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College Algebra - 5th edition
Summary: Learn to think mathematically and develop genuine problem-solving skills with Stewart, Redlin, and Watson's COLLEGE ALGEBRA, Fifth Edition. This straightforward and easy-to-use algebra book will help you learn the fundamentals of algebra in a variety of practical ways. The book features new tools to help you succeed, such as learning objectives before each section to prepare you for what you're about to learn, and a list of formulas and key concepts after each section that help reinf...show moreorce what you've learned. In addition, the book includes many real-world examples that show you how mathematics is used to model in fields like engineering, business, physics, chemistry, and biology. ...show less
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TH214 Modeling of the Environment
Course Description
A course offering a thorough and useful beginning-level understanding of mathematical modeling. Examines diverse applications from the physical, biological, business, social and computer sciences. Discusses the limitations, as well as the capabilities, of models applied in understanding the real world and its inhabitants.
Learning Outcomes
Enter data into the computer and plot equations. The student will also determine variable dependency and characterize data.
Solve equations on the computer, substitute data points into equations, and manipulate the equations, and manipulate the equations on the screen.
Understand the concept of a linear model on the computer to describe appropriate data.
Formulate an appropriate quadratic model to describe carbon dioxide emissions from autos and power consumption in the U.S. or other appropriate problems.
Understand the concept of a quadratic relationship and the difference to other models. The student will be able to develop a quadratic model on a computer to describe appropriate data.
Understand the concepts of exponential and logarithmic relationships and their difference from other models.
Summarize models to environmental or economic problems such as those connecting carbon dioxide emissions, people and money, models of the gross national product, population growth, and the efficiency of alternative energy sources developed.
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An excellent, very comprehensive study guide containing all the maths you should be able to do before moving on to your final year in school. A MUST-HAVE FOR ALL GRD. 11 & 12 LEARNERS. The book includes lots of questions and answers on Number Patterns; Increase & decrease (Growth and Decay) Graphs of functions; Mathematical modelling; Length, area & volume; Co-ordinate Geometry; Transformation Geometry; Data handling & Probability. Optional work will be published in a separate book.
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Algebra I Workbook for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
From signed numbers to story problems — calculate equations with ease Practice is the key to improving your algebra skills, and that's what this workbook is all about. This hands-on guide focuses on helping you solve the many types of algebra... (read more)
Elementary and Intermediate Algebra (11 Edition)
by Jay Lehmann Publisher Comments
Unique in its approach, the Lehmann Algebra Series uses curve fitting to model compelling, authentic situations, while answering the perennial question "But what is this good for?" Lehmann begins with interesting data sets, and then uses the... (read more)
Practical Algebra
by Peter H. Selby Publisher Comments
Practical Algebra If you studied algebra years ago and now need a refresher course in order to use algebraic principles on the job, or if you're a student who needs an introduction to the subject, here's the perfect book for you. Practical... (read more)
Basic Algebra II: Second Edition
by Nathan Jacobson Publisher Comments
Volume II of a pair of classic texts — and standard references for a generation — this book comprises all of the subjects of first-year graduate algebra. In addition to the immediate introduction and constant use of categories and functors,... (read more)
The Math Problems Notebook
by Valentin Boju Publisher Comments
This volume offers a collection of non-trivial, unconventional problems that require deep insight and imagination to solve. They cover many topics, including number theory, algebra, combinatorics, geometry and analysis. The problems start as simple... (read more)
Algebra (Graduate Texts in Mathematics #73)
by Thomas W. Hungerford Publisher Comments
Finally a self-contained, one volume, graduate-level algebra text that is readable by the average graduate student and flexible enough to accommodate a wide variety of instructors and course contents. The guiding principle throughout is that the material... (read more)
College Algebra (5TH 11 Edition)
by Mark Dugopolski Publisher Comments
This package consists of the textbook plus an access kit for MyMathLab/MyStatLab. Dugopolski's College Algebra, Fifth Edition gives readers the essential strategies to help them develop the comprehension and confidence they need to be... (read more)
Algebra II Workbook for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
From radical problems to rational functions -- solve equations with ease Do you have a grasp of Algebra II terms and concepts, but can't seem to work your way through problems? No fear -- this hands-on guide focuses on helping you solve the many types of... (read more)
Algebra Workbook for Dummies (05 - Old Edition)
by Mary Jane Sterling Publisher Comments
From signed numbers to story problems — calculate equations with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear — this hands-on-guide focuses on... (read more)
Vector and Tensor Analysis with Applications
by A. I. Borisenko Publisher Comments
Concise and readable, this text ranges from definition of vectors and discussion of algebraic operations on vectors to the concept of tensor and algebraic operations on tensors. It also includes a systematic study of the differential and... (read more)
College Algebra (4TH 07 - Old Edition)
by Mark Dugopolski Publisher Comments
Dugopolski's College Algebra, Fifth Edition gives readers the essential strategies to help them develop the comprehension and confidence they need to be successful in this course. Readers will find enough carefully placed learning aids and review... (read more)
Painless Pre-Algebra (Barron's Painless)
by Amy Stahl Publisher Comments
(back cover) Really. This won't hurt at all . . . The thought of having to learn pre-algebra once turned brave students into cowards . . . but no more! THE PAIN VANISHES WHEN YOU TRANSFORM PRE-ALGEBRA INTO FUN-- Learn how to solve fun number puzzles by... (read more)
The Manga Guide to Linear Algebra
by Shin Takahashi Publisher Comments
Reiji wants two things in life: a black belt in karate and Misa, the girl of his dreams. Luckily, Misa's big brother is the captain of the university karate club and is ready to strike a deal: Reiji can join the club if he tutors Misa in linear algebra
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iTech Troubleshooter | 2010-12-09 | Software In the algebraic specification technique an object class or type is specified in terms of relationships existing between the operations defined on that type. It was first brought into prominence by Guttag [1980, 1985] in specification of abstract data types. Various notations of algebraic specifications have evolved, including those based on OBJ and Larch languages. Representation of algebraic specification Essentially, algebraic specifications define a system as a heterogeneous algebra. A het read more
By: Joe Pagano | 2010-03-27 | Reference & Education Now that we understand some key algebraic terminology, we are prepared to recognize some special products and to be able to factor them accordingly. Herein we master how to recognize and factor both differences of perfect squares and perfect square trinomials Jacker Martyn | 2011-04-09 | Business Algebra is one of the most important fields of mathematics. Majority of the students find algebra tough. This is a result of their disinterest towards the subject or due to the way in which they are t... read more
By: timcyhood | 2010-12-22 | Science Pre Algebra, a course in middle school curriculum of mathematics is introduced, to brush the skills, and prepare the student for the study of Algebra, that would be helpful for student. read more
By: Richard Morrisson | 2011-01-22 | Reference & Education Pre algebra being the foundation of algebra can help one in climbing the ladder of success without coming across any hurdles in the way. read more
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JMAPís Lesson Plans are intended as foundation documents for teachers who wish to develop their own lesson plans. JMAP's Lesson Plans summarize many of the big ideas associated with the various strands of mathematics covered in New York Stateís
Integrated Algebra, Geometry and Algebra 2 & Trigonometry curricula and provide recommendations for classroom dialogue and student activities. They were developed primarily for Regents prep and review classes, but can be adapted to a variety of teaching needs. Their primary advantages are:
1) They are grounded in past Regents examinations and include problem sets taken directly from previous examinations, thus facilitating the alignment of instruction and statewide assessment; and
2) They are
provided free of charge in DOC format, thus facilitating distribution and
revision to suit individual teaching and learning styles.
JMAP's Lesson Plans may be adapted for use with a wide variety of texts and pedagogical styles, and JMAP encourages teachers to revise them in ways that will be meaningful to their individual students and classes. JMAP's Lesson Plans may be used for any non-profit purposes including classroom instruction, curriculum development, home schooling, and teacher education courses. When appropriate, we ask only that you cite the Jefferson Math Project as a resource. All materials used in the development of these lesson plans are believed to belong to the public domain.
The Jefferson Math Project promotes free distribution of high quality
mathematics education resources as a matter of social justice.
JMAP resources in Word and Adobe format may be accessed
using a Mac computer that has Microsoft Word, Adobe Reader or comparable
programs installed. JMAP resources in tst format are compatible with
the Mac version 6.x of ExamView.
The Worksheet Builder and Saxon programs
downloadable from JMAP cannot be installed on a Macintosh computer.
Therefore JMAP resources in Worksheet Builder format may not be accessed
using a Mac computer.
However, you may purchase a program,
Parallels Desktop, that allows you to run
Windows on a Mac computer. JMAP is edited with a MacBook OS X Tiger
donated by
Electra Information Systems with Parallels Desktop installed using Windows XP.
The new version of Mac's operating system, Leopard, includes a free
application called
Boot Camp, which also allows you to run Windows on a Mac.
The RCTs in all subjects are subject to strict control requirements that prevent their being copied and published. The
State Math Tests, CSTs and SATs are protected by copyright law.
JMAP does offer
SAT Prep Exercises based upon the chapters in
Amsco's Integrated Algebra textbook.
EXAMVIEW (TST) -
(INTEGRATED) - These files are the most concise versions of the NY Regents exams available, and include
the original graphs, charts and diagrams.
WORKSHEET BUILDER (WS) - (INTEGRATED) - Each exam fits on 5-6 pages (2 columns and 1/2" margins), and includes the original graphs, charts and diagrams. Worksheet Builder can automatically create a separate answer sheet, and the answers keys are integrated as well.
The designations of JMAP represent the number of Regents Exam/Test
Sampler questions
contained in that iteration of JMAP. For example, the first iteration was
JMAP 611, with 611 questions from Math A Regents Exams up to and including
January 2005.
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This class is designed to provide a review of the concepts and applications of polynomials. This class will use examples commonly found in a college Beginning Algebra course.
This class will be beneficial to students currently enrolled or planning to enroll in a beginning, intermediate, or college algebra course. This class will also be advantageous for anyone preparing for an assessment or standardized exam or simply looking to refresh their algebra skills.
After this class the student will be able to: 1) use the rules of exponents to simplify; 2) compute scientific notation; 3) convert between standard form and scientific notation; 4) identify the term and degree of a polynomial
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Computational Mathematics Colleges
A program that focuses on the application of mathematics to the theory, architecture, and design of computers, computational techniques, and algorithms. Includes instruction in computer theory,cybernetics, numerical analysis, algorithm development, binary structures, combinatorics, advanced statistics, and related topics
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Investigate logistic functions in a world population setting. High schoolers will create a scatter plot of the world population from 1950 to 2050 to find a logistic function to model the data. They then discuss the end behavior of their logistic model. Graphing calculators are needed.
Throught this subscription-based sight, learners explore different aspects of the parabola by changing equations from standard to vertex form. Next, find the general form of the vextex based on the values of a, b, and c, and investigate the minimum and maximum points of a real-world example. Students can gain further insight by looking closer at the process of completing the square.
High schoolers explore the vector model of projectile motion and then derive the parametric model from the vector model. Many formulas will be explored and used to solve a real-world situation. The class will be guided through solving a similar problem on the TI-89.
Students investigate and identify the maximum and minimum of their graphs. In this calculus lesson, students graph their equations and find the highest and lowest points. They relate the max and min to the real world.
Students explore the various uses of the logistic function. Students use the internet to collect world population data and find a logistic model for their data and use their chosen model to predict future populations.
Students investigate inverse functions. In this calculus activity, students use the horizontal line test to determine if a functions inverse is a function. They define functions given a graph or an equation.
Learners calculate the revolution and circumference of circles. In this calculus lesson, students derive the formulas for their given shape. They use the d=rt to calculate the distance and rate of travel.
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Using Textbook Outlining to Empower Students to Become More Active Learners
Laura Graff: lgraff@collegeofthedesert.edu
Dustin Culhan: dculhan@collegeofthedesert.edu
Felix Marhuenda-Donate: fmarhuenda@collegeofthedesert.edu
When they enroll in a History, Government, or Psychology class, students expect to attend lectures and take notes, read the textbook, and study questions the instructor provides to help them prepare for exams. They do not expect the exam questions to be identical to the study questions. Rather, they expect the exam to ask new questions that allow them to show how they have incorporated the information from the lecture and the textbook along with their own ideas to make their own connections.
Yet, for some reason, these same attitudes and expectations do not seem to apply to math class. Students approach the math textbook as little more than an (extremely expensive) problem set, expecting to get all of the information they need to prepare for tests simply by attending lecture. A typical college math course requires a great deal of homework, and students are expected to spend many hours outside of class studying. When students lack the ability to use their textbook as a learning tool, the results -- low test scores and poor retention and success rates -- can be frustrating for students and teachers alike.
In addition to the above difficulties are these depressing facts:
At College of the Desert, 92 percent of all incoming students place into a remedial level mathematics course (Intermediate Algebra or below).
Also shocking is the fact that a full 67 percent of students start their college mathematics careers seated in a basic arithmetic course.
Retention rates are dismal enough to reduce even the most hardened classroom veteran to tears.
In an effort to turn back this wave of despair, a trio of math professors at College of the Desert has incorporated the idea of outlining math textbooks into their courses. By getting students in the habit of really using their textbooks, outlining helps them gain a deeper knowledge of the material that, in turn, enables them to make their own connections between ideas. From passive listeners, students become independent and active learners.
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Beaver Springs Calculus. To me, Algebra 2 was basically the more complex, more intricate following to Algebra 1. The same subject, but with longer, complex numbers and "drawn out" equations, primarily d...
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0840048149
9780840048141
113317101X
9781133171010, delivering matchless flexibility to both traditional and modern practitioners. The Tenth Edition, as with previous editions, is suitable for both majors and non-majors alike. The text abounds with helpful examples, exercises, applications, and features to motivate further exploration. Portfolio profiles highlight the way actual professionals use math in their day-to-day business. Newly enhanced technology sections provide step-by-step instructions for solving examples and exercises in Excel 2010. Supported by a powerful array of supplements including Enhanced WebAssign, the new Tenth Edition enables students to make full use of their study time and maximize their chances of success in class. «Show less,... Show more»
Rent Finite Mathematics for the Managerial, Life, and Social Sciences 10th Edition today, or search our site for other Tan
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Dublin, GA Geometry...In this subject, the student will learn probability in terms of the basic definition, the binomial distribution, and the normal distribution. Probability is used in inferential statistics. ACT Math basically consists of every branch of Mathematics except for CalculusMost five.
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Fr...
For courses in secondary or middle school math. This text focuses on all the complex aspects of teaching mathematics in today's classroom and the most current NCTM standards. It demonstrates how to creatively incorporate the standards into teaching along with inquiry-based instructional strateg...
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Gary Rockswold teaches algebra in context, answering the question, ?А?Why am I learning this??А? By experiencing math through applications, students see how it fits into their lives, and they become motivated to succeed. Rockswold focus on conceptual understanding helps students make connections between the concepts and as a result, students see the bigger picture of math and are prepared for future courses.
COLLEGE ALGEBRA AND CALCULUS: AN APPLIED APPROACH, the newest addition to the Larson Applied math family, allows your students to feel confident moving from college algebra to applied calculus. The mathematical concepts and applications are consistently presented in the same tone and pedagogy. The combination of these two subjects into one text will not only save you time in your course, but it will also save your students the cost of an extra textbook.
Elementary and Intermediate Algebra: A Combined Approach, 5th Edition Master algebraic fundamentals with Kaufmann/Schwitters Elementary and Intermediate Algebra 5e. Learn from clear and concise explanations, multiple examples and numerous problem sets in an easy-to-read format. The text's 'learn, use & apply' formula helps you learn a skill, use the skill to solve equations, then apply it to solve application problems. With this simple, straightforward approach, you will grasp and apply key problem-solving skills necessary for success in future mathematics courses.
Algebra Readiness Made Easy: Grade 2 2.
Algebra Readiness Made Easy: Grade 1 1.
Every lesson in McDougal Littell Math Course 3 has both skill practice and problem solving, including multi-step problems. These type of problems often appear on standardized test and cover a wide variety of math topics. To help students prepare for standardized tests, McDougal Littell Math Course 3 provides instruction and practice on standardized test questions in many formats – multiple choice, short response, extended response, and so on. The number and variety of problems, ranging from basic to challenging, give students the practice they need to develop their math skills.
King Arthur was a good ruler, but in this math adventure he needs a good ruler. Geometry is explained with humor in Sir Cumference and the First Round Table, making it fun and accessible for beginners. What would you do if the neighboring kingdom were threatening war? Naturally, you'd call your strongest and bravest knights together to come up with a solution. Enter Sir Cumference, his wife Lady Di of Ameter, and their son Radius. Thanks to them, even the most hesitant will be romancing math. Reading Level: Grade 2-5
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Weekly quizzes will be given on the following days. (Other "pop" quizzes will also be given)
Tuesday: Geometry
Wednesday: Trigonometry/Pre-Calculus
Thursday: AP Calculus
We will use graphing Calculator technology. We currently use TI-84's in the classroom. Unfortunatley the students are not allowed to remove these calculators from the classroom. (A TI-84 would make a great gift!)
In AP Calculus I will issue a TI-89 to students to take come and care for for the semester. (If you are headed for a math or science major it would be worth your investment to buy your own TI-89)
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Introductory & Intermediate Algebra for College Students, 4Chapter 3 Group Project
Chapter 3 Summary
Chapter 3 Review Exercises
Chapter 3 Test
Cumulative Review Exercises (Chapters 1–3)
4. Systems of Linear Equations Equations in Three Variables Long Division of Polynomials; Synthetic Division
Mid-Chapter Check Point Section 7.1–Section 7.4
7.5 Complex Rational Expressions
7.6 Solving Rational Equations
7.7 Applications Using Rational Equations and Proportions
7.8 Modeling Using Variation
Chapter 7 Group Project
Chapter 7 Summary
Chapter 7 Review Exercises
Chapter 7 Test
Cumulative Review Exercises (Chapters 1–7)
8. Basics of Functions
8.1 Introduction to Functions
8.2 Graphs of Functions
8.3 The Algebra of Functions
Mid-Chapter Check Point Section 8.1–Section 8.3
8.4 Composite and Inverse Functions
Chapter 8 Group Project
Chapter 8 Summary
Chapter 8 Review Exercises
Chapter 8 Test
Cumulative Review Exercises (Chapters 1–8)
9. Inequalities and Problem Solving
9.1 Reviewing Linear Inequalities and Using Inequalities in Business Applications
9.2 Compound Inequalities
9.3 Equations and Inequalities Involving Absolute Value
Mid-Chapter Check Point Section 9.1–Section 9.3
9.4 Linear Inequalities in Two Variables
Chapter 9 Group Project
Chapter 9 Summary
Chapter 9 Review Exercises
Chapter 9 Test
Cumulative Review Exercises (Chapters 1–9)
10. Radicals, Radical Functions, and Rational Exponents
10.1 Radical Expressions and Functions
10.2 Rational Exponents
10.3 Multiplying and Simplifying Radical Expressions
10.4 Adding, Subtracting, and Dividing Radical Expressions
Mid-Chapter Check Point Section 10.1–Section 10.4
10.5 Multiplying with More Than One Term and Rationalizing Denominators
10.6 Radical Equations
10.7 Complex Numbers
Chapter 10 Group Project
Chapter 10 Summary
Chapter 10 Review Exercises
Chapter 10 Test
Cumulative Review Exercises (Chapters 1–10)
11. Quadratic Equations and Functions
11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
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Applications of Calculus and Numerical Methods
Applications of Calculus and Numerical Methods
Hi, i'm a new user of this forum and i wanted to know how i can use calculus and interpolation in my projects.
I'm a student in engineering and i usually do home made electronics and computer programming projects, i would like to include calculus to my projects and understand the practical side of what I'm studying but sometimes in calculus and linear algebra for me it's diffcult to understand the practical aspects of some arguments or theorems.
Math for me is really enjoyable when i understand how i can apply the concepts to my projects and my work, i wanted to know : how can i use calculus for my electronics and computer engineering projects ?
And why scientists and engineers use interpolation instead of keeping tables ?
Does anybody know a good book or web source where i can really learn the full application of the concepts of calculus that i'm studying?
Thank you ! You gave me a very good explanation, i understand now! Anyway I tried and understood the technique to make a polynomial interpolation, but i still don't understand how to make a linear or spline interpolation... can somebody explain me how to do it or link me a website where this is explained? Because I didn't find anything but the definitions and wikipedia page, but can't imagine the method.
Applications of Calculus and Numerical Methods
Well let us say you have a table of data. That is the y values for a series of x values. Each data point is therfore a pair of values (y,x)
Well polynomial interpolation of degree N means that you take a polynomial of degree N and make it 'fit' your data at each of the data points. That is you make the value of your polynomial at each x equal to your y data at that point.
So the polynomial equation is
y = axN + bxN-1 + cxN-2.......dx + e
If you substitute each pair of x and y values in turn you will obtain a set of simultaneous equations you can solve for the constants a, b, c etc.
Since you have N+1 constants ypu will require N+1 data points to give N+1 equations.
Linear interpolation is just using a polynomial of degree one. This requires N+1 points ie 2 points to fit a straight line.
OK so your polynomial 'fits' the data at each point and you interpolate by using it to calculate y at some x you haven't measured at.
You asked about calculus and splines.
Well your polynomial will fit the y values at each measured x, but it will not in genreal have the same slope as the real curve at these points.
For instance there are many many functions that are zero at x = 0 but have vastly different slopes at x = 0.
Enter splines.
Differentiate the polynomial and do the same matching for the slopes at selected or all points.
The process of differentiation will lower all the equations by one degree so you will loose the cosntant and have one less point and simultaneous equation.
If you have more data points than you need for a given polynomial there are many possible solutions and you then need statistical techniques to chose the best one.
Thank you very much for the explanation! But if i have my table of data what is the procedure to get a a linear function with the interpolation ? I didn't understand. With polynomial interpolation i use systems but with linear ? what is the procedure ?
the shape of the first one is a parabola and the second one is like a turned straight S.
For the first one, the sign of y is always positive and for the second one the sign of y is positive if x is positive and negative if x is negative.
Very often we break the full range of the datset into shorter sections.
Each section is approximated by its own cubic spline.
These will in general have different constants, but the same form.
When we are using a cubic spline we are approximating our data by a general cubic
y = ax3 + bx2 + cx + d
There are four constants, but instead of making this cubic match the data points we make the first and second derivatives at the end of the first spline match the beginning of the spline in the second section and so on.
That way the splines for each sections blend smoothly one into the next.
Incidentally the word spline comes from the way an old fashioned draftsman would have used a flexible steel strip to bend round the plotted data points to provide a curved ruler to draw the plotted line. A bent steel strip, fixed at four points naturally takes up a cubic profile.
ok thanks a lot! anyway which is a nice way that i can use calculus in my electronics and comupter engineering projects ? what can i analyze and how can i include functions and plots ? Sorry for a too general question, but i'm trying to get a better understing on how i can apply calculus and numerical methods.
