The topics studied will include techniques and applications of set theory, counting techniques, matrices, linear systems, statistics and probability, and linear programming. Prerequisite: passing grade in the Algebra Placement exam.
MATH 150 Architectural Mathematics (3)
The principles of mathematics relating to architecture and building design. Topics include plane and solid geometry, coordinate systems, vectors, isometrics, the golden ratio, conic sections, tilings, fractals, and concepts in topology. Prerequisite: passing grade in the Algebra Placement exam.
MATH 155 Statistics for the Behavioral and Social Sciences (3) (Formerly MATH 216)
Surveys the basic statistical concepts applicable to problems in the behavioral and social sciences. Includes descriptive statistics, regression and correlative, hypothesis testing, nonparametric methods, and analysis of variance. Computer software will be utilized for calculations. Prerequisite: passing grade in the Algebra Placement exam.
MATH 160 Analysis of Functions (Pre-calculus) (3) (Formerly MATH 115)
Topics include polynomial, rational, exponential, logarithmic and trigonometric functions, as well as conic sections. Prerequisite: passing grade in the Algebra Placement exam.
MATH 170 Applications of Mathematics to Biology (3)
Examines problems in biology through the use of a variety of mathematical tools and models. Topics are chosen from linear, exponential, and logarithmic functions, set theory, linear systems, probability, and an introduction to calculus. Prerequisites: algebra and trigonometry. Prerequisite: C or better in MATH 160, or passing the Calculus placement exam.
MATH 204 Calculus with Analytic Geometry IV (3)
MATH 219 History of Mathematics (3)
The study of mathematical concepts from arithmetic to calculus in their historical perspective. This study will be supplemented by historical background material, biographies of mathematicians and translations of source manuscripts in which mathematical discoveries were first announced. Attention will be given to the relationship of mathematics to other disciplines. For Mathematics majors and minors.
MATH 271 Transition to Advanced Mathematics (3)
A transition from lower level mathematics courses to higher level courses. Emphasis will be placed on correct reading, understanding, and writing of proofs. Topics will include logic, direct proofs, proof by contra-positive, proof by contradiction, equivalence relations, functions, and mathematical induction.
MATH 311 Differential Equations (3)
The study of differential equations and first-order linear systems through a combination of analytical, numerical, and qualitative techniques. Topics include the standard analytical methods of solving nth-order linear equations, eigenvalues and eigenvectors for systems, phase-plane trajectories, the Laplace transform, and numerical approximations. Technology is used in conjunction with theory to approximate and analyze solutions. Modeling physical phenomena is emphasized through a rich variety of applications. Prerequisite: MATH 204, MATH 271.
MATH 324 College Geometry (3)
The study of axiomatic systems and the notions of proof and consistency. Examines finite, elliptical, and hyperbolic geometries, and advanced topics in Euclidean Geometry. Software is used to enhance exploration and discovery of theorems. Prerequisite: MATH 202, MATH 271.
MATH 411B Curriculum Methods and Materials in Mathematics (3)
See EDUC 411.
MATH 420 Discrete Mathematics (3)
An introduction to the algebraic concepts, methods and techniques that form the theoretical basis for computer science, including relevant areas of logic, set theory, relations and functions, and Boolean algebra. Prerequisite: MATH 202, MATH 271.
MATH 430 Real Analysis (3)
An introduction to the analysis of real numbers, variables, and functions. Topics include topology of the real numbers, sequences and series, limits, continuity and uniform continuity, differentiation, the Riemann integral, and sequences of functions. Prerequisite: MATH 204, MATH 271.
MATH 456 Mathematical Statistics (3)
MATH 495 Senior Seminar (1)
Analysis of the underlying foundational questions of mathematics including the notions of proof and consistency within a specific mathematical framework. Examination of the considerable impact of mathematics on culture and society from ancient to modern times. | 677.169 | 1 |
In linear algebra, Gaussian elimination is an algorithm for solving systems of
linear equations. It is usually understood as a sequence of operations performed
on the corresponding matrix of coeffic... | 677.169 | 1 |
Enumerative Combinatorics: Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of Volume 1 includes ten new sections and more than 300 new exercises, most with solutions, reflecting numerous new developments since the publication of the first edition in 1986. The author brings the coverage up to date and includes a wide variety of additional applications and examples, as well as updated and expanded chapter bibliographies. Many of the less difficult new exercises have no solutions so that they can more easily be assigned to students. The material on P-partitions has been rearranged and generalized; the treatment of permutation statistics has been greatly enlarged; and there are also new sections on q-analogues of permutations, hyperplane arrangements, the cd-index, promotion and evacuation and differential posets.
Richard P. Stanley is a professor of applied mathematics at the Massachusetts Institute of Technology. He is universally recognized as a leading expert in the field of combinatorics and its applications to a variety of other mathematical disciplines. In addition to the seminal two-volume book Enumerative Combinatories, he is the author of Combinatories and Commutative Algebra (1983) and more than 100 research articles in National Academy of Sciences (elected in 1995), the 2001 Leroy P. Steele Prize for mathematical exposition, and the 2003 Schock Prize. | 677.169 | 1 |
There are a lot of threads asking about math courses to take, so I figured it was time to put together a blog entry on this. I'm making the assumption that you, the reader, are some sort of an IT or CS major, so I'm starting at Calculus and moving my way up. This isn't a complete or comprehensive list of math courses commonly available to undergraduates, but this is my opinion of a set of course that a well-rounded computer scientist will find of interest.
Calc I-II: These classes cover differential and integral calculus, as well as series and sequences on most occasions. These courses should provide a good foundation to understanding rates, changes, motion, basic optimization on the real numbers, and some basic modeling. When working with a lot of dynamical systems or moving objects, it's important to understand calculus.
Multivariable Calculus- This class covers calculus on two or more variables. Topics covered include partial derivatives and multiple integrals. Weaker courses may cover series and sequences, and stronger courses focus on topics like Stokes Theorem, Green's Theorem, and Divergence Theorem. Economists, Mathematicians, and Engineers need this course, and it is helpful for computer scientists focusing on graphics or scientific computing.
Differential Equations- This class covers solving equations with derivatives (usually ordinary differential equations), which is extremely helpful for modeling pretty much any system with changing variables. The topics are very applicable to graphics programming, scientific computing, numerical methods, and dynamical systems. This is a very important class for computer scientists, amongst other majors.
Linear Algebra- I highly recommend a strong linear algebra class to anyone. Linear Algebra has applications everywhere, including numerical methods, scientific computing, mathematical modeling, graph theory, and graphics programming. Make sure to take a strong, abstract and proofs-based version to really get a good flavor. It has been my experience that classes solely focused on number crunching don't encourage a lot of thinking, which isn't as beneficial to a computer scientist (or anyone really) as a class that forces one to think. The other benefit of Linear Algebra is that it introduces rigorous proof-writing, for those who may not have previously taken a formal proof-writing course.
Discrete Math- This class is often times taught as a survey class, based on the professor's interests. You will almost always learn something new repeating this course at a different school or with a different professor. Topics are generally relevant to computer science, and may include logic, set theory, Boolean algebras, computational complexity, number theory, graph theory, combinatorics, languages, proof by induction, and discrete probability. This is another staple course for computer scientists.
Number Theory- Number Theory is the study of the integers. Topics include primality, primitive roots, quadratic reciprocity, factoring, and more. Number Theory is applied in the context of cryptography, so this is a very relevant course for computer scientists. This course is similar to Linear Algebra in the fact that it introduces proof-writing to those who may not have seen it before. Unlike Linear Algebra, however, Number Theory stresses proof by induction. | 677.169 | 1 |
This course involves statistics, numbers, algebra, geometry, trigonometry and functional maths. The syllabus is modular with 4 exam modules, which may be taken in March or June depending on student...more
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An essential subject for all students, IGCSE Mathematics is a fully examined course which encourages the development of mathematical knowledge as a key life skill, and as a basis for more advanced study evening GCSE course will equip you with the knowledge required to pass the Edexcel GCSE Mathematics examination and gain skills in applying this knowledge. This course is only open to studentsMath isn't about plugging numbers into formulas. It's about knowing enough to make the numbers and formulas work for you. Math can be incredibly useful - but only if you understand how and when to apply...more
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This course will help you to improve your GCSE grade in mathematics, which is required for entry into various professional courses and higher education. This course is taught using a mixture of groupDo you need to obtain a recognised qualification for work, university entrance or a vocational course? Have you recently taken your GCSE maths and achieved a grade D? Have you recently achieved a Level 2...more
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Overview Our Online Mathematics Course is the international version of IGCSE. The objective of this course is to develop your understanding of mathematical concepts and techniques offers adult learners the opportunity to complete their GCSE Maths and English. The entry requirements are to successfully pass a GCSE suitability test when you come in for your interview. This...more
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This course is ideal for anyone looking to gain a level 2 qualification in maths. This is one of the qualifications you will need for entry to level 3 courses and for a course at university. As a...more
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This course is for adults who are wishing to gain a recognised qualification in Maths. The level studied will depend on a diagnostic assessment at interview. Courses run in the Fashion Retail Academy (The)
We train people aged 16+ who are interested in a career in fashion through funded vocational courses in areas including merchandising & buying
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You will have 3 hours of teaching per week for 32 weeks. This will be one evening per week and will consist of the following:
• Algebra
• Data Handling
• Fractions, Decimals and Ratios
• Shape, Space and | 677.169 | 1 |
8.79
FREE
Used Like New(1 Copy):
Like New
Ohio Valley Goodwill
OH, USA
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About the Book
The resource math teachers have been waiting for is finally here! Volume One of the Van de Walle Professional Mathematics Series provides practical guidance along with proven strategies for practicing teachers of kindergarten through grade 3. In addition to many of the popular topics and features from John Van de Walle's market-leading textbook, "Elementary and Middle School Mathematics," this volume offers brand-new material specifically written for the early grades. The expanded grade-specific coverage and unique page design allow readers to quickly and easily locate information to implement in the classroom. Nearly 200 grade-appropriate activities are included. The student-centered, problem-based approach will help students develop real understanding and confidence in mathematics, making this series indispensable for teachers!
Big Ideas provide clear and succinct explanations of the most critical concepts in K-3 mathematics.
Problem-based activities in every chapter provide numerous engaging tasks to help students develop understanding.
Assessment Notes illustrate how assessment can be an integral part of instruction and suggest practical assessment strategies.
Expanded Lessons elaborate on one activity from each chapter, providing examples for creating step-by-step lesson plans for classroom implementation.
A Companion Website ( provides access to more than 50 reproducible blackline masters to utilize in the classroom.
The NCTM Content Standards are provided for teachers' reference in the appendix.
About the Authors John Van de Walle is Professor Emeritus at Virginia Commonwealth University. He is a co-author of "Scott Foresman-Addison Wesley Mathematics," a K-to-6 textbook series and the author of "Elementary and Middle School Mathematics: Teaching Developmentally," the best-selling text and resource book on which this series is based. LouAnn Lovin is a former classroom teacher and is currently an assistant professor in mathematics education at James Madison University, where she teaches mathematics methods and mathematics content courses for Pre-K-8 prospective teachers and is involved in the mathematical professional development of teachers in grades 4-8. Collect all three volumes in the Van de Walle Professional Mathematics Series! Each volume provides in-depth coverage at specific grade levels. Learn more about the series at Teaching Student-Centered Mathematics: Volume One, Grades K-3, 1/e 0-205-40843-5 Teaching Student-Centered Mathematics: Volume Two, Grades 3-5, 1/e 0-205-40844-3 Teaching Student-Centered Mathematics: Volume Three, Grades 5-8, 1/e 0-205-41797-3 | 677.169 | 1 |
Introductory Algebrais typically a 1-semester course that provides a solid foundation in algebraic skills and reasoning for students who have little or no previous experience with the topic. The goal is to effectively prepare students to transition into Intermediate Algebra.
Table of Contents
Tools to Help Students Succeed
ix
Additional Resources to Help You Succeed
xi
Preface
xiii
Applications Index
xxiii
Prealgebra Review
1
(1)
Factors and the Least Common Multiple
2
(7)
Fractions
9
(10)
Decimals and Percents
19
Group Activity: Interpreting Survey Results
28
(1)
Vocabulary Check
29
(1)
Highlights
29
(3)
Review
32
(2)
Test
34
Real Numbers and Introduction to Algebra
1
(88)
Tips for Success in Mathematics
2
(6)
Symbols and Sets of Numbers
8
(11)
Exponents, Order of Operations, and Variable Expressions
19
(10)
Adding Real Numbers
29
(9)
Subtracting Real Numbers
38
(10)
Integrated Review--Operations on Real Numbers
46
(2)
Multiplying and Dividing Real Numbers
48
(12)
Properties of Real Numbers
60
(8)
Simplifying Expressions
68
(21)
Group Activity: Magic Squares
77
(1)
Vocabulary Check
78
(1)
Highlights
78
(5)
Review
83
(4)
Test
87
(2)
Equations, Inequalities, and Problem Solving
89
(89)
The Addition Property of Equality
90
(9)
The Multiplication Property of Equality
99
(9)
Further Solving Linear Equations
108
(10)
Integrated Review--Solving Linear Equations
116
(2)
An Introduction to Problem Solving
118
(12)
Formulas and Problem Solving
130
(12)
Percent and Mixture Problem Solving
142
(12)
Solving Linear Inequalities
154
(24)
Group Activity: Investigating Averages
164
(1)
Vocabulary Check
165
(1)
Highlights
165
(3)
Review
168
(5)
Test
173
(2)
Cumulative Review
175
(3)
Exponents and Polynomials
178
(76)
Exponents
179
(12)
Negative Exponents and Scientific Notation
191
(9)
Introduction to Polynomials
200
(10)
Adding and Subtracting Polynomials
210
(7)
Multiplying Polynomials
217
(7)
Special Products
224
(9)
Integrated Review--Exponents and Operations on Polynomials
231
(2)
Dividing Polynomials
233
(21)
Group Activity: Modeling with Polynomials
240
(1)
Vocabulary Check
241
(1)
Highlights
241
(3)
Review
244
(5)
Test
249
(2)
Cumulative Review
251
(3)
Factoring Polynomials
254
(70)
The Greatest Common Factor
255
(10)
Factoring Trinomials of the Form x2 + bx + c
265
(7)
Factoring Trinomials of the Form ax2 + bx + c
272
(7)
Factoring Trinomials of the Form ax2 + bx + c by Grouping
279
(4)
Factoring Perfect Square Trinomials and the Difference of Two Squares
283
(10)
Integrated Review--Choosing a Factoring Strategy
291
(2)
Solving Quadratic Equations by Factoring
293
(9)
Quadratic Equations and Problem Solving
302
(22)
Group Activity
311
(1)
Vocabulary Check
312
(1)
Highlights
312
(3)
Review
315
(4)
Test
319
(2)
Cumulative Review
321
(3)
Rational Expressions
324
(82)
Simplifying Rational Expressions
325
(10)
Multiplying and Dividing Rational Expressions
335
(9)
Adding and Subtracting Rational Expressions with the Same Denominator and Least Common Denominators
344
(9)
Adding and Subtracting Rational Expressions with Different Denominators | 677.169 | 1 |
The LV courses provide extensive coverage of the elementary school math standards in arithmetic, algebra, geometry, measurement, probability, and statistics. The ALEKS calculator is available for selected topics.
The Arithmetic courses focus almost exclusively on arithmetic, with some coverage of geometry and measurement. The ALEKS calculator is never allowed in the ALEKS Arithmetic courses.
Pre-Algebra provides standards-based coverage of all of Grade 8 Math, including a robust introduction to the basic concepts of algebra and its prerequisites.
Algebra Readiness also provides robust coverage of the basic concepts of algebra, algebra prerequisites, and related math curriculum standards. Algebra Readiness does not provide coverage of non-algebra middle school mathematics topics, such as probability, statistics, and geometry.
Essentials for Algebra provides the necessary prerequisite topics that are central for success in Algebra 1, including standards-based geometry, probability, and statistics conceptsTX Algebra 1 is a new course which comprehensively covers the Texas Essential Knowledge and Skills (TEKS) for Mathematics outlined in Proclamation 2015. The new TEKS for Algebra 1 are scheduled for implementation in 2015-2016.
The Integrated Mathematics I-III courses offer an alternative to the Algebra 1, High School Geometry, and Algebra 2 course sequence. Either course sequence can be used to prepare students for courses in higher level mathematics. Each Integrated Mathematics I-III course offers comprehensive, standards-based coverage and reporting against the Common Core Standards.
GPS Integrated High School Math I (GA) includes course topics from algebra, geometry, probability, and statistics, and provides students with an especially rigorous 9th grade integrated mathematics course. This course offers comprehensive coverage conforming to the Georgia Performance Standards (GPS), and includes reporting against the GPS and the Common Core Standards.
High School Preparation for Algebra 1 focuses on critical prerequisite topics that are central for success in Algebra. This is the best course to prepare students to advance as quickly as possible into Algebra 1 or Algebra 1A.
Foundations of High School Math is a broad, standards-based course offering comprehensive coverage of the middle school mathematics curriculum, including many topics that are not prerequisites for success in Algebra. This course is intended to develop mastery of the full breadth of middle school math concepts to facilitate success in high school mathematics, including algebra and geometry courses.
Mastery of SAT Math is designed to help the student achieve mastery of the math topics on the SAT Reasoning Test.
We recommend the following use of this course by students preparing for the SAT:
Master 100% of the material in this ALEKS course; and
Take a number of SAT practice tests (widely available from other sources).
ALEKS individualized assessment and learning enables students to efficiently refresh and fill gaps in their knowledge of the mathematics tested on the SAT. The course works best when supplemented with SAT practice tests, so that students achieve fluency in the particular style and format of the SAT test questions.
Prep for TX - STAAR Algebra 1 prepares students for the 2014-2015 STAAR Algebra 1. This course is based on the Texas Essential Knowledge and Skills (TEKS) for Mathematics outlined in Proclamation 2004. The Proclamation 2004 TEKS were implemented in the 2006-2007 school year and are anticipated to be superseded by the scheduled adoption of Proclamation 2015 in 2015-2016.
College Preparedness prepares students for college math success by providing thorough coverage of the essential math through intermediate algebra topics necessary for students to progress into a credit-bearing college math course.
AP Chemistry provides rigorous coverage of chemistry topics that are typically included in a university-level General Chemistry course. This course includes the built-in ALEKSpedia, which is a General Chemistry Primer, making the ALEKS AP Chemistry course your chemistry solution. This course can also be used to help students achieve better results on the AP Chemistry exam.
Prep for AP Chemistry is designed to prepare high school students for an AP Chemistry course. This course covers prerequisite and foundational material necessary for success in AP Chemistry.
Florida Math 0018 provides 100% coverage of the Lower portion of the FL Developmental Education Mathematics Competencies. See the correlation of the topics in ALEKS Florida Math 0018 with the Florida Competencies - Lower.
Florida Math 0022 provides 100% coverage of all of the FL Developmental Education Mathematics Competencies (Lower and Upper). See the correlation of the topics in ALEKS Florida Math 0022 with the Florida Competencies.
Florida Math 0028 provides 100% coverage of the Upper portion of the FL Developmental Education Mathematics Competencies. See the correlation of the topics in ALEKS Florida Math 0028 with the Florida Competencies - Upper. | 677.169 | 1 |
Mathematics Standards and Technology Goals
Selected Common Core State Standards: Mathematics
Grade 7 Geometry
Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Grade 7 Statistics & Probability
Use random sampling to draw inferences about a population.
7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
7.SP.A.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Draw informal comparative inferences about two populations.
7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
7.SP.C.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7.SP.C.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
7.SP.C.7a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
7.SP.C.7b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7.SP.C.8a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
7.SP.C.8b Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
7.SP.C.8c Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
Grade 8 Algebra (Expressions & Equations; Functions)
8.EE.A.2 Use8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.
8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology
Understand the connections between proportional relationships, lines, and linear equations.
8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
8.EE.C.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
8.EE.C.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Define, evaluate, and compare functions.
8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.A.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For
Use functions to model relationships between quantities.
8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Standards for Mathematical Practice
MP1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
MP2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
MP3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
MP4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
MP5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
MP6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
MP7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
MP8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. | 677.169 | 1 |
Browse by
Computational Methods for Solving Differential
Equations
• Euler's method is a numerical or computational method for
solving a differential equation.
• It is called numerical, because the result is a list of numbers, not
the formula for a function.
• Euler's method is simple conceptually and gets at the heart of
what differential equations are.
• However, Euler's method is not very efficient, and so is not used
much in practice.
Improving on Euler's Method
Two improvements:
1. Runge-Kutta methods: Sample the rate of change at several
points along the interval ∆t and average them.
2. Adaptive step size: Automatically adjust ∆t on the fly. Make
∆t small (large) when the derivative changes rapidly (slowly).
• Most numerical programming environments (matlab, octave,
maple, mathematica, python with numpy, etc.) have a built-in
RK, adaptive step size solver.
Looking Ahead
• Almost all of the solutions to differential equations that I will
discuss in this course will be numerical solutions.
• You will not need to solve differential equations on your own.
• However, it is important that you have a sense of where
numerical solutions come from and what they mean. | 677.169 | 1 |
#1 Engineering Mathematics, Second Edition
This book is the sequel to Stroud's excellent "EngineeringMathematics", which focused on the undergraduate engineer and the math that he/she should know by graduation. This book continues on with crystal-clear discussions of numerical methods, linear algebra including the singular value decomposition and its uses, linear programming methods, multiple integration, and partial differential equations, to name a few of the topics covered. Just because the mathematics is more advanced in this book does not mean that it is any less clear than its less advanced predecessor
#2 Engineering Mathematics Pocket Book, 4 Edition (with Solutions)
This compendium of essential formulae, definitions, tables and general information provides the mathematical information required by students, technicians, scientists and engineers in day-to-day engineering practice. A practical and versatile reference source, now in its fourth edition, the layout has been changed and the book has been streamlined to ensure the information is even more quickly and readily available - making it a handy companion on-site, in the office as well as for academic study. It also acts as a practical revision guide for those undertaking BTEC Nationals, Higher Nationals and NVQs, where engineeringmathematics is an underpinning requirement of the course.
#3 Oxford Users' Guide to Mathematics
The Oxford User's Guide to Mathematics in Science and Engineering represents a comprehensive handbook on mathematics. It covers a broad spectrum of mathematics including analysis, algebra, geometry, foundations of mathematics, calculus of variations and optimization, theory of probability and mathematical statistics, numerical mathematics and scientific computing, and history of mathematics. This is supplemented by numerous tables on infinite series, special functions, integrals, integral transformations, mathematical statistics, and fundamental constants in physics. The book offers a broad modern picture of mathematics starting from basic material up to more advanced topics. It emphasizes the relations between the different branches of mathematics and the applications of mathematics in engineering and the natural sciences. The book addresses students in engineering, mathematics, computer science, natural sciences, high-school teachers, as well as a broad spectrum of practitioners in industry and professional researchers. A comprehensive table at the end of the handbook embeds the history of mathematics into the history of human culture. The bibliography represents a comprehensive collection of the contemporary standard literature in the main fields of mathematics.
#4 Handbook of Mathematics, 5th Edition
This guidebook to mathematics contains in handbook form the fundamental working knowledge of mathematics which is needed as an everyday guide for working scientists and engineers, as well as for students. Easy to understand, and convenient to use, this guidebook gives concisely the information necessary to evaluate most problems which occur in concrete applications.
#5 Production Factor Mathematics
Mathematics as a production factor or driving force for innovation? Those, who want to know and understand why mathematics is deeply involved in the design of products, the layout of production processes and supply chains will find this book an indispensable and rich source. Describing the interplay between mathematical and engineering sciences the book focusses on questions like How can mathematics improve to the improvement of technological processes and products? What is happening already? Where are the deficits? What can we expect for the future? 19 articles written by mixed teams of authors of engineering, industry and mathematics offer a fascinating insight of the interaction between mathematics and engineering
#6 Elements of Mathematics: General Topology, Pt.1
Most branches of mathematics involve structures of a type different from the algebraic structures (groups, rings, fields, etc.) which are the subject of the book Algebra of this series: namely structures which give a mathematical content to the intuitive notions of limit, continuity and neighborhood. These structures are the subject matter of the present book.
#9 U.G. Mathematics ; Short Questions and Answers
This book titled U. C. Mathematics (Short Questions and Answers) has been written for the students of B.A., B.Sc. general course for all Indian Universities. All the efforts have been made to make this book useful for other competitive examinations. Though the book is of general nature but an effort has been made to cover up most of the topics prescribed for B.A.,B.Sc. general courses.
#10 Shelly Frei - Teaching Mathematics Today
Enhance your professional resource library with this up-to-date, research-based guide featuring best practices based on solid research and proven methodology. This resource equips teachers with sound educational strategies and resources, and provides interactive elements while promoting a thorough understanding of mathematics and its importance. Designed for new teachers, pre-service educators, or anyone interested in current educational theory and best practices, this book is perfect for staff development sessions.
#12 Embracing Mathematics
This alternative textbook for courses on teaching mathematics asks teachers and prospective teachers to reflect on their relationships with mathematics and how these relationships influence their teaching and the experiences of their students. Applicable to all levels of schooling, the book covers basic topics such as planning and assessment, classroom management, and organization of classroom experiences; it also introduces some novel approaches to teaching mathematics, such as psychoanalytic perspectives and post-modern conceptions of curriculum.
'Total Recording' is the complete and comprehensive guide to audio production and engineering musical recordings in all genres. Written by Grammy-nominated recording engineer/composer/author/industry consultant/researcher/guru Dave Moulton, it's the product of over three decades worth of professional experience.
Along with its companion CD of audio examples, this thoroughly current package is chock-full of in -depth information for professionals and students alike about the entire recording process, including:
#16 The Language of Mathematics: Making the Invisible Visible
Keith Devlin is trying to be the Carl Sagan of mathematics, and he is succeeding. He writes: "Though the structures and patterns of mathematics reflect the structure of, and resonate in, the human mind every bit as much as do the structures and patterns of music, human beings have developed no mathematical equivalent of a pair of ears. Mathematics can be seen only with the eyes of the mind." All of his books are attempts to get around this problem, to "try to communicate to others some sense of what it is we experience--some sense of the simplicity, the precision, the purity, and the elegance that give the patterns of mathematics their aesthetic value."
This textbook offers an accessible and highly-effective approach which is characterised by the combination of the textbook with a detailed study guide on an accompanying CD. This study guide divides the whole learning task into small units which the student is very likely to master successfully. Thus he or she is asked to read and study a limited section of the textbook and then to return to the study guide. Through interactive learning with the study guide, the results are controlled, monitored and deepened by graded questions, exercises, repetitions and finally by problems and applications of the content studied. Since the degree of difficulties is slowly rising, the students gain confidence and experience their own progress in mathematical competence thus fostering motivation. Furthermore in case of learning difficulties, he or she is given supplementary explanations and, in case of individual needs, supplementary exercises and applications. So the sequence of the studies is individualized according to the individualís performance and needs and can be regarded as a total tutorial experience.
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Synopsis
Employing a practical and empathetic approach, this mathematics resource advocates for a new teaching methodology that removes any anxiety associated with math. Covering topics such as addition, multiplication tables, fractions, probabilities, algebra, and ratios, this book enables readers to feel in control and to understand, for the first time, how math can be used in one's daily life. With techniques that link facts, procedures, and ideas, both teachers and students will find this easily accessible work provides a stable foundation upon which an advanced understanding of mathematics can be built. | 677.169 | 1 |
Given composite figures (combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles), students will be able to determine expressions for the area as well as calculate the area of the figure.
Given information about composite fiugres, the student will determine the area of composite 2-dimensional figures comprised of a combination of triangles and parallelograms using appropriate units of measure.
Given a graph and/or verbal description of a situation (both continuous and discrete), the student will identify mathematical domains and ranges and determine reasonable domain and range values for the given situations.
Given a real-world situation that can be modeled by a linear function or a graph of a linear function, the student will determine and represent the reasonable domain and range of the linear function using inequalities. | 677.169 | 1 |
Description
This calculator supports 40 diffused mathematical functions. It is also able to record historical inputs and results and evaluate complicated expressions, which are either not supported or hard to input in most traditional calculators. This calculator fully supports all the android phones and tablets from Android 1.5 to Android 4.0.
Input:
User is able to input an expression using keyboard or function buttons, or copy a historical record from history screen or by clicking the output box. An expression is made up of operands, operators, functions and parameters. An operand or a parameter may not be a number, it can be an expression. Blank characters between expression elements, i.e. operands, operators, functions and parameters, do not affect calculation result. Capitalized and uncapitalized characters are both supported. An example of expression is pow(4.01,3.1) *0.0731 + sin(toRAD(sum(17, 21, avg(3.71, log(198.2), -9.99,112.7),abs(-11.2))))/(2!) .
