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A First Course in Analysis by Donald Yau
This e-book is an introductory textual content on genuine research for undergraduate scholars. The prerequisite for this booklet is an exceptional history in freshman calculus in a single variable. The meant viewers of this booklet comprises undergraduate arithmetic majors and scholars from different disciplines who use genuine research. because this e-book is geared toward scholars who do not need a lot earlier adventure with proofs, the speed is slower in past chapters than in later chapters. There are countless numbers of routines, and tricks for a few of them are incorporated.
The results of DSP has entered each part of our lives, from making a song greeting playing cards to CD avid gamers and cellphones to clinical x-ray research. with out DSP, there will be no net. lately, each element of engineering and technology has been motivated by way of DSP end result of the ubiquitous machine computing device and on hand sign processing software program.
§1 confronted through the questions pointed out within the Preface i used to be triggered to put in writing this e-book at the assumption regular reader can have definite features. he'll most likely be accustomed to traditional bills identical token) to be fake.
B) Is {an } necessarily a Cauchy sequence if C = 1? (16) Let {an } be a bounded sequence with α = inf{an } and β = sup{an }. (a) If α =/ an for any n, prove that there exists a decreasing subsequence of {an } that converges to α. (b) If β =/ an for any n, prove that there exists an increasing subsequence of {an } that converges to β. (17) In each case, construct a sequence {an } with the given properties. (a) (b) (c) (d) There exist subsequences converging to 1, 2, and 3. For every positive integer M , there exists a subsequence converging to M . | 677.169 | 1 |
Omtale
Basic College Mathematics with Early Integers with MyMathLab, Global Edition
This package includes MyMathLab (R). Objective: Guided Learning The The new edition supports students with quality applications and exercises, a new MyMathGuide workbook and video program, and an updated MyMathLab course that brings it all together! Teaching and Learning Experience This program will provide a better teaching and learning experience for you and your students. Here's how: Guide Students' Learning: The Bittinger team helps today's math students stay on task by guiding them to understand what to do and when. Reinforce Study Skills: The Bittinger program is equipped with tools and resources to help students develop effective study and learning habits that will help them in their college careers and beyond. Improve Results: MyMathLab (R) delivers proven results in helping students succeed and provides engaging experiences that personalize learning. This package includes MyMathMathLab should only be purchased when required by an instructor. Please be sure you have the correct ISBN and Course ID. Instructors, contact your Pearson representative for more information. | 677.169 | 1 |
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Cooperative group problem solving is a deductive reasoning activity where the solution cannot be found without everyone's contribution. This approach has the potential to benefit all students in a class, both mathematically and socially.
Oxford Mathematics Study Dictionary
Barbara Lynch, R. E. Parr
This is a unique and comprehensive reference book for secondary school mathematics students. The 'Dictionary' contains an extensive list of mathematical words and their meanings, organised into topic sections typically included in mathematics syllabi around the country: arithmetic, number, complex numbers, matrices, circles, symmetry and transformations, functions and relations, coordinate geometry, differential calculus, vectors… Specific words can be found through the index; the actual entries contain appropriate definitions, theorems, formulae, diagrams and worked examples of major applications designed to make the meanings clear. | 677.169 | 1 |
Course: Number Theory
David Pollack
Assignments
Course Description
Number Theory is one of the oldest and most extensively studied
branches of mathematics. At its core number theory studies properties
of the integers and their arithmetic, including questions about primes
and divisibility, and solutions to polynomial equations. For example,
one famous problem we'll study is to determine which prime numbers
can be written in the form x2 + y2 where x and y are integers.
There are many number theory questions which are easy to state,
and which involve familiar objects, but whose solutions turn out to
involve beautiful and interesting ideas. In that spirit, our motto for the
course will be to \Think Deeply of Simple Things." We'll approach the
material in a spirit of exploration, with students being encouraged to
look for patterns, make conjectures, and develop proofs. Our goal will
be to learn to think like mathematicians as we delve into the subject
and uncover its hidden structure.
Topics to be covered include congruences, continued fractions, unique
factorization, and extensions to systems beyond the integers.
The only background required for the course is high school algebra,
and an eagerness to explore. Homework will be assigned regularly
during the course, and there will be a research project that expands
on the ideas we will be discussing. Mathcampers will meet daily with
our coach to work on the homework and project, and are very much
encouraged to work on the material with each other as well. | 677.169 | 1 |
Algebra II Syllabus
Holy Family Academy
Ad Veritatem Per Fidem Et Rationem
Grade 10 Instructor: Mr. Forrester Course Title: Algebra II
Course Description: In this course, you will sharpen and expand your ability to work and reason with numbers and quantities. After becoming fluent in the language that is Algebra, you will be equipped to dive into studies of trigonometry, conic sections, logarithms, and other more advanced mathematical notions.
Course Objectives: 1. To master algebraic concepts and know of their origin 2. Be able to articulate an account of key operations and functions; not just the 'what' but also the 'how' and 'why.' | 677.169 | 1 |
Aims :
1. To explore the concepts of local and global solutions and of blow up for ordinary and partial differential equations.
2. To introduce the relevant sets of functions to study nonlinear evolution equations and show how they are used.
3. To construct solutions to nonlinear wave equations.
1. Understand local wellposedness.
2. Ability to calculate conserved quantities using Noether's theorem.
3. Ability to use contraction mapping theorem.
4. Familiarity with function spaces.
5. Understand global existence and blow up and the ability to determine which in common cases. | 677.169 | 1 |
COURSE DESCRIPTION
WHAT IS MATH 104A? This is the first
undergraduate course in number theory. In this course we examine topics
from
elementary number theory and we focus on five major themes: 1)
Divisibility; 2) Multiplicative Functions; 3) Congruence Theory; 4)
Quadratic Residues; 5) Algebraic Numbers and Integers (time permitting).
TEXT Rose, H. E., A Course in Number Theory, Second Edition
(Oxford Science Publ., 1994). You
are expected to read the text BEFORE each lecture.
EXAMS
Midterm I - Monday, October 22, 10:00am, in AP&M B412. Topics:
Chapter 1. Midterm
1 has been postponed to Monday, October 29 ! | 677.169 | 1 |
Study Tips
Keep Up
Mathematics is different from other disciplines. You need to know yesterday's material to understand today's.
Quality Time
This is about 2 hours at home for every hour in class. Time thinking counts. Time staring at the wall doesnít: get help! Your math sessions should be short and frequent. Donít save it all for one long night of cramming.
Cooperate
Ask questions of your classmates, listen and respond to theirs. Form a study group. The more ways you look at something, the clearer it will become.
Get Interested
For some folks this may sound hard. But an ounce of understanding is worth a pound of formulas. You only learn what you really want to learn.
Get Feedback
Talk to me! In class and during office hours. Ask questions, answer questions, donít be afraid to guess. Correct quiz and test mistakes right away. Check homework answers after each session. Join a study group. Go to the Math Clinic.
Practice
There are two very different skills involved:
1) Practice each technique we learn on homework problems.
2) Practice deciding which technique to use. Write one problem per index card and shuffle.
Make your own study guide. Making it yourself is half the benefit.
Categorize problems by the question asked and by the key words.
If you can't perfect all the techniques, be sure to excel at the ones you do know.
There are two kinds of test questions:
1) familiar problems similar to homework;
2) concept questions that ask you to put your knowledge and understanding together in a different way.
Memorization is not sufficient for concept questions — you must understand the material.
Test Yourself
Before you take my test, take your own. Try to duplicate the test conditions as much as possible: quiet, time limit, no book.
Stay Healthy
Eat and sleep wisely. Don't use alcohol or drugs. Your brain and body need a lot of TLC. Similarly your psyche — check out the College Happiness Guide.
The Academic Success Center can help you improve your study skills through assessment and seminars. Their web page has more information and materials for self help.
How to Read Mathematics
Find a comfortable time and place, not too much noise or distracting traffic, no deadlines weighing on your mind. Allow 45-90 minutes. Have plenty of paper and a pencil.
Be an active reader. Take charge. The book will lead you, but wonít tell you everything. Ask yourself questions, fill in missing steps. Do the examples before the book does. Reading mathematics is slow.
Summarize each section.
Context.
Is this the beginning of a chapter? What is the chapter topic? How does the section topic fit with the rest of the chapter?
Is this section later in the chapter? Again, how does this topic fit, and how does it expand your view of the chapter topic?
Does the new material look familiar? Might you confuse new techniques with earlier ones? Can you replace old techniques with new and easier ones?
Key Words.
Write your version of their meanings. Paraphrase, but use technical terms correctly.
Facts.
What theorems and formulas did the book state? Write and explain them in your own words. Do you need to memorize any of them?
Do the homework exercises. Check your answers. Look at the examples for hints. How do the problems relate to the topic?
Keep up and keep your work. If you learned to solve equations in chapter 1, the book will assume you can do this in chapter 2. The sentence, "Then we solve this equation for x and observe that ..." expects you to solve the equation without further help. The phrase, "Building on example 1.5 ..." expects you to look up this example and remember it.
Special Strategies for Oral Tests
Avoid Silence
Silence implies ignorance. If you know or suspect an answer, say it right away. If you're thinking, think out loud. There are no penalties for wrong turns as you think; trying out different ideas is normal. It's what you eventually say you are satisfied with that counts.
Write
Again, no penalty for initial missteps. Seeing, as opposed to imagining, can help you think. If you can't talk and write at the same time, alternate.
Big Ideas First
Worrying over proper notation, or what you are "allowed" to know in this particular course, can distract you. If I want more precision or a different strategy, I will ask.
Simplify
Always a good strategy for hard problems: take them in smaller steps. If you can't get a hard problem at all, mention a similar easier problem to show your knowledge.
Calm Down
Prepare well: know the material and practice with a friend. If you're stuck on one problem, make a short list of pertinent results or definitions that might help. Ask for a hint, or to skip to another problem. Manage your overall stress; see Stay Healthy in the Study Tips above. Visit the Academic Success Center. | 677.169 | 1 |
MULTIVARIABLE CALCULUS & DIFFERENTIAL EQUATIONS
Students in this course start the year by strengthening skills in the calculus of a single variable (including the calculus of parametric and polar equations, and advanced integration techniques). They learn how to describe lines, planes, and a variety of other surfaces in space. They then apply the tools of calculus to functions in multidimensional spaces. They master the vector-calculus skills in a typical college-level Calculus III course, including vectors and vector-valued functions; partial derivatives, directional derivatives, and gradients; multiple integration; and line and surface integrals. Students learn to identify and solve a variety of differential equations, including exact first-order equations, second-order homogeneous and nonhomogeneous linear equations, and partial differential equations. Students apply what they're learning to various scientific fields. Built on a foundation of sophisticated problem solving, the course also features meaty mathematical discussions, a major project, and exploratory activities that will help students develop their advanced math skills. | 677.169 | 1 |
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when solving a rational equation why is it okay to remove the denominator by multiplying both sides by the LCD | 677.169 | 1 |
Geometry for Elementary School
Description: This book is intended for use by a parent (or a teacher) and a child. It is recommended that the parent have some familiarity with geometry, but this is not necessary. The parent can simply read the chapter before teaching the child and then learn it together.
Higher Elementary Geometry by Venugopal This geometry text is written for the Pre-university students to serve as an introduction to Higher Elementary Geometry, either as a subsidiary or a main Subject. The aim is to give to the Students certain Elementary ideas about the subject. (7274 views)
Solid Geometry, with Problems and Applications by H. E. Slaught, N. J. Lennes - Allyn and Bacon From the table of contents: Introduction; Axioms and Theorems from Plane Geometry; Properties of the Plane; Regular Polyhedrons; Prisms and Cylinders; Pyramids and Cones; The Sphere; Portraits and biographical sketches; and more. (8715 views)
The Elements of Solid Geometry by William C. Bartol - Leach, Shewell & Sanborn The author gives a number of theorems for demonstration and many illustrative examples. A section on Mensuration is introduced with the design of calling special attention to all the important rules for finding volumes and surfaces of solids ... (1736 views) | 677.169 | 1 |
Variation: Direct, Inverse, Joint and Combined Lesson
Be sure that you have an application to open
this file type before downloading and/or purchasing.
5 MB|6 pages + complete solutions
Share
Product Description
In this lesson, students cover the following topics:
• Direct, Inverse, Joint, and Combined Variation problems
• Identify the constant given points, tables, or a graph
• Identify the type of variation given points, a table, a graph, or a real world scenario
• Write an equation based on the variation relationship
• Understand the similarities and differences between variations
Students preview the lesson by watching a short video on YouTube and then come to class with some prior knowledge. This lesson includes a video link, a warm-up, notes and homework.
Please look at the preview to find out more information about this resource. | 677.169 | 1 |
Edexcel AS and A Level Modular Mathematics: Mechanics 5
•Student-friendly labored examples and strategies, best as much as a wealth of perform questions.
•Sample examination papers for thorough examination coaching.
•Regular assessment sections consolidate studying.
•Opportunities for stretch and problem awarded during the path.
•'Escalator section' to step up from GCSE.
Quick preview of Edexcel AS and A Level Modular Mathematics: Mechanics 5 PDF
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Homework
Two kinds of homework
Problem based homework. These
assignments focus on learning the mathematical
content of the course.
Writing based homework. These
assignments focus on the communication of
mathematics.
Eventually the expectation will be that the techniques
you develop on your writing homeworks will carry over
onto the problem based assignments. That is, by the
end of the semester the writing on the problem based
assignments should be at the same level as those of
the writing based assignments | 677.169 | 1 |
This edition: The Language of Algebra
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This program provides a survey of basic mathematical terminology. Content includes properties of the real number system and the basic axioms and theorems of algebra. Specific terms covered include algebraic expression, variable, product, sum term, factors, common factors, like terms, simplify, equation, sets of numbers, and axioms. Definitions of these terms lay a foundation for working with the concepts.
About this series
In this series, host Sol Garfunkel explains how algebra is used for solving real-world problems and clearly explains concepts that may baffle many students. Graphic illustrations and on-location examples help students connect mathematics to daily life. The series also has applications in geometry and calculus instruction. College Algebra: In Simplest Terms is also valuable for teachers seeking to review the subject matter. | 677.169 | 1 |
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Precalculus: An Investigation of Functions
Precalculus: An Investigation of Functions by David Lippman, Melonie Rasmussen
Publisher: Lulu.com2011 ISBN/ASIN: B005J2KM16 Number of pages: 568
Description: A textbook covering a two-quarter precalculus sequence including trigonometry. An emphasis is placed on modeling and interpretation, as well as the important characteristics needed in calculus. Exercises are included in the book.
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Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965-1996
Begun in Hungary in the nineteenth century, Mathematical Olympiads are now held for high school students throughout the world. They feature problems which, though they require only high school mathematics, seem very difficult because they are unpredictable and have no obvious starting point. This book introduces readers to these delightful and challenging problems and aims to convince them that Olympiads are not just for a select minority. The book contains problems from the British Mathematical Olympiad (BMO) competitions between 1965 and 1996. It includes hints and solutions for each problem from 1975 on, a review of the basic mathematical skills needed, and a list of recommended reading, making it an ideal source for enriching one's experience in mathematics.
Product Details
Table of Contents
Problems and Problem Solving How to Use this Book A Little Useful Mathematics Introduction Numbers Algebra Proof Elementary number theory Geometry Trigonometric formulae Some Books for Your Bookshelf The Problems Hints and Outline Solutions Appendix: The International Mathematical Olympiad: UK teams and results 1967 - 1996
Editorial Reviews
"I would highly recommend this book for anyone interested in this sort of problem solving. It would be a particularly valuable resource for those who participate in mathematics competitions at the high school or college level."—Mathematics Association of America Online
"Gardiner uses the problems from the first 32 British Mathematical Olympiads as a vehicle for encouraging mathematical problem-solving skills. The problems only require basic mathematical knowledge at the high-school level, but the fact that several problems were to be solved in a three-hour period adds a requirement for quick ingenuity to the problem-solving approaches." —Choice
"This valuable reference for mathematics teachers contains problems from the first thirty-two years of the British Mathematical Olympiad, 1965-1996. Although they do not depend on any particular prerequisite skills, the problems are intended for high school students. The book is clearly arranged, with a section on refresher mathematics, a section containing the problems, and a section on hints and solutions. The hints are distinguished from the solutions, so that the reader can read only the hint and continue solving the problem. The author recommends these challenging problems as practice for students who are preparing for competitions. Teachers could also use these problems in middle and high school courses to add rigor to the mathematics curriculum. The book contains a bibliography of other problem-solving books and a listing of the British team members who participated in the International Mathematical Olympiad."—The Mathematics Teacher
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Similar ThisThe 10th edition of Elementary Differential Equations, like its predecessors, In addition to expanded explanations, the 10th edition includes new problems, updated figures and examples to help motivate students. The book is written primarily for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for reading the book is a working knowledge of calculus, gained from a normal two?, or three? semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.
An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.
"Written in an admirably cleancut and economical style." — Mathematical Reviews. This concise text offers undergraduates in mathematics and science a thorough and systematic first course in elementary differential equations. Presuming a knowledge of basic calculus, the book first reviews the mathematical essentials required to master the materials to be presented. The next four chapters take up linear equations, those of the first order and those with constant coefficients, variable coefficients, and regular singular points. The last two chapters address the existence and uniqueness of solutions to both first order equations and to systems and n-th order equations. Throughout the book, the author carries the theory far enough to include the statements and proofs of the simpler existence and uniqueness theorems. Dr. Coddington, who has taught at MIT, Princeton, and UCLA, has included many exercises designed to develop the student's technique in solving equations. He has also included problems (with answers) selected to sharpen understanding of the mathematical structure of the subject, and to introduce a variety of relevant topics not covered in the text, e.g. stability, equations with periodic coefficients, and boundary value problems.
This book has been widely acclaimed for its clear, cogent presentation of the theory of partial differential equations, and the incisive application of its principal topics to commonly encountered problems in the physical sciences and engineering. It was developed and tested at Purdue University over a period of five years in classes for advanced undergraduate and beginning graduate students in mathematics, engineering and the physical sciences. The book begins with a short review of calculus and ordinary differential equations, then moves on to explore integral curves and surfaces of vector fields, quasi-linear and linear equations of first order, series solutions and the Cauchy Kovalevsky theorem. It then delves into linear partial differential equations, examines the Laplace, wave and heat equations, and concludes with a brief treatment of hyperbolic systems of equations. Among the most important features of the text are the challenging problems at the end of each section which require a wide variety of responses from students, from providing details of the derivation of an item presented to solving specific problems associated with partial differential equations. Requiring only a modest mathematical background, the text will be indispensable to those who need to use partial differential equations in solving physical problems. It will provide as well the mathematical fundamentals for those who intend to pursue the study of more advanced topics, including modern theory.
This revised and updated text, now in its second edition, continues to present the theoretical concepts of methods of solutions of ordinary and partial differential equations. It equips students with the various tools and techniques to model different physical problems using such equations. The book discusses the basic concepts of ordinary and partial differential equations. It contains different methods of solving ordinary differential equations of first order and higher degree. It gives the solution methodology for linear differential equations with constant and variable coefficients and linear differential equations of second order. The text elaborates simultaneous linear differential equations, total differential equations, and partial differential equations along with the series solution of second order linear differential equations. It also covers Bessel's and Legendre's equations and functions, and the Laplace transform. Finally, the book revisits partial differential equations to solve the Laplace equation, wave equation and diffusion equation, and discusses the methods to solve partial differential equations using the Fourier transform. A large number of solved examples as well as exercises at the end of chapters help the students comprehend and strengthen the underlying concepts. The book is intended for undergraduate and postgraduate students of Mathematics (B.A./B.Sc., M.A./M.Sc.), and undergraduate students of all branches of engineering (B.E./B.Tech.), as part of their course in Engineering Mathematics. New to the SECOND Edition • Includes new sections and subsections such as applications of differential equations, special substitution (Lagrange and Riccati), solutions of non-linear equations which are exact, method of variation of parameters for linear equations of order higher than two, and method of undetermined coefficients • Incorporates several worked-out examples and exercises with their answers • Contains a new Chapter 19 on 'Z-Transforms and its Applications'.
A self-contained text for an introductory course, this volume places strong emphasis on physical applications. Key elements of differential equations and linear algebra are introduced early and are consistently referenced, all theorems are proved using elementary methods, and numerous worked-out examples appear throughout. The highly readable text approaches calculus from the student's viewpoint and points out potential stumbling blocks before they develop. A collection of more than 1,600 problems ranges from exercise material to exploration of new points of theory — many of the answers are found at the end of the book; some of them worked out fully so that the entire process can be followed. This well-organized, unified text is copiously illustrated, amply cross-referenced, and fully indexed.
Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, the book is appropriate for courses for majors in mathematics, science, and engineering that study systems of differential equations. Because of its emphasis on linearity, the text opens with a full chapter devoted to essential ideas in linear algebra. Motivated by future problems in systems of differential equations, the chapter on linear algebra introduces such key ideas as systems of algebraic equations, linear combinations, the eigenvalue problem, and bases and dimension of vector spaces. This chapter enables students to quickly learn enough linear algebra to appreciate the structure of solutions to linear differential equations and systems thereof in subsequent study and to apply these ideas regularly. The book offers an example-driven approach, beginning each chapter with one or two motivating problems that are applied in nature. The following chapter develops the mathematics necessary to solve these problems and explores related topics further. Even in more theoretical developments, we use an example-first style to build intuition and understanding before stating or proving general results. Over 100 figures provide visual demonstration of key ideas; the use of the computer algebra system Maple and Microsoft Excel are presented in detail throughout to provide further perspective and support students' use of technology in solving problems. Each chapter closes with several substantial projects for further study, many of which are based in applications. Errata sheet available at: Convenient Organization of Essential Material so You Can Look up Formulas Fast
Containing a careful selection of standard and timely topics, the Pocket Book of Integrals and Mathematical Formulas, Fourth Edition presents many numerical and statistical tables, scores of worked examples, and the most useful mathematical formulas for engineering and scientific applications. This fourth edition of a bestseller provides even more comprehensive coverage with the inclusion of several additional topics, all while maintaining its accessible, clear style and handy size.
New to the Fourth Edition
• An expanded chapter on series that covers many fascinating properties of the natural numbers that follow from number theory
• New applications such as geostationary satellite orbits and drug kinetics
• An expanded statistics section that discusses nonlinear regression as well as the normal approximation of the binomial distribution
• Revised format of the table of integrals for easier use of the forms and functions
Easy to Use on the Go
The book addresses a range of areas, from elementary algebra, geometry, matrices, and trigonometry to calculus, vector analysis, differential equations, and statistics. Featuring a convenient, portable size, it is sure to remain in the pockets or on the desks of all who use mathematical formulas and tables of integrals and derivatives.
Many textbooks on differential equations are written to be interesting to the teacher rather than the student. Introduction to Differential Equations with Dynamical Systems is directed toward students. This concise and up-to-date textbook addresses the challenges that undergraduate mathematics, engineering, and science students experience during a first course on differential equations. And, while covering all the standard parts of the subject, the book emphasizes linear constant coefficient equations and applications, including the topics essential to engineering students. Stephen Campbell and Richard Haberman--using carefully worded derivations, elementary explanations, and examples, exercises, and figures rather than theorems and proofs--have written a book that makes learning and teaching differential equations easier and more relevant. The book also presents elementary dynamical systems in a unique and flexible way that is suitable for all courses, regardless of length.
This exceptionally well-written and well-organized text is the outgrowth of a course given every year for 45 years at the Chalmers University of Technology, Goteborg, Sweden. The object of the course was to give students a basic knowledge of Fourier analysis and certain of its applications. The text is self-contained with respect to such analysis; however, in areas where the author relies on results from branches of mathematics outside the scope of this book, references to widely used books are given. Table of Contents: Chapter 1. Fourier series 1.1 Basic concepts 1.2 Fourier series and Fourier coefficients 1.3 A minimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem 1.4 Convergence of Fourier series 1.5 The Parseval formula 1.6 Determination of the sum of certain trigonometric series Chapter 2. Orthogonal systems 2.1 Integration of complex-valued functions of a real variable 2.2 Orthogonal systems 2.3 Complete orthogonal systems 2.4 Integration of Fourier series 2.5 The Gram-Schmidt orthogonalization process 2.6 Sturm-Liouville problems Chapter 3. Orthogonal polynomials 3.1 The Legendre polynomials 3.2 Legendre series 3.3 The Legendre differential equation. The generating function of the Legendre polynomials 3.4 The Tchebycheff polynomials 3.5 Tchebycheff series 3.6 The Hermite polynomials. The Laguerre polynomials Chapter 4. Fourier transforms 4.1 Infinite interval of integration 4.2 The Fourier integral formula: a heuristic introduction 4.3 Auxiliary theorems 4.4 Proof of the Fourier integral formula. Fourier transforms 4.5 The convention theorem. The Parseval formula Chapter 5. Laplace transforms 5.1 Definition of the Laplace transform. Domain. Analyticity 5.2 Inversion formula 5.3 Further properties of Laplace transforms. The convolution theorem 5.4 Applications to ordinary differential equations Chapter 6. Bessel functions 6.1 The gamma function 6.2 The Bessel differential equation. Bessel functions 6.3 Some particular Bessel functions 6.4 Recursion formulas for the Bessel functions 6.5 Estimation of Bessel functions for large values of x. The zeros of the Bessel functions 6.6 Bessel series 6.7 The generating function of the Bessel functions of integral order 6.8 Neumann functions Chapter 7. Partial differential equations of first order 7.1 Introduction 7.2 The differential equation of a family of surfaces 7.3 Homogeneous differential equations 7.4 Linear and quasilinear differential equations Chapter 8. Partial differential equations of second order 8.1 Problems in physics leading to partial differential equations 8.2 Definitions 8.3 The wave equation 8.4 The heat equation 8.5 The Laplace equation Answers to exercises; Bibliography; Conventions; Symbols; Index Written on an advanced level, the book is aimed at advanced undergraduates and graduate students with a background in calculus, linear algebra, ordinary differential equations, and complex analysis. Over 260 carefully chosen exercises, with answers, encompass both routing and more challenging problems to help students test their grasp of the material.