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Mathematics - AS/A2 AQA
Why choose Mathematics?
This is a highly regarded and enjoyable A level. Many students will find it a much more stimulating and rewarding course than at GCSE level, with greater focus on imaginative problem solving skills. The skills and knowledge which you develop will be useful in a wide range of other subjects and careers, particularly in the areas of science and business.
What can I expect to learn?
The maths course has a modular structure with three modules leading to an AS level in the first year and six modules to an A level. The core of the subject is the Pure Maths modules (Core 1, 2, 3 and 4). The first module extends certain topics you have studied at GCSE (equations, graphs, geometry) and introduces new ones (calculus) so that you have the basic tools needed to study maths at a more advanced level in Core 2, 3 and 4. In addition students will study some Applied Maths modules. These modules are concerned with how maths can solve reallife problems. In the first year everyone studies Decision 1. In the second year students have a choice of studying Statistics 1 or Mechanics 1.
How will I learn?
Maths is a 'doing' subject. Your teacher will explain new concepts using a mixture of demonstration and exploration, with regular use of ICT, but most of all studying Maths is about students investigating and tackling problems themselves.
How will I be assessed?
Throughout the course your progress will be monitored with regular homework assignments and tests. Your final assessment in each module is by examination. In each year you will take one examination in January and the remaining two examinations in June. There is no coursework involved.
What subjects combine well Mathematics?
The study of mathematics is particularly valuable for any students who wishes to study physics or chemistry. Students studying biology, psychology or geography will also benefit from following a mathematics course and should consider the AS level if they do not wish to take the full A level.
Where does this lead? What can I do next?
As well as being a subject which can be studied in its own right, mathematics will support most scientific or business related courses and areas of employment.
What background, skills and achievements do I need?
You need to get at least a B grade, at the Higher Level, at GCSE in order to have the confidence and algebraic skills to move on to AS/A2. Success in mathematics requires determination in solving problems and commitment. Above all, you must enjoy doing maths.
*Further Maths will be available for students who achieve grade A* or A at GCSE.
The oldest, shortest words - "yes" and "no" are those which require the most thought."
Pythagoras
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Provides the math background and uses practical examples, real data, and a different approach to life the ''myst'' from algebra. This book presents general concepts first - and the details follow. In order to make the process simple, long computations are presented in a logical, layered progression with one execution per step.Packed with practical examples, graphs, and Q&As, this complete self-teaching guide from the best-selling author of Algebra Demystified covers all the essential topics, including: absolute value, nonlinear inequalities, functions and their graphs, inverses, proportion and ratio, and much more.
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Your step-by-step solution to mastering precalculus. Understanding precalculus often opens the door to learning more advanced and practical math subjects, and can also help satisfy college requisites. Precalculus Demystified , is your key to mastering this sometimes tricky subject. This self-teaching guide presents general precalculus concepts first,... more...
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calculus, this applied Calculus text provides students with the solid background they need in the subject with a thorough understanding of its applications in a wide range of fields ? from biology to economics.Key features of this innovative text include:The text is problem driven and features exceptional exercises based on real-world applications.The authors provide alternative avenues through which students can understand the material. Each topic is presented four ways: geometrically, numerically, analytically, and verbally.Students are encouraged to interpret answers and explain their reasoning throughout the book, which the author considers a unique concept compared to other books.Many of the real-world problems are open-ended, meaning that there may be more than one approach and more than one solution, depending on the student's analysis. Solving a problem often relies on the use of common sense and critical thinking skills. Students are encouraged to develop estimating and approximating skills.The book presents the main ideas of calculus in a clear, simple manner to improve students' understanding and encourage them to read the examples.Technology is used as a tool to help students visualize the concepts and learn to think mathematically. Graphics calculators, graphing software, or computer algebra systems perfectly complement this book but the emphasis is on the calculus concepts rather than the technology. (Textbook ISBN: 0471207926)Student Solutions Manual: Provides complete solutions to every odd exercise in the text. These solutions will help you develop the strong foundation you need to succeed in your Calculus class and allow you to finish the course with the foundation that you need to apply the calculus you learned to subsequent courses.(Solutions Manual ISBN: 0471213624) «Show less
... Show more»
Rent Applied Calculus 2nd Edition today, or search our site for other Gleason
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Introduction to Calculus
Calculus is the study of change and motion, in the same way that geometry
is the study of shape and algebra is the study of rules of operations and relations. It is the culmination of algebra, geometry, and trigonometry,
which makes it the next step in a logical progression of mathematics.
Calculus defines and deals with
limits,
derivatives, and
integrals of
functions. The key ingredient in calculus is the notion of infinity.
The essential link to completing calculus and satisfying concerns about infinite
behavior is the concept of the limit, which lays the foundation for both derivatives
and integrals.
Calculus is often divided into two sections: Differential Calculus (dealing with
derivatives, e.g. rates of change and tangents), and Integral Calculus (dealing
with integrals, e.g. areas and volumes). Differential Calculus and Integral Calculus
are closely related as we will see in subsequent pages. It is important to have
a conceptual idea of what calculus is and why it is important in order to understand
how calculus works.
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Yes, it is bad. And yes, it is new. If I am hindered in my progress and can't get a fix on how to adapt, I've wasted my time in this course. I was hoping to hear constructive advice from others who experienced the same. No guarantee, I know, that it will apply to my particular case, but one can always try.
Why am I struggling with Differential Equations??
How can you expect constructive advice when the explanation for your situation is literally one sentence? Maybe you should expound a bit on what you're facing, why you feel like you're struggling, what exactly you're struggling with, etcI think the best differential equations book is Ross, Differential Equations.
There are common themes of what give people trouble but it's still helpful to know what is hindering you.
What helped me a lot was Schaum's Outline of Diff Eq (I used the 3rd Edition). With the book we were using, it would often just present the theory, but not an example of the theory in action. What also helped me was to understand how to identify basic forms of differential equations such seperable, homogenous, etcThere isn't always one way to get the correct answer. Also, I have seen quite a bit of mistakes on that site.
I've never known anybody having troubles in an ODE course if they were adequately prepared for it, it's pretty cut and dry algorithmic solutions and whatnot ... and if you did fine in calc 1-3 not sure what the problem would be ... linear algebra maybe? If that's the problem, then just start reviewing the most important computational sections of LA for ODEs (determinants, wronskian, eigenvalues/vectors).
I guess some profs may put some 1st order problems on their exams that involve solving tricky integrals ... if that has been a problem, spend time reviewing integration by parts, trig substitutions, and partial fractions decomposition.
Other than that, my best advice would be to make an outline for yourself ... maybe a page long on the categories of DEs (with examples) you're dealing with and then their solution methods. Examples:
x^2y'=Ay >>>>>>>> separate to get x's on one side and y's on the other then integrate and solve for y.
so yeah just do that for everything you've learned so you have a cheat sheet to use until it's all second nature.
to make it second nature, just practice loads of problems, and try to do it randomly (possibly by having a friend pick problems for you from various sections of your book and writing them down for you to solve in a jumbled order), so you won't already know the solution method ahead of time due to the section of the book you're in (since all the problems in section 3.2 or whatever are the same solution method = just repeating the same thing over and over rather than having to analyze which solution method to use like you'll have to do on exams).
I thought ODEs wasn't too bad but there were some tricky problems in the class. One of the problems was taking this higher order differential equation and turning into a 4 x 4 system of differential equations. That was tricky.
You are suppose to be struggling, therefore you are right on track. Congratulations.
The only thing that is going to help you.....is hours and hours and hours of studying. There is no shortcut. Eventually, the bell will ring and you will pass a test. Keep in mind too....your fellow students are also struggling...you are not alone!
I disagree--ODE is NOT one of the harder courses in engineering--did you have to do dynamics? Mass and heat transfer? Mechanical vibrations? ODE is a joke compared to those--IMO, of course.
Hell--calculus 3 (for engineers, with all the application problems) was harder than ODE.
Difference of opinion. I thought dynamics was simple.....and I thought calc 3 was a cake walk because it was identical to calc II with just the extra z dimension that followed the same rules. Didn't take the other two because I'm an EE.
I agree Dynamics is not the hardest but it is harder than ODE simply because it involves thinking--ODE does not IMO. As someone pointed out--it's a cookbook type class. Oh--Bernoulli--apply method. Oh, Cauchy-Euler--apply method.
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Foundations of Mathematical & Computational Economics
9780324235838
ISBN:
0324235836
Edition: 1 Pub Date: 2006 Publisher: Thomson Learning
Summary: Economics doesn't have to be a mystery anymore. FOUNDATIONS OF MATHEMATICAL AND COMPUTATION ECONOMICS shows you how mathematics impacts economics and econometrics using easy-to-understand language and plenty of examples. Plus, it goes in-depth into computation and computational economics so you'll know how to handle those situations in your first economics job. Get ready for both the test and the workforce with this ...economics textbookMt. Morris, ILShipping:Standard. [
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6th Grade Honors Math
For students: the Important Files (documents and power points) are located at the bottom of the page
Overview of 6th Grade Honors Math
HONORS: Since this is an honors class, this means that the 6th graders will actually be using the 7th grade textbook and will move at a faster pace than the other grade level class of 6th grade. At Students Students will review basic geometry concepts and vocabulary. They will then use this foundation to work at a more complex algebraic level. They will make conjectures as to how various formulas are derived. These include formulas to find areas of various shapes--such as rectangles, trapezoids, triangles, circles--as well as the Pythagorean Theorem. GRAPHING:Students will be introduced functions and Since it is an honors class, there are several projects that are completed as well.
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Looking for book with good general overview of math and its various branches
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Book Description: This text takes an emerging approach to physics, emphasizing important geometrical structures that lay the foundations of much of modern theory. The subject of Clifford algebras is presented in efficient geometric language - common concepts in physics are clarified, united and extended. The text serves as a pedgogical tool for either self-study or in undergraduate/graduate courses. Topics covered include: history of teaching algebras; linear algebra; gravity; spinors; applications in engineering; spacetime algebra and line geometry; and Clifford algebra with Maple.
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Inequality1 This is a basic foundation about inequalities. It is intended for students studying algebra at elementary-to intermediate level. If you want to UNDERSTAND inequalities from the scratch, this program is for you. It involves the principle adopted by the company of UNDERSTANDING rather than memorizing mathematics. It involves the stepwise fashion of solving inequalities so that you can understand each step in the solution and the principle why we gave to do each step. Inequality1 by Ika Company. Software Home & Education Mathematics.
Music Organizer Deluxe Music Inventory Software for Music Collectors.. Music Organizer Deluxe by PrimaSoft PC, Inc.. Software Home & Education Hobbies
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Scheduled Class Times: =
span>Monday/Wednesday
11:00-12:15
COURSE
DESCRIPTION
Essentials of Algebra<=
/span>
is a one-term introductory course in algebra that is intended for students =
who
have a firm background in arithmetic but need to improve their algebra
skills.This course contains =
work
with real and rational numbers, algebraic expressions, solving equations and
inequalities, polynomials, graphs, systems of equations, radicals, and
quadratic equations.
Prerequisite=
:MATH009 or placement by the Mathem=
atics
Department
OBJECTIVES OF THE COURSE
The Essentials of Algebra course aims to:=
1)Significantly raise the
level of algebraic skills in students with deficiencies in these skills.
2)Prepare students to be
successful in general education mathematics courses.
3)Help students overcome=
a
"fear of mathematics".
INSTRUCTIONAL MATERIALS
TEACHING STRATEGIES
During class time, there will be a lecture and a discussion on new
concepts.All concepts will be
practiced and reviewed in class as necessary.Students are encouraged to ask the
instructor and their peers for help.Exercises assigned in class are to be finished as homework.
COURSE OUTLINE
Pre-algebra review
Real numbers and algebraic expressions (Chapter =
7)
Solving equations and inequalities (Chapter 8)
Graphs of equations and inequalities (Chapter 9)=
Operations with polynomials (Chapter 10)
Factoring polynomials (Chapter 11)
REQUIREMENTS
Students are expected to attend all classes and read the textbook.<=
span
style=3D'mso-spacerun:yes'> The student who attends class regu=
larly
and is prepared will benefit.The
instruction for each class is built upon the concepts from the previous
classes.Therefore, attendanc=
e and
participation in class discussions is essential.If you are absent for any reason, =
it is
your responsibility to get the notes and assignments that you missed.
You m=
ust be
in class to take a quiz/exam or submit homework.There are no make up exams or quiz=
zes.
Quizz=
es and
collection of homework need not be announced.Be Prepared!
The l=
owest
quiz/homework score will be dropped.
Class
participation plays a large part in determining your success in this
course.Be an active particip=
ant in
the course and ask questions.
If at=
anytime
you are having difficulty with the material, please see me during office ho=
urs
or make an appointment.
If yo=
u are
passing the course with a C or better, the comprehensive final is optional =
and
will replace your lowest exam score.If for any reason you miss an exam, you must take the comprehensive
final and it will replace the missed exam score.If you have taken all three exams =
but do
not have a score of C or better, you must take the comprehensive final.Warning:Do not view the final exam as a
"mulligan".The
comprehensive final encompasses all material learned throughout the semester
and is very difficult.
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How to Study for a Mathematics Degree
Synopsis study and to learning from lectures. Other struggles, however, are more fundamental: the mathematics shifts in focus from calculation to proof, so students are expected to interact with it in different ways. These changes need not be mysterious - mathematics education research has revealed many insights into the adjustments that are necessary - but they are not obvious and they do need explaining. This no-nonsense book translates these research-based insights into practical advice for a student audience. It covers every aspect of studying for a mathematics degree, from the most abstract intellectual challenges to the everyday business of interacting with lecturers and making good use of study time. Part 1 provides an in-depth discussion of advanced mathematical thinking, and explains how a student will need to adapt and extend their existing skills in order to develop a good understanding of undergraduate mathematics. Part 2 covers study skills as these relate to the demands of a mathematics degree. It suggests practical approaches to learning from lectures and to studying for examinations while also allowing time for a fulfilling all-round university experience. The first subject-specific guide for students, this friendly, practical text will be essential reading for anyone studying mathematics at university
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Applied Mathematics
THE FIELD
Mathematics is one of the oldest human disciplines dating back to the earliest civilizations. Since its origins, it has proved to be an indispensable tool for understanding the world around us. Mathematics is the language of modern science and basic training in the discipline. It is essential for those who want to understand the important scientific developments of our time.
CAREER OPPORTUNITIES
Excellent career opportunities for bi-lingual and multi-lingual applied mathematics graduates exist. Even in areas where the application of mathematics may not be obvious, a mathematical education provides training in logical and analytical skills, which are invaluable in many industries. As well as the obvious careers in teaching and science, opportunities exist in insurance companies, industry and commerce, economics, genetics, meteorology and forestry.
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Abstract
Mathematical problem solving has been the subject of substantial and often controversial research for several decades. We use the term, problem solving, here in a broad sense to cover a range of activities that challenge and extend one's thinking.
In this chapter, we initially present a sketch of past decades of research on mathematical problem solving and its impact on the mathematics curriculum. We then consider some of the factors that have limited previous research on problem solving.
In the remainder of the chapter we address some ways in which we might advance the fields of problem-solving research and curriculum
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Textbook Algebra I, McDougal Littell, 2007. Whatever ... Our normal routine will consist of 5-10 minute warm-ups, which will be on the board or overhead ... Chapter Tests will be worth approximately 100 points and will occur at the end of each ...
Textbook The textbook we will use is Algebra 1 published by McDougal Littell. Every student may ... Summative Assessments (40%) Each chapter will include a chapter test. Students will not be ... Chapter 10 Quadratic Equations and Functions Students will be able to graph and solve quadratic equations while comparing ...
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Mathematics Textbooks is an introduction to linear algebra. The main part of the book features row operations and everything is done in terms of the row reduced echelon form and specific algorithms. At the end, the more abstract notions of vector spaces and linear transformations on vector spaces are presented. This is intended to be a first course in linear algebra for students who are sophomores or juniors who...
This book provides a compendium of selected important topics covered in any finance course. The main subject on time value of money and its computational application are explained and demonstrated. This follows other subjects as cost of capital, capital budgeting and securities valuation that used time value factor in their computational analysis textbook provides a comprehensive collection of examples of the topics of Linear equations, Matrices and Determinants.
The reader will obtain the necessary routine of handling these topics by working through these examples.
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8
Total Time: 1h 43m
Use: Watch Online & Download
Access Period: Unlimited
Created At: 07/29/2009
Last Updated At: 07/20/2010
This 8-lesson series will give you an introduction to Calculus I and will walk you through a review of some Pre-Calculus material that you need to master for success in Calculus. Calculus is used to find instantaneous rates of change and the areas of exotic shapes.While we can find average rates of change, like velocity, with set formulas, we cannot find instantaneous rate of change with calculus because dividing by a 0 gives us an undefined answer. Our review of Pre-Calculus will cover functions, the graphing of lines, parabolas, and an intro to Non-Euclidean Geometry.
A function pairs one object with another. A function will produce only one object for any pairing. A function can be represented by an equation. To evaluate the function for a particular value, substitute that value into the equation and solve. You can evaluate a function for an expression as well as for a number. Substitute the entire expression into the equation of the function. Be careful to include parentheses where needed.
A graph is a way of illustrating a set of ordered pairs. One of the easiest objectsto graph is the line. Lines have direction, but no thickness. The slope-intercept form, y = mx + b, and the point-slope form, ( y - y1 ) = m( x - x1 ), are two means of describing lines. When writing the equation of a line, the point-slope form is easier to use than the slope-intercept form, because you can use any point.
The graph of a second-degree polynomial expression is a parabola. A parabola consists of the set of all points in a plane that are equidistant from a fixed line (the directrix) and a fixed point not on the line (the focus). When graphing functions, start by looking for ways to simplify their expressions. Always promise that the denominator will not equal zero when you cancel. The distance formula is an application of the Pythagorean theorem. It states that d = [(x2-x1)^2 + (y2-y1)^2]^(1/2)
In Euclidean geometry, the shortest distance between two points is inevitably going to be a straight line. In Non-Euclidean geometry, however, this is not always the case..Below are the descriptions for each of the lessons included in the
series:
Calculus: An Introduction to Thinkwell's Two Questions of Average Rates of Change antid How to Do Math Functions Graphing Lines Parabolas
Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can Some Non-Euclidean Geometry
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BOOKS: Finding the Path: Themes and Methods for the Teaching of Mathematics in a Waldorf School
Title:
Finding the Path: Themes and Methods for the Teaching of Mathematics in a Waldorf School
BookID:
9
Authors:
Bengt Ulin
ISBN-10(13):
0962397814
Publisher:
AWSNA Publications
Publication date:
1996-10
Number of pages:
318
Language:
English
Rating:
Picture:
Description:
Themes and Methods for the Teaching of Mathematics in a Waldorf School • translated by Archie Duncanson • The presentations in this book are built on experiences from mathematics teaching in grades 7-12. The book does not offer a pedagogical collection of recipes but rather how one might engage the pupils. The presentation is a pedagogical handbook. It addresses such topics as the history of mathematics, Fibonacci numbers, nature's geometry, the step from arithmetic to algebra, and gaining confidence in thinking. Mr. Ulin takes the reader into the wonders of mathematics. Translated from the Swedish.
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Non-HP RPN Scientific Calculators?
Non-HP RPN Scientific Calculators?
I'm a freshman now in college and I'm looking to buy a scientific calculator, since graphing calculators aren't allowed in my exams. I've been running through high school with my TI-89 in class, though in senior year I became interested in RPN and have been using an HP-48 emulator for Android in RPN mode just for kicks
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Introductory Algebra For College Students - 6th edition
Summary: TheBlitzer Algebra Seriescombines mathematical accuracy with an engaging, friendly, and often fun presentation for maximum appeal. Blitzer's personality shows in his writing, as he draws readers into the material through relevant and thought-provoking applications. Every Blitzer page is interesting and relevant, ensuring that students will actually use their textbook to achieve success!
Absolutely BRAND NEW ORIGINAL US HARDCOVER STUDENT 6th Edition / Mint condition / Never been read / ISBN-13: 9780321758958 / Shipped out in one business day with free tracking.
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Hard cover New. No dust jacket. 100% BRAND NEW ORIGINAL US HARDCOVER STUDENT 6th Edition / Mint condition / Never been read / ISBN-13: 9780321758958 / Shipped out in one business day with free trac...show moreking
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Purchasing Options
Features
Analyzes algorithms in generation, enumeration, search, and optimization as well as basic algorithmic paradigms
Examines topics not found in other texts, including group algorithms, graph isomorphism, hill-climbing, and heuristic search algorithms
Provides accessible reading of modern combinatorial techniques
Unifies diverse scientific and mathematical research into one volume
Includes pseudocode description of all algorithms
Summary
This textbook thoroughly outlines combinatorial algorithms for generation, enumeration, and search. Topics include backtracking and heuristic search methods applied to various combinatorial structures, such as:
Combinations
Permutations
Graphs
Designs
Many classical areas are covered as well as new research topics not included in most existing texts, such as:
Group algorithms
Graph isomorphism
Hill-climbing
Heuristic search algorithms
This work serves as an exceptional textbook for a modern course in combinatorial algorithms, providing a unified and focused collection of recent topics of interest in the area. The authors, synthesizing material that can only be found scattered through many different sources, introduce the most important combinatorial algorithmic techniques - thus creating an accessible, comprehensive text that students of mathematics, electrical engineering, and computer science can understand without needing a prior course on combinatorics.
Editorial Reviews
"…book serves as an introduction to the basic problems and methods…style is clear, transparent…The algorithmic problems are always considered and they are in the center of the discussion…has a fresh approach to combinatorics that is available for readers, students in computer science, electrical engineering without any background in mathematics." - Péter Hajnal, Acta Science Math
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A fun little book which contains no exercises but rather simply explains the concepts, strategies, and vocabulary of algebra. What are polynomials ... more » and why do I care? What's a quadratic equation and how do I solve it? How do you multiply polynomials? What is the slope of a line? What are functions? The language is clear and often amusing. The authors assume the reader has no knowledge of algebra, and also assume he or she has forgotten how to multiply fractions and other preliminary processes. Not a stand-alone text, but a book to read on a Saturday afternoon before taking your first algebra class, it touches nearly all the concepts of a first year course. It has been used and praised by eighth grade students as well as college students. A wonderful refresher before taking a college entrance exam or graduate admissions test Pro Quo Books Bookstore Rating: 4(of 5) Book Location: Toledo, OH, U.S.A. Quantity: 1 Textbook Description: Book is clean and tight, and has minimal or no Better World Books Quantity: 7 Textbook Description: Great condition for a used book! Minimal wear 7 items left Textbook Description: Great condition for a used book! Minimal wear. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy! Textbook Edition: Normal
Other marketplace offers: 28 New from $5.25 and 50 river-city-books (Rating: 99.6 % positive) Standart shipping: 3.49 Expedited delivery available: no Book Location: Tualitan, OR Book Availability: 1 items left Textbook Description: Cover is bent, but the pages, text, and binding are in Good condition. GOODBook Seller: River City Books, LLC Book Location: USA Textbook Description: Clearwater Publishing. Used - Good. Cover62781576/9780962781575 (0-962-78157-6/978-0-962-78157-5
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Puy, WA SAT MathThe use of numbers and symbols, which may be frightening to students, has already begun in the use of numerals, for example, 1, 2, 3, etc., in arithmetic. Algebra uses additional symbols, which can easily learned by using the basic rules of arithmetic, such as addition, subtraction, multiplication, and division. Algebra has these same rules and also others to be learned.