If user is not sure how to use a function or a operator, he/she can press QuickHelp button and then press the function or operator button or type the function name or the operator by keyboard.
Output:
The output text box shows user the result of calculation. If syntax of the input expression is incorrect, output box shows error message. User can also place last calculation result in the input text box by clicking the output box. If the last calculation result includes error, expression will be put into input box instead of result.
Settings:
1. Bits of precision. This setting determines how many effective bits after decimal points should be shown. For example, if bits of precision is 4, 0.003204876 will be rounded to 0.003205.
2. Scientific notation. This field sets the value range that scientific notation, e.g. 2.1 [10 -37], should be used to show result.
3. Record length. The record length gives out the number of historical calculations recorded by Exp Scientific Calculator.
History:
The history screen shows user the recent calculation history. By clicking one of the records, user is able to place either expression or result into input box, which saves typing time. Note that if result includes error, expression will be put into input by default.
My review
Review from
Reviews
4.4
49 total
5 34
4 8
3 2
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1 3
A Google User
Adverts OK expression calculator but the interface is clunky and adverts make it worse. Calc Etc is much better and is free (no ads either)
A Google User
Well designed calculator. Best to me comparing to others.
A Google User
Expect to see graph support.
A Google User
Good tool for students
A Google User
Great tool. Like it
A Google User
Good but can be improved Says it does complex numbers - but although exp(i) works, log(i) sin(i) etc do not. I have moved to using addi instead.
User reviews
A Google User June 22, 2012
Adverts OK expression calculator but the interface is clunky and adverts make it worse. Calc Etc is much better and is free (no ads either) | 677.169 | 1 |
Developed for test-takers who need a refresher, GMAT Foundations of Math provides a user-friendly review of basic math concepts crucial for GMAT success. Designed to be user-friendly for all students, GMAT Foundations of Math provides: * Easy-to-follow explanations of fundamental math concepts * Step-by-step application of concepts to example... more...
For parents who want to be fully informed about essential information that high school students need to know in order to plan for, apply to, and be successful in college, this guide details all stages of the process. It explains the terminology, policies, and procedures for successful college application, and will empower parents to manage their relationship... more... | 677.169 | 1 |
Details about Mathematics:
MATHEMATICS: ITS POWER AND UTILITY, 9e, combines a uniquely practical focus on real-world problem solving with a thorough and effective grounding in basic concepts and skills, allowing even the least-interested or worst-prepared student to appreciate the beauty and value of math while mastering course material. The first section of the book explores the power and historic impact of mathematics and helps you harness that power by developing an effective approach to problem solving. The second section builds on this foundation by applying math concepts to a wide variety of real-life situations--including money management; handling of credit cards; inflation; purchase of a car or home; use of probability, statistics, and surveys; and many more topics of interest. MATHEMATICS: ITS POWER AND UTILITY, 9e, assumes only a basic working knowledge of arithmetic, making it effective even if you have no exposure to algebra or little confidence in your current math skills.
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Rent Mathematics 9th edition today, or search our site for other textbooks by Karl J. Smith. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Brooks Cole.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our tutors now. | 677.169 | 1 |
Taalman and Kohn's Calculus offers a streamlined, structured exposition of calculus combining the clarity of classic textbooks with a modern perspective on concepts, skills, applications, and theory. Its uncluttered design... | 677.169 | 1 |
The first edition of this book sold more than 100,000 copies-and this new edition will show you why! Schaum's Outline of Discrete Mathematics shows you step by step how to solve the kind of problems you're going to find on your exams. And this new edition features all the latest applications of discrete mathematics to computer science!
New Senior Mathematics Extension 1 for Years 11&12 covers all aspects of the Extension 1 Mathematics course for Year 11&12. We've completely updated the series for today's classrooms, continuing the much-loved approach to deliver mathematical rigour with challenging student questions.
This book provides a complete course for first-year engineering mathematics. Whichever field of engineering you are studying, you will be most likely to require knowledge of the mathematics presented in this textbook. Taking a thorough approach, the authors put the concepts into an engineering context, so you can understand the relevance of mathematical techniques presented and gain a fuller appreciation of how to draw upon them throughout your studies.
Expert Guidance on the Math Needed for 3D Game Programming. Developed from the authors' popular Game Developers Conference (GDC) tutorial, Essential Mathematics for Games and Interactive Applications, Third Edition illustrates the importance of mathematics in 3D programming.
Ideal for courses that require the use of a graphing calculator, ALGEBRA AND TRIGONOMETRY: REAL MATHEMATICS, REAL PEOPLE, 6th Edition, features quality exercises, interesting applications, and innovative resources to help you succeed. Retaining the book's emphasis on student support, selected examples include notations directing students to previous sections where they can review concepts and skills needed to master the material at hand. The book also achieves accessibility through careful writing and design including examples with detailed solutions that begin and end on the same page, which maximizes readability. Similarly, side-by-side solutions show algebraic, graphical, and numerical representations of the mathematics and support a variety of learning styles. Reflecting its new subtitle, this significant revision focuses more than ever on showing readers the relevance of mathematics in their lives and future careers. | 677.169 | 1 |
Study Guide for Stewart/Redlin/Watson's College Algebra, 5thReinforces student understanding with detailed explanations, worked-out examples, and practice problems. Lists key ideas to master and builds problem-solving skills. There is a section in the Study Guide corresponding to each section in the text. | 677.169 | 1 |
Product Description
Singapore Math's Primary Mathematics, Standards Edition Workbooks are consumable and should be used in conjunction with the corresponding textbooks; these Standard Edition workbooks are not compatible with other Singapore editions (e.g. U.S. 3rd Ed.). Containing independent student exercises, workbooks provide the practice essential to skill mastery; each chapter in the textbook has corresponding problems in the workbook while review exercises help students to consolidate concepts they've learned previously. A variety of exercises are presented in the concrete> pictorial>abstract approach, covering decimals, measures and volume, percentage, angles, average & rate, data analysis, and algebra. 160 pages, softcover. Grades 5-6. | 677.169 | 1 |
Synopsis
Mathematical Topics: Mathematics with Understanding, Book 1 focuses on the approaches in teaching mathematics. The book first offers information on the aims of modern approaches in teaching mathematics. The text discusses the language of sets. Set notation, empty, disjoint, and universal sets; union and intersection of two sets; Venn diagrams; and complements of sets are clarified. The selection also reviews relations and sorting, including equivalence relations, equivalent sets, partitioning sets, and number games. Recording of numbers and use of different bases are also discussed. Topics include multiplication in different bases, decomposition, equal addition, bicimals, and operations using bases other than 10. The text also focuses on open sentences, number facts, and pictorial representations. The number line, collection of data, bar charts, block graphs, pi graphs, tally charts, and line graphs are discussed. The book also takes a look at the processes of addition, subtraction, multiplication, and division. The selection is a reliable reference for readers interested inMathematics with Understanding: The Commonwealth and International Library: Mathematical Topics, Volume 1
by Fletcher | 677.169 | 1 |
Au sujet du livre
Description :
Language: English . Brand New Book. This text is designed to provide a mathematically rigorous, comprehensive coverage of topics and applications, while still being accessible to students. Calter/Calter focuses on developing students critical thinking skills as well as improving their proficiency in a broad range of technical math topics such as algebra, linear equations, functions, and integrals. Using abundant examples and graphics throughout the text, this edition provides several features to help students visualize problems and better understand the concepts. Calter/Calter has been praised for its real-life and engineering-oriented applications. The sixth edition of Technical Mathematics has added back in popular topics including statistics and line graphing in order to provide a comprehensive coverage of topics and applications--everything the technical student may need is included, with the emphasis always on clarity and practical applications. WileyPLUS, an online teaching and learning environment that integrates the entire digital text, will be available with this edition. N° de réf. du libraire BZV97804704647 | 677.169 | 1 |
Find a Concord taught students and taken them through both the theoretical and practical applications. Theory is important to understand how to visualize. Applications are critical for hands-on understanding and to build the mathematics into our physical world | 677.169 | 1 |
Topics in Contemporary Mathematics for the Math for Liberal Arts course, Topics in Contemporary Mathematics helps students see math at work in the world by presenting problem solving in purposeful and meaningful contexts. Many of the problems in the text demonstrate how math relates to subjects--such as sociology, psychology, business, and technology--that generally interest students. | 677.169 | 1 |
Details about Graphing Calculator Guide for the TI-83 to accompany Functions Modeling Change: A Preparation for Calculus, 2nd Edition:
Work more effectively using this Graphing Calculator Guide for the TI-83! This guide is designed to accompany Connally's Functions Modeling Change: A Preparation for Calculus, 2nd Edition. It instructs students on how to utilize their TI-83/82 calculators with this textbook. The TI-86/85 and TI-89 are also discussed. This guide contains samples, tips, and trouble shooting sections to answer students' questions. From the Calculus Consortium based at Harvard University, Functions Modeling Change, 2nd Edition prepares readers for the study of calculus, presenting families of functions as models for change. These materials stress conceptual understanding and multiple ways of representing mathematical ideas. The focus throughout is on those topics that are essential to the study of calculus and these topics are treated in depth.
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Rent Graphing Calculator Guide for the TI-83 to accompany Functions Modeling Change: A Preparation for Calculus, 2nd Edition 2nd edition today, or search our site for other textbooks by Jerry Morris. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Wiley.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our Calculus tutors now. | 677.169 | 1 |
A general overview of the Singapore model method of math instruction--- and an invaluable resource for teachers! This thorough introduction covers all levels of teaching for ages 8 to 12. Learn how the innovative model approach helps students first visualize and conceptualize problems so that they can better formulate the algebraic expressions necessary to solve them. 136 pages, softcover.
This book focuses exclusively on K-8 mathematics, developing elementary mathematics at the level of teacher knowledge. Themes focus on how the nature of a mathematics topic suggests an order for developing it in the classroom, how topics are developed through 'teaching sequences', and how math builds on itself. Originally designed as a textbook for teachers, this book is divided into short sections, each with a single topic and homework set. The homework sets were designed with the intention that all or most of the exercises will be assigned; many of the questions involve solving problems in actual elementary school textbooks. Others involve studying the textbook - carefully reading a section of the book and answering questions about the mathematics being presented, with attention to the 5A U.S. Edition. 140 pages, spiralbound softcover.A U.S. Edition. 137 pages, spiralbound softcoverB U.S. Edition. 146 pages, spiralbound softcover.
This Student Packet kit contains the Math in Focus books needed by the 2nd Grade student, and includes: Workbook 2A, 256 pages, softcover. Workbook 2B, 264 pages, softcover. Textbook 2A, 305 pages, hardcover. Textbook 2B, 328 pages, hardcover. Assessments Book, 182 Grades 7-10; 444 pages, hardcover. Teacher involvement is generally required. Textbook 1 is recommended for 7th The consumable student workbooks...
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The New Elementary Mathematics Solution Manuals provide step-by-step solutions to the textbook's Exercises, Revision Exercises, Miscellaneous Exercises and Assessment Papers. It does not provide answers or solutions to the Class Activities, the Challenger and Problem Solving Exercises, or the Investigation sections, to answers are found in the Teacher's Guide/Teacher's Manual. The New Elementary Mathematics solution manuals do not cover exercises found in the workbooks. 247 pages, softcover.
This Student Packet kit contains the Math in Focus books needed by the student in 1st Grade, and includes: Workbook 1A, 263 pages, softcover. Workbook 1B, 263 pages, softcover. Textbook 1A, 278 pages, hardcover. Textbook 1B, 318 pages, hardcover. Assessments Book, 184...
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This Student Packet kit contains the Math in Focus books needed by the 3rd Grade student, and includes: Workbook 3A, 200 pages, softcover. Workbook 3B, 264 pages, softcover. Textbook 3A, 286 pages, hardcover. Textbook 3B, 428 pages, hardcover. Assessments Book, 191 4th Grade student, and includes: Workbook 4A, 193 pages, softcover. Workbook 4B, 170 pages, softcover. Textbook 4A, 300 pages, hardcover. Textbook 4B, 266 pages, hardcover. Assessments Book, 151 5th Grade student, and includes: Workbook 5A, 264 pages, softcover. Workbook 5B, 218 pages, softcover. Textbook 5A, 348 pages, hardcover. Textbook 5B, 334 pages, hardcover. Assessments Book, 178 Textbook 2 is recommended for Grade 8; 444 pages, hardcover. Teacher involvement is generally required. New...
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Accompanying the Math in Focus Grade 3 Tests cover multiplication tables, bar models, money, bar graphs, fractions, measurement, angles, area, and other topics covered in the third grade curriculum. Answers and a student record sheet are included. 191 pages, paperback; tests are perforated, three-hole-punched , and reproducible. Please Note: This product isAccompanying the Math in Focus Grade 2 Answers and a student record sheet are included. 182 pages, paperback; tests are perforated, three-hole-punched , and reproducible. Please Note: This product is only available for purchase by homeschools, consumers, and public institutions.
This teacher's edition is designed to be used during the second half of the Grade 1 year. It features objectives, materials needed, and a list of page numbers; teaching instructions are provided in a step-by-step, listed format that's incredibly easy to see and read. Reduced student pages are given with the correct answers overlaid. 304 pages, indexed, spiralbound, soft front-cover, hard back-cover. Please Note: This product is only available for purchase by homeschools, consumers, and public i nstitutions. second | 677.169 | 1 |
97803879875ifolds play an important role in topology, geometry, complex analysis, algebra, and classical mechanics. Learning manifolds differs from most other introductory mathematics in that the subject matter is often completely unfamiliar. This introduction guides readers by explaining the roles manifolds play in diverse branches of mathematics and physics. The book begins with the basics of general topology and gently moves to manifolds, the fundamental group, and covering spaces | 677.169 | 1 |
288762847625855
Publication Year:
2011
ISBN-13:
9780547625850
Language:
English
Author:
Stephen Douglas Hake
Educational Level:
Trade
ISBN:
9780547625850
Detailed item info
Synopsis
Students who are interested in taking Saxon Homeschool Geometry course may chose the 4th edition Algebra 1 and Algebra 2 courses, which are designed to accompany Geometry. Featuring the same incremental approach that is the hallmark of the Saxon program, the 4th Edition Algebra 1 and Algebra 2 textbooks feature more algebra and precalculus content and fewer geometry lessons than their 3rd edition counterparts | 677.169 | 1 |
This is the course homepage that also serves as the syllabus
for the course. Here you will find
our weekly schedule and updates on scheduling matters.
The Mathematics Department also has a general
information page
on this course. Deadlines from the Registrar's page.
Course description
Math 431 is an introduction to
probability theory, the
part of
mathematics that studies random phenomena. We model simple random experiments mathematically
and learn techniques for studying these models. Topics covered
include methods of counting
(combinatorics), axioms of probability,
random variables, the most important discrete and
continuous probability distributions, expectations, moment
generating functions,
conditional probability and conditional expectations,
multivariate distributions, Markov's and Chebyshev's inequalities,
laws of large numbers, and the central limit theorem.
Probability theory is ubiquitous
in natural science, social science and engineering,
so this course can be valuable in conjunction with many different
majors. 431 is not a course in statistics.
Statistics
is a discipline mainly concerned
with analyzing and representing data. Probability theory forms the mathematical
foundation of statistics, but the two disciplines are separate.
From a broad intellectual perspective, probability is one of the
core areas of mathematics with its own distinct style of reasoning.
Among the other core areas are analysis, algebra,
geometry/topology, logic and computation.
To go beyond 431 in probability you should take next
521 Analysis for basic mathematical groundwork. After that you have several options for subsequent undergraduate probability courses.
531 Probability Theory is a proof-based introduction to probability that covers the material of 431 in a deeper way and tackles some additional topics. There are two courses on stochastic processes:
632 Introduction to Stochastic Processes and
635 Introduction to Brownian Motion and Stochastic Calculus.
Prerequisites
To be technically prepared for Math 431 one needs to be
comfortable with the language of sets and calculus, including multivariable
calculus, and be ready for abstract reasoning. Probability theory can seem
very hard in the beginning, even after success in past math courses.
Textbook
The course follows lecture notes authored by David Anderson, Timo Seppäläinen, and Benedek Valkó. These will be provided to the students at no cost.
Learn@UW
We use Learn@UW to post homework assignments, solutions to homework, and lecture notes.
Library reserves
The following books are on reserve in the library. Going over different presentations of the material can be helpful. Especially the first two are good sources for additional practice problems.
A First Course in Probability, by Sheldon Ross.
Probability, by Jim Pitman.
Probability, Statistics, and Stochastic Processes, by Peter Olofsson.
Probability and Random Processes, by Geoffrey Grimmett.
You can find old 431 exams in the
Math Library Course Reserves. But be aware that old exams are not necessarily a perfect fit for this semester's course.
Piazza
We use Piazza for class discussion.
Post your math questions on Piazza. Students and instructors
from all sections of 431 have access to the same
page and can ask and answer questions. If you have any problems or
feedback for the developers, email team@piazza.com.
Quizzes
There will be quizzes at the end of Wednesday classes during weeks 2-5, to get used to problem solving under time pressure. We can have quizzes later on in the semester too in order to have some testing in the long period between the two midterm exams.
Instructions for homework
Homework is collected at the beginning of the class period on the due date, or alternately can be brought to the instructor's office or mailbox by 2 PM on the due date.
No late papers will be accepted. You can bring the homework earlier to
the instructor's office or mailbox.
Observe rules of academic integrity.
Handing in plagiarized
work, whether copied from a fellow student or off the web, is not acceptable.
Plagiarism cases will lead to sanctions.
Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Answers should be simplified as much as possible.
If the answer is a simple fraction or expression,
a decimal answers from a calculator is not necessary. But for some
exercises you need a calculator to get the final answer.
Answers to some exercises are in the back of
the book, so answers alone carry no credit. It's all in the reasoning
you write down.
Put problems in the correct order and staple your pages together.
Do not use paper torn out of a binder.
Be neat. There should not be text crossed out.
Recopy your problems. Do not hand in your rough draft or first attempt.
Papers that are messy, disorganized or unreadable cannot be graded. | 677.169 | 1 |
Browse by
BASIC MATHEMATICS FOR BUSINESS
OBJECTIVES : MATHEMATICS &
STATISTICS
Develop Mathematical skills needed for • To talk about problems that are formulated in mathematical terms • Not to make you expert Mathematicians but help you in the understanding and interpretation of the results given by experts. • Project Work : All areas of Management
• Have access to large amounts of information – Use available information to make better decisions • Mathematical models are used in decision making , so the practitioner must be able to think and write in mathematical terms • In management environment it will be necessary to translate detailed mathematical reasoning into something closer to everyday language – Able to translate to and from Mathematics
MEANING OF QUANTITATIVE TECHNQUES • Group of statistical and operations research techniques. • Requires preliminary knowledge of mathematics . • Quantitative approach in decision making requires that problems be defined , analyzed and solved in a conscious , rational, systematic and scientific manner based on data , facts ,information and logic , not on guesses
• Common characteristic of all types of quantitative techniques is that numbers , symbols or mathematical formulae are used to represent the models of reality. • Quantitative techniques have made valuable contribution towards arriving at an effective decision in various functional areas of management - Marketing , Finance , Production and Personnel. | 677.169 | 1 |
Math Concepts for Food Engineering / Edition 2/b>
Paperback
Overview illustrates the importance of mathematical concepts and relates them to the study of food engineering.
New to the Second Edition
· Straightforward explanations of basic balance and transport principles used in food engineering
· Various exercises throughout that use spreadsheets, which are available on the publisher's website
· A chapter on mass transfer
· A mathematical skills screening quiz
· A simple units-conversion page
This new edition is student testedWhat students have to say"... a must-have for any student in food science engineering ... teaches students how to think like an engineer. Each chapter provides meaningful applications ... shows students both the approach and the mathematical solution needed to solve example problems." "This workbook not only taught me which mathematical equations are needed to solve various food engineering problems, it helped me understand the analysis and approach needed when solving any engineering problem. The practice questions helped me gain confidence in my problem-solving skills, and they make the coursework more interesting by relating it to real-world problems."
Builds Mathematical Confidence
This text helps assess the mathematical reasoning skills of food science and engineering students and offers assistance for those who need a refresher. It supplies the necessary material to solve simple engineering problems so that students are prepared to face more rigorous challenges in class. | 677.169 | 1 |
.
"Geometry, Projective"@en .
.
"Symbolic and Algebraic Manipulation." .
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"2016-06-18" .
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"Lord" .
"Eric A." .
"Eric A. Lord" .
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"enk" .
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"Mathematics." .
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"Projective Geometry." .
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"9781447146315" .
"144714631X" .
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"Springer" .
.
"London" .
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"Algebra -- Data processing." .
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"Symmetry and pattern in projective geometry"@en .
"823241937" .
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"en" .
"2013" .
.
.
.
"Electronic books"@en .
.
.
.
"Symmetry and Pattern in Projective Geometry is a self-contained study of projective geometry which compares and contrasts the analytic and axiomatic methods. The analytic approach is based on homogeneous coordinates. Brief introductions to Plücker coordinates and Grassmann coordinates are also presented. This book looks carefully at linear, quadratic, cubic and quartic figures in two, three and higher dimensions. It deals at length with the extensions and consequences of basic theorems such as those of Pappus and Desargues. The emphasis throughout is on special configurations that have particularly interesting symmetry properties. The intricate and novel ideas of H S M Coxeter, who is considered one of the great geometers of the twentieth century, are also discussed throughout the text. The book concludes with a useful analysis of finite geometries and a description of some of the remarkable configurations discovered by Coxeter. This book will be appreciated by mathematics undergraduate students and those wishing to learn more about the subject of geometry. Subject and theorems that are often considered quite complicated are made accessible and presented in an easy-to-read and enjoyable manner."@en .
.
.
"823241937" .
.
.
"2013" .
.
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. | 677.169 | 1 |
Maple Tutorials
Maple is a computer algebra system (CAS)
available for
Windows like
Maxima or Mathematica.
There are various tutorials out there on how to use Maple, but not
many about using it for circuit analysis. I've attempted to change
that. | 677.169 | 1 |
This activity is an introduction to the concept of convergent infinite series using an iterative geometric construction. This activity has been adapted from the following article: Choppin, J. M. (1994... More: lessons, discussions, ratings, reviews,...
The applet shows graphically and numerically consecutive terms of a sequence or consecutive partial sums of a series. The user enters a formula for a sequence or a series and the terms are plotted. Ma... More: lessons, discussions, ratings, reviews,...
A very powerful graphing program that is also especially easy to use. You can graph functions in two or more dimensions using different kinds of coordinates. You can make animations and save as movies... More: lessons, discussions, ratings, reviews,...
The NA_WorkSheet Demo (beta version) is a collective aggregation of algorithms coded in Java that implements various Numerical Analysis solutions/techniques in one easy to use open source tool. The to... More: lessons, discussions, ratings, reviews,...
This is a Java graphing applet that can be used online or downloaded. The purpose it to construct dynamic graphs with parameters controlled by user defined sliders that can be saved as web pages or em... More: lessons, discussions, ratings, reviews,...
Guided activities with the Graph Explorer applet, designed to let students learning about quadratic functions explore: the parabolic shape of the graphs of quadratic functions; how coefficients affect... More: lessons, discussions, ratings, reviews,...
A page that allows you to print rectangular (cartesian) graph paper. You can control if grid lines are printed and
the position of the origin. By dragging the origin into any corner a single ... | 677.169 | 1 |
This invaluable collection of memoirs and reviews on scientific activities of the most prominent theoretical physicists belonging to the Landau School — Landau, Anselm, Gribov, Zeldovich, Kirzhnits, Migdal, Ter-Martirosyan and Larkin — are being published in English for the first time.The main goal is to acquaint readers with the life... more...
Resource allocation and power optimization is a new challenge in multimedia services in cellular communication systems. To provide a better end-user experience, the fourth generation (4G) standard Long Term Evolution/Long Term Evolution-Advanced (LTE/LTE-Advanced) has been developed for high-bandwidth mobile access to accommodate today?s data-heavy... more...
Teach lessons that suit the individual needs of your classroom with this SQA endorsed and flexibly structured resource that provides a suggested approach through all three units. National 4 Maths Teacher's Book, Answers and Assessment provides detailed answers to all questions contained in National 4 Maths, a book specifically written to meet the... more...
With this book, children can unlock the mysteries of maths and discover the wonder of numbers. Readers will discover incredible information, such as why zero is so useful; what a googol really is; why music, maths and space are connected; why bees prefer hexagons; how to tell the time on other planets; and much much more. From marvellous measurements... more... | 677.169 | 1 |
Linear Algebra: A First Course presents an introduction to the fascinating subject of linear algebra. As the title suggests,...
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Linear Algebra: A First Course presents an introduction to the fascinating subject of linear algebra. As the title suggests, this text is designed as a first course in linear algebra for students who have a reasonable understanding of basic algebra. Major topics of linear algebra are presented in detail, with proofs of important theorems provided. Connections to additional topics covered in advanced courses are introduced, in an effort to assist those students who are interested in continuing on in linear algebra. Each chapter begins with a list of desired outcomes which a student should be able to achieve upon completing the chapter. Throughout the text, examples and diagrams are given to reinforce ideas and provide guidance on how to approach various problems. Suggested exercises are given at the end of each section, and students are encouraged to work through a selection of these exercises.A brief review of complex numbers is given, which can serve as an introduction to anyone unfamiliar with the topic.Linear algebra is a wonderful and interesting subject, which should not be limited to a challenge of correct arithmetic. The use of a computer algebra system can be a great help in long and difficult computations. Some of the standard computations of linear algebra are easily done by the computer, including finding the reduced row-echelon form. While the use of a computer system is encouraged, it is not meant to be done without the student having an understanding of the First Course in Linear Algebra to your Bookmark Collection or Course ePortfolio
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This lesson uses WebImage, a Web-based customized version of ImageJ, to investigate the relationship between the...
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This lesson uses WebImage, a Web-based customized version of ImageJ, to investigate the relationship between the circumference and diameter of a circle. Images of circular objects are provided, and, by measuring diameter and circumference, students are able to obtain an approximate value of Pi. They also explore the history behind Pi and how to find its value Slice of Pi to your Bookmark Collection or Course ePortfolio
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This module comes from Visionlearning, an educational resource funded by the National Science Foundation. This particular...
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This module comes from Visionlearning, an educational resource funded by the National Science Foundation. This particular module introduces the history of wave theories, basic descriptions of waves and wave motion, and the concepts of wave speed and frequency. The module, available in Spanish, also includes illustrations, embedded definitions of key terms, additional links, and questions & Waves and Wave Motion: Describing Waves to your Bookmark Collection or Course ePortfolio
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This online exercise lets students practice vector addition. They choose the precision of the test by selecting a target...
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This online exercise lets students practice vector addition. They choose the precision of the test by selecting a target size, then estimate the sum of the two vectors by dragging and dropping a third arrow. Points are awarded; a higher degree of precision scores more points Adding Vectors to your Bookmark Collection or Course ePortfolio
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Think Stats is an introduction to Probability and Statistics for Python programmers.Think Stats emphasizes simple techniques...
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Think Stats is an introduction to Probability and Statistics for Python programmers.Think Stats emphasizes simple techniques you can use to explore real data sets and answer interesting questions. The book presents a case study using data from the National Institutes of Health. Readers are encouraged to work on a project with real datasets.If you have basic skills in Python, you can use them to learn concepts in probability and statistics. Think Stats is based on a Python library for probability distributions (PMFs and CDFs). Many of the exercises use short programs to run experiments and help readers develop Think Stats: Probability and Statistics for Programmers to your Bookmark Collection or Course ePortfolio
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Created by Lang Moore for the Connected Curriculum Project, the purpose of this module is to study solutions of...
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Created by Lang Moore for the Connected Curriculum Project, the purpose of this module is to study solutions of initial/boundary value problems for the one-dimensional wave equation One-Dimensional Wave Equation to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material The One-Dimensional Wave Spring Motion to your Bookmark Collection or Course ePortfolio
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Created by Lang Moore and David Smith for the Connected Curriculum Project, this module develops and explores the...
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Created by Lang Moore and David Smith for the Connected Curriculum Project, this module develops and explores the Lotka-Volterra model for predator-prey interactions as a prototypical first-order system of differential equations. This is part of a larger collection of Predator-Prey Models to your Bookmark Collection or Course ePortfolio
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Created by Lang Moore of the Connected Curriculum Project, the purpose of this module is to provide a qualitative...