Elementary Differential Equations, 11th Edition
In addition to expanded explanations, the 11th edition includes new problems, updated figures and examples to help motivate students. The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two?] or three?] semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.
This systematically-organized text on the theory of differential equations deals with the basic concepts and the methods of solving ordinary differential equations. Various existence theorems, properties of uniqueness, oscillation and stability theories, have all been explained with suitable examples to enhance students' understanding of the subject. The book also discusses in sufficient detail the qualitative, the quantitative, and the approximation techniques, linear equations with variable and constants coefficients, regular singular points, and homogeneous equations with analytic coefficients. Finally, it explains Riccati equation, boundary value problems, the Sturm–Liouville problem, Green's function, the Picard's theorem, and the Sturm–Picone theorem. The text is supported by a number of worked-out examples to make the concepts clear, and it also provides a number of exercises help students test their knowledge and improve their skills in solving differential equations. The book is intended to serve as a text for the postgraduate students of mathematics and applied mathematics. It will also be useful to the candidates preparing to sit for the competitive examinations such as NET and GATE.
Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory. Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent study will particularly appreciate the worked examples that appear throughout the text.
Although the Partial Differential Equations (PDE) models that are now studied are usually beyond traditional mathematical analysis, the numerical methods that are being developed and used require testing and validation. This is often done with PDEs that have known, exact, analytical solutions. The development of analytical solutions is also an active area of research, with many advances being reported recently, particularly traveling wave solutions for nonlinear evolutionary PDEs. Thus, the current development of analytical solutions directly supports the development of numerical methods by providing a spectrum of test problems that can be used to evaluate numerical methods.
This book surveys some of these new developments in analytical and numerical methods, and relates the two through a series of PDE examples. The PDEs that have been selected are largely "named'' since they carry the names of their original contributors. These names usually signify that the PDEs are widely recognized and used in many application areas. The authors' intention is to provide a set of numerical and analytical methods based on the concept of a traveling wave, with a central feature of conversion of the PDEs to ODEs.
The Matlab and Maple software will be available for download from this website shortly.
Includes a spectrum of applications in science, engineering, applied mathematicsPresents a combination of numerical and analytical methodsProvides transportable computer codes in Matlab and Maple
In the first edition of his seminal introduction to wavelets, James S. Walker informed us that the potential applications for wavelets were virtually unlimited. Since that time thousands of published papers have proven him true, while also necessitating the creation of a new edition of his bestselling primer. Updated and fully revised to include the latest developments, this second edition of A Primer on Wavelets and Their Scientific Applications guides readers through the main ideas of wavelet analysis in order to develop a thorough appreciation of wavelet applications.
Ingeniously relying on elementary algebra and just a smidgen of calculus, Professor Walker demonstrates how the underlying ideas behind wavelet analysis can be applied to solve significant problems in audio and image processing, as well in biology and medicine.
Nearly twice as long as the original, this new edition provides
· 104 worked examples and 222 exercises, constituting a veritable book of review material
· Two sections on biorthogonal wavelets
· A mini-course on image compression, including a tutorial on arithmetic compression
· Concise yet complete coverage of the fundamentals of time-frequency analysis, showcasing its application to audio denoising, and musical theory and synthesis
· An introduction to the multiresolution principle, a new mathematical concept in musical theory
· Expanded suggestions for research projects
· An enhanced list of references
· FAWAV: software designed by the author, which allows readers to duplicate described applications and experiment with other ideas.
To keep the book current, Professor Walker has created a supplementary website. This online repository includes ready-to-download software, and sound and image files, as well as access to many of the most important papers in the field.
This text develops the theory of systems of stochastic differential equations, and it presents applications in probability, partial differential equations, and stochastic control problems. Originally published in two volumes, it combines a book of basic theory and selected topics with a book of applications. The first part explores Markov processes and Brownian motion; the stochastic integral and stochastic differential equations; elliptic and parabolic partial differential equations and their relations to stochastic differential equations; the Cameron-Martin-Girsanov theorem; and asymptotic estimates for solutions. The section concludes with a look at recurrent and transient solutions. Volume 2 begins with an overview of auxiliary results in partial differential equations, followed by chapters on nonattainability, stability and spiraling of solutions; the Dirichlet problem for degenerate elliptic equations; small random perturbations of dynamical systems; and fundamental solutions of degenerate parabolic equations. Final chapters examine stopping time problems and stochastic games and stochastic differential games. Problems appear at the end of each chapter, and a familiarity with elementary probability is the sole prerequisite.
In this third volume of his modern introduction to quantum field theory, Eberhard Zeidler examines the mathematical and physical aspects of gauge theory as a principle tool for describing the four fundamental forces which act in the universe: gravitative, electromagnetic, weak interaction and strong interaction.
Volume III concentrates on the classical aspects of gauge theory, describing the four fundamental forces by the curvature of appropriate fiber bundles. This must be supplemented by the crucial, but elusive quantization procedure.
The book is arranged in four sections, devoted to realizing the universal principle force equals curvature:
Part I: The Euclidean Manifold as a Paradigm
Part II: Ariadne's Thread in Gauge Theory
Part III: Einstein's Theory of Special Relativity
Part IV: Ariadne's Thread in Cohomology
For students of mathematics the book is designed to demonstrate that detailed knowledge of the physical background helps to reveal interesting interrelationships among diverse mathematical topics. Physics students will be exposed to a fairly advanced mathematics, beyond the level covered in the typical physics curriculum.
Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosmos), and in the world of elementary particles (microcosmos). broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fields
Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible, combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger's equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems.
The Third Edition is organized around four themes: methods of solution for initial-boundary value problems; applications of partial differential equations; existence and properties of solutions; and the use of software to experiment with graphics and carry out computations. With a primary focus on wave and diffusion processes, Beginning Partial Differential Equations, Third Edition also includes:
Proofs of theorems incorporated within the topical presentation, such as the existence of a solution for the Dirichlet problem The incorporation of Maple™ to perform computations and experiments Unusual applications, such as Poe's pendulum Advanced topical coverage of special functions, such as Bessel, Legendre polynomials, and spherical harmonics Fourier and Laplace transform techniques to solve important problems
Beginning of Partial Differential Equations, Third Edition is an ideal textbook for upper-undergraduate and first-year graduate-level courses in analysis and applied mathematics, science, and engineering.
Theoretically, multiwavelets hold significant advantages over standard wavelets, particularly for solving more complicated problems, and hence are of great interest. Meeting the needs of engineers and mathematicians, this book provides a comprehensive overview of multiwavelets. The author presents the theory of wavelets from the viewpoint of general multiwavelets, which includes scalar m-band and standard wavelets as special cases, provides a more coherent approach, and provides alternative proofs and new insights even for standard wavelets. The treatment includes complete MATLAB routines that allow readers to implement and experiment with multiwavelet algorithms.
This monograph deals with the quantitative overconvergence phenomenon in complex approximation by various operators. The book is divided into three chapters. First, the results for the Schurer-Faber operator, Beta operators of first kind, Bernstein-Durrmeyer-type operators and Lorentz operator are presented. The main focus is on results for several q-Bernstein kind of operators with q > 1, when the geometric order of approximation 1/qn is obtained not only in complex compact disks but also in quaternion compact disks and in other compact subsets of the complex plane. The focus then shifts to quantitative overconvergence and convolution overconvergence results for the complex potentials generated by the Beta and Gamma Euler's functions. Finally quantitative overconvergence results for the most classical orthogonal expansions (of Chebyshev, Legendre, Hermite, Laguerre and Gegenbauer kinds) attached to vector-valued functions are presented. Each chapter concludes with a notes and open problems section, thus providing stimulation for further research. An extensive bibliography and index complete the text.
This book is suitable for researchers and graduate students working in complex approximation and its applications, mathematical analysis and numerical analysis.
This text presents numerical differential equations to graduate (doctoral) students. It includes the three standard approaches to numerical PDE, FDM, FEM and CM, and the two most common time stepping techniques, FDM and Runge-Kutta. We present both the numerical technique and the supporting theory.The applied techniques include those that arise in the present literature. The supporting mathematical theory includes the general convergence theory. This material should be readily accessible to students with basic knowledge of mathematical analysis, Lebesgue measure and the basics of Hilbert spaces and Banach spaces. Nevertheless, we have made the book free standing in most respects. Most importantly, the terminology is introduced, explained and developed as needed.The examples presented are taken from multiple vital application areas including finance, aerospace, mathematical biology and fluid mechanics. The text may be used as the basis for several distinct lecture courses or as a reference. For instance, this text will support a general applications course or an FEM course with theory and applications. The presentation of material is empirically-based as more and more is demanded of the reader as we progress through the material. By the end of the text, the level of detail is reminiscent of journal articles. Indeed, it is our intention that this material be used to launch a research career in numerical PDE.
The purpose of this textbook is to present an array of topics in Calculus, and conceptually follow our previous effort Mathematical Analysis I.The present material is partly found, in fact, in the syllabus of the typical second lecture course in Calculus as offered in most Italian universities. While the subject matter known as `Calculus 1' is more or less standard, and concerns real functions of real variables, the topics of a course on `Calculus 2'can vary a lot, resulting in a bigger flexibility. For these reasons the Authors tried to cover a wide range of subjects, not forgetting that the number of credits the current programme specifications confers to a second Calculus course is not comparable to the amount of content gathered here. The reminders disseminated in the text make the chapters more independent from one another, allowing the reader to jump back and forth, and thus enhancing the versatility of the book. On the website: 2, the interested reader may find the rigorous explanation of the results that are merely stated without proof in the book, together with useful additional material. The Authors have completely omitted the proofs whose technical aspects prevail over the fundamental notions and ideas. The large number of exercises gathered according to the main topics at the end of each chapter should help the student put his improvements to the test. The solution to all exercises is provided, and very often the procedure for solving is outlined.
"Optimal control theory is concerned with finding control functions that minimize cost functions for systems described by differential equations. The methods have found widespread applications in aeronautics, mechanical engineering, the life sciences, and many other disciplines. This book focuses on optimal control problems where the state equation is an elliptic or parabolic partial differential equation. Included are topics such as the existence of optimal solutions, necessary optimality conditions and adjoint equations, second-order sufficient conditions, and main principles of selected numerical techniques. It also contains a survey on the Karush-Kuhn-Tucker theory of nonlinear programming in Banach spaces. The exposition begins with control problems with linear equations, quadratic cost functions and control constraints. To make the book self-contained, basic facts on weak solutions of elliptic and parabolic equations are introduced. Principles of functional analysis are introduced and explained as they are needed. Many simple examples illustrate the theory and its hidden difficulties. This start to the book makes it fairly self-contained and suitable for advanced undergraduates or beginning graduate students. Advanced control problems for nonlinear partial differential equations are also discussed. As prerequisites, results on boundedness and continuity of solutions to semilinear elliptic and parabolic equations are addressed. These topics are not yet readily available in books on PDEs, making the exposition also interesting for researchers. Alongside the main theme of the analysis of problems of optimal control, Tr'oltzsch also discusses numerical techniques. The exposition is confined to brief introductions into the basic ideas in order to give the reader an impression of how the theory can be realized numerically. After reading this book, the reader will be familiar with the main principles of the numerical analysis of PDE-constrained optimization."--Publisher's description.
differential equations in the field – the heat equation, the wave equation, and Laplace's equation. The most common techniques of solving such equations are developed in this book, including Green's functions, the Fourier transform, and the Laplace transform, which all have applications in mathematics and physics far beyond solving the above equations. The book's focus is on both the equations and their methods of solution. Ordinary differential equations and PDEs are solved including Bessel Functions, making the book useful as a graduate level textbook. The book's rigor supports the vital sophistication for someone wanting to continue further in areas of mathematical physics.Examines in depth both the equations and their methods of solutionPresents physical concepts in a mathematical frameworkContains detailed mathematical derivations and solutions— reinforcing the material through repetition of both the equations and the techniques Includes several examples solved by multiple methods—highlighting the strengths and weaknesses of various techniques and providing additional practice
This text records the problems given for the first 15 annual undergraduate mathematics competitions, held in March each year since 2001 at the University of Toronto. Problems cover areas of single-variable differential and integral calculus, linear algebra, advanced algebra, analytic geometry, combinatorics, basic group theory, and number theory. The problems of the competitions are given in chronological order as presented to the students. The solutions appear in subsequent chapters according to subject matter. Appendices recall some background material and list the names of students who did well.
The University of Toronto Undergraduate Competition was founded to provide additional competition experience for undergraduates preparing for the Putnam competition, and is particularly useful for the freshman or sophomore undergraduate. Lecturers, instructors, and coaches for mathematics competitions will find this presentation useful. Many of the problems are of intermediate difficulty and relate to the first two years of the undergraduate curriculum. The problems presented may be particularly useful for regular class assignments. Moreover, this text contains problems that lie outside the regular syllabus and may interest students who are eager to learn beyond the classroom.Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity. Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and differential equations in the complex field. The author discusses only ordinary differential equations, excluding coverage of the methods of integration and stressing the importance of reading the properties of the integrals directly from the equations. An extensive bibliography and helpful indexes conclude the text.
1001 Calculus Practice Problems For Dummies takes you beyond the instruction and guidance offered in Calculus For Dummies, giving you 1001 opportunities to practice solving problems from the major topics in your calculus course. Plus, an online component provides you with a collection of calculus problems presented in multiple-choice format to further help you test your skills as you go.
Gives you a chance to practice and reinforce the skills you learn in your calculus course Helps you refine your understanding of calculus Practice problems with answer explanations that detail every step of every problem
The practice problems in 1001 Calculus Practice Problems For Dummies range in areas of difficulty and style, providing you with the practice help you need to score high at exam time.
In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula—long regarded as the gold standard for mathematical beauty—and shows why it still lies at the heart of complex number theory. In some ways a sequel to Nahin's An Imaginary Tale, this book examines the many applications of complex numbers alongside intriguing stories from the history of mathematics. Dr. Euler's Fabulous Formula is accessible to any reader familiar with calculus and differential equations, and promises to inspire mathematicians for years to come.
This book deals with the mathematical analysis and the numerical approximation of eddy current problems in the time-harmonic case. It takes into account all the most used formulations, placing the problem in a rigorous functional frameworkThe present monograph, based mainly on studies of the authors and their - authors, and also on lectures given by the authors in the past few years, has the following particular aims: To present basic results (with proofs) of optimal stopping theory in both discrete and continuous time using both martingale and Mar- vian approaches; To select a seriesof concrete problems ofgeneral interest from the t- ory of probability, mathematical statistics, and mathematical ?nance that can be reformulated as problems of optimal stopping of stochastic processes and solved by reduction to free-boundary problems of real analysis (Stefan problems). The table of contents found below gives a clearer idea of the material included in the monograph. Credits and historical comments are given at the end of each chapter or section. The bibliography contains a material for further reading. Acknowledgements.TheauthorsthankL.E.Dubins,S.E.Graversen,J.L.Ped- sen and L. A. Shepp for useful discussions. The authors are grateful to T. B. To- zovafortheexcellenteditorialworkonthemonograph.Financialsupportandh- pitality from ETH, Zur ̈ ich (Switzerland), MaPhySto (Denmark), MIMS (Man- ester) and Thiele Centre (Aarhus) are gratefully acknowledged. The authors are also grateful to INTAS and RFBR for the support provided under their grants. The grant NSh-1758.2003.1 is gratefully acknowledged. Large portions of the text were presented in the "School and Symposium on Optimal Stopping with App- cations" that was held in Manchester, England from 17th to 27th January 2006.
This classroom-tested volume offers a definitive look at modern analysis, with views of applications to statistics, numerical analysis, Fourier series, differential equations, mathematical analysis, and functional analysis. Upper-level undergraduate students with a background in calculus will benefit from its teachings, along with beginning graduate students seeking a firm grounding in modern analysis. A self-contained text, it presents the necessary background on the limit concept, and the first seven chapters could constitute a one-semester introduction to limits. Subsequent chapters discuss differential calculus of the real line, the Riemann-Stieltjes integral, sequences and series of functions, transcendental functions, inner product spaces and Fourier series, normed linear spaces and the Riesz representation theorem, and the Lebesgue integral. Supplementary materials include an appendix on vector spaces and more than 750 exercises of varying degrees of difficulty. Hints and solutions to selected exercises, indicated by an asterisk, appear at the back of the book.
Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan, Second Edition provides an accessible approach for conducting Bayesian data analysis, as material is explained clearly with concrete examples. Included are step-by-step instructions on how to carry out Bayesian data analyses in the popular and free software R and WinBugs, as well as new programs in JAGS and Stan. The new programs are designed to be much easier to use than the scripts in the first edition. In particular, there are now compact high-level scripts that make it easy to run the programs on your own data sets.
The book is divided into three parts and begins with the basics: models, probability, Bayes' rule, and the R programming language. The discussion then moves to the fundamentals applied to inferring a binomial probability, before concluding with chapters on the generalized linear model. Topics include metric-predicted variable on one or two groups; metric-predicted variable with one metric predictor; metric-predicted variable with multiple metric predictors; metric-predicted variable with one nominal predictor; and metric-predicted variable with multiple nominal predictors. The exercises found in the text have explicit purposes and guidelines for accomplishment.
This book is intended for first-year graduate students or advanced undergraduates in statistics, data analysis, psychology, cognitive science, social sciences, clinical sciences, and consumer sciences in business.
Accessible, including the basics of essential concepts of probability and random samplingExamples with R programming language and JAGS softwareComprehensive coverage of all scenarios addressed by non-Bayesian textbooks: t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis)Coverage of experiment planningR and JAGS computer programming code on websiteExercises have explicit purposes and guidelines for accomplishment
Provides step-by-step instructions on how to conduct Bayesian data analyses in the popular and free software R and WinBugs
This lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks—a sort of potato-stamp method—Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling art images where the beauty of nature meets the precision of mathematics.
Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you'll learn how to create breathtaking art images of your own.
Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.
Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI explores and illustrates this process. Each section of the first six chapters is motivated by a specific application. The author applies a model, selects a numerical method, implements computer simulations, and assesses the ensuing results. These chapters include an abundance of MATLAB code. By studying the code instead of using it as a "black box, " you take the first step toward more sophisticated numerical modeling. The last four chapters focus on multiprocessing algorithms implemented using message passing interface (MPI). These chapters include Fortran 9x codes that illustrate the basic MPI subroutines and revisit the applications of the previous chapters from a parallel implementation perspective. All of the codes are available for download from www4.ncsu.edu./~white.
This book is not just about math, not just about computing, and not just about applications, but about all three--in other words, computational science. Whether used as an undergraduate textbook, for self-study, or for reference, it builds the foundation you need to make numerical modeling and simulation integral parts of your investigational toolbox.
"This is quite a well-done book: very tightly organized, better-than-average exposition, and numerous examples, illustrations, and applications." —Mathematical Reviews of the American Mathematical Society
An Introduction to Linear Programming and Game Theory, Third Edition presents a rigorous, yet accessible, introduction to the theoretical concepts and computational techniques of linear programming and game theory. Now with more extensive modeling exercises and detailed integer programming examples, this book uniquely illustrates how mathematics can be used in real-world applications in the social, life, and managerial sciences, providing readers with the opportunity to develop and apply their analytical abilities when solving realistic problems.
This Third Edition addresses various new topics and improvements in the field of mathematical programming, and it also presents two software programs, LP Assistant and the Solver add-in for Microsoft Office Excel, for solving linear programming problems. LP Assistant, developed by coauthor Gerard Keough, allows readers to perform the basic steps of the algorithms provided in the book and is freely available via the book's related Web site. The use of the sensitivity analysis report and integer programming algorithm from the Solver add-in for Microsoft Office Excel is introduced so readers can solve the book's linear and integer programming problems. A detailed appendix contains instructions for the use of both applications.
Additional features of the Third Edition include:
A discussion of sensitivity analysis for the two-variable problem, along with new examples demonstrating integer programming, non-linear programming, and make vs. buy models
Revised proofs and a discussion on the relevance and solution of the dual problem
A section on developing an example in Data Envelopment Analysis
An outline of the proof of John Nash's theorem on the existence of equilibrium strategy pairs for non-cooperative, non-zero-sum games
Providing a complete mathematical development of all presented concepts and examples, Introduction to Linear Programming and Game Theory, Third Edition is an ideal text for linear programming and mathematical modeling courses at the upper-undergraduate and graduate levels. It also serves as a valuable reference for professionals who use game theory in business, economics, and management science.
This book presents a twenty-first century approach to classical polynomial and rational approximation theory. The reader will find a strikingly original treatment of the subject, completely unlike any of the existing literature on approximation theory, with a rich set of both computational and theoretical exercises for the classroom. There are many original features that set this book apart: the emphasis is on topics close to numerical algorithms; every idea is illustrated with Chebfun examples; each chapter has an accompanying Matlab file for the reader to download; the text focuses on theorems and methods for analytic functions; original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment. This textbook is ideal for advanced undergraduates and graduate students across all of applied mathematics.
Linear systems theory is the cornerstone of control theory and a well-established discipline that focuses on linear differential equations from the perspective of control and estimation. In this textbook, João Hespanha covers the key topics of the field in a unique lecture-style format, making the book easy to use for instructors and students. He looks at system representation, stability, controllability and state feedback, observability and state estimation, and realization theory. He provides the background for advanced modern control design techniques and feedback linearization, and examines advanced foundational topics such as multivariable poles and zeros, and LQG/LQR.
The textbook presents only the most essential mathematical derivations, and places comments, discussion, and terminology in sidebars so that readers can follow the core material easily and without distraction. Annotated proofs with sidebars explain the techniques of proof construction, including contradiction, contraposition, cycles of implications to prove equivalence, and the difference between necessity and sufficiency. Annotated theoretical developments also use sidebars to discuss relevant commands available in MATLAB, allowing students to understand these important tools. The balanced chapters can each be covered in approximately two hours of lecture time, simplifying course planning and student review. Solutions to the theoretical and computational exercises are also available for instructors.
Easy-to-use textbook in unique lecture-style format Sidebars explain topics in further detail Annotated proofs and discussions of MATLAB commands Balanced chapters can each be taught in two hours of course lecture Solutions to exercises available to instructors
For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus. Proofs are given in full, and the large number of well-chosen examples and exercises range from routine to challenging.
The second edition preserves the book's clear and concise style, illuminating discussions, and simple, well-motivated proofs. New topics include material on the irrationality of pi, the Baire category theorem, Newton's method and the secant method, and continuous nowhere-differentiable functions.
Review from the first edition:
"This book is intended for the student who has a good, but naïve, understanding of elementary calculus and now wishes to gain a thorough understanding of a few basic concepts in analysis.... The author has tried to write in an informal but precise style, stressing motivation and methods of proof, and ... has succeeded admirably."
Designed for students familiar with abstract mathematical concepts but possessing little knowledge of physics, this text focuses on generality and careful formulation rather than problem-solving. Its author, a member of the distinguished National Academy of Science, based this graduate-level text on the course he taught at Harvard University. Opening chapters on classical mechanics examine the laws of particle mechanics; generalized coordinates and differentiable manifolds; oscillations, waves, and Hilbert space; and statistical mechanics. A survey of quantum mechanics covers the old quantum theory; the quantum-mechanical substitute for phase space; quantum dynamics and the Schrödinger equation; the canonical "quantization" of a classical system; some elementary examples and original discoveries by Schrödinger and Heisenberg; generalized coordinates; linear systems and the quantization of the electromagnetic field; and quantum-statistical mechanics. The final section on group theory and quantum mechanics of the atom explores basic notions in the theory of group representations; perturbations and the group theoretical classification of eigenvalues; spherical symmetry and spin; and the n-electron atom and the Pauli exclusion principle.
Incorporating an innovative modeling approach, this book for a one-semester differential equations course emphasizes conceptual understanding to help users relate information taught in the classroom to real-world experiences. Certain models reappear throughout the book as running themes to synthesize different concepts from multiple angles, and a dynamical systems focus emphasizes predicting the long-term behavior of these recurring models. Users will discover how to identify and harness the mathematics they will use in their careers, and apply it effectively outside the classroom. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Due to the rapid expansion of the frontiers of physics and engineering, the demand for higher-level mathematics is increasing yearly. This book is designed to provide accessible knowledge of higher-level mathematics demanded in contemporary physics and engineering. Rigorous mathematical structures of important subjects in these fields are fully covered, which will be helpful for readers to become acquainted with certain abstract mathematical concepts. The selected topics are:
This book is essentially self-contained, and assumes only standard undergraduate preparation such as elementary calculus and linear algebra. It is thus well suited for graduate students in physics and engineering who are interested in theoretical backgrounds of their own fields. Further, it will also be useful for mathematics students who want to understand how certain abstract concepts in mathematics are applied in a practical situation. The readers will not only acquire basic knowledge toward higher-level mathematics, but also imbibe mathematical skills necessary for contemporary studies of their own fields.
A coherent introduction to the techniques for modeling dynamic stochastic systems, this volume also offers a guide to the mathematical, numerical, and simulation tools of systems analysis. Suitable for advanced undergraduates and graduate-level industrial engineers and management science majors, it proposes modeling systems in terms of their simulation, regardless of whether simulation is employed for analysis. Beginning with a view of the conditions that permit a mathematical-numerical analysis, the text explores Poisson and renewal processes, Markov chains in discrete and continuous time, semi-Markov processes, and queuing processes. Each chapter opens with an illustrative case study, and comprehensive presentations include formulation of models, determination of parameters, analysis, and interpretation of results. Programming language–independent algorithms appear for all simulation and numerical procedures.
