text stringlengths 6 976k | token_count float64 677 677 | cluster_id int64 1 1 |
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E-Z Tr self-instruction book explains angles and triangles, and demonstrates the solutions to right triangle problems. Chapters that follow deal with trigonometric functions of sine, cosine, and tangent, radian measures, Pythagorean and otherMore...
This self-instruction book explains angles and triangles, and demonstrates the solutions to right triangle problems. Chapters that follow deal with trigonometric functions of sine, cosine, and tangent, radian measures, Pythagorean and other trigonometric identities, graphs of trigonometric functions, waves, polar coordinates, complex numbers, conic sections, spherical trigonometry, polynomial approximation for sin x and cos x, and more. Exercises follow every chapter with answers given at the back of the book. Barron's continues its ongoing project of updating, improving, and giving handsome new designs to its popular list ofEasy Waytitles, now re-namedBarron's E-Z Series.The new cover designs reflect the books' brand-new page layouts, which feature extensive two-color treatment, a fresh, modern typeface, and more graphic material than ever. Charts, graphs, diagrams, instructive line illustrations, and where appropriate, amusing cartoons help to make learning E-Z.Barron's E-Zbooks are self-teaching manuals focused to improve students' grades across a wide array of academic and practical subjects. For most subjects, the skill level ranges between senior high school and college-101 standards. In addition to their self-teaching value, these books are also widely used as textbooks or textbook supplements in classroom settings.E-Zbooks review their subjects in detail, using both short quizzes and longer tests to help students gauge their learning progress. All exercises and tests come with answers. Subject heads and key phrases are set in a second color as an easy reference aid | 677.169 | 1 |
Professional Learning
Mathematica Pedagogy – 4 sessions for those who are familiar with Mathematica. This would be suitable for those who attended the Hands-On sessions with Craig Bauling.
Polycom options available.
Mathematica Skills – Beginners
These sessions will give participants the skills required to tackle all required Mathematics for VCE Mathematical Methods Units 1-4. You will be shown how to use the software to answer exam-style questions, how to introduce Mathematica to beginning students, and how to create worksheets in Mathematica for your students to use.
This is a repeat of the session offered in Term 1.
Thursdays 3:40pm – 4:40pm
Mode – Online
Mathematica Pedagogy
This set of four lessons are designed to demonstrate how CAS (Mathematica) can be used to introduce and explore complex mathematical ideas. Activities will be designed in such a way that will encourage student-centred learning. Sample activities and corresponding Mathematica files will be demonstrated. Participants will write and share their own resources as part the course.
Participants who have completed the Term 1 Virtual Program or any of the face-to-face Hands-on training with Craig Bauling should consider this program. | 677.169 | 1 |
VCE Specialist Mathematics Units 3 & 4
Specialist Mathematics incorporates a range of skill areas including, but not limited to, functions, relations, algebra, graphs, differential and integral calculus, trigonometry, vectors, vector calculus, complex numbers and mechanics. A good understanding of the mathematical concepts and skills completed in both General Mathematics (Methods) and Mathematical Methods Units 1 and 2 is assumed in addition to the concurrent knowledge of the skills and concepts covered in Mathematical Methods Units 3 and 4.
Students are expected to be able to apply techniques, routines and processes, involving rational, real and complex arithmetic, algebraic manipulation, diagrams and geometric constructions, solving equations, graph sketching, differentiation and integration related to the areas of study as applicable, both with and without the use of technology.
Note: Students undertaking Specialist Mathematics are expected to have access to either a TI-84Plus graphics calculator or a TI-Nspire(CAS+) calculator. These calculators will be available for purchase through the school as applicable.
Structure
It expected that students will have successfully completed Advanced Mathematics (Year 10), General Mathematics (Methods) Units 1 and 2, Mathematical Methods (Units 1 and 2) and be studying Mathematical Methods (Units 3 and 4) concurrently with Specialist Mathematics. Students that have not completed General Mathematics (Methods) will need to consult with the mathematics coordinator prior to attempting this subject. There exists a large range of prerequisite material, including, but not limited to, familiarity with sequence and series notation and related applications, use of the sine and cosine rules in non-right angled triangles in a variety of different contexts; a variety of geometrical properties including, but not limited to, the two tangents to a circle from an exterior point are equal in length, the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference and the sum of the opposite angles of a cyclic quadrilateral is 180°.