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Created by Lang Moore of the Connected Curriculum Project, the purpose of this module is to provide a qualitative introduction to a simple second-order linear partial differential equation. In this module we will concentrate on graphical representations of the solutions of appropriate initial and boundary value problems; we examine symbolic solutions in other modules. This is one lesson within a the One-Dimensional Heat Equation to your Bookmark Collection or Course ePortfolio
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Created by Lang Moore for the Connected Curriculum Project, the purpose of this module is to study the Laplace transform and...
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Created by Lang Moore for the Connected Curriculum Project, the purpose of this module is to study the Laplace transform and use it to examine both an ordinary differential equation problem and a problem for the one-dimensional heat equation on a semi-infinite rod With the Laplace Transform to your Bookmark Collection or Course ePortfolio
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books.google.com - Basic Mathematics for Economists /FONT Economics students will welcome the new edition of this excellent textbook. Given that many students come into economics courses without having studied mathematics for a number of years, this clearly written book will help to develop quantitative skills in even... Mathematics for Economists
Basic Mathematics for Economists
Basic Mathematics for Economists /FONT Economics students will welcome the new edition of this excellent textbook. Given that many students come into economics courses without having studied mathematics for a number of years, this clearly written book will help to develop quantitative skills in even the least numerate student up to the required level for a general Economics or Business Studies course. All explanations of mathematical concepts are set out in the context of applications in economics.This new edition incorporates several new features, including new sections on: financial mathematics continuous growth· matrix algebraImproved pedagogical features, such as learning objectives and end of chapter questions, along with an overall example-led format and the use of Microsoft Excel for relevant applications mean that this textbook will continue to be a popular choice for both students and their lecturers. Mike Rosser is Principal Lecturer in Economics in the Business School at Coventry University. | 677.169 | 1 |
WolframAlpha
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Transcript of WolframAlpha
What exactly is WolframAlpha? Being a computational knowledge engine, WolframAlpha takes information that has been put into its system and is able to compute almost anything based on that information (Wolfram Alpha LLC, 2013). Goals "To make all systemic knowledge immediately computable and accessible to everyone (Wolfram Alpha LLC, 2013)." They are trying to gather all possible pieces of information so that WolframAlpha will be able to compute anything based on data. What WolframAlpha is NOT ESPN.COM CNN.COM Discovery.com Clarion.edu History Geography Mathematics Shouldn't we be hiding this from our students? Doesn't this ruin math homework? What about chemistry homework? And our students need to learn long division don't they???? WolframAlpha: The Search Engine That Does Not Search James D'Annibale May 12, 2013 Why do we assign homework? Which picture do we want? Cannot figure out the answer *Frustration helps no one. (Jezbeck, 2012) Able to check WolframAlpha *After trying on her own (Comas, 2007). I understand...but what exactly can WolframAlpha do? It really depends on your given topic. Topics on WolframAlpha include: Mathematics Weather Statistics & Data Analysis Places & Geography Physics People & History Chemistry Culture & Media Materials Music Money & Finance Many More WolframAlpha can help drive your instruction. When teaching the steps of a math problem, the step-by-step instructions could be right in front of your students. Also, if a student is "stuck" on a problem for homework, he/she can check to see on which step they made their mistake. Current weather and historic weather patterns can be displayed. We can also compare weather and climate for two locations Statistics could be as simple as the most common total when 5 dice are rolled. Statistics could be as hard as calculating sample sizes for a given population with a .03 margin of error. Finding Standard Deviations could also be double checked after students work on it themselves. Comparing cities for Geography class is a breeze. Latitude and Longitude Coordinates can easily be practiced with WolframAlpha. Compare oceanic information in a heartbeat. It would take a Google search of each ocean and a website for each ocean to get this data elsewhere. Students can view relief information of landforms like on the right and then compare two landforms such as the image on the left. Social Studies Teachers always want students to compare countries. If not using WolframAlpha, a student would need to go to CIA World Factbook twice. Students can check homework problems. Circuitry can now easily be explained. Image shows the flow of liquid around a cylinder. Now images like this don't have to come from the textbook. Pro Features Data Input Features
Image Input Features
CDF Interactivity Try it out and see what you find! Inputting data from a spreadsheet will allow WolframAlpha to analyze your data. This is merely a sampling of what was done with this data about fundraising by members of the House of Representatives This data being analyzed is from a consumer study looking at differences between male vs. female and consumers of three different states as well. Flight Prices to different locations WolframAlpha can edit the ladybug picture in any of these ways. WolframAlpha analyzes many pieces of information about the picture. Picture Editing Capabilities 3-D objects on WolframAlpha can be manipulated. Use the right and left buttons at the bottom of the presentation. Throughout the presentation, you can zoom in and out to see details by moving your mouse to the right of the screen and clicking on the + and = magnifying glasses. To view full-screen, click on the button at the bottom-right of the presentation. Conclusions: WolframAlpha should be used in three different ways. It can be used to help teachers deliver effective instruction. It can be used by students to help check their homework. Finally, it can be used to gain new information for projects. (Wolfram Alpha LLC, 2013) References Jazbeck. (Photographer). (2012). Untitled. [Digital Image]. Retrieved from Page, Jeremy. (Photographer). (2010). Finishing Homework. [Digital Image]. Retrieved from Wolfram Alpha LLC. (2013). About WolframAlpha. Retrieved from Wolfram Alpha LLC. (2013). Examples by topic. Retrieved from *Prezi does not allow websites to go without being hyperlinked Place your mouse in this image and press play. Place your mouse in this image and press play. | 677.169 | 1 |
28
Total Time: 4h 49m
Use: Watch Online & Download
Access Period: Unlimited
Created At: 07/29/2009
Last Updated At: 06/01/2011
In this extensive 28-lesson series, we'll work on techniques of integration. We'll first cover the usage of integration tables and u-substitution. Then, we'll move on to integration techniques and identities to use when approaching integration problems that contain trigonometric functions (in combination with each other, raised to a power, or raised to various powers and combined). After this, we'll move on to several lessons on partial fraction decomposition, integration by parts, trigonometric substitution and numerical integration with the trapezoidal rule.
Taught by Professor Edward Burger, this seriesmy test score went up a WHOPPING 40 points once i started watching Professor burgers videos! i do have one thing i did not get out of this and it is how to change the limits of the integral to radians in the very beginning...
I am an AP Calculus student and I was having the hardest time learning u substitution. Because of this video, I learned the same thing in 10 minutes that my teacher would have tried (unsuccessfully) to teach me for 45 minutes. :)
Great use of examples to explain how to find the derivative of a trig function. I hate taking the derivative of sec(x) and really appreciate this video for helping me learn how to live with it. :)
Below are the descriptions for each of the lessons included in the
series:
Calculus: An Introduction to the Integral Table Making u-Substitutions
In this lesson, you will learn about the u-substitution expression, how to use it, and how to use it under a radical when doing integration problems. The u-substitution is a technique to use when evaluation integrals or antidifferentiation problems. U-substitution is effectively the integral version of the chain rule given that it is the chain rule in reverse. Professor Burger will walk you through what to look for when choosing the u-substitution expression anytime you are faced with integration substitutionIN & COS Intro vectorine and Cosine
In this lesson, you will review the integration of sin^3(x) and sin^3(x)*cos^3(x). By substituting for equivalent trigonometric identities, we are able to use the Pythagorean identity for sine and consine (sin^2X+cos^2X = 1) and u-substitution (u-sub) to arrive at the antiderivative of both of these trig expressions. Professor Burger will walk you through the proof of the associated identities derived from this type of manipulation and explain how to recognize other problems which can be solved in the same mannerrate Even & Odd Powers of SIN, COS of Other Trig Functions
In this lesson, you will learn how to integrate tangent, cotangent and secant. You will see how solving tangent and cotangent antidifferentiation problems will generally involve expressing them in terms of sine and cosine and then applying u-substitution to the problem. Integrating secant and cosecant, on the other hand, involves multiplication by a specific fraction that is equal to one. Professor Burger will walk you through what the integrals are of these trigonometric identities as well as how one would arrive at them Odd TAN Power, Any SEC Even SEC Power, Any TAN Finding Partial Fraction Decompositionsractions including Long Division One Two Distinct and Repeated Quadratic Factors Universityraction of Transcendental Fctn. Introduction to Integration by Parts & the Natural Log Examples Williams Application Manipulation, Converting Radicals to Trig Expressions to Integrate Radicals differentialitutions on Rational Powers Overview of Trig Substitution Strategy 1 2 Deriving Example ofSupplementary Files:
Once you purchase this series you will have access to these files: | 677.169 | 1 |
Product details
ISBN-13: 9781932410303
ISBN: 1932410309
Publication Date: 2002
Publisher: American Book Company
AUTHOR
by Unknown Author
SUMMARY
REA's new Mathematics test prep for the Ohio Graduation Test (OGT) provides all the instruction and practice that students need to excel. Passing this exam is required to receive a high school diploma. The book's review covers the areas articulated in Ohio's Academic Content Standards for Mathematics: Number, Number Sense, and Operations; Measurement; Geometry and Spatial Sense; Patterns, Functions, and Algebra; and Data Analysis and Probability . Includes two full-length practice tests and complete explanations of all answers. Details: - All materials in this book are aligned with Ohio's Academic Content Standards - Two full-length practice tests- Lessons enhance all skills necessary for the exam- Confidence-building tips reduce test anxiety and boost test-day readiness"REA ... Real review, Real practice, Real results."Ohio Graduation Test Mathematics Review, was published 2002 under ISBN 9781932410303 and ISBN 1932410309 | 677.169 | 1 |
Details about Mathematical Olympiad Treasures:
Mathematical Olympiad Treasures aims at building a bridge between ordinary high school exercises and more sophisticated, intricate and abstract concepts in undergraduate mathematics. The book contains a stimulating collection of problems in the subjects of algebra, geometry, trigonometry, number theory and combinatorics. While it may be considered a sequel to "Mathematical Olympiad Challenges," the focus is on engaging a wider audience to apply techniques and strategies to real-world problems. Throughout the book students are encouraged to express their ideas, conjectures, and conclusions in writing. The goal is to help readers develop a host of new mathematical tools that will be useful beyond the classroom and in a number of disciplines.
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Rent Mathematical Olympiad Treasures 2nd edition today, or search our site for other textbooks by Titu Andreescu. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Birkhauser Verlag GmbH.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our tutors now. | 677.169 | 1 |
Assignments
"Reasonable'' collaboration is permitted, but you should not just copy someone else's solution or look up a solution from an outside source. We will (subjectively) indicate the difficulty level of each problem as follows: | 677.169 | 1 |
Browse related Subjects
These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs.
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These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs | 677.169 | 1 |
2. Large Determinants
Evaluating large determinants can be tedious and we will use computers wherever possible (see box at right). But if you have to do large determinants on paper, here's how.
Expanding 4×4 Determinants
Computers and Determinants
Now that we have powerful tools like MathCad,
Scientific Notebook, Mathematica, Matlab, Maple, etc, we
should concentrate on understanding the uses of mathematics and
not so much on its mechanics.
Students spend hours multiplying out large determinants and a lot of the time is spent figuring out where errors were made. And for what? That time is better spent using a computer (or calculator) to solve it directly for us and using the remaining time learning how to apply it.
Mostly, we will use Computer Algebra Systems to find
large determinants. | 677.169 | 1 |
Thinking algebraically helps us to develop different ways of representing real-world situations. You may have chosen to use a table to represent the situation with Eric the Sheep, for example, or you may have tried to describe the process in words or as an equation. Representations of mathematical ideas enable us to use mathematics as a way of communicating with others. Note 6
The next set of problems involves qualitative graphs, representations that focus on the important general features of a situation. Looking at qualitative graphs helps us to make sense of a situation and allows us to make predictions and draw conclusions. In this way, even a simple qualitative graph can communicate a great deal of information.
Making sense of graphs and drawing conclusions from them make it possible for us to understand our world and the information around us. If you look at a newspaper, a financial report, or virtually any statistical information, you'll find a graph. The ability to interpret these graphs is essential to understanding the information contained within it. Note 7 | 677.169 | 1 |
umerical Methods for Ordinary Differential Equations is a self-contained introduction to a fundamental field of numerical analysis and scientific computation. Written for undergraduate students with a mathematical background, this book focuses on the analysis of numerical methods without losing sight of the practical nature of the subject. It covers the topics traditionally treated in a first course, but also highlights new and emerging themes. Chapters are broken down into 'lecture' size pieces, motivated and illustrted by numerous theoretical and computational examples. Over 200 exercises are provided and these are starred according to their degree of difficulty. Solutions to all exercises are available to authorized instructors | 677.169 | 1 |
GRE and GMAT Math - So Easy a Child Could Do It
Quantitative Problem Solving: So Easy a Child Could Do It
4.6Not a child genius? Not a problem. My name is Corey and I'm here to guide you through an interactive course on quantitative problem solving for mastering the math section of the GRE and GMAT exams. These problems are not duplicates of the long, tedious, subpar GRE and GMAT course books that you've already purchased (such as the Barron's or Manhattan series which deviate from the actual GRE given by ETS and assume that you have infinite preparation time to work through their long lists of questions), so you will be getting brand new problems modeled after REAL GRE and GMAT questions that will challenge you in a unique way. This course is vitally important for anyone planning to apply to most graduate programs and the GRE and GMAT math questions come fully annotated solutions. This course is also ideal for high school students preparing for the SAT, as many of the harder math questions on the SAT will seem simple after mastering these questions. Feel free to preview sample solutions so that you have an idea of what to expect in the rest of the course. Thank you for your time and I look forward to helping you maximize your math proficiency.
Feel free submit questions through the course or directly to my email (hokiesalum[AT_symbol]]gmail[DOT]com ; include Udemy in the subject line). I want to do my best to make sure that you succeed! :-)
What are the requirements?
High School Math
What am I going to get from this course?
Master GRE and GMAT math for quantitative problem solving
What is the target audience?
Prospective graduate students
Prospective business school students
Prospective college students cover the derivation of the formulas for
permutations and combinations.
Permutations represent selecting objects from a set of objects and the
order of selection IS important.
Combinations represent selecting objects from a set of objects and the
order of selection is NOT important.
Prime factorization is when you split a number into its prime factors. A prime number is a number that is only divisible by 1 and itself. 2 is defined as the first prime number. The factors of a number are the numbers that evenly divide into that number. For example, the factors of 10 are 1, 2, 5, and 10. The prime factorization can be used to find the greatest common factor (GCF) or the least common multiple (LCM) of two or more numbers. The GCF can be used to simplify fractions. The LCM can be used to find a common denominator to add or subtract fractions.
The most important exponent rules are covered in this lecture. The most important rules are: 1. Add the exponents when you have two numbers with the same base multiplied together. 2. Subtract the exponents when you have two numbers with the same base divided by each other. 3. Multiply the exponents when you have an exponent raised to an exponent. 4. Negative exponents can be made positive by moving the number from the numerator to the denominator or vice versa. 5. Radicals can be changed into fractional exponents and vice versa.
Combined rate / work problems are used to add or subtract the rates of different people when they are doing the same task.
Remember that when you add or subtract the fractions, the time component has to be in the denominator, on the bottom. For example, 5 miles per hour + 10 miles per hour = 15 miles per hour. Do NOT do: 1 hour per 5 miles + 1 hour per 10 miles = 3 hours per 15 miles.
This lecture covers the most important subtopics for dealing with triangles. 1. A triangle has three sides and the sum of its angles is 180 degrees. 2. The area of a triangle is 1/2 * base * height. 3. Pythagorean's Theorem for right triangles: c^2 = a^2 + b^2. 4. An exterior angle for a triangle is equal to the sum of the other two angles. 5. There are two special triangles in terms of angles:
This lecture covers fraction arithmetic. 1. To multiply fractions, multiply the numerators and denominators straight across. 2. To divide fractions, change the fraction being divided into its inverse, and then multiply the fractions straight across. 3. To add and/or subtract fractions, find the LCM (least common multiple) of the denominators, change all fractions so that they share this LCM, then perform the addition and subtraction operations only on the numerators, not the denominators.
1. Always use your pencil and paper. 2. Add or subtract any number from both sides to simplify the problem. Usually this is useful if you have the same number, variable, or equivalent expression on both sides. 3.
Multiply of divide any positive number, variable, or expression from
both sides to simplify the problem. Sometimes this is optional, but
other times this is essential for solving a problem. Remember that it
must be a positive number, variable, or expression, as doing this with a
negative number change the problem being asked. 4. Have
confidence. The GRE test is also a psychological test of your math
ability and confidence level. Not being confident in your math skills
leads to double and triple checking answers, thereby taking away time
from the other questions in the test.
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Instructor Biography
Graduated summa cum laude from Virginia Tech with a major in chemical engineering (in-major GPA: 3.94) and a minor in chemistry. Completed written and oral doctoral qualifiers in chemical engineering at MIT. Extensive research experience with the US Army Countermining Division, DuPont, and MIT.
Adjunct Professor (New England College of Business). Tutoring prodigy...I love my job and fully invest myself in my clients, as I love their success even more. Extensive tutoring experience (300+ clients in the past 4 years), particularly in math (with a specialty in calculus through differential equations), physics, chemistry (general and organic), statistics (high school through upper graduate level, with a specialty in SPSS), academic and research writing, and test preparation (GRE, GMAT, MCAT, PCAT, DAT, SAT, ACT, ASVAB, SSAT, and MTEL math). I also create electronic flashcards to help my clients study, compatible with smartphones and iPads. | 677.169 | 1 |
Find an American CanyonMy goal is to impart skills, not just information. I break down Prealgebra into the major classes of fundamental problems that will be presented. From each fundamental problem, we will work from the base easiest example and then increase in complexity. | 677.169 | 1 |
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Set Theory
Semester:
Spring
Year:
2013
Subject Name:
Mathematics
Course Number:
380
Institution:
Amherst College
Most mathematicians consider set theory to be the foundation of mathematics, because everything that is studied in mathematics can be defined in terms of the concepts of set theory, and all the theorems of mathematics can be proven from the axioms of set theory. This course will begin with the axiomatization of set theory that was developed by Ernst Zermelo and Abraham Fraenkel in the early part of the twentieth century. We will then see how all of the number systems used in mathematics are defined in set theory, and how the fundamental properties of these number systems can be proven from the Zermelo-Fraenkel axioms. Other topics will include the axiom of choice, infinite cardinal and ordinal numbers, and models of set theory. Four class hours per week. | 677.169 | 1 |
Summer Math Tournament
Latest Contest Results
Taking Math Contests Online
Math League Bookstore
Schools Register for Contests
Articles
Grades 4,5 and 6 Contest Books
Math League Contest Problem Books contain the actual math contests given to students participating in Math League contests. The contests are designed to build student interest and confidence in mathematics through solving worthwhile problems. The Math League has Math Contest Books for Grades 4, 5, & 6; Grades 7 & 8; High School, and Algebra Course 1 students. Over 1 million students from the United States and Canada participate in Math League Contests each year. Every contest has questions from different areas of mathematics, suitable for the grade level listed. Many students first develop an interest in mathematics through problem-solving activities such as these contests.
There are 6 volumes for Math Contests-Grades 4, 5, & 6.
Volume 1 contains the contests given in the school years 1979-80 through 1985-86.
$12.95 USD
Volume 2 contains the contests given in the school years 1986-87 through 1990-91.
$12.95 USD
Volume 3 contains the contests given in the school years 1991-92 through 1995-96.
$12.95 USD
Volume 4 contains the contests given in the school years 1996-97 through 2000-01.
$12.95 USD
Volume 5 contains the contests given in the school years 2001-02 through 2005-06.
$12.95 USD
Volume 6 contains the contests given in the school years 2006-07 through 2010-11.
$12.95 USD
Volume 7 contains the contests given in the school years 2011-12 through 2015-16. To be published in September, 2016
$12.95 USD
These books are divided into three sections for ease of use by students and teachers. You'll find the contests in the first section. Each contest consists of 30 or 40 multiple-choice questions that you can do in 30 minutes. On each 3-page contest, the questions on the 1st page are generally straight-forward, those on the 2nd page are moderate in difficulty, and those on the 3rd page are more difficult. In the second section of the book, you'll find detailed solutions to all the contest questions. In the third and final section of the book are the letter answers to each contest. In this section, you'll also find rating scales you can use to rate your performance.
Many people prefer to consult the answer section rather than the solution section when first reviewing a contest. We believe that reworking a problem when you know the answer (but not the solution) often leads to increased understanding of problem-solving techniques.
Each year we sponsor an Annual 4th Grade Mathematics Contest, an Annual 5th Grade Mathematics Contest, and an Annual 6th Grade Mathematics Contest. A student may participate in the contest on grade level or for any higher grade level. For example, students in grades 4 and 5 (or below) may participate in the 6th Grade Contest. Starting with the 1991-92 school year, students have been permitted to use calculators on any of our contests. | 677.169 | 1 |
20 Dec 2012
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views:17401712 Sep 2012
views:137413published:11 Oct 2013
views:447published:15 Jan 2007
views:608431
conversion tables us customary unit capacity
published:17 Mar 2016
views:25
Conversion of units
Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors.
Techniques
Process
The process of conversion depends on the specific situation and the intended purpose. This may be governed by regulation, contract, technical specifications or other published standards. Engineering judgment may include such factors as:
Some conversions from one system of units to another need to be exact, without increasing or decreasing the precision of the first measurement. This is sometimes called soft conversion. It does not involve changing the physical configuration of the item being measured.
conversion tables
14:11
Review of the metric system (and how to convert)
Review of the metric system (and how to convert)1:26
PDF book of conversion tables
PDF book of conversion tablesUnderstanding Conversion FactorsConverting Units using Multiple Conversion Factors2:44
How to Learn Metrics Without Conversion Tables : Applied Mathematics
How to Learn Metrics Without Conversion Tables : Applied Mathematics9:17
Unit conversion within the metric system | Pre-Algebra | Khan Academy
Unit conversion within the metric system | Pre-Algebra | Khan Academyconversion tables
published: 27 Aug 2014published: 14 Jul 2013
Calculator Achievement-Weights and Measures: Conversion Tables 20 Dec 2012 10 Feb 2012 16 Sep 2012 12 Sep 2012 f...
published: 11 Oct 2013 wil... thin... lives, even if you don't necessarily realize it. Get tips that will make mathematics easier than ever before with help from a longtime math teacher in this free video series.Understanding Conversion Factors
To see all my Chemistry videos, check out
Even if you can write conversion factors and cancel units, that doesn't necessarily mea...Converting Units using Multiple Conversion Factors
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published: 12 Aug 2016
Converting ER-diagram to Database tables
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Mod 7 Lesson 3 converting units of time
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MSA Exposé Part 3: The Downfall and the Redefinition of Success.
This is the link to the article I briefly quote:
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Muzzle velocity is NOT a constant! It varies considerable due to a host of factors. In order to successfully engage targets at long range it is CRUCIAL that y... first varia p...Converting Future Pinball tables to Visual Pinball X - Part 9
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back ServerThe Cookbook - California Strawberry Pie
This video, part of the The Cookbook - Recipes App and Cookbook, demonstrates the technique for making a butter crust past...To see all my Chemistry videos, check out
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Converting Future Pinball tables to Visual Pinball X - Part 1
This is the first part of a quick conversion of a FP table to VPX. The goal here is to con...46:00
Converting Future Pinball tables to Visual Pinball X - Part 9
I think I'll call this done, the rest of the stuff is even more boring - tweaking gameplay...25:29
Convert a table-based layout into a CSS-based layout
Use CSS and HTML5 tags using Dreamweaver to convert a table-based HTML layout into a CSS-b...20:36
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The Cookbook - California Strawberry Pie
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MSA Exposé Part 3: The Downfall and the Redefinition of Success.
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MSA Exposé Part 3: The Downfall and the Redefinit This crafted conversation isn't distressing ... With all my love Marion". ....
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Greeted warmly with breakfast favourites and delicious beverages, guests embarked on meeting and greeting new friends and old friends alike, sharing personal and professional conversations. The unique nature of the 'Round the Table Networking' ensures that all guests are able to network within a friendly and supportive business community....
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Being a sponsor means that you 'buy' a table - you get however many tickets are at that table - at most fundraisers that means 8-10 seats for you to fill with whomever you would like. For the Bridge dinner, one table sold to a sponsor raises significantly more funds to cover the festival than 10 individual tickets sold ... I sat at a table with a farmer from Yolo County and another from Sonoma County and had great conversation and food....
Most of South Africa's universities have dropped down these ranking tables. Some people argue that the protests - which relate to fees, access and transformation and have occurred on and off for the past 18 months - are having a direct effect on universities' global standing on rankings tables... This article first appeared in The Conversation, a ......
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Offers a flexible organization, enabling instructors to adapt the book to their particular courses. This book gives emphasis on algorithms and applications. Including exercises, it features numerous computer science applications.
Read More
Offers a flexible organization, enabling instructors to adapt the book to their particular courses. This book gives emphasis on algorithms and applications. Including exercises, it features numerous computer science applications Intended for professional and scholarly audience.
All Editions of Discrete and Combinatorial Mathematics
Customer Reviews
Discrete and Combinatorial Mathematics
by Ralph P. Grimaldi
A different kind of math
I'm new to this area of mathematics, and it is requiring me to work at it. The text is full of meaningful and helpful examples, and there is an associated student solutions manual that provides worked solutions to the odd-numbered problems. It is a very complete work, so there may be some topics that could be left out on a first | 677.169 | 1 |
Saxon Math Homeschool 8/7 teaches math with a spiral approach, which emphasizes incremental development of new material and continuous review of previously taught concepts.
Building upon the principles taught in Saxon Math 7/6, the Saxon 87 textbook reviews arithmetic calculation, measurements, geometry and other skills, and introduces pre-algebra, ratios, probability and statistics. Students will specifically learn about adding/subtracting/multiplying fractions, equivalent fractions, the metric system, repeating decimals, scientific notation, Pi, graphing inequalities, multiplying algebraic terms, the Pythagorean Theorem, the slope-intercept form of linear equations, and more .
The Tests and Worksheetsbook provides supplemental "facts practice" tests for each lesson, as well as 23 cumulative tests that cover every 5-10 lessons. The included "activity sheets" are designed to be used with the activities given in the student worktext. Five optional, reproducible, recording forms are also included.
The Solutions Manual provides answers for all problems in the lesson (including warm-up, lesson practice, and mixed practice exercises), as well as solutions for the supplemental practice found in the back of the student text. It also includes answers for the facts practice tests, activity sheets, and tests in the separate tests & worksheets book.
Saxon Math 8/7 is designed for students in grade 7, or for 8th grade students who are struggling with math.
Focusing on algebraic reasoning and geometric concepts, Saxon Math Homeschool 8/7 teaches math with a spiral approach, which emphasizes incremental development of new material and continuous review of previously taught concepts. The back of the book contains supplemental problems for selected lessons and concepts.
This Saxon Math Homeschool 8/7 Tests and Worksheets book is part of the Saxon Math 8/7 curriculum for 7th grade students, and provides supplemental "facts practice" tests for each lesson, as well as 23 cumulative tests that cover every 5-10 lessons. The included "activity sheets" are designed to be used with the activities given in the (sold-separately) student worktext.
A testing schedule and five optional, reproducible, recording forms are also provided; three forms allow students to record their work on the daily lessons, mixed practice exercises, and tests, while the remaining two forms help teachers track and analyze student performance.
Solutions to all problems are in the (sold-separately) Solutions Manual.
This Saxon Math Homeschool 8/7 Solutions Manual provides answers for all problems in the textbook lesson (including warm-up, lesson practice, and mixed practice exercises), as well as solutions for the investigations and supplemental practice found in the back of the student text. It also includes answers for the facts practice tests, activity sheets, and tests in the tests & worksheets book. Answers are line-listed, and are organized by type (lessons & investigations, facts practice tests, tests, etc.).
Give your Saxon Math 8/7 students support and reinforcement! isLearning--and teaching!--math does not have to be difficult! Give your students and yourself the tools to succeed with this Saxon Teacher and Saxon 8/7 kit combination! Introduce your middle-schoolers to the concepts they'll need for upper-level algebra and geometry, including functions and coordinate graphing; integers; multiplying decimals and fractions; radius, circumference, and pi; and more. This kit includes Saxon's 3rd Edition Math 8/7 textbook, solutions manual, and tests/worksheets book.
Saxon Teacher provides support and reinforcement. are:MacMac | 677.169 | 1 |
Most colleges and universities now require their non-science majors to take a one- or two-semester course in mathematics. Taken by 300,000 students annually, finite mathematics is the most popular. Updated and revised to match the structures and syllabuses of contemporary course offerings, Schaum's Outline of Beginning Finite Mathematics provides... more...
Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately, there's Schaum's. This all-in-one-package includes more than 750 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 20 detailed videos featuring Math instructors who explain how to solve the most commonly | 677.169 | 1 |
Description
If you're looking for a graphing calculator app that works smoothly and seamlessly, you've found it! Graphing Calculator by Mathlab is a scientific graphing calculator integrated with algebra and is an indispensable mathematical tool for students in elementary school to those in college or graduate school, or just anyone who needs more than what a basic calculator offers. It is designed to replace bulky and costly handheld graphing calculators and works on virtually any Android phone or tablet.
Furthermore, Graphing Calculator by Mathlab displays calculations as it performs them on the high-quality display of the Android device, making it easier for the user to understand the calculations and see them clearly. This app has two great strengths. First, it acts as a fine scientific calculator, but, more than that, it displays the intermediate steps of the calculations as you type. It allows the students to both watch and learn how the calculations are made and how to find the final answer. Second, the graphing ability is absolutely stunning! Not only does the calculator beautifully display the graphs, but it automatically generates the x- and y- values and displays them as well.
Video: Help site with instructions and examples: If you have a question, send email to calc@mathlab.us
PRO FEATURES * 3D graphs * Full screen * 9 workspaces * Longer input and history * Save constants, functions and expressions in the library * Internet is not required * No advertisements
Reviews
4.8
3,123 total
5 2,592
4 445
3 46
2 14
1 26
James MooreJacob DionneDaniel Smith
Best on the market This has been my favorite calculator for a couple years now and decided I should pony up the money to show my appreciation. Always nice to get rid of the adds too...
Victoria ruano
Not the best but not the worst I can't scatterplot which is what I need to do for my class work. Other than that I like it
Nicholas Paraskevas
One of the most useful apps. Very clear and intuitive interface. Can do things I never expected from a phone app.
Jynx Momma
Best math app I use this math app to study my college math. Works in all departments of math. Works great!
User reviews
James Moore September 10, 2016Hi James :) Thank you very much for your input. I've issued the task for developers to solve this problem.
Jacob Dionne September 25, 2016Hi Daniel :)
Thank you very much for your appreciation and purchasing the calculator. We really need more users and purchases to afford its further development instead of just maintaining and developing slow. It would be nice if you invite your friends and classmates to try the calculator also:
Victoria ruano September 21, 2016
Not the best but not the worst I can't scatterplot which is what I need to do for my class work. Other than that I like it
19.8.3. Scatterplot and Regression Line:
When you need something, open the Help:
Make "custom search", for example, by the word "scatterplot".
The result is above.
Video tutorial:
English:
Spanish:
Nicholas Paraskevas September 12, 2016
One of the most useful apps. Very clear and intuitive interface. Can do things I never expected from a phone app.
Change the result of division by zero. A number divided by zero is not defined with the "infinite" symbol, in fact it is "undefined". This may cause confusion amongst the general public so please consider re-justifying the definition of division by zero.
Suggestions * please improve the result-display interface, the grid line is so messy, should make the current display result in different colors from the historic display of previous one * first startup of the app, please immediately pop up the keyboard, without having to tab the input field.
Excellent design This is a great calculator. The graphing functions alone make it worth buying. Easy to use interface, and a clear display of the formula. I also like that it instantly displays the answer to whatever you have typed.
Hi Randy :)
We are going to improve the graphing part making the surface even, showing intersections, calculating extremes etc. Based on user's feedback, the list to do consists of about 100 tasks now, and many of them have pretty high priority. We have been improving the calculator all the time and issuing updates almost every month. Read Help!
Danish Javed August 5, 2016
Great application, worth the pay The app works perfectly, multiple displays and the 3d graph is fun to use 😁 10/10 recommend | 677.169 | 1 |
Meet the world's top Mathematics freelancers
Find the World's Best Mathematics Freelancers
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What our customers are saying
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At A Glance: Mathematics
Mathematics is the language of science, data, and logic. Businesses and organizations can find themselves working with a wide variety of mathematical techniques and methods no matter what industry they are in. Mathematics, then, provides the equations that define and represent the physics and mechanics of physical products. Proper use of mathematics generates analytics that can be used to take the enormous amount of data available to modern businesses and provide commercial predictive power, which can drive and generate sales. Or mathematics can be used to create a host of other solutions, from security needs to general timeline extrapolation for projects and businesses.
Mathematics specialists on Upwork are highly-skilled professionals from a variety of backgrounds. Mathematics specialists can be highly educated, holding any number or type of advanced degrees, or they can be self-taught through experience. Your particular needs determine the type of specialist you will look for on Upwork. If you are seeking to develop highly theoretical predictive models, or just develop simple algorithms that can automate sales and inventory processes, a mathematics specialist can provide you with efficient and cost-effective solutions. As the nature of mathematics is generally abstract, a specialist can work remotely, either independently or with a team, and provide you and your organization with highly-flexible and dependable results. No matter what need you have in your organization, a math specialist can work with you to provide precise solutions that will maximize and streamline any aspect of production or | 677.169 | 1 |
Why Should We Take Math
Why Should We Take Math?
Math is power that grows your brain.
Math gives you credit to get your certificate or diploma.
As Georgia Highlands College student now and a professional later you will need to be able to obtain information, process that information into knowledge, and turn your knowledge into a tool to solve problems and tasks.
Mathematics is the universal language. The word mathematics comes from the Greek word meaning "something that is learned."
Most companies do a math test before hiring.
Math can help you learn other subjects faster.
Math is essential for everyday task.
Math might help you find a good paying job.
Math keeps your options open.
Math is about connections and relationships in everything we do.
Math is important in the advancement of science and our understanding of the workings of the universe | 677.169 | 1 |
New Math and Science Learning Solution
TI-nspire™ CAS+ learning solution benefits
students with different learning styles and increases
comprehension by providing the ability to simultaneously see
math concepts presented in multiple
representations.
(Wellington,NZ) Friday 12 May 2006 –
Texas Instruments announced today the New Zealand
introduction of the TI-nspire™ CAS+, a classroom learning
solution developed specifically for teachers and students in
secondary school mathematics . In addition to improved
computer algebraic system (CAS) functionality, the
TI-nspire™ CAS+, solution provides the ability for users to
view different representations of a mathematical concept at
the same time. These representations are dynamically
linked, so that any changes the user makes to one
representation is automatically made to the others to help
students see connections and build understanding. This is
the first time these representations have been this tightly
integrated, and its development was the result of extensive
educator feedback requesting this functionality.
"In
developing TI-nspire™ CAS+,, we worked with educators around
the world to better understand the classroom challenges they
faced, and how our technology could help," said Melendy
Lovett, president, Texas Instruments, Educational &
Productivity Solutions. "Having the ability to teach a child
based on what would best help him or her understand the
subject matter can make a dramatic difference in the success
of that child in maths and science classes, and inspire them
to continue studying these subjects."
The TI-nspire™
CAS+,solution brings together representations including
graphing, interactive geometry, and mathematical
spreadsheets. Graphs, geometric sketches, spreadsheets of
problem-based data, mathematical figures/symbols and text
are organized and linked. Up to four representations can be
viewed at one time. For instance, the student can view the
problem's description, the values and potential solutions, a
graph plotting the data and the equations the student used
to solve the problem.
Peter Fox, an educator involved in
the New Zealand Ministry of Education CAS Pilot programme
and who has worked in the development of TI-nspire™ CAS+,
says "When computers changed from text to a more graphical
interface it opened up a whole new world for the less
technologically literate. The development of TI-nspire™ CAS+
followed a similar path. As an educator involved in the
development of this product, much of the focus was on
creating an environment that is as user-friendly as
possible. If you want to see how an equation changes as the
graph is shifted, drag it. As you would expect, the table of
values will change too."
Another key teaching benefit of
TI-nspire™ CAS+ is the ability it gives students and
teachers to create and save their work, including text,
graphs and mathematical equations, for use in classroom
discussions or as homework. These electronic records
provide teachers with the opportunity to better understand a
student's thought process and diagnose problem areas.
To
maximize learning, TI-nspire™ CAS+ will be available in both
a portable handheld device and a computer software version
under the TI-nSpire name. Students can work seamlessly in
both environments or use one as their primary platform.
"Educators and students differed in terms of the platform
they preferred, so we developed the handheld and computer
software with the intent of providing them with virtually
the same maths and science teaching and learning experience
regardless of their preference," said Lovett.
Activities
from leading textbook publishers specific to the new product
will be available for TI-nspire™ CAS+ to help teachers
integrate the product into existing classroom practice. In
addition, an international team of educators from Europe,
Canada and Australasia are creating activities that will be
included with the TI-nspire™ CAS+ solution.
Teachers
Teaching with Technology™ (T3), TI's professional
development organization for math and science instructors,
will begin integrating TI-nspire™ CAS+ into their
Professional Development Programme early in
2007.
TI-nspire™ CAS+ will be available in August through
School Supplies
About Texas Instruments Educational &
Productivity Solutions, a business of Texas Instruments,
provides a wide range of advanced tools connecting the
classroom experience with real-world applications and
enabling students and teachers to explore math and science
interactively. Designed with leading educators, Texas
Instruments educational technology and services are tested
against recognized third-party research on effective
instruction and improved student learning. Such research
shows that use of graphing calculators and wireless
collaborative technology in the classroom helps teachers
implement instructional strategies that lead to higher
student interest, engagement and achievement in mathematics.
For more than 15 years, TI has worked closely with educators
and administrators to develop student-focused curricular and
supplemental classroom materials, and it supports the
world's largest professional development organization for
the appropriate use of educational technology. More
information is available at
Texas Instruments Incorporated is the world leader in
digital signal processing and analog technologies, the
semiconductor engines of the Internet age. In addition to
Semiconductor, the company's businesses also include Sensors
& Controls, and Educational & Productivity Solutions. TI is
headquartered in Dallas, Texas, and has manufacturing or
sales operations in more than 25 countries.
Texas
Instruments is traded on the New York Stock Exchange under
the symbol TXN | 677.169 | 1 |
Based on new research that proves repeated practice is more effective than repeated study, this Edexcel Maths all-in-one revision and practice book is guaranteed to help you achieve the best results. Containing clear and accessible explanations of all the GCSE content, there are lots of practice opportunities for each topic throughout the book.
For the 2017 exams and beyond
Suitable for the new Edexcel Maths GCSE, this Maths foundation tier revision and practice book includes clear and concise revision notes for every topic covered in the curriculum. Seven practice opportunities ensure the best results on exam day and Q&A flashcards can be downloaded for free online. More topic-by-topic practice and a complete exam-style paper can be found in the added workbook.
Included in this book:
• quick tests to check understanding • end-of-topic practice questions • topic review questions later in the book • mixed practice questions at the end of the book • free Q&A flashcards to download online • an ebook version of the revision guide | 677.169 | 1 |
How to Help Your Child Excel in Math 8vo-over 7¾"-9¾" tall. Wraps have only light wear, spine unbent. Pages are clean, text is unmarked.
Top Notch Books16
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$21.19The book is an alphabetical dictionary and handbook that gives parents of elementary, middle school, and high school students what they need to know to help their children understand the math they're learning. The book can also be used by students themselves and is suitable for anybody who is reviewing math to take standardized tests or other exams. Foreign students, whose English-language mathematics vocabulary needs to be strengthened, will also benefit from this book. | 677.169 | 1 |
A:While people struggle with math for many reasons, one of the most common is that math is a sequential subject and missing one concept along the way leaves the student in a growing state of confusion. A child who struggles with basic arithmetic doesn't have the foundation for algebra or advanced math classes. Other math-haters struggle to connect math to real life or suffer from a learning disability with numbers.
A:A formula equation is a visual representation of a reaction using chemical formulas. A chemical formula is an expression that states the number and types of atoms that make up any given molecule using the symbols for the elements and sub-scripted numbers.
A:A kilogram is approximately 2.20 pounds. The kilogram is the base unit the metric system uses for mass. Beginning in July 1959, the internationally accepted standard for the avoirdupois pound became exactly 0.45359237 kg.
A:The set of all points is called space. Space is taken as the largest possible set in geometry and algebra. Points, lines, line segments, rays, angles and planes are all considered as subsets of space.
A:A factor is a number or expression that divides another number or expression evenly without a remainder. For instance, two, four, five and 10 are factors of 20 because 20 can be divided by all those smaller numbers an exact number of times. Prime numbers are positive integers greater than one that have exactly two factors. A composite number has more than two factors.
A:Math word problems are important for high school students because they help develop a conceptual understanding of the processes involved in arriving at solutions and enhance students' abilities to communicate mathematically. Word problems also may help students become more interested in math by showing how concepts they have been studying abstractly can be put to use in real-world situations.
A:Math is useful for economics, science, sports, social fields and a wide range of other areas for making decisions and analyzing information. Basic math skills are essential for many jobs. Math has been a core component of many human advances.
A:To calculate bulk density, simply weigh the sample and divide its mass by its volume. Bulk density is commonly used when referring to solid mixtures like soil. Just like particle density, bulk density is also measured in mass per volume.
A:A nonlinear function in math creates a graph that is not a straight line, according to Columbia University. Three nonlinear functions commonly used in business applications include exponential functions, parabolic functions and demand functions. Quadratic functions are common nonlinear equations that form parabolas on a two-dimensional graph.
A:According to class notes from Bunker Hill Community College, calculus is often used in medicine in the field of pharmacology to determine the best dosage of a drug that is administered and its rate of dissolving. Usually, the drug is slowly dissolved in the stomach.
A:A few examples of how logarithms are used in the real world include measuring the magnitude of earthquakes or the intensity of sound and determining acidity. A logarithm explains how many times a number is multiplied to a power to reach another number. It is expressed as loge(x) and is commonly written as ln(x).
A:The definition of a limit in calculus is the value that a function gets close to but never surpasses as the input changes. Limits are one of the most important aspects of calculus, and they are used to determine continuity and the values of functions in a graphical sense.
A:Graphs are beneficial because they summarize and display information in a manner that is easy for most people to comprehend. Graphs are used in many academic disciplines, including math, hard sciences and social sciences. They make appearances in corporate settings, serving as useful tools to convey financial information and facilitate data analysis.
A:According to the BBC, data is transformed into information after being imported into a database or spreadsheet. Information is defined as a collection of facts or data, whereas data is defined as information organized for analysis or used to reason.
A:To make a tally chart, single lines are drawn next to one another until reaching five, where the fifth line crosses the four other lines diagonally. This is a simple charting method that can be used quickly for surveys or other needs.
A:A line graph is a graph that charts the relationship between two variables or a progression of a single quantity through time. Without the lines connecting points of data, it would be difficult to make sense of the data being presented if there was no overall trend between the dots. A bar graph, on the other hand, is meant to compare two or more statistics, such as population figures.
A:A segmented bar graph is similar to regular bar graph except the bars are made of different segments that are represented visually through colored sections. A segmented bar graph is sometimes known as a stacked bar graph, and it offers greater detail about data sets.
A:A bar graph is used to compare items between different groups and track changes over a period of time. Bar graphs are best used for changes that happen over a large amount of time instead of just months or weeks. | 677.169 | 1 |
0201895's Solutions Manual: Intermediate Algebra
Intermediate Algebra is designed to provide your students with the algebra background needed for further college-level mathematics courses. The unifying theme of this text is the development of the skills necessary for solving equations and inequalities, followed by the application of those skills to solving applied problems. The primary goal in writing the fourth edition of Intermediate Algebra has been to retain the features that made the third edition so successful, while incorporating the comments and suggestions of third-edition users. As always, the author endeavors to write texts that students can read, understand, and enjoy, while gaining confidence in their ability to use mathematics | 677.169 | 1 |
A Gentle Introduction to the Art of Mathematics
A Gentle Introduction to the Art of MathematicsIt is written in an informal, conversational style with a large number of interesting examples and...
More
It is written in an informal, conversational style with a large number of interesting examples and exercises, so that a student learns to write proofs while working on engaging Gentle Introduction to the Art of Mathematics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material A Gentle Introduction to the Art of Mathematics
Select this link to open drop down to add material A Gentle Introduction to the Art of Mathematics to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
SYMMETRIES
Content
In the first week the mathematics course 'Symmetries' treats the mathematical concepts 'groups' and 'symmetries' that form the basis of the deepening that then follows. The entire group is then split into smaller groups of about 7 students, with a maximum of 10, by choosing one of the following topics (listed in increasing difficulty and abstraction):
·
Finite groups in daily life (music, nature, art, math,…);
·
General relativity and black holes;
·
The fundaments of quantum mechanics.
All students read every week parts of Roger Penrose's book "The Road to Reality" (and additional material) and they make assignments. Each week the material is discussed in these smaller groups and new targets are set.
At the end of the course each group tells the other groups what they have learned.
Prerequisites: Linear Algebra (MathC1).
General information
This course will take place in the first quartile of your second year. It will be taught by Ruud van Damme.
Teachers
Ruud van Damme
He was born in 1955 in the Southern Dutch Town of Breda and obtained his master's degree in theoretical physics at the University of Utrecht after completing his studies there. In 1984, he added a PhD to this achievement, with Nobel Prize winner Gerard 't Hooft as his supervisor. Afterwards, he worked as a postdoc at the University of Leiden, before moving to Twente. His premier interest lays with methods of calculation, which he applies to the physics of fluids, the theory of graphs, quantum mechanics, and technical processes. His research concentrates on numerical techniques that deal with these matters and their applications. Overall, he has written over 100 papers on these topics, and acted as a co-auteur for a book about "splines and wavelets".
Ruud van Damme is head of the research group of numerical analysis and computational mechanics (NACM) at the department of Applied Mathematics at the University of Twente. Besides this, he also teaches algorithms and programming, mathematical modelling, introduction to engineering and several courses for the mathematical track of the Honours programme. | 677.169 | 1 |
ii Copyright c 2007 David Anthony SANTOS
Preface
There are very few good Calculus books, written in English, available to the American reader. Only [Har], [Kla], [Apo], [Olm], and [Spi] come to mind. The situation in Precalculus is even worse, perhaps because Precalculus is a peculiar American animal: it is a review course of all that which should have been learned in High School but was not. A distinctive American slang is thus called to describe the situation with available Precalculus textbooks: they stink! I have decided to write these notes with the purpose to, at least locally, for my own students, I could ameliorate this situation and provide a semi-rigorous introduction to precalculus. I try to follow a more or less historical approach. My goal is to not only present a coherent view of Precalculus, but also to instill appreciation for some elementary results from Precalculus. Thus I do not consider a student (or for that matter, an √ instructor) to be educated in Precalculus if he cannot demonstrate that 2 is irrational;1 that the equation of a non-vertical line on the plane is of the form y = mx + k, and conversely; that lines y = m1 x + k1 and y = m2 x + k2 are perpendicular if and only if m1 m2 = −1; that the curve with equation y = x2 is a parabola, etc. I do not claim a 100% rate of success, or that I stick to the same paradigms each semester,2 but a great number of students seem genuinely appreciative what I am trying to do. I start with sets of real numbers, in particular, intervals. I try to make patent the distinction between rational and√ irrational numbers, and their decimal representations. Usually the students reaching this level have been told fairy tales about 2 and π being irrational. I prove the irrationality of the former using Hipassus of Metapontum's proof.3 After sets on the line, I concentrate on distance on the line. Absolute values are a good place (in my opinion) to introduce sign diagrams, which are a technique that will be exploited in other instances, as for example, in solving rational and absolutevalue inequalities. The above programme is then raised to the plane. I derive the distance formula from the Pythagorean Theorem. It is crucial, in my opinion, to make the students understand that these formulæ do not appear by fiat, but that are obtained from previous concepts. Depending on my mood, I either move to the definition of functions, or I continue to various curves. Let us say for the sake of argument that I have chosen to continue with curves. √ Once the distance formula is derived, it is trivial to talk about circles and semi-circles. The graph of y = 1 − x2 is obtained. This is the first instance of the translation Geometry-to-Algebra and Algebra-to-Geometry that the students see, that is, they are able to tell what the equation of a given circle looks like, and vice-versa, to produce a circle from an equation. Now, using similar triangles and the distance formula once again, I move on to lines, proving that the canonical equation of a non-vertical line is of the form y = mx + k and conversely. I also talk about parallel and normal lines, proving4 that two non-vertical lines are perpendicular if and only if the product of their slopes is −1. In particular, the graph of y = x, y = −x, and y = |x| are obtained. The next curve we study is the parabola. First, I give the locus definition of a parabola. We use a T-square and a string in order to illustrate the curve produced by the locus definition. It turns out to be a sort-of "U"-shaped curve. Then, using the distance formula again, we prove that one special case of these parabolas has equation y = x2 . The graph of x = y2 is obtained, √ and from this the graph of y = x. Generally, after all this I give my first exam. We now start with functions. A function is defined by means of the following five characteristics:
1 Plato's dictum comes to mind: "He does not deserve the appellative man who does not know that the diagonal of a square is inconmensurable with its side. 2 I don't, in fact, I try to change emphases from year to year. 3 I wonder how many of my colleagues know how to prove that π is irrational? Transcendental? Same for e, log 2, cos 1, etc. How many tales are the students told for which the instructor does not know the proof? 4 The Pythagorean Theorem once again!
vi 1. a set of inputs, called the domain of the function; 2. a set of all possible outputs, called the target set of the function; 3. a name for a typical input (colloquially referred to as the dummy variable); 4. a name for the function; 5. an assignment rule or formula that assigns to every element of the domain a unique element of the target set. All these features are collapsed into the notation f: Dom( f ) → x → Target ( f ) . f (x)
Defining functions in such a careful manner is necessary. Most American books focus only on the assignment rule (formula), but this makes a mess later on in abstract algebra, linear algebra, computer programming etc. For example, even though the following four functions have the same formula, they are all different: a: R x R x → R ; → x2 b: [0; +∞[ → R ; x → x2 [0; +∞[ → [0; +∞[ ; x → x2
c:
→ [0; +∞[ ; → x2
d:
for a is neither injective nor surjective, b is injective but not surjective, c is surjective but not injective, and d is a bijection. I first focus on the domain of the function. We study which possible sets of real numbers can be allowed so that the output be a real number. I then continue to graphs of functions and functions defined by graphs.5 At this point,√ course, there are very functional of √ curves of which the students know the graphs: only x → x, x → |x|, x → x2 , x → x, x → 1 − x2, piecewise combinations of them, etc., but they certainly can graph a function with a finite (and extremely small domain). The repertoire is then extended by considering the following transformations of a function f : x → − f (x), x → f (−x), x → V f (Hx + h) + v, x → | f (x)|, x → f (|x|), x → f (−|x|). These last two transformations lead a discussion about even and odd functions. The floor, ceiling, and the decimal part functions are also now introduced. The focus now turns to the assignment rule of the function, and is here where the algebra of functions (sum, difference, product, quotient, composition) is presented. Students are taught the relationship between the various domains of the given functions and the domains of the new functions obtained by the operations. Composition leads to iteration, and iteration leads to inverse functions. The student now becomes familiar with the concepts of injective, surjective, and bijective functions. The relationship between the graphs of a function and its inverse are explored. It is now time for the second exam. The distance formula is now powerless to produce the graph of more complicated functions. The concepts of monotonicity and convexity of a function are now introduced. Power functions (with strictly positive integral exponents are now studied. The global and local behaviour of them is studied, obtaining a catalogue of curves y = xn , n ∈ N. After studying power functions, we now study polynomials. The study is strictly limited to polynomials whose splitting field is R.6 We now study power functions whose exponent is a strictly negative integer. In particular, the graph of the curve xy = 1 is deduced from the locus definition of the hyperbola. Studying the monotonicity and concavity of these functions, we obtain a catalogue of curves y = x−n , n ∈ N.
last means, given a picture in R2 that passes the vertical line test, we derive its domain and image by looking at its shadow on the x and y axes. used to make a brief incursion into some ancillary topics of the theory of equations, but this makes me digress too much from my plan of AlgebraGeometry-Geometry-Algebra, and nowadays I am avoiding it. I have heard colleagues argue for Ruffini's Theorem, solely to be used in one example of Calculus I, the factorisation of a cubic or quartic polynomial in optimisation problems, but it seems hardly worth the deviation for only such an example.
6I 5 This
figures. David A. Santos
. The problem of graphing them is reduced to examining the local at the zeroes and poles. usually during the last week of classes. n ∈ Z \ {0}. and it is time for the third exam.. A comprehensive final exam is given during final-exam week. I now introduce formulæ of the type x → x1/n . n ∈ Z. whose graphs I derived by means of inverse functions of x → xn . etc. These notes are in constant state of revision. and their global behaviour. additions.Preface
vii
Rational functions are now introduced. I would greatly appreciate comments. This concludes the story of Precalculus I as I envision it. exercises. but only those whose numerators and denominators are polynomials splitting in R. in order to help me enhance them.
The motivation or informal ideas of looking at a certain topic. I will allow you time in the lecture to do so. read the definitions. some needle thread. Here are more recommendations: • Read a section before class discussion. Questions of Understanding: I don't get it! Admitting that you do not understand something is an act requiring utmost courage. as well as gain confidence by providing your insights and interpretations of a topic. are given in class. Don't wait till the end of the class to point out an error.To the Student
These notes are provided for your benefit as an attempt to organise the salient points of the course. for example. The questions on assignments and exams will be posed in such a way that it will be of no advantage to have a graphing calculator. take a fresh look at the notes of the lecture topic. • Don't fall behind! The sequence of topics is closely interrelated. but the approach presented here is at times unorthodox and finding alternative sources might be difficult. you may try to emulate the style presented in the scant examples furnished in these notes. etc. and a compass. with one topic leading to another. in particular. The order of the notes may not necessarily be the order followed in the class. Don't be absent! • I encourage you to form study groups and to discuss the assignments. ask. a ruler (preferably a T-square). Clearly outline your ideas. especially in the occasional lengthy calculations. Again. outline major steps and write in complete sentences. • Try to understand a single example well. other books may help. But if you don't. it may be that you need extra help. rather than ill-digest multiple examples. when graphing. The best way to ask a question is something like: "How did you get from the second step to the third step?" or "What does it mean to complete the square?" Asseverations like "I don't understand" do not help me answer your queries. Questions of Correction: Is that a minus sign there? If you think that. it is likely that many others in the audience also don't. However. the ideas linking a topic with another. Do it when there is still time to correct it! 2. When writing solutions.. If at any stage you stumble in Algebra. you will need to provide algebraic/analytic/geometric support of your arguments. • You will need square-grid paper. • Ask questions during the lecture. the worked-out examples. There is a certain algebraic fluency that is necessary for a course at this level. If I consider that you are asking the same questions too many times. There are two main types of questions that you are likely to ask. These algebraic prerequisites would be difficult to codify here.
7
My doctoral adviser used to say "I said A. but bear in mind that whoever tutors you may not be familiar with my conventions. No one likes to carry an error till line XLV because the audience failed to point out an error on line I. The number of examples is minimal. As a guide. Discuss among yourselves and help each other but don't be parasites! Plagiarising your classmates' answers will only lead you to disaster! • Once the lecture of a particular topic has been given. and you will profit from the comments of your classmates. On the same vein.7 then by all means. Hence these notes are not a substitute to lectures: you must always attend to lectures. • The use of calculators is allowed. I wrote B. I have missed out a minus sign or wrote P where it should have been Q. in which case we will settle what to do outside the lecture. seek help! I am here to help you! Tutoring can sometimes help. • Presentation is critical. I am here to help! On the same vein. I meant C and it should have been D!
viii
. as they vary depending on class response and the topic lectured. They are a very terse account of the main ideas of the course. if you feel you can explain a point to an inquiring classmate. • Start working on the distributed homework ahead of time. and are to be used mostly to refer to central definitions and theorems. • Class provides the informal discussion. 1.
4 Example The sets A = {x ∈ Z : x2 ≤ 9}. B = {x ∈ Z : |x| ≤ 3}. then we write a ∈ A. will be denoted by ∅. . 1.
3 Definition Let A be a set. A containment we will simply write A B. or we may list its elements individually. ∅ Empty set. How many elements does it have? Is 401 ∈ A? Is 514 ∈ A? What is the sum of the elements of A?
1 There is no agreement relating the choice. that is. which is precisely the second set.}.1 Some sets of numbers will be referred to so often that they warrant special notation. Q The Rational Numbers. Is not an element of. 16. 2. 9. −2. −1. We will review some of the properties of real numbers as a way of having a handy vocabulary that will be used for future reference. . −1. 0. . Belongs to. ∀ For all (Universal Quantifier). C = {−3. 1. ∈ Is in.}. read "a is not an element of A. 3. 1. ." The set that has no elements. In the case when we want to denote strict
1
. The focus of this course will be the real numbers. We denote that B is a subset of A by the notation B A or sometimes B ⊂ A. 2. that is empty set. . . 2 Example −1 ∈ Z but
1 2
∈ Z. borrowed from set theory and logic.