This incredibly useful guide book to mathematics contains the fundamental working knowledge of mathematics which is needed as an everyday guide for working scientists and engineers, as well as for students. Now in its fifth updated edition, it is easy to understand, and convenient to use. Inside you'll find the information necessary to evaluate most problems which occur in concrete applications. In the newer editions emphasis was laid on those fields of mathematics that became more important for the formulation and modeling of technical and natural processes. For the 5th edition, the chapters "Computer Algebra Systems" and "Dynamical Systems and Chaos" have been revised, updated and expanded.
Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This second edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.
In the late 1950s, many of the more refined aspects of Fourier analysis were transferred from their original settings (the unit circle, the integers, the real line) to arbitrary locally compact abelian (LCA) groups. Rudin's book, published in 1962, was the first to give a systematic account of these developments and has come to be regarded as a classic in the field. The basic facts concerning Fourier analysis and the structure of LCA groups are proved in the opening chapters, in order to make the treatment relatively self-contained.
A half-century ago, advanced calculus was a well-de?ned subject at the core of the undergraduate mathematics curriulum. The classic texts of Taylor [19], Buck [1], Widder [21], and Kaplan [9], for example, show some of the ways it was approached. Over time, certain aspects of the course came to be seen as more signi?cant—those seen as giving a rigorous foundation to calculus—and they - came the basis for a new course, an introduction to real analysis, that eventually supplanted advanced calculus in the core. Advanced calculus did not, in the process, become less important, but its role in the curriculum changed. In fact, a bifurcation occurred. In one direction we got c- culus on n-manifolds, a course beyond the practical reach of many undergraduates; in the other, we got calculus in two and three dimensions but still with the theorems of Stokes and Gauss as the goal. The latter course is intended for everyone who has had a year-long introduction to calculus; it often has a name like Calculus III. In my experience, though, it does not manage to accomplish what the old advancedcalculus course did. Multivariable calculusnaturallysplits intothreeparts:(1)severalfunctionsofonevariable,(2)one function of several variables, and (3) several functions of several variables. The ?rst two are well-developed in Calculus III, but the third is really too large and varied to be treated satisfactorily in the time remaining at the end of a semester. To put it another way: Green's theorem ?ts comfortably; Stokes' and Gauss' do not.
"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."
(David Parrott, Australian Mathematical Society)
"The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community."
(European Mathematical Society)
"Since Stillwell treats many topics, most mathematicians will learn a lot from this book as well as they will find pleasant and rather clear expositions of custom materials. The book is accessible to students that have already experienced calculus, algebra and geometry and will give them a good account of how the different branches of mathematics interact."
(Denis Bonheure, Bulletin of the Belgian Society)
This third edition includes new chapters on simple groups and combinatorics, and new sections on several topics, including the Poincare conjecture. The book has also been enriched by added exercises.
The purpose of this book is to isolate and draw attention to the most important problem-solving techniques typically encountered in undergradu ate mathematics and to illustrate their use by interesting examples and problems not easily found in other sources. Each section features a single idea, the power and versatility of which is demonstrated in the examples and reinforced in the problems. The book serves as an introduction and guide to the problems literature (e.g., as found in the problems sections of undergraduate mathematics journals) and as an easily accessed reference of essential knowledge for students and teachers of mathematics. The book is both an anthology of problems and a manual of instruction. It contains over 700 problems, over one-third of which are worked in detail. Each problem is chosen for its natural appeal and beauty, but primarily to provide the context for illustrating a given problem-solving method. The aim throughout is to show how a basic set of simple techniques can be applied in diverse ways to solve an enormous variety of problems. Whenever possible, problems within sections are chosen to cut across expected course boundaries and to thereby strengthen the evidence that a single intuition is capable of broad application. Each section concludes with "Additional Examples" that point to other contexts where the technique is appropriate.
The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners.
* Focuses on the most often used features of Mathematica for the beginning Mathematica user * New applications from a variety of fields, including engineering, biology, and physics * All applications were completed using recent versions of Mathematica
Fast Fourier Transform - Algorithms and Applications presents an introduction to the principles of the fast Fourier transform (FFT). It covers FFTs, frequency domain filtering, and applications to video and audio signal processing.
As fields like communications, speech and image processing, and related areas are rapidly developing, the FFT as one of the essential parts in digital signal processing has been widely used. Thus there is a pressing need from instructors and students for a book dealing with the latest FFT topics.
Fast Fourier Transform - Algorithms and Applications provides a thorough and detailed explanation of important or up-to-date FFTs. It also has adopted modern approaches like MATLAB examples and projects for better understanding of diverse FFTs.
Fast Fourier Transform - Algorithms and Applications is designed for senior undergraduate and graduate students, faculty, engineers, and scientists in the field, and self-learners to understand FFTs and directly apply them to their fields, efficiently. It is designed to be both a text and a reference. Thus examples, projects and problems all tied with MATLAB, are provided for grasping the concepts concretely. It also includes references to books and review papers and lists of applications, hardware/software, and useful websites. By including many figures, tables, bock diagrams and graphs, this book helps the reader understand the concepts of fast algorithms readily and intuitively. It provides new MATLAB functions and MATLAB source codes. The material in Fast Fourier Transform - Algorithms and Applications is presented without assuming any prior knowledge of FFT. This book is for any professional who wants to have a basic understanding of the latest developments in and applications of FFT. It provides a good reference for any engineer planning to work in this field, either in basic implementation or in research and development.
This incisive text deftly combines both theory and practical example to introduce and explore Fourier series and orthogonal functions and applications of the Fourier method to the solution of boundary-value problems. Directed to advanced undergraduate and graduate students in mathematics as well as in physics and engineering, the book requires no prior knowledge of partial differential equations or advanced vector analysis. Students familiar with partial derivatives, multiple integrals, vectors, and elementary differential equations will find the text both accessible and challenging. The first three chapters of the book address linear spaces, orthogonal functions, and the Fourier series. Chapter 4 introduces Legendre polynomials and Bessel functions, and Chapter 5 takes up heat and temperature. The concluding Chapter 6 explores waves and vibrations and harmonic analysis. Several topics not usually found in undergraduate texts are included, among them summability theory, generalized functions, and spherical harmonics. Throughout the text are 570 exercises devised to encourage students to review what has been read and to apply the theory to specific problems. Those preparing for further study in functional analysis, abstract harmonic analysis, and quantum mechanics will find this book especially valuable for the rigorous preparation it provides. Professional engineers, physicists, and mathematicians seeking to extend their mathematical horizons will find it an invaluable reference as well.
The goal of this text is to help students learn to use calculus intelligently for solving a wide variety of mathematical and physical problems. This book is an outgrowth of our teaching of calculus at Berkeley, and the present edition incorporates many improvements based on our use of the first edition. We list below some of the key features of the book. Examples and Exercises The exercise sets have been carefully constructed to be of maximum use to the students. With few exceptions we adhere to the following policies. • The section exercises are graded into three consecutive groups: (a) The first exercises are routine, modelled almost exactly on the exam ples; these are intended to give students confidence. (b) Next come exercises that are still based directly on the examples and text but which may have variations of wording or which combine different ideas; these are intended to train students to think for themselves. (c) The last exercises in each set are difficult. These are marked with a star (*) and some will challenge even the best students. Difficult does not necessarily mean theoretical; often a starred problem is an interesting application that requires insight into what calculus is really about. • The exercises come in groups of two and often four similar ones. | 677.169 | 1 |
Mathematics
Learning Goals & Mission
Mission
Quantitative literacy and critical thinking are necessary for the grounding in the liberal arts that our diverse community of learners experience as preparation for them to excel intellectually … in our ever-changing urban and global environment. Mathematics has a rich history of providing the language for understanding natural phenomena, including occurrences of both permanence and change in the world. The precise communication of mathematics, in any medium, is a challenging example of expressing thoughts.
Learning Goals
(These may be read as completions of the sentence "The student, upon completion of the major, will be able to…") | 677.169 | 1 |
Algebra Teacher's Activities Kit: 150 Activities that Support Algebra in the Common Core Math Standa
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Help your students succeed with classroom-ready, standards-based activities!
Algebra Teacher's Activities Kit: 150 Activities That Support Algebra in the Common Core Math Standards helps you bring the standards into your algebra classroom with a range of engaging activities that reinforce fundamental algebra skills.
This newly updated second edition is formatted for easy implementation, with teaching notes and answers followed by reproducibles for activities covering the algebra standards for grades 6 through 12. Coverage includes whole numbers, variables, equations, inequalities, graphing, polynomials, factoring, logarithmic functions, statistics, and more, and gives you the material you need to reach students of various abilities and learning styles. Many of these activities are self-correcting, adding interest for students and saving you time.
This book provides dozens of activities that:
• Directly address each Common Core algebra standard
• Engage students and get them excited about math
• Are tailored to a diverse range of levels and abilities
• Reinforce fundamental skills and demonstrate everyday relevance
Algebra lays the groundwork for every math class that comes after it, so it's crucial that students master the material and gain confidence in their abilities. The Algebra Teacher's Activities Kit helps you face the challenge, well-armed with effective activities that help students become successful in algebra class and beyond. 352 pages. | 677.169 | 1 |
Title: Teaching School Mathematics: Algebra | Author(s): Hung-Hsi Wu | Publisher: American Mathematical Society | Year: 2016 | Language: English | Pages : 274 | Size: 2 MB | Extension: pdf
This is a systematic exposition of introductory school algebra written specifically for Common Core era teachers. The emphasis of the exposition is to give a mathematically correct treatment of introductory algebra. For example, it explains the proper use of symbols, why "variable" is ...
Author(s): A K Mallik | Publisher: World Scientific | Year: 2017 | Language: English | Pages: 196 | Size: 3 MB | Extension: pdf
This book is not a mathematics text. It discusses various kinds of numbers and curious interconnections between them. Without getting into hardcore, difficult mathematical technicalities, the book lucidly introduces all kinds of numbers that the mathematicians have created. Interesting anecdotes involving great mathematicians and their marvelous ... | 677.169 | 1 |
Foundations of Algebra
by Lynn Marecek and MaryAnne Anthony-Smith
Hi! We are excited to introduce Foundations of Algebra to you!
We wrote the Foundations of Algebra manuscript for the Prealgebra classes at Santa Ana and Santiago Canyon Colleges. It was used for nearly ten years and provided students with a bridge between arithmetic and beginning algebra. Student understanding of algebraic concepts was supported through the use of manipulative activities called Manipulative Mathematics. Students' literacy in written English and fluency with mathematical vocabulary were addressed through Links to Literacy activities. And every lesson in Foundations of Algebra included a pro-active study skills activity, which we called Strategies for Success.
Our Foundations of Algebra manuscript evolved into three textbooks that have been published and are available to teachers and students anywhere.
The mathematics content of Foundations of Algebra was expanded to become two textbooks, Prealgebra and Elementary Algebra, published by OpenStax. Both books maintain our philosophy of fostering conceptual understanding of mathematics through the use of manipulatives. Working on her own, Lynn went on to author Intermediate Algebra, which has also been published by OpenStax. The texts can be read online or downloaded at no cost from OpenStax.org.
Strategies for Success: Study Skills for the College Math Student, is available from Pearson Higher Education in workbook or electronic format.
The Manipulative Mathematics or Links to Literacy links on the menu bar above take you to pages with more information about, and samples of, these unique features we included in Foundations of Algebra. We invite you to use them with your students!
I would love the password, please! Georgia is teaching a Foundations of Algebra course that focuses on the use of manipulatives to develop a conceptual understanding of mathematics. Any additional help is much appreciated! | 677.169 | 1 |
Students are expected to be able to solve logarithmic and exponential equations using their exponent and log rules. Natural log and base e is included. Application questions on the 2nd page remind students of real world context questions. These questions involve carbon dating and continuous compound interest.
See other items on the Making Math Matter MMM site for other in-context assignments, activities and assessments. | 677.169 | 1 |
Mathematics Placement
John A. Logan College is committed to offering each of its students the opportunity for a successful and rewarding educational experience. The Board of Trustees, Administration, faculty, and professional and nonprofessional staff are dedicated to this endeavor.
The purpose of the Mathematics Placement Review is to provide you with assistance in preparing for the mathematics placement test at the College. In it, you will find information regarding our placement process, course descriptions, objectives, and practice problems for our developmental courses. You will also find resources that are available to help you prepare for the placement exam.
The Mathematics Department desires that every student be initially enrolled in the highest level mathematics course that he/she would have a reasonable chance for success. Assessment into mathematics courses is based on the placement score and a review of the student's high school background in mathematics. At a minimum, the student should have the high school courses Algebra I, Geometry and Algebra II with grades of at least "C," along with the appropriate placement test score in order to be ready to begin with a transfer level mathematics course. Also, a student's chance to begin at a higher level mathematics course is increased by taking a mathematics course during the senior year.
If you have questions about the booklet or desire some additional information or advice about what mathematics course you should begin with, feel free to contact the department chair, Kathirave Giritharan at kathiravegiritharan@jalc.edu or extension 8458. | 677.169 | 1 |
a comprehensive presentation of the mathematics required to tackle problems in economic analysis. To give a better understanding of the mathematical concepts, the text follows the logic of the development of mathematics rather than that of an economics course. After a review of the fundamentals of sets, numbers, and functions, the book covers limits and continuity, the calculus of functions of one variable, linear algebra, multivariate calculus, and dynamics. To develop the student's problem-solving skills, the book works through a large number of examples and economic applications. The second edition includes simple game theory, l'Hopital's rule, Leibniz's rule, and a more intuitive development of the Hamiltonian. An instructor's manual is available. | 677.169 | 1 |
The Physics Philes, lesson 2: Math for Miles!
In which math is relearned, more books are purchased, and heads are banged.
Last week was my first full week of lessons. It nearly killed me. OK. Not really. But I have been sleeping really well.
As I wrote last week, I'm really not a math person. Which makes my choice of physics as a topic a bit curious. But I was convinced that, provided with an explanation. I was humming along fine until I came across this phrase:
"…on a graph of position as a function of time…"
Whut?
I also ran across some problems that required me to know and execute derivative equation rules. I had no idea what to do. I thought I could make it work. But I can't. Not on my own. So I did what any self-respecting student would do. I bought a study book.
It turns out that the warnings issued by many a math teacher are true: Math is, indeed, cumulative knowledge. If you don't grasp a basic concept, it's hard to move on to more advanced topics. Since it's been about 10 years since I've had a math class (and I've never had calculus) I've forgotten most of what I used to know. So I have to start from the beginning. And by "the beginning," I mean algebra. This project has morphed into my own part-time class schedule: physics and calculus. So in addition to discussing a physics concept every week I'll discuss some math that I am either relearning or learning for the first time.
Let me apologize if this is incredibly boring for you. I didn't fully realize when I started how much algebra, geometry, and trigonometry I had forgotten. The examples I use are taken from my study book, because I want to make sure: a. I got the answer write and b. the problem is set up correctly. I'll get to the physics in a bit.
Graphs seem to be an important part of physics so far, likewise with calculus. So I needed to brush up on my linear equations.
Linear equations always have three components: two variable terms and a constant term. Standard form of a linear equation will look like this:
Ax + By = C
OK. That looks familiar. I seem to remember solving equations like that years and years ago. I wonder if I can still hack it. I'll put this equation in standard form:
3x – 4y – 1 = 9x + 5y – 12
I'll first need to subtract 9x from both sides:
(3x – 9x) – 4y – 1 = (9x – 9x) + 5y – 12
-6x – 4y – 1 = 5y – 12
Next, I'll need to subtract 5y from both sides:
-6x (-4y – 5y) – 1 = (5y – 5y) – 12
-6x – 9y – 1 = -12
-6x – 9y – 1 = -12
Hey Mr. 1! What are you doing on that side of the equation? Get over there with Ms. 12:
-6x – 9y – (1 + 1) = (-12 + 1)
-6x – 9y = -11
Whoops! See that -6x? That's supposed to be a positive number in standard form. I better multiply each side by -1:
-1 (-6x – 9y) = -1 ( -11)
6x + 9y = 11
Ah yes. It's coming back to me. Slowly, and I need a lot more practice, but it's there. A rich vein of algebra-ore waiting to be mined. But there is oh-so-much more. Next I need to relearn the two major ways to create the equation of a line.
The first is the slope-intercept form: y = mx + b, where m represents the slope of the line and b represents the y-intercept of the line. Solve for y if the slope of the line is -3 and the y-intercept is 5:
y = -3x + 5
See what I did there? I just plugged in the values. Easy peasy lemon squeezy. The second major way to create the equation of a line is point-slope form. You can use this form when you only have a point and a slope, but no y-intercept. If we have point (x1, y1) and slopem, the equation will be:
y – y1 = m(x – x1)
The x and y with subscripts represent the point coordinates. Let's try one. Find the equation of the line through point (0, -2) with slope 2/3. First, let's plug in the values we have into the above equation:
y – (-2) = 2/3(x – 0)
y + 2 = 2/3(x – 0)
Oh noes! Fractions. It's possible that I got the mechanics of this next part wrong. If I did, please tell me in the comments. Now, we multiply each side by 3 to get rid of the fraction:
3(y + 2 = 2/3(x – 0)
3y + 6 = 2(x – 0)
Let's get that right side simplified.
3xy + 6 = 2x
Subtract 3x from both sides:
3y + 6 -3y = 2x – 3y
6 = 2x – 3y
Ta-da! I did it! I think. Right? Someone please check my work.
That's not quite all the math I retaught myself this week, but you came here for some physics. So that's what you're going to get.
This week was dominated by velocity and acceleration. This is how we can talk about the motion of an object. Today I'll be discussing average velocity and average acceleration (as opposed to instantaneous velocity and instantaneous acceleration. Mostly because the instantaneous stuff completely confounds me.) First, velocity!
Average velocity is the change in position of an object divided by the change in time. To find the average velocity (represented by v), all we have to do is divide the change in position (represented by x) by the change in time (represented by t). Or, in math-speak:
v = Δx/Δt
Whoa there. What's with those shapes? That's the Greek letter delta, used in algebra to represent "change in." Let's say we have a starting place of x0 and an ending point of xf. To find Δx, we just subtract xf from x0. We do the same with Δt. We subtract the final time (tf) from the initial time (t0). Like this:
v = xf – x0 / tf – t0 = Δx/Δt
Hm. That might not be very clear. Let's do an example. Let's say you're on your bike. You start from rest and ride for 10 s. In that 10 s you go 50 m. What is your velocity? All we have to do is plug the values into equation:
v = 50 m/10 s = 5 m/s
Pretty easy, right? Once you get past all the Greek letters and subscripts, it's a pretty straight forward concept. We can also use this equation to solve for the position of the object by solving for x.
Now that we have average acceleration down, let's figure out average acceleration. Acceleration is the change in velocity divided by the time in which that change takes place. We come across our good friend Δ again:
a = vf – v0/tf – t0 = Δv/Δt
OK! Let's plug in some values! Let's say that you're at a stop light. When the light turns green you accelerate. After 10 s the car is moving at 50 m/s. What is the acceleration?
a = (50 m/s – 0 m/s) / (10 s – 0 s)
a = 50 m/s / 10 s = 5 m/s2
Notice the difference between the answers for velocity and acceleration? For velocity, the unit is m/s. For acceleration, the unit is m/s2. That's how you know when we're talking about velocity and when we're talking about acceleration.
I mentioned above that I was having a little trouble with instantaneous acceleration and instantaneous velocity. But my existing math experience does not encompass derivative equations, which is evidently necessary to calculate instantaneous acceleration and velocity. This is cause for great concern; I'm afraid I won't be able to catch up on the math in time to figure out the physics. I know we have a bunch of smart-pants readers. You will get 100 gold stars if you can explain the concept of limits and derivative equation rules to me in a useful way. The challenge is issued. Will you accept? (Please accept!)
This is, of course, only a fraction of what I actually covered this week. Did I make any mistakes? Should I have gone about something in a different manner? Let me know in the comments!
8 Comments
If you wanted to use the point-slope form to find the standard form of the equation then 2x-3y=6 is right, also you could have just multiplied 2/3 to x and 0, would have been shorter.
I never studied physics so I didn't look at that.
If you're looking for some educational resources to help you relearn some math here some AWESOME sites. Everything I list is absolutely FREEEEEEEE, and HIGH QUALITY education!
(Over 3200 FREE videos on Math, science, and even Art! Literally goes from 1+1=2 to differential calculus, and everything in between. He teaches the math very informally, focusing alot on intuitively understanding what you're doing. Very entertaining, Highly Recommend! Plus I suspect Salman Khan is one of us, i.e. an atheist/skeptic.)
(150 Free textbooks! Typically people want the most up to date textbooks which I don't think these are, but it should be okay for Math textbooks, because, you know, Math doesn't really change.)
There's also stuff like "MIT opencourseware" (I never bothered with this because i don't know how to use it, maybe you'll have better luck lol.) and which has a free beginners physics class beginning June 25th, not sure if you need that though.
Anyways hope you find something useful in all of that. (Alternatively you could just torrent a thousand textbooks on Math and Physics. Just kidding…. Or am I? Dun dun dun)
Challenge accepted!
So, limits.
Limits are a useful way of looking at equations when they get really big or really small. So generally, the way we use limits is by letting x get as close as possible to a value without actually letting it reach that value and seeing what happens. For instance, in the equation y=x, if we let x get really really big, then y gets really really big as well. So we can assume than when x goes to infinity, so does y. (Although assuming things in math does get a bit tricky sometimes.) What happens with the equation y=1/x? Well, in this example, as x gets big, y gets small. So if x goes to infinity, y goes to zero. But what happens if x gets really small? X can never equal zero. The fancy math Powers That Be decided that 1/0 is undefined. That is, it doesn't exist. But what if x were to simply *approach* zero? There's nothing in the rules about that. So we let x approach 0, and we notice that y gets really big. So we can conclude that in the *limit* as x approaches zero, y goes to infinity.
Limits are useful in other ways, too. Take for example the equation y=1/(x-2). X can never equal two, because then the equation would be y=1/0, and that's not allowed. But suppose we wanted to find out what happens around x=2? Well, we could calculate the value of y in the limit as x approaches 2. What we would find is that the value of y approaches infinity as x approaches 2. In this way, we can use the concept of limits to understand certain things about equations that we wouldn't be able to understand otherwise.
And now, on to derivatives. This explanation might seem a bit roundabout, but stick with me.
Let's say you have an equation and you need to find the slope. Easy, right? You pick two points (x1,y1) and (x1,y2) and then use the slope equation m=(y2-y1)/(x2-x1). Problem solved! Well, not quite. You see, that equation only works for lines, and I never said that this equation was a line. In fact, it's a very curvy curve. "But wait!" you say. "If it's a curve, then the slope changes with the value of x. How am I supposed to find the slope when it's not constant?" Very good question. The answer is you can, but it's a little complicated, so we'll get to it later. For now, let's consider the slope at a single point. Now you might think that you still can't find the slope because it's not a line, but you can at least pretend that it is one. I'm going to let that sink in for a moment. Pretend that a curve is a line. This probably violates a dozen math rules or something, but maybe if we pick two points that are fairly close together, nobody will notice. So we'll let (x1,y1) be the point that we're trying to find the slope at, and we'll let (x2,y2) be a point not too far away. We plug them into the slope equation, and voila! We have a slope that might not be too far away from the actual value!
But surely we could do better. What if we moved our second point a little closer to the first? That would certainly make our approximation a little more accurate. But why stop there? Why not move the second point as close as possible? Why not, indeed? So how close can we get? The answer is infinitely close. And now (hopefully) you see where I'm going with this. If we use the concept of limits, we can calculate the value of the slope as the limit as (x2,y2) approaches (x1,y1). And what we get is something called the derivative.
The derivative can be thought of as the slope of an equation at a particular point, but more generally, it's the rate of change of the y variable at a specific x value. Now granted, both of those definitions mean pretty much the same thing, but the second is more useful when you're talking about instantaneous velocities and accelerations. If velocity is the change in position over time, then instantaneous velocity is the derivative of the position equation at a specific point. Likewise, acceleration is the change in velocity over time so the instantaneous acceleration is the derivative of the velocity equation.
Now, it turns out that it's easier to apply derivatives to general equations than to specific numbers, so that's what we'll be dealing with here. First, this next section will involve some actual math but most of it shouldn't be very complicated. Second is a bit of terminology. We indicate the derivative of a function by putting an apostrophe after it. For example, the derivative of the function y is denoted y' (pronounced "y prime"). You may also see it written dy/dx. That means a derivative of y with respect to x. There's also one other way to denote derivatives that involves putting a dot over the y, but I'll only use the apostrophe method. Now that that's out of the way, time to actually do some derivatives.
There are four main rules that you need to learn in order to be able to do derivatives. They're not really all that hard, but they will require a bit of practice. The first is known as the "Power Rule." Let's say you're trying to take the derivative of the function: x^3+x^2+x. That may look complicated, but notice that each term is of the form x^n, where n is some number. This is where the Power Rule comes into play. The Power Rule says that if you encounter a term of the form x^n, you simply multiply the whole term by n and reduce the power by 1. The derivative of x^n is nx^(n-1). So in the example above, the derivative of x^3+x^2+x^1 is 3x^2+2x+1x^0. Anything raised to the 0th power is 1, so this simplifies to 3x^2+2x+1. Not bad, right?
This also works for negative powers. Say you have x^-2. Multiply the term by -2, then reduce the power by 1. -2-1 is -3, so the derivative of x^-2 is -2 x^-3. Similarly, this works for non-integer powers as well. What if we had x^(3/2)? Then the derivative would be (3/2)x^(1/2). Simple!