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MATH 210: Calculus III
Navigation:
Contents
COURSE DESCRIPTION
In this course we learn how to extend the ideas of calculus to two and three dimensions.
The concepts of 1-variable calculus arise in studying the motion of a particle along a
line. For a particle moving through space, not just along a line, the position, velocity,
and acceleration at each moment are described by vectors, not just by single real
numbers. Many other physical quantities, such as force and angular velocity, are also
modeled mathematically as vectors. We begin by studying the algebra of vectors (linear
algebra), which allows us to describe the relationships between vector quantities in
physics and also forms the basis of analytic geometry in 3-dimensional space.
Vector-valued functions of a single real variable (time) are used to represent,
for example, the velocity of a moving particle and also to study the geometry of space
curves. We learn how to generalize the concepts of derivative and integral to
vector-valued functions.
A real-valued function of 2 variables can be used to model quantities such as the
temperature on the surface of the earth, which varies from one location to another.
The graph of such a function is a surface in space. At a point of such a graph, one
has a tangent plane, not just a tangent line. We learn how to describe the tangent
plane in terms of ideas of calculus, and learn how the concepts of derivative and
integral generalize to functions of several variables.
In the last part of the course we learn the 2-dimensional version of the Fundamental
Theorem of Calculus, Green's Theorem. This is the mathematics behind the physical
notions of work and potential energy, and is a big step toward understanding electric
and magnetic fields.
TEXTBOOK
Calculus, Early Transcendentals, by W. Briggs and L. Cochran.
NOTE: This textbook has been used since Fall 2011. It is the same textbook used in Math 181. We will cover Chapters 11 through 14.
LABS, QUIZZES
Math 210 includes a weekly computer lab that helps students to visualize and
develop intuition about the concepts being taught in the course. The lab meets either Tuesdays or Thursdays, depending on the lecture section.
Each lab contains problems to be worked with the assistance of the computer; students will submit written lab reports
at the end of each lab unit.
The lab is a required part of the course, a component of the course grading.
It will meet every week, starting in the FIRST week of the course.
Starting in the second week, you will take quizzes according to schedule on the Labs page. The quizzes will be based on the lectures and will be written by your instructor.
Your instructor may also give additional quizzes during the lecture.
More information about the computer lab and the quiz schedule is available here.
HOMEWORK
Homework will consist of problems assigned from the textbook (see the link below)
as well as (possibly) additional problems provided by your instructor. Each
lecturer will announce that section's policy on collection and grading of homework.
You are encouraged to discuss homework problems with your fellow students.
Working in groups makes the explanation of approaches and solutions a part of
the process and helps you learn. Your goal is to find solutions and to communicate
your work in a convincing manner.
GRADES
The course grade is based on the total number of points from hour exams,
homework, quizzes, computer labs, and the final exam.
Quizzes/Homework
100 points total
Computer labs
50 points total
Two Midterm Exams
100 points each
Final Exam
200 points
PREREQUISITE
Grade of C or better in MATH 181. Please make sure that you have met the prerequisite. Students without the prerequisite will not be allowed to take the course.
Please take a minute to do the survey about the ALEKS placement test:
STUDENTS WITH DISABILITIES
Students with disabilities who require special accommodations for access and
participation in this course must be registered with the
Office of Disability Services (ODS). Students who need exam accommodations
must contact ODS in the first week of the term to arrange a
meeting with a Disability Specialist.
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June 12, 2013 will be a Release Time Day.
June 27th is the last day of school (1/2 day).
Dartmouth High School requires that all students have a minimum of three semester credits in mathematics for graduation. A typical high school sequence of 3 courses includes: Algebra 1, Geometry, and Algebra 2. This is a minimum requirement, therefore students are strongly urged to take additional mathematics courses. We recommend that students talk to their guidance counselors and math teachers to determine which mathematics sequence will fulfill their long-term goals. To ensure that all material necessary to pass the tenth grade MCAS exam has been covered, students are to enroll in Algebra I and Geometry by the end of their Sophomore year. Past experience has shown that students who are not proficient with the basic concepts of Algebra 1 and Geometry have not been successful on the MCAS exam. As of 2008, students taking the Mathematics MCAS need to score Proficient or Advanced to fulfill graduation requirements. Students who receive a Failing (F) or Needs Improvement (NI) score will be required to participate in the MCAS Re-Test and enroll in additional math courses.
Depending on their course placement and grade in that course, 8th grade students should choose from the following courses only: Algebra 1 in 2 Semester or Algebra 1 Honors.
The Mathematics Department believes that all students are entitled to a rich mathematical curriculum which is developmentally appropriate. Students will achieve mathematical power through mathematical connections, communications, reasoning and problem solving.
A graphing calculator is required for Algebra I Honors, Algebra 2 and all subsequent courses since a considerable amount of work will be done using one. Classroom instruction will be given only on the above models. Ordering information and the policy on loans will be available the first day of class each semester.
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Calculus : Easy Way - 4th edition
Summary: This ingenious, user-friendly introduction to calculus recounts adventures that take place in the mythical land of Carmorra. As the story's narrator meets Carmorra's citizens, they confront a series of practical problems, and their method of working out solutions employs calculus. As readers follow their adventures, they are introduced to calculating derivatives; finding maximum and minimum points with derivatives; determining derivatives of trigonometric functions; ...show morediscovering and using integrals; working with logarithms, exponential functions, vectors, and Taylor series; using differential equations; and much more. This introduction to calculus presents exercises at the end of each chapter and gives their answers at the back of the book. Step-by-step worksheets with answers are included in the chapters. Computers are used for numerical integration and other tasks. The book also includes graphs, charts, and whimsical line illustrations. Barron's Easy Way books focus on both practical and academic topics, presenting fundamental subject matter in clear, understandable language. Equally popular as self-teaching manuals and supplementary texts for classroom use, they are written to help students improve their grades and review subject matter before tests. They are also useful for introducing general readers to a new career-related skill. Easy Way titles cover virtually all subjects that are taught on advanced high school and college-101 levels. New subjects are periodically added, and existing titles are frequently updated to keep them timely and relevant to students' needs. Subject heads and key phrases are set in a second color5.16 +$3.99 s/h
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Personal Math Companion
Synopsis
This quick reference guide is suitable for students in middle school, grades 6-8. The book will serve as a true companion by providing quick access to the metric system, basic algebra, geometry, exponents, probability, mathematical formulas, intro to statistics, and much more. This is truly the ultimate companion for students who wishes to excel in math. The Personal Math Companion will have all the answers to the tough math questions that are frequently asked by students in the 6th, 7th, or 8th grades. "« Do you want to know how to find the third angle of a triangle? "« How to use and understand a number line for better assistance when handling real numbers? "« Classifying the different types of polygons? Well, it has all these answers and more, plus helpful tips that will help your children to understand the different concepts of mathematics
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Course Syllabus for Math 241:
Honors Calculus III (Fall 2012)
COURSE OVERVIEW:
Calculus is the mathematics of quantities that vary in time,
with a wealth of applications to many different branches of
science. It's also a marvelous synthesis of algebra and
geometry, tying together pretty much everything you learned
in high school math, and serving as a bridge to higher
mathematics.
In Math 241, we'll take a more in-depth study of calculus
than is offered in Math 231,
with more of a focus on the
rigorous underpinnings of the subject: not just what's
true, but why it's true. Also, since the class
is small (fewer than 20 students), class discussions can
play a big role in helping you learn to view the material
from many different angles, and you'll get more direct
contact with a faculty member.
You'll also learn the answer to the questions:
What do you get when you mate a mountain-climber and a mosquito?
And: Why do pirates like polar coordinates?
I want all of you to succeed in this class; below you will find
some tips for how you can help make this happen.
USEFUL LINKS:
Course
overview (the lecture notes for the first day;
these notes will probably answer most of your questions
about the structure of the course)
CONTACTING THE INSTRUCTOR:
Professor James Propp
Email: jpropp at cs dot uml dot edu
(note: I also have a James_Propp account but I don't read it very often).
Phone: (978) 934-2438. I'll leave a message on my voice mail if the
university is open but I'm unable to attend class. To check whether the
university has been closed because of weather, call (978) 934-2121.
Fax: (978) 934-3053 ("Attn: James Propp").
Office: Olney 428C.
Consultation Hours: TBA.
Meetings at times other than my office hours can be arranged by
appointment; see me after class, call me on the phone, or send me
an email message.
Suggestions about how the course is being run are welcome at any time.
If something isn't working for you, please don't wait until the end of the
semester to tell me!
GENERAL COURSE INFORMATION:
Expectations: You're expected to attend classes, do the reading
in advance, ask questions, and make serious attempts to answer
questions raised by me or by other students during class.
If you miss a class, it's your responsibility to make
sure you obtain all information (course material, assignments,
changes in exam dates, etc.) presented that day.
TEXT:
James Stewart, Essential Calculus: Early Transcendentals (2nd edition),
2012. A copy of this book will be placed on reserve at Lydon Library.
(On the other hand, the Stewart book Calculus: Early Transcendentals ---
note the absence of the word ``Essential'' --- is structured differently
and cannot be used as a textbook for this class.)
During the Fall semester, we'll cover Chapters 10 through 13.
GRADING POLICY:
Course grades
Course grades will be based on three numbers: your Homework score,
your score on the in-class Midterm, and your score on the Final.
Your average score for the course will be computed as a weighted average
of your Homework, Midterm, and Final scores in which the highest
of the three scores is assigned weight 40% and the other two scores
are assigned weight 30%. (For instance, if your highest score was
on the Midterm, your average score for the course would be 30% of
your Homework score plus 40% of your Midterm score plus 30% of your
Final score.) Since this is an Honors class with challenging
problems, the scheme for computing letter-grades is on the lenient
side, and is determined from your weighted average score according
to the following table:
Average
[85, 100)
[82, 85)
[80, 82)
[75, 80)
[72, 75)
[70, 72)
Grade
A
A-
B+
B
B-
C+
Average
[65, 70)
[62, 65)
[60, 62)
[55, 60)
[0, 55)
Grade
C
C-
D+
D
F
(I may raise your grade above what's shown in the table
if your class participation is strong: one more reason
to come to class. Also, coming to office hours counts
as a form of class participation.)
Exam dates: Midterm TBA; final exam TBA.
Exam Policy
It's important that everyone take the same exams under the same
conditions for maximum fairness and reliability of testing. I therefore
don't give makeup exams unless you have a valid reason for missing
the scheduled exam (for example, illness or a religious holiday), and I
don't allow extra time on exams unless you have a note from Disability
Services (see below).
If you have to miss a scheduled exam, please let me know
ahead of time if at all possible; I'm much more likely to be
sympathetic if you call me the morning of the exam and say "I have the
flu and can't take the exam" than if you come in two days after the
exam and say "I missed the exam. When can I take a makeup?"
You may not use a cell phone in any way during an exam.
Use of calculators is prohibited during exams.
You can always reschedule an exam
that falls on a day that is a religious holiday for you, but you must
make these arrangements ahead of time.
Tips on Preparing for Exams
Start studying for an exam at least one week ahead of time.
Begin by reviewing the homework problems for the sections that will
be covered on the exam. Make sure you know how to solve each problem.
If you can't solve a particular problem, make a note of the problem
number and move on to the next problem; you can go back to the problem
later with a fresh head (yours or someone else's!).
You can test your knowledge by trying odd-numbered problems
for which the answer is given at the back of the book.
Try the review problems that appear at the end of each chapter.
Ask me or someone else for help on any homework problem that gave
you trouble, then try to solve a similar problem from the textbook.
Get a good night's sleep the night before the exam. You'll
perform better if you are fresh and able to think clearly.
Tips on Taking Exams
Read every question on the exam before you start working. This will
give you a feel for how long the exam is and how you should pace
yourself. It'll also give your subconscious mind a chance to start
working on the questions.
If you're not sure what a question means, please ask me. I'm
trying to see how well you know the material, not to trick you with
ambiguous wording.
Show as much of your work as possible, in as clear a way as possible.
Even if you get the wrong answer, I'll try to award you as much partial
credit as I feel I can conscientiously give you, but it's hard for me to
do this if you don't show your thought-processes.
Look at the point value of each question. Obviously, it's more
important to do well on the questions that count the most than the ones
that count the least.
It's generally best to do the easiest problem first, then the next
easiest, and so on. You don't have to do the problems in the order
they appear on the exam.
If you get stuck on one question, move on to the next. Come back
later to the question that is giving you trouble.
Be aware of how much time you have left. Don't spend too much time
on a single question. It's generally better to get partial credit on
every question than full credit on a small number of questions.
If you have extra time, use it to check your work! Better still,
if there's more than one natural approach to the problem, try to
solve the problem with a different method; this can be a better way
to catch mistakes than just re-reading your calculations. If you get
the wrong answer with one approach but the right answer with the
other approach, I'll give you nearly full credit (especially if
you speculate intelligently on where you might have made an error).
If you get an answer that doesn't make sense but don't have time to
trace where your error came from, don't just cross out your answer;
explain why you think the answer you got looks wrong,
and you may get some extra points for having good instincts.
Never be afraid to ask for extra paper. (If you want to write
on the reverse side of a page, please write "see other side".)
Homework
Typically there'll be one homework assignment per week,
due one week after it is assigned. (We may deviate from this
schedule at the beginning of the term and around the time of
the midterm.)
In order for you to understand the material in this course, it's
extremely important that you do the assigned homework problems.
Working with your classmates can be a great help, and I strongly
encourage it, subject to certain provisos (see below).
I also urge you to ask questions about any problems that give you
trouble.
Homework will usually be due each week on Friday
(except during the week of an exam).
Your grade will be based on clarity as well as correctness, so
neatness, grammar, and punctuation should not be neglected.
Harder problems will in general be worth more points.
You are required to include an estimate of how much time you spent
on each and every assigned problem; this will help me assess
which of the problems are the harder ones. (I reserve the
right to throw out a problem entirely if it turns out to be
too hard.)
Barring unusual circumstances, late homeworks will not be accepted.
Each student will be allowed to skip two assignments without penalty;
additional skipped homeworks will only be permitted if a valid
excuse is presented, preferably ahead of time rather than afterwards.
Don't use up your "free skips" too early in the semester!
If you skip just one assignment, your lowest homework score gets dropped.
If you don't skip any assignments, your two lowest homework scores get dropped.
While you can discuss the exercises with classmates, the work you hand in
should be your own write-up and not copied from someone else. When leaving a
joint homework-solving session, don't carry away anything that doesn't fit
in your own brain. Also, you must acknowledge who you worked with.
(If you didn't work with anyone, please write "I worked alone on this
assignment".)
Academic honesty in homeworks is expected. (E.g., if you use web-resources
or tutors or collaborators of any kind, the role of their contribution must
be acknowledged; you won't receive a lower grade for using such resources,
but if the grader and I feel you're relying on them too heavily, we may
require you to change your way of doing homework.) My expectations for
appropriate ways of doing the homework will be discussed in class; in
case you are in any doubt about what is expected, it is your responsibility
to contact me for clarification. See
the UMass Lowell catalogue
for a definitive statement of UMass Lowell's academic honesty policy.
It is not required that you submit your solutions in
LaTeX,
but if you are planning to be a mathematician,
scientist, or engineer, it's never too early to learn!
LaTeX is free software that lets you typeset formulas
about as fast as you can write them (with some practice).
Composing your homework in LaTeX will help you pay attention
to your communication of mathematics, and make it much easier
to edit your work as you go along. There will be an initial
hump of getting started, but after a couple of problem sets,
using LaTeX will become quite natural. You'll probably still
want to draw your diagrams and figures free-hand, but knowing
how to write equations in LaTeX is a life-skill that will
serve you well in later courses in which homeworks involve
fewer pictures and more formulas.
Also, if you want to use Mathematica as an aid to your learning, check out
Effective Fall 2011, students will be able to download Mathematica
as part of the campus license, so using it for classes will be more
convenient than in the past.
You shouldn't use Mathematica as a substitute for being able
to do the work yourself the old-fashioned way,
but it's a great way to check your work.
Also, Mathematica features many demonstrations
(see
that can bring course material to life in a vivid way.
You will be expected to fill out a time sheet
that tells me how much time you spent on each problem
(rounded to the nearest minute, or the nearest five minutes;
there's no need to be super-precise).
This helps me improve the course from year to year
by spreading out the work-load more evenly from week to week.
Points may be deducted from students who repeatedly
fail to submit legible time sheets.
Many students find it profitable to read the solutions to all the
problems in the current assignment (posted on the web each week).
Even if you got a problem correct, you may learn something from
reading the posted solution, such as an alternate approach to
the problem or a good clear way to express the main ideas.
Attendance
Regular attendance is expected. It is not part of the grading scheme,
but it may be used to adjust grades upward in the event of a borderline
grade. Class participation that shows that you have read the assigned
material may also be helpful in borderline cases.
SPECIAL NEEDS:
If you have any special needs, e.g., you need more time on exams
because of a disability, I'll do my best to accomodate you.
Please notify me at least two weeks in advance.
|
MERLOT Search - materialType=Collection&category=2513&sort.property=dateCreated
A search of MERLOT materialsCopyright 1997-2013 MERLOT. All rights reserved.Sat, 18 May 2013 16:40:39 PDTSat, 18 May 2013 16:40:39 PDTMERLOT Search - materialType=Collection&category=2513&sort.property=dateCreated
4434Rob Beezer's Free Texts
From this site there are links to a variety of free math textbooks that can be downloaded in various formats.New York University - Free Textbooks
This site has a list of different topics in math and physics. When one clicks on the link, a list of free textbooks is provided along with a link to the book.Math Formulas Reference App for iOS
'Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry, probability and statistics. Most of the formulas come with examples for better understanding. Use the powerful search function to find what you are looking for and mark your favorites for easier access. A convenient tool for students and teachers and a handy reference for anyone interested in math!'This app costs $0.99The Teacher's Guide
Provides free worksheets, printouts, lesson plans, SMARTBoard templates, and more. Also provides free Math Interactive Sites.Ampersand Academic Press
Ampersand Academic Press is a section within Free-ebooks.org that offers free educational articles books, and textbooks that you can download to digital devices such as eReaders. Free textbooks are currently offered in Business, Computer Science, Engineering, Law, Mathematics, and Science. You can also submit your own eBook that you have written for review by their staff. Standard Membership is free.The Futures Channel
The Futures Channel provides students and educators with an excellent resource collection of inspirational Educational videos about current trends and advancements in Science, Engineering, and Technology. It also showcases conversations with visionaries that are helping to change our future perspectives about the world we live in it by providing viewers with realistic real-world applications of their work within their field of study. An excellent Multimedia resource for K-12 STEM and Science teachers to encourage and inspire their students to pursue careers in Math, Science, and Technology.DNATube Scientific Video Site
DnaTube Scientific Video Site is a collection of video-based studies, lectures, seminars, animations, and slide presentations that explain biological concepts. Entries can be easily searched by scientific categories as well by "Recently Added" and "Most Watched" categories. The collection of educational resources is appropriate for middle school, high school, and college classrooms.Elementary Algebra Free Youtube Modules
This site is the entry list to a collection of short, engaging text/graphic modules on standard topics in Elementary Algebra. They are available as Youtube videos for free classroom or download by instructors and students. Basic formulas, sample problems and standard applications are included. Format is text, audio and graphics.Sector matemáticas
recopilación de variados programas matemáticos que he encontrado en mis paseos por la red y que puedes descargar gratuitamente.ארכיון שווה
ארכיון עלון המתמטיקה לבית הספר היסודי מבית מט״ח. כל גיליון עוסק בנושא שונה מהעולם הרחב, אשר מוצג באמצעות קטעי קריאה וחידות.
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This is a module framework. It can be viewed online or downloaded as a zip file. It is as taught in 2009-2010. This module introduces mathematical analysis building upon the experience of limits of sequences and properties of real numbers and on calculus. It includes limits and continuity of functions between Euclidean spaces, differentiation and integration. A variety of very important new concepts are introduced by investigating the properties of numerous examples, and developing the associated theory, with a strong emphasis on rigorous proof. This module is suitable for study at undergraduate level 2. Dr Joel Feinstein, School of Mathematical Sciences Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area
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Self-Check
Quizzes randomly generates a self-grading quiz
correlated to each lesson in your textbook. Hints are available
if you need extra help. Immediate feedback that includes specific
page references allows you to review lesson skills. Choose
a lesson from the list below.
The student will solve practical problems involving rational numbers, percents, ratios, and proportions. Problems will be of varying complexities and will involve real-life data, such as finding a discount and discount prices and balancing a checkbook.
The student will apply transformations (rotate or turn, reflect or flip, translate or slide, and dilate or scale) to geometric figures represented on graph paper. The student will identify applications of transformations, such as tiling, fabric design, art, and scaling.
The student will make comparisons, predictions, and inferences, using information displayed in frequency distributions; box-and-whisker plots; scattergrams; line, bar, circle, and picture graphs; and histograms.
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This chapter introduces the fundamental concepts and terminology of options. The relationships between options and the underlying securities are intuitively explained and how these relationship must be maintained to eliminate arbitrage is used to then motivate and setup the Black-Scholes PDE
This textbook provides an introduction to financial mathematics and financial engineering for undergraduate students who have completed a three or four semester sequence of calculus courses. It introduces the theory of interest, random variables and probability, stochastic processes, arbitrage, option pricing, hedging, and portfolio optimization. The student progresses from knowing only elementary calculus to understanding the derivation and solution of the Black–Scholes partial differential equation and its solutions. This is one of the few books on the subject of financial mathematics which is accessible to undergraduates having only a thorough grounding in elementary calculus. It explains the subject matter without "hand waving" arguments and includes numerous examples. Every chapter concludes with a set of exercises which test the chapter's concepts and fill in details of derivations.
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Rent Textbook
Buy Used Textbook
Buy New Textbook
In Stock Usually Ships in 24 Hours.