! Observe that N ⊆ Z ⊆ Q ⊆ R ⊆ C. . Here are some of the most common ones. P⇔Q P if and only if Q. ∈ Is not in. where the elements are in arithmetic progression. 0. . N The Natural Numbers {0.
B but A = B.1 Sets and Notation
1 Definition We will mean by a set a collection of well defined members or elements. 2. −3.
From time to time we will also use the following notation. The first set is the set of all integers whose square lies between 1 and 9 inclusive. of which we assume the reader has passing familiarity. A subset is a sub-collection of a set. . ∃ There exists (Existential Quantifier). R The Real Numbers. Is an element of. We may use a description. Some use ⊂ to denote strict containment. which again is the third set. If a belongs to the set A. 716}. . −2. Does not belong to. P =⇒ Q P implies Q. 3}
are identical.." If a does not belong to the set A. . 5 Example Consider the set A = {2. 3. .
1. Z The Integers {. C The Complex Numbers.1
The Line
This chapter introduces essential notation and terminology that will be used throughout these notes. we write a ∈ A. There are various ways of alluding to a set. read "a is an element of A.
1 shews the various types of intervals. π .4 . π . −10[.
Figure 1. and if s < x < t. Observe that we indicate that the endpoints are included by means of shading the dots at the endpoints and that the endpoints are excluded by not shading the dots at the endpoints. The following result will be used later. 1 + 2 ≈ 2. since 0 does not have left neighbours in the interval and 1 does not have right neighbours on the interval. π ≈ 1.571. 2 √ B \ A = 1 + 2. then x ∈ I.732. but think that in [a. A \ B = [1. "Hugging" is thus equivalent to including the endpoint. . is a neighbourhood of all of its points. B = π . We say that the set Na R is a neighbourhood of a if there exists an open interval I centred at a such that I Na .4: Neighbourhood of a. B = ]−∞. π .b[ the "arms" are repulsed. but in ]a. We say that the set V R is a dextral neighbourhood or right-hand neighbourhood of a if there exists a δ > 0 such that [a. If Na is a neighbourhood of a. 2] . Table 1. In other words. then A ∩ B = [−10. on the contrary. We say that the set V ′ R is a sinistral neighbourhood or left-hand neighbourhood of a if there exists a δ ′ > 0 such that ]a − δ ′ . π ≈ 3.142. 13 Example The interval ]0. a + δ [ Na . 14 Definition Let a ∈ R. a + δ [ V . 1 + 2 .
√ √ 11 Example Let A = 1 − 3. Thus 2 A∩B = √ π . 2 √ A ∪ B = 1 − 3.
Chapter 1
9 Definition An interval I is a subset of the real numbers with the following property: if s ∈ I and t ∈ I. This means that Na is a neighbourhood of a if a has neighbours left and right.b] the brackets are "arms" "hugging" a and b. By approximating the endpoints to three decimal places. 1[ is neighbourhood of all of its points. In other words. with the exception of its endpoints 0 and 1.
We may now extend the definition of neighbourhood.1 + 2 . 12 Definition Let a ∈ R.
We conclude this section by defining some terms for future reference.5: Sinistral neighbourhood of a. 1[. 3 10 Example If A = [−10. then we say that Na \ {a} is a deleted neighbourhood of a.
a−δ
a
a+δ
a−δ
a
a
a+δ
Figure 1. This last condition may be written in the form {x ∈ R : |x − a| < δ } Na . Since there are infinitely many decimals between two different real numbers. 2].
Figure 1. 1]. intervals with distinct endpoints contain infinitely many members. B \ A = ]−∞. A ∪ B = ]−∞. 1[. √ π A \ B = 1 − 3.414.6: Dextral neighbourhood of a. and "repulsing" is equivalent to excluding the endpoint. intervals are those subsets of real numbers with the property that every number between two elements is also contained in the set. The interval [0. a] V ′ . we find 1 − 2 √ √ 3 ≈ −0. Na is a neighbourhood of a if there exists a δ > 0 such that ]a − δ . 2] .
3 It may seem like a silly analogy.
.
they were able to make one-to-one correspondences. it is believed that the introduction of negative quantities arose as an accounting problem in Ancient India. . fractioning it) justifies the creation of the positive rational numbers.14 as 3.345454545 . a real number with a terminating or repeating decimal expansion must be a rational number." 6 That this cancellation is meaningful depends on the concept of convergence.14 = 314 157 = . in other words. we can write the decimal 3.7 we prefer to call it a 18 Fact Every rational number has a terminating or a repeating decimal expansion. in other words. the word Calculus comes from the Latin for "stone. b = 0 . cancelling them out. and then subtract these tails. Since we are too cowardly to prove the next statement. but that. Fair enough. subtract. For example. 16 Definition The set of rational numbers Q is the set of quotients of integers where a denominator 0 is not allowed. 1000x = 345. and Derision. most ancient societies did very well with just the strictly positive rational numbers. a rational number has a terminating decimal expansion if and only if its denominator is of the form 2m 5n . "and the different branches of Arithmetic–Ambition.45454545 . 990 55
. Thus we have constructed N. for example [HarWri]
342 19 = . of course. we obtain as a result a rational number. Uglification. Notice that every finite decimal can be written 1 as a fraction. In fact. a very elegant system of numbers which allows us to perform four arithmetic operations (addition. multiply or divide any two rational numbers (with the exclusion of division by 0). to begin with. without actually carrying out the long division. 1024 2 1 decimal expansion. of which we may talk more later. b ∈ Z. that say. . so far. Z ⊆ Q.454545 . =⇒ 1000x − 10x = 342 =⇒ x = ◭ By mimicking the above examples.. Write 0 if you are rupeeless. 100 50
What about non-finite decimals? Can we write them as a fraction? The next example shews how to convert an infinitely repeating decimal to fraction from. the need for new numbers arose. as the quotient of two natural numbers. where m and n are natural numbers. which we will discuss in a latter section. Write −1 if you owe one rupee. . Distraction. the following should be clear: decimals whose decimal expansions terminate or repeat are rational numbers. 6
5 "Reeling and Writhing. . Solution: ◮ The trick is to obtain multiples of x = 0. Moreover. does not. meaning that if we add.6 So observe that 10x = 3. In other words: a : a ∈ Z. The problems of counting and of counting broken pieces were solved completely with these numbers. a Since a = . we stay within the same set. .345454545 . . subtraction. and division)5and that has the notion of "order". In Q we have. . In fact. so that they have the same infinite tail. that is. 7 The curious reader may find a proof in many a good number theory book.6
Chapter 1
following abstraction: add to a pile one pebble (or stone) for every sheep. every integer is also a rational number. . Z and Q.345 = 0. say." Breaking an object into almost equal parts (that is. A formal definition of the rational numbers is the following. 1 1 = 10 has a terminating From the above fact we can tell. "the Mock Turtle replied. 17 Example Write the infinitely repeating decimal 0. As societies became more and more sophisticated. for example. Conversely. write +1 if you have a rupee—or whatever unit that ancient accountant used—in your favour. Q= b Notice also that Q has the wonderful property of closure. multiplication.
since n is not a perfect square. and hence. n2 s2 = m2 . But then we have a contradiction. of the following. 6. 1. Hipassos of Metapontum. n such that this fraction be in least terms. to study numbers in the abstract. n is irrational. This means that m2 is even. it was irrational. u
−2
−1
0
1
√ 2
2
Figure 1. But their lunacy went even farther.
!The above theorem says that the set R \ Q of irrational numbers is non-empty. We can find integers m. This contradicts unique factorisation. Rather than studying numbers to solve everyday "real world problems"—as some misguided pedagogues insist—they tried to understand the very essence of numbers. b such that n = . if you want to be called mathematically literate. m Proof: Assume there is s ∈ Q such that s2 = 2. the product of a rational and an irrational giving an irrational number. but the sinistral side does not. Fortunately. n = 0 such that s = . that they drowned him. Pythagoras lived 582 to 500 BC. since the product of two odd numbers is odd. was able to prove that the length of hypotenuse of a right triangle whose legs8 had unit length could not be expressed as the ratio of two integers and hence. if at all possible. 3. But then m itself must be even. the sum of two irrationals giving an irrational number. Solution: ◮
8
The appropriate word here is "cathetus.Rational Numbers and Irrational Numbers
7
Is every real number a rational number? Enter the Pythagorean Society in the picture. At the beginning it seems that they thought that the "only numbers" were rational numbers."
.
! From now on we will accept the result that √n is irrational whenever n is a positive non-square integer. √ 19 Theorem [Hipassos of Metapontum] 2 is irrational. the sum of two irrationals giving a rational number. Now. The dextral side of this equality has an even number of b √ prime factors. the product of a rational and an irrational giving a rational number. This means that 2n2 = (2a)2 = 4a2 =⇒ n2 = 2a2 . This implies that nb2 = a2 . that is 2n2 = m2 . For. But one of them. mathematicians have matured since then and the task of burning people at the stake or flying planes into skyscrapers has fallen into other hands. 20 Example Give examples.7: Theorem 19. The crucial part n of the argument is that we can choose m. the product of two irrationals giving a rational number. Thus m = 2a for some non-zero integer a (since m = 0). and so n must be irrational. This loony sect of Greeks forbade their members to eat beans. the sum of two rational numbers giving an irrational number. 2. there would exist two strictly positive √ a natural numbers a. whose founder. This is one of the very first
theorems ever proved.
Suppose that we knew that every strictly positive natural number has a unique factorisation into primes. 7. 5. n cannot be both even. since m and n were not both even. This means once again that n is even. the product of two irrationals giving an irrational number. m. It befits you.
The shock caused to the other Pythagoreans by Hipassos' result was so great (remember the Pythagoreans were a cult). in general. Then if n is not a √ √ perfect square we may deduce that. if n were rational. dear reader. 4. to know its proof.
4 Problem Suppose that you are given a finite string of integers.2 Problem Prove that √ 8 is irrational. Do we have a pattern or do we not? 21 Example We expect a number like 0. 4. . Your calculator probably gives about 9 decimal places when you try to compute 2. The "irrationalities" of 2 and π are of two entirely "different flavours.
√ 4 2 were rational. .. √ √ 1 1 7.2.123456789101112 . 16. Irrational numbers. √ √ √ 6 is irrational. b such that √ √ a a2 4 2 = =⇒ 2 = 2 . . say. Take both numbers to be 2. a. suspicion arose that there were other irrational numbers. which is rational. . 22 Example Prove that Solution: ◮ If √ 4 2 is irrational. 12345. Take one irrational number to be 2 and the other to be √ . This says that 2 is rational. Their product is 1 · 2 = 2. For the same reason.
1. then there would be two non-zero natural numbers. This number is known as the Champernowne-Mahler number. zeroes between consecutive ones.1 Problem Write the infinitely repeating decimal 0. b b
Since
√ a2 a a a is rational.414213562. take the rational number to be 1 and the irrational to be 2. The rational numbers are closed under addition and multiplication.100100001000000001 . Their sum is 0. as many lifetimes as a cat..2. Archimedes √ suspected that π was irrational.3 Problem Assuming that must be irrational. it says 2 ≈ 1. that is. since the gaps between successive 1's keep getting longer. the set R \ Q. Their product is 2 · √ = 1. . In fact.123123123 . ◭ b b b b
Homework
1.6 Problem Find an irrational number between the irrational √ √ numbers 2 and 3. Take one irrational number to be 2 and the other to be 3.5√ Problem Find a rational number between the irrational num√ bers 2 and 3. to be irrational. . and so the decimal does not repeat.2. .2.8 1. Take one number to be 2 and the other − 2.2.
9
Or in the case of people in the English and the Social Sciences Departments. 1. Can you find an irrational number whose first five decimal digits after the decimal point are 12345? 1. What happens after the final 2 is the interesting question. where there are 2. √ √ √ √ √ 6. which consists of enumerating all strictly positive natural numbers after the decimal point. by simply "looking" at the decimal expansion of a number we can't tell whether it is irrational or rational without having √ √ more information. are those then having infinite non-repeating decimal expansions. √ √ √ 4. as the quotient of two positive integers. Their sum is 2 2 which is also irrational. √ √ 2. . 2 = · must also be rational. Their product is 2 · 3 = 6. This is impossible. prove that 2 + 3 1.123 = 0. 2 2 ◭
Chapter 1
√ After the discovery that 2 was irrational. Of course. is irrational. say. . a fact that wasn't proved till the XIX-th Century by Lambert." but we will need several more years of mathematical study9 to even comprehend the meaning of that assertion. √ √ 3. Their product is 0 · 2 = 0. Take the rational number to be 0 and the irrational to be 2.
1.2. contradicting Theorem 19. 8. √ √ 5.
. the number 0.
say.
This axiom tells us that if we add or multiply two real numbers. 10 9
9
bers
1. called the opposite of x.7 Problem Find an irrational number between the rational num1 1 and . The above axioms allow us to obtain various algebraic identities. as in xy. so. z. This gives 0 = 0 · 0−1 = 1. say. Observe that this axiom does not hold for division. This is also not √ true of taking square roots. 0 and 1. for. but 1 ÷ 0 is not a real number. (1 − 1) − 1 = 1 − (1 − 1). 25 Axiom (Associativity) x ∈ R. Observe that subtraction is not associative. it is immaterial where we put the parentheses. called the multiplicative inverse of y. with 0 = 1. For all y ∈ R \ {0} ∃y−1 ∈ R \ {0}. such that ∀x ∈ R. y. and 1 · x = x · 1 = x. in contradiction to the assumption that 0 = 1. 1 ÷ 0 is the division of two real numbers.Operations with Real Numbers
1. such that y · y−1 = y−1 · y = 1. say 0−1 . First. 28 Axiom (Distributive Law) For all real numbers x.
. if we multiply any real number by 0 we get 0. omitting the product symbol ·.3 Operations with Real Numbers
The set of real numbers is furnished with two operations + (addition) and · (multiplication) that satisfy the following axioms.
! It is customary in Mathematics to express a product like x · y by juxtaposition. 24 Axiom (Commutativity) x∈R and y ∈ R =⇒ x + y = y + x and xy = yx. there holds the equality x · (y + z) = x · y + x · z. that is. and hence. since.
This axiom tells us that order is immaterial when we add or multiply two real numbers. for example. Notice that this is not true of division.
27 Axiom (Existence of Opposites and Inverses) For all x ∈ R ∃ − x ∈ R. y ∈ R and z ∈ R =⇒ x + (y + z) = (x + y) + z and (xy)z = x(yz). 0 · 0−1 = 1. 26 Axiom (Additive and Multiplicative Identity) There exist two unique elements. division by 0 is not allowed. 1 ÷ 2 = 2 ÷ 1. in particular. for example. notice that 0 does not have a multiplicative inverse. We will obtain a contradiction as follows. of which we will demonstrate a few. by writing together the
letters. 0 · 0−1 = 0. −1 is a real number but −1 is not. that is. because. if we multiply a number by its multiplicative inverse we should get 1.2. Why? Let us for a moment suppose that 0 had a multiplicative inverse. From now on we will follow this custom. then we stay within the realm of real numbers. Also. for. such that x + (−x) = (−x) + x = 0. In the axiom above.
This axiom tells us that in a string of successive additions or multiplications. 0 + x = x + 0 = x. 23 Axiom (Closure) x∈R and y ∈ R =⇒ x + y ∈ R and xy ∈ R.
7) (1. (1. essentially. which is smaller than any real number.9.
−∞
−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7
+∞
Figure 1. the positive numbers increasing towards the right from 0 and the negative numbers decreasing towards the left of 0. Any set of real numbers which is bounded below has a infimum. that this line has no "holes.9) (1.
Observe that the rational numbers are not complete. each real number can be viewed as a point on a straight line. 0(±∞).9: The Real Line.10) (1.8) (1. Letting x ∈ R. there is no largest rational number in the set {x ∈ Q : x2 < 2} √ √ since 2 is irrational and for any good rational approximation to 2 we can always find a better one. For example.12) (1.
23
65 Axiom (Completeness of R) Any set of real numbers which is bounded above has a supremum. Geometrically.11) (1. which further distinguishes them from the rational numbers.6) (1. (+∞) + (+∞) = +∞ (−∞) + (−∞) = −∞ x + (+∞) = +∞ x + (−∞) = −∞ x(+∞) = +∞ if x > 0 x(+∞) = −∞ if x < 0 x(−∞) = −∞ if x > 0 x(−∞) = +∞ if x < 0 x =0 ±∞ (+∞) + (−∞). The Completeness Axiom says. We make the convention that we orient the real line with 0 as the origin. and the object −∞.13) (1. which is larger than any real number.14)
Observe that we leave the following undefined: ±∞ . as in figure 1." We append the object +∞.Completeness Axiom The real numbers have the following property. ±∞
. we make the following conventions.
the set of all ordered pairs whose elements belong to the given sets. that is. (2. Their Cartesian Product A × B is defined and denoted by A × B = {(a. −2} and B = {−1. 2) ∈ Z2 but (−1. y) ∈ R2 : y = b} is a horizontal line. −2). The points x and y are the coordinates of P. y) on the plane R2 . b ∈ B}.
1
B × A = {(−1. and y is the ordinate. 2) ∈ Z × R but (−1. √ 68 Example (−1. B. (−1. It is therefore sufficient to have two numbers x and y to completely characterise the position of a point P = (x. −1). (2. Intersect perpendicularly two copies of the real number line. 2) ∈ Z2 . 2))}. Notice that these sets are all different. Here x is the abscissa1 . 2)}. √ √ 69 Example (−1. be subsets of real numbers. This manner of dividing the plane and labelling its points is called the Cartesian coordinate system. 72 Definition Let b ∈ R be a constant. The set {(x. (−1. (2.2
The Plane
2. 2) ∈ R × Z.
! In the particular case when A = B we write
A × A = A2 . These two lines are the axes. To each point P on the plane we associate an ordered pair P = (x. −1). 67 Example If A = {−1. We represent the elements of R2 graphically as follows. (−2. (−2. b) : a ∈ A. −2). y) of real numbers.
24
. −1).is the set of all ordered pairs (x. 0)— is the origin. −1). 70 Definition R2 = R × R—the real Cartesian Plane—. −2)}. 2} then A × B = {(−1. which measures the horizontal distance of our point to the origin. (−1. 2).
From the Latin linea abscissa or line cut-off. y) ∈ R2 : x = a} is a vertical line. (−1. The set {(x. 2). (−2. 71 Definition Let a ∈ R be a constant. (−2. A2 = {(−1. y) of real numbers. −1). (2. −1). −2)}. B2 = {(−1. which measures the vertical distance of our point to the origin. −1). −1). Their point of intersection—which we label O = (0. The horizontal axis is called the x-axis and the vertical axis is called the y-axis.1 Sets on the Plane
66 Definition Let A. even though some elements are shared.
First we consider the interval [a. consider the projections of this point onto the line segments AC and BC. y1 ) and B(x2 . its midpoint is at MB = ( x1 +x2 . 2 2
lies on the line joining A(x1 . y1 ) and B(x2 .9: Example 80. its midpoint is at a + 2 2 Let (x. y) must have the same abscissa 2 as MB and the same ordinate as MA . By Thales' Theorem. y1 ) and B(x2 .
Proof: The proof proceeds along the lines of Theorem 81. With C(x2 . into two line segments in the ratio m : n has coordinates nx1 + mx2 ny1 + my2 . right-angled at C. since 3 is irrational.28 From BC = CA we deduce that −2bm + b2 = 0 =⇒ b(b − 2m) =⇒ b = 2m. n)
(b. b]. form the triangle △ABC. y2 ). y1 +y2 ). m+n m+n . has length b − a. y1 and the coordinates of R and x2 . as we are assuming that b > 0. From (x. ◭
2 2 2
Chapter 2
(m. Hence. AC and BC in the ratio m : n. u In general. y2 ). Suppose that x−a m na + mb a < x < b and that = . Notice that these projections are parallel to the legs of the triangle and so these projections pass through the midpoints of the legs. b−x n m+n Form now △ABC.
x1 + x2 y1 + y2 . y2 ).
81 Theorem (Midpoint of a Line Segment) The point and it is equidistant from both points. with A(x1 . The interval [a. From P. its midpoint is MA = (x2 . n cannot be an integer. right-angled at C. √ b2 3 2 + n =⇒ n = b. As BC is a horizontal 2 segment. By what was just demonstrated ny1 + my2 nx1 + mx2 . respectively. Q and R divide.
Proof: First observe that it is easy to find the midpoint of a vertical or horizontal line segment. b = m +n = 4 2 √ Since we are assuming that b = 0. y1 ). the coordinates of Q are m+n m+n the result. y1 ). y). Hence. 0)
Figure 2. y) be the midpoint of the line segment joining A(x1 . giving . b] b−a a+b = . consider the projection Q on AC and the projection R on BC. u
Homework
. The result is obtained on noting that (x. we have the following result. Since AC is a horizontal segment. 82 Theorem (Joachimstal's Formula) The point P which divides the line segment AB. about intervals. This gives x = .
2. . 2. 2)? 2. If (ak .15 Problem Find the coordinates of the point symmetric to (−a. b) with respect to the point (b. . a). 2.
2.2. is an integer. Consider
P
Sn = min ∑
n
k=1
(2k − 1)2 + a2 .. a2 .8 Problem A bug starts at the point (−1. (iii) the origin. except in quadrant II. . 1/4 unit down.2. 1/2 unit right. 1/2). a). This is called the canonical equation of the circle with centre ((x0 .. Which route should the bug take in order to minimise its time? The answer is not a straight line from (−1. the car at point A travels downwards at constant speed. (4. Sn . 2. −1) and wants to travel to the point (2.2. . 2. b). 2. it moves with unit speed. 0) and (0. . (ii) the y-axis.3 Circles
The distance formula gives an algebraic way of describing points on the plane.13 Problem A fly starts at the origin and goes 1 unit up. 1) at distance 2 del from the point (0. k
where the minimum runs over all such partitions P. ad infinitum. an } be a collection of points with 0 < a1 < a2 < · · · < an < 17. b1 b2 bn 2. a) and (b. 1 ≤ k ≤ n.2. (1. 0). 1)! 2.2. Starting at time t = 0. In each quadrant.17 Problem Prove the following generalisation of Minkowski's Inequality.2. Shew that exactly one of S2 . 2. 1/8 unit left. . −3) . the car at point B travels upwards at constant speed.2. 2 2
29
2. −5). Find C. In what coordinates does it end up? 2. b2 + b) and (b + a. y0 ) is (x − x0)2 + (y − y0)2 = R2 .4 Problem Demonstrate by direct calculation that d (a. and on the axes. . a+c b+d .2 Problem If a and b are real numbers. b + a). 2. etc.2. and find which one it is. (c.2.11 Problem Demonstrate that the diagonals of a rectangle are congruent. .2.2. find the distance between the points (a.18 Problem (AIME 1991) Let P = {a1 . 2 2 =d a+c b+d .12 Problem Prove that the diagonals of a parallelogram bisect each other. at a rate of b > 0 units per second.2.
Equality occurs if and only if a1 a2 an = = ··· = . How many units apart are these cars after t > 0 seconds? 3 of the distance from A(1. .2.2. 0) and an identical car is located at point (x.5 Problem A car is located at point A = (−x. then
k=1
∑
n
a2 + b2 ≥ k k
k=1
∑ ak
n
2
+
k=1
∑ bk
n
2
. bk ) ∈ (R \ {0})2 .1 Problem Find d (−2.1)
. . d) ∈ R2 .2.6 Problem Point C is at 2. y0 )) and radius R. 2. (2. b) to (b.2.7 Problem For which value of x is the point (x.14 Problem Find the coordinates of the point which is a quarter of the way from (a. 1). . 0). . −1) to (2.3 Problem Find the distance between the points (a2 + a. d) . b) with respect to: (i) the x-axis.
Equality occurs if and only if ad = bc. 10) on the segment AB (and closer to B than to A). 2.16 Problem (Minkowski's Inequality) Prove (a. where it moves with half the speed. b).10 Problem Find the coordinates of the point symmetric to (a. b). 83 Theorem The equation of a circle with radius R > 0 and centre (x0 . (c.Circles
2.9 Problem Find the point equidistant from (−1. 1/16 unit up.
c2 + d 2 . at a rate of a > 0 units per second and simultaneously. then (a + c)2 + (b + d)2 ≤ a2 + b2 + that if
2.2. S3 . 5) to 5 B(4.2.
0). y0 ) belonging to Lt no matter which real number t be chosen? 2. 5. 2. 1.6. Lt is parallel to the line of equation 3x − 2y − 6 = 0. b) ∈ (R \ {0})2 . 4. 2. Is there a point (x0 .6. 1 5. Lt is parallel to the x-axis. u
Homework
2. a) is symmetric to the point (a. Lt has gradient −2.6.5 Problem Find the equation of the line passing through (12. Find the equation of the line passing through (a. See figure 2. 0). b) and parallel to the line a − b = 1. establishing the result. −1). b) to (b.9 Problem For any real number t. 2) and (−3. b) with respect to the line y = x. 1). P =
1 |− 2 − 1 − 3| 3
1 + ( 2 )2 3
√ 6 13 . 6). Is there a point (x0 .6 Problem Find the equation of the line passing through (12. 1. 2. x = −x + a + b =⇒ x = 2 a+b Then.6. a).8 Problem Consider the line L passing through (a. 4. In each of the following cases.36. 0) and normal to the line joining (1. −1). 3. This line is perpendicular to the line y = x and intersects it when a+b . 2. 8. Find the equations of the lines L1 parallel to L and L2 normal to L.4 Problem Let a.13 Problem Let a ∈ R. Lt is normal to the line of equation y = − 4 x − 5. 0) and parallel to the line joining (1. Find the distance from the point (a. y0 ) belonging to Lt no matter which real number t be chosen?
2. 2. C = (5. 1).
. Lt passes through (−2. Lt passes through (1.6. 0) to the line L : y = ax + 1. since y = x = . if L1 and L2 must pass through (1. b2 ). 2. Lt is normal to the line of equation y = 4x − 5. Lt passes through the origin (0. In each of the following cases.6. 2. find an t satisfying the stated conditions.6. 3.10 Problem For any real number t.12 Problem Find the distance from the point (1.42
Chapter 2 Solution: ◮ The equation of the line L can be rewritten in the form L : y = 2 x − 1 . 3) form the vertices of a rectangle. and D = (−1. Lt is parallel to the x-axis. b be strictly positive real numbers. 1) to the line y = −x.6.6. a+b ). Lt is parallel to the line of equation x − 2y − 6 = 0. Lt is parallel to the y-axis. 6.11 Problem Shew that the four points A = (−2. which means that both (a.6. 4) and tangent to the line x − 2y + 3 = 0.1 Problem Find the equation of the straight line parallel to the line 8x − 2y = 6 and passing through (5. a) to (a. 2.2 Problem Let (a. But this point is the midpoint of the line segment 2 2 2 joining (a.7 Problem Find the equation √ the straight line tangent to the of 1 circle x2 + y2 = 1 at the point ( 2 . Lt is parallel to the y-axis. 6. 2.3 Problem Find the equation of the straight line normal to the line 8x − 2y = 6 and passing through (5. 2. the point of intersection is ( a+b .14 Problem Find the equation of the circle with centre at (3. = 13
◭ 107 Theorem The point (b. associate the straight line Lt having equation (2t − 1)x + (3 − t)y − 7t + 6 = 0. 6). −2). 2.6. 3). 1). B = (4. find an t and the resulting line satisfying the stated conditions. a) are equidistant from the line y = x.6. b) and perpendicular to the y x line a − b = 1. associate the straight line Lt having equation (t − 2)x + (t + 3)y + 10t − 5 = 0. 2. b) has equation y = −x + a + b. 2) and (−3. a2 ) and (b.6. 2.6. 7. 23 ). Find the equation of the line y x passing through (a. we have 3 3 d L. Using Theorem 105. b) and (b. Proof: The line joining (b.
we obtain the graphs of y = 2. For the parabola y = 115 Example Draw the parabola y = x2 . we may draw the graph of the curve x = y2 . We have |y + d| = x2 + (y − d)2 =⇒ =⇒ =⇒ =⇒ as wanted. the vertex is clearly (0. Then the distance of (x. d = . while letting the other end of the ruler to pivot around F1 . y) to the point (0. 4d
! Observe that the midpoint of the perpendicular line segment from the focus to the directrix is on the parabola.