This next rule is probably the most difficult to understand. It's called the "Chain Rule." What if we had the equation (x^3)^2? Notice that it's a power raised to a power. We might be tempted to simply say that (x^3)^2 equals x^6 (algebra) and use the Power Rule to get 6x^5, but we're not going to do that. Instead, we're going to use the Chain Rule. The Chain Rule says that when you have an equation nested inside another equation like those weird Russian dolls, you evaluate them as separate equations and multiply them all together. So if we were to set x^3 equal to some other variable, let's say y, then the equation would look like y^2. Here you have two equations, the y^2 equation, and the equation the y represents. So first you take the derivative of each equation separately, and then you multiply them all together. The derivative of y^2 is 2y, that's just the Power Rule. The derivative of y is y', and that's equal to 3x^2. So the answer is (2y)(3x^2). But remember that y=x^3, so our final answer is (2x^3)(3x^2), which simplifies to 6x^5. This was the answer we got before, and it shows that math does, indeed, work.
Let's look at another example. Try finding the derivative of the equation (x^4+x^2+2)^3. Well, the first thing to do is substitute y for x^4+x^2+2. The derivative of y^3 is 3y^2, and the derivative of y is y' is 4x^3 +2x. (The derivative of a constant is always zero.) Multiplying those together, we get (3y^2)(4x^3 +2x). Plugging in x^4+x^2+2 for y gives us 3(x^4+x^2+2)^2(4x^3 +2x). It may take some practice, but soon you'll be doing these almost instinctively.
The next two rules are rather simple, and deal with what happens if you take the derivative of two functions that are being multiplied or divided together. The first is called the "Product Rule." Let's say we have two functions, y and z. Y is equal to 3x^2+2, and z is equal to x^3. What's the derivative of yz? Well, the Product Rule says that the derivative is equal to y'z+z'y. So, it's the derivative of the first times the second, plus the derivative of the second times the first. We can calculate y' as 6x, and z' as 3x^2. Therefore, the derivative of yz is equal to (6x)(x^3)+(3x^2)(3x^2+2). That's not too complicated.
What happens if we want to calculate the derivative of y/z? Then we use the "Quotient Rule." The Quotient Rule says that the derivative of y/z is equal to (y'z-z'y)/z^2. Because it involves subtraction, the order is really important on this one. Many a math test has been failed because of mixing up the order. Probably. Honestly, I rarely use this one, because it's often easier to rewrite y/z as y(z^-1) and use the Product Rule. Still, it's helpful to know. If we use the values from our last example, the derivative of y/z is [(6x)(x^3)-(3x^2)(3x^2+2)]/(x^3)^2. That's a lot of math. I'll give you a moment to read through it. Now perhaps you can see why nobody likes the Quotient Rule.
You'll notice that all of these equations are in general forms. That is, taking the derivative of a general equation produces another general equation. The equation y' is the value of the slope, not at one value of x, but for all values of x. Pretty cool, right? If you wanted to find the slope at a specific point, just plug in the value of that point for x.
But this raises another question. If the derivative of an equation is another equation, is it possible to take the derivative of a derivative? In fact it is. This is called the second derivative, and is denoted by two apostrophes: y". Or like this: (d^2)y/dx^2. Or with a y with two dots above it. Second derivatives have a physical meaning too, but it's much more complicated. Although, you may notice that acceleration is the second derivative of position. So there's that. You can also get third derivatives too, and fourth derivatives, and so on, sometimes to infinity.
And that's it! Well, almost. There are other functions besides powers of x. Functions like exponentials, and logarithms, and trig functions, and other wonky stuff, and all of them have their own derivative rules. If you want me to go through them, I can, but you could also just look them up in the back of your calculus textbook.
Derivatives are complicated little things. The amount of math in this comment will probably be a little overwhelming, but I'm sure you'll get used to it soon. In addition to limits and derivatives, I'd recommend learning more about higher order polynomials, especially quadratics (if you haven't already!) which take the form y=ax^2+bx+c. From there, you might try learning a bit about trig. You're going to need it in a few months.
BTW, all of your math was right, and your understanding of the concepts of velocity and acceleration was very good. You'll make a great physics student!
Lastly, if there's anything you'd like clarified, if you'd like something covered in more depth, or you want me to go over anything else, don't be afraid to ask!
"To find Δx, we just subtract xf from x0. We do the same with Δt. We subtract the final time (tf) from the initial time (t0). Like this:v = xf – x0 / tf – t0 = Δx/Δt"
In your equation you I believe you are subtracting initial from final | 677.169 | 1 |
I: RELATIONS AND FUNCTIONS
1. Relations
2. Functions
3. Binary Operations
4. Inverse Trigonometric Functions
UNIT — II: ALGEBRA
1. Determinants
2. Matrices
3. Multiplication of Matrices
4. Adjoint and Inverse of a Matrix
5. Solutions of Simultaneous Linear Equations
| UNIT — III: CALCULUS
1. Continuity
2. Differentiability
3. Differentiation
4. Second Order Derivatives
5. Rolle's Theorem and Mean Value Theorem
6. Rate of Change of Quantitives
7. Tangents and Normals
8. Differentials, Errors and Approximations
9. Increasing And Decreasing Functions
10. Maxima and Minima | 677.169 | 1 |
Contents
Overview
"this document does not treat in detail all of the material studied" - At this point I'm not trying very hard to understand, since I'm sure they're about to get into territory I barely remember and they're not going into much detail. I'll understand it when I get there in the curriculum, and the faster I get through the progressions, the sooner I can start.
"reasonable in the context" - It's kind of annoying that you can't figure everything out just by looking at the calculations, but it could be interesting to think about the relationship between the math and its context. And are mathematicians missing something by divorcing math from any context and thinking of it as a symbol manipulation game?
"algebraic expressions may not be suitable" - I'm looking forward to learning the other options.
Grade 8
"a linear function does not have a slope" - Why not? Just because it's not visual in itself? Is the function different from whatever you use to calculate the slope of its graph?
"describe the relationships qualitatively" - Yes, it's good to remember that people are still human when they think about math.
High School
Interpreting Functions
"Although it is common to say" - These kinds of language distinctions are important to me, so I want to come back to this when I get here in the curriculum. Actually I'm thinking of revisiting all the progressions as I go through the curriculum.
"the vertical line test is problematic" - I don't know what this is about, but it sounds like the kind of thing I want to know. The discussion distinguishes between a flawed method and a better one, and it tries to get down to the real issue in the mathematical task it's addressing.
"The square root function" - I've read that +/- 3 isn't the right solution but not why, so I'm glad they cover this. There's so much useful, in-depth information in the Common Core that I think people who dismiss it are cheating themselves. Unless they don't care to know math, in which case they may still be cheating themselves.
"all students are expected to develop fluency" - At this point I do wonder why we make everyone learn so much math. Most people don't ever need these functions after school. How does it benefit them? If they're relevant to the kinds of statistics that inform public policy, that would be a good reason, but otherwise the only reasons I can come up with sound like rationalizations.
"looking for and making use of structure" - This seems to be what some people mean when they talk about patterns, rather than simply noting and interpreting ambiguous, surface patterns like sequences.
"To avoid this problem" - I'm glad these exercises are on Illustrative Mathematics. It might make me more likely to remember to come back to them when I'm in the curriculum.
Building Functions
"subtleties and pitfalls" - Sounds like fun. :) Until I get into it and feel the strain.
"from scratch ... special recipes" - Yes, that's what I'm hoping this time, to learn the principles behind the recipes, so I can make my own math.
"Some students might" - I'm glad I figured out early that it's a basic feature of math that operations can be converted into each other. It helped me understand some other reading I was doing today. It's important to look at math from different angles like that one--what math means, how it works, etc. I'm sure there are other angles I haven't learned about yet.
My notes are starting to get repetitive, so I'm going to speed through the rest of the progressions.
Linear and Exponential Models
Trigonometric Functions
I like that mathematicians have so fully studied circles. It's nice to completely understand something, at least the things you consider important about it. I wonder if it's possible to design a good, single diagram that displays all of a circle's important mathematical features.
"Prove and apply trigonometric identities" - I'd like to try translating mathematical proofs into some other logical notation, just to clarify how the grammar of math relates to the grammar of logic. They're not the same thing. | 677.169 | 1 |
CALCULUS III Advice
Showing 1 to 2 of 2
I would recommend this course because it is important for all STEM majors, and is taught very well by Dr. Allen. She is one of the top professors in the Mathematics & Statistics department at Texas Tech, and I learned exactly what I needed to learn in the course.
Course highlights:
I gained further skills in mathematical analysis and derivation of very useful formulas used within all STEM majors. Most of this course is easy if you did well in Calculus I, and the very last part is the most important part.
Hours per week:
6-8 hours
Advice for students:
Be sure to review the material you learned in Calculus I, as this course mirrors that almost completely for a few weeks. The difference is, Calculus I was one and two dimensional, whereas Calculus III is three-dimensional and extensions of Calculus to higher dimensions.
Course Term:Spring 2016
Professor:LindaAllen
Course Required?Yes
Course Tags:Math-heavyBackground Knowledge ExpectedGo to Office Hours
Feb 17, 2017
| Would recommend.
This class was tough.
Course Overview:
Prof. Allen loves teaching and is good at it.
Course highlights:
It solidified the knowledge I gained in Calc 2 and built upon it.
Hours per week:
6-8 hours
Advice for students:
make sure to read the book and do the homework. Prof will help you a lot if she knows you are dedicated. | 677.169 | 1 |
Question: The use of
"x" and other letters near the end of the alphabet to represent an
"indeterminate" is
due to__________. He is also responsible for the first publication of the
Factor Theorem in his work The Geometry, which appeared as an appendix to his Discourse on Method.
Read Section 35.
Thursday February 22
Study Section
35. The division algorithm
Do problems # 1-10, 12, 13,
17.
What was the contribution of
Girolamo Cardano to the solution of polynomial equations?
Study Section 24
Introduction to rings-understand all definitions and examples.
Read Handout-Amazing secrets
to be successful in any mathematics advanced class! Write a summary!
Do problems # 1-6, 12, 15-18
pages 124-125.
For extra challenge do
Problems # 19, 22.
Question: Who was the first
mathematician to use the term "ring"?
Read Section 25.
Thursday January 18
Study Section
25-Integral domains. Subrings.
Do problems # 1-5, 9, 10, 12,
14, 17, 22 pages 127-128.
Give an example of a ring R
with unity e that has a subring S with unity e' not equal to e.
For extra challenge do
Problems # 16, 19, 23, 24 pages 127-128.
Question: Which mathematician
is responsible for the development of axiomatic ring theory?
Read Section 26.
Tuesday January 23
Study Section
26-Fields
Do problems # 1-10, 14, 18,
19, 20, 23, 24 pages 130-131.
Finish the problems in the
handout given in class.
For extra challenge do
Problems # 12, 13, 16, 17 pages 130-131.
Question: Is it possible for
the unity element in a subfield of a field to be different than the unity of
the whole field?
Provide a proof or give a counterexample. Compare to Exercise 3 (Jan 18)
above.
Read Section 27. Study the
definitions of ring isomorphism and characteristic of a ring. | 677.169 | 1 |
Students may manipulate vectors but do they understand them? Teaching using the approach described here should
ensure that they do.
During many years of teaching vectors to sixth-formers
and undergraduates I felt, perhaps more than in any other topic of physics, that I was teaching manipulation rather
than understanding. The better students grasp vectors intuitively but the weaker students - or even those bright
students who demand to understand physics - are often mystified. Vector graphics is easier to teach than vector
algebra, but there are difficulties with both.
Early in my career I had the good fortune to encounter
a student who told me that he had understood Ohm's law until I explained it. I learned then that subtle explanations
must be avoided. With difficult subjects we also need to phrase our explanations very carefully, and pace them
well - otherwise they may crash in the minds of our students! Clear explanation is, of course, insufficient. Different
teachers use different classroom aids and approaches. My own favoured way of promoting active learning is to develop
as much of the subject as possible through challenging questions addressed to the class.
The teaching of vectors provides the opportunity to illustrate
several important strategies used by the physicist such as simplification, the role of fictional entities, the
fertility of the geometrical representation of non-geometrical quantities and the usefulness of geometrical algorithms.
About 15 years ago I found a way of teaching vector graphics which my students seemed to understand. I tried approach
after approach with vector algebra but without success. Then, quite suddenly, about six years ago, I began to feel
I was making some progress. Here, I attempt to prepare the ground which leads on to a mathematical treatment of
vectors.
Vector graphics is more than 2000 years old. It was originally
used to compound velocities. Vector algebra is a little more than a century old: it grew out of fragments of William
Rowan Hamilton's quaternion notation, which had a strong geometrical interpretation. Many physicists late last
century searched through various branches of physics looking for physical applications of the existing formalism,
but geometrical applications remained paramount. I believe that physics is still too heavily reliant on the geometrical
model of the vector - the displacement vector. Indeed, I have not found that the distinctive vector properties
of important physical quantities such as force or angular momentum have been carefully worked out anywhere in the
literature of physics.
Vector graphics
I find it helpful to begin vector graphics in terms of
a series of concrete examples: abstraction and generalization can follow later. The displacement vector is easiest
to explain, perhaps too easy. It is very different from other vectors such as force. Displacements are applied
successively, they occupy different intervals of time, they involve a displacement through space and they are directly
descriptive. Forces, on the other hand, can be applied at the same time and at the same point in space. Furthermore,
the line representing a force is a geometrical analogue of the actual force and is not directly descriptive. These
are important differences and there are many others. I have chosen to concentrate on the force vector here because
of its importance. I choose tension in a wire or cord as the standard example of a force because it can be regarded
as acting at a point. Other forces, such as surface friction and weight, are forces distributed over a surface
or over a body, respectively.
Suppose the force to be represented is a tension of 20
N applied by a marquee cord to a peg. The force can be represented geometrically by an arrow drawn to a suitable
scale, pointing in the correct direction and applied to a definite point. I believe it is important to emphasize
that the point of application of the force vector on your diagram represents a physical body.
I also introduce the concept of the resultant, and the
parallelogram of forces, in terms of an example such as tractors on opposite banks of a canal towing a barge. Our
students will know intuitively that the combined effect of the tensions in the hauling ropes will be directed along
the line of motion of the barge. It is also helpful to emphasize that both vectors represented in the graphical
parallelogram are applied to the same body and at the same time.
I have found that the most important point to make is that
the resultant of two given forces is an imaginary force which is equivalent to the actual forces in the sense that
acting alone it would have the same effect as the real forces combined. If we do not say this some may think that
there are three real forces acting. We need to tell them that they must make a choice: either they deal with the
original pair and ignore the resultant or they deal with the resultant and ignore the others. It is illegitimate
to do both. This point can be emphasized by representing the resultant by a light line and the original forces
by heavy lines (figure 1).
The resultant of two forces. T 1 and T
2 are represented in scale
and direction by the lines shown. The resultant of the two tensions is represented by R and is obtained by completing
the parallelogram. R is equivalent to T 1 and
T 2 , but it does not have an independent existence.
It is challenging to ask the class why the parallelogram
algorithm works for forces. The line of action of a force can be introduced as an imaginary line of indefinite
length coinciding with the force. On a rigid body a force can be applied with equal effect at any point along its
line of action. The concept of the line of action is useful in simplifying representations (figure 2) and it is particularly helpful when calculating moments.
The line of action of a force. Although
the ropes are attached at A and B the forces may be represented as acting at O. This is because a force acts equally
at every point along its line of action.
A common complaint about physics is that we do not explain
why we are introducing certain concepts: we simply introduce them mathematically, without any justification. We
may introduce the resultant as a tool of graphical calculation in an engineering drawing to find the combined effect
of a set of forces, for example, or to simplify a problem. The most important example of the latter is weight thought
of as a single force acting through the centre of gravity. The resultant weight is more convenient, both in explanation
and in calculation, than the real weight, which is a force distributed over the whole body. This difference is
brought out dramatically, for example, by the weight of a horizontal ring (figure 3). Another
example is the reaction of a plane on the body it supports: in reality this is a force distributed over the undersurface
of that body. We commonly replace it in thought, however, and on our diagrams, by its resultant represented as
a single force acting on a point of the surface (figure 4).
Distributed weight and resultant weight.
Weight is a distributed force, but it may be replaced by its resultant for explanatory and mathematical convenience.
Notice here, however, that the resultant pull of gravity on the ring 'acts' on empty space. C is the centre of
gravity.
Surface reaction and resultant surface
reaction. Surface reaction is a distributed force but it may be replaced for convenience by its resultant Rn.
The components of a single force are quite a different
matter. A force such as weight, although it has a definite direction, is able to act in every other direction except
a perpendicular direction: weight is able to roll a sphere along a plane with any inclination except the horizontal.
The effective force in a given direction is called its component in that direction. It is calculated by multiplying
the full force by the cosine of the angle made with the relevant direction. To avoid confusing our weaker students
I believe it is important to introduce only one mathematical rule here and to calculate all components using the cosine rule.
Any pair of perpendicular components is equivalent to the
full force since, in combination, they have the same effect as the latter. In some cases a given force seems to
divide naturally into such a pair. For example, the perpendicular (or normal) reaction and friction (the shear
reaction) seem to be a more natural description of the action of an inclined plane on the undersurface of a sliding
body than the total reaction. Components are sometimes introduced, therefore, to explain a physical process and
not simply for mathematical convenience.
Artificial vectors
The position of the Moon with respect to the Earth may
be specified by giving distance and orientation only. However, the position vector of the Moon provides distance,
orientation and sense. The sense is chosen by convention to point from the Earth to the Moon. This artifice allows
the position of a body to be treated as a vector and incorporated into the vector calculus. Area treated as a vector
also has a directional sense by convention, only. The most subtle of all the artificial vectors are the axial vectors.
Take the vector representing torque. This vector points along the axis of the torque. Our students know perfectly
well intuitively that the action of a torque is in the plane of the torque and that, although it takes place around
the axis, nothing actually points along the axis itself. If we tell them without qualification that the torque
points along the axis, as is often done, we perplex them.
I believe we should emphasize that axial vectors, such
as angular velocity, torque and angular momentum, represent something going on around an axis and not along the
axis. Processes of this sort, strictly speaking, cannot be represented by vectors since they do not point in just
one direction. However, provided the students are willing to agree on a pure convention, they can be represented
by a made-up vector. Notice that there are only two senses of rotation around a given axis, and only two arrow
directions along that axis. This allows us to set up a one-to-one correspondence: we can agree by convention that
an up arrow means a counterclockwise rotation - that of a right-handed screw - and a down arrow means the opposite.
To represent a torque fully by graphics we then construct an arrow whose length represents the magnitude of the
torque and whose arrow direction is linked by our convention to the sense of rotation of the torque.
To verbalize this has taken me more than 30 years of frustrated
groping. I am confident it can be further simplified. It is interesting to note that both the axial vector and
the area vector are perpendicular to a plane and both have direction by convention only.
Vector algebra
It was common before the late nineteenth century to treat directed quantities using scalar algebra. Cartesian components
were used and also the polar representation employing magnitude and angle.
Scalar algebra is still widely used, of course. I advise
students to use scalar algebra to represent vectors where this is most convenient: the mathematics of physics is
highly flexible and they should learn to use that form of mathematics which best suits the problem in hand. I emphasize,
however, that vector algebra is often more economical and intuitive and is more convenient where the geometrical
imagination is in trouble: in three-dimensional physics such as occurs in wave propagation, rotational dynamics
and electromagnetism..
Vector algebra is often built upon sketch graphics but
it is more abstract since graphics provide an analogue representation while algebra only provides a symbolic representation.
If we return to the example of the barge, we can invite the class to symbolize the relationship between the two
real forces and the resultant. Suppose they call the tensions T1 and T2 and the resultant R. Bold is used in print to
emphasize that symbols represent directed quantities. We can verbalize the relationship as 'T1 combined with T2 is equivalent to R'. I then tell them that 'combined with'
is commonly abbreviated by + in vector algebra ('bold plus' in printed
notation) and 'equivalent
to' is abbreviated to = ('bold equals'). So, the relationship becomes T1+
T2= R.
Do bold plus and bold equals
here have the same meaning as the plain plus and plain equals of ordinary algebra? No. Meaning, as ever, is determined
by context. I encourage the students to read + and = in their minds here as 'combined with' and 'equivalent
to', respectively, until the idea sinks
in. This helps them to avoid the usual ambiguities.
I find with other teachers that the subtraction of vectors
is best explained as the combination of a vector with a reversed vector. For example, v1-v2=vr can be read as 'the actual velocity of the ferry ahead, when combined with the reversed velocity of our
hovercraft, gives the relative velocity of the ferry'.
We can interpret T1+
T2= R, or any similar expression, as a statement of the quantity calculus symbolizing the relationship between
the physical quantities themselves, and then it represents something concrete. We may equally well interpret it
as an abstract statement relating graphical analogues or directed numerical values. All three interpretations are
valid and we commonly shift from one to another.
I find it difficult to introduce the scalar product and
the vector product clearly. There is a tendency in some texts, which I can sympathize with, to introduce them abstractly
as algorithms and then, much later, perhaps, to find physical examples to which they apply. In class this is unnecessary
and I find it unhelpful.
I may use the example of a body sliding a certain distance
down an inclined plane. The work done by gravity is measured by multiplying the displacement d by the component of the weight along the plane, W cos(d^W), where d^W
represents the angle between d and the weight W. Notice that the measure
of the work, dW cos d^W, is the product of the magnitudes of two vector quantities and the cosine of the angle between them.
This kind product turns up so frequently in physics that
it is given a special name and a special notation. It is defined as the scalar product of two vectors. It is scalar
because no direction is assigned or required in the outcome. It is symbolized by dW, where dW = dWcos d^W. Whenever we see a bold dot between
two vectors we mean this product. This needs to be generalized, of course.
Torque and the vector product. yoz is the plane of action of the torque and ox
its axis.The
magnitude of the torque is given by the moment of the force F,
which is equal to pF = rF sin rˆF.
The conventional direction of the torque T is
ox. This is symbolized by r
XF = T in vector notation.
The vector product is more difficult to introduce. When a wrench exerts a torque on a wheel nut the moment of the
force is measured by the product of the effective lever length and the magnitude of the force F. Now the effective lengthpof the lever is the perpendicular distance
from the nut to the line of action of the force. But p = r sin r^F where r
is the length of the wrench and r^F is the angle between r and F (figure 5). The magnitude of the torque T is, therefore, T =rF sin r^F. Now, r
can be made into a vector r - the position vector - by giving it a direction from the origin
to the point of application of F. We have again multiplied the magnitudes of two vectors together
but this time we have multiplied them by the sine of the angle between them. It has already been explained to the
class that torque itself is a vector T, directed by convention along its axis. In this case, therefore,
the product of two vectors can be given both a magnitude and a direction. The usefulness of this algorithm recommends
that we devise a special notation for it.
The vector product r X F (= T) is then defined as a vector of magnitude rF sin r^F and direction given by the following rule: curl the fingers of the right hand in the direction of rotation
of the torque. The thumb will point in the direction of the vector product. This needs to be tightened up, of course,
and generalized.
None of this is easy for students to grasp. At this stage many will have that characteristic dazed look which I
know only too well. I am sure a much better explanation can be found, but I have yet to find it. (I might add that
there is, of course, a third kind of product of vectors, the tensor product, but this is introduced at a more advanced
level and I have rarely tried to make physical sense of it before a class.)
Unit vectors
In practice much of the advanced use we make of vectors involves unit vectors. The use of such vectors gives us
a symbolism which is less abstract than general vector algebra. I may choose a two or three-dimensional framework
of reference based on a nearby corner of the classroom. Suppose a cord with a tension of 10 newton is pulling along
the positive x-axis on a nail hammered into the corner. To show its direction explicitly in our notation I attach
the directional flag i and write T = 10i newton.
For mathematical purposes, I then argue, i cannot simply be placed beside 10: it must be interpreted as multiplied by 10 and so we have to turn
it into a mathematical object and give it a magnitude. The most convenient magnitude is the abstract and dimensionless
number 1, since multiplying 10 newton by such a number changes neither the magnitude nor the dimensions of the
force. Because i has this magnitude it is called a unit vector. The unit vector,
therefore, is an artificial vector which provides directional information only. We can put it anywhere in the diagram
- it is a free vector - and attach it to any physical quantity: it always gives the same information. j represents the unit vector in the y-direction and k that in the z-direction.
The weight of the teacher standing anywhere on the floor can be represented, for example, by W = -850k newton and the reaction of the floor on the teacher by R = 850k newton. From these beginnings we can build up the addition and
subtraction of vectors represented by their components. We can also use our definition of the general scalar product
and the vector product to prove that i.i = 1, i.j = 0, etc, and also that i Xj = k etc. This allows us to introduce scalar and vector products in
terms of unit vectors - and much else.
We can, of course, introduce unit vectors for intermediate directions: for example, r/r is a unit vector in the direction
of the position vector r, and F/F a unit vector in the direction
of the force F. We can even introduce unit vectors which rotate, but this involves
a more advanced treatment. Interestingly, some texts still state incorrectly that the unit vector has unit length,
revealing the origins of the concept in the unit displacement vector.
Conclusions
When we first introduce vectors I believe we should try
to spell out what they mean. With this background our students can go on to manipulate them more confidently. I
have attempted to give verbal expression here to what every experienced teacher already knows intuitively, if not
consciously, about vectors. Although educated intuition in physics is not infallible or easily accessible it is
immensely rich and should be treated with the greatest respect: it has, after all, built up over several millennia.
Indeed, it is very optimistic to claim that one has faithfully articulated some elements of it. If one's efforts
provoke controversy then one should think again. I would even go so far as to say, and I intend all of this to
be self-referring, that the usual mathematical way of presenting vectors - which relies on a subsequent build-up
of intuitive understanding - may be safer than an introduction using poorly thought-out explanations. Nevertheless,
the attempt to articulate the tacit meaning of the concepts of physics is highly important, and not only for teaching.