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Starting at $19Normal 0 false false false KEY BENEFIT: TheBittinger Concepts and Applications Seriesbrings proven pedagogy to a new generation of students, with updates throughout to help todayrs"s students learn. Bittinger transitions students from skills-based math to the concepts-oriented math required for college courses, and supports students with quality applications and exercises to help them apply and retain their knowledge. New features such as Translating for Success and Visualizing for Success unlock the way students think, making math accessible to them. KEY TOPICS: Algebraic and Problem Solving; Graphs, Functions, and Linear Equations; Introduction to Graphing; Inequalities and Problem Solving; Polynomials and Polynomial Functions; Rational Expressions, Equations, and Functions; Exponents and Radicals; Quadratic Functions and Equations; Exponential and Logarithmic Functions; Conic Sections; Sequences, Series, and the Binomial Theorem MARKET: For all readers interested in algebra.
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Mathematics
Mathematics is a useful, exciting, creative and vital area of study that is taught and explored by faculty who foster an appreciation and sense of enjoyment for all students.
The rigorous and varied curriculum provides opportunities for students to develop their abilities to solve problems and reason logically.
The curriculum offers courses of study, which will enable students to explore and make sense of their world. The study of mathematics is an ongoing process that will extend to and enhance every facet of the students' lives.
As students reach the junior high years, their course placement will be determined by a variety of assessment tools. In 7th grade, students take Pre-Algebra at either the college placement or honors level. Occasionally 7th grade students are placed in Algebra I Honors. Most eighth grade students follow the progression to Algebra I or Algebra I Honors, while those students who successfully complete Algebra I Honors in 7th grade move to Geometry Honors.
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Peer review is a process of self-regulation by a profession or a process of evaluation involving qualified individuals within the relevant field. Peer review methods are employed to maintain standards, improve performance and provide credibility...
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background.
Paid circulation in 2008 was 9,500 and total circulation was 10,000.
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Designed for undergraduate students, A Mathematics Companion for Science and Engineering Students provides a valuable reference for a wise variety of topics in pre-calculus mathematics. The presentation is brief and to-the-point, but also precise, accurate and complete.
Learn how to read mathematical discourse, write mathematics appropriately, and think in a way that is conducive to solving mathematical problems. Topics covered include: Logic, sets numbers, sequence, functions, powers and roots, exponentials and logarithms, possibility, matrices, Euclidean geometry, analytic geometry, and the application of mathematics to experimental data. The epilogue introduces advanced topics from calculus and beyond. A large appendix offers 360 problems with fully detailed solutions so students can assess their basic mathematical knowledge and practice their skills.
Here are just some of the questions answered in this book: How can
a) Logarithm be converted from one base to another?
b) Simultaneous linear equations be solved by hand painlessly?
c) Some infinities be bigger then other?
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Search Course Communities:
Course Communities
Lesson 21: Variation
Course Topic(s):
Developmental Math | Variation
The lesson begins with a comparison of data tables and graphs of two functions, one directly proportional (cost of gas) and the other exponential (population), before a definition for direct variation is introduced. Direct variation is then linked to linear function (f(x)= kx) and the scaling property of direct variation is examined (i.e. a multiple of the independent variable will always correspond to that same multiple of the dependent variable). Direct variation with a power of (x) follows with a test for direct variation before indirect variation and indirect variation with a power of (x) are introduced.
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Advanced undergraduates and graduate students studying quantum mechanics will find this text a valuable guide to mathematical methods. Emphasizing the unity of a variety of different techniques, it is enduringly relevant to many physical systems outside the domain of quantum theory. Concise in its presentation, this text covers eigenvalue problems in classical physics, orthogonal functions and expansions, the Sturm-Liouville theory and linear operators on functions, and linear vector spaces. Appendixes offer useful information on Bessel functions and Legendre functions and spherical harmonics. This introductory text's teachings offer a solid foundation to students beginning a serious study of quantum mechanics. Reprint of the W. A. Benjamin, New York, 1962$18
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Hardcover
Click on the Google Preview image above to read some pages of this book!
"Numerical Methods in Engineering: Theories with MATLAB, Fortran, C and Pascal Programs" presents a clear, easy-to-understand manner on introduction and the use of numerical methods. The book contains nine chapters with materials that are essential for studying the subject. The book starts from introducing the numerical methods and describing their importance for analyzing engineering problems. The methods for finding roots of linear and nonlinear equations are presented with examples. Some of these methods are very effective and implemented in commercial software. The methods for interpolation, extrapolation and least-squares regression are explained. Numerical integration and differentiation methods are presented to demonstrate their benefits for solving complicate functions. Several methods for analyzing both the ordinary and partial differential equations are then presented. These methods are simple and work well for problems that have regular geometry. For problems with complex geometry, the finite element method is preferred. The finite element method for analyzing one- and two-dimensional problems is explained in the last chapter.
Numerous examples are illustrated to increase understanding of these methods for analyzing different types of problems. Computer programs corresponding to the computational procedures of these methods are provided. The programs are written in MATLAB, Fortran, C and Pascal, so that readers can use the preferred language for their study. These computer programs can also be modified to use in other courses and research work.
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LIFEPAC® Mathematics Curriculum Grades 7 to 12
LIFEPAC Mathematics emphasizes mastery of concepts through
drill and practice activities and hands-on activities. The middle school grades
focus on numbers, operations, structures, geometry, applications, use of
calculators, the metric system, and more. From 9th to 12th grade, LIFEPAC math
follows the traditional math program: Algebra 1 in grade 9, Geometry in grade
10, Algebra 2 in grade 11, and Pre-Calculus, Advanced Math (including
trigonometry, probability and special functions) in grade 12. Teacher
involvement is considered necessary since math builds on concepts. The teacher
needs to make sure the concepts are mastered.
If you need to mix and match grade levels or to find out more about the
individual subjects, then you can check out our LIFEPAC subject pages:
Bible,
Language Arts,
Math, Science,
and Electives.
Each LIFEPAC Mathematics homeschool curriculum set from Alpha Omega is
composed of ten LIFEPACs (consumable textbook/workbook combinations) and a
Teacher's guide for that grade. The LIFEPACs include the subject text,
exercises, projects, review, and tests.
The Teacher's Guide includes answers and solutions for the math problems.
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Whether teaching remedial, mainstream, or honors classes,
in a segregated or integrated program, these 179 dynamic animations bring to
life algebraic topics from pre-algebra through pre-calculus. Packaged on a
CD-ROM (with a basic license for 4 on-site computers to use at a variety of
levels), Algebra In Motion™
animations perform equally well on either the Windows or Macintosh platform.
They must be run by The Geometer's Sketchpad v4 or v5 (no prior versions),
owned and sold by Key Curriculum (
on either Windows or Macintosh platforms.
Although a detailed instruction manual is included on the CD-ROM (PDF
format), most of the animations can be run successfully using only the
on-screen information.
THE BASICS Visually explore or review fractions (meanings,
comparisons, improper, LCD, adding, decimals, percents), signed number
operations, absolute value, and introduce the concept of an equation as a
balance of values and use that balance to solve preset equations or create
your own.
INTRODUCING THE COORDINATE PLANE
& GRAPHING Introduce students to the basic vocabulary and
characteristics of the coordinate plane (+ history). Dynamically display the
definition of slope. Practice graphing lines from y = mx+b, Ax+By = C, and
y-y1 = m(x-x1) forms.
Explore the relationship of parallel and perpendicular lines to slope.
Develop the formulas for midpoint and distance. Test relations using an
animated vertical line test
Present 4 different graphing grids on the same
screen.
axes~quadrants~coordinates
vertical line test, domain/range
exploring domain/range
with any function
visualizing slope
graph y =
mx + b
graph Ax +
By = C
graph y-y1
= m(x-x1)
parallel
and perpendicular
developing
the midpoint formula
developing
the distance formula
4 grids on
1 screen
evolution
of a polynomial
parabola evolution
parabola 3 graphing forms
transform
f(x) to f(x) + a
transform f(x) to
af(x)
transform f(x) to f(x-a)
transform f(x) to f(ax)
transformation practice
(line)
transformation practice
(absolute value)
transformation example
(parabola)
transformation practice
(sine)
transformation practice
(exponential)
MULTIPLICATION & FACTORING The distributive property is geometrically demonstrated
for products of all combinations of monomials, binomials, and trinomials.
The factored form of a2 – b2 is developed and proved.
Special emphasis is given to (x+h)2 ≠ x2+h2
and (x+h)3 ≠ x3+h3. "Completing
the square" is modeled physically.
FOIL (a+a)(x+b)
(a+b)(c+d) = ac+ad+bc+bd
(a+b+c)(d+e+f)
a2 - b2 = (a+b)(a-b)
(x+h)2 = x2 + 2xh + h2
(x+h)3 = x3 + 3x2h + 3xh2 +
h3
completing
the square
CONNECTING SOLUTIONS TO GRAPHS Reinforce meaningful understanding of solutions to
sentences with absolute value, systems of 2 linear equalities or
inequalities, and finding the roots of a quadratic equation (including
complex roots) using its related parabola.
|ax+b| ≥ c
system of
linear equalities
system of
linear inequalities
complex
roots of quadratic equations
polynomial
root dragging
(set of 7 animations)
ADVANCED GRAPHING Dynamically graph points in 3D space or on the complex
number plane (history included). Control coefficients of a polynomial to
"morph" it from a constant function up through a 5th degree polynomial.
Similarly "morph" graphs of logarithmic & exponential functions, parametric
& polar graphs, greatest integer functions, and inverses. Explore how
composites of functions are created, and create linear programming examples.
points in
xyz-space
points in
complex plane
"morphing
polynomials"
"morph"
exponential functions
"morph"
logarithmic functions
parametric
graphs
polar
graphs
your
choice
greatest
integer functions
more
greatest integer
inverses
creating
composites
(adapts to any example)
composite
ex. 1
composite
ex. 2
composite
ex. 3
linear
programming
(create your own example)
linear
programming
(preset example 1)
linear
programming
(preset example 2)
CONICS &
THEIR APPLICATIONS Dynamically create each conic section from its
definition. Graph and "morph" all features of each. Alter coefficients of
the general equation or eccentricity to "morph" one conic into another.
Explore applications to satellite dishes, elliptical pool tables, whisper
chambers, & falling objects.
overview
construct
circle by def. & graph
construct parabola by def.
construct
ellipse by definition
construct
hyperbola by definition
graphing
parabolas
graphing
ellipses
graphing
hyperbolas
"morphing"
from general equation
parabaloid,
ellipsoid, hyperboloid
family of
hyperbolas & ellipses
mutually
orthogonal
reflections & collections
falling
projectile
altering
eccentricity
TRIGONOMETRY In an environment where rotation is real, not merely
imagined, thoroughly investigate the unit circle's angles, coordinates, and
ratios. Literally unwrap the unit circle to form sine and cosine waves.
Dilate and translate trigonometric graphs to explore amplitude, period, and
shift. Explore and prove Pythagorean identities, the Law of Sines, and the
Law of Cosines. Convincingly demonstrate that sin (a+b) can't be (sin a +
sin b).
unit
circle angles
sine,
cosine, tangent, definitions
sin, cos, tan, sec, cec, cot on the unit circle
special angles of
the unit circle
unwrapping the unit circle
"morphing" trig graphs
Pythagorean Identities
Law of
Sines
Law of Sines- ambiguous case
Law of
Coines
sin (a+b),
Q1
sin (a+b),
Q2 both acute
sin (a+b), Q2 acute+obtuse
cos (a+b),
Q1
sin
(a+b), cos (a+b)
geometric approach
THEOREMS Dynamically explore conventional theorems such as the
Pythagorean Theorem (+ history) along with 7 different visual proofs of it.
In addition, discover a large selection of unusual and unexpected theorems
concerning tangents to parabolas, cubics, and quartics that will amaze and
fascinate your students while laying an excellent foundation for more
advanced mathematical study.
Pythagorean Theorem
More
Pythagorean Theorem
any 3
tangents to any parabola
3 tangent
proof (part 1)
3 tangent
proof (part 2)
parallel
tangent at midpoint
inflection
point at midpoint
use 2
roots to find 3rd (+ extend to imag. rts)
ratio of
areas in cubic
more
ratios of areas in cubic
ratios of
areas in quartic
PT –
shearing proof
PT –
Chinese proof
PT –
Pythagoras proof
PT –
Bhaskara proof
PT –
DaVinci proof
PT –
Garfield proof
PT –
Generalized
any 2
tangents to any parabola
2 tangent
proof (part 1)
2 tangent
proof (part 2)
MORE APPLICATIONS & VERTICAL
TEAMING Build basic graphing sense using intriguing questions
about real-world situations that animate at the click of a button.
Thoroughly explore classic problems such as the "open box" (vary the size of
squares removed from the corners of the original rectangle), the sliding
ladder, flying kite, etc. As the level of the class increases, more and more
features can be explored at the click of a button. Finally, even calculus
students will benefit by experiencing the varying rates of change in
familiar favorites – perfect for vertical teaming!
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Students progressing to advanced calculus are frequently confounded by the dramatic shift from mechanical to theoretical and from concrete to abstract. This text bridges the gap, offering a systematic development of the real number system and careful treatment of mappings, sequences, limits, continuity, and metric spaces. The first five chapters consist of a systematic development of many of the important properties of the real number system, plus detailed treatment of such concepts as mappings, sequences, limits, and continuity. The sixth and final chapter discusses metric spaces and generalizes many of the earlier concepts and results involving arbitrary metric spaces. An index of axioms and key theorems appears at the end of the book, and more than 300 problems amplify and supplement the material within the text. Geared toward students who have taken several semesters of basic calculus, this volume is an ideal prerequisite for mathematics majors preparing for a two-semester course in advanced calculus. Reprint of the John Wiley & Sons, Inc., New York, 1963$23.95
Introduction to Analysis by Maxwell Rosenlicht Written for junior and senior undergraduates, this remarkably clear and accessible treatment covers set theory, the real number system, metric spaces, continuous functions, Riemann integration, multiple integrals, and more. 1968 edition. read moreGroup Theory by W. R. Scott Here is a clear, well-organized coverage of the most standard theorems, including isomorphism theorems, transformations and subgroups, direct sums, abelian groups, and more. This undergraduate-level text features more than 500 exercises. read more
$21Intermediate Mathematical Analysis by Anthony E. Labarre, Jr. Focusing on concepts rather than techniques, this text deals primarily with real-valued functions of a real variable. Complex numbers appear only in supplements and the last two chapters. 1968 edition. read more
$15Introduction to the Theory of Sets by Joseph Breuer
Howard F. Fehr This undergraduate text develops its subject through observations of the physical world, covering finite sets, cardinal numbers, infinite cardinals, and ordinals. Includes exercises with answers. 1958 edition. read more
$11.95Combinatorial Enumeration by Ian P. Goulden
David M. Jackson Graduate-level text presents mathematical theory and problem-solving techniques associated with enumeration problems, from elementary to research level, for discrete structures and their substructures. Full solutions to 350 exercises. read more
$34.95
Counterexamples in Analysis by Bernard R. Gelbaum
John M. H. Olmsted These counterexamples deal mostly with the part of analysis known as "real variables." Covers the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, functions of 2 variables, plane sets, more. 1962 edition. read more read moreInfinite Sequences and Series by Konrad Knopp Careful presentation of fundamentals of the theory by one of the finest modern expositors of higher mathematics. Covers functions of real and complex variables, arbitrary and null sequences, convergence and divergence, Cauchy's limit theorem, more. read more
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Algebra Buster is simply amazing. Who knew that such an inexpensive program would make my sons grades improve so much. John Davis, TX
I was afraid of algebra equations. After using Algebra Buster, the fear has vanished. In fact, I have almost started enjoying doing my algebra homework (I know, it is hard to believe!) Franklin Bradley, AK
I use this great program for my Algebra lessons, maximize the program and use it as a blackboard. Students just love Algebra Buster presentations Oscar Peterman29:
what is a negative radical
how we use this keyword in java stastic
surface area of triangular prism
latest math trivia with answers
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Why is it important to simplify radical expressions before adding or subtracting?
Why is it important to simplify radical expressions before adding or subtracting? How is adding radical expressions similar to adding polynomial expressions? Use examples to illustrate the concept. Use √ for radical, along with proper grouping symbols.
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This Pre-Algebra course emphasizes the skills necessary to manipulate numbers, solve equations, and understand the general principles that allow mathematical processes. Students will explore topics in Number Theory, Scientific Notation, Linear Functions, Pythagorean Theorem, Transformations, and Bi-variant dataALGEBRA I (COLLEGE PREP) 9-11
This Algebra I course emphasizes the study of functions with tables, graphs, verbal rules and symbolic rules. Students will explore topics in Systems of Linear Functions and Inequalities, Exponential Functions, Quadratic Functions, and Piece-Wise Functions Pre-Algebra.
GEOMETRY (COLLEGE PREP) 9-12
This Geometry course emphasizes the study of congruence and similarity among classes of two and three dimensional geometric objects. Through the introduction of the point, line, plane, and space the students develop an understanding of theorems and postulates which form the foundation of geometry. Students will explore topics in Quadrilaterals, Triangles, Surface Area and Volume, Circles, Trigonometry, and topics on ProbabilityGEOMETRY (HONORS) 9-12COLLEGE PREP)HONORS) 11-12
This course in intermediate algebra reviews the principles studied in Algebra I and further develops these ideas with more advanced material. Topics presented are polynomials and factoring, rational numbers, complex numbers, quadratic functions, exponential, logarithmic, rational and trigonometric functions. Relations and functions are thoroughly explored and used to unify the course. Graphing calculators are implemented in each unit to combine technology into the lessons and it is recommended that each student have one. This course is at a faster pace then the college prep and covers extensions on each unit of study Geometry.
STATISTICS AND DATA GATHERING (COLLEGE PREP) 10-12
This course is a general-purpose introduction to the field of statistics. The content of the course is designed for students with a wide variety of vocational and educational interests. Topics will include: data gathering, frequency distributions, summarizing data, understanding scores, probability, sampling, and simulations. Classroom projects and computer activities will be utilized.
PRE-CALCULUS (COLLEGE PREP) 11-12
Pre-calculus is an advanced form of algebra. The course is intended to prepare students for the study of Assessment will be based on tests, quizzes, homework, and class workPRE-CALCULUS (HONORS) 11-12
Honors Pre-calculus is an accelerated form of Pre-Calculus. The course is intended to rigorously prepare students for the study of AP Honors Pre-calculus includes additional topics not covered in College Prep. Assessment will be based on tests, quizzes, and homework, with a higher emphasis on testsA.P. CALCULUS AB (11-12)
A.P. Calculus is designed to teach students how to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations; understand the meaning of the derivative in terms of a rate of change and local linear approximation and they should be able to use derivatives to solve a variety of problems; understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems; understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus; communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems; model a written description of a physical situation with a function, a differential equation, or an integral; use technology to help solve problems, experiment, interpret results, and verify conclusions; determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement; and develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment, Students are required to take the A.P. test.
A.P. CALCULUS BC (11-12)
A.P. Calculus BC is an extension of A.P. Calculus AB rather than an enhancement; common topics require a similar depth of understanding. Both courses are intended to be challenging and demanding. The content of Calculus BC is designed to qualify the student for placement and credit in a course that is one course beyond that granted for Calculus AB. Students are required to take the A.P. test.
A.P. STATISTICS 11-12
The purpose of this course is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: Exploring data by describing patterns and departures from patterns; sampling and experimentation by planning and conducting a study; anticipating patterns by exploring random phenomena using probability and simulation; and statistical inference by estimating population parameters and testing hypotheses. Students are required to take the A.P. test.
MATH LAB 9
This is a course that is required for 9th grade students who have been identified as in need of reading intervention based upon middle school performance. This elective course will provide students with a second course in pre-algebra to ensure that students have the foundational skills required to be successful in subsequent math courses.
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Linear Algebra, 2nd Edition
This popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been ...Show synopsisThis popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been carefully revised to improve upon its already successful format and approach. In particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory texts on linear algebra.Hide synopsis288 pages) this popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. the second edition has been carefully revised to improve upon its already successful format and approach. in particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory texts on linear algebra. a system of vectors. matrices. elementary row operations. an introduction to determinants. vector spaces. linear mappings. matrices from linear mappings. eigenvalues, eigenvectors, and diagonalization. euclidean spaces. quadratic forms. appendix: mappings. index. (Paperback)
Description:New. 0751401595 ***BRAND-NEW*** FAST Fedex shipping, so you'll...New. 0751401595
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Here we describe the Natural Math program. It is easy to use.
Start with a file whose extension is .nat, for example,
test.nat. This tutorial was created by the file
tutor.nat.
Each line of your file xxx.nat is written in what we call ``natural
math,'' that is, math written as you might naturally express it if you
only had a simple typewriter.
You will use numbers, letters, and symbols, although anything that
can be expressed in symbols can also almost always be expressed in letters.
What the program will do is to convert the natural math file into a LaTEX file. You run it like this:
naturalmath xxx.nat
This will create a file xxx.tex.
Here is an example of lines of input, followed by the output that would
be created.
integral from 0 to infinity of e ^ (-x^2/2) dx
= sqrt (pi over 2)
Each formula is created by a sequence of such lines, terminated by a
a blank line. Let us first give an example, where we attempt to solve
a homework problem. First we give the input, then the output.
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Intro Math for Engineering applications
By: SAC PR Friday, May 18, 2012 Tag: Archived
WHAT San Antonio College to offer free ENGR 1377 – Intro Math for
Engineering Applications. This prototype, noncredit course provides an
application-oriented, hands-on introduction to math topics widely used in core
freshman and sophomore-level engineering and physics courses. Selected topics from Algebra, Calculus,
and Trigonometry are presented within the context of physical applications and
reinforced with extensive examples of their use in core physics and engineering
(e.g., civil, mechanical, electrical) courses. Concepts presented in lecture and recitation sessions are
reinforced through hands-on applications in a laboratory.
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Overview - PM PRE-ALGEBRA STUDENT TEXT
Provide a strong foundation for future math learning.
Designed as a foundation for algebra, this comprehensive program motivates
students as they build the important skills and confidence they need to take
on algebra. Correlated to the NCTM Standards, Pacemaker Pre-Algebra features
an attractive, full-color design that offers predictable and manageable two-page
lessons that promote student success.
Written at a controlled reading level of grades 3-4, students of all abilities
are provided with essential preparation for a variety of testing situations,
including the most widely used standardized tests. This program teaches the
essentials of problem solving using the Polya 4-step approach which provides
step-by-step guidance for building successful problem-solving skills.
Student Edition - Improves student comprehension and retention with
short, predictable, and manageable two-page lessons, with frequent opportunities
for review, margin notes that highlight important points, and guided practice
to build student confidence. It teaches key vocabulary in each chapter, aids
goal-setting with clear learning objectives, and helps students see the relevance
of algebra with real-life examples.
Student Workbook - Fosters content understanding and skill development
and retention through practice worksheets and critical-thinking questions that
are correlated to selected lessons in the Student Edition.