We call this point the vertex. Following Theorem 114. 4d
3 2 1 −1 −3−2−1 −2 −3 1 2 3
3 2 1 −1 −3−2−1 −2 −3 1 2 3
3 2 1 −1 −3−2−1 −2 −3 1 2 3
Figure 2.
√ Figure 2.
2
Foci is the plural of focus. where |F1 D − F2D| = |F1 D′ − F2 D′ |.
116 Example Using Theorem 107.49: y = − x.47: x = y2 . Attach piece of thread to one end of the ruler.
. See figure 2. y) be an arbitrary point on the parabola. Their graphs appear in figures
118 Definition A hyperbola is the collection of all the points on the plane whose absolute value of the difference of the distances from two distinct fixed points F1 and F2 (called the foci2 of the hyperbola) is a positive constant. 1 1 Solution: ◮ From Theorem 114. Put tacks on F1 and F2 and measure the distance F1 F2 .50.
Figure 2. Its graph appears in figure 2.49. we locate the focus 4d 4 1 1 at (0.48: y =
√
x.47. y) to the line y = −d is |y + d|. The lengths of the ruler and the thread must satisfy length of the ruler − length of the thread < F1 F2 . 4 ) and the directrix at y = − and use a T-square with these references.48 and 2. u (|y + d|)2 = x2 + (y − d)2 y2 + 2yd + d 2 = x2 + y2 − 2yd + d 2 4dy = x2 y= x2 . ◭ x2 of Theorem 114. The distance of (x.46. The graph is in figure 2. We can draw a hyperbola as follows.46
Chapter 2 Proof: Let (x. √ √ x and of y = − x. d) is x2 + (y − d)2. we want = 1. 0). that is. The vertex of the parabola is at 4 (0. and the other to F2 . 117 Example Taking square roots on x = y2 . 0).
6 Problem The points A(0. 2. 2.4 Problem Draw the curve x2 + 2x + 4y2 − 8y = 4. Hyperbolas. 0) and directrix x = −d is x = 4d 2. determine 2 the coordinates of B and C. 2. If △ABC is equilateral. 1). 2.3 Problem Find the equation of the parabola with directrix y = −x and vertex at (1. Find A the equation of the curve it describes.54. of a parabola with focus at (d. and Ellipses
49
Homework
2. C B
Figure 2.8.8.8. and C lie on the parabola x2 y= as shewn in figure 2. 3) and the line x = −4.2 Problem Find the focus and the directrix of the parabola x = y2 . 0) .5 Problem The point (x.8.Parabolas. Prove that the equation y2 .
.8. B.8.1 Problem Let d > 0 be a real number.6.54: Problem 2. y) moves on the plane in such a way that it is equidistant from the point (2.8.
3. 50
. 5. 3. a set of real number inputs—usually an interval or a finite union of intervals—called the domain of the function.3
Functions
This chapter introduces the central concept of a function. that is. The domain of f is denoted by Dom( f ). called the target set of the function. assigning to every input a unique output. we refer to the "function f " when all the other descriptors of the function are understood.
Dom ( f ) 125 Definition By a (real-valued) function f : x dients: 1. The output of x under f is also referred to as the image of x under f .1 Basic Definitions
f Im ( f ) Target ( f ) Dom ( f )
Figure 3. We will only concentrate on functions defined by algebraic formulæ with inputs and outputs belonging to the set of real numbers.
→ Target ( f ) → f (x)
we mean the collection of the following ingre-
2. The target set of f is denoted by Target ( f ). We usually denote a typical input by the letter x. an assignment rule or formula.
Dom( f ) 127 Definition The image of a function f : x
→ Target ( f ) → f (x)
is the set
Im ( f ) = { f (x) : x ∈ Dom ( f )}. See figure 3. 4. and is denoted by f (x). a name for the function. an input parameter . We will introduce some basic definitions and will concentrate on the algebraic aspects. This assignment rule for f is usually denoted by x → f (x). as they pertain to formulæ of functions. also called independent variable or dummy variable. The subject of graphing functions will be taken in subsequent chapters.1. Usually we use the letter f . the collection of all outputs of f .1: The main ingredients of a function. a set of possible real number outputs—usually an interval or a finite union of intervals—of the function. 126 Definition Colloquially.
then the number of functions from A to B is mn . Since the square of every real number is positive. Observe that Im ( f1 ) = {c. not the actual outputs.
→ x2
4. b} and target set {c. so a ∈ Im ( f ). +∞[ ⊆ Im ( f ). For. We conclude that Im ( f ) = [0. an are the elements of A.
2. +∞[. This means that [0.Basic Definitions
51
! Necessarily we have Im ( f ) ⊆ Target ( f ). giving a total of m · · · m = mn .
It is easy to see that if A has n elements and B has m elements. . f (− 2) √ 3. m choices for the output of a2 .
128 Example Find all functions with domain {a. . C++. Observe that Im ( f1 ) = {c. one defines functions by statements like int f(double). What is Im ( f )? Solution: ◮ We have 1. f3 (b) = d. if a1 .
n times
!
possibilities. . f (− 2) = (− 2)2 = 2 √ √ √ √ √ 3. . f4 given by f4 (a) = d. namely: 1. this calculation is in fact very difficult. and Java. f2 given by f2 (a) = f2 (b) = d. f (0) √ 2. . a2 . d}. Now. +∞[. f3 given by f3 (a) = c. . . +∞[. Observe that Im ( f1 ) = {c}. f (0) = 02 = 0 √ √ 2. let a ∈ [0. m choices for the output of an . there are 22 = 4 such functions. and that the output will be allocated enough memory to carry an integer variable. Then √ √ a ∈ R and f ( a) = a. d}. f (1 − 2) = (1 − 2)2 = 12 − 2 · 1 · 2 + ( 2)2 = 3 − 2 2 → R . f1 given by f1 (a) = f1 (b) = c. d}. then there are m choices for the output of a1 . In most cases. 3. 129 Example Consider the function f: R x Find the following: 1. Solution: ◮ Since there are two choices for the output of a and two choices for the output of b. Observe that Im ( f2 ) = {d}. f (1 − 2) 4.
. f4 (b) = c. but we will see later on that these two sets may not be equal. we have Im ( f ) ⊆ [0. The target set must be large enough to accommodate all the possible outputs of a function. ◭
4. In some computer programming languages like C.
◭ In the above example it was relatively easy to determine the image of the function. This is the reason why in the definition of a function we define the target set to be the set of all possible outputs. . This tells the computer that the input set is allocated enough memory to take a double (real number) variable.
The functions f and h are different functions. at least in Calculus. we will see the importance of choosing an appropriate target set. every input must have a defined output. In a function. this is wasteful. The√ above rule is telling us that every output belongs to Z. Their target sets are identical. 2. Since f (0) is undefined. This means that the only two things that can be different are the names of the functions and the name of the input parameter. f (1 − 2) = 3 − 2 2 ∈ Z. ◭ Upon consideration of the preceding example. the "input" may not be part of the domain of the function. 133 Example Consider the functions Z → x Z
2
→ →
R 1 x2
. 131 Example Does f: R x define a function? Solution: ◮ No.
→ x
→ s
→ x
Then the functions f and g are the same function. this is not a function. Their assignment rules are identical.
f:
Determine:
. From the point of view of Computer Programming. since for example. The target set is not large enough to accommodate all the √ outputs. the reader may wonder why not then. We must pay special attention to the fact that although a formula may make sense for a "special input".
f:
g:
Z → s
Z
2
h:
Z → x
R
2
. ◭ 132 Definition (Equality of Functions) Two functions are equal if 1. as we would be allocating more memory than really needed. 134 Example Consider the function N \ {0} → x → Q 1 1 x+ x . Their domains are identical.
Chapter 3
→ x2
Solution: ◮ No. as their target sets are different.52 130 Example Does f: R x define a function? → Z . But this is not true. This is in fact what is done in practice. select as target set the entire set R. When we introduce the concept of surjections later on in the chapter. 3.
f (2) 3.
Figure 3. For example.3 does not represent a function. that is −1 is not part of the domain. Also important in the definition of a function is the fact that the output must be unique. if A ⊆ R.2: Not a function.
! In general. f (1) 2. f (2) = = = 1 5 5 2+ 2 2 1 1 1 2 = 3. f (−1) Solution: ◮ 1 = 1 2 1+ 1 2 1 1 2. since the last element of the domain is assigned to two outputs.
.2 does not represent a function.Basic Definitions 1. Thus Id (−1) = −1. we will give some miscellaneous examples on evaluation of functions. Id (0) = 0. f = = 1 1 1 2 5 + +2 2 1 2 2 4. etc. the identity function on the set A is defined and denoted by
Id A : A → A x → x .3: Not a function. f (1) = 1 ◭ It must be emphasised that the exhaustion of the elements of the domain is crucial in the definition of a function. f (−1) is undefined.
Figure 3. the diagram in figure 3.
This function assigns to every real its own value. 1. 135 Example (The Identity Function) Consider the function R x → R → x
Id :
. as some elements of the domain are not assigned. For example. the diagram in 3.
To conclude this section. as −1 ∈ N \ {0}. f 1 2
53
4. Id (4) = 4.
Figure 3.
→ x
This function assigns to every real its square. the graph of the square root function is the half parabola that appears in figure 3.1]. the graph of the square function is a parabola.
Figure 3. the formula for Sc only makes sense in the interval [−1. and it is presented in in figure 3.8.5: Fails the vertical line test. 144 Example (Absolute Value Function) Consider the function R x → R
AbsVal :
.7: AbsVal
145 Example (The Square Function) Consider the function R x → R
2
Sq :
.6: Id
Figure 3. 146 Example (The Square Root Function) Consider the function [0. the graph of the identity function is a straight line.
Figure 3.10. We will examine this more closely in the next section.
1 Since we are concentrating exclusively on real-valued functions. 147 Example (Semicircle Function) Consider the function1 [−1. By Theorem 114. 1] → x → R 1 − x2
Sc :
.
By Example 89.Graphs of Functions and Functions from Graphs 143 Example (Identity Function) Consider the function R x → R → x
57
Id :
. Not a function. the graph of Sc is the upper unit semicircle. which is shewn in figure 3.9. +∞[ → x → R √ x
Rt :
.4: Fails the vertical line test.
By Example 117.
.
→ |x|
By Example 108. the graph of the absolute value function is that which appears in figure 3.7.
By Theorem 93. Not a function.
12: Example 149.58 148 Example (The Reciprocal function) Consider the function2 R \ {0} → R x → 1 x
Chapter 3
Rec :
. the graph of the reciprocal function is the hyperbola shewn in figure 3.8: Sq
Figure 3. 1} → R with assignment rule −x if x < −1 2 f (x) = if − 1 < x < 1 x x if x > 1
Its graph appears in figure 3.9: Rt
Figure 3. and the image of the functional curve is its shadow on the y-axis. 149 Example Consider the function f : R \ {−1. 150 Definition A functional curve on the plane is a curve that passes the vertical line test. The alert reader will notice that. for example.
.12.
2
The formula for Rec only makes sense when x = 0.11. We will now present a related concept in order to alleviate this problem. the two different functions R x → R R x → [0.11: Rec
We can combine pieces of the above curves in order to graph piecewise defined functions. It is then difficult to recover all the information about a function from its graph. The domain of the functional curve is the "shadow" of the graph on the x-axis.
Figure 3. in particular.
Figure 3. it is impossible to recover its target set. +∞[ → x2
f:
g:
→ x2
possess the same graph.
By Example 120.10: Sc
Figure 3.
Figure 3. 7}. If f is bijective. and hence. . f (x2 ). . 6.21: Bijective.21 that if the domain and the target set of a function are finite. and 4 for the image of 3.20: Neither injective nor surjective. .18: Injective.
. If f were injective then f (x1 ). and and B m elements. there are m choices for f (x1 ). then the number of injections from A to B is m(m − 1)(m − 2) · · ·(m − n + 1). 5. m − n + 1 choices for f (xn ). not surjective. If n ≤ m.21 present various examples. and let A and B be finite. .Injections and Surjections
73
f is said to be surjective or onto if Target ( f ) = Im ( f ). xn } and B = {y1 . If f is surjective then m ≤ n. In this case. we refer the reader to any good book in Combinatorics. A function is thus injective if different inputs result in different outputs. ym }. and it is surjective if every element of the target set is hit. . Similarly. . there is an xi with f (xi ) = yk . and (iii) those that map to exactly one element of A. then m = n. and among the yk . . there are 4 · 3 · 2 = 24 injections from A to B. then n ≤ m.
Figure 3. 187 Theorem Let f : A → B be a function. if (∀b ∈ B) (∃a ∈ A) such that f (a) = b. Thus there are m(m − 1)(m − 2) · · ·(m − n + 1) injections from A to B. Hence n ≤ m. .
Figure 3. there are 3 · 3 · 3 · 3 = 81 functions from B to A.
It is apparent from figures 3. since there are 4 possibilities for the image of 1. We make the precise statement in the following theorem. How many functions are there from A to B? How many functions are there from B to A? How many injections are there from A to B? How many surjections are there from B to A? Solution: ◮ There are 4 · 4 · 4 = 64 functions from A to B. and so n ≥ m. . . surjective or bijective. Thus there are at least m different images.18 through 3. m − 1 choices for f (x2 ). f (xn ) are all distinct. u To find the number of surjections from a finite set to a finite set we need to know about Stirling numbers and inclusionexclusion. . Figures 3. (ii) those that map to exactly two elements of A. y2 . not injective. If f is injective. with A having n elements. . x2 .19: Surjective.
Figure 3. 2. 188 Example Let A = {1. By Theorem 187. . Proof: Let A = {x1 . then there are certain inequalities that must be met in order for the function to be injective. . If f were surjective then each yk is hit. . .18 through 3. f is bijective if it is both injective and surjective. The number a is said to the the pre-image of b. That is. and for each. 4 for the image of 2. The 34 functions from B to A come in three flavours: (i) those that are surjective. 3} and B = {4.
The next few examples shew how to find the image of a formula in a few easy cases. Since there is nothing holy about choosing 1 ∈ A. See figures 3.23: Fails horizontal line test: not. say 1 ∈ A. +∞[ → x R
is no x ∈ R with a(x) = −1. we have taken out all the functions that miss one or two elements of A. Since there are three 2-element subsets in A—namely {1. we conclude that there are 3 · 24 from B to A that skip either one or two elements of A. and there R x → [0.
!
Figure 3. +∞[ x → x
2
is bijective. we have taken out twice those that miss one element. Now take two particular elements of A.
Figure 3. 2}. There are 14 functions from B to {3}.injective. 3} or they may skip an element. {1. Hence we put those back in and we obtain 34 − 3 · 24 + 3 · 14 = 36 surjections from B to A. In considering the difference 34 − 3 · 24 . this means that there are 24 functions from B to A that skip the 1 and may or may not skip the 2 or the 3. There are 24 functions from B to {2. This is why when we talk about function. 3}—this means that there are 3 · 14 functions from B to A that map precisely into one element of A. it is particularly difficult to know in advance what it set of outputs is going to be. Determine Im ( f ). 190 Example Let f : R → R. The function d :
[0. The function b :
is surjective but not injective. +∞[ → x2 [0.
Given a formula. Coupling this with the 1 ∈ A.22 and 3. 3}. Notice that some of these may map to the whole set {2. f (x) = x2 + 2x + 3. we specify the target set to be a canister for every possible value. 3}.22: Passes horizontal line test: injective. ◭ It is easy to see that a graphical criterion for a function to be injective is that every horizontal line crossing the function must meet it at most one point.74
Chapter 3 Take a particular element of A. and {2.23.
189 Example The a :
R x
→
R
2
→ x
is neither injective nor surjective. but in so doing. a(−2) = a(2) = 4 but −2 = 2. 2} ⊆ A. Solution: ◮ Observe that x2 + 2x + 3 = x2 + 2x + 1 + 2 = (x + 1)2 + 2 ≥ 2.
. The function c :
→ x2
is injective but not surjective. +∞[ → [0. For example. To find the number of surjections from B to A we weed out the functions that skip elements. say {1.
Inversion 202 Theorem A function f : A → B is invertible if and only if it is a bijection. Proof: Assume first that f is invertible. Then there is a function f −1 : B → A such that f ◦ f −1 = Id B and f −1 ◦ f = Id A . (3.3)
79
Let us prove that f is injective and surjective. Let s,t be in the domain of f and such that f (s) = f (t). Applying f −1 to both sides of this equality we get ( f −1 ◦ f )(s) = ( f −1 ◦ f )(t). By the definition of inverse function, ( f −1 ◦ f )(s) = s and ( f −1 ◦ f )(t) = t. Thus s = t. Hence f (s) = f (t) =⇒ s = t implying that f is injective. To prove that f is surjective we must shew that for every b ∈ f (A) ∃a ∈ A such that f (a) = b. We take a = f −1 (b) (observe that f −1 (b) ∈ A). Then f (a) = f ( f −1 (b)) = ( f ◦ f −1 )(b) = b by definition of inverse function. This shews that f is surjective. We conclude that if f is invertible then it is also a bijection. Assume now that f is a bijection. For every b ∈ B there exists a unique a such that f (a) = b. This makes the rule g : B → A given by g(b) = a a function. It is clear that g ◦ f = Id A and f ◦ g = Id B . We may thus take f −1 = g. This concludes the proof. u
b a c
u
y x z
y x z
v
b a c
Figure 3.24: A function and its inverse. We will now give a few examples of how to determine the assignment rule of the inverse of a function. 203 Example Assume that the function f: R \ {−1} → R \ {1} x is a bijection. Determine its inverse. Solution: ◮ Put → x−1 x+1
Since by Theorem 107, (x, f (x)) and ( f (x), x) are symmetric with respect to the line y = x, the graph of a function f is symmetric with its inverse with respect to the line y = x. See figures 3.25 through 3.27.
209 Example Figures 4.2 through 4.4 shew various translations of f : [−4; 4] → [−2; 1] in figure 4.1. Its translation a : [−4; 4] → [−1; 2] one unit up is shewn in figure 4.2. Notice that we have simply increased the y-coordinate of every point on the original graph by 1, without changing the x-coordinates. Its translation b : [−5; 3] → [−2; 1] one unit left is shewn in figure 4.3. Its translation c : [−5; 3] → [−1; 2] one unit up and one unit left is shewn in figure 4.4. Notice how the domain and image of the original curve are affected by the various translations. 210 Example Consider f: R x → R
2
Homework
4.1.1 Problem Graph the following curves: 1. y = |x − 2| + 3 3. y = 1 4.1.2 Problem What is the equation of the curve y = f (x) = x3 − x after a successive translation one unit down and two units right? 4.1.3 Problem Suppose the curve y = f (x) is translated a units vertically and b units horizontally, in this order. Would that have the same effect as translating the curve b units horizontally first, and then a units vertically?
6. y = 2x2 + 8x 1 4.2.2 Problem The curve y = experiences the following succesx sive transformations: (i) a translation one unit left, (ii) a vertical dilatation by a factor of 2, (iii) a translation one unit down. Find its resulting equation and make a rough sketch of the resulting curve.
1. f (+2+) Solution: ◮ 2.53: b : x →
x . but not precisely at x = a.54. x = 0.54: c : x → . 5. ◭
3.53? −1 + 1 b(0) = = 0? In figure 4. What value can we reasonably assign in figure 4. f (+2) 6. Thus f (−2+) = 2 + 2(−2) = −2. To find f (−2+) we look at the definition of f just to the right of −2. Which value can we reasonably assign to f (a)? Consider the situations depicted in figures 4. To find f (−2−) we look at the definition of f just to the left of −2. but not necessarily exhaustive. Thus f (+2+) = 6. It is continuous on the interval I if it is continuous on every point of I.52: a : x → |x|. Thus f (+2−) = 2 + 2(2) = 6.
97
if − 4 ≤ x < −2 if x = −2 if − 2 < x < +2 if + 2 ≤ x ≤ 4
Determine
4. A function f is said to be right continuous at the point x = a if f (a) = f (a+). f (−2−) 2. f (−2+) 4. what value would it be reasonable to assign? c(0) = 0?. f (−2) 3.Behaviour of the Graphs of Functions 237 Example Let f : [−4.54. |x|
1 Figure 4.
Let us consider the following situation. f (+2−) 5. To find f (+2+) we look at the definition of f just to the right of +2.
. x = 0.52 it seems reasonably to assign a(0) = 0. Thus f (−2−) = (−2)2 + 1 = 5. 4] → R be defined as follows: 2 x +1 2 f (x) = 2 + 2x 6 1. Let f be a function and a ∈ R. To find f (+2−) we look at the definition of f just to the left of +2. A function f is said to be continuous at the point x = a if f (a−) = f (a) = f (a+). f (+2) = 6. x
238 Definition A function f is said to be left continuous at the point x = a if f (a−) = f (a). c(0) = +∞?. In figure 4.
Figure 4.
Figure 4.52 through 4. 6. c(0) = −∞? The 2 situations presented here are typical. f (−2) = 2. Assume that f is defined in a neighbourhood of a. x = 0.
An inflexion point is a point where a graph changes convexity. Thus there exist α . Consider now a continuous function in a closed interval [a. b] such that f (α ) ≤ f (x) ≤ f (β ). −2]
2 3x + xa if x ∈] − 2. The maxima and the minima of a function are called its extrema.3 Extrema
242 Definition If there is a point a for which f (x) ≤ f (M) for all x in a neighbourhood centred at x = M then we say that f has a local maximum at x = M. a < b =⇒ g(a) < g(b)). since otherwise it would be unbounded and hence not continuous. if there is a point m for which f (x) ≥ f (m) for all x in a neighbourhood centred at x = m then we say that f has a local minimum at x = m. b. 1]. Similarly. +∞[
Solution: ◮ Since f (−2−) = f (−2) = 6 − 2 = 4 and f (−2+) = 3(−2)2 − 2a = 12 − 2a we need f (−2−) = f (−2+) =⇒ 4 = 12 − 2a =⇒ a = 4.
! If the function f is (strictly) increasing.
f (λ a + (1 − λ )b) ≤ f (a)λ + (1 − λ ) f (b). 243 Definition A function f : A → B is convex in A if ∀(a.6.98 Heuristically speaking. β in [a. Unless it is a horizontal line there. λ ) ∈ A2 × [0. b−a
4. it cannot go down forever. f reaches maxima and minima in [a. b. b].
. A function is monotonic if it is either (strictly) increasing or decreasing. its graph goes up and down in [a. 241 Theorem A function f is (strictly) increasing if for all a < b for which it is defined f (b) − f (a) ≥0 b−a (respectively. Similarly.6. strictly increasing) if a < b =⇒ f (a) ≤ f (b) (respectively. and viceversa. a continuous function is one whose graph has no "breaks.
The following theorem is immediate. It cannot go up forever. a function g : A → B is concave in A if ∀(a.4 Convexity
We now investigate define the "bending" of the graph of a function. find a.2 Monotonicity
240 Definition A function f is said to be increasing (respectively." 239 Example Given that f (x) = is continuous. b].
g(λ a + (1 − λ )b) ≥ g(a)λ + (1 − λ )g(b). 6+x
Chapter 4
if x ∈] − ∞. By the intervals of monotonicity of a function we mean the intervals where the function might be (strictly) increasing or decreasing. ◭
4. its opposite − f is (strictly) decreasing.6.
By the intervals of convexity (concavity) of a function we mean the intervals where the function is convex (concave). f (b) − f (a) > 0). λ ) ∈ A2 × [0. strictly decreasing) if a < b =⇒ g(a) ≤ g(b) (respectively.
4. a function g is (strictly) decreasing if for all a < b for which it is defined g(b) − g(a) ≤0 b−a (respectively. 1]. b−a
Similarly. A function g is said to be decreasing (respectively. b].
g(b) − g(a) < 0). a < b =⇒ f (a) < f (b)). Similarly. that is.
x → {x}
99
By Lemma 15. If n ∈ Z and if 3.6. between 0 and 1 it will have output 0. ± 3 4. always taking the smaller of the two consecutive integers.5 Problem √ Give √ example √ a function discontinuous at the an √ of √ points ± 3 1. the graph of the function bends upwards. 1} but continuous everywhere else. For example.56: A concave curve. or the integer just to the left.55: A convex curve
Figure 4. find a. −π = −4. x → x .
Homework
4.. In other words. x is x if x is an integer. 245 Definition The ceiling x of a real number x is the unique integer defined by the inequality x −1 < x ≤ x .6.
. For example 3 = 3. then its opposite − f is concave. 0. geometrically speaking. Its graph has the staircase shape found in figure 4..7 The functions x → x .
is continuous.3 Problem Given that 2 x −1 f (x) = 2x + 3a
4.6.
then x = n.
4. x → x . if x is not an integer. This means that the function x → x is constant between two consecutive integers. ± 3 3.4 Problem Let n be a strictly positive integer. 4. find a. b] for 0 ≤ λ ≤ 1. Hence.6. it will have output 1. x → {x}
244 Definition The floor x of a real number x is the unique integer defined by the inequality x ≤ x < x + 1.2 Problem Give an example of a function which is discontinuous on the set {−1. . .57. ± 3 5. a convex function is one such that if two distinct points on its graph are taken and the straight line joining these two points drawn. etc.9 = 3.
n ≤ x < n + 1.
4. Given that if x = 1 xn − 1 x−1 f (x) = a is continuous.1 Problem Given that x2 − 1 x−1 f (x) = a if x = 1 4. Notice that if f is convex.
Figure 4. In other words. λ a + (1 − λ )b lies in the interval [a. if x ≤ 1 if x > 1 if x = 1 if x = 1 is continuous. ± 3 2. between 1 and 2.6.The functions x → x . find a. then the midpoint of that straight line is above the graph. .
where α ∈ R. 253 Example What is the degree of the polynomial identically equal to 0? Put p(x) ≡ 0 and. It has no roots. In this chapter we will only study the case when α is a positive integer. 2π 2π A polynomial of degree 2 is also called a quadratic polynomial or quadratic function. of degree 0. ak ∈ R. The degree n of the polynomial p is denoted by deg p.1 Power Functions
254 Definition A power function is a function whose formula is of the form x → xα .
where the ak are constants. By the quadratic formula b has the two roots √ √ −1 + 1 + 4π 3 −1 − 1 + 4π 3 x= and x= . is a polynomial of degree 1. 250 Example Here are a few examples of polynomials. In symbols. and we write p(x) ∈ Z[x]. We attach to it. p(x)q(x) = (an xn + an−1xn−1 + · · · + a1 x + a0)(bm xm + bm−1 xm−1 + · · · + b1 x + b0) = an bm xm+n + · · · +. if the ak are real numbers then we say that p has real coefficients and we write p(x) ∈ R[x]. √ • b(x) = π x2 + x − 3 ∈ R[x]. since it is never zero. is a constant polynomial.
5. 102
. an = 0. u 252 Example The polynomial p(x) = (1 + 2x + 3x3)4 (1 − 2x2)5 has leading coefficient 34 (−2)5 = −2592 and degree 3 · 4 + 2 · 5 = 22. Thus the 0-polynomial does not have any finite degree. • C(x) = 1 · x0 := 1. nonsense. Then by Theorem 251 we must have deg pq = deg p + deg q = deg p + 1. But if deg p were finite then deg p = deg pq = deg p + 1 =⇒ 0 = 1. with non-vanishing leading coefficient an bm .5
Polynomial Functions
249 Definition A polynomial p(x) of degree n ∈ N is an expression of the form p(x) = an xn + an−1xn−1 + · · · + a1x + a0. But pq is identically 0. In this chapter we learn how to graph polynomials all whose roots are real numbers. deg pq = deg p + degq. is a polynomial of degree 2 and leading coefficient π . 1 • a(x) = 2x + 1 ∈ Z[x]. with an = 0 and bm = 0 then upon multiplication. 251 Theorem The degree of the product of two polynomials is the sum of their degrees. A polynomial 2 of degree 1 is also known as an affine function. if p. etc. and q(x) = bm xm + bm−1 xm−1 + · · · + b1 x + b0. The coefficient an is called the leading coefficient of p(x). q are polynomials. and hence deg pq = deg p. and leading coefficient 2. It has x = − as its only root. A root of p is a solution to the equation p(x) = 0. q(x) = x + 1. Proof: If p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 . degree −∞. by convention. say. If the ak are all integers then we say that p has integer coefficients.
Affine Functions
103
If n is a positive integer. If. A function of the form x → mx + k is called an affine function. Hence every real number is an image of f meaning that Im ( f ) = R. then we call the function x → mx a linear function. we have the square function x → x2 whose graph is the parabola y = x2 encountered in example 115. if m = 0. the function x → 1 is a constant function. and convexity. monotonicity. an affine function is everywhere continuous. This means that it does not bend upwards or downwards (or that it bends upwards and downwards!) always. it must be a straight line. whose graph is the straight line y = 1 parallel to the x-axis. For n = 1. 1]. If m = 0 then x → mx + k has a k unique zero x = − . This method of obtaining graphs of functions is quite limited.