If parts of the meaning of many of the concepts of physics are grasped intuitively, only, can physics be said to
have full rational control over its concepts?
Received 4 February 1997, in final form 7 March 1997 NEW APPROACHES PII: S0031-9120(97) 81541-9 | 677.169 | 1 |
Whether you are looking to learn a new technology this summer or want to refresh your math skills, we have the course for you with Summer Learning @ edX! From planes, to the cloud, smartphones, sports, and more, here are some interactive math and technology courses to keep your mind curious all summer long! Enroll today!
Data is rapidly changing businesses, how we interact with each other, and the future. In this course, you will apply analytics to real world situations to better work with data and improve your problem solving skills. Engage with examples of analytics making a difference in a course that will give you an edge in your career!
Uncover new ways to use cloud computing to solve all types of problems. Get acquainted with different cloud computing service and deployment models, cloud infrastructure, cloud scenarios, and more. Gain a solid understanding of the issues surrounding cloud computing, as you learn to make smart decisions while using this popular technology.
Linear algebra is at the root of scientific computing and many of today's real world problems. Dive into the development of mathematical theory in a visual course designed to connect mathematics with programming. Gain a deeper understanding of this essential discipline in this interactive course applicable all over the world!
Chance, strategy, and physics all have an impact on the athlete's performance. Learn how the application of math can give us better insight into the sports we love. Use probability, model physical systems, game theory, and more to analyze the game. You will become an expert in the technicalities of sport, so get off the bench and take this course!
In today's world, multisystem issues, like global energy reform, are not easily defined, and do not have simple answers. Using analytics, learn to weigh the costs and benefits for creating and implementing potential solutions for a wide range of challenges. Improve your decision-making with a course that will make you more confident, as you think rationally about complex problems!
Imagine the astonishment of the Wright brothers if they could see what planes can do today! Delve into the history of and map the advances made in aeronautic engineering from the beginning of aviation to the development of the current Airbus A380. Come aboard this engaging course and your knowledge of aeronautics will be sure to take off!
Statistics is all about turning data into group decisions. This course teaches you the basics of R, a powerful open source programming language used for data inquiry around the world. Conduct your own experiments to solve real life problems by exploring specific examples in a course that will ready you for the world of statistical analysis.
From robots to smartphones, we interact with the digital world everyday. Learn the basics of engineering electronic devices and use these skills to build a robot from scratch. You will explore the building blocks of electronics as you learn by doing in this interactive course!
* Advanced Placement and AP are registered trademarks of the College Board, which was not involved in the production of, and does not endorse, these offerings. | 677.169 | 1 |
Enter your keyword
Mathematics is tough in itself, but it gets even more difficult when students have to deal with arithmetic, geometry and algebra. Suddenly, it seems ten times more complicated and awhole lot more confusing. This is where many students tend to lose their balance. It is very rare that every child is able to master all these subdivisions.
Some are good in arithmetic and algebra but not in geometry. This is understandable, but there are many more confusions with algebra simply because of the variables that are introduced. Given enough time and practice, anyone can master mathematics and algebra to attain the best grades in the class. The tricks mentioned below may not be new, but going through them will allow you to find the motivation to do better.
Algebra Hacks
Practice Regularly
The trick to studying any mathematics based subject is to practice regularly.
As they say, practice makes permanent.
Habits are simply a form of repetition and by repeating certain formulas and systems of working, solving a problem will not be very difficult.
An equation or how to solve it cannot be learned simply by going through the steps the night before an exam and trying to cram in the information. Nothing can be remembered this way.
A good way to practice algebra regularly and stay in touch with old and new information is to do your homework.
Homework is assigned specifically for this purpose.
If you need to include more practice into your daily routine, buy a separate notebook that has nothing to do with class or home assignments.
Write down difficult sums that you come across and save it to be practice during free time or on a day when there are no assignments.
This will ensure that there is at least an hour of time spent solving algebraic problems daily.
While practicing, you will have the opportunity to identify problem areas. There may be some theories or solving methods that are difficult to grasp.
Being able to note these issues you're facing will make it simpler to find a resolution since you know where you're failing.
This will also ensure that you're well prepared for tests of any sort.
To make sure that you stay on track with regular practice, you must develop an efficient routine.
This routine should be a daily one, but for a more general overview, a weekly or monthly routine also works.
By planning out the day, you will be able to have a fair idea of the amount of hours at your disposal and how to utilize them efficiently.
This way by battling the clock, you'll be less likely to waste time procrastinating or focusing on elements of the day that are not a priority.
Pay Attention in Class
The easiest and most overlooked piece of advice given to most students is to pay attention in class.
It is crucial to understand the building blocks of every subject; this is what is taught in class.
Teachers help you comprehend the basics, and you are then required to apply these to the problems that are assigned.
If you're not familiar with the basics then going further will obviously become an issue.
This will also increase your reputation with the teacher as someone who is interested in learning and puts in thenecessary
It is in class that one is able to ask any and all questions that may crop up. Teachers are more than willing to answer any questions without discriminating against the students of their proposed doubts.
It's an amazing opportunity to clarify any issues. This, in turn, makes homework simple, since you already know what needs to be done and how to go about doing so.
Listening to the professor is not nearly enough; take down detailed notes as well.
This is a form of practice in itself. Once you've gone home and sit down to study, going over these notes will help keep the information fresh.
Teachers also help in identifying easier ways to solve equations, to remember all the tips and tricks, it is wise to have it noted down somewhere for easy reference.
Get a Tutor
If all else fails and it becomes difficult to stick to a schedule or routine and finish assignments on your own, hire a reliable tutor.
Hire someone who has the time to teach you in depth and help you build a better foundation of knowledge.
Tutors are exceptionally useful, especially when preparing for tests and examinations.
They offer that extra boost of motivation to work hard.
If tutors in the area are unavailable, study with a willing friend who could also use the company and motivation.
This also creates a comfortable environment to ask questions and solve them together.
However, the best person to go to is a teacher. They can explain anything you're having issues with and you'll also get a better idea of what they expect from a certain assignment or tests.
Do not hesitate to ask them for some extra coaching in their free time after class or school.
This will show your interest in their class, and they will be more understanding of any predicament you may be facing.
Prepare for Tests
It is not sufficient to study for an examination the night before.
You need to be sleeping and reserving energy for the exam or at most, revising.
Start preparation well in advance.
Doing so will help in identifying and solving problems well in advance and will also help in correcting any unforeseen mistakes.
There should be more than enough time for revision, and each topic should be covered
Just skimming through information will not be enough for a test.
No matter how much you dislike a subject, it isn't necessary to fail at any cost. While a liking is required to gain a better understanding of thematerial, passion is also developed from hard work and enthusiasm to do better. Employ these methods and make it a part of your routine for the best results! | 677.169 | 1 |
Concrete treatment of fundamental concepts and operations, equivalence, determinants, matrices with polynomial elements, and similarity and congruence. Each chapter has many excellent problems and optional related information. No previous course in abstract algebra required.
Basic textbook covers theory of matrices and its applications to systems of linear equations and related topics such as determinants, eigenvalues, and differential equations. Includes numerous exercises.
This unique text provides students with a basic course in both calculus and analytic geometry — no competitive editions cover both topics in a single volume. Its prerequisites are minimal, and the order of its presentation promotes an intuitive approach to calculus. Algebraic concepts receive an unusually strong emphasis. Numerous exercises appear throughout the text. 1951 edition.
The Handbook of Mathematics for Engineers and Scientists covers the main fields of mathematics and focuses on the methods used for obtaining solutions of various classes of mathematical equations that underlie the mathematical modeling of numerous phenomena and processes in science and technology. To accommodate different mathematical backgrounds, the preeminent authors outline the material in a simplified, schematic manner, avoiding special terminology wherever possible. Organized in ascending order of complexity, the material is divided into two parts. The first part is a coherent survey of the most important definitions, formulas, equations, methods, and theorems. It covers arithmetic, elementary and analytic geometry, algebra, differential and integral calculus, special functions, calculus of variations, and probability theory. Numerous specific examples clarify the methods for solving problems and equations. The second part provides many in-depth mathematical tables, including those of exact solutions of various types of equations. This concise, comprehensive compendium of mathematical definitions, formulas, and theorems provides the foundation for exploring scientific and technological phenomena.
This outstanding text offers undergraduate students of physics, chemistry, and engineering a concise, readable introduction to matrices, sets, and groups. Concentrating mainly on matrix theory, the book is virtually self-contained, requiring a minimum of mathematical knowledge and providing all the background necessary to develop a thorough comprehension of the subject. Beginning with a chapter on sets, mappings, and transformations, the treatment advances to considerations of matrix algebra, inverse and related matrices, and systems of linear algebraic equations. Additional topics include eigenvalues and eigenvectors, diagonalisation and functions of matrices, and group theory. Each chapter contains a selection of worked examples and many problems with answers, enabling readers to test their understanding and ability to apply concepts. | 677.169 | 1 |
Akira Suzuki sakira@kobe-u.ac.jp
Graduate School of Science and Technology Kobe University Japan
Abstract
We have been using computer algebra systems such as Mathematica,
Maple and Risa/Asir in a class of mathematics major students
to teach a course of elementary computer algebra.
In this course, we teach basic algorithms of computer algebra
such as polynomial GCD calculation, polynomial factorization or
Groebner bases of polynomial ideals as standard topics of
computer algebra. Besides them, we also teach some classical
topics such as elementary Galois theory or the fundamental
theorem of algebra in order to give students minimum
mathematical background of the course. We have been trying to
use computer algebra systems even for teaching those topics.
Through our experience, we found that computer algebra systems
are extremely useful to make students understand essential
ideals of even non-algorithmic advanced mathematics.
The most succesful one is the fundamental theorem of algebra.
Using graphics tools and programing tools of computer algebra
systems, we sufficiently succeeded to make almost all students
understand why the fundamental theorem of algebra holds.
The essence is understanding homotopy intuitively.
In this paper, we will report on our attempt. | 677.169 | 1 |
Math 109 Summer Academy 2016
Clicker Questions
Updated 9/1/16
The clicker questions presented in class are intended to exercise your conceptual understanding of the underlying ideas in this course. Below is a table of the clicker questions organized by the date they were presented in class. The correct response is indicated by * (if you disagree or don't follow why it's correct, please post on our Math 109 Piazza page).
"The dictionary is the only place where success comes before work."
(Vince Lombardi)
Clicker Questions
Most clicker questions will be posted here after light editing to correct typographical errors. | 677.169 | 1 |
Explaining Logarithms by Dan Umbarger
Description: These materials show the evolution of logarithmic ideas over 350 years. I do believe that a quick review of mathematics as it was practiced for hundreds of years would be helpful for many students in understanding logarithms as they are still used today. I see three potential audiences for this material: 1.) students who have never studied logarithms, 2.) students who have studied logarithms but who did not master the concepts or have forgotten key ideas, or 3.) summer school reading for students taking calculus in the fall.
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The New A Level Further Maths
Further Maths is a difficult subject. It follows on from the A Level Maths course, but usually is taken at the same time as A Level Maths. The Further Maths course assumes that the student already knows much of the A Level Maths Syllabus. This is where we can help. Instead of teaching the Futher Maths straight away we teach some of the essential Maths skills needed from A level Maths to ensure a thorough understanding of the Further Maths concepts. The New A level Further Maths is a difficult undertaking and we can provide full access to all the Integral Maths Suite to help Maths students achieve the very best possible grades using the latest technolgy and suppport.
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Author: Sharon Weiner Greene, M.A. Publisher: Barron's Educational Series ISBN: 1438065078 Size: 16.51 MB Format: PDF, ePub, Docs View: 2208 DownloadRead Online
For additional practice on SAT-type questions, see Barron's Math Workbook for
the SAT. ARITHMETIC To do well on the SAT, you need to be comfortable with
most topics of basic arithmetic. The first five sections of this chapter provide you
with a review of the arithmetic concepts you need to know. Note that you should
do almost no arithmetic by hand. In particular, you should never do long division,
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noncalculator ... | 677.169 | 1 |
J.V., Maryland Gus Taylor, AZ
No Problems, this new program is very easy to use and to understand. It is a good program, I wish you all the best. Thanks!11:
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College
This course for junior and senior math majors uses mathematics from ordinary differential equations, to analyze and understand a variety of real-world problems. Among the civic problems explored are specific instances of population growth and over-population, over-use of natural resources leading to extinction of animal populations and the depletion of natural resources, genocide, and the spread of diseases, all taken from current events. While mathematical models are not perfect predictors of what will happen in the real world, they can offer important insights and information about the nature and scope of a problem, and can inform solutions.
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This course will establish the relevance of calculus to medicine and
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EOC Griddable Practice Algebra I, Algebra II, Biology
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This product includes two pages of notes (one with examples) and two worksheets that may serve as a "crash course" on filling in griddable answers on the Algebra I, Algebra II, and Biology End of Course tests. A key is included to help fill out the blank spots found in the notes.
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Algebra: Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological algebra: basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-references.
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Meet the Authors
Richard N. Aufmann,
Palomar College
Richard Aufmann is the lead author of two bestselling developmental math series and a bestselling college algebra and trigonometry series, as well as several derivative math texts. He received a BA in mathematics from the University of California, Irvine, and an MA in mathematics from California State University, Long Beach. Mr. Aufmann taught math, computer science, and physics at Palomar College in California, where he was on the faculty for 28 years. His textbooks are highly recognized and respected among college mathematics professors. Today, Mr. Aufmann's professional interests include quantitative literacy, the developmental math curriculum, and the impact of technology on curriculum development.
Joanne S. Lockwood,
Nashua Community College
Joanne Lockwood received a BA in English Literature from St. Lawrence University and both an MBA and a BA in mathematics from Plymouth State University. Ms. Lockwood taught at Plymouth State University and Nashua Community College in New Hampshire, and has over 20 years' experience teaching mathematics at the high school and college level. Ms. Lockwood has co-authored two bestselling developmental math series, as well as numerous derivative math texts and ancillaries. Ms. Lockwood's primary interest today is helping developmental math students overcome their challenges in learning math.
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Prep Tests now appear at the start of each chapter. The Prep Tests will help students determine those skills in which they may already be proficient versus those skills which may need to be reviewed further before they can successfully complete the work in the upcoming chapter.
The end of chapter Summary has been revised. The Key Words and Essential Rules and Procedures are now mapped to specific page numbers, objectives, and examples for review.
Easily write, edit and update your syllabus with the Aufmann/Lockwood Syllabus Creator. This software program allows you to create your new syllabus in 6 easy steps: select the required course objectives, add your contact information, course information, student expectations, the grading policy, dates and location, and your course outline. And now you have your syllabus!
Focus on Success appears at the beginning of each chapter and offers practical tips for improving study habits and performance on tests and exams.
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The definition/key concepts boxes are newly revised. They now contain examples to illustrate how each definition or key concept applies in practice.
Try Exercise prompts are given at the end of each Example/Problem pairing and correspond with exercises in the section-ending Exercises. By following the prompts, students can immediately apply techniques presented in worked examples to homework exercises.
Concept Check exercises appear at the beginning of each section-ending exercise set. They promote conceptual understanding. Completing these exercises will help students deepen their understanding of the concepts being addressed.
In the News exercises are application exercises found throughout most sections of the text. They are based on newsworthy data and facts and are often drawn from current events.
Projects or Group Activities appear at the end of each set of exercises. These may be assigned to students to complete individually, or they may be assigned as group activities.
Features
Getting Ready exercises provide students with guided practice on the underlying principles of various objectives.
Think About It Exercises are conceptual in nature. They ask students to think about a concept, make generalizations, and apply them to more abstract problems. The focus is on mental mathematics, not calculation or computation, and help students synthesize concepts.
Important Points are highlighted to capture students' attention. With these signposts, students are able to recognize what is most important and study more efficiently.
Supplements
Cengage provides a range of supplements that are updated in coordination with the main title selection. For more information about these supplements, contact your Learning Consultant.
FOR INSTRUCTORS
FOR STUDENTS
Chapter Test Videos Instant Access Code, Multimedia Ed.
ISBN: 9780840035950
Chapter Test Videos will help you prepare for your next test.
Every end-of-chapter test question from the textbook has an accompanying video, featuring actual instructors.
Step-by-Step solutions follow the problem-solving methods used in the text.
Rewind, play and pause buttons allow you to review at your own pace.
The solutions include interactive questions with immediate feedback on your answer. | 677.169 | 1 |
Welcome
to M337 Elementary Differential Equations. Ordinary Differential
Equations or ODE's are equations that involve functions of one
independent variable and their derivatives. They arise
naturally when modeling many different types of phenomena in
Physics, Biology, Chemistry, Engineering, and Ecomonics. Thus
our goal will be to learn different analytic and geometric techniques
to solve ODE's and understand the behavior of the actual phenomenon
they may represent.
Specific examples from interdisciplinary
areas will be covered as time allows it. Mathematical software
such as MATLAB will be used to enhance basic concepts. The course
is intended for senior undergraduate and first year graduate
students in Applied Mathematics, Computational Science, Engineering,
Physics, Chemistry, Biology, etc. | 677.169 | 1 |
About the course:
Introductory Abstract Algebra consists mainly of a study of mathematical
structures-specifically groups, rings, fields, and a few others. This semester
we will concentrate on what is in some ways the simplest of these and in other
ways the richest and most complex: the theory of groups. We start the semester
by studying the most fundamental concept in abstract algebra, that is,
operations. We will cover chapters one through five in the textbook.
·Chapter I. Mappings and
Operations
·Chapter II. Introduction
to Groups
·Chapter III. Equivalence
relations, Congruence, Divisibility
·Chapter IV. Groups
·Chapter V. Group
Homomorphisms
Evaluation:
Three in-class examinations, homeworks and a comprehensive final examination on
the following dates (subject to change): | 677.169 | 1 |
Mathematical systems theory is concerned with problems related to dynamic phenomena in interaction with their environment. These problems include:
Modeling. Obtaining a mathematical model that reflects the main features. A mathematical model may be represented by difference or differential equations, but also by inequalities, algebraic equations, and logical constraints.
Analysis and simulation of the mathematical model.
Prediction and estimation.
Control. By choosing inputs or, more general, by imposing additional constraints of the system may be influenced so as to obtain certain desired behavior.
The aim of the course is to become familiar with the basic concepts and more advanced notions of the mathematical theory of systems and control.
Learning Goals
After completion of the course the student is able to:
Work with dynamical systems in which no distinction between input and outputs is made and to work with system theoretic notions in terms of such models. In particular to obtain full row rank or minimal representations.
To understand and analyze controllability and observability of dynamical systems
Derive a manifest model description from a model with latent variables
To analyze stability properties and to synthesize a stabilizing controller through pole placement.
To synthesize observers in combination with controller design.
Present a topic from the literature in oral and written form.
Content
Notice: National course - Mastermath Utrecht
Course description
Linear time-invariant differential systems, algebraic representation of dynamical systems using polynomial matrices. Minimal representations. Autonomous systems. State space models and the Markov property. Nonlinear systems and linearization. Controllability and observability. Latent variable models. Stability of state space models. Stabilization by state feedback and by dynamic feedback. Basic observer theory and its relation to filter theory. Transfer matrices and the connection with state space models and behaviors. Poles of transfer matrices and the connections with internal stability and input-output stability. Injective and surjective dynamic systems and the connection to invertibility. Zeros of transfer matrices and the connection to invertibility of dynamic systems. Relationship with tracking problems. Zero-dynamics and minimum-phase systems. Connections with unstable pole-zero cancellation and the problems of infinite zeros.
This course is part of the MasterMath program. Information about the course (description, organization, examination and prerequisites) can be found on | 677.169 | 1 |
Textbook. Zalman Usiskin, Anthony L. Peressini, Elena Marchisotto, and Dick Stanley. Mathematics for High School Teachers - An Advanced Perspective. Prentice Hall. The textbook is required at all class meetings, and the parts covered in class are intended to be read in full.
Course Requirements.
Quizzes, etc.(15%): I will give regular, but unannounced quizzes in class. Quiz problems will be identical to prior homework assignments. There will also be some other assignments (worksheets, writing assignments). Your worst two grades will be dropped.
Exams (25% total): You will have two in-class exams on the following days: Thursday, March 8 and Tuesday, May 1.
Class Presentations (25%): Small groups of students will each design and conduct all classroom activities for a class session and will be responsible for the content covered in that session. Each group will also create homework assignments.
The groups will meet with me two weeks before their presentation for a trial run so that I will know that you are prepared. This is not optional. If you do not meet with me, you will lose half of your possible points.
Final Project (20%): There are mathematics problems that require more attention than just one day. Some of these problems are, for example, found at the end of the chapters in the textbook. Student groups will complete such a problem and present the results in class and in a written report at the end of the semester.
Class Participation (15%): Mathematics is not a spectator sport. During class I expect you to participate. This is an active class where students daily present solutions to their peers. The participation grade will be based both on the quality and frequency of your contributions.
Grades. Your grade will be based on the percentage of the total points that you earn during the semester. You need at least 90% of the points to earn an A, at least 80% for a B, at least 70% for a C, and at least 60 % for a D.
Teacher Certification. In order to student teach in Fall 2018, you must pass your TExES state teacher certification exams by May 31, 2018.
Make-up Exams. Make-up tests will only be given under extraordinary circumstances, and only if you notify the instructor prior to the exam date. There will be no make-up quizzes.
Time Requirement. I expect that you spend an absolute minimum of six hours a week outside of class on preparing your group activities, reading the textbook, preparing for the next class, reviewing your class notes, and completing homework assignments. Not surprisingly, it has been my experience that there is a strong correlation between class grade and study time.
Attendance. Due to the nature of the course you are strongly encouraged to attend class every day. I expect you to arrive for class on time and to remain seated until the class is dismissed. Students with five or more absences (excused or unexcused) will be dropped from the course with a grade of "F".
Drop Policy. The class schedule lists Thursday, March 29, as the last day to drop with an automatic "W". The College of Science will not approve any drop requests after that date.
Students with Disabilities. If you have a disability and need classroom accommodations, please contact The Center for Accommodations and Support Services (CASS) at 747-5148, or by email to cass@utep.edu, or visit their office located in UTEP Union East, Room 106. For additional information, please visit the CASS website at
Academic Integrity. All students must abide by UTEP's academic integrity policies, see for details. Please note that you may not leave the classroom during a test. | 677.169 | 1 |
Edexcel GCSE Mathematics
4.11 - 1251 ratings - Source
This text provides additional exercises written to complement those in the Edexcel GCSE mathematics course textbooks. The Foundation text is targeted towards lower ability students.This exercise is linked to exercises 1A and IB in the new edition of the course
textbook. Please note that the answers to the questions are provided in a
separate booklet, available free when you order a pack of 10 practice books. You
can buyanbsp;... | 677.169 | 1 |
43.24Calculus teachers recognize Calculus as the leading resource among the "reform" projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. The Sixth Edition uses all strands of the Rule of Four - graphical, numeric, symbolic/algebraic, and verbal/applied presentations - to make concepts easier to understand. The book focuses on exploring fundamental ideas rather than comprehensive coverage of multiple similar cases that are not fundamentally unique. | 677.169 | 1 |
Learning Fourier Analysis
Is learning Fourier analysis useful for a high school student? If so, which book should I refer for learning the basics of fourier analysis? This topic is not in my syllabus. But will it be useful for solving problems? (even if its not, it seems interesting to me).
I have learnt single variable calculus.
What micromass said. That said, Fourier analysis can be quite beautiful and its physical consequences are fascinating. Perhaps a short introduction and overview of what Fourier analysis is about would be interesting to you, but save the full treatment for when you're more mathematically able.
Fourier analysis can be quite beautiful and its physical consequences are fascinating.
Oh, most definitely. I guess many physics and engineering majors learn about Fourier analysis when solving applications. But as a mathematics major, I learned about Fourier analysis in a very sterile pure math context, with no eye for applications whatsoever. It was still extremely beautiful: it's amazing what kind of pure math one can do with Fourier stuff. But seeing the application of Fourier analysis in quantum mechanics was a real eye-opener to me. I think that's one of the most beautiful things I've ever seen. From then on, I really hated the idea of separating pure math from its applications.
Fourier analysis can be a valuable tool in the toolbox. I mentored an elementary school science project last year where the student used the Fourier transform feature in Audacity to measure the frequency of a guitar note played at different temperatures to test a hypothesis regarding frequency vs. temperature. I also worked with a high school student on a project demonstrating accuracy advantages of computing Fourier transforms by explicit integration instead of fast Fourier transform methods. The student is now beginning work on using the more accurate Fourier transform on environmental applications.
Fourier analysis will provide a tool in your tookbox that very few high school and early college students have, which may be an advantage if you want to land a job in a college lab or other research environment. Applications can pop up in many areas of science and engineering, but without the tool in your toolbox, you might not think of it.
Here are links to a few of our papers. A couple are introductory and educational:
It is surprising how much you can do and learn with a few musical instruments and Audacity.
Download our Fourier transform program (link in my sig) and you can do even more. Oscillatory data sets are widely available all over the net: temperatures, tides, sounds, greenhouse gas concentrations, etc. | 677.169 | 1 |
CALCULUS Advice
Showing 1 to 1 of 1
The professor treats the students like adults, not children that need to be persuaded or forced to learn. He does his best to impart his understanding of the material and make it interesting with relevant situations. He grades on students' understanding of the material, not how much homework you can do, and he doesn't try to trick students on the tests.
Course highlights:
Vectors, Gradients, Lagrange method; interesting ways to solve difficult problems. When there is no way to directly come to a final solution, there is always approximation. How to solve equations with multiple variables.