Teacher's Answer Edition - Provides a Getting Started activity before
each lesson, easily accessible answers at point-of-use, instructions on how
to use the program, helpful teacher notes and tips, and ESL notes for vocabulary
words.
Classroom Resource Binder - Extend and support lessons with a wealth
of reproducible activities that review, reinforce, and enrich key skills and
concepts covered in the Student Edition. This comprehensive resource includes
a variety of assessment opportunities, as well as organization and planning
charts to ease classroom management.
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Key to Geometry offers a non-intimidating way to prepare students for formal geometry as they do step-by-step constructions.
Students begin by drawing lines, bisecting angles, and reproducing segments using only a pencil, compass, and straightedge.
Later they do sophisticated constructions involving more than a dozen steps and are prompted to form their own generalizations.
When they finish, students have been introduced to 134 geometric terms and are ready to tackle formal proofs
Key To… When it comes to higher math, if either you (teaching) or your student (learning) lack confidence, then this curriculum may be your answer. These consumable work booklets are so well laid out and easy to understand, he'll finally be able to say, "I got it!" Work through each at your own pace. What you won't find: Lots of wordy explanations that leave you going, "Huh??" What you will find: Short, simple explanations with lots of examples, and a handful of well-designed problems on each page.
The Key To Geometry Answers and Notes are available separately or purchased as a complete set.
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Modern Algebra: A Historical Approach
Presenting a dynamic new historical approach to the study of abstract algebra Much of modern algebra has its roots in the solvability of equations ...Show synopsisPresenting a dynamic new historical approach to the study of abstract algebra Much of modern algebra has its roots in the solvability of equations by radicals. Most introductory modern algebra texts, however, tend to employ an axiomatic strategy, beginning with abstract groups and ending with fields, while ignoring the issue of solvability. This book, by contrast, traces the historical development of modern algebra from the Renaissance solution of the cubic equation to Galois's expositions of his major ideas. Professor Saul Stahl gives readers a unique opportunity to view the evolution of modern algebra as a consistent movement from concrete problems to abstract principles. By including several pertinent excerpts from the writings of mathematicians whose works kept the movement going, he helps students experience the drama of discovery behind the formulation of pivotal ideas. Students also develop a more immediate and well-grounded understanding of how equations lead to permutation groups and what those groups can tell us about multivariate functions and the 15-puzzle. To further this understanding, Dr. Stahl presents abstract groups as unifying principles rather than collections of "interesting" axioms. This fascinating, highly effective alternative to traditional survey-style expositions sets a new standard for undergraduate mathematics texts and supplies a firm foundation that will continue to support students' understanding of the subject long after the course work is completed. An Instructor's Manual presenting detailed solutions to all the problems in the book is available upon request from the Wiley editorial department.Hide synopsis
Description:Good. Discoloration, Tanning or Foxing on cover and pages. -Dog...Good. Discoloration, Tanning or Foxing on cover and pages. -Dog Eared Pages
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MyMathLab
MyMathLab Algebra 1 and Algebra 2 (Martin-Gay)
MyMathLab Algrebra 1 and MyMathLab Algebra 2 by Elayn Martin-Gay provide teachers with rich and flexible course materials that allow them to teach in a traditional classroom, a lab setting, or completely online. An Implementation Guide of mini lessons and online course management tools make it easier to teach Algebra 1 and Algebra 2 to students of all levels.
Access codes are available in two ways: printed access cards/booklets or digitally-delivered via email. Please use the ISBNs below to order printed access cards, or visit k12oasis.pearson.com to order codes for digital delivery.
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Geometry : From Euclid To Knots - 03 edition
Summary: The main purpose of this book is to inform the reader about the formal, or axiomatic, development of Euclidean geometry. It follows Euclid's classic text Elements very closely, with an excellent organization of the subject matter, and over 1,000 practice exercises provide the reader with hands-on experience in solving geometrical problems. Providing a historical perspective about the study of plane geometry, this book covers such topics as other geometries, the neutr...show moreal geometry of the triangle, non-neutral Euclidean geometry, circles and regular polygons, projective geometry, symmetries, inversions, informal topology, graphs, surfaces, and knots and links. ...show less
2002 Hard cover New in very good dust jacket. New! Very Clean Hardcover! No Markings! Glued binding. Paper over boards. 458 p. Contains: Illustrations. Audience: General/trade. The main purpose of t...show morehis book is to inform the reader about the formal, or axiomatic, development of Euclidean geometry. It follows Euclid's classic text Elements very closely, with an excellent organization of the subject matter, and over 1, 000 practice exercises provide the reader with hands-on experience in solving geometrical problems. Providing a historical perspective about the study of plane geometry, this book covers such topics as other geometries, the neutral geometry of the triangle, non-neutral Euclidean geometry, circles and regular polygons, projective geometry, symmetries, inversions, informal topology, graphs, surfaces, and knots and links
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East Candia Prealgebra Linux systems programming, Linux shells, common Linux graphics systems (X-Windows), etc. am eager to share this experience with beginning to intermediate users. Most students find geometry to be more abstract and challenging than algebra. While the outcomes may be easily understood for someone with a good sense of spatial relationships, formulating a proof requires careful discipline not previously expected.
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Online Companion: Technical Mathematics with Calculus, 3e
Tech Math Web Links
Where can
you go to get more applications? The World Wide Web is a source
of information in many areas. Listed below are a few of the links
that you can use as a starting point for improving your technical
mathematics course. The links are grouped in the categories of applications,
calculators, mathematics,
aerospace, and construction.
APPLICATIONS
Teachers of technical mathematics courses are always looking for
good examples that show how mathematics is used.
British Columbia Institute of Technology (BCIT)
This table at BCIT
applications page has links to examples of how various areas
of mathematics are applied to various areas of technology. The examples
have been created by members of the BCIT mathematics department.
More examples are added each year so return to this site often.
The AMATYC web page has links to many online resources. Among them
is this link to Real
World Applications which has links to 16 web pages.
The MathWorks Project
The MathWorks
Project has developed a resource and guide that can be used
to create an interdisciplinary course or club in science, technology,
engineering, and mathematics. Even if you do not want to create
a course, student will find the applications interesting.
Graphing Calculator Program (not free)
A much more comprehensive graphing calculator program for your Windows
or Macintosh computer, that is not free, is available from graphing
calculator. This program graphs functions in two, three, and
four dimensions, in explicit, implicit, or parametric formats. It
will also graph inequalities, contour plots, density plots and vector
fields. You may use rectangular, polar, cylindrical, or spherical
coordinates. The program can be used to solve equations numerically,
graphically, or symbolically as well as create mathematical movies
and web pages.
Casio
Information about Casio calculators can be found at Casio.
For example, by clicking on "Support" you can get the
latest software designed to augment Casio products as well as technical
support or replacement manuals. Clicking on "Education"
leads to other items such as programs you can download for your
Casio Calculator to solve a variety of mathematical and statistical
problems.
Hewlett Packard
The HP
calculators web page is used to connect to the specific web
page for your model of HP calculator. Among the things you will
find at these web pages you will find how to solve a problem with
your calculator and software & drivers updates.
Texas Instruments
To obtain information about TI calculators go to TI
Cares. From this initial page you can go to pages that have
programs and software updates for your model of TI calculator.
MATHEMATICS
AMATYC
The American Mathematical Association of Two-Year Colleges (AMATYC)
is the only organization exclusively devoted to providing a national
forum for the improvement of the instruction of the mathematics
in the first two years of college. Its Online
Resources page has links to many other web pages on mathematics
education.
Careers in Mathematics
The Association for Women in Mathematics has a short phamphlet,
Careers
in Mathematics, that is available on-line. The organization
has another publication, Careers
That Count, that discusses mathematics career opportunities
and "gives you the opportunity to meet some real mathematicians
and to read about the work they do."
The Vocational
Biographies materials can help is by providing you with real-life
career success stories to inspire and enlighten your students. You
can show your students careers through the eyes of real people in
nearly every walk of life. The 75 biographies for mathematics range
from from Assistant Chief Flight Instructor to Transportation Planner.
AEROSPACE
NASA and AMATYC
These two publications, Mathematics
Explorations I & II were developed through a joint effort
of NASA and AMATYC and a desire to create exciting mathematics classroom
materials based on NASA space activities.
NASA
The main NASA
home page allows you to explore many on-line resources.
CONSTRUCTION
Buildings
Great
Buildings Online has a searchable database or photos, history
and statistics of some of the world's most famous buildings like
the Chrysler Building and the Eiffel Tower. Drawings and 3-D models
of many of the building are available for free and can be viewed
on-line.
Bridges
Information about bridge project, past, present, and future can
be found at Great
Bridges.
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Mathematics Course Descriptions
The following course descriptions are from the Clarke University 2012-2013 Academic Catalog.
COURSE DESCRIPTIONS: MATHEMATICS
MATH 005 ELEMENTARY ALGEBRA 3 hours Students learn the numeric and algebraic skills necessary for future mathematics work. The focus is on the development of the real number system with emphasis on relationships and applications. Credit for this course does not count toward the 124 credits required for graduation. Students who are enrolled in this course may use it to fulfill athletic and financial aid eligibility.
MATH 090 INTERMEDIATE ALGEBRA 3 hours Explores linear, quadratic, exponential and logarithmic functions. This course may include polynomial functions of higher order, rational functions and systems of equations. Students develop their skills by approaching functions numerically, algebraically, graphically and verbally. Credit for this course does not count toward the 124 credits required for graduation. Students who are required to enroll in this course may use it to fulfill athletic and financial aid eligibility.
MATH 105 FOUNDATIONS OF MATHEMATICS I 3 hours Designed for strengthening the mathematical backgrounds of elementary teachers, this course may cover topics such as numeration systems, number systems, problem solving, and topics in number theory, statistics and geometry.
MATH 110 MATH AS A LIBERAL ART 3 hours Enables students to appreciate mathematics in the world around them. The emphasis is on reading, writing and conceptual understanding as opposed to rote skills. Topics may include networks, voting, games, statistics, coding, tiling, symmetry and patterns, infinity, personal finance, and the fourth dimension. This course is designated as a mathematics and natural science division general education course, and fulfills the university mathematics requirement.
MATH 117 PRECALCULUS WITH ALGEBRA 3 hours Oriented to preparation for calculus, this course continues the exploration of functions, including algebraic, exponential, and trigonometric functions. Students learn about functions through symbolic, numerical, graphical and verbal techniques. This course may not be taken for credit if a grade of C or above was achieved in a higher mathematics course (with the single exception of MATH 220 Statistics). This course is designated as a mathematics and natural science division general education course, and fulfills the university mathematics proficiency requirement. Prerequisites: Three years of high school mathematics or equivalent and appropriate placement.
MATH 180 TOPICS IN MATHEMATICS 1-3 hours A study of basic concepts in various areas of mathematics.
MATH 220 STATISTICS 3 hours Using technology and real-world data, this course explores descriptive and inferential statistics in preparation for research in various fields of study. A TI-83 graphing calculator or equivalent may be required. This course is designated as a mathematics and natural science division general education course, and fulfills the university mathematics proficiency requirement. Prerequisite: MATH 090 or equivalent or appropriate placement.
MATH 225 CALCULUS I 4 hours Includes the study of functions via rates of change. The main tool is the derivative, and it is approached from algebraic, numerical and graphical points of view. There are applications of differentiation and an introduction to integration. Meets five days per week, which may include time in the computer lab. A TI-83 graphing calculator or equivalent is required. This course is designated as a mathematics and natural science division general education course, and fulfills the university mathematics proficiency requirement. Prerequisite: Four years of high school mathematics or equivalent or MATH 117.
MATH 226 CALCULUS II 4 hours Sequel to MATH 225 Calculus I, in which functions are studied via integration. Topics include applications of the definite integral and an introduction to infinite series and differential equations. Meets five days per week, which may include time in the computer lab. A TI-83 graphing calculator or equivalent is required. Prerequisite: MATH 225 or consent.
MATH 230 STATISTICS FOR MAJORS 3 hours This course explores the statistical concepts in the MATH 220 course at a depth more appropriate for mathematics majors and minors. Prerequisite: permission of department.
MATH 280 TOPICS IN MATHEMATICS 1-3 hours Students will study basic concepts in various areas of mathematics.
MATH 336 GEOMETRY SEMINAR 3 hours Topics include Euclidean and non-Euclidean geometries. Emphasis is on student exploration, communication and research skills. This course is offered every other year. Prerequisite: MATH 226 or consent.
MATH 395 INTERNSHIP, UPPER DIVISION CV A professional experience in mathematics as arranged with department or off-campus supervisors.
MATH 443 ABSTRACT ALGEBRA 3 hours Includes the study of abstract algebraic structures, including groups, rings and fields. This course is offered every other year. Prerequisites: MATH 226 and MATH 333 or consent.
MATH 487 RESEARCH 1-4 hours Students read and conduct research or do creative work on a problem in mathematics and/or computer science.
MATH 499 CAPSTONE: MATHEMATICS SEMINAR 3 hours This course focuses on discipline-specific topics and expands to include breadth of knowledge and synthesis. Interdisciplinary integration of knowledge and research is emphasized. General education and major outcomes are integral to course assessment. Prerequisites: Ordinarily, a student must have senior standing with a minimum of 42 credit hours in general education completed, MATH 333 Linear Algebra, MATH 336 Geometry Seminar, and consent.
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Preface: There are many
different kinds of numbers and ways to express, compare, and relate them.
Wherever we turn, whatever we read, we find numbers used to describe situations
while the relations between numbers help us understand, plan, and control
these situations. The special kind of relation between numbers that
is called a function, first recognized explicitly by Leibniz, evolved
through the work of the Swiss mathematician Leonhard Euler (1707-1783)
( pronounced "Oiler") and the French mathematician Joseph Lagrange (1763-1813)
to become a cornerstone for mathematics in the 20th century. Though Newton,
Leibniz and the other early developers of the calculus did not have the
current concept of a function to assist them, we would handicap ourselves
seriously if we did not recognize and use the function concept today.
Most
students in a precalculus course have had some experience with functions in
their previous studies. A complete and detailed introduction and review of
function concepts and examples early in this course would delay our reaching
the object of our studies, the analysis of the functional relations that are
the background for a calculus course. For this reason, our review
in this section will be limited to an exploration of the basic nature of
functions and ways to think about functions that make sense for describing,
visualizing and analyzing them. Thus our attention is directed primarily
at functions and change. Throughout this course we will develop a catalog
of definitions and fundamental properties of functions used in the further
study of mathematics, especially in the calculus.
The Trip: One simple and very useful context for understanding
functions is that of a car traveling on a road, what we might call "the trip."
Numbers that are interesting here are often found on the dashboard of the
car and by the roadside. They measure such quantities as the amount of gas
in the gas tank, the distance traveled during this trip, the distance traveled
by the car since it was "new," the temperature
of the engine, the mileage markers on the side of the road, the speed limit
markers, the speedometer giving the car's speed, the tachometer indicating
the rate at which the engine is turning, and the clock.
For
sure when we are considering any trip in a car we are aware that time is changing
whether the car is standing still or moving very quickly. Thinking about the
relations between these quantities can help us gain some informal understanding
of functions and how they can be used to understand the trip. For example,
the distance the car has traveled is related to the gas the car's engine
has consumed. The temperature of the engine is related to the rate at which
the engine is turning. All the measurements can be thought of as related
to time, since when we make a measurement we do that at a particular time.
The concept of function can be applied in
many contexts beyond that of real numbers. For the purpose of beginning
our study of functions, we will use the concept only for examining functions
that relate real numbers. These functions are usually described as real
valued functions of a real variable. Other function concepts studied
in mathematics treat relations between three or more variable numerical
quantities as well as variables that are not numbers but geometric points
and vectors.
Static Functions as Relations:
The mathematical concept of function connects quantities while suggesting
informally something stronger than just a relation between the quantities.
In one sense when we describe a function relationship between variable
quantities, we are establishing a priority or an order to the information.
When two variables are related we often can identify one as being an independent,
governing, or controlling variable while describing the other as a
dependent, regulated, or controlled variable. These descriptives indicate
an important quality of a function, namely that knowledge, assignment,
or specification of one variable's value will determine the value of the other
variable. In this sense we can consider each of the variables in
the trip context as functions controlled by the time variable. It is not that
there is any necessary scientific or causal relation between time and the
position of the car (or any of the car variables). The function language merely
indicates that knowing the time should allow us to determine the position
of the car.
A function connects the information about the variables by pairing the data
and assigning a priority to the pairs.Knowing the first number of
a pair uniquely determines the pair, and thus the second number of the pair.
Dynamic Functions -
paired changes. There is a second way to
think of functions relating variables. We consider the function as
a mechanism or interpreter that transforms one measurement or number
into a second number. [It helps to avoid thinking of this process
as being strictly causal in physical situations, even though there is
some strong connection to causality with a priority to the order of
the relation.]
In the trip context we can think of the car dashboard as a mechanism.
Knowing the time by reading the clock allows us to determine what the
reading on the odometer is. The distance our car has traveled is a
function of time. One can also connect the amount of gas in the
gas tank, measured by the gas gauge, to the distance the car has traveled
during a trip, thus the amount of gas is a function of the distance
traveled.
The history of clocks and watches and the
measurement of time are a fascinating part of the development of science
and mathematics. Time certainly plays an important role in our lives
today, one that we sometimes overlook too easily. We take for granted
our ability to measure time with precision to the microsecond, but in
1773 the British government paid a 20,000 pound sterling prize to John
Harrison for developing a very accurate navigation chronometer with
which to calculate longitude.
This mechanistic view of functions goes particularly well
with the use of calculators. You enter a number on the display, push
a single button and the (resulting) number on the display is usually changed.
The terms "input" and "output" are used here to
describe the values of the controlling and the controlled variables respectively.
When thought of as a machine, a function will process some numbers while
being unable to process others. For example, if the process returns the multiplicative
inverse (reciprocal) of a number, then the process will work well on all numbers
with the single exception of 0. Or if the process returns the real number
square root of a given number, then the process will operate on non negative
real numbers but will fail to return a result for negative real numbers. [Remember,
the square root of a negative number is not a real number, but an imaginary
number.]
Figure illustrating a function machine.
We
refer to the collection of numbers which the function can process as
the source or domain of the function. Thus the function that returns
the reciprocal of a number has a source of the set of all non zero real numbers,
while the source for the real valued square root function is the set of non negative
real numbers. The source of a function when not described explicitly is assumed
to contain all the inputs that work.
It is useful to describe the source of a function so you can avoid errors
that sometimes arise from applying the function in meaningless situations.
The most common algebraic restrictions on the source of a function arise from
division by 0 and finding square roots of negative numbers.
For example,
the function which returns the number `1/{1-x^2}` for an input of the number x can have a source that includes all real
numbers except 1 and -1.
As a second
example, the function that returns the number `sqrt{1-x^2}`
for the input number x can have a source that includes only numbers
in the interval [-1,1].
Notation
for Describing Functions: As these last examples demonstrate, it would
be convenient to have a notation that allows us to describe a function either
from the static view of a collection of ordered pairs or from the dynamic
view by designating what the function yields as output for a specified, yet
arbitrary, input. The notation and vocabulary that has evolved (and is still
evolving) must distinguish at least three things: the first number of
the pair, called the argument or the input number, the second number of the
pair, called the result, value, or the output number, and the designation
or name for the function.
Variable
Relations: A traditional approach to this
notation considers 2 variable names, say x and y, with
x representing the first number for the function and y
representing the second number. In this approach there is no name
for the function, only a statement that y is a function of
x and an equation describing an explicit relation between the
variables, such as y = 2x+3, or an implicit relation
such as `x = y^2` and `y>=0`. This approach
abuses the notation since the letter y refers both
to the number and the function relation of that number to the number
x.
For example, when the function y is described by the equation
y = 2x+3 we can determine the value of y
when x = 6 by substituting 6 for x in the formula 2x+3
to find y = 2 (6)+3=15.
The ordered pairs determined by this function have
the form `(x,y)` where `y=2x+3` or `(x,2x+3)`. For `x = 6`, the information
from the function is that (6,15) is one of the pairs of the function.
The evaluation of y when x=6 has a traditional notation
of
`y| _{x=6}=2x+3| _{x=6}=2(6)+3=15`
Beware: A common error is confuse the
parentheses in this notation for the value of a function for
the parentheses used frequently to collect terms in an expression
to be treated as a single number in some algebraic calculation like
multiplication.
By convention, when f is a function, the symbols f(t+5)
means the value of f for the number described by t+5,
it does not mean the product of the number f by the number
(t+5).
Function Values: A more contemporary
approach to the notation (based on the notation used by Lagrange in his work
at the end of the 18th century), with which you are no doubt somewhat familiar,
assigns to the function a specific letter or symbol that suggests what
the function does or at least gives the function a recognizable name.
For example, the function might be the square root function. The square
root function is assigned the shorthand name sqr, or, as with less
familiar functions, it might be named temporarily for the purpose of discussion
with the symbol f or g. Once the name has been given, the relation
between input and output, argument and result, can be described in a number
of ways. A variable name is given, or presumed, for the input, like x
or t, and then the output is described as the value of the function
at x or t, which is denoted most often as f(x)
or f(t). Notice that this last notation for the value
of the function at x has four symbols: f denotes the name of
the function, x denotes the name of the number to be input, the two
parentheses separate the name of the function from the name of the variable.
A description of the function can then be accomplished by giving a
procedure for finding the value of f(x), such as f(x)=2x+3.Once `f` has been described and thus defined for the purpose of
discussion, the notation allows us to denote the value of the function
that corresponds to the number 6 as `f(6)`. This number can be computed
using the defining equation: `f(6)=2(6)+3=9`. Likewise the
value of the function for the number `-4` is denoted `f(-4)`, and can be
computed to be `f(-4)=2(-4)+3=-5`. For the expression `t+5`,
which can represent a number, the value of the function is denoted `f(t+5)`,
which can be simplified by using the definition of `f` so that `f(t+5)=2(t+5)+3
= 2t+13`.
The function
value notation is sometimes combined with the two variable notation in statements
such as "suppose `y` is a function of `x`, with `y=f(x)=2x+3`." Then evaluation can be expressed by the equation: `y| _{x=6}=f(6)=2(6)+3=15`.
Transformation Notation: Another
current approach to describing the input/output information of a function
is to give the name of the function, then a name for an input variable followed
by an arrow and an expression describing the output that results from the
input. For example, `f : x -> 2x+3`
or more generally `f : x -> y
= f(x)`.
The arrow in the notation helps convey visually the dynamic aspect of a function
that transforms a number. This notation helps underscore the active nature
of those functions that in some way do require a construction of the resulting
number values by some conceptual (and perhaps even mechanical or electronic)
process. Multi-case functions: In
some contexts a function cannot be described by a single simple algebraic
formula using well known conventional functions. This can be for many reasons:
There
may be no algebraic formula that captures that the relation;
The function
may arise from a non algebraic context;
The function
may piece together some more simple function.