Unit 3 and 4 - Assessment
Note: The outcomes for Units 3 and 4 are the same for Mathematical Methods.
Outcome 1
On completion of each unit the student should be able to define and explain key concepts as specified in the content from the areas of study, and apply a range of related mathematical routines and procedures.
Outcome 2
On completion of this unit the student should be able to apply mathematical processes in non-routine contexts, and analyse and discuss these applications of mathematics.
Outcome 3
On completion of this unit the student should be able to select and appropriately use technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem solving, modeling or investigative techniques or approaches.
Levels of Achievement
The Victorian Curriculum and Assessment Authority will supervise the assessment of all students undertaking Units 3 and 4. In Mathematics: Further Mathematics the student's level of achievement will be determined by school-assessed coursework and two end-of-year examinations. Percentage contributions to the study score in Mathematics are as follows: | 677.169 | 1 |
Composite Functions, An Intro and 4 Assignments for PDF
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
17.72 MB | 54 pages
PRODUCT DESCRIPTION
Included in this zip folder are 5 PDF files. 1 is an introduction to composite functions and 4 are assignments. A brief description of each:
The intro is a 54 page file. In this file the following is discussed:
1. The domain and range of composite functions
2. Composite values given written rules
3. Composite values on the x,y coordinate plane
4. Written rules for f[g(x)] and g[f(x)]
The 1st 10 pages are free in the preview.
Assignment #1 is a 20 question multiple choice and short answer file. The student finds values from written rules and graphs of composite functions. It is on 1 page for easy printing. It can also be used at socrative if you wish.
Assignment #2 is a 20 question multiple choice file. The student finds values of composite functions and writes rules given 2 functions. It is on 1 page for easy printing. It can also be used at socrative if you wish.
Assignment #3 is a 20 question multiple choice file. The student writes rules given 2 functions. It is on 1 page for easy printing | 677.169 | 1 |
Resource Type: Lesson Plan
Sep 07, 2016 - Using Area Models to Understand Polynomials (lesson)
The lesson plan featured in this unit models a progression for using area models, starting with arrays and building all the way up to multiplying...
Sep 07, 2016 - Developing Algebraic Reasoning Through Visual Patterns (lesson)
At the core of this unit is the fact that algebra is far more interesting than just solving for x. By working through the two lessons...
Sep 07, 2016 - Equality (teacher support)
Equality is a fundamental concept in algebra. It is noted through use of an equal sign, represents a relationship of equivalence, and can be conceptualized by the idea of...
Sep 07, 2016 - Modeling Exponential Growth (teacher support)
Now that students have had a chance to explore functions that grow at linear and quadratic rates, they have all the tools to analyze exponential growth...
Sep 07, 2016 - Nonlinear Functions (teacher support)
In this unit, students are introduced to nonlinear functions. The core problem in this unit helps students see how the graph of a quadratic or polynomial...
Sep 07, 2016 - Systems of Equations: Making and Justifying Choices (teacher support)
Systems of equations appear frequently on the HSE exam, as well as almost all college-entrance exams. Our students really...
Sep 07, 2016 - Rate of Change/Starting Amount (lesson)
Rate of change is a fundamental concept when working with functions. This lesson focuses students on function tables, looking for patterns and making...
Sep 07, 2016 - Three Views of a Function (teacher support)
The problems in this unit introduce students to the three views of a function: a rule, a table, and a graph. At the heart of the unit is the Commission...
Sep 07, 2016 - Introducing Functions (lesson)
The 5 activities in this lesson build off each other to develop a conceptual introduction to functions. The lesson begins with a conversation about appropriate age...
Sep 07, 2016 - This lesson picks up on volume and introduces the huge scope of the size of matter. In looking at matter, we could be talking about the size of an atom, a cell, a human, an ocean, the atmosphere, the...
Sep 07, 2016 - In this lesson, we focus on what matter is. Matter has a formal science definition as anything that has mass and takes up space (or has volume). This lesson covers matter, mass, and volume in some Through maps and graphs, students learn about westward expansion, and the role canals and railroads played in expanding the western frontier of the newly independent United States. Canals and...