Hours per week:
3-5 hours
Advice for students:
Read the book along with the lectures. When the book isn't making sense, the Professor will help clear it up. When the professor's lecture isn't enough, the book usually has clear equations outlined and explanations to help your understanding. | 677.169 | 1 |
Discrete mathematics thoery graphics kaldırıldı, görüntülemek için giriş yapın] contains graphics like diagram, graphs etc should be... | 677.169 | 1 |
A Quick Introduction to Maple 16Maple is a computer algebra system that can do many computationsfor you. It also has good graphics capability. Among the toolsavailable are student LinearAlgebra, MultivariateCalculus and VectorCalculus packages. (The bluish sections are links to help topics. Try one to see a help topic.)There are also standard LinearAlgebra and VectorCalculus packages.The student routines are meant specifically for learning. The regular packages are usually more flexible and contain more commands.Maple worksheets have two modes, document mode and worksheet mode.Document mode is good for documenting work. The worksheet modeis nice for computing. This is a document block in a worksheet mode document.This is a very short introduction. To get a more complete interoduction to theworksheet interface, see Chapter 3 of the User's Manual. Between these youwill have seen all of the basics of Maple 16.Doing some simple computations.Most users eventually use the Maple mathematics format for input. Change to this modeby going into the tools>options>display menu. Set the input display to "Maple Notation." Here is how one does addition. Enter "3+4;" and hit the enter key. Note thatthe semicolon is required in multiline computations. Using a colon will suppress output.QyQtSSIrRyUqcHJvdGVjdGVkRzYkIiIkIiIlIiIiTo store a quantity in a named location use ":=". Here 10! is storeda := 10!;There is no implicit multiplication in Maple 11, when using Maple Notation for input.(Some implicit multiplication is allowed in the document mode and in 2-D input mode.)a := 3;
b := 4;
5*a;
a*b;Most of the standard functions are accessed in the same manner as on a calculator.Here are sin and cos of 1.1. (In this case a <shift><enter> was used to put theexpressions on different lines.)QyYtSSRzaW5HNiI2IyQiIzYhIiIiIiItSSRjb3NHRiVGJkYqA couple problem spots for people new to Maple are the exponential function andpi. You cannot use "e^x" for the exponential function or "pi" for the value of pi.The exponential function is exp(x) and the constant pi is in Pi.sin(pi);
sin(Pi);
e^2.;
exp(2.);One can use the evalf command to evaluation an expression numerically. Hereare a couple examples. Maple usually uses 10 decimal digits for numerical computations.a := cos(2);
evalf(a);
evalf(a,5);
b := exp(2);
evalf(b);Some calculusMaple can do many calculus operations. It can take derivatives or do integralsusing the diff and int commands.diff(exp(x)-x/(x^2+1),x);
int(exp(x)-x/(x^2+1),x);To do a definite integral one simply adds a range for the variable of integration.int(exp(x)-x/(x^2+1),x=0..5);To get a numerical answer, one uses the evalf command.evalf(int(exp(x)-x/(x^2+1),x=0..5));VectorsThere are two formats for vectors in Maple 10. They are incompatable with each other.The newer version is the one used in this worksheet. It is the Vector format. One can enter a vector in two basic ways. Here they are.a := Vector([1,2,3]);
b := <1,2,3>;The operations of addition, subtration, and scalar multiplication are the same as for numbers.u := <2,-5,6>;
v := <-3,4,-1>;
u+v;
v-u;
5*u;To compute vector products one needs a vector package. Here theLinearAlgebra package is loaded using the with command.with(LinearAlgebra):
DotProduct(u,v);
CrossProduct(u,v);Simple PlotsPloting a function is fairly simple in Maple. One normally uses the plot command.Here is aJJW1zdXBHRiQ2JS1GLDYlUSJ4RidGL0YyLUYjNiQtSSNtbkdGJDYkUSIyRidGOUY5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y5 from x=-4 to x=4.plot(x^2,x=-4..4);There are many options that one can use for the plot command. They arelist in plot,options. Here is the same plot with a different color, axes labels, and a title.plot(x^2,x=-4..4,title=(y=x^2),labels=[x,y],color="DarkGreen");Sometimes is is useful to limit the vertical range or to constrain the scaling by making the axes use the same scale.plot(x^2,x=-4..4,title=(y=x^2),labels=[x,y],color="DarkGreen",view=[-4..4,-1..10],scaling=constrained);ExercisesTo do these exercises, open a new worksheet using File>New>Worksheet Mode. Put your name at the top of the worksheet using a text block (CTRL-T). Insert a section, Insert>Section, for each problem. To insert a new prompt, [>, use CTRL-J.To insert a new line without executing, use CTRL-ENTER.1Findkc2luGLDYlUScmIzk2MDtGJ0YvRjItRiM2JC1JI21uR0YkNiRRIjRGJ0YyRksvJSliZXZlbGxlZEdGMUYyRjItRiw2I1EhRidGMg==kY29zGIzYmLUkjbW5HRiQ2JFEiNUYnRjItSSNtb0dGJDYtUSJ+RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlctRiw2JVEnJiM5NjA7RidGL0YyRjItRiM2JC1GQDYkUSI0RidGMkhby8lKWJldmVsbGVkR0YxRjJGMkYy.2Find the sum of the vectors (1,2,3,4) and (-3,4,-5,1)using Maple.3Find the square root of 141 to 5 decimal places.(If you do not know how to get the square root,go the Help>Maple Help and search for sqrt.)4Find the derivative and antiderivativeGQDYtUSIuRidGPUZDRkZGSEZKRkxGTkZQRmBvRmJvRj0=5Find the integral 0 to 10.6Plot -5 to 5.7Plot LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiaEYmbWZyYWNHRiQ2KC1JJW1zdXBHRiQ2JUY6LUYjNiQtSSNtbkdGJDYkUSIyRidGPUY9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GIzYmRlotRkA2LVEoJm1pbnVzO0YnRj1GQ0ZGRkhGSkZMRk5GUC9GU1EsMC4yMjIyMjIyZW1GJy9GVkZmby1Gam42JFEiMUYnRj1GPS8lLmxpbmV0aGlja25lc3NHRmpvLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl9wLyUpYmV2ZWxsZWRHRkVGPQ== from -2 to 2 withoutvertical lines as vertical asymptotes and a vertical range of -20 to 20. (Hint: Lookat plot,options.)8910 | 677.169 | 1 |
Rough syllabus information for
MATH 561, ``The Role of History in the Teaching of Mathematics''
New Mexico State University
Dec. 31, 2001, David Pengelley
Information for students and instructors
Advance Information for students:
This course will be valuable to anyone who plans to teach or is already
teaching mathematics. The course description is:
An in depth study of selected mathematical topics through examination
of their historical development, with special emphasis on studying original
sources. Pedagogical aspects of using history and original sources in teaching
mathematics. Research and preparation of classroom materials based on original
sources.
Prerequisite: A good undergraduate mathematics background.
This will be a seminar-style course including individual student investigations,
discussion, preparation, and presentation of teaching materials based on
historical sources, particularly original sources. The topics of study,
and of student projects, will be based largely on the individual interests
of students, both mathematically and in terms of teaching level. Much of
our focus may be on aspects of nineteenth or twentieth century mathematics
relevant to the undergraduate and graduate curricula, including analysis
of that relevance. This can include applications to courses individual
students have been taking, teaching, or expect to teach. The intent is
to shed historical light on what we teach and why we teach it, to enrich
the mathematical learning experience itself, and to improve our teaching
through the various ways of incorporating history. At the same time, students
will learn some history of mathematics in the context of the mathematics
they study and work with. Much more information about teaching with original
sources is available at our web site
There is a collection in the Mathematics Reading Room of all the original
source teaching projects developed by students in this course in past semesters,
which you are welcome to look at. You will find it on the top shelf of
the curriculum bookcase on the south wall. There is a black-spiral bound
blue cover collection from Spring 95, and a manila folder collection inside
containing projects from later semesters.
Advance information for instructors:
This seminar style course, with discussion format and regular student
presentations, introduces graduate students to the many pedagogical facets
of using history in teaching mathematics worldwide, via numerous resource
materials, with special emphasis on using original historical source material
in the teaching classroom at all levels, from elementary school to graduate
school. Graduate students in the course are expected to learn enough to
be able each to create, write, polish, present, and in some cases publish,
at least one major teaching module for a teaching setting of their choice,
usually based on original historical source material. Many of the course
resources used are provided or referenced at
including a list of all the teaching modules developed by students in previous
semesters of MATH 561. The actual teaching modules are archived in the
Mathematics Reading Room (top shelf of teaching bookcase on south wall),
from which they may be copied by students and instructors.
Instructor's advance preparation for the course:
Advertise course to mathematics and education graduate students (ask
D. Pengelley for his advertising material)
Order books at bookstore: V. Katz, A History of Mathematics, second
edition, required text for all students; Carl Boyer & Uta Merzbach,
A History of Mathematics, 2nd ed., recommended; R. Calinger, Classics of
Mathematics, 2nd ed, recommended; J. Fauvel & J. Gray, The History
of Mathematics: A Reader, recommended.
Put things on reserve at the library as needed, e.g. Mathematical Expeditions,
Vita Mathematica, Learn from the Masters, History in Mathematics Education,
and the recommended books above.
Find out dates for submission for student presentations at NMSU Graduate
Research and Arts Symposium.
Books and resources for course:
`see web' means available online at
especially good is the BSHM website linked here.
Teaching modules developed by students in previous semesters of MATH
561 are in the MRR, with a list at `see web'.
R. Laubenbacher & D. Pengelley, Some Selected Resources for Using
History in Teaching Mathematics (see web under A selective bibliography
for using history in teaching mathematics)
J. Fauvel and J. Van Maanen, The ICMI Study discussion document on The
role of the history of mathematics in the teaching and learning of mathematics
(1997-2000), in
which led to the book
Critique of Modules Questions for MATH 561 (from D. Pengelley,
and also at back of bound teaching modules in MRR)
Instructions for daily diary (from D. Pengelley)
Course activities:
Day 1: Give history of course, goals, current worldwide viewpoint on
using history in teaching mathematics, course combines history/mathematics/education,
introduce everyone, find out backgrounds, interests, make contact info.
sheet for everyone. Explain what we'll do, and that project modules can
be tailored to individual student interests and level. Seminar style: presentations/discussion/written
assignments. Show previous course student modules, and various resource
materials. Hand out various resource materials, or tell students to print
them from the web, especially Some Selected Resources for Using History
in Teaching Mathematics. Students should write about their background
in mathematics, history. Students should join the Historia Matematica web
discussion list at Students
should keep a daily journal (see handout from D. Pengelley).
Weeks 1-2: Each student should read in Katz, pick a topic, read/write/present
on it in the next week or two, to get started learning how to read and
think about history. Students should discuss together and pick jointly
2-3 of the previous course modules, read and write critiques of them using
the critique questions. Explore web and other resources.
Weeks 2-3: Revise Katz and module critiques after class presentation
and instructor feedback. Read and write critiques of Guzman and UME Trends
articles. Then discuss in class. Students select/write some ideas for topics
of first (primary) course project, get instructor feedback. Students read/write
on sections of Mathematical Expeditions, class discussion of pedagogical
use.
Week 3: Flesh our project ideas further. Read/write on Siu's ABCD article,
then discuss in class over the next few weeks.
Week 4: Refine project ideas, make individual timetables for project
work and completion.
Week 6: Students report on projects every week from now on. Instructor
meets individually with students to help with projects.
Week 7: Read/write on ICMI discussion document, then discuss. Set dates
for preliminary project presentations in class, and for rough drafts for
everyone to read and critique.
Week 8: Read/write to compare and contrast corresponding material in
Katz with Calinger's Contextual History of Mathematics (from David; this
is not Calinger's source book). The two books reflect very different approaches
to history. Start class presentations based on preliminary project drafts.
Week 9: Continue project presentations, work with students individually,
begin to refine drafts. Student projects will go through numerous written
draft revisions with instructor feedback. Plan for public presentations
of final projects at conference (NMSU Graduate Research and Arts Symposium)
or dept. seminar or colloquium.
Week 10: Start reading Sfard article, and discuss over next several
weeks. Heavy and lengthy, deep and intellectually rewarding; worth emphasizing
for several weeks.
Weeks 11 - end: Continue project revisions, instructor feedback and
individual assistance, class presentations, practice for public presentations,
continue detailed discussion of Sfard article. Final student projects are
copied and added to materials in MRR, with copy to D. Pengelley for further
archiving and addition to web list.
List of student written assignments:
Students write about their own background in relation to the course
Each student writes critiques of three previous teaching modules using
historical sources developed in the course in previous years (see web for
listing; actual modules are in Math Reading Room); use developed list of
questions to consider in critiquing modules
Each pick a historical thread from Katz and write on it, present and
discuss
Write critique of Guzman and UME Trends articles (see web), discuss
Read/write on part of Mathematical Expeditions as teaching material
Individual project ideas due
Write on ABCD article (see web)
Individual project plans due
Each student chooses and critiques three more previous teaching modules
developed in the course
Critique ICMI document (this is now evolved to a book History in
mathematics education; see NMSU library) | 677.169 | 1 |
Books : reviews
Mathematicians have always used experiments and visualization to explore new ideas and ways to prove them.
Using examples that truly represent the experimental methodology,
this book provides the historical context of, and rationale behind, experimental mathematics.
It shows how today, the use of advanced computing technology provides mathematicians with
an amazing, previously unimaginable "laboratory," in which examples can be analyzed,
new ideas tested, and patterns discovered.
The last twenty years have been witness to a fundamental shift in the way mathematics is practiced.
With the continued advance of computing power and accessibility,
the view that " real mathematicians don't compute" no longer has any traction for
a newer generation of mathematicians that can really take advantage of computer-aided research,
especially given the scope and availability of modern computational packages such as Maple, Mathematica, and MATLAB.
The authors provide a coherent variety of accessible examples of modern mathematics subjects
in which intelligent computing plays a significant role. | 677.169 | 1 |
ALEGEBRA B/TRIGONOMETRY
This course deals with operations and equations, graphing linear equations, solving systems of linear equations (graphically and elimination of variables) products, factoring, solving quadratic equations, graphic conic sections, and rational number calculations. The trig portion of this course deals with an introduction to the six trig functions solving triangles, radian measure, graphing the trig functions, and solving fundamental identities using the six trigonometric functions.
Geometry
A course offered to students in grades 11 and 12, geometry deals with problems involving two dimensional aspects of a line, angles, rectangles, triangles, and circles. Through these problems the deductive method is also studied.
Calculus
Calculus is an advanced level math course open to juniors and seniors with extensive math background. This course deals with functions, limits, differentiation, integration, maximum and minimum values of a function, methods of differentiation, trigonometric, logarithmic and exponential functions as well as applications of calculus.
Particular Topics in Foundation Math
This course may be taken by a 12th grade student. This course is a review of basic math skills. Topics will include but are not limited to adding, subtracting, multiplying, dividing, fractions, equations, check writing, conversions, percentages, and other pre-algebra problems | 677.169 | 1 |
Ti-83 plus инструкция
Ti-83 plus инструкция Как использовать графический калькулятор TI 83 Plus. 2 метода:Основные инструкцииДополнительные инструкции. На первый взгляд множество кнопок. TI graphing calculators are learning tools designed to help students visualize concepts and make. The TI-83 Plus is approved for use on the following exams. TI-83 Plus графический калькулятор принадлежит серии калькуляторов TI-83. Этот калькулятор является модернизированной версией TI-82. This manual describes how to use the TI.83 Graphing Calculator. Getting. Started is an overview of TI.83 features. Chapter 1 describes how the TI.83 operates. Путем гугления выяснилось, что TI-84 Plus это не только ценный. TI-84+ программируемый графический калькулятор с. Драйвера есть: wikiti. /?title=83Plus:Software:usb8x. Только. 10 Aug 2011. TI-83/84 Graphing Calculator Tutorial by mathmyway. How to Graph Equations on the TI-83 Plus and TI-84 Plus | Calcblog - Duration: 6:21. TI-83 Plus instruction Manual (Texas Instruments Graphing Calculator TI83 plus MANUAL) [Steven Kelly] on . *FREE* shipping on qualifying offers. This Guide is designed to offer step-by-step instruction for using your TI-83, TI-83. Plus graphing calculator with the fourth edition of Calculus Concepts: An. TI-83-Plus Calculator Basics. Tips and Techniques. TI-83 Plus Keyboard. Generally, the keyboard is divided into these zones: graphing keys, editing keys. Most of the instructions also apply to the TI-83 Plus. Calculator key strokes are shown in brown: ON Features that appear above the keys have the complete key. This tutorial is designed with the student in mind. The topics selected are those that students will use in college algebra, college trigonometry, and freshman. Introduction to the TI-83. Before learning how to use the TI83 as a graphing calculator, an introduction to. On the TI83 Plus, it is above the. APPS button as. Statistics with the TI-83 Plus (and Silver Edition) | 677.169 | 1 |
This collection of lessons is due primarily to
the efforts of Mike Mills of the Rowland Unified
School District. Mike who
has believed in the power of this technology early
on, realized that many teachers don't
have the time or inclination to write PowerPoint
lessons
themselves. In an effort to help the Algebra
2 teachers in his own district, he has
spent
2 years
writing
lessons to complement
Holt's Algebra 2 by Burger, Chard,
Hall,
Kennedy, and Leinwand.
These lessons can be viewed slide by slide under
the Previews tab. There you will see the vast
majority
of lessons
are
completely
original and
written by Mike.
Deliver lessons to students that meet
rigorous state standards in a time efficient manner.
These lessons are specifically designed to
fill in gaps in understanding and go beyond the
book. Many concepts that are tough to learn or are
inadequately explained in the book are elaborated on
in these PowerPoint lessons. Most lessons consist
of about 5 to 8 slides and are easily presented in a
30 to 50 minute period.
Significantly raise your student's
year-end test scores. Engage your students interest like never
before. Most of the presentations have some
animated effects that add to the effectiveness of
the lesson. You can personalize the presentations
by adding your own backgrounds, sounds, and clipart
to make them even more powerful.
Integrate technology into your
pre-algebra classroom on a daily basis. PowerPoint
is versatile and provides you with the ability to
hyperlink to other applications and web-sites while
still in PowerPoint. At the touch of a button you
can switch from delivering your presentation to
using graphing calculator software and back again.
Free yourself from the overhead or
whiteboard. Using a remote mouse and laser
pointer, you will be able to circulate around the
room as you deliver the lesson. You can monitor
note taking, see where students are getting hung up,
and control unwanted behavior.
Supply new teachers with a complete set
of well designed lessons that can be shared directly
with students. Created and written by a
public school algebra teacher with over 16 years
experience, these lessons could serve as a valuable
base on which to build.
Add, delete, or change any part of any
lesson you wish. Unlike videos, software,
or other similar media, PowerPoint lessons can be
modified. All teachers are encouraged to make
changes to these lessons and make them their own.
Share your own lessons with colleagues.
These lessons are copyrighted and any
duplication or transfer of these lessons or
modification of these lessons is illegal unless all
involved parties have purchased Algebra 2
Slide Show: Teaching Made Easy As Pi. However,
your own creations can be easily sent to colleagues
through e-mail.
Absent students can access these
PowerPoint lessons from a home computer through the
"Previews Page" of this web-site or by visiting
Notes can also be
easily printed for those without home computers.
You will be pleasantly surprised how many students
return from an absence and immediately request a
printout of the PowerPoint lesson they missed. | 677.169 | 1 |
The first book is our main text . The second is important for
learning AMPL, a modeling language for linear and nonlinear
programming
Required work
There will be 6 or 7 homework sets
during the semester. They will constitute 50% of your final grade. .
Also you will not pass this course if you
fail to hand in two homework assignments or more regardless of your
total score!
We will have a final exam. worth 40% of your grade.
The other 10% is based on class participation, and
performance
Our main purpose in this course is to cover a broad
spectrum of topics on linear programming. We cover the basic material
such as the simplex method and duality theory. But a healthy part of
the course will be devoted to the modeling power of linear
programming and its applications particularly in Business disciplines
such as supply chain, finance, marketting etc. We will also use AMPL
or related software for modeling projects. Some linear algebra
computations may be done using Matlab or Matlab like languages (such
as Octave).
Topics
Below is a list of topic to be covered in the course. Topics with
an asterisk * are optional and may be covered at the discretion of
the instructor if there is time.
Week 1 &
2: Basics and definitions.
Definition of linear programming
Some real world models: the diet
problem, the transportation problem, foreign exchange arbitrage
problem, some economics and financial models for LP
Geometry of LP: polytopes and
polyhedra, vertices, edges, facets, the two and three dimensional
case.
Optimality condition in both
Geometric and algebraic sense
The basic simplex algorithm
Use of MATLAB and MATLAB like languages for linear algebra
calculations | 677.169 | 1 |
GCSE Mathematics: Number [Higher Tier]
By Nelson Thornes
Release Date : 2013-10-04
Genre : Mathematics
FIle Size : 68.36 MB
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GCSE Mathematics: Number [Higher Tier] Number is a key component of your Mathematics GCSE. This provides the perfect combination of explanation and practice, offering the opportunity for you to make connections between mathematical concepts and apply them in everyday, real-life situations.
Plenty of worked examples, interactive explanations, exam-style questions and quizzes with instant feedback will help to your test knowledge and understanding and prepare you for the assessment.
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·Represent relations and functions with graphs, tables, and sets of ordered pairs.
Introduction
Algebra is a potent tool for describing and exploring relationships.
Imagine tossing a ball straight up into the air, watching it rise, stop, and fall back down into your hand. As time passed, the height of the ball changed, creating a relationship between the amount of time the ball was in the air and its height.
In mathematics, a relationship between variables that change together (such as time and height) is called a relation.
Functions Defined
There are many kinds of relations. Among the most important algebraic relations are functions. A function is a relation in which one variable specifies a single value of another variable. For example, when you toss a ball, each second that passes has one and only one corresponding height. Time only goes forward, and never repeats itself. The height of the ball depends on how much time has passed since it left your hand. This is a one way relationship—although each moment of time is unique, it is possible for the ball to be at a particular height more than once as it goes up and then down. Knowing the time will tell you the height, but knowing the height won't give you an exact time.
The parts of a function are called inputs and outputs. An input is the independent, non-repeating quantity. The output quantity is the dependent quantity. The value of the output depends on the value of the input. For each input, there is a single output. In the case of tossing a ball in the air, time is the input and height is the output.
Let's look at a few more examples to get comfortable recognizing what is a function and what is not. Remember the last time you were in a parking lot? You won't be surprised to know that there's a relation between the number of cars and the number of tires in a lot—the number of cars and the number of tires are linked. Is this relation a function? Can you use the number of cars to correctly figure out the number of tires?
Yes, you can. Every single car has 4 tires, so the number of tires depends on how many cars are in the parking lot. Every input of cars specifies a single possible output of tires. (In this example, the relation of tires to cars is also a function—the number of tires also specifies the number of cars.)
Now consider a different relation, between houses and the people who live in them. If an address is the input, and the output is the occupants, is this relation also a function? Think of your own house or apartment—are the people staying there always the same?
Nope. That time you went to camp, the occupancy changed. Every time you had a friend stay over, it changed again. Because a single address can produce more than one set of occupants, the relation is not a function.
Here's a good rule of thumb to use to recognize functions: If you put the input in more than once, are you guaranteed to always get the same output? With the cars and wheels, the answer is yes. For an input of 25 cars we always get an output of 100 tires, no matter which 25 cars drive into that parking lot or when they arrive. The relation is a function.
With the houses and occupants, the input of an address is not guaranteed to always produce the same output, because a house stays put while people come and go. The relation is not a function.
Which of the following situations describes a function?
A) Your age and your weight on your birthday each year.
B) The name of a course and the number of students enrolled in that course.
Correct. Age only increases while weight can fluctuate. On each birthday you have just one weight (not counting the cake and ice cream), so for every input, there is only one output.
B) The name of a course and the number of students enrolled in that course.
Incorrect. A course will have different enrollment in different semesters and in different schools. A single input of the course name will produce different outputs of enrollment. The correct answer is a person's age and his weight on his birthday each year.
C) The diameter of a cookie and the number of chocolate chips in it.
Incorrect. Although bigger cookies can hold more chips, the exact number in any size of cookie will vary with the recipe and how evenly the batter is mixed and rolled out. A single input of cookie size will produce different outputs of chips. The correct answer is a person's age and his weight on his birthday each year.
Graphing Functions
When both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a coordinate graph. The independent value is plotted on the x-axis and the dependent value is plotted on the y-axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input (x-coordinate) on the graph, there will be exactly one output (y-coordinate).
For example, the graph of this function, drawn in blue, looks like a semi-circle. We know that y is a function of x because for each x-coordinate there is exactly one y-coordinate.
If we draw a vertical line across the plot of the function, it only intersects the function once for each value of x. That is true no matter where the line is drawn. Placing or sliding such a line across a graph is a good way to determine if it shows a function.
Compare that graph with this one, which looks like a blue circle. This relationship cannot be a function, because each x-coordinate has two corresponding y-coordinates.
When a vertical line is placed across the plot of this relation, it intersects the graph more than once for some values of x. If a graph shows two or more intersections with a vertical line, then an input (x-coordinate) can have more than one output (y-coordinate), and y is not a function of x.
Yes, this graph is a function. Every x-coordinate has exactly one y-coordinate.
Functions in Table Form
Tables can also be used to describe functions. Let's compare tables of functions with tables of non-functions.
This table represents a function. None of the independent values (x) are repeated and each has only one corresponding dependent value (y).
x
y
-1
3
-2
5
-3
3
-5
-3
The next table does not represent a function. The x column has two values that are 3, and they correspond to two different values for y. Remember, when a single input can produce multiple outputs, the relation is not a function.