To piece together simple functions, the most
common method is to establish some tests on the source numbers. These tests
determine precisely what the appropriate method is for determining the function's
value.
Here is an example of a function F defined by cases.
Case 1: if `x <= 0`, then `F(x)
= 0`;
Case 2: if `0< x <4` then `F(x)
= {x^2}/16, and
Case 3: if `4 <= x` then `F(x)=1`.
To
use this definition of `F` we need to first check under which case the
argument for evaluation falls, then follow the function's rule as appropriate.
So, to
find `F(-3)` we see first that `-3` falls under Case 1, so `F(-3)
= 0`.
To
find `F(3)` we check that `3` is under Case 2, so `F(3) = {3^2}/16
= 9/16`.
Finally
to find `F(5)` we use Case 3, so `F(5) = 1`.
The conventional notation for this function's definition is expressed by
a multi-lined equation:
Functions
and Tables: In some cases we do not know
any formula that precisely matches the pairings for a function. This
is in fact mot common when we look at any function that arises
from real world phenomena. We may have a formula that provides
a good estimate or we have only some data . Sometimes this data is
measured and recorded at selected numbers (or times). In this case
the information we have for the function may be displayed in the form
of a table.
For example if f is the function
recording the room temperature we may know only that f(0)
= 60, f(4) = 55, f(8) = 65, f(12) = 68, f(16)
= 68, f(20) = 68, f(24) = 60. It is convenient to
put this data in a table. See Table 1.
Be careful not to assume too much about a function that is represented
only by a table of values at selected numbers. For example, although
f(0)= 60 and f(4)=55, you cannot assume that f(2)
is between 60 and 55.
x
f(x)
0
60
4
55
8
65
12
68
16
68
20
68
24
60
Table 1
Non algebraic [Geometric] Functions: If we think of the room temperature
function again, we can imagine a device recording the temperature graphically
with a sheet of paper moving at a steady rate under a marking pen attached
to a temperature sensitive device that moves the pen depending on the room
temperature. This would give a rather different record of the room's temperature
as a function of time. Once scales are established on the paper for the time
and temperature, we can give a better estimate of the temperature at many
different times by being able to read the graph. This graphical presentation
of a function gives much more information than a table, but you should be
careful here as well not to infer too much from the graph, especially since
the mechanism, the recording instrument, and the scales all can contribute
to the imprecision of the information.
Curves and
coordinates in Cartesian geometry. Using numbers to study figures in geometry
is not a very recent part of mathematics. And using figures to study connections
between numbers is also not new to math. In your previous course work you have
studied the algebraic relations between coordinates of points on lines, circles,
and other curves and conversely have graphed figures to illustrate the relation
between numbers involved in an equation. The key in these correspondences
has been the connection usually attributed to Rene Descartes in analyzing
curves with numbers through use of common measurements and variables related
by equations determined by the geometry, called analytic (or Cartesian) geometry.
Coordinate system analysis applied to planar curves can lead to function
relations that are commonly encountered in science. Measuring devices record
the changes during some experiment as a curve. The variables connected to
the curve are associated with the experiment using rectangular coordinates
and then the curve is interpreted as representing the relation between these
variables. Without any prior knowledge of the relation of the variables,
neither can be assumed to be controlling, but there is a general convention
to consider the horizontal, first coordinate variable (X) as controlling
with the vertical and second coordinate (Y) as the controlled variable.
But for a curve to give a function relation between these variable there
needs to be more either understood implicitly or made explicit.
In particular, when given a value for X we need a way to determine a value
for the Y variable using the curve as the mechanism for that determination.
This is easy enough in many familiar cases. For the given value of X, say
a, find a point P on the curve with that value as its first coordinate,
i.e., P has coordinates (a,b) for some number b. When there
is only one point P with first coordinate a on the curve, then the
value b is uniquely determined by the curve, and b is the value
of the Y variable corresponding to a. In this case we say that the
curve has determined Y as an explicit function of X and assume
we are using the coordinate convention just described.
The Slope
of the Tangent to a Curve as a Function: There are other functions we
can associate with curves in analytic geometry. For many curves we can determine
at each point on the curve a line that very close to the point looks indistinguishable
from the curve and yet close to the point meets the curve only once. As mentioned
in section 0.A, these lines are sometimes referred to as touching or tangent
to the curve at the point, or tangent lines. To repeat the example from 0.A,
to find the tangent line to a circle at a point P, you need only draw a radius
from the center of the circle to the point P and then construct the line perpendicular
to the radius at P, which by Euclidean geometry must be the tangent line.
So how does this give rise to a function using numbers?
Consider the case when the curve determines Y as an explicit function of X.
For a given value of X, say a, we again find a unique point P
on the curve with that value as its first coordinate, i.e., P has coordinates
(a,b) for a unique number b. Now it sometimes turns out that
we can find a unique, non vertical tangent line to the curve at P and determine
the slope of this line, which we will call m. Since the value of
m is uniquely determined using the curve from the value of X, m
is a function of X. The value of m is derived geometrically from the
original curve using the measurement of the slope of the tangent line at the
point P determined by a.
Area of Geometric
Figures as Functions: One of the most frequently encountered problems
in geometry is that of finding a general method for determining the area
enclosed by a class of planar figures. You have learned formulae for areas
of squares, rectangles, triangles, trapezoids, circles, and perhaps two or
three other general shapes. The measurement of these areas is usually based
on other measured features of the figures, such as lengths of sides or relevant
line segments and sometimes even the size of angles.
The relationships between area and these other variables of geometric figures
can often be described as functions. For example we can consider rectangles
that have a base of length 20 centimeters and determine the area when the
altitude has length l centimeters. Not too hard.
The area is 20 l square centimeters. Or in the same setup, we can determine
the length of the altitude when the rectangle has area A square centimeters
with almost as little effort to be `A/20` centimeters. It requires a
little more thought to determine the length of the diagonal of the rectangle
with this setup based on the area A to be `sqrt{A^2+1600}/20`.
Another familiar
area relation is found in the circle where the equation `A=pir^2` allows
us to determine either the area or the radius of a circle by knowing the other
measurement. A slightly more subtle area relation was described in section
0.A. As the example there demonstrated there are many ways to measure a region
in the plane, like a triangle, and passing a line across that region can give
an area function determined by the position of the line.
Implicit
functions: Let's consider the equation 3X + 4Y = 24 where X and
Y represent real numbers. There are many possible choices for X and
Y, some of which will make the equation true (say X = 4 and
Y = 3 ) and some of which will not (say X = 1 and Y = 2). Given a
value for X there is one and only one value for Y which will
make the equation true. Thus if X = 1 then for the equation to be
true we have 3 + 4Y = 24 , so 4Y = 21 and Y = 5.25. Since the equation
determines a unique value for Y from any choice of X, we can say that
the equation has determined Y implicitly as a function of X.
A more
subtle relation is presented by the equation X2 +
Y2 = 25. This equation can be satisfied by many functions.
For example `f(x)=sqrt{25-x^2}`,
`g(x)=-sqrt(25-x^2)`,
or .
Here the equation does not determine a single value of one variable
from the choice of another value but allows many possible functions
which will satisfy the equation when the Y is determined by one of
these functions. In this case the functions are described as
being defined implicitly by the equation.
Comment: With only a limited list of function
values there is no way to tell what the function might have
for its graph without assuming some more restrictive qualities.
For example, suppose the graph of the function is known
to pass through two points with coordinates (0,0) and (1,1).
You might think this is the function with `f(x) = x`, but in fact
it could be `f(x)=x^2` or `f(x) = x^3`
or
`f(x) = x^2 + x - x^3`.
In fact if we consider all functions with f(0)= 0
and separately those functions with f(1)=1, we are only asking
for those functions that satisfy both conditions, which doesn't
seem like a large restriction when we think of all the different
ways we could connect the points with coordinates (0,0) and (1,1)
in a Cartesian plane.
Visualizing
Functions and Transformation Figures: The key idea
in visualizing functions with mapping diagrams or transformation figures is
to have two parallel number lines representing the source (domain) and
the target (range). The function can be thought of as a process relating
points (numbers) on these two lines.
A point element on the source line is
chosen which corresponds to a number. The function is applied to that number,
and the resulting value is found represented on the target line. An arrow
drawn from the point on the source line to the corresponding point on the
target line visualizes the relation between the corresponding numbers.
In
one sense, the transformation figure is a visualization of a function
table. The numbers in the two columns of the table are represented
by points on the two lines in the figure. The function relation
that the table displays implicitly by having corresponding numbers
in the same row is visualized in the figure by the arrow.
While the relative size of the numbers in the target column of the
table is not represented in the display, the transformation figure
uses the number line order to represent this aspect of the function's
values.
Here is an illustration that should help you see
some of these features. You can work on other examples after this
one to begin to see some of the power of this visualization. [ To
see a dynamic example of a transformation figure for linear functions,
follow
this link.]
Example: Suppose f(x)=2x+
3. Table 1 shows a selection of the values this function relates, while
this same information is visualized in Figure 4. Notice that larger
numbers in the source column of the table correspond to larger values
in the target column. On the transformation figure this feature can
be seen by the fact that the lines connecting the corresponding points
on the source and target lines do not cross. This is evidence of a function
with increasing values.
x
f(x)=2x+3
5
13
4
11
3
9
2
7
1
5
0
3
-1
1
-2
-1
-3
-3
-4
-5
-5
-7
Table 1
Figure 4
Graphs
of Functions and Other Relations: In your
previous work with functions and equations you have worked extensively with
the graphical visualization using Cartesian coordinates for the plane to identify
the function pairing of numbers.
In
the graph of a function f we identify the pair of numbers a
and f(a) with the point in the plane with coordinates (a,f(a)).
We can plot marks at many of these points but when the domain of the function
is an interval or as is more common all real numbers, we cannot hope to plot
all the points. Instead we try to give a sense of how the points are related
by drawing a curve that passes through some points that are known to be on
the graph of the function. In doing this we are drawing figures much as students
in elementary school draw figures by connecting the dots in order, or as economists
graph the hour to hour price of some stock on the stock market or as a chemist
would visualize the minute by minute temperature reading on a laboratory thermometer
during an experiment.
Here
are some examples of transformation figures and graphs. On the left are the
tables of values for the functions at selected points, while on the right
are the corresponding figures and graphs. [Graphs and Figures made using Winplot.]
Example
0.B.2 f(x) = x
x
f(x)=x
2
2
1
1
0
0
-1
-1
-2
-2
Example
0.B.3 f(x) = -x
x
f(x)=-x
2
-2
1
-1
0
0
-1
1
-2
2
Example
0.B.4 f(x) = |x|
x
f(x)=|x|
2
2
1
1
0
0
-1
1
-2
2
Example
0.B.5 f(x) = 2x
x
f(x)=2x
2
4
1
2
0
0
-1
-2
-2
-4
Example
0.B.6 f(x) = x+1
x
f(x)=x+1
2
3
1
2
0
1
-1
0
-2
-1
Example
0.B.7 f(x) = x 2
x
f(x)=x 2
2
4
1
1
0
0
-1
1
-2
4
Example
0.B.8 f(x) = 1/x
x
f(x)=1/x
2
0.5
1
1
0
??
-1
-1
-2
-0.5
Example
0.B.9 f(x) = 2x
x
f(x)=2x
2
4
1
2
0
1
-1
0.5
-2
0.25
Example
0.B.10 f(x) = 3
x
f(x)=3
2
3
1
3
0
3
-1
3
-2
3
Example
0.B.11 f(x) =-2x + 1
x
f(x)=-2x+1
2
-3
1
-1
0
1
-1
3
-2
5
Exercises: The exercises for this section cover material
which you may recall from previous course work in algebra. You may not recall
all of these topics or how to do the problems precisely. You may want to refer
to the texts or notes from your previous courses in mathematics if you find
these difficult. The skills needed to solve these problems will be important
in the work ahead- so be careful to identify any difficulties you have with
these problems and try to remedy any misunderstandings as you proceed. 1. For this problem let f be defined by f (x) = 5x
2 + 3. a) Find the following. Simplify you answer when possible.
i) f(1)
iii) f(1+h)
ii) f(h)
iv) [f(1+h) - f(1)]/h
b) Find any
number(s) z where f(z) = 23. c) For which values of x is
f(x) < 23? Express your answer as an interval.
2. USING INTERVAL
NOTATION, express the largest set of real numbers that can serve as the
domain of each of the following functions:
a)
f(x) = (4 - x 2)/(x + 2)
b) g(x) = 1/[(4-x 2)]
3. Suppose that F is defined by .
Sketch a transformation figure and a complete graph of f.
Determine the domain and the range of f.
4. Solve for x:
a)
3x-2 = 3 7-2x b) 4
3x = 8 c) 1/3(x
- 5) = 2 d) 1/30 - 1/x = 1/6
5. Boyle's law states that, for a certain gas P*V = 320, where P is
pressure and V is volume.
(a) Draw a complete
graph representing this situation. Label your axes and write an equation for
each asymptote.
(b) If `8 <= V <= 40`, what are the corresponding values of
P?
6. Let f(x) = x 2 + 4x
- 5. A. Find the axis of symmetry and the vertex of f. B. Sketch
a graph of f labeling clearly the coordinates of the vertex and the
X- and Y- intercepts.
7. Old McDonald has a farm ,and on that farm she has some sheep and a
pasture with a 200 meter long stone wall. She wants to enclose a rectangular
section of the pasture for a small sheep pen using the wall for one side and
140 meters of fencing she was given by her uncle Milo for the other three
sides.
A. Let x denote the length of the fence that will be attached to the
wall used as a side for the pen. Which of the following equations express
the area of the pen, A, as a function of x?
a. A = x ( 70 - x)
b. A = 2 x ( 140 - x)
c. A = x ( 140 - x)
d. A = x ( 200 - (1/2)x)
d. A = 2 x ( 70 - x)
e. A = 2 x ( 200 - x)
17. Write a short story about cooking dinner. Discuss briefly
those aspects of the meal's preparation that might be related by functions.
18. Write a short story about going shopping. Discuss briefly
those aspects of the shopping that might be related by functions.
19. Write a short description of an ecological system.
Discuss briefly those aspects of the system that might be related by functions.
20. Write a short description of the human body. Discuss
briefly those aspects of the body that might be related by functions.
21. Write a short story about an athletic event or sports
competition. Discuss briefly those aspects of the story that might be related
by functions.
|
Gladewater Calculus worked two years on a Carl Perkins federal grant project to support using applied learning techniques in teaching algebra and made presentations on the project at state-wide TechPrep conventions in Austin. Algebra is like arithmetic on steroids! Its power comes from making math apply to general situations instead of only specific problems
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Description
Pre-calculus introduces the concepts that define calculus and provides the tools needed to tackle this branch of mathematics. This course focuses on non-trigonometry pre-calculus topics such as limits, parametric equations, and function inverses. A solid understanding of algebra will help you get the most out of this series—and set you on your way to taking on calculus itself.
Note that given the length of these lessons, you may want to adjust your settings to receive one or two lessons a week.
Opening Lines (Experimental)
Today's Pre-calculus lesson (in video) from the Khan Academy is: Introduction to Limits: To view other Khan Academy videos, you can find them at their website here: Enjoy! P.S. Note that given the length of these lessons, you may want to adjust your settings to receive one or two lessons a ...
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We will be studying vertex-edge graphs and using them to solve a variety of special problems. The topic has few mathematical prerequisites and will be presented through activities and student presentations as well as written and oral discussions. The students will be doing the mathematics and explaining the mathematics not just listening to the instructor.
Through experiences with new and exciting mathematics problems, students will increase their ability to reason and explain mathematics, develop their problem solving skills, and learn how to incorporate discrete mathematics into K-8 classrooms.
Required materials:
Discrete Mathematics for K-8 Teachers by Valerie Debellis and Joseph Rosenstein. This interactive book includes discussion, activities, and classroom guides.
Information on how to purchase this book will be available at the first class meeting.
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Linear Applicatins 1 v1,0
Version: 1.0
it is basic that every Algebraic student shous have. it inckudes the aookicatiu of linear relationshio between two variables in real life situations. Each example, each practice is involved in step-by-step to UNDERSTAND not to have it already toasted by somebody else. It includes different fomats of word problems in different styles and formated. highly recommended it to students from high school up to universirt. Also for internationa exam tyes. It includes percents, woed problems, formulas, Age and integer, problems, Grade, coin, and investment problems, Solution problems in Chemistry and distance provblems and finally a little bit of geometry.
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understanding an assignment
prepping for a quiz or exam
analyzing data
writing computer code
solving a problem
calculating and interpreting statistics
proving or applying a theorem
using quantitative software
designing an experiment
Credit for
photo: Albert Einstein Institute at the Max Planck Institute for
Gravitational Physics and Konrad-Zuse-Zentrum, Berlin. Visualization by
Werner Benger and Edward Seidel, director of Louisiana State University's
Center for Computation and Technology
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Chapter 31: The Complex Fourier Transform
Although complex numbers are fundamentally disconnected from our reality, they can be used
to solve science and engineering problems in two ways. First, the parameters from a real world
problem can be substituted into a complex form, as presented in the last chapter. The second
method is much more elegant and powerful, a way of making the complex numbers
mathematically equivalent to the physical problem. This approach leads to the complex Fourier
transform, a more sophisticated version of the real Fourier transform discussed in Chapter 8. The
complex Fourier transform is important in itself, but also as a stepping stone to more powerful
complex techniques, such as the Laplace and z-transforms. These complex transforms are the
foundation of theoretical DSP.
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Linear Equations: Explain Transformation The learner will be able to
explain the transformations of the graph that exist when coefficients and/or constants of the corresponding linear equations are changed.
Expressions: Evaluate/Exponents The learner will be able to
determine an answer for an algebraic expression when given values for one or more variables applying grouping symbols and/or exponents less than four.
Functions: Explain The learner will be able to
explain the domain and range of functions and describe restrictions imposed by either the operations or by the real world scenario which the functions illustrate.
Operations: Use/Order of Operations The learner will be able to
use the order of operations when completing computations with integers that apply no more than two sets of grouping symbols and exponents 1 and 2.
Patterns/Functions: Real World The learner will be able to
explain, continue, study, and develop a large variety of patterns and functions applying suitable materials and illustrations in real world problem solving.
Probability/Statistics: Concepts The learner will be able to
gather, organize, illustrate, and interpret data; formulate, present, and evaluate inferences and predictions; present and evaluate arguments based on analysis of data; and model situations to find theoretical and experimental probabilities.
Graphing: Select/Create/Study The learner will be able to
select, create, and study suitable graphical illustrations for a set of data including pie charts, histograms, stem and leaf plots, scatterplots and/or box and whisker plots.
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Quoted from ``Teaching at the University Level" by Steven Zucker. The article was published in Notices of the American Mathematical Society, August 1996 (Vol. 43, No. 8).
You are no longer in high school. The great majority of you , not having done so already, will have to discard high school notions of teaching and learning and replace them by university-level notions. This may be difficult, but it must happen sooner or later, so sooner is better. Our goal is more than just getting you to reproduce what was told to you in the classroom.
Expect to have material covered at two to three times the pace of high school. Above that, we aim for greater command of the material, especially the ability to apply what you have learned to new situations (when relevant).
Lecture time is at a premium, so it must be used efficiently. You cannot be
``taught" everything in the classroom. It is your responsibility to learn the material.
Most of this learning must take place outside the classroom. You should be willing to put in (at least) two hours outside the classroom for each hour of class.
The instructor's job is primarily to provide a framework, with some of the particulars, to guide you in doing your learning of the concepts and methods that comprise the material of the course. It is not to ``program" you with isolated facts and problem types nor to monitor your progress.
You are expected to read the textbook for comprehension. It gives the detailed account of the material of the course. It also contains many examples of problems worked out, and these should be used to supplement those you see in the lecture. The textbook is not a novel, so the reading must often be slow-going and careful. However, there is the clear advantage that you can read it at your own pace. Use pencil and paper to work through the material and to fill in omitted steps.
As for when you engage the textbook, you have the following dichotomy:
[recommended for most students] Read for the first time the appropriate section(s) of the book before the material is presented in lecture. That is, come prepared for class. Then the faster-paced college-style lecture will make more sense.
If you haven't looked at the book beforehand, try to pick up what you can from the lecture (absorb the general idea and take thorough notes) and count on sorting it out later while studying from the book outside of class.
Ask questions in class. Even if it appears as though everyone
else is ``getting it," chances are that they are actually about in the same
position as you and will appreciate, not resent, your question (providing, of course,
that you are prepared enough to ask a vaguely reasonable question).
Study with other students in the class. This greatly decreases your chances of "learning the material wrong" and gives you the opportunity to explain what you know to others, which greatly clarifies your knowledge.
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Figure It Out: A Review of Virtual Pencil Arithmetic and Virtual Pencil Algebra From Henter Math
Article provides a review of Virtual Pencil Arithmetic and Virtual Pencil Algebra from Henter Math, both of which are software programs that create a workspace that looks like a piece of paper. The computer's cursor functions as a pencil, as it allows the user to solve problems, show one's work, and put numbers in columns when necessary. Virtual Pencil Arithmetic covers basic operations such as addition, subtraction, multiplication, division, and fractions, while Virtual Pencil Algebra allows the user to solve both simple problems as well as equations. The programs work with a number of applications that make them accessible for people with visual disabilities, including JAWS for Windows and Window-Eyes screen readers, as well as ZoomText and MAGic screen magnification programs. Both products are evaluated in terms of the following features and criteria: (1) setup, (2) installation, (3) getting started and getting help, (4) customizing, and (5) mathematical operation performance. The author contends that both problems are easy to use after minimal practice, and that they both offer a variety of customization options. The author recommends that users begin by using simple operations in order to become familiar with the software
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Algebraic Formalism within the Works of Servois and Its Influence on the Development of Linear Operator Theory
Introduction
Before the nineteenth century, algebra usually referred to the theory of solving equations. However, the field of algebra experienced an extensive transformation during the nineteenth century, a time period referred to by many historians as the Golden Age of mathematics. Consequently, by 1900 algebra encompassed the study of algebraic structures. One contributor to the advancement of algebra was François-Joseph Servois (1767-1847). Servois was a priest, artillery officer, professor of mathematics, and museum curator. Not only did he battle to defend Paris in 1814, but he also fought for an algebraic foundation for calculus. As we will see, Servois was an advocate of "algebraic formalism," and the majority of his contributions to the field of mathematics fall under this category. In an "Essai" written in 1814, Servois attempted to provide a rigorous foundation for the calculus by introducing several algebraic properties, such as "commutativity" and "distributivity." Essentially, he presented the notion of a field, an idea far ahead of his time. Although Servois was not successful in providing calculus with a proper foundation, his work did have an impact on the field of algebra, and influenced several mathematicians, including the English mathematicians Duncan Gregory and Robert Murphy. Many English mathematicians of this period used the works of French mathematicians to aid in their development of linear operator theory and abstract algebra [Koppleman 1971].