Sep 07, 2016 - The lesson focuses on the government--forms of government around the world--and the structure of the US government. The lesson begins with a review of the world map. Students identify key countries...
Sep 07, 2016 - The lesson focuses on reading strategies for different text types, asking students to consider the different ways that information can be organized and expressed, depending on the author's purpose,...
Sep 07, 2016 - Students write summaries of key events leading to the American Revolution. They read about the Enlightenment, particularly the ideas of John Locke, stopping to paraphrase key concepts such as...
Sep 07, 2016 - Students review a timeline of US history so that they can place what they are learning about the colonial period in a larger context. They then read about triangular trade, linking it to the economy...
Sep 07, 2016 - In this lesson, students look at contemporary and historic maps to "visualize" colonization. They then look at a map of the New England, Middle and Southern colonies, learning about the economies of...
Jul 29, 2015 - Coda: What Gives My Story Power? Celebrating Student Work
In Lessons 11 and 12, students return to the guiding question that launched this module: What gives stories and poems their enduring power?...
Jul 29, 2015 - Coda: What Gives This Story Power? Re-examining Powerful Stories
In Lessons 11 and 12, students return to the guiding question that launched this module: What gives stories and poems their enduring...
May 21, 2015 - The Developing Core Proficiencies Curriculum is an integrated set of English Language Arts/Literacy units spanning grades 6-12. Funded by the USNY Regents Research Fund, the free curriculum is...
Jan 26, 2016 - Student Outcomes
Students use probability to learn what it means for a game to be fair.
Students determine whether or not a game is fair.
Students determine what is needed to make fair an unfair game... | 677.169 | 1 |
Coral Gables, FL AlgebraReimsky T.
...It is very important that a student comes to Algebra 2 with advanced understanding of many concepts and notions seen before, including: operation tables (memorized); sign operations; linear functions and linear patterns; going from decimals, fractions, and percentages; fraction operations; expone...
Praise A | 677.169 | 1 |
Mary Winter
MATH 106: Applications of Algebra
Dr. Winter mixes and matches diverse technologies to address her learning goals, including weekly in-person tutoring and help, PowerPoint lectures with video and printable handouts, LON-CAPA homework problems, and interactive Excel spreadsheet problems. A goal was to show math-phobic students that there exists mathematics which is interesting, relevant, and that they could learn to do. Virtually all topics are related to the real world.
Each week has variety, coherence, and depth of student experience. In its current form, the course has been offered and refined over many semesters, serving at least 175 or more students per semester since 2003. Evidence of effectiveness of this approach includes noticeably higher course grades and passing percentages. The Math department hopes to keep the course in the hybrid format. Weekly contact with students at this level is essential. | 677.169 | 1 |
What knowledge of mathematics do secondary school math teachers need to facilitate understanding, competency, and interest in mathematics for all of their students? This unique text and resource bridges the gap between the mathematics learned in college and the mathematics taught in secondary schools.
From addition and subtraction to plane and space geometry, simultaneous linear equations, and probability, this book explains middle school math with problems that kids want to solve: "Seventy-five employees of a company buy a lotto ticket together and win $22.5 million.
Math Without Numbers - The Mathematics of Ideas, Vol. 1 Foundations. This Volume 1 in the "Math Without Numbers" series explores the nature of Ideas, from both a practical and an abstract mathematical point of view. | 677.169 | 1 |
The ability to construct proofs is one of the most challenging aspects of the world of mathematics. It is, essentially, the defining moment for those testing the waters in a mathematical career. Instead of being submerged to the point of drowning, readers of Mathematical Thinking and Writing are given guidance and support while learning the language... more...
Constructing concise and correct proofs is one of the most challenging aspects of learning to work with advanced mathematics. Meeting this challenge is a defining moment for those considering a career in mathematics or related fields. A Transition to Abstract Mathematics teaches readers to construct proofs and communicate with the precision necessary... more... | 677.169 | 1 |
Hands-On Math Projects with Real-Life middle school as well as functional and applied high school mathematics courses, this fully revised edition includes 60 project-based activities. New sections address technology-based applications and e-learning, standards and testingMore...