. Also. for n = 2. Also given any a ∈ R we have f (x) = a =⇒ mx + k = a =⇒ x =
a−k . 256 Theorem (Graph of an Affine Function) The graph of an affine function R x → R
f:
→ mx + k
is a continuous straight line. k In particular. however. This means that f is a strictly increasing function for m > 0 and strictly decreasing for m < 0. we will derive the shape of their graphs once more. which bisects the first and third quadrant. Clearly.
Figure 5. It is strictly increasing if m > 0 and strictly decreasing if m < 0.1 through 5.3: x → x2 . we call x → k a constant function.3 for easy reference. These graphs were not obtained by fiat. We have already encountered a few instances of power functions. then m Im ( f ) = {k}. Let a < b. the function x → x is the identity function. Since f (λ a + (1 − λ )b) = m(λ a + (1 − λ )b) + k = mλ a + mb − mbλ + k = λ m f (a) + (1 − λ )m f (b). Then f (b) − f (a) mb + k − ma − k = = m. and hence.1: x → 1.2 Affine Functions
255 Definition Let m. For n = 0. m Proof: Since for any a ∈ R. we are interested in how to graph x → xn . then x = − is the only solution to the equation f (x) = 0.2: x → x. f (a+) = f (a) = f (a−) = ma + k. We reproduce their graphs below in figures 5. If m = 0 then Im ( f ) = R. whose graph is the straight line y = x. if a = 0. as a view of introducing a more general method that argues from the angles of continuity.
Figure 5. k be real number constants. m which is a real number as long as m = 0. and hence.
5. b−a b−a which is strictly positive for m > 0 and strictly negative for m < 0. we demonstrated that the graphs are indeed straight lines in Theorem 93.u This information is summarised in the following tables. k = 0 and m = 0.
Figure 5. Let λ ∈ [0. an affine function is both convex and concave.
The graphs above were obtained by geometrical arguments using similar triangles and the distance formula. In the particular case that m = 0.
we study how to graph polynomials of the form p(x) = a(x − r1)m1 (x − r2 )m2 · · · (x − rk )mk . we must investigate the global behaviour of the polynomial. Figure 5. x x x
since as x → ±∞. and that the polynomial is said to split in R if all the solutions to the equation p(x) = 0 are real. Then p(−∞) = (signum (an ))(−1)n ∞. Figure 5. be a polynomial with real number coefficients.8 Graphs of Polynomials
Recall that the zeroes of a polynomial p(x) ∈ R[x] are the solutions to the equation p(x) = 0. Proof: If x = 0 then p(x) = an xn + an−1xn−1 + · · · + a1x + a0 = an xn 1 + a0 a1 an−1 + · · · + n−1 + n ∼ an xn . In this section we study how to graph polynomials that split in R. that is. • crosses the x-axis at x = ri if mi is odd. Then the graph of the polynomial p(x) = a(x − r1)m1 (x − r2 )m2 · · · (x − rk )mk .Graphs of Polynomials
113
x
−∞
0
+∞
ր f (x) = xn ր Figure 5.
Thus a polynomial of odd degree will have opposite signs for values of large magnitude and different sign. • is tangent to the x-axis at x = ri if mi is even.26: y = x5 . with n ≥ 3 odd. that is. the quantity in parenthesis tends to 1 and so the eventual sign of p(x) is determined by an xn . 273 Theorem Let a = 0 and the ri are real numbers and the mi be positive integers. and we must also investigate the local behaviour around each of the roots ri . 272 Theorem Let p(x) = an xn + an−1xn−1 + · · · + a1 x + a0 an = 0.24: x → xn .25: y = x3 . where a ∈ R \ {0} and the ri are real numbers and the mi ≥ 1 are integers. We start with the following theorem. p(+∞) = (signum(an ))∞. u We now state the basic result that we will use to graph polynomials. Figure 5. what happens as x → ±∞.
. which gives the result. 271 Theorem A polynomial function x → p(x) is an everywhere continuous function. which we will state without proof. 0
5.27: y = x7 . and a polynomial of even degree will have the same sign for values of large magnitude and different sign. To graph such polynomials.
The graph is shewn in figure 5. so the graph crosses the x-axis changing convexity at x = −2.
.29: 275. Solution: ◮ The dominant term of (x + 2)2x(1 − x)2 is x2 · x(−x)2 = x5 . x = 0. Hence if p(x) = (x + 2)2x(1 − x)2 then p(−∞) = (−∞)5 = −∞ and p(+∞) = (+∞)5 = +∞. Example
5 4 3 2 1 0 −1 −2 −3 −4 −5
−5 −3 −10 1 2 3 4 5 −4 −2 Figure 5. u
5 4 3 2 1 0 −1 −2 −3 −4 −5
−5 −3 −10 1 2 3 4 5 −4 −2 Figure 5. By Theorem 273. By Theorem 273. the theorem follows at once from our work in section 5. and 2 16 so the graph crosses the x-axis at x = 1 . You may also assume that the graph of the polynomial changes concavity at x = 2. Solution: ◮ We have p(x) = (x + 2)x(x − 1) ∼ (x) · x(x) = x3 . This means that for large negative values of x the graph will be on the negative side of the y-axis and that for large positive values of x the graph will be on the positive side of the y-axis. The graph is shewn in figure 5. Example
274 Example Make a rough sketch of the graph of y = (x + 2)x(x − 1). Hence p(−∞) = (−∞)3 = −∞ and p(+∞) = (+∞)3 = +∞. ◭ 275 Example Make a rough sketch of the graph of y = (x + 2)3x2 (1 − 2x). The polynomial in figure ??. the graph crosses the x-axis changing convexity at x = −2. You may assume that the points marked below with a dot through which the polynomial passes have have integer coordinates. The graph is shewn in figure 5.31: 277. in a neighbourhood of x = −2. p(x) ∼ 20(x + 2)3 . it is tangent to the x-axis 1 at x = 0 and it crosses the x-axis at x = 2 . has degree 5. Determine where it achieves its local extrema and their values. In a neighbourhood of x = . the graph crosses the x-axis at x = −2. which means that for large negative values of x the graph will be on the negative side of the y-axis and for large positive values of x the graph will be on the positive side of the y-axis. Example
5 4 3 2 1 0 −1 −2 −3 −4 −5
−5 −3 −10 1 2 3 4 5 −4 −2 Figure 5.29. Example
5 4 3 2 1 0 −1 −2 −3 −4 −5
−5 −3 −10 1 2 3 4 5 −4 −2 Figure 5. p(x) ∼ (1 − 2x). ◭ 277 Example . and x = 1. By Theorem 273. Solution: ◮ We have (x + 2)3 x2 (1 − 2x) ∼ x3 · x2 (−2x) = −2x6 .114 • has a convexity change at x = ri if mi ≥ 3 and mi is odd. Hence if p(x) = (x + 2)3 x2 (1 − 2x) then p(−∞) = −2(−∞)6 = −∞ and p(+∞) = −2(+∞)6 = −∞.1.
Chapter 5
Proof: Since the local behaviour of p(x) is that of c(x − ri )mi (where c is a real number constant) near ri .28.28: 274. Determine where it changes convexity.. which means that for both large positive and negative values of x the graph will be on the negative side of the y-axis. In a neighbourhood of 0. as x → +∞.30: 276.30. ◭ 2 276 Example Make a rough sketch of the graph of y = (x + 2)2x(1 − x)2 . 25 1 p(x) ∼ 8x2 and the graph is tangent to the x-axis at x = 0.
p(x) = an xn + an−1xn−1 + · · · + a1x + a0. s = an t sn tn + an−1 sn−1 t n−1 + · · · + a1 s + a0 . concluding the proof. then it must have at most its degree number of roots. It does not say that a polynomial must possess a root. then s divides a0 and t divides an . 288 Theorem (Fundamental Theorem of Algebra) A polynomial of degree at least one with complex number coefficients has at least one complex root. without a proof. then it has at most n roots. The Fundamental Theorem of Algebra implies then that a polynomial of degree n has exactly n roots (counting multiplicity).Polynomials 285 Example Find the value of a so that the polynomial t(x) = x3 − 3ax2 + 2 be divisible by x + 1. That all polynomials have at least one root is much more difficult to prove. 289 Corollary If the polynomial p with integer coefficients. 287 Corollary If a polynomial of degree n had any roots at all. from where we deduce that t divides an . 0 = an sn + an−1sn−1t + · · · + a1 st n−1 + a0t n . Proof: If it had at least n + 1 roots then it would have at least n + 1 factors of degree 1 and hence degree n + 1 at least. a contradiction. Since both sides are integers.
Proof: We are given that 0= p Clearing denominators. and since s and t have no factors in common. u 290 Example Factorise a(x) = x3 − 3x − 5x2 + 15 over Z[x] and over R[x]. This last equality implies that −a0t n = s(an sn−1 + an−1sn−2t + · · · + a1t n−1 ). we must have 1 0 = t(−1) = (−1)3 − 3a(−1)2 + 2 =⇒ a = . Solution: ◮ By Ruffini's Theorem 284. u Notice that the above theorem only says that if a polynomial has any roots. 3 ◭
117
286 Definition Let a be a root of a polynomial p. We will quote the theorem. A more useful form of Ruffini's Theorem is given in the following corollary. then s must divide a0 . We also gather that −an sn = t(an−1 sn−1 + · · · + a1st n−2 + a0t n−1 ). This means that p can be written in the form p(x) = (x − a)m q(x) for some polynomial q with q(a) = 0. We say that a is a root of multiplicity m if p(x) is divisible by (x − a)m but not by (x − a)m+1 .
s has a rational root t ∈ Q (here s t
!
is assumed to be in lowest terms). t
.
what is the balance at the end of a full year?2 Suppose a dollars are deposited.
n
2 "Quæritur. 1. hence a proving midpoint convexity. Recall that f is convex if for arbitrary 0 ≤ λ ≤ 1 we have f (λ s + (1 − λ )t) ≤ λ f (s) + (1 − λ ) f (t). We will not be able to prove this quickly. A similar argument proves that for 0 < a < 1.1. f would be decreasing. we will just content with proving midpoint convexity: we will prove that f This is equivalent to a which in turn is equivalent to 2≤a
s−t 4 s−t 2 s+t 2
127
s+t 2
≤
1 1 f (s) + f (t). after the n-th time period. b1 = 1 + n After the second time period. If a > 1. x → 2x+3
7. quantum ipsi finito anno debeatur?"
.
! The line y = 0 is an asymptote for x → a . then
ax
→ +∞ as x → −∞ and
ax
→ 0 as x → +∞.1 Problem Make rough sketches of the following curves. After the first time period. x → 2x 2. the balance is b2 = 1 + x x b1 = 1 + n n x n
n 2
a. To prove convexity will be somewhat more arduous. x → 2−|x| 4. ea lege. a straight line joining any two points of the curve lies above the curve. If
x x x
0 < a < 1. x → 2x + 3 5. si creditor aliquis pecuniam suam fœnori exponat. and the interest is added n times a year at a rate of x. that is.
Proceeding recursively. then a → 0 as x → −∞ and a → +∞ as x → +∞.
But the square of a real number is always non-negative. 2 2
1 1 ≤ as + at .
Homework
7.Homework whence f is increasing for a > 1. x → 2|x| 3. 2 2 +a
t−s 2
. u −a
t−s 4
2
≥ 0 =⇒ a
s−t 2
+a
t−s 2
≥ 2. ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur.2 The number e
Consider now the following problem. the balance will be bn = 1 + The study of the sequence en = 1 + 1 n a. first studied by the Swiss mathematician Jakob Bernoulli around the 1700s: Query: If a creditor lends money at interest under the condition that during each individual moment the proportional part of the annual interest be added to the principal. the balance is x a.
Hence a unit circle.. . We put an endpoint 0. −4π . 365 Definition A radian is a 1 th part of the circumference of a unit circle. We start again. 2π
1
Figure 9.. (2k + 2)π [. . 146
. −2π [∪[−2π . as shewn in figure 9. A radian is simply a real number! 2. 4π [∪[4π . this time going to the left and marking off intervals with endpoints at −2π . mark off intervals to the right of 0 with endpoints at 2π .
We have decomposed the real line into the union of disjoint intervals .16. . etc. . that is. If a central angle of a unit circle cuts an arc of x radians. −8π −6π −4π −2π 0π 2π 4π 6π 8π
Figure 9.9
Goniometric Functions
9. 6π [∪ .1: A radian.e. 3. Suppose now that we cut a unit circle into a "string" and use this string to mark intervals of length 2π on the real line. . has circumference 2π . ∪ [−6π .1 The Winding Function
Recall that a circle of radius r has a circumference of 2π r units of length. Since
1 2π
≈ 0. 4π . A semicircle has arc length 22 = π radians. .2. . a radian is about
π 4
length of
4 of the circumference of the unit circle. 2π [∪[2π . Observe that each real number belongs to one. −6π . −4π [∪[−4π .2: The Real Line modulo 2π . For example 100 ∈ [30π . −2π [. 32π [ and −9 ∈ [−4π . there is a unique integer k such that if x ∈ R then x ∈ [2π k. A quadrant or quarter part of a circle has arc 25 π radians. one with r = 1.
!
1. then the central angle measures x radians. i. .. . and only one of these intervals.. The sum of the internal angles of a triangle is π radians. etc. 0[∪[0. 6π . .
This shews that a + 2π k1 ≡ a + 2π k2 mod 2π . modulo 2π .
π 7π 5π 13π 11π . ◭
. .The Winding Function
147
a−b 366 Definition Given two real numbers a and b. 370 Definition We will call the procedure of finding a canonical representative for the class of x. we simply look for the integer k such that 2kπ ≤ x < (2k + 2)π . However. which is not an integer. If is not an integer. written a ≡ b mod 2π . 2π which being the difference of two integers is an integer. say. all the numbers of the form a + 2π k. k ∈ Z belong to the same residue class modulo 2π .
Given a real number x. However. We also say that a and b are representatives of the same residue class modulo 2π . Then (a + 2π k1) − (a + 2π k2) = k1 − k2 . we say that a and b belong to the same residue class mod 2π .− 3 3 3 3 3
! If a ≡ b
mod 2π then there exists an integer k such that a = b + 2π k. k2 . 2π 5π − (−7π ) 12π For example. exactly one representative x0 lies in the interval [0. Then then 0 ≤ x − 2kπ < 2π and so x − 2π k is the canonical representative of the class of x. 5π ≡ −7π mod 2π . 371 Example Reduce 5π mod 2π . reduction modulo 2π . an integer. as we can add any integral multiple of 2π to x and still lie in the same class. 5π ≡ 2π mod 2π as 2π 2π 3π 3 5π − 2π = = . 2π [. it is clear that there are infinitely many representatives of the class to which x belongs. since = = 6. Proof: Take two numbers of this form. we say that a is congruent to b modulo 2π . To find the canonical representative of the class of x. as we saw above. a + 2π k1 and a + 2π k2. we say that a and b are incongruent modulo 2π and we write a ≡ b mod 2π . u 369 Example Take x = π . Thus π is the canonical representative of the class to which 5π belongs. We call x0 the canonical representative of the class (to which x belongs modulo 2π ). 2π 2π 2 367 Definition If a ≡ b mod 2π . we have 5π ≡ 5π − 4π ≡ π mod 2π . with integers k1 . Then 3
π 3
≡ ≡ ≡ ≡
π 3 π 3 π 3 π 3
+ 2π − 2π + 4π − 4π
≡ ≡ ≡ ≡
7π 3
mod 2π mod 2π mod 2π mod 2π
π − 53
13π 3
− 11π 3
Thus all of belong to the same residue class mod 2π . if 2π a−b is an integer. 368 Theorem Given a real number a. Solution: ◮ Since 4π < 5π < 6π .− . .
n N ⌊ ε + 1⌋
t − 1 < ⌊t⌋ ≤ t... . . . has no limit (diverges). . . 4. (−1)n n. −1. xn . −1. 1. 1. 2 3 n
Appendix C
1 1 → 0 as n → +∞.1: Theorem 515. . . . 16. . no matter how small. 4. 514 Example The sequence 0. The same argument works for any distance. we need only to look at the terms after N = ⌊ ε + 1⌋ to see that. . . then the terms after N = ⌊ M⌋ + 1 satisfy (n > N) √ tn = n2 > N 2 = (⌊ M⌋ + 1)2 > M. . . .
| s
Figure C.00001 to 0. so we can eventually get arbitrarily close to 0.
6A 1 rigorous proof is as follows. .
.
7A
√ rigorous proof is as follows. −3. . has no limit (diverges). −1. .7 513 Example The sequence 1. . If M > 0 is no matter how large. as it bounces back and forth from −1 to +1 infinitely many times. . diverges to +∞. We claim that 512 Example The sequence 0. We had to wait a long time—till after the 100000-th term—but the sequence eventually did get closer than .00001 = 5 . . indeed. .
When is it guaranteed that a sequence of real numbers has a limit? We have the following result.204 510 Example The constant sequence 1.6 . 9. . if n > N. as the sequence gets arbitrarily large. converges to 1. 511 Example Consider the sequence 1 1 1 1. −5. . . .. 1. 2. then
sn = Here we have used the inequality
1 1 1 < = 1 < ε. .00001 of 0. any term 100000 10 after this one will be within . 1. ∀t ∈ R.. . . . . . 1.
| x0
| x1
| x2
| | | . If ε > 0 is no matter how small. −1. . . . Since the terms of the sequence get smaller and smaller. as it is unbounded and alternates back and forth positive and negative values. We only need to n 10 1 1 look at the 100000-term of the sequence: = 5 . 1. (−1)n . Suppose we wanted terms that get closer to 0 by at least . n2 . . . .
2. say an0 with s − ε ≤ an0 by virtue of the Approximation Property Theorem ??. Find its second term.
516 Definition A geometric sequence or progression is a sequence of the form
that is. . there must be a term of the sequence.
!
1. a rather uninteresting case. 2n → 0 as n → +∞ 2. diverges as n → +∞ 8. . Hence the 35-th term is √ √ 51 1 √ (−2 2)34 = 2 = 1125899906842624 2. . We claim that eventually all the terms of the sequence are closer to s than a preassigned small distance ε > 0. . .
.Homework 515 Theorem Every bounded increasing sequence {an }+∞ of real numbers converges to its supremum. then every term of the progression is 0. Every term of the sequence satisfies an ≤ s.1. 517 Example Find the 35-th term of the geometric progression 8 1 √ . ar. we then have s − ε ≤ an0 ≤ an0 +1 ≤ an0 +2 ≤ an0 +2 ≤ . ◭ √ 2 2 518 Example The fourth term of a geometric progression is 24 and its seventh term is 192. 3. −2. 1. Trivially. we get to within ε of s. ar3 . By virtue of Axiom ??. diverges as n → +∞
C. ar3 . 9. . If ar = 0. . the sequence has a supremum s. Similarly. √ . . n = 0. 5. u n=0
Homework
C. if a = 0. 1. The n-th term of the progression a. n! → 0 as n → +∞ 4. . ar. ( 2 )n → +∞ as n → +∞ n 2n
7. . 2.
To obtain the second half of the theorem.3
Finite Geometric Series
a. . n = 0. 2 2 √ 1 Solution: ◮ The common ratio is −2 ÷ √2 = −2 2. then the common ratio can be found by dividing any term by that which immediately precedes it. Since the sequence is increasing.1 Problem Give plausible arguments to convince yourself that 1 1.
2n n
→ 0 as n → +∞ → +∞ as n → +∞
10. . . the sequence 1 + (−1)n . . ar4 . is arn−1 . . every term is produced from the preceding one by multiplying a fixed number. . we simply apply the first half to the sequence {−an}+∞ . ar2 . Proof: The idea of the proof is sketched in figure C. 2n → +∞ as n → +∞ 1 3. the sequence (−2)n . every n=0 bounded decreasing sequence of real numbers converges to its infimum. The number r is called the common ratio. . ar2 . Since s − ε is not an upper bound for the sequence. ≤ s. . .
n+1 → 1 as n → +∞ n ( 2 )n → 0 as n → +∞ 3 3 6. . ar4 .
205
which means that after the n0 -th term.
2 on the second square." 521 Example Without using a calculator. 4. 4 16
are amoebas bad mathematicians? Because they divide to multiply! Depending on your ethnic preference. · · · . 1 + y + y2 + y3 + · · · + y100 . 4 on the third square. of wheat needed?. A geometric progression with positive terms and common ratio 0 < r < 1 has a sum that grows rather slowly. The next second. 1.3.
!
Homework
C. are needed in order to satisfy the inventor (assume that production of wheat stays constant)9 . Observe that 9 < 8 2: for.4 Problem A colony of amoebas8 is put in a glass at 2 : 00 PM. it does not answer the√ question addressed in the problem. .2 Problem The 6-th term of a geometric progression is 20 and the 10-th is 320. 2 3
! The sum of the first two terms of the series in example 520 is
To close this section we remark that the approximation 210 ≈ 1000 is a useful one. 2750 = (23 )250 < (32 )250 = 3500 . but never an American businessman!!!
.
C. Later in this chapter we will see how to solve this problem using logarithms. 2. − 15 . Now. in kg. are needed in order to satisfy the greedy inventor?.6 Problem In this problem you may use a calculator. If y = 1.Homework
207
2 + 32 = 8 .3. (4) Given that the annual production of wheat is 350 million tonnes. the glass is full. One readily sees that 23 = 8 < 9 = 32 . how many years. 1 − y + y2 − y3 + y4 − y5 + · · · − y99 + y100 .3.1 Problem Find the 17-th term of the geometric sequence − 2 2 2 . the ruler in this problem might be an Indian maharajah or a Persian shah. etc. C. (2) How many grains. (1) How many grains of wheat are to be put on the last (64-th) square?. Solution: ◮ The idea is to find a power of 2 close to a power of 3. if 9 ≥ 8 2. If y = 1. After one minute. Legend says that the inventor of the game of chess asked the Emperor of China to place a grain of wheat on the first square of the chessboard. determine which number is larger: 2900 or 3500 . One second later each amoeba divides in two. It is nowadays used in computer lingo. etc.3. 8 on the fourth square.3. total. what is the approximate weight. approximately. √ However.3.3 Problem Find the sum of the following geometric series. though close to 1 is not as 9 close as the sum of the first 99 terms. 3.5 Problem Without using a calculator: which number is greater 230 or 302 ?
8 Why 9
C. squaring both sides √ we would obtain 81 > 128.7 Problem Prove that 1 + 2 · 5 + 3 · 52 + 4 · 53 + · · · + 99 · 5100 = C.3. 317 316 3 C. The inequality just obtained is completely useless. raising both sides to the 250-th power.3. Raising 9 < 8 2 to the 250-th power we obtain √ 3500 = (32 )250 < (8 2)250 = 2875 < 2900 . we may go around this with a similar idea.. C. When was the glass half-full? C. 99 · 5101 5101 − 1 − . 1 + y2 + y4 + y6 + · · · + y100 . ◭ You couldn't solve example 521 using most pockets calculators and the mathematical tools you have at your disposal (unless you were really clever!). a contradiction. C.3.9 Problem Prove that 1+x+x2 +· · ·+x80 = (x54 +x27 +1)(x18 +x9 +1)(x6 +x3 +1)(x2 +x+1).8 Problem Shew that 1+x+x2 +· · ·+x1023 = (1+x)(1+x2 )(1+x4 ) · · · (1+x256 )(1+x512 ). If y = 1.. which. whence 2900 is greater. 1 + 3 + 32 + 33 + · · · + 349 . Find the absolute value of its third term. where a kilobyte is 1024 bytes—"kilo" is a Greek prefix meaning "thousand. the present generation divides in two again. (3) Given that 15 grains of wheat weigh approximately one gramme.
For which value of u is Lu parallel to the line 2x − y = 2?
A 5
B 0
C −3
D
1 3
E none of these
53. For which value of u is Lu parallel to the x-axis?
A −2
B) 2
C −1
D 1
E none of these
51. 2)
B (2. Which of the following points is on every line Lu regardless the value of u?
A (−1.218 46. For which value of u is Lu perpendicular to the line y = 2x − 1? 1 A u = −5 B u=0 C u=− 2 For a real number parameter u consider the line Lu given by the equation Lu : (u − 2)y = (u + 1)x + u. 3 ) 3
E none of these
. For which value of u is Lu parallel to the line y = 2x − 1? B u=2
C u=5
D u=
E none of these
48. For which value of u is Lu parallel to the y-axis?
A −2
B 2
C −1
D 1
E none of these
52.−2) 3
1 D (− 2 . For which value of u does Lu pass through the point (−1. For which value of u is Lu perpendicular to the line 2x − y = 2?
A 5
B 0
C
1 3
D −
1 3
E none of these
54. Questions 49 to 54 refer to Lu . 1)?
D u=5
E none of these
A 1
B −1
C 2
D 3
E none of these
50. 49. For which value of u is Lu vertical? A u = −1 A u=0 B u=2 C u=
Appendix D
1 3
D u=
2 3 2 3
E none of these
47. −1)
1 C (3.
−4).70: C D D
Figure D. Which one most resembles the graph of q? Notice that there are four graphs but five choices.67: D E none of these
143.
Figure D. How many of the following assertions is (are) true? (a) q is convex.69: B C C
Figure D. 5) and x-intercepts (−1.
Appendix D
141. (d) the graph of q has y-intercept (0.64: A A A B B
Figure D. A none B exactly one C exactly two D exactly three E all four (c) the graph q has vertex (−3. Which one most resembles the graph of y = q(|x|)? Notice that there are four graphs but five choices.
142.65: B C C
Figure D.
Figure D.232 Problems 141 through 143 refer to the quadratic function q : R → R with assignment rule given by q(x) = x2 − 6x + 5.71: D E none of these
.68: A A A B B
Figure D.66: C D D
Figure D. 0) and (5. 0). (b) q is invertible over R.
find: (a) the distance between P and Q. that is. Sketch the graphs of the curves in the order given. If the points (1. 8. If all the units are inhabited. 3.
Figure D. b. c]. find the value of t. the rent for each unit is $700 per unit. and that its range can be written in the form [u. Explain.3
Essay Questions
(x − 1)(x + 2) ≥ 0.180: Problem 4. the greatest integer less than or equal to x. where a. Draw a rough sketch of the graph of y = x − x . (x − 3)
1. management increases the rent of the remaining tenants by $25. Consider the graph of the curve y = f (x) in figure D. by which transformations (shifts. An apartment building has 30 units. 3). 2). Show that if the graph of a curve has x-axis symmetry and y-axis symmetry then it must also have symmetry about the origin. v]. 6. c. You may assume that the domain of f can be written in the form [a. b[ ∪ ]b.258
Appendix D
D.
(b) the midpoint of the line segment joining P and Q. 3). c are integers. For every empty unit. Find a. 2. with u and v integers. 4. For the points P(−1. (−1. b. elongations.180.) how one graph is obtained from the preceding one. u and v. Find the solution set to the inequality
and write the answer in interval notation. (c) if P and Q are the endpoints of a diameter of a circle. where x is the the floor of x.
5.t) all lie on the same line. compressions. What will be the profit P(x) that management gains when x units are empty? What is the maximum profit? 7. (2. (a) y = x − 1
(b) y = (x − 1)2 (d) y = |x2 − 2x| 1 (e) y = 2 |x − 2x| (c) y = x2 − 2x
. 2) and Q(2. etc. find the equation of the circle. squaring. reflections.
Figure D. 20. If not. Find all the rational roots of x5 + 4x4 + 3x3 − x2 − 4x − 3 = 0. 18. graph x+1
(b) y = f (|x|).
16. The cost of the material for the top and bottom of the box is 20 cents per square foot. (d) y = f (−|x|). find it.
.182 has degree 3.
(a) y = | f (x)|. A rectangular box with a square base of length x and height h is to have a volume of 20 ft3 . 17. assuming it is an integer. Also. the cost of the material for the sides is 8 cents per square foot. (c) y = | f (|x|). and (c) the variable h only. Given f (x) = 1 . The polynomial p in figure D. You may assume that all its roots are integers. Situation: Questions 15a to 15e refer to the straight line Lu given by the equation Lu : (u − 2)y = (2u + 4)x + 2u.260 15.182: Problems 16a to 16b. prove that there is no such point. x+1
19. Sketch the graph of the curve y = 1−x and label the axis intercepts and asymptotes. (b) the variable x only. (a) For which value of u is Lu a horizontal line? (b) For which value of u is Lu a vertical line? (d) For which value of u is Lu perpendicular to the line y = −2x + 1? (c) For which value of u is Lu parallel to the line y = −2x + 1?