This article further illustrates that while the disciplines of algebra and analysis are studied separately today, mathematicians of the eighteenth century (and before) made little, if any, distinction between them. Finally, instructors can use the original sources found in this article to demonstrate to students the connection between classical and modern day mathematics.
A Concise History of Abstract Algebra
According to Piccolino [1984], the nineteenth century is considered by many historians to be a Golden Age in the development of mathematics. Advancements in several branches of mathematics, such as geometry and analysis, occurred during this revolutionary time period. Another area that experienced change was algebra [Piccolino 1984]. Prior to the nineteenth century, algebra usually referred to the theory of solving equations; however, by 1900 it involved the study of mathematical structures, such as groups, rings, and fields [Katz 2009]. Mathematicians found that these structures often did not share properties found in the real and complex number systems, such as commutativity. This fundamental change in algebraic thought is characteristic of modern or abstract algebra.
One of the earliest instances where we see a mathematician's work on the solvability of algebraic equations is in the writings of Mohammed ibn Mûsâ al-Khowârizmî (ca. 790 CE - ca. 850 CE). Al- Khowârizmî provided solutions to linear (first-degree) equations and quadratic (second-degree) equations, but his results were presented verbally, without the use of algebraic symbols [Dunham 1990]. Al-Khowârizmî did not recognize either negative coefficients or negative solutions in his general solution to the quadratic equation \(ax^2 + bx + c = 0\), which he broke up into six cases. According to David Eugene Smith [1958], the first significant treatment of negative numbers was by Girolamo Cardano (1501-1576) in his 1545 book on algebra, Ars Magna. The first consideration of imaginary solutions occurred a few years later when Rafael Bombelli (1526-1572) used imaginary numbers as a "tool" for solving cubic equations [Dunham 1990, pp. 150-151]. According to Victor Katz, Bombelli's work "provided mathematicians with the first hint that there was some sense to dealing with" imaginary numbers in their algebraic work [2009, p. 407].
Figure 2. Joseph-Louis Lagrange (public domain).
Initial developments in abstract algebra occurred in Continental Europe during the late eighteenth and early nineteenth centuries. These changes were driven by problems in classical algebra, such as the solvability of third, fourth, and higher degree equations [Piccolino 1984]. The works of Joseph-Louis Lagrange (1736-1813), Augustin-Louis Cauchy (1789-1857), Paolo Ruffini (1765-1822), Niels Henrik Abel (1802-1829), and Evariste Galois (1811-1832) were of central importance during this time period and contained several concepts associated with modern group theory.
Lagrange explored the solvability of equations via the theory of permutations. Cauchy also made contributions to the theory of permutations by introducing concepts such as the identity permutation, a permutation that does not change a given arrangement of objects. Furthermore, Ruffini made several attempts at proving that the general equation of degree five is unsolvable in terms of radicals. Although Ruffini's efforts were not successful, his work provided a foundation for Abel's proof that such a solution cannot exist [Katz 2009]. Finally, Galois made significant contributions to the theory of solvability of algebraic equations by studying the structure of algebraic equations, particularly what he called "the group of the equation" [Katz 2009, p. 726]. We also owe to Galois the first known use of the term "group" in mathematics, which appeared in 1830 [Boyer 1989].
Although the work on algebraic solvability was carried out on the Continent, it was the British school of algebra that was primarily responsible for the shift in algebraic thinking towards abstract structural properties. As we shall see, this was not necessarily done by building on continental work on algebraic solvability, but rather by extending properties of ordinary arithmetic and of what we would call functions from analysis. Important figures in this movement included George Peacock (1791-1858), Duncan Farquharson Gregory (1813-1844), and William Rowan Hamilton (1805-1865). Peacock introduced the notions of arithmetical algebra and symbolical algebra. He defined arithmetical algebra as a universal arithmetic (using letters instead of numbers) of positive numbers [Katz 2009]. In this system, the term \(a - b\) had meaning only if \(a\) was greater than or equal to \(b\). On the other hand, symbolic algebra referred to the study of operations that were defined through arbitrary laws. In Peacock's symbolic algebra, \(a - b\) was valid regardless of the relationship between the symbols \(a\) and \(b\) [Piccolino 1984]. However, his laws in symbolic algebra were derived using principles found in his arithmetical algebra [Katz 2009]. Peacock was on the cusp of formulating an internally consistent algebra and his efforts in that direction were extended by Gregory.
Gregory, founder of the Cambridge Mathematical Journal, focused on algebraic structure. In his works, he often referred to the ideas of commutativity, distributivity, index operations (a sort of law of exponents for operators), and inverses, which he described as "circulating operations." He also mentioned the principle of the separation of symbols of operation, crediting the French mathematician François-Joseph Servois (1767-1847) as the first to "correctly give" the procedure [Allaire and Bradley 2002, p. 410]. Gregory appears to have been one of the first mathematicians to establish a connection between differentiation in calculus and the ordinary symbols of algebra, noting that the commutative and distributive laws hold true for what he referred to as the symbols of differentiation. Despite his contributions to the development of abstract algebra, Gregory, like Peacock, maintained the stance that results in symbolic algebra had to suggest results in arithmetical algebra [Piccolino 1984].
Figure 4. Duncan Farquharson Gregory (public domain).
A new, internally consistent algebraic system was finally introduced by Hamilton with his discovery of quaternions on October 16, 1843. Hamilton extended the algebra of number pairs to ordered quadruples of numbers, \((a, b, c, d)\), and defined the quaternions as ordered quadruples of numbers that followed several rules [Katz 2009]. Most notably, Hamilton's quaternions did not satisfy the commutative postulate for multiplication [Boyer 1989]. His system was the first algebra that did not follow all of the laws established by Peacock [Katz 2009]. The freedom and structure present in Hamilton's system was unprecedented and, as a result, many historians consider his discovery of the quaternions as the beginning of abstract algebra [Piccolino 1984].
Fundamental structures in abstract algebra, such as groups and fields, were formally defined in the later part of the nineteenth century. Heinrich Weber (1842-1913) was the first mathematician to present detailed, axiomatic definitions of groups and fields [Katz 2009]. Weber's definition of a finite group was slightly different than the one that most mathematicians are familiar with today. His definition included three axioms analogous to the modern day ideas of closure, associativity, and left- and right-hand cancellation laws. The terminology of closure under an operation is first found in Saul Epsteen and J. H. Maclagan-Wedderburn's "On the Structure of Hypercomplex Number Systems," which appeared in Transactions of the American Mathematical Society, Vol. 6, No. 2. in April of 1905 (see [Epsteen and Maclagan-Wedderburn 1905] and [Miller 2010]). Weber then showed that his three laws imply the existence of a unique identity element, and for each element, the existence of a unique inverse. He also incorporated his notion of group in the definition of a field. He defined a field as a set with two operators, addition and multiplication. In Weber's field, the entire set forms a commutative group under addition and the nonzero elements form a commutative group under multiplication as well. Weber also noted several properties of fields including the distributive law, which states that \(a\cdot (b + c) = a\cdot b + a\cdot c\) for all elements in the field [Katz 2009].
It is clear that mathematicians of the nineteenth century were concerned with foundational issues that spanned across several different areas of mathematics. Servois was no different and concentrated mainly on the foundational issues of the calculus. His attempts to settle the foundational issues of calculus were not successful [Bradley and Petrilli 2010]; however, as we will see, his work had a direct influence on the development of abstract algebra, and in particular, linear operator theory.
Francois-Joseph Servois
François-Joseph Servois was born on July 19, 1767, in the village of Mont-de-Laval, located in the Department of Doubs close to the Swiss border. Throughout his youth, Servois attended several religious schools in Mont-de-Laval and Besançon, the capital of Doubs, aspiring to become a priest. He was ordained a priest at Besançon shortly before the start of the French Revolution. He then left the priesthood in 1793 and became an officer in the Foot Artillery (sometimes referred to as the Heavy Artillery) with the outbreak of the revolutionary wars. In his leisure time Servois studied mathematics and his mathematical talents were apparent when he made improvements to one of the cannons, increasing its firing range significantly [Boyer 1895]. He suffered from poor health during his military career and, as a result, requested a non-active military position in the field of academia. He was assigned his first academic position on July 7, 1801, as a professor at the artillery school in Besançon, by virtue of a recommendation from the great mathematician Adrien-Marie Legendre (1752-1833). Throughout his academic career, Servois was on faculty at numerous artillery schools, including Besançon (1801), Châlons (March 1802 - December 1802), Metz (December 1802 - February 1808, 1815-1816), and La Fère (February 1808-1814, 1814-1815). His research spanned several areas, including mechanics, geometry, and calculus; however, he is best known for first introducing the words "distributive" and "commutative" to mathematics. On May 2, 1817, Servois was assigned to what would be his final position, as Curator of the Artillery Museum, which is currently part of the Museum of the Army in Paris. Servois retired to his hometown of Mont-de-Laval in 1827 and lived for another twenty years with his sister and his two nieces. He died on April 17, 1847. Readers interested in a more extensive biography and a review of Servois' other mathematical works can refer to Petrilli [2010].
It would be customary to include a painting or photograph of Servois in this biographical section, but there are no known images of him. However, due to Anne-Marie Aebischer and Hombeline Languereau [2010], there is now a photograph of his signature available to the public.
Servois' Belief in Algebraic Formalism
During the years 1811 to 1817, the majority of Servois' works were published in Joseph Diaz Gergonne's (1771-1859) Annales des mathématiques pures et appliquées. Much of his work focused on what Taton [1972a and 1972b] called "algebraic formalism." In 1814, we witness Servois' first defense of algebraic formalism, when he began a heated debate with Jean Robert Argand (1768-1822) and Jacques Français (1775-1833). In 1813, Français published a paper based on the work of Argand, in which he viewed complex numbers geometrically. In modern day mathematics, the view of complex numbers in the plane is known as the Argand Plane. Servois highly criticized the work of these two mathematicians, saying: "I had long thought of calling the ideas of Messrs. Argand and Français on complex numbers by the odious qualifications of useless and erroneous ...." [Servois 1814b, p. 228]. For instance, Servois argued against Français' geometric "demonstration," given in the preceding issue of Gergonne's journal [1813], that the quantity \(a\sqrt{-1}\) can be seen as the geometric mean of \(-a\) and \(a\). (In the field of complex numbers, we define the geometric mean of two real numbers \(a\) and \(b\) as \(\sqrt{ab}\).) Furthermore, he went on to state that it was in the best interest of the science to express his personal view, because in this work he saw nothing but "a geometric mask applied to analytic forms ...." [Servois 1814b, pp. 228-230].
Servois' fight for "algebraic formalism" continued in 1814 with the publication of his "Essai sur un nouveau mode d'exposition des principes du calcul différential" [Servois 1814a] ("Essay on a New Method of Exposition of the Principles of Differential Calculus"). This work was an extension of Lagrange's research on the foundations of the differential calculus. In his "Essai," Servois stated his belief that the differential calculus could be unified through algebraic generality:
In the preceding article, we have sketched the set of laws that brings together and unites all the differential functions, that is, the most general theory of the differential calculus. The practice of this calculus, which is nothing other than the execution of the operations given in the definitions .... [Servois 1814a, p. 122].
The notion of algebraic generality is apparent in the opening sections of his "Essai," where Servois essentially defined a field for his set of functions under the operations of addition and composition [Bradley and Petrilli 2010]. However, it was not Servois' intention to create formal structures within algebra, but rather he "was concerned above all else to preserve the rigor and purity of algebra" [Taton 1972a].
Algebraic Structure within Servois' 'Essai'
In his "Essai," Servois attempted to provide a rigorous foundation for the calculus through algebra. In light of what we know today, Servois did not fully succeed in putting calculus on a mathematically correct foundation. It was Cauchy [1821] who, through his approach to calculus by means of limits and inequalities, moved the subject into the modern age. Although Servois' efforts were ultimately unsuccessful, we find several ideas associated with abstract algebra in his "Essai."
Figure 6. Title page of Servois' "Essai" (public domain).
In Sections 1-4 of the "Essai," Servois presented several definitions that would be crucial to his work. He began by introducing his notation for a function as \(f\,z\), where a modern reader would understand this as \(f(z)\). The formal definition of a function that we use today would not be introduced until 1837 by Lejeune Dirichlet (1805-1859). A reader will notice that Servois used the term function not only to describe ordinary functions of an independent variable, but also to describe operators, such as the difference and differential operators. After he presented his preliminary definitions, Servois introduced two functions or operators with special properties, namely the identity \(f^0\) and inverse \(f^{-1}\). Undergraduate students will notice that these operators and properties are analogous to the modern day notions of an identity element and inverse element. For instance, Servois explained that when the identity function or operator is applied to \(z\), "\(z\) does not undergo any modification" [Servois 1814a, p. 96]. Additionally, Servois provided many examples of inverse functions. For instance, he considered the inverse of the sine function, noting that \[z = \mbox{sin}(\mbox{sin}^{-1} (z)) = \mbox{sin}(\mbox{arcsin}(z)).\] For the difference operator \(\Delta\), he noted that \[\Delta^n (\Delta^{-n} (z)) = \Delta^{-n} (\Delta^n (z)) = z.\]
In Section 3, Servois defined a function or operator \(\varphi\) to be distributive if it satisfied \[\varphi (x + y + ...) = \varphi(x) + \varphi(y) + \cdots.\] He presented several examples of functions and operators that are distributive and others that are not. One such distributive function was \(f(x) = ax\), because \[a(x + y + \cdots) = ax + ay + \cdots.\] Later, Cauchy [1821] showed that this is the only continuous function that satisfies this property. Conversely, Servois provided as a function that does not satisfy the distributive property \(f(x) = \mbox{ln}\,x\). He also demonstrated that the differential and integral operators are distributive.
Actually, undergraduates are exposed to specific examples of Servois' distributive property in a first-year calculus course, namely differentiation and integration under the operation of addition. Additionally, students see a generalized version of Servois' distributive property in a beginning linear algebra course. One of the properties of a linear transformation is that it preserves the addition operation.
Finally, in Section 4, Servois stated that two functions or operators \(f\) and \(\varphi\) are commutative between themselves if \[f(\varphi (z)) = \varphi (f (z)).\] For example, he stated that \(z\) commutes with any constants \(a\) and \(b\) because \[abz = baz;\] however, the sine function is not commutative with any constant \(a, a\not= -1,0,1\), because \[\mbox{sin}(az) \neq a \mbox{sin}(z).\]
If we consider the commutative operators \(f(z)\) and \(\varphi (z) = kz\), where \(k\) is any scalar, \[f(kz) = kf(z),\] then we have the familiar scalar multiplication property that undergraduates would see in a first-year calculus course for vectors or beginning linear algebra course for linear transformations.
In modern day mathematics, we speak of an algebraic structure (ring, field, etc.) as being commutative when \(a \cdot b = b \cdot a\) and distributive when \(a \cdot (b + c) = a \cdot b + a \cdot c\), for all elements \(a\), \(b\), and \(c\) in the structure. The hallmark of Servois' calculus was his examination of the set of all functions and operators that satisfy these properties. Interestingly, the words "commutative" and "distributive" were medieval legal terms [Bradley 2002] and Servois was the first to use them in a modern mathematical sense.
In Sections 5-9 of his "Essai," Servois examined the "closure" properties of distributive and commutative functions or operators. Servois demonstrated that distributivity is closed under composition and addition, and if \(f\) and \({\rm f}\) are commutative functions or operators, then each commutes with the inverse of the other. An examination of Servois' proof of the latter theorem reveals a law from abstract algebra. By the definition of the inverse function, we have \[f( \mbox{f(f}^{-1}(z))) = \mbox{f(f}^{-1}(f(z))),\] and by virtue of the commutativity of \(f\) and \({\rm f}\), we get \[f( \mbox{f(f}^{-1}(z))) = \mbox{f}(f(\mbox{f}^{-1}(z))).\] Now, substitute \(\mbox{f}(f(\mbox{f}^{-1}(z)))\) for \(f( \mbox{f(f}^{-1}(z)))\) in equation (1), and we get \[\mbox{f}(f(\mbox{f}^{-1}(z))) = \mbox{f(f}^{-1}(f(z))).\] Finally, apply \(\mbox{f}^{-1}\) to both sides of equation (2) and we arrive at the desired result that: \[f(\mbox{f}^{-1}(z)) = \mbox{f}^{-1}(f(z)).\] Essentially, when Servois applied \(\mbox{f}^{-1}\) to both sides of equation (2), he invoked a familiar theorem from group theory, the left-hand cancellation law.
After he considered the properties of commutativity and distributivity separately, Servois examined the collection of functions or operators that satisfy both of these properties. He used this as a launching point to introduce his theory for the differential calculus.
From a modern standpoint, the first twelve sections of Servois' "Essai" constitute the creation of an algebraic structure. In them, he showed that the set of invertible, distributive, and pairwise commutative functions or operators forms a field with respect to the operations of addition and composition. Servois never discussed the associative property with respect to these two operations. However, the importance of associativity was being uncovered during the nineteenth century. For instance, Carl Friedrich Gauss (1777-1855) did prove an associative law in 1801 [Gauss 1801, Section 240] and Hamilton stated the importance of associativity in 1843 after his discovery of the quaternions [Crilly 2006, p. 102]. Hamilton's statement was actually the first appearance of the term [Miller 2010]. Additionally, Servois assumed the existence of inverses for all of his functions. Again, a reader must keep in mind that Servois worked only with functions that were well-behaved and he did not examine the issue of domain.
Servois' Influence on the Development of Linear Operator Theory
Servois' influence on the development of Linear Operator theory can be traced through the works of several well-known mathematicians, including Murphy and Gregory. Although the formalization of symbolic algebra is generally credited to these two mathematicians, we see several ideas associated with Linear Operator Theory in by a lesser-known academic, Thomas Jarrett (1805-1882). Jarrett was an English cleric, Professor of Arabic at the University of Cambridge, and a linguist. According to the biography by E. J. Rapson [1892], he knew at least twenty different languages and would translate Chinese characters into Roman characters using a system that he devised himself. There is no biographical information that indicates that he had any formal mathematical training.
In the Preface to his book, Jarrett stated that part of his work was taken from the following mathematicians: Servois, Louis François Antoine Arbogast (1759-1803), John Frederick William Hershel (1792-1871), Carl Friedrich Hindenburg (1741-1808), Sylvestre François Lacroix (1765-1843), Pierre-Simon Laplace (1749-1827), Ferdinand Franz Schwiens (1780-1856), and Josef-Maria Hoëné-Wronski (1776-1853). Interestingly, Wronski's calculus was based on infinitesimals and Jarrett used no such foundations. Jarrett went on to say that some of the material was partly original; however, according to his biographers [Rapson 1892], the original contributions could simply have been new notation. Their contention is supported by Jarrett himself: "In the present Work [Algebraic Notation] is applied to the demonstration of the most important series in pure Analysis" [Jarrett 1831, p. III].
Jarrett's work is similar to Servois' "Essai" in that they both used algebra as a foundation for calculus; however, Jarrett presented Servois' material in a more structured format. Interestingly, when Jarrett discussed the summation operator he distinguished between operators (which he called operations) and functions, but when he presented his theory of the calculus he made no such distinction and classified both as functions. He derived many of the same results as Servois, only using different notation. Jarrett's calculus was based on the concept of the separation of symbols, which he credited to Servois in his Preface: "The demonstration of the legitimacy of the separation of the symbols of operation and quantity, with certain limitation, belongs to Servois ..." [Jarrett 1831, p. III]. At the heart of Jarrett's theory were Servois' distributive and commutative properties:
If \(\varphi (u)\) is such a function of \(u\) that \(\varphi (u + v) = \varphi (u) + \varphi (v)\), then \(\varphi (u)\) is called a distributive function of \(u\)
If \(\varphi (u)\) and \(\psi (u)\) are such functions of \(u\) that \(\varphi \psi (u) = \psi \varphi (u)\), then the functions \(\varphi (u)\) and \(\psi (u)\) are said to be commutative with each other.
From a modern standpoint, Jarrett defined a field in a fashion similar to Servois' by introducing the notions of an identity, inverses, and closure, in addition to these two properties. Using properties of this field he derived his theory of the differential and integral calculus.
We also see Servois' ideas in the work of another mathematician, Robert Murphy (1806-1843). Murphy's [1837] "First Memoir on the Theory of Analytic Operations" is a detailed exposition on the theory of operators. Murphy clearly distinguished between functions and operations, and called the objects on which operations are performed subjects [Allaire and Bradley 2002]. With respect to his notation, if Murphy wanted to discuss the operator \(\psi\) applied to the function \(f(x)\), then he denoted it as \([f(x)] \psi\), where the subject is contained within brackets.
Murphy [1837] began his paper by examining special types of operators. He considered the operators \(p\) and \(q\) as fixed or free, where "in the first case a change in the order in which they are to be performed would affect the result, in the second case it would not do so" [Murphy 1837, p. 181]. To relate this to Servois' work, a free operator would be one that satisfied Servois' commutative property. Now, let \(a\) and \(b\) be subjects and \(p\) be the operation of multiplying by the quantity \(p\). Then \[\left[a \pm b\right]p = \left[a\right]p \pm \left[b\right]p,\] which makes \(p\) (or multiplication by \(p\)) a linear operator according to Murphy's definition. Thus, a linear operator is one that satisfies Servois' distributive property. In modern day mathematics, in order for \(p\) to be a linear operator it would also have to be free with respect to a constant \(k\). Additionally, Murphy was the first mathematician to use the term linear to describe a special class of operators [Allaire and Bradley 2002].
Bradley and Allaire [2002] state that Murphy derived many of the same results as Servois, only with greater clarity and brevity. However, Murphy also expanded on the theory of linear operators. For example, he defined the appendage of a linear operator as "the result of its action on zero" [Murphy 1837, p. 188]. Here, "action" refers to the inverse image of the operator. In modern day mathematics, the appendage would be referred to as the kernel of a linear transformation, and this was the first time that the kernel of an operator had been considered [Allaire and Bradley 2002]. The kernel is an important concept in understanding the behavior of linear transformations.
Unlike Jarrett, Murphy did not acknowledge a debt to Servois nor is there solid evidence that he actually read his work. Murphy's opening sections are devoted to the important properties of operators – that is, Servois' commutativity and distributivity – but Murphy gave these properties different names. However, Servois began the study of linear operators and his work was read in England, as demonstrated by Jarrett's use of his research in 1831. As we will see, Duncan Gregory gave Servois the credit he was due.
According to Piccolino [1984], Gregory played a vital role in the development of symbolic algebra and was a key contributor to the overall advancement of mathematics in England during the late 1830s and early 1840s. In addition to his mathematical demonstrations, Gregory often provided philosophical insights to reinforce his views on algebra. For example, Gregory considered symbolic algebra as "the science which treats of the combinations of operations defined not by their nature, that is, by what they are or what they do, but by the laws of combination to which they are subject" [Gregory 1865, p. 2]. Essentially, Gregory believed that the general principles of algebra must fit a certain structure, which he called a class.