Ideal for middle school as well as functional and applied high school mathematics courses, this fully revised edition includes 60 project-based activities. New sections address technology-based applications and e-learning, standards and testing preparation.
Muschla received her B.A. in Mathematics from Douglass College at Rutgers University and is certified K-12. She has taught mathematics in South River, New Jersey for the last 25 years. At South River High School she has taught math at various levels, ranging from basic skills through Algebra II. She has alto taught at South River Middle School. While there, in her capacity as a Team Leader, she hepled revise the mathematics curriculum to reflect the standards of the NCTM, coordinated interdisciplinary units, and conducted mathematics workshops for teacher and parents. She was recipient of the 1990-91 Governor's Teacher Recognition Program in New Jersey.
About the Authors
Acknowledgments
About This Book
Alignment to National Council of Teachers of Mathematics Standards
How to Use This Resource
Implementing Projects in the Math Class
Overview of Projects in the Math Class
Your Role
Supporting the Standards of the National Council of Teachers of Mathematics | 677.169 | 1 |
Operational Systems course covers modular arithmetic using secret codes and online
games. Learn about operational systems
and their properties (commutativity, associativity, neutral
elements, invertibility) by building interactive machines
and evaluating non-numeric operations. Get a solid
introduction to the concepts of least common multiple
and greatest common divisor, as well as to the geometric
notions of midpoint and reflection.
It is a self-study online course, where suitably talented students are able to work independently and at their
own pace while still developing a deep understanding of the material.
Elements of Mathematics: Foundations (EMF) is a series of courses designed for gifted, mathy middle school students, and it enables them to cover all of middle and high school math except calculus and much, much more by the time they finish middle school.
Here's some more information from the provider of these courses, The Institute for Mathematics and Computer Science (IMACS):
The EMF curriculum was designed from scratch specifically for talented children to leverage their advanced capacity for learning and to engage their unique ways of thinking.
EMF provides a deep, intuitive, and lasting understanding of mathematics as a cohesive body of knowledge that opens the door to scientific discovery and technological advancement.
EMF focuses on the powerful and elegant ideas of mathematics, the kind that talented children find deeply satisfying and inspiring.
Motivated students can learn all of middle and high school math except calculus and much, much more by the time they finish middle school.
EMF courses are modestly priced, and the first one is free for a limited time | 677.169 | 1 |
Description
Calculus may not seem very important to you but the lessons and skills you learn will be with for your whole lifetime!
Calculus is the mathematical study of continuous change. It helps you practice and develop your logic/reasoning skills. It throws challenging problems your way which make you think. Although you may never use calculus ever again after school or college, you will definitely hold on to the lessons that calculus teaches you.
Things like time management, how to be organized, how to accomplish things on time, how to perform under pressure, how to be responsible are just some of the things Calculus helps you become proficient in. Traits that will help you succeed.
Calculus plays a big role in most universities today as students in the fields of economics, science, business, engineering, computer science, and so on are all required to take Calculus as prerequisites.
Our Pre-Calculus guide is a preliminary version of Calculus containing over 300 rules, definitions, and examples that provides you with a broad and general introduction of this subject. A valuable reference guide to have on your phone.