The correct answer is A. None of the values for x is repeated, and each x-value has only one corresponding y-value.
Functions as Sets of Ordered Pairs
Functions can also be represented by sets of ordered pairs of x and y values, inputs and outputs. We can pull pairs from tables or graphs, and use parentheses to keep them together.
Let's go back to this table of a function:
x
y
-1
3
-2
5
-3
3
-5
-3
Each row in the table describes an ordered pair, like this: an x of -1 corresponds to a y of 3, so that's the ordered pair (-1, 3). An x of -2 corresponds to a y of 5, so that's the ordered pair (-2, 5). The whole table gives us a set of ordered pairs:
{(-1, 3), (-2, 5), (-3, 3), (-5, -3)}
To show that the four ordered pairs belong together in a set, we list them with commas in between each and brackets around the group. As with other methods of representing relations, we can check the characteristics of a set of ordered pairs to determine if it is a function. Since the first value in each pair is the input and the second is the output, we can scan the set to see if each input is associated with a single, consistent output. If it is, the set is a function.
Or we can plot the points on a coordinate grid, for a visual check. Here, we can see that in the set of pairs we just listed, every x/input/independent value has one and only one y/output/dependent value:
In another set of ordered pairs: {(3,-1),(5,-2),(3,-3),(-3,5)} one of the inputs, 3, can produce two different outputs, -1 and -3. You know what that means—this set of ordered pairs is not a function. A plot confirms this:
Notice that a vertical line passes through two plotted points. One x-coordinate has multiple y-coordinates. That too means that this relation is not a function.
Which of the following is a set of ordered pairs representing a function?
Incorrect. These numbers are not grouped into ordered pairs. Without proper notation, it is impossible to know if which values are the inputs and which are the outputs. The correct answer is {(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)}.
B) {(0, 0), (1, 1), (1, -1), (2, 2), (2, -2)}
Incorrect. Some x-coordinates are repeated and have different y-coordinates. This is not a function. The correct answer is {(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)}.
C) (4, 2), (5, 1), (6, 0), (7, -1), (8, -2)
Incorrect. Although no x-coordinate is repeated and each has exactly one y-coordinate, the group is not enclosed in brackets. Without proper notation, this group of numbers is not considered a set. The correct answer is {(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)}.
D) {(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)}
Correct. No x-coordinate is repeated and each has exactly one y-coordinate, so this is a function.
Horizontal and Vertical Lines—Functions or Not?
Two special types of relations are those of horizontal and vertical lines. Are they functions?
Let's begin with a horizontal line. A line on the coordinate plane is horizontal when every x-coordinate has the same y-coordinate. No x-coordinates have more than one y-coordinate, and each input always produces the same output. Therefore, all horizontal lines represent a function.
Now consider a vertical line. In this situation, every y-coordinate has the same x-coordinate. The input never changes, but the output changes constantly. Since the same value for x has many values for y, a vertical line cannot represent a function.
Summary
In real life and in algebra, different variables are often linked. When variables change together, their interaction is called a relation. When one variable determines the exact value of a second variable, their relation is called a function. Functions can be recognized, described, and examined in a variety of ways, including graphs, tables, and sets of ordered pairs. | 677.169 | 1 |
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Combinatorics is the mathematics of counting and enumeration. It is typically concerned with finite or discrete mathematical structures (unlike calculus, which deals with the infinite). This class is an introduction to some of the main ideas of combinatorics: counting methods, binomial theorems, permutations and combinations, recurrence relations, generating functions, and basic graph theory.
This course assumes that you have a solid working knowledge of the topics in VCU's MATH 300, including sets, set operations, logic, methods of proof (including induction), methods of disproof, relations and functions.
Your grade will be determined by two tests, weekly homework assignments and a final exam. Details follow.
GRADED WORK:
Tests: There are two tests, each closed book. Tests are written under the assumption that you are studying the material AT LEAST 6 hours per week outside
of class.
Assignments: Weekly written assignments are collected, graded and returned.
Papers are collected at the beginning of class on appointed days.
Papers submitted after the beginning of class may not be graded.
If you must miss class when an assignment is due, please give it to me early or have a classmate turn it in for you.
You may email an assignment to me, but it must arrive in my inbox no later than the beginning of class on the day it is due. I sometimes don't print emailed assignments, so you may not get any written feedback from me. If the scan or photo is of poor quality I will not be able to grade it.
Exceptionally sloppy or disorganized work is not graded.
I encourage you to work together, though the work you turn
in must be your own.
Resist the temptation to hunt for solutions on line. I do not grade work that I recognize as copied.
In addition to the work you hand in, you should work lots of extra
problems for practice.
Some assigned problems are intended to make you think about ideas not discussed
in class.
Final Exam: The final exam is cumulative, covering all material discussed in class. It is scheduled for Thursday May 12, 8:00–10:50 am, in our usual classroom.
The exam is written under the
assumption that you have been studying the material AT LEAST 6 hours per week outside
of class for the entire semester.
Dropped Scores: A small number of low homework grades will be droppedTest 1:
25%
Test 2:
25%
Homework:
25%
Final Exam grade:
25%
Total:
100%
COURSE POLICIES:
Attendance:
Attendance is not taken. You are responsible for all material covered in class.
Etiquette: Put away all phones for the entire duration of class. Please do not text in class or leave to take a call.
YouA make-up test can be arranged in the event of a documented illness or emergency.
The final exam cannot be given early. If you miss the final exam because of a documented illness or emergency, then I can give you a grade of incomplete (I) for the course and you will have to make up the final exam by the date set by the University.
Honor System:
Any instance of cheating on tests and exams is considered an honor offence and is dealt with according to University policy.
You are expected to work lots of extra problems for practice.
LAST DAY TO WITHDRAW: Friday March 25
BOILERPLATE INFORMATION: The following is required on all VCU syllabi:
Email Policy Electronic mail or "email" is considered an official method for communication at VCU because it delivers information in a convenient, timely, cost effective and environmentally aware manner. Students are expected to check their official VCU e-mail on a frequent and consistent basis in order to remain informed of university-related communications. The University recommends checking e-mail daily. Students are responsible for the consequences of not reading, in a timely fashion, university-related communications sent to their official VCU student e-mail account. This policy ensures that all students have access to this important form of communication. It ensures students can be reached through a standardized channel by faculty and other staff of the university as needed. Mail sent to the VCU email address may include notification of university-related actions, including disciplinary action. Please read the policy in its entirety:
VCU Honor System: Upholding Academic Integrity The VCU honor system policy describes the responsibilities of students, faculty and administration in upholding academic integrity, while at the same time respecting the rights of individuals to the due process offered by administrative hearings and appeals. According to this policy, "members of the academic community are required to conduct themselves in accordance with the highest standards of academic honesty and integrity." Also, "All members of the VCU community are presumed to have an understanding of the VCU Honor System and are required to:
Agree to be bound by the Honor System policy and its procedures;
Report suspicion or knowledge of possible violations of the Honor System;
Support an environment that reflects a commitment to academic integrity;
Answer truthfully when called upon to do so regarding Honor System cases; and,
Maintain confidentiality regarding specific information in Honor System cases."
The Honor System in its entirety can be reviewed on the Web at or it can be found in the current issue of the VCU Insider at
Student Conduct in the Classroom According to the Faculty Guide to Student Conduct in Instructional Settings "The university is a community of learners. Students, as well as faculty, have a responsibility for creating and maintaining an environment that supports effective instruction. In order for faculty members (including graduate teaching assistants) to provide and students to receive effective instruction in classrooms, laboratories, studios, online courses, and other learning areas, the university expects students to conduct themselves in an orderly and cooperative manner." Among other things, cell phones and beepers should be turned off while in the classroom. Also, the university Rules and Procedures prohibit anyone from having "in his possession any firearm, other weapon, or explosive, regardless of whether a license to possess the same has been issued, without the written authorization of the President of the university..." For more information, visit the VCU Insider online at
Students with Disabilities SECTION 504 of the Rehabilitation Act of 1973 and the Americans with Disabilities Act of 1990 as amended, require that VCU provides "academic adjustments " or "reasonable accommodations" to any student who has a physical or mental impairment that substantially limits a major life activity. To receive accommodations, students must request them by contacting the Disability Support Services Office on the Monroe Park Campus (828-2253) or the Division for Academic Success on the MCV campus (828-9782). More information is available at the Disability Support Services webpage: ; or the Division for Academic Success webpage at If you have a disability that requires an academic accommodation, please schedule a meeting with me at your earliest convenience.
Excused Absences for Students Representing the University Students who represent the university (athletes and others) do not choose their schedules. Student athletes are required to attend games and/or meets. All student athletes should provide their schedule to the instructor at the beginning of the semester. The Intercollegiate Athletic Council (IAC) strongly encourages faculty to treat missed classes or exams (because of a scheduling conflict) as excused absences and urges faculty to work with the students to make up the work or exam.
Class Registration Required for Attendance Please remember that students may only attend those classes for which they have registered. Faculty may not add students to class rosters. Therefore, if students are attending a class for which they have not registered, they must stop attending. | 677.169 | 1 |
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This publication offers in a concise, but particular approach, the majority of the probabilistic instruments pupil operating towards a sophisticated measure in statistics,probability and different similar parts, can be outfitted with. The process is classical, warding off using mathematical instruments no longer worthwhile for accomplishing the discussions.
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1. 19 challenge 19 (Bundeswettbewerb Mathematik, 1976 1st around [90]) allow alb be a fragment in its easiest phrases, that's, the best universal divisor of a and b is 1. within the tree diagram proven in Fig. 1. 20 the fraction alb has successors: a a+b and every one of those successors has successors of its personal, developed within the related approach (the successors of al(a + b) are Problems for research 17 ~ b b a+b b a+b 2a+b a+2b a+b a+2b Fig. 1. 20 a 2a +b and a+b 2a + b' and the successors of b/(a + b) are b a + 2b and a+b a + 2b· discover a and b such that the tree diagram beginning with alb involves all confident fractions below 1.
Think that E; = C;_I for all i = 2, three, . . . , n. this means that therefore En = C n- I for all traditional numbers n > 2. difficulties according to recognized issues within the historical past of arithmetic a hundred seventy five challenge ninety five A zigzag permutation p = a l a2 a3 • • ·an will be represented through a zigzag line z becoming a member of the issues P 1(1, a l ), P 2(2, a2 ), • • • , Pn(n, an) in a Cartesian coordinate procedure, as proven in Fig. three. 18. y ~ a2 a1 a3 I -P'I I I I I I I , I I _-+_~_ P' II eleven" II I" I I zero I P3 \ P3 1 I I I I I \1 I ", V P' 1 four 1 ;p2 : I I three four 2 I I I II " 1/ ~P' I I /',P; I .....
Then the instantly traces A I A 2, B} B2 and C I C2 are both all parallel to each other, or all meet in some degree s. evidence. AIBI and A2B2 belong to a few aircraft 'Y. BICI and B 2C2 belong to a airplane a, and C I A I and C2 A 2 belong to a aircraft {3. those 3 planes intersect pairwise in 3 strains: a and {3 in C I C2 , (3 and 'Y in AIA2' and 'Y and a in B I B 2. 3 special traces alongside which 3 planes intersect pairwise are both all parallel in any other case they meet in a few element S. This proves our assertion.
Denote via n. the sum am + am-. + · · · + a2 + a l + ao. From 10 m - 1am - I -> am - I it follows that hence no > n •. equally, if n l has a couple of digit, then the sum of its digits n2 is lower than n. ; if n2 has multiple digit, then the sum of its digits n3 is below n2 , etc. despite the fact that, a lowering series of confident numbers can't be endured ceaselessly. So, after a undeniable quantity ok of steps a one-digit quantity nk has to be reached. (b) the adaptation d = no - n. should be expressed within the shape within the above expression every one summand (10; - l)a; is divisible through nine.
Thus at the very least one vertex of T' is separated from R through the other aspect of T' (Le. the road section, becoming a member of this vertex to R, intersects the other part of T'). with no lack of generality allow us to consider that M'N' separates R from L'. this means that the space d R of MNfrom R is bigger than its distance d L from L. The distances d R and d L are the altitudes of the triangles RMN and LMN, equivalent to their universal part MN. hence which suggests that region RMN = ±MNdR > ±MNdL = quarter LMN. | 677.169 | 1 |
Promo video
Complete Course Description
The following series of videos will familiarize you with the mathematical skills and concepts that are important to know, for the purpose of solving problems on the quantitative reasoning section of the GRE revised General Test.
The revised GRE Quantitative reasoning sections are designed to measure your problem-solving ability, and focuses on basic concepts from arithmetic, algebra, geometry, and data analysis. | 677.169 | 1 |
Course: Pre-Algebra GT (Grade 6) Unit 1: The Number System and Exponents (8.NS/8.EE/7.RP) In this unit, students. They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.
Unit 2: Geometry (8.G/7.G) Students understand the statement of the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons.
Students continue their work with area, solving problems involving the area and circumference of a circle and surface area of three-dimensional objects. In preparation for work on congruence and similarity, they reason about relationships among two-dimensional figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.
Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angle in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. Unit 3: Analyzing Functions & Linear Equations (8.F/7.RP/8.EE/7.G) Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx+b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount (m)(A). Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems. Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.
Unit 4: Statistics &Students | 677.169 | 1 |
In this lesson we apply the skill of solving exponential and logarithmic equations by translating growth and decay application problems into such an equation and then solving. The growth and decay applications include problems such as viruses spreading exponentially, nuclear waste decaying exponentially, and logarithmic learning curves. These will be important processes in calculus where our primary focus will be on the rate of change of this growth or decay.
Objectives
By the end of this topic you should know and be prepared to be tested on:
4.5.1 Use the formula A=Iekt to solve exponential growth and decay application problems
4.5.2 Understand the growth constant k and be able to solve for it exactly when solving an exponential growth/decay application problem
4.5.3 Solve growth and decay problems including doubling time and carbon dating
Be sure to practice a variety of applications exponential and logarithmic, growth and decay. You are expected to memorize the exponential growth model formula. You are not expected to memorize other application formulas (e.g. Newton's Law of Cooling, Logistic Growth Model), but you should know how to use them when applicable.
You may SKIP any "curve fitting" and "regression" examples/problems in this section and throughout the course. | 677.169 | 1 |
CURATOR
EXTRAS
Vector algebra (12th Grade Mathematics)
12th Grade Mathematics
At the end of this lesson, you will be able to:
• Differentiate between scalar and vector
quantity.
• Define poistion vector.
• Find direction cosines of the position
vector.
• Find the additions of vectors.
• Find components of a vector.
• State section formula for internal and
external division.
• Find dot product of two vectors.
• Cross product of two vectors. | 677.169 | 1 |
Homework help math help
Welcome to Free Math Help. Homework Help: Math Branches of Math. Udents, teachers, parents, and everyone can find solutions to their math. A resource provided by Discovery Education to guide students and provide Mathematics Homework help to students of all grades. Gebra; Arithmetic; Calculus; Geometry; Statistics; Trigonometry. Vering pre algebra through algebra 3 with a variety of introductory and advanced lessons. Vering pre algebra through algebra 3 with a variety of introductory and advanced lessons! Cent Math Questions. R K 12 kids, teachers and parents. Ease select a topic for math help: Algebra. Welcome to Free Math Help. Udents, teachers, parents, and everyone can find solutions to their math. Ease select a topic for math help: Algebra. Tmath explains math textbook homework problems with step by step math answers for algebra, geometry, and calculus. Select your textbook and enter the page you are working on and we will give you the exact lesson you need to finish your math homework!Tutorvista provides Online Tutoring, Homework Help, Test Prep for K 12 and College students. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. Mathematics homework help, lessons, questions, worksheets, and quizzes in arithmetic, algebra, geometry, trigonometry and more. A resource provided by Discovery Education to guide students and provide Mathematics Homework help to students of all grades. Math homework help. Th The illustration at right shows. Nnect to a Tutor Now for Math help, Algebra help, English, ScienceFree math lessons and math homework help from basic math to algebra, geometry and beyond. Math Tutor DVD provides math help online and on DVD in Basic Math, all levels of Algebra, Trig, Calculus, Probability, and Physics. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Need math homework help. Line tutoring. The best multimedia instruction on the web to help you with your homework and study.
Ease select a topic for math help: Algebra. Welcome to Free Math Help. Tmath explains math textbook homework problems with step by step math answers for algebra, geometry, and calculus. Vering pre algebra through algebra 3 with a variety of introductory and advanced lessons. Homework Help: Math Branches of Math. Udents, teachers, parents, and everyone can find solutions to their math. eNotes Homework Help is a way for educators to help students understand their school work. R K 12 kids, teachers and parents. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. R K 12 kids, teachers and parents. Webmath is a math help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment. Th The illustration at right shows. Free math lessons and math homework help from basic math to algebra, geometry and beyond! This introduction will be great math homework help for fractions. A resource provided by Discovery Education to guide students and provide Mathematics Homework help to students of all grades. eNotes Homework Help is a way for educators to help students understand their school work. Line tutoring. Free math lessons and math homework help from basic math to algebra, geometry and beyond? Math homework help. R experts are here to answer your toughest academic questions!Need math homework help. R K 12 kids, teachers and parents. Math Tutor DVD provides math help online and on DVD in Basic Math, all levels of Algebra, Trig, Calculus, Probability, and Physics. Gebra; Arithmetic; Calculus; Geometry; Statistics; Trigonometry. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum! Can help with your online class. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum! Webmath is a math help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment. Ull get a quick refresher on fraction fundamentals and the other concepts needed to do your. Mework exam help by email, Skype, Whatsapp. R experts are here to answer your toughest academic questions!Math Tutor DVD provides math help online and on DVD in Basic Math, all levels of Algebra, Trig, Calculus, Probability, and Physics. Ease select a topic for math help: Algebra. Select your textbook and enter the page you are working on and we will give you the exact lesson you need to finish your math homework!Tutoring homework help for math, chemistry, physics. Welcome to Free Math Help. Cent Math Questions. Udents, teachers, parents, and everyone can find solutions to their math. Ee study guides, cheat. Webmath is a math help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment. Vering pre algebra through algebra 3 with a variety of introductory and advanced lessons. | 677.169 | 1 |
Review for Thursday's final. Chapters 1-5, 7 and 8. Solving for a single variable, finding the slope of a line and standard line form, graphing linear inequalities. "
★★★★
"This week, the student and I focused on completing all of his practice problems before his test on Thursday. We began by solving polynomials that factored easily (either by grouping or by quadratic-type factoring). The student has completely mastered these types of problems and had no trouble completing them. The next set of problems were a mixture of simple factoring and those that require the use of the rational root theorem. I was worried about his progress on this, but after one example, he did rather well. Once we worked through these problems, we went over his last quiz to correct minor mistakes he had made. Our next session will focus on a review for his upcoming test." | 677.169 | 1 |
This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – so-called phase portraits – which visualize functions as images on their domains.
More college students use Amos Gilats MATLAB: An Introduction with Applications than any other MATLAB textbook. This concise book is known for its just-in-time learning approach that gives students information when they need it. The 6th Edition gradually presents the latest MATLAB functionality in detail. The book includes numerous sample problems in mathematics, science, and engineering that are similar to problems encountered by new users of MATLAB. MATLAB: An Introduction with Applications is intended for students who are using MATLAB for the first time and have little or no experience in computer programming. It can be used as a textbook in first-year engineering courses or as a reference in more advanced science and engineering courses where MATLAB is introduced as a tool for solving problems.
Equally effective as a freshmen-level text, self-study tool, or course reference, the book is generously illustrated through computer screen shots and step-by-step tutorials, with abundant and motivating applications to problems in mathematics, science, and engineering.
MATLAB: An Introduction with Applications 4th Edition walks readers through the ins and outs of this powerful software for technical computing. The first chapter describes basic features of the program and shows how to use it in simple arithmetic operations | 677.169 | 1 |
Experiments with MATLAB by Cleve Moler
Description: It started out in the late 1970s as a simple "Matrix Laboratory". We want to build on this laboratory tradition by describing a series of experiments involving applied mathematics, technical computing, and Matlab programming. We will introduce Matlab by way of examples. Many of the experiments involve understanding and modifying Matlab scripts and functions that we have already written. You should have access to Matlab and to our exm toolbox, the collection of programs and data that are described in Experiments with MATLAB. We hope you will not only use these programs, but will read them, understand them, modify them, and improve them.
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MATLAB Programming - Wikibooks MATLAB is a numerical computing environment and fourth-generation programming language. It started out as a matrix programming language where linear algebra programming was simple. It can be run both under interactive sessions and as a batch job. (3405 views) | 677.169 | 1 |
physics
I am having trouble understanding the theory and how to apply it into the equations. People have told me I should not just punch in numbers, but I do not know what to do besides that; I treat it like math.
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Why are variables useful in algebra? I feel the use of variables make it difficult for many people to understand because it is subconcious that math is numbers, not letters.... Thank you! By using a variable, one may make use of a
I am having trouble using modus ponens and all the rules of inherence. By any chance do you guys have a link that can explain these concepts in more detail. I've tried multiple books and links...and haven't found a good website that | 677.169 | 1 |
Welcome to my KOHS Algebra I class. I am very excited to be in the Freshman Campus and teaching Algebra I. We are off to a great start and hope you are ready to jump right in and start working. Parents create your "Gradespeed" account so you can keep up with your students grades.
Jones Tutoring Sessions
Before and after school most days. Check with Mrs. Jones the day before you attend.
Links Page again
The site that has interactive "Gizmos" for Algebra I. You will be assigned lessons to do from this site.
aleks.com **Note** If you need tutoring--parents, here is your answer. Instead of paying $30 an hour, this site is $19 a month and parents can monitor the online time.
PURPLE MATH -- Your Algebra Resource Help with Algebra homework. A good site that explains algebra concepts. Be sure you check this out to review, if you are absent, or if you are having trouble.
Class Procedures and Grading Guidelines Algebra I with Mrs. A. Jones Please see "Classwork/Homework/Tests" tab for the Alg. I Syllabus.
Bring your charged tablet each day along with the following. a) Daily notes of each day's lesson b) All assignments of each six week period c) All quizzes Tests - Math test days are Mondays and Thursdays. If you are absent the day of a test, be prepared to schedule a time to make up the test before or after school. Please bring 2 pencils with erasers and a hand held sharpener to class each day. Sharpening of pencils is restricted to the time BEFORE the tardy bell rings and AFTER homework has been assigned. Please bring your charged tablet and stylus every day. | 677.169 | 1 |
default (namely Firefox, Chrome, Internet Explorer, Opera, and S... read more
The program allows you to solve algebraic equations in the automatic mode. You just enter an equation in any form without any preparatory operations. Step by step Equation Wizard reduces it to a canonical form performing all necessary operations. After that it determines the order of the equation, which can be any - linear, square, cubic or, for instance, of the 7-th power. The program finds the roots of the equation - both real and imaginary.
You just enter an equation you see in your textbook or notebook and click one button! In an instant, you get the step-by-step solution of the equation with the found roots and the description of each step. The solution is completely automatic and does not require any math knowledge from you. Then just print the result or save it to a file.
Besides, the program allows you to simplify math expressions with one variable. Use this feature to speed up your calculations. Equation Wizard is an indispensable assistant for students at university and at school allowing them to save their time and make their learning easier. | 677.169 | 1 |
math
The new edition of Preparation for College Mathematics now covers even more intermediate-level algebraic topics and increases focus on application, conceptual understanding, and the development of the academic mindset. Request an examination copy.
The goal of this newly enhanced title is to develop holistic learners who are adequately prepared for subsequent, higher-level math courses on their path to college success.
2. Integers
Introduction to Integers
Addition with Integers
Subtraction with Integers
Multiplication, Division, and Order of Operations with Integers
Simplifying and Evaluating Expressions
Translating English Phrases and Algebraic Expressions
Solving Equations with Integers (ax + b = c)
3. Fractions, Mixed Numbers, and Proportions
Introduction to Fractions and Mixed Numbers
Multiplication with Fractions
Division with Fractions
Multiplication and Division with Mixed Numbers
Least Common Multiple (LCM)
Addition and Subtraction with Fractions
Addition and Subtraction with Mixed Numbers
Comparisons and Order of Operations with Fractions
Solving Equations with Fractions
Ratios and Rates
Proportions
Probability
4. Decimal Numbers
Introduction to Decimal Numbers
Addition and Subtraction with Decimal Numbers
Multiplication and Division with Decimal Numbers
Estimating and Order of Operations with Decimal Numbers
Statistics: Mean, Median, Mode, and Range
Decimal Numbers and Fractions
Solving Equations with Decimal Numbers
5. Percents
6. Measurement and Geometry
US Measurements
The Metric System: Length and Area
The Metric System: Weight and Volume
US and Metric Equivalents
Angles and Triangles
Perimeter
Area
Volume and Surface Area
Similar and Congruent Triangles
Square Roots and the Pythagorean Theorem
8. Graphing Linear Equations and Inequalities
The Cartesian Coordinate System
Graphing Linear Equations in Two Variables
Slope-Intercept Form
Point-Slope Form
Introduction to Functions and Function Notation
Graphing Linear Inequalities in Two Variables
10. Exponents and Polynomials
Rules for Exponents
Power Rules for Exponents
Applications: Scientific Notation
Introduction to Polynomials
Addition and Subtraction with Polynomials
Multiplication with Polynomials
Special Products of Binomials
Division with Polynomials
Synthetic Division and the Remainder Theorem
Below is information about the Hawkes materials regarding the new Missouri Math Pathways Initiative. We know this is an incredibly important topic of conversation across the state, and our goal is to deliver a curriculum uniquely designed to better prepare students for college-level math in Missouri.