Throughout his Mathematical Writings [1865], Gregory provided several examples illustrating the "laws of combination" to which operations are subject. For example, he considered two classes of operations \(F\) and \(f\), which are connected by the following laws:
\(FF(a) = F(a)\)
\(ff(a) = F(a)\)
\(Ff(a) = f(a)\)
\(fF(a) = f(a)\)
His most general interpretation of these laws was multiplication for positive and negative numbers [Allaire and Bradley 2002]. For instance, (2.) shows that a negative number multiplied by a negative number yields a positive number. In this case, \(F\) and \(f\) should not be interpreted as functions, but rather as operations.
Next, he considered a general class of operations, which satisfy the following laws:
\(f(a) + f(b) = f(a + b)\)
\(f_1 f(a) = f f_1 (a)\)
Gregory credited Servois with the classification of these laws, writing, "Servois, in a paper which does not seem to have received the attention it deserves, has called them, in respect of the first law of combination, distributive functions, and in respect to the second law, commutative functions" [Gregory 1865, pp. 6-7]. Whereas Murphy [1837] considered the special class of functions satisfying (1.) to be linear operators, Gregory noticed that these two laws together constitute a special class of operations, which are called linear transformations or linear operators in modern mathematics.
Gregory then provided an example to demonstrate his first law, where \(f\) is taken to be the operation of multiplying by a constant \(a\): \[a(x) + a(y) = a(x + y).\] In his Mathematical Writings, Gregory stated that Cauchy sometimes utilized the "laws of combination" [p. 7], so Gregory may have been familiar with the fact that this is the only continuous function that satisfies the distributive law.
Finally, Gregory defined a class of operations by the law \[f(x) + f(y) = f(xy).\] This is the first time we see a general law for a class in which two different operations are considered. Gregory related this definition to a familiar law of logarithms, that \(\ln (x) + \ln (y) = \ln (xy)\), saying "when \(x\) and \(y\) are numbers, the operation is identical with the arithmetical logarithm" [Gregory 1865, p. 11].
Figure 7. Augustus De Morgan (public domain).
Petrova [1978] stated that linear operator theory began with Servois and was continued by Murphy. According to Allaire and Bradley [2002], Gregory was the next key figure in the development of this theory. We wondered, however, if any other mathematicians were influenced by the work of Servois. The authors examined numerous works written by mathematicians during the Golden Age of mathematics, including Peacock, Augustus De Morgan (1806-1871), and George Boole (1815-1864). These mathematicians made great advances in algebra; however, they appear to have made no significant contributions to the theory of linear operators. After examining several of their works, we now make some observations regarding the influence Servois may have had on these mathematicians. It should be noted that this analysis is highly subjective, because even though some of these mathematicians used methods similar to Servois', none gave him direct credit. Additionally, a majority of these works began to appear in the 1840s, and the works of Murphy and Gregory were already available by this time.
According to O'Connor and Robertson [1996], Peacock was interested in making reforms to Cambridge mathematics and he aided in the creation of the Analytical Society in 1815 as a result. The society was intended to bring the continental methods of the calculus to Cambridge. The reform began when Peacock translated Lacroix's calculus text, Traité élémentaire de calcul differéntiel et du calcul intégral [1802]. Using Lacroix's ideas, Peacock published his Collection of Examples of the Applications of the Differential and Integral Calculus [1820]. Lacroix's work was based on the calculus of Lagrange. Consequently, Peacock adopted many of the methods presented by Lacroix. Peacock did not explicitly use Servois' methods in his work and made no claim about the algebraic properties of operators. However, because Peacock was interested in the continental calculus, it is possible that he was familiar with the works of Servois.
Now, De Morgan, who was a student of Peacock's, presented the algebraic definitions for distributivity and commutativity in his work, Trigonometry and Double Algebra. These definitions are very similar to the ones that students would learn in a high school algebra course today. For instance, De Morgan stated, "A symbol is said to be distributive over terms or factors when it is the same thing whether we combine that symbol with each of the terms or factors, or whether we make it apply to the compound term or factor" [De Morgan 1849, pp. 102-103]. Being a student of Peacock's, De Morgan could have been familiar with the works of the continental mathematicians. This is further supported by the fact that he used the terms distributive and commutative in a fashion similar to Servois'.
Figure 8. George Boole (public domain).
Finally, in his Treatise on the Calculus of Finite Differences [1860], Boole gave the laws for the symbols \(\Delta\) and \(\frac{d}{dx}\). For instance, he stated:
Since Boole was a student of Gregory's, it is reasonable to conjecture that he was introduced to the works of Servois via Gregory's teachings. Servois' influence can be seen in Boole's own statements about \(\Delta\) and \(\frac{d}{dx}\) being distributive and commutative operators.
Conclusions and Recommendations
Although Servois was not successful in providing calculus with a proper foundation, his work did have an influence on the field of algebra, where he was a pioneer ahead of his time. He knew that, when performing algebraic manipulations on quantities, he needed to have a structure consisting of a set that obeyed certain axioms. Additionally, his work on analysis spread to England and significantly influenced mathematicians such as Duncan Gregory and Robert Murphy in their efforts to establish the foundations of linear operator theory. Servois' main contribution to this development was recognizing the "distributive" and "commutative" properties of operators, terms that he coined, and the method of separating symbols and their operations. Koppleman [1971] stated that the English mathematicians found the tools for the calculus of operations in the works of French mathematicians, such as Servois'. She went on further to say that "it was the English who developed this work in the calculus of operations both in extending the scope of its applications and in relating it to the theory of abstract algebra" [Koppleman 1971, p. 175].
The material discussed in this paper can aid teachers and students of abstract algebra, linear algebra, and the history of mathematics. By presenting the history of mathematics, instructors can illustrate the idea that mathematics is a constantly evolving field. Besides providing a readable account of the history of algebraic structures and the beginnings of linear operator theory, this paper contains many explanations of and references to original sources. Additionally, this article highlights the fact that while the disciplines of algebra and analysis are studied separately today, mathematicians of the eighteenth century (and before) made little, if any, distinction between them. When analyzing the original sources, the student co-author of this article, Anthony, was initially somewhat shocked to see so many calculus-like and algebraic ideas presented together. As an undergraduate, he had rarely seen ideas from both analysis and algebra meshed so closely together.
In the spirit of Victor Katz, we would encourage instructors to incorporate original sources within their classrooms. This paper provides references to original sources that can easily be found on the internet. With a little consideration on the part of the instructor, it is easy to create historical activities that can fit in any mathematics course. For instance, pages 1-13 of Gregory's The Mathematical Writings contain many examples of symbolical algebra that can be incorporated into any course, such as a first-year calculus course. An instructor could provide a copy of Gregory's discussion of operators that satisfy the property \(f(x) + f(y) = f(xy)\) and ask students to write a list of functions that satisfy this property.
Furthermore, these sources provide students with an opportunity to conduct research on the history of mathematics. Open questions include, for instance:
Who was Thomas Jarrett? Did he receive any formal mathematical training? If so, from whom did he receive training? Are there any other mathematical works published by him?
Jarrett's work is taken from a few mathematicians [Jarrett 1831, pp. III-V]. Many are well-known, but who was Ferdinand Franz Schwiens? Other than the analysis textbook mentioned by Jarrett, what did Schwiens write?
Many unanswered questions still remain regarding Servois' mathematical career. For instance, was there correspondence between Servois and any English mathematicians? If so, it would be interesting to use it to explore the extent of Servois' influence on these mathematicians.
Acknowledgments / About the Authors
Acknowledgments
The authors are extremely grateful to referees for their many helpful suggestions and corrections.
About the Authors
Anthony J. Del Latto is an undergraduate mathematics major at Adelphi University. He is currently a senior and serves as a tutor for the Department of Mathematics and Computer Science. After completing his bachelor's degree, he wishes to continue his studies in mathematics at the graduate level. His research interests include abstract algebra, history of mathematics, and applied statistics.
Salvatore J. Petrilli, Jr. is an assistant professor at Adelphi University. He has a B.S. in mathematics from Adelphi University and an M.A. in mathematics from Hofstra University. He received an Ed.D. in mathematics education from Teachers College, Columbia University, where his advisor was J. Philip Smith. His research interests include history of mathematics and mathematics education.
[Boyer 1989] Boyer, C. B. (1989). A History of Mathematics. New York: John Wiley and Sons.
[Bradley 2002] Bradley, R. E. (2002). "The Origins of Linear Operator Theory in the Work of François-Joseph Servois," Proceedings of Canadian Society for History and Philosophy of Mathematics14, 1 - 21.
[Jarrett 1831] Jarrett, T. (1831).. London: Cambridge University.
[Petrova 1978] Petrova, S. S. (1978). "The Origin of Linear Operator Theory in the Works of Servois and Murphy." History and Methodology of the Natural Sciences20, 122-128. (Unpublished translation by Valery Krupkin.)
[Piccolino 1984] Piccolino, A. V. (1984). A Study of the Contributions of Early Nineteenth Century British Mathematicians to the Development of Abstract Algebra and Their Influence on Later Algebraists and Modern Secondary Curricula. Doctoral thesis: Columbia University Teachers College.
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ACAFree sugars in fruits and vegetablesby C. Y. Lee, R. S. Shallen
NEW YORK'S FOOD AND LIFE SCIENCES BULLETINNO. 6, JANUARY 1971
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Laboratory: Soils in Agricultural SystemsThis weeks lab will take place at the Dilmun Hill, the Cornell student-run organic farm. Your task will be to make measurements of some soilthe agricultural systems being employed at Dilmun Hill.character
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Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician. less
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Mathematics Courses
Grades 1 and 2
ELEMENTARY MATHEMATICS
This course is offered once a week in one and a half hour sessions. The course is designed to help the youngest students to develop the basic arithmetic and analytical skills. The main emphasis is on word problems, logical reasoning, and measurements.
Grades 3 and 4
MATHEMATICS
This course is offered once a week in two-hour or three-hour sessions. The main objective of this course is to lay mathematical background for the future success in studying mathematics. Students are provided with the necessary learning experience to develop computational and problem solving skills. Elements of algebra and geometry are first introduced in this course.
Grades 5 and 6 *
PRE-ALGEBRA
This course is offered once a week in two and a half hour or three-hour sessions. This course provides a concrete approach to algebraic concepts while reinforcing numeric skills. After an intensive review of decimals, fractions and percents, students focus on linear equations, polynomials and inequalities.
* 6th grade students are recommended to take GEOMETRY concurrently with PRE-ALGEBRA course.
Grades 6, 7, 8 and 9 *
ALGEBRA 1
A three-year comprehensive algebra course is offered once a week in two-hour or three-hour sessions. During the first year students learn algebraic skills in the area of basic operations with polynomials, algebraic fractions, and irrational numbers. The extensive treatment of word problems and other applications aid students in developing good problem solving techniques. The second year of this course emphasizes the solution of equations and application to problem situations. Topics covered include intensive treatment of radicals, systems of linear and non-linear equations and inequalities, graphs. Students will use higher order thinking to strengthen their problem solving skills. Theory of functions studied during the third year completes this course. Other topics studied include permutations, arithmetic and geometric sequences, probability.
Prerequisite: PRE-ALGEBRA, or consent of the teacher.
GEOMETRY
A three-year complete course of geometry (two hours a week) is offered concurrently with our PRE-ALGEBRA and ALGEBRA I courses. First year topics include angles, polygons, parallel lines and congruence. Students will use deductive reasoning to write direct proofs in two-column format. Second year topics include similarity, circles, areas, and volumes. Students will be introduced to inductive reasoning, formal logic, and indirect proofs. Third year topics include constructions with straight edge and compass, trigonometry of right triangle, coordinate geometry, and solids.
** Successful completion of ALGEBRA I and GEOMETRY courses should prepare students for SAT I.
Grades 9, 10 and 11:
ALGEBRA II with PRE-CALCULUS
This course is offered two times a week in two-hour and two and a half hour sessions. Topics include but are not limited to a comprehensive study of trigonometry, rational, exponential and logarithmic expressions and functions. Vector algebra, polar coordinates, sequences and series are also studied in this course.
Prerequisite: ALGEBRA I and GEOMETRY, their equivalents, or consent of the teacher.
TRIGONOMETRY
This 16-weeks Trigonometry course is offered once a week in three-hour sessions. Topics include comprehensive study of unit circle, and radian measure; trigonometric equations and inequalities; trigonometric functions, their properties and graphs; formulas and trigonometric identities.
Prerequisite: ALGEBRA I and GEOMETRY, their equivalents, or consent of the teacher.
SAT I and SAT II PREPARATION
These courses are offered once a week in three-hour sessions. Students can take SAT I and SAT II prep courses concurrently with their regular math classes. SAT topics are thoroughly studied during the sessions, homework assignments are given after each class. Students are tested on the covered material each week, real SAT tests are given periodically to develop test taking skills.
Grades 10, 11 and 12:
CALCULUS
This one year course is offered once a week for three-hour sessions. The course begins with a review of functions, limits, and continuity as well as derivatives and integration and their applications. The course will focus on the study of the calculus of polynomial, rational, exponential, trigonometric functions. Calculus of vectors, infinite series and introduction to differential equations are also covered in this course.
Prerequisite: ALGEBRA II, its equivalent, or consent of the teacher.
Physics
PHYSICS
A conceptually oriented course where the development of basic laws are discussed and the concepts involved are used to explain common phenomena. Topics covered are motion, Newtonian dynamics, waves, heat, electricity and magnetism. This course is designed to prepare students for Subject SAT II Physics test.
Chemistry
Chemistry
This course is designed to help student to do well in their study of chemistry and prepare them for Subject SAT II Chemistry test. Topics covered include: The Periodic Table, chemical reactions, chemical bonding, organic compounds, acids and bases, oxidation-reduction reactions, equilibrium.
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When I first saw the ClassPad (CP)
demonstrated by HideshiFukaya
and Diane Whitfield in 2002, I was impressed by how much it can offer in
Dynamic Geometry (DG) and Computer Algebra System (CAS). I had to admit that it
took me a while to learn how to use the CP and later incorporate it into my
regular teaching. One of the best ways to learn how to use software or a
hand-held device is to see what other people have done. This is precisely why
we are launching this project by providing a soft copy of each eActivity, which learners can modify to suit their needs
using the ClassPad Manager software or the trial
version. In the mean time, we are also including a brief video clip for each
corresponding eActivity.
In this project, you will find 24 eActivities
and corresponding video clips. It is a fact that evolving technological tools
have prompted us to rethink the way we teach and learn mathematics. The eActivities cover topics from Pre-Calculus to Advanced
Calculus. Each demonstrates how CP can help us learn traditional contents with
ease and allow us to experiment and explore more mathematics along the way.
We often find ourselves lacking the time needed to apply why
we learn Calculus. In this project, you will find many applications on
optimization, finding limits and etc. For my general approach, I use graphical
and geometrical animations to bring out the natural conjectures from a learner
and later prove or disprove conjectures analytically with the help of CAS.It is natural to use graphical
representations to make mathematics more accessible to more learners, which in
turn gives motivation to solve a problem analytically. Once learners capture
the essence of solving a problem, complicated manipulations can be done with
the help of CAS.
I am fortunate to have Professor Jonathan Lewin giving me advice on this project and helping with the
creation of some of the sound tracks. I would also like to sincerely thank
CASIO staffs from MRD in Portland, USA and Tokyo, Japan for much assistance.
Please excuse my not so perfect spoken English (in the video clips) and some
errors you might find in the contents. I claim to not be an expert using CP
here; my main goal of this project is to share how much a technological tool
like CP has helped us in teaching and learning.
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This course emphasizes further development of mathematical knowledge and skills to prepare students for success in their everyday lives, in the workplace, in the Grade 10 LDCC course (MAT 2L1), developing and consolidating key foundational mathematical concepts and skills by solving authentic, everyday problems. Students have opportunities to further develop their mathematical literacy and problem-solving skills and to continue developing their skills in reading, writing, and oral language through relevant and practical math activities.
MFM1P - Mathematics: Foundations (Applied):
This course enables students to develop understanding of mathematical concepts related to introductory algebra, proportional reasoning, and measurement and geometry through investigation, the effective use of technology, and hands-on activities. Students will investigate real-life examples to develop various representations for linear relationships, and will determine the connections between the representations. They will also explore certain relationships that emerge from the measurement of three-dimensional objects and two-dimensional shapes. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. Learning through hands-on activities and the use of concrete examples is an important aspect of this course.
MPM1D - Mathematics: Principles (Academic):
This course enables students to develop relationship. They will also explore relationships that emerge from the measurement of three-dimensional objects and two-dimensional shapes. Students will reason mathematically and communicate their thinking as they solve multistep problems. Learning through abstract reasoning is an important aspect of this course.
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Seventh Grade Mathematics
Mathematics is indispensable for understanding our world. In addition to providing the tools of arithmetic, algebra, geometry and statistics, it offers a way of thinking about patterns and relationships of quantity and space and the connections among them. Mathematical reasoning allows us to devise and evaluate methods for solving problems, make and test conjectures about properties and relationships, and model the world around us.
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FL Students
Course Name:
Advanced Algebra with Financial Applications
Course Code:
1200500
Honors Course Code:
AP Course Code:
Description:
This course walks students through the information needed to make the best decisions with money. Advanced Algebra with Financial Applications is an advanced course incorporating real-world applications, collaboration, and calculations using technology. Students learn the formulas used to determine account balances, monthly payments, total costs, and more. They examine budgeting, spending, saving, investment, and retirement. Students explore mortgages and other debt structures and how to make good decisions about borrowing money. This knowledge will propel students into the future with a good foundation on how to handle finances.
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What about "Mathematics with Computers"? Having modern computer algebra and symbolic computation tools available, one can use them to present and explore nontrivial examples in various fields of mathematics. Part of the course could also present basic algorithms and other techniques used.
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culus 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 fifth 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. Readers will also gain access to WileyPLUS, ... MOREan online tool that allows for extensive drills and practice. Calculus teachers will build on their understanding in the field and discover new ways to present concepts to their students.
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ID: 334 | Video: Medium | Audio: None | Animation: None
Prepare for the ACT (American College Testing) math exam successfully
The ACT (American College Testing) is a standardized test for high school achievement and college admissions in the United States produced by ACT, Inc. The ACT test has historically consisted of four tests: English, Mathematics, Reading, and Science Reasoning. The 60-question math test consists of 14 questions covering pre-algebra, 10 elementary algebra, 9 intermediate algebra, 14 plane geometry, 9 coordinate geometry, and 4 elementary trigonometry. This free online course from ALISON contains 60 sample problems similar to the ones you will find in your own. This course is ideal for any learner studying for the ACT math exam.
After completing this course you will be well prepared for the ACT maths exam. You will gain a good knowledge of the questions that are asked in this exam and you will know the methods that are used to solve the problems successfully.
Comments & Reviews
Madjid Belkaid - United Kingdom
2012-09-01 11:09:41
Course Module: ACT Math Exam Part 1 Course Topic: ACT Sample Questions Part 1 Comment: This how I saw the explanation on the video
A 0.25 . 80 = 20
B 0.35 . 80 = 28
C .4 x 80= 32 and 0.4 x 100 = 40
if you compute A and B by 80 why not computing C by 80
how do we know there is a total of 80
if 80 is a result of 100-20
people may think that 40cows minus 100 is 60
Mohamed Saad - Egypt
2012-08-30 18:08:42
Course Module: ACT Math Exam Part 7 Course Topic: ACT Sample Questions Part 7 Comment: Q35:the ratio of the volumes of the two.... i think it should be the ratio of the area of the ....
crystal brownhartman - United States of America
2012-07-13 04:07:10
Course Module: ACT Math Exam Part 1 Course Topic: ACT Sample Questions Part 1 Comment: i dont get the last one
Hevedar Yousif - United States of America
2012-02-13 00:02:33
Course Module: ACT Math Exam Part 1 Course Topic: ACT Sample Questions Part 1 Comment: So great to see free service that actually works. Thumbs up.
Roweisha Graham - United Kingdom
2011-09-18 10:09:10
I recieved a "B" at gcse maths 3 years ago and some of the stuff I easily understand but stiff like this, I find he doesn't explain well.
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MATH 548: Introduction to Ring Theory
Course ID
Mathematics 548
Course Title
MATH 548: Introduction to Ring Theory
Credits
3
Course Description
A ring is an algebraic system described by a set equipped with addition and multiplication operations. Rings arise naturally as generalized number systems. The integers, for example, form a ring with the usual addition and multiplication operations. Ring theory has applications in diverse areas such as biology, combinatorics, computer science, physics, and topology. Topics include rings of matrices, integers modulo n, polynomials, and integral domains. Some of the important theorems covered are the Fundamental Theorem of Algebra, the Division and Euclidean Algorithms, and Eisenstein's Criterion. 348/548
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Algebra Problems – Effective Teaching
Algebra problems are one among the significant aspects of mathematics. But, most of the time, people fail to be good at them due to their apparently confusing nature. And the difficulty is that one cannot even avoid them. If you are one among those who fall into this group, there is a good news for you, subliminal messages.
Subliminal messages, a surefire way to make Algebra problems enjoyable
Generally, people turn away from algebra problems because they consider them to complicated to be handled by themselves. But the fact is just the opposite. Often, it is your own negative feelings that make algebra problems tough. If you manage to come out of your negative frame of mind and program your brain about positive thoughts about algebra, you are sure to be good at them. Subliminal messages are messages sent to your subconscious mind so as to get rid of any unwanted habit or feeling. And the same can be applied to assist your brain to get rid of the preconceived idea that algebra problems are beyond your capabilities.
A few more tips to make algebra problems enjoyable
Refrain from pressurizing your brain. In the instance of your failure to solve an algebra problem, do not blame yourself for your incapability. Continue with your positive thoughts about the subject and keep on practicing. Practice helps you get rid of your innate fear for algebra problems. Do not ever adopt the attitude of procrastination. Always feel confident about your potentials and Keep on trying to convince your brain of its capabilities and continue with your attempts. A time is sure to come when you start enjoying algebra problems and treat them as fun games.
Do not simply turn away from algebra problems, they would be there to help you in life with wise logic in times of trouble.
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This book gives a clear rational explanation of some basic mathematical concepts, dispelling the usual rote and how-to approaches that make too many people dislike mathematics. Just as the artist wishes people to enjoy his art, I wish people to enjoy the beauty of rational thinking and mathematics. This very useful, easy-to-read book will help students and the public understand and appreciate mathematics.
show more show less
Edition:
N/A
Publisher:
CreateSpace Independent Publishing Platform
Binding:
Trade Paper
Pages:
192
Size:
5.75
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