Appendix D
(e) Is there a point which is on every line Lu regardless the value of u? If so. Problems 16a to 16b refer to it. where u is a real parameter. (a) Find p(−2). Express the cost of the box in terms of (a) the variables x and h. (b) Find a formula for p(x).
type the following. − for subtraction. ∗ for multiplication.
E. The commands used here can run on any version of Maple (at least V through X).E
Maple
The purpose of these labs is to familiarise you with the basic operations and commands of Maple. / for division. pressing ENTER after the semicolon:
268
. Maple also has other useful commands like expand and simplify.1 Basic Arithmetic Commands
Maple uses the basic commands found in most calculators: + for addition. For example. as Maple distinguishes between capital and lower case letters. Be careful with capitalisation. and ∧ for exponentiation. to expand the algebraic expression √ ( 8 − 21/2)2 .
4 and so the equation of the line containing this line segment is of the form y = x + k1 . Since (−1, 1) is on the line, 3 4 7 4 7 1 = − + k1 =⇒ k1 = , so this line segment is contained in the line y = x + . The second line segment L2 has slope 3 3 3 3 slope L2 = 1−1 = 0, 2 − (−1)
and so this line segment is contained in the line y = 1. Finally, the third line segment L3 has slope slope L3 = −5 − 1 = −3, 4−2
and so this line segment is part of the line of the form y = −3x + k2 . Since (1, 2) is on the line, we have 2 = −3 + k2 =⇒ k2 = 5, and so the line segment is contained on the line y = −3x + 5. Upon assembling all this we see that the piecewise function required is 4x+ 7 3 3 f (x) = 1 −3x + 5 if x ∈ [−4; −1] if x ∈ [−1; 2] if x ∈ [2; 4] 6. R 7. ] − ∞; −1[∪] − 1; 0] √ 8. ] − 1; 1[ 9. {0}
Our choice of a works and hence the function is surjective. 3.6.4 We must shew that there is a solution x for the equation f (x) = b, b ∈ R \ {2}. Now f (x) = b =⇒ b 2x = b =⇒ x = . x+1 2−b
Thus as long as b = 2 there is x ∈ R with f (x) = b. Since there is no x such that g(x) = 2 and 2 ∈ Target (g), g is not surjective. 3.6.5 2. surjective, f (1) = f (−1) so not injective. 1. neither, f (−1) = f (1) so not injective. There is no a with f (a) = −1, so not surjective.
290
3. surjective, not injective. 5. neither, |1| = | − 1| so not injective, there is no a with |a| = −1, so not surjective. 4. injective, as proved in text, there is no a with f (a) = −1, so not surjective.
Since these grey rectangles do not intersect with the green squares on the corners, their collective area is less than the area of the unit 1 4 95 square minus these smaller squares: 1 − − = . We thus conclude that 100 100 100 1 10 1 10
2
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Product Description
This teacher's guide accompanies Horizons' sold-separately Horizons Pre-Algebra Student Book. Each lesson includes the main concepts, lesson objectives, materials needed, teaching tips, the assignment for the day, and the reduced student pages with the correct answers supplied. Each lesson will take approximately 45-60 minutes, and is designed to be teacher-directed. 400 | 677.169 | 1 |
There is an increasing need for undergraduate students in physics to have a core set of computational tools. Most problems in physics benefit from numerical methods, and many of them resist analytical solution altogether. This textbook presents numerical techniques for solving familiar physical problems where a complete solution is inaccessible using traditional mathematical methods. The numerical techniques for solving the problems are clearly laid out, with a focus on the logic and applicability of the method. The same problems are revisited multiple times using different numerical techniques, so readers can easily compare the methods. The book features over 250 end-of-chapter exercises. A website hosted by the author features a complete set of programs used to generate the examples and figures, which can be used as a starting point for further investigation. A link to this can be found at | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
Synopsis
Excellent intro to basics of algebraic number theory. Gausian primes; polynomials over a field; algebraic number fields; algebraic integers and integral bases; uses of arithmetic in algebraic number fields; more. 1975 edition. | 677.169 | 1 |
Modify Your Results
An ideal program for struggling students "Glencoe Algebra: Concepts and Applications" covers all the Algebra 1 concepts. This program is designed for students who are challenged by high school mathematics.
Glencoe Algebra: Concepts and Applications includes lessons that will help students prepare for the Texas Essential Knowledge and Skills assessed on the Texas state test. This textbook contains a special section of practice problems specifically for the Texas state test.
Algebra: Concepts & Applications, is a comprehensive Algebra 1 program that is available in full and two-volume editions. Algebra: Concepts & Applications uses a clean lesson design with many detailed examples and straightforward narration that make Algebra 1 topics inviting and Algebra 1 content understandable. Volume 1 contains Chapters 1-8 of Algebra: Concepts & Applications plus an initial section called Chapter A. Chapter A includes a pretest, lessons on prerequisite concepts, and a post test. Designed for students who are challenged by high school mathematics, the 2006 edition has many new features and support components.<P>
Advisory: Bookshare has learned that this book offers only partial accessibility. We have kept it in the collection because it is useful for some of our members. To explore further access options with us, please contact us through the Book Quality link on the right sidebar. Benetech is actively working on projects to improve accessibility issues such as these.
Algebra: Concepts & Applications, is a comprehensive Algebra 1 program that is available in full and two-volume editions. Algebra: Concepts & Applicationsuses a clean lesson design with many detailed examples and straightforward narration that make Algebra 1 topics inviting and Algebra 1 content understandable. Volume 1 contains Chapters 1-8 ofAlgebra: Concepts & Applicationsplus an initial section called Chapter A. Chapter A includes a pretest, lessonson prerequisite concepts, and a posttest. Designed for students who are challenged by high school mathematics, the 2007 edition has many new features and support components | 677.169 | 1 |
About
Overview
The new edition of INTERMEDIATE ALGEBRA is an exciting and innovative revision that takes an already successful text and makes it more compelling for today's instructor and student. The authors have developed a learning plan to help students succeed in Intermediate Algebra and transition to the next level in their coursework. Based on their years of experience in developmental education, the accessible approach builds upon the book's known clear writing and engaging style which teaches students to develop problem-solving skills and strategies that they can use in their everyday lives. The authors have developed an acute awareness of students' approach to homework and present a learning plan keyed to Learning Objectives and supported by a comprehensive range of exercise sets that reinforces the material that students have learned setting the stage for their success.
Additional Product Information
Features and Benefits
The design incorporates a tabular structure identifying features that guide students through the textbook, providing for increased readability.
Chapter Openers showcase the many career paths available in the world of mathematics and include a brief overview of each career, as well as job outlook, growth and annual earnings potential statistics from the U.S. Department of Labor.
Reach for Success highlights study skills and is featured at the beginning and end of each chapter. This feature is designed as an activity to help the student prepare for a successful semester.
Learning Objectives at the beginning of each section are mapped to content, as well as to exercises in the Guided Practice section, helping students identify specific concepts that may need reinforcement, and helping instructors tie problems to objectives and their own learning outcomes.
Section vocabulary feature help students speak the language of mathematics. Not only are vocabulary words identified at the beginning of each section, these words are also bolded within the section. Exercises include questions on the vocabulary words, and a glossary has been developed to facilitate the students' reference to these words.
Getting Ready questions appear at the beginning of each section, linking past concepts to the upcoming material.
Examples are worked out in each chapter, highlighting the concept being discussed. "Notes" in many of the text's examples, give students insight into the process adopted when approaching a problem and working toward a solution. Each Example also ends with a Self Check problem, so that students may immediately apply concepts.
The Now Try This feature at the end of each section focuses on developing transitional group-work exercises and increase conceptual understanding through active classroom participation. These problems can be worked independently or in small groups and transition to exercises in the following section.
The Exercise Sets at the end of each section help transition students through progressively more difficult homework problems. Students are initially asked to work quick, simple problems on their own, proceed to working exercises keyed to examples, and finally complete application and critical thinking questions on their own. These sets include a variety of problem types, including "Warm Ups," "Review," "Vocabulary and Concept," "Guided Practice," "Additional Practice," "Applications," "Writing About Math," and "Something to Think About."
What's New
NEW! Reach for Success highlights study skills and is featured at the beginning and end of each chapter. This feature is designed as an activity to help the student prepare for a successful semester.
NEW! The Warm Ups have been significantly revised to include skills needed for the development of section concepts.
NEW! Self Checks are now included with each Example so that students can check reading comprehension.
NEW! A tear-out Basic Calculator Keystroke Guide for the Casio 9750GII has been provided, in addition to the existing TI-83 and TI-84 guide, to assist students with their calculator functionality and to serve as a quick reference.
NEW! Additional Teaching Tips have been included at the request of reviewers.
Learning Resource Bundles
Choose the textbook packaged with the resources that best meet your course and student needs.
Contact your Learning Consultant for more information.
Each section of the main text is discussed in uniquely designed Teaching Guides containing instruction tips, examples, activities, worksheets, overheads, assessments, and solutions to all worksheets and activities5846257 | ISBN-13: 9781285846255)
Get a head-start and go beyond the answers to improve your grade! The Student Workbook contains all of the studentRosemary Karr
Rosemary Karr graduatedRosemary Karr graduatedMarilyn MasseyR. David Gustafson | 677.169 | 1 |
Algebra Applications: Variables and Equations
DVD Features:
Rated: G
Run Time: 20 minutes
Released: June 9, 2009
Originally Released: 2009
Label: am productions, llc
Encoding: Region 1 (USA & Canada)
Audio:
Dolby Digital 2.0 Stereo - English
Product Description:
An engaging teaching aid for algebra teachers, this program explores the practical application of variables and equations with the use of a graphic calculator, leading viewers through a series of real-world examples where the concepts can be used, like the biology of honey bee colonies, and the forging of rivers through geological landscapes. The program leads viewers through the keystrokes involved in each example, and uses animations to illustrate ideas. | 677.169 | 1 |
Thomas W. Baumgarte
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Teaching this semester
PHYS 2510. Astrophysics
Thomas Baumgarte A quantitative discussion that introduces the principal topics of astrophysics, including stellar structure and evolution, planetary physics, and cosmology.
PHYS 3000. Methods of Theoretical Physics
Thomas Baumgarte Mathematics is the language of physics. Similar mathematical techniques occur in different areas of physics. A physical situation may first be expressed in mathematical terms, usually in the form of a differential or integral equation. After the formal mathematical solution is obtained, the physical conditions determine the physically viable result. Examples are drawn from heat flow, gravitational fields, and electrostatic fields. | 677.169 | 1 |
In this powerful and dramatic biography Sylvia Nasar vividly recreates the life of a mathematical genius whose career was cut short by schizophrenia and who, after three decades of devastating mental illness, miraculously recovered and was honored with a Nobel Prize. ?How could you, a mathematician, believe that extraterrestrials were sending you... more...
This comprehensive textbook introduces readers to the principal ideas and applications of game theory, in a style that combines rigor with accessibility. Steven Tadelis begins with a concise description of rational decision making, and goes on to discuss strategic and extensive form games with complete information, Bayesian games, and extensive form... more...
Anyone with a knowledge of basic mathematics will find this an accessible and informative introduction to game theory. It opens with the theory of two-person zero-sum games, two-person non-zero sum games, and n-person games, at a level between nonmathematical introductory books and technical mathematical game theory books. Succeeding sections focus... more...
Many illuminating and instructive examples of the applications of game theoretic models to problems in political science appear in this volume, which requires minimal mathematical background. 1975 edition. 24 figures. more...
Definitive work draws on game theory, calculus of variations, and control theory to solve an array of problems: military, pursuit and evasion, athletic contests, many more. Detailed examples, formal calculations. 1965 edition. more...
This volume lays the mathematical foundations for the theory of differential games, developing a rigorous mathematical framework with existence theorems. It begins with a precise definition of a differential game and advances to considerations of games of fixed duration, games of pursuit and evasion, the computation of saddle points, games of survival,... more... | 677.169 | 1 |
Fun Self-Discovery Tools
Basic Properties of Math and Order of Operations
Rating:
Description:
Since not everyone understands everything to know about math, including myself, I just wanted to post this packet in order to help everyone who may not understand the subject of algebra very well and wants to get that knowledge.
I made a powerpoint explaining each basic property with examples and listed the order of opertations used in algebraic equations. Also, there is a video that shows a live demonstration of a couple examples from each property. | 677.169 | 1 |
If you have aspirations to becoming a mathematician or physicist, learn LaTeX! The best way is to simply turn in all your homework in LaTeX format. This advice is particularly useful for an undergraduate who wants an advanced degree with a thesis, but I think it couldn't hurt to learn LaTeX in high school. | 677.169 | 1 |
The fascinating world of graph theory goes back several centuries and revolves around the study of graphs—mathematical structures showing relations between objects. With applications in biology, computer science, transportation science, and other areas, graph theory encompasses some of the most beautiful formulas in mathematics—and some of its most famous problems. For example, what is the shortest route for a traveling salesman seeking to visit a number of cities in one trip? What is the least number of colors needed to fill in any map so that neighboring regions are always colored differently? Requiring readers to have a math background only up to high school algebra, this book explores the questions and puzzles that have been studied, and often solved, through graph theory. In doing so, the book looks at graph theory's development and the vibrant individuals responsible for the field's growth.
Introducing graph theory's fundamental concepts, the authors explore a diverse plethora of classic problems such as the Lights Out Puzzle, the Minimum Spanning Tree Problem, the Königsberg Bridge Problem, the Chinese Postman Problem, a Knight's Tour, and the Road Coloring Problem. They present every type of graph imaginable, such as bipartite graphs, Eulerian graphs, the Petersen graph, and trees. Each chapter contains math exercises and problems for readers to savor.
An eye-opening journey into the world of graphs, this book offers exciting problem-solving possibilities for mathematics and beyond.
Arthur Benjamin is professor of mathematics at Harvey Mudd College. His books include Secrets of Mental Math and Proofs That Really Count. Gary Chartrand is professor emeritus of mathematics at Western Michigan University. Ping Zhang is professor of mathematics at Western Michigan University. Chartrand and Zhang are the coauthors of several books, including A First Course in Graph Theory and Discrete Mathematics.
Reviews:
"The Fascinating World of Graph Theory is readable and 'student-friendly'--more so than the typical math textbook."--Robert Schaefer, New York Journal of Books
"[The authors] have set out to make graph theory not only accessible to people with a limited mathematics background, but also to make it interesting. They have--by virtue of very clear writing, combined with a greater-than-usual emphasis on the historical and personal side of the subject--succeeded admirably."--Mark Hunacek, MAA Reviews
"The book is written masterfully; the narrative in each chapter flows naturally, engagingly. . . . [I]t's a popular but also comprehensive introduction into graph theory."--Alexander Bogomolny, Cut the Knot blog
"This book is a fun and interesting tour of graph theory, leaving each visitor with a feeling of accomplishment and a satisfying understanding of this unusual mathematical world. . . . This is an entertaining book for those who enjoy solving problems, plus readers will learn about some powerful mathematical ideas along the way!"--Choice
"Here is a book with an enjoyable mix of mathematics and its applications, spiced with liberal amounts of history and anecdote. . . . The value of books like this is that they make mathematics come alive to a broad range of readers who might not look twice at a textbook or monograph."--Norman Biggs, London Mathematical Society Newsletter | 677.169 | 1 |
Learn about this topic in these articles:
discussed in biography
...his
In artem analyticem isagoge (1591; "Introduction to the Analytical Arts") closely resembles a modern elementary algebra text. His contribution to the theory of equations is
De aequationum recognitione et emendatione (1615; "Concerning the Recognition and Emendation of Equations"), in which he presented methods for solving equations of second, third, and... | 677.169 | 1 |
Details about Complex Analysis For Mathematics And Engineering:
Intended For The Undergraduate Student Majoring In Mathematics, Physics Or Engineering, The Sixth Edition Of Complex Analysis For Mathematics And Engineering Continues To Provide A Comprehensive, Student-Friendly Presentation Of This Interesting Area Of Mathematics. The Authors Strike A Balance Between The Pure And Applied Aspects Of The Subject, And Present Concepts In A Clear Writing Style That Is Appropriate For Students At The Junior/Senior Level.Through Its Thorough, Accessible Presentation And Numerous Applications, The Sixth Edition Of This Classic Text Allows Students To Work Through Even The Most Difficult Proofs With Ease. New Exercise Sets Help Students Test Their Understanding Of The Material At Hand And Assess Their Progress Through The Course. Additional Mathematica And Maple Exercises Are Available On The Publishers Website.
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Rent Complex Analysis For Mathematics And Engineering 6th edition today, or search our site for other textbooks by Mathews. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Jones & Bartlett.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our Set-Theory tutors now. | 677.169 | 1 |
Painless Geometry
Paperback
Item is available through our marketplace sellers.
Overview find the relationships that exist between parallel and perpendicular lines, and discover the characteristics of shapes such as triangles, quadrilaterals, and circles. Real-world geometry experiments will help make the concepts you learn less abstract. You'll also find advice on study strategies, and see how mini-proofs are the first step toward understanding formal geometry proofs.
Most Helpful Customer Reviews
I'm a 7th grader who decided to get ahead in geometry for next year. There aren't many cheap, good textbooks out there, so I had to settle for this. I figured it would be okay... But it really is PAINLESS. Just for kicks I took a released SOL (Standards of Learning) test to see if I could pass is just from what I'd learned in this book. 97%. That score speaks for itself.
prettypastrychef
More than 1 year ago
This book is wonderful for anyone who struggles with math. I struggled and it sure helped me! | 677.169 | 1 |
Algebraic equations are theoretical support for solving many problems of current life. Understanding concepts and solving techniques is very important for studying concepts that follow the school work in math or other subjects.
In this fun, light-hearted and easy to understand book we explore one of the most well-known problems in Number Theory. Throughout the journey of trying to understand a problem which has taxed mathematicians since the 17th century, we are joined by some inquisitive children who ask many questions along the way.
This workbook is aimed at aiding the lecturer in economizing study time while teaching the foundations of mathematical logic. Each topic includes short illustrated theoretical material, samples of solving problems, and tasks for the students.
Sections: Natural and Whole Numbers,Rules of Divisibility,Factors and Multiples,Highest Common Factor and Lowest Common Denominator,Squares and Cubes,Square Root and Cube Root,Exponents,Algebraic Expressions,Integers,Equations,Fractions (Rational Numbers),Decimals,Ratio and Rate,Scientific Notation and Percentages.
This Mathematics Guide replaces the presence of a teacher.
"Introduction to Gambling Theory – Know the Odds!" covers applications of probability theory to coin tossing, dice and playing cards. These three forms of gambling are important to gain insight into how probability can be used to predict possible successful outcomes when gambling. Questions and answers are listed in exercises at the end of each sectionwhat do you want to known about how to learn multiplication and division in china , you may look this .Just a multiplication table. It will make your ability improve in math .You will find calculation is geting easier.
This book contains the fundamental trigonometric and hyperbolic functions, 25 challenging problems, allong with their solutions and analysis.
The readers of this book should be familiar with Trigonometry, Algebra, Equations and Complex Numbers, and will be able to challenge and evaluate their knowledge and understanding of the subject Long Answer Type Questions and Additional Problems. | 677.169 | 1 |
Focus in High School Mathematics: Reasoning and Sense Making in Algebra (Teaching and learning mathematics) Paperback April 9, 2010
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B010EVPARC Focus in High School Mathematics: Reasoning ...
Publisher: national council of teachers of mathematics (april 9, 2010 | 677.169 | 1 |
Friday, January 14, 2011
Microsoft Mathematics
Heaven knows I don't need another graphing software or mathematical package. But still, when I came across this, I thought that this might be very useful to students, especially those in high school or early undergraduate years. I know that there's probably several freeware or shareware mathematics packages that one can afford without paying the hundreds (or thousands) of dollars for the high-powered mathematics packages. Still, this might be "free" enough that it is interesting.
It appears that Version 4 of Microsoft Mathematics is available for download. A brief description of it can be found here. I haven't used it before, and before I recommend it to people I know who might need it, does anyone else have a review of this or an earlier version of it? And if you have used it for physics purposes, I would like to hear your opinion even more.
1 comment:
I just downloaded this and played around with it a bit. It has a bit of a Mathematica feel but nice in that it offers explanations (up front) of what you need to input and even has a quick blurb of how to do some calcualtions by hand, like derivatives. | 677.169 | 1 |
Splung Physics ...designed for students who are studying A-level physics and beyond. Also covers basic calculus and mathematics that students frequently find difficult. Flash animations explain key ideas. Also a physics forum for discussing and answering physics problems. | 677.169 | 1 |
Mathematics For Common Schools. A Manual For Teachings, Including Definitions, Principles, And Rules And Solutions Of The More Difficult Problems
About the Book
We're sorry; this specific copy is no longer available. Here are our closest matches for Mathematics For Common Schools. A Manual For Teachings, Including Definitions, Principles, And Rules And Solutions Of The More Difficult Problems by Walsh, John H. (John Henry).
Description:
B007SPFLCW7860002156035
Book Description ReInk Books, 2015. Paperback. Book Condition: New. Reprinted from 1900 336 pages. Bookseller Inventory # 451825930002156035
Book Description ReInk Books, 2015. Hardback. Book Condition: New. Reprinted from 1900 Boston, D.C. Heath & co.) 336 pages. Bookseller Inventory # HB451825930
Book Description Hardcover. Book Condition: New. Hardcover reprint of the original 1900 Walsh, John H. (John Henry). Mathematics For Common Schools. A Manual For Teachings, Including Definitions, Principles, And Rules And Solutions Of The More Difficult Problems. Indiana: Repressed Publishing LLC, 2012. Original Publishing: Walsh, John H. (John Henry). Mathematics For Common Schools. A Manual For Teachings, Including Definitions, Principles, And Rules And Solutions Of The More Difficult Problems, . Boston, D.C. Heath & Co., 1900. Subject: Arithmetic Study And Teaching Elementary. New. Bookseller Inventory # RP164776981 | 677.169 | 1 |
Learn more with other tutorials...
In all electronic computation environments the equality symbol requires special treatment because of the different roles this symbol plays in mathematical communication (definition, evaluation, equality). In a Mathcad worksheet definition (or assignment), numeric evaluation, and symbolic evaluation (CAS) provide users with a great deal of power in a computational document. This brief video compares and introduces Mathcad's numeric evaluation (=) and symbolic evaluation (->) symbols by comparing results from the evaluation or two simple algebraic expressions.... (Show more)(Show less)
PTC Mathcad provides users with the capability to simultaneously perform and document calculations. Nonetheless, there is always just a little bit of editing required at the end of the process before sharing a worksheet with colleagues. This tutorial provides a basic introduction to the worksheet formatting capabilities in Mathcad Prime. ... (Show more)(Show less) | 677.169 | 1 |
Mathematics
Mathematics
Solve problems with competency, care and creativity
A degree in mathematics, mathematics education, or mathematics and systems from Taylor University will prepare you to be a competent, caring and creative problem solver working in education, government, business or ministry in areas like web design, consulting, computer analysis, engineering, graphic design, statistics, and actuarial services. You will also be prepared for graduate school at places like MIT, the Mayo Clinic, Rutgers University, and Indiana University.
Our faculty engage you in a dynamic community that expands beyond academics to enrich your spiritual and social development. You will have the opportunity to participate in State Math competitions, a math club, summer research and internships, Honors Guild, and take trips with other students during J-term and Spring Break.
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Interested? Watch these clips from Dr. Matt DeLong and Aaron. Check out our degrees and majors. Better yet, schedule a campus visit and see for yourself how our faculty is second to none when it comes to student engagement and academic excellence.
Listen to Dr. Matt DeLong, Professor of Mathematics, describe how the Mathematics Department develops math majors into creative problem solvers who can impact the world.
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An estimated 150 students from five area high schools are expected at the Second Annual High School Math Field Day, scheduled Thursday, March 17, at the Euler Science Complex at Taylor University. The event is jointly hosted by Taylor and Indiana Wesleyan University; professors and students from both schools will staff the event. | 677.169 | 1 |
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"The mathematical methods that physical scientists need for solving substantial problems in their fields of study are set out clearly and simply in this tutorial-style textbook. Students will develop problem-solving skills through hundreds of worked examples, self-test questions and homework problems. Each chapter concludes with a summary of the main procedures and results and all assumed prior knowledge is summarized in one of the appendices. Over 300 worked examples show how to use the techniques and around 100 self-test questions in the footnotes act as checkpoints to build student confidence. Nearly 400 end-of-chapter problems combine ideas from the chapter to reinforce the concepts. Hints and outline answers to the odd-numbered problems are given at the end of each chapter, with fully-worked solutions to these problems given in the accompanying Student Solutions Manual. Fully-worked solutions to all problems, password-protected for instructors, are available at www. cambridge. org/essential"--
This tutorial-style textbook develops the basic mathematical tools needed by first and second year undergraduates to solve problems in the physical sciences. Students gain hands-on experience through hundreds of worked examples, self-test questions and homework problems. Each chapter includes a summary of the main results, definitions and formulae. Over 270 worked examples show how to put the tools into practice. Around 170 self-test questions in the footnotes and 300 end-of-section exercises give students an instant check of their understanding. More than 450 end-of-chapter problems allow students to put what they have just learned into practice. Hints and outline answers to the odd-numbered problems are given at the end of each chapter. Complete solutions to these problems can be found in the accompanying Student Solutions Manual. Fully-worked solutions to all problems, password-protected for instructors, are available at www. cambridge. org/foundation.
After reviewing the basic concept of general relativity, this introduction discusses its mathematical background, including the necessary tools of tensor calculus and differential geometry. These tools are used to develop the topic of special relativity and to discuss electromagnetism in Minkowski spacetime. Gravitation as spacetime curvature is introduced and the field equations of general relativity derived. After applying the theory to a wide range of physical situations, the book concludes with a brief discussion of classical field theory and the derivation of general relativity from a variational principle www. cambridge. org/9780521679718.
This Student Solution Manual provides complete solutions to all the odd-numbered problems in Essential Mathematical Methods for the Physical Sciences. It takes students through each problem step-by-step, so they can clearly see how the solution is reached, and understand any mistakes in their own working. Students will learn by example how to select an appropriate method, improving their problem-solving skills.
This Student Solution Manual provides complete solutions to all the odd-numbered problems in Foundation Mathematics for the Physical Sciences. It takes students through each problem step-by-step, so they can clearly see how the solution is reached, and understand any mistakes in their own working. Students will learn by example how to arrive at the correct answer and improve their problem-solving skills | 677.169 | 1 |
A searchable book on the Web that addresses the need to establish standards for the mathematics programs that bridge the gap between high school mathematics and college calculus. It also aims to satisfy the needs of students who do not plan to study calculus. Chapters on: Standards for Introductory College Mathematics; Interpreting the Standards; Implications; Implementation; Looking to the Future. With illustrative examples and references. | 677.169 | 1 |
Multivariable Calculus. This consists of 35 video lectures given by Professor Denis Auroux, covering vector and multi-variable calculus. Topics covered in this course include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space.
Partial derivatives are so called because they're the derivatives of multivariable functions. When a function is defined in terms of two or more variables, the function's derivative is actually a collection of partial derivative equations.
In the same way that partial derivatives allow you to take the derivatives of multivariable functions, multiple integrals let you take the integrals of multivariable functions. In this course, calculus tutor Krista King discusses iterated, double, double polar, and triple integrals, including the mathematical applications of each. | 677.169 | 1 |
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Accompanying BJU Press' Math 3 Student Text, 3rd Edition, this teacher's guide features reduced, full-color, student pages with the correct answers overlaid, as well as extensive teaching notes. Each chapter is introduced with a mini scope-and-sequence, as well as a bulletin board and theme. Lesson notes display related worktext and math review page numbers at the top of each lesson, as well as objectives; needed teacher/student materials; practice & review activities; a scripted introduction; and teaching exercises that include questions and answers. | 677.169 | 1 |
Details about A First Course in Differential Equations with Modeling Applications:
A First Course in Differential Equations with Modeling Applications, 9th Edition strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. This proven and accessible text speaks to beginning engineering and math students through a wealth of pedagogical aids, including an abundance of examples, explanations, "Remarks" boxes, definitions, and group projects. Using a straightforward, readable, and helpful style, this book provides a thorough treatment of boundary-value problems and partial differential equations.
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Rent A First Course in Differential Equations with Modeling Applications 9th edition today, or search our site for other textbooks by Dennis G. Zill. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Brooks Cole.
Need more help with A First Course in Differential Equations with Modeling Applications ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our Differential-Equations tutors now. | 677.169 | 1 |
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