Hawkes Courseware
Chapter projects, simulations, and real-world games promote collaboration and show students the practical side of mathematics through activities using real-world applications of concepts taught. Offerings include new corequisite-ready courses that integrate foundational skills necessary for success in curriculum content.
Check out these two quick videos to learn more:
Mastery Learning:
Explain Error:
Quick Links discussionsSometimes, getting students excited about math isn't easy. Nearly every math instructor has heard "When will I use this in real life?" at least once during their teaching career. Many students don't see right away that they use math just about every day, and you can lose their interest in the subject if you don't connect your course objectives to their lives outside of class. Thankfully, math applies to more fields than most students realize. Here are just a few ways to connect mathematical concepts to other areas and to get students more motivated to learn.
1. Create art with math.
Not all students see how subjects in STEM connect with the liberal arts. Some people mistakenly think the fields are separate and never the two shall meet. One great way to get rid of this misconception is to show how art can be created by using math. Creative Bloq shows eight examples of beautiful fractal art with suggestions on programs to use in order to create your own fractal masterpieces, such as Mandelbulb 3D and FraxHD.
2. Show students how to be fiscally responsible.
Chances are you have some students who don't know much about personal finance beyond having a checking and savings account. Teaching them about budgeting, loans, interest, and more will benefit them now and in all the years to come. Students can start with concepts such as calculating tip and figuring out how much money they save when they buy discounted items before moving on to long-term financial decisions, such as putting a down payment on a house and paying a mortgage.
3. Calculate sports statistics.
Have students who want to be professional athletes, coaches, sports announcers, agents or just die-hard fans of the game? They'll benefit from learning how much math goes into any sport. Everything from calculating batting averages in baseball to knowing touchdowns per pass attempt in football to determining the probability of winning a point in tennis can connect the concepts learned in class to some students' favorite extracurricular activities. Plus, fantasy sports are especially popular, so you may even consider having your class join a fantasy league and see who wins!
Fantasy Sports and Mathematics is a website that includes the latest scores and injuries lists for various sports and sample math problems to use in class. This NYT blog post lists out ways to use sports analytics to teach math and includes additional resources ranging from a video demonstrating what it's like to return a serve in professional tennis to a graphic showing how often football teams go for the fourth down.
4. Delve into the history of mathematics.
Students gain a deeper appreciation of the subject when they know who's behind all those theories, formulas, and discoveries. Plus, they just might connect with the subject more when they know that people from similar demographics advanced the field.
A Buzzle article introduces readers to several achievements of African American mathematicians, ranging from those in the 18th century like Benjamin Banneker to the present day like Dr. William A. Massey.
This Smithsonian.com post highlights five influential female mathematicians throughout history, including Ada Lovelace and Emmy Noether. It gives a little background into these women's lives, explains their accomplishments, and kicks the blatantly false stereotype that women aren't good at math to the curb!
5. Have students write about how they think they'll use math in their future careers.
Are your students still not feeling connected with the course content? Dedicate some class time to brainstorming how they'll use math in the careers they're planning to pursue. While at first some may assume they won't use math at all in their chosen professions, they might surprise themselves once they think a little harder and dig deeper into a job's tasks and expectations. They may want to interview someone in their field via email or phone to get an insider's perspective into the kind of math skills needed to excel in the workplace.
On the blog Math for Grownups, author Laura Laing interviewed several professionals—including writers, academic advisors, and artists—asking them how they use math in their jobs. Her books Math for Grownups and Math for Writers delve into more detail on these topics and encourage folks who are hesitant about math or think they're bad at it to rethink their perspective.
What are some lessons you've taught that encouraged students to apply math to other subjects and think outside the box? Let us know in the comments!
We've updated the default curriculum to include new questions just in time for the spring term. The updated questions provide a more uniform level of difficulty to give the overall lesson more consistency. Most do not require a calculator for your students to complete. In many cases, these questions cover lesson topics more comprehensively than before.
*Please note that these new questions are only available to students using the web platform at learn.hawkeslearning.com.*
To see the new questions, please log into your Assignment Builder to view your curriculum. These questions are marked as new and are always located at the very bottom of the lesson so that they're easy to identify:
Here is a list of the new questions and where to find them:
Developmental Mathematics
Lesson 2.2
Questions 45-50
Lesson 2.3
Questions 10-16
Prealgebra and Introductory Algebra
Lesson 2.4
Questions 21-27
Lesson 2.6
Questions 18-21
Beginning Statistics Plus Integrated Review
Lesson 4.R.1
Questions 45-50
Lesson 4.R.2
Questions 10-16
Viewing Life Mathematically Plus Integrated Review
Lesson 7.R.1
Questions 45-50
Lesson 7.R.2
Questions 10-16
Developmental Math – North Carolina Curriculum
Lesson 2.2
Questions 21-27
Lesson 2.4
Questions 44-50
Foundations of Mathematics for Virginia
Lesson 1.4
Questions 45-50
Lesson 3.3
Based on instructor feedback, we've added one question on the topic of estimating square roots and replaced a few questions with new, more refined problem cases
If you have any questions about these updates, please contact your Training and Support Specialist or call 1-800-426-9538.
According to the National Assessment of Educational Progress (NAEP), high school seniors in the United States haven't improved their reading skills, and their math skills have declined since 2013.
Emma Brown reports, "Eighty-two percent of high school seniors graduated on time in 2014, but the 2015 test results suggest that just 37 percent of seniors are academically prepared for college coursework in math and reading — meaning many seniors would have to take remedial classes if going on to college." | 677.169 | 1 |
0201525Student Solutions Manual for Algebra and Trigonometry: Unit Circle
This paperback text is designed specifically to motivate students to participate--actively and immediately--in the learning process. The text is crafted to meet the varied skill levels of students--giving them solid content coverage in a supportive format. This text also fosters conceptual thinking with exercises, computer/graphing calculator exercises, and a thoroughly integrated five-step problem solving approach. This worktext features a right triangle introduction to trigonometry. | 677.169 | 1 |
BASIC MATH AND PRE-ALGEBRA WORKBOOK FOR DUMMIES
ZEGARELLI M
MATHEMATICS
WILEY
2017
9781119357513 - Tag: 93922
$ 19.99 - Pak Rs. 2,540.33
Basic Math & Pre-Algebra Workbook For Dummies is your ticket to finally getting a handle on math! Designed to help you strengthen your weak spots and pinpoint problem areas, this book provides hundreds of practice problems to help you get over the hump. Each section includes a brief review of key concepts and full explanations for every practice problem, so you'll always know exactly where you went wrong. The companion website gives you access to quizzes for each chapter, so you can test your understanding and identify your sticking points before moving on to the next topic. You'll brush up on the rules of basic operations, and then learn what to do when the numbers just won't behave-negative numbers, inequalities, algebraic expressions, scientific notation, and other tricky situations will become second nature as you refresh what you know and learn what you missed. Each math class you take builds on the ones that came before; if you got lost somewhere around fractions, you'll have a difficult time keeping up in Algebra, Geometry, Trigonometry, and Calculus-so don't fall behind! This book provides plenty of practice and patient guidance to help you slay the math monster once and for all. Make sense of fractions, decimals, and percentages. Learn how to handle inequalities, exponents, square roots, and absolute values. | 677.169 | 1 |
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Learn algebra 1 for free—linear equations, functions, polynomials, factoring, and more full curriculum of exercises and videos. Interactive solvers for algebra word problems homework help by free math tutors, solvers, lessons each section has solvers (calculators), lessons. An adaptive learning system features games and awards help your students embark on a virtual treasure hunt as you tackle math challenges and reveal colorful. If you are in need of help with math instructional videos that offer help with math problems students who are looking for math help with algebra should. Calculus calculator in math calculators precalculus calculator trigonometry calculator what do you need help with statistics calculator. | 677.169 | 1 |
Reviewer's Comments This textbook is recommended for upper level undergraduates and advanced classes at the community college level. It is appropriate for math and science majors. This text is well designed, innovative in its approach, and very adaptable. In contrast to the usual approach, the author introduces examples first, followed by proofs and formal definitions. The exercises are well chosen to expand on the concepts, and answers are provided. Each chapter includes sections on history and applications of the theory. The author also introduces the excellent open source math software Sage. A strong math background is necessary to follow the examples, including some understanding of calculus, linear algebra and number theory.
Reviewer's Comments I recommend this book for undergraduates. The content is especially useful for those in finance, probability statistics, and linear programming. The course material is consistent, and the use of money, coins, cards, and marbles make it relevant for all races and ethnicities.
Reviewer's Comments I think this resource is extremely appropriate as an adjunct teaching aid for a course in geometry. It is a robust resource for the exploration of some very interesting topics in geometry. "Dimensions" is not actually a textbook but a multimedia presentation using two formats--a video format and a text that is presented in a web format, which is very readable.
Reviewer's Comments I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job in presenting the concepts of terms and factors, as well as other important algebraic concepts. The PDF file should include links.
Reviewer's Comments I recommend this book for junior and senior level students with advanced mathematical maturity majoring in mathematics, physics, and all branches of engineering. The core of the text is excellent. The writing style is friendly and conversational. The common sense segues into abstract material which aids comprehension without sacrificing rigor. Archetypes are used throughout. The author did an outstanding job designing those. They serve as rich examples that guide students in investigating the major concepts of linear algebra. The core is modular and can be modified based on class level.
Reviewer's Comments I would recommend this text for a basic math course for students moving on to elementary algebra. The information in most chapters is useful, very clear, and easily comprehended by most students. The reading level is appropriate to a first-year college student. While I recommend some organizational changes, something as simple as including ethnically diverse first names in word problems can indeed appeal to students of many races
Reviewer's Comments I recommend this book, but not as the primary text. The text is well written and mathematically accurate, and provides in-depth coverage of the essential materials of most intermediate algebra courses. I would suggest it as a reference/supplementary text in an online class.
Reviewer's Comments I recommend this book for community or technical college, year 2, non-math, liberal arts majors. This text provides rich discussion of many topics where students may not consider mathematical thinking is involved. It is good primary text for various topics in mathematics and mathematical decision-making for novice and liberal arts students in community and technical colleges. The exercise sets contain "Skills", "Concepts", and "Exploration" divisions--which adds to clarity and comprehensibility.
Reviewer's Comments I recommend this textbook for beginning algebra students. The author's writing style is very readable, making the chapters very clear and comprehensible. Content is very appropriate and useful. While I recommend some organizational and content changes in several of the chapters, I would also include text problem examples that use names of students from diverse ethnicities.
Reviewer's Comments I would definitely use this excellent book to teach an undergraduate course. It is very well written, makes good use of exercises, and its use of visuals to bring the concepts to life is just magnificent. The PDF format is very readable. The author does a very good job of smoothly extending what the student already knows in one dimension to two and three-dimensional Euclidean spaces. The content gets more sophisticated and complex so it should be geared towards college juniors or seniors. | 677.169 | 1 |
Mathematics
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Image Caption
Page Content
Mathematics provides students with essential skills and
knowledge.It develops numeracy
capabilities that all students need in their personal, work and social life,
and provides the fundamentals on which mathematical specialties and
professional applications of mathematics are built. | 677.169 | 1 |
Following on the success of the Algebra Survival Guide, the Algebra Survival Guide Workbook presents thousands of practice problems (and their answers ... more ») to help children master algebra. The problems are keyed to the pages of the Algebra Survival Guide, so that children can find detailed instructions and then work the sets. Each problem set focuses like a laser beam on a particular algebra skill, then offers ample practice problems. Answers are conveniently displayed in the back. This book is for parents of schooled students, homeschooling parents and teachers. Parents of schooled children find that the problems give their children a "leg up" for mastering all skills presented in the classroom. Homeschoolers use the Workbook - in conjunction with the Guide - as a complete Algebra 1 curriculum. Teachers use the workbook's problem sets to help children sharpen specific skills - or they can use the reproducible pages as tests or quizzes on specific topics. Like the Algebra Survival Guide, the Workbook is adorned with beautiful art and sports a stylish, teen-friendly design Former Library bookReturns Policy
Returns are accepted within 14 days of the date that you received your order. Books, DVDs and CDs (in their original, unopened, shrink-wrapped condition), and Toys & Games should be posted with the original packing slip | 677.169 | 1 |
Los Angeles City College Math Contest for High School and Middle School Students, since 1951
The next LACC Math Contest will be held on March 3, 2018.
About the Contest
LACC invites high school students to enter our Annual Mathematics Contest to be held on our campus in March. This is an opportunity for you to win money, a scholarship, and recognition.
The problems to be solved in this competition require a solid algebra and geometry foundation. These problems challenge the student's ability to think and synthesize rather than recall standard solution techniques from their courses. We believe this will give students with different backgrounds an equal footing. Knowledge of trigonometry and calculus is not required for almost all questions. There may occasionally be a problem requiring knowledge of trigonometry or calculus.
Please understand that calculators are NOT allowed to be used during the contest.
Minimum Qualifications to Enter the Contest
The contest is open to all junior high and high school students in Southern California. | 677.169 | 1 |
My College Algebra students can also benefit from my project structures. I have more flexible goals with this group of students, and a main feature is letting them have as much class time as possible to get work done. I set up very brief lessons and let them spend time working. Most of the material is not new and students in here could use more reps.
We spent the first month of the school year talking about linear systems, quadratic systems, and radical equations. In all cases my goal was to show them the importance of a graph and how it related to work they do by hand. Students were to create 3 problems for a set of 4 possibilities: a linear system, a quadratic system with real solutions, a quadratic system with non-real solutions, and a radical equation. In all cases they stated the problem, did the work by hand, and graphed the equation to prove their work. Then they explained their process.
Students were able to use previous classwork as a starting point if they weren't sure how to make up a problem. For most of them they had rarely, if ever, been asked to do something like this. As we have progressed, students have been persevering through their work because checking themselves is so accessible. They don't need me to share an answer key, they have the ability to do it on their own.
As with Pre-Cal and Calculus, I got a lot of variety in the kind of work students turned in and all of them had great conversations along the way thinking about how to represent the situations they chose. | 677.169 | 1 |
Description
"Math Principals for Food Service Occupations, 4th Edition" teaches students that the understanding and application of mathematics is critical for all food service jobs, from a salad person to an executive chef. All the mathematics problems and concepts presented are done so in a simplified, logical, step-by-step manner. In this 4th edition, "Chef Sez," quotes from chefs and managers, have been added to show students just how applicable these skills are to food service professionals. TIPS - To Insure Perfect Solutions - have been included to provide hints on how to make problem solving simple. Learning objectives have also been added at the beginning of each chapter to identify the key information to be learned. The content, including the answer key in the Instructor's Manual, has been completely revised to ensure accuracy and relevancy.show more
Table of contents
Pretest: Math Skills. PART I: Using the Calculator. PART II Review of the basic math fundamentals: Numerals, Symbols of Operations, and the Mill. Addition, Subtraction, Multiplication, and Division. Fractions, Decimals, Ratios. (Part contents). | 677.169 | 1 |
Finite Mathematics for the Managerial, Life, and Social Sciences, 11th Edition
Math plays a vital role in our increasingly complex daily life. Finite Mathematics for the Managerial, Life, and Social Sciences attempts to illustrate this point with its applied approach to mathematics. Students have a much greater appreciation of the material if the applications are drawn from their fields of interest and from situations that occur in the real world. This is one reason you will see so many exercises in my texts that are modeled on data gathered from newspapers, magazines, journals, and other media. In addition, many students come into this course with some degree of apprehension. For this reason, I have adopted an intuitive approach in which I try to introduce each abstract mathematical concept through an example drawn from a common life experience. Once the idea has been conveyed, I then proceed to make it precise, thereby ensuring that no mathematical rigor is lost in this intuitive treatment of the subject.
The only prerequisite for understanding this text is one to two years, or the equivalent, of high school algebra. This text offers more than enough material for a onesemester or two-quarter course. The following chapter dependency chart is provided to help the instructor design a course that is most suitable for the intended audience.
The approach
Presentation
Consistent with my intuitive approach, I state the results informally. However, I have taken special care to ensure that mathematical precision and accuracy are not compromised. Motivation
Illustrating the practical value of mathematics in applied areas is an objective of my approach. Concepts are introduced with concrete, real-life examples wherever appropriate.
These examples and other applications have been chosen from current topics and issues in the media and serve to answer a question often posed by students: "What will I ever use this for?" Problem-solving emphasis
Special emphasis is placed on helping students formulate, solve, and interpret the results of applied problems. Because students often have difficulty setting up and solving word problems, extra care has been taken to help them master these skills:
■ Very early on in the text, students are given practice in solving word problems.
■ Guidelines are given to help students formulate and solve word problems.
■ One entire section is devoted to modeling and setting up linear programming problems. Modeling
One important skill that every student should acquire is the ability to translate a real-life problem into a mathematical model. In Section 1.3, the modeling process is discussed, and students are asked to use models (functions) constructed from real-life data to answer questions. Additionally, students get hands-on experience constructing these models in the Using Technology sections. | 677.169 | 1 |
The 1947 paper by John von Neumann and Herman Goldstine, "Numerical Inverting of Matrices of High Order" ( Bulletin of the AMS , Nov. 1947), is considered as the birth certificate of numerical analysis. Since its publication, the evolution of this domain has been enormous. This book is a unique collection of contributions by researchers... more...
Introduction to Mathematical Proofs helps students develop the necessary skills to write clear, correct, and concise proofs. Unlike similar textbooks, this one begins with logic since it is the underlying language of mathematics and the basis of reasoned arguments. The text then discusses deductive mathematical systems and the systems of natural... more...
A solutions manual to accompany An Introduction to Numerical Methods and Analysis, Second Edition An Introduction to Numerical Methods and Analysis, Second Edition reflects the latest trends in the field, includes new material and revised exercises, and offers a unique emphasis on applications. The author clearly explains how... more...
Plot graphs, solve equations, and write code in a flash! If premier data tool, and MATLAB For Dummies... more...
Topics in Multivariate Approximation contains the proceedings of an international workshop on multivariate approximation held at the University of Chile in Santiago, Chile, on December 15-19, 1986. Leading researchers in the field discussed several problem areas related to multivariate approximation and tackled topics ranging from multivariate splines... more...
A New Approach to Scientific Computation is a collection of papers delivered at a symposium held at the IBM Thomas J. Watson Research Center on August 3, 1982. The symposium provided a forum for reviewing various aspects of an approach to scientific computation based on a systematic theory of computer arithmetic. Computer demonstration packages for... more...
Computer Science and Applied Mathematics: Iterative Solution of Nonlinear Equations in Several Variables presents a survey of the basic theoretical results about nonlinear equations in n dimensions and analysis of the major iterative methods for their numerical solution. This book discusses the gradient mappings and minimization, contractions and... more...
Basic Numerical Mathematics, Volume II: Numerical Algebra focuses on numerical algebra, with emphasis on the ideas of "controlled computational experiments" and "bad examples". The existence of an orthogonal matrix which diagonalizes a real symmetric matrix is highlighted, and partitioned or block matrices are discussed, along with induced norms and... more...
Numerical Analysis is an elementary introduction to numerical analysis, its applications, limitations, and pitfalls. Methods suitable for digital computers are emphasized, but some desk computations are also described. Topics covered range from the use of digital computers in numerical work to errors in computations using desk machines, finite difference... more... | 677.169 | 1 |
14 Images of Printable Worksheets Algebra 1
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Trigonometry: A Graphing Approach
As part of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, Trigonometry: A Graphing Approach, 4/e, provides both ...Show synopsisAs part of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, Trigonometry: A Graphing Approach, 4/e, provides both students and instructors with a sound mathematics course in an approachable, understandable format. The quality and quantity of the exercises, combined with interesting applications, cutting-edge design, and innovative resources, make teaching easier and help students succeed in mathematics. This edition, intended for trigonometry courses that require the use of a graphing calculator includes a moderate review of algebra to help students entering the course with weak algebra skills | 677.169 | 1 |
Angel, Allen RAn emphasis on the practical applications of algebra motivates readers and encourages them to see algebra as an important part of their daily lives. Strongly emphasizes good problem-solving skills, uses real-world applications. For anyone interested in Algebra | 677.169 | 1 |
Introduction to Difference Equations by Samuel Goldberg
"The highest standards of logical clarity are maintained." — Bulletin of The American Mathematical Society Written with exceptional lucidity and care, this concise text offers a rigorous introduction to finite differences and difference equations-mathematical tools with widespread applications in the social sciences, economics, and psychology. The exposition is at an elementary level with little required in the way of mathematical background beyond some facility with standard algebraic techniques and the essentials of trigonometry. Moreover, the author explains when needed such relevant ideas as the function concept, mathematical induction, binomial formula, de Moivre's Theorem and more. The book begins with a short introductory chapter showing how difference equations arise in the context of social science problems. Chapter One then develops essential parts of the calculus of finite differences. Chapter Two introduces difference equations and some useful applications in the social sciences: compound interest and amortization of debts, the classical Harrod-Domar-Hicks model for growth of national income, Metzler's pure inventory cycle, and others. Chapter Three treats linear differential equations with constant coefficients, including the important question of limiting behavior of solutions, which is discussed and applied to a variety of social science examples. Finally, Chapter Four offers concise coverage of equilibrium values and stability of difference equations, first-order equations and cobweb cycles, and a boundary-value problem. More extensive coverage is devoted to the relatively advanced concepts of generating functions and matrix methods for the solution of systems of simultaneous equations. Throughout, numerous worked examples and over 250 problems, many with answers, enable students to test their grasp of definitions, theorems and applications. Ideal for an undergraduate course or self-study, this cogent treatment will be of interest to all mathematicians, and especially to social scientists, who will find it an excellent introduction to a powerful tool of theory and research.
First published in 1914, as the second edition of a 1909 original, this book forms
number ten in the Cambridge Tracts in Mathematics and Mathematical Physics series. It was written to provide readers with 'the main portions of the theory ...
Everything you need to understand both Laruelle's critique of difference and his project of non-philosophyGilles
Deleuze described Laruelle's thought as 'one of the most interesting undertakings of contemporary philosophy'. Now, Rocco Gangle - who translated Laruelle's philosophy into English - ...
This book by a prominent mathematician is appropriate for a single-semester course in applied numerical
analysis for computer science majors and other upper-level undergraduate and graduate students. Although it does not cover actual programming, it focuses on the applied topics ...
Written for the undergraduate who has completed a year of calculus, this clear, skillfully organized
text combines two important topics in modern mathematics in one comprehensive volume.As Professor Dettman (Oakland University, Rochester, Michigan) points out, Not only is linear algebra ...
The ultimate aim of the field of numerical analysis is to provide convenient methods for
obtaining useful solutions to mathematical problems and for extracting useful information from available solutions which are not expressed in tractable forms. This well-known, highly respected ...
This volume offers an excellent undergraduate-level introduction to the main topics, methods, and applications of
partial differential equations.Chapter 1 presents a full introduction to partial differential equations and Fourier series as related to applied mathematics. Chapter 2 begins with a ... | 677.169 | 1 |
When I started giving tuition for A Maths, I went to Popular bookstore to find a book for revision. I was overwhelmed by the large selection of assessment books available and couldn't find the 'right' book to buy. All of them looked the same!
Over time, after more research and receiving recommendations from my students, I have built up a fairly large collection of A Maths books.
The books listed here are the 'recommended' ones from my collection. They are categorised as follows:
When going through the books listed, ask yourself if you need the book. No point in buying one that you wouldn't use!
Disclaimer: I have no relationship with any publisher and opinions expressed are entirely my own.
Quick Revision Guide - To Prepare for Tests & Exams
A revision guide summarizes all concepts covered, such as formulas, theorems and shapes of graphs. Examples are provided to show students how to apply the concepts. Since the guide is designed for quick reference and revision, no practice questions are provided.
I recommend students to have a copy as there's a lot of concepts to remember for A Maths. If you have the time and want to save money, you can make your own revision guide.
Alternatively, two options are available for purchase. Some schools may buy in bulk for all students, so check with your teacher before purchasing.
Ten-Year-Series (TYS) - Past-year O Level Questions
The TYS is a collection of O Level examination questions from the past ten years. Full solutions to the question are provided.
All students should practice questions in the TYS once to familiarise themselves with the concepts. However, do not memorise model answers as questions are unlikely to be repeated. Focus on understanding the question and applying the right concepts.
For A Maths, TYS come in two versions: Yearly and Topical. In the Topical version, questions are grouped by topic rather than the year of examination.
Topical TYS From Shing Lee Publishers
Schools usually require students to purchase the TYS (sometimes both the Topical and Yearly version). Check with your teacher before buying it on your own.
If you don't have one from school, I recommend the Topical version as you can choose your weak topics to focus on.
Look for the version published by Shing Lee Publishers as the questions are furthered grouped according to their difficulty.
Topical Assessment Book - For More Practice!
If you need more practice, look for topical assessment books. The following books are available in Popular and comes with full worked solutions, sometimes in a separate booklet.
Before purchasing, I suggest heading to Popular and look through the books. It is important to find one that you are comfortable with.
#1: Mentor Additional Mathematics
Shinglee Mentor Series
This book provides a lot of practice questions. On average, there's a minimum of 20-30 questions for each topic.
Furthermore, questions are arranged in levels of increasing difficulty: Basic, Intermediate and Advanced. Start with the 'Basic' questions to improve your understanding of the concepts. Once confident, attempt the 'Intermediate' and 'Advanced' questions.
Revision notes and examples are provided at the start of each chapter as well. In particular, examples are annotated with comments to explain the reasoning behind key steps.
Finally, I don't recommend purchasing assessment books that compile mock examination papers as school teachers usually provide many past-year exam papers from other schools. If you need more papers to practice, you can find them online (I have compiled some links here). | 677.169 | 1 |
Key Study Guide Grade 9
6 2 study guide solving quadratic equations by factoring answers
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