Topics include:
1. Matrix Definition
2. Matrix Addition, Subtraction and Scalar Multiplication
3. Matrix Multiplication
4. Matrix Multiplication Example
5. Augmented Matrix for a System of Equations
6. Solving Augmented Matrices
7. Solving by Gauss-Jordan Elimination
8. Gauss-Jordan Elimination (continued)
9. Special Types of Matrices
10. 2 x 2 Matrix Determinant
11. 3 x 3 Matrix Determinant: Expansion by Minors
12. Determinant of a 3 x 3 Diagonal Multiplication
13. Cramer's Rule for Solving 2 Linear Equations
14. Cramer's Rule - Example of solving 2 Equations
15. Cramer's Rule for 3 Equations in 3 Unknowns
16. Cramer's Rule - Example of solving three Equations
17. Inverse of a 2 x 2 Matrix
18. System of Equations by Inverse Matrices
19. Area of a Triangle Using Matrices
20. Test for Collinear Points Using Matrices
21. Finding Equation of a Line Given Two Points
22. Conic Sections
23. Ellipses
24. Ellipse Whose Center is at the Origin
25. Ellipse with Center at the Origin Example
26. Ellipse Translation
27. Ellipse Translation Example
28. Equation of an Ellipse in Standard Form
29. Hyperbola
30. Hyperbola Standard Form
31. Hyperbola Standard Form (continued)
32. Hyperbola Centered at the Origin Example
33. Hyperbola Translation
34. Hyperbola Centered at the Origin Example
35. Parabolas
36. Parabola Equation with Vertex at the Origin
37. Diagrams of Previous Page Parabolas
38. Parabola Equation with Vertex (h, k)
39. Polar Coordinate Plane
40. Polar Coordinate System - Plotting Points
41. Multiple Representation of points
42. Coordinate Conversion
43. Examples of Conversion
44. Equation Conversion
45. Polar Equations Graphing
46. Special Graphs – Limaçons
47. Special Graphs - Rose Curves
48. Special Graphs - Circles & Lemniscates
49. About Pre-Calculus
50. Support Page
Even if you dont use Calculus, this app sure is a cool way to show-off some high IQ!
Like all our 'phoneflips', this fast and lightweight application navigates quick, has NO Adverts, NO In-App purchasing, never needs an internet connection and will not take up much space on your iPhone!
Helped with my exams
Paulie132 write: This puts everything out in an easy to read format with the formulas. Instead of going through 100's of pages to make my cheat sheet for my finals and it was all readly available in moments
5/5 stars
This is comprehensive
BrieMarin write: This is everything you need in a short easy to sort list. I took pre cal already (I'm on my second time around) and this is easier than any text book that I paid over 100 for. Now I need my physics app lol
Users Community
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The Pre Calculus flashcard package has 140 flashcards. Definitions, theorems, formulas, examples are all included in this package and highlighted in different colors, such as theorems are highlighted in green, definitions are highlighted in light blue.... It's very easy and fun for anyone to… more
GoLearningBus: A complete education journey from school, college to professional life More than 4 million paying customers from 175 countries.Get on the GoLearningBus today and take a complete education journey from school, college to professional learning.Access all 300 apps forI have tutored many, many people in Math through Calculus, and I have found that if you start off with the basics and take things one step at a time - anyone can learn complex Math topics. This 5 hours video tutorial series is full of worked example problems in Trig and Pre-CalculusYou will learn 1 Ended and is full 2 ended and is fullDoes calculus have you dazed and confused? Calculus FTWhas hundreds of detailed step-by-step solutions to example problems from first year calculus Written by a college professor, Calculus FTW is an app that helps students learn calculus from a problem based approach. Calculus FTW is not a mere… more
Does calculus have you dazed and confused? Calculus FTW Free has detailed step-by-step solutions to example problems from first year calculus NOTE: This is the FREE version of Calculus FTW. It includes all concept pages from the FULL version and approximately 25% of the detailed solutions to… more | 677.169 | 1 |
Calculus: Graphical, Numerical, Algebraic
Learners explore the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function. They write a paragraph describing any difficulties they encountered while graphing parametric equations. Students discuss the notation f-1, which represents the inverse function of f. | 677.169 | 1 |
Looking to cover a concept that students really have a hard time with? Distributive Property is truly an abstract concept for MOST students and this lesson will help you build a resource for each of them.
Included you will find:
- Full Teacher
Are you ready to start using an Interactive Notebook with your Algebra 1 classes? This unit on Algebraic Foundations is the perfect unit to start your school year with review from Pre-Algebra. Students will review concepts to build the foundational
Domain and Range is a necessary skill for student's to learn in Pre-Algebra and Algebra. Understanding the concepts of x and y coordinates as well as their relationships on a graph is crucial as they progress in Algebra
Domain and Range Task Cards
Are you ready to start using an Interactive Notebook with your Algebra 1 classes? This unit on Linear Functions is the perfect unit to continue working in an Algebra 1 Interactive Notebook. Students will continue to build on prior knowledge to form
Are you ready to start using an Interactive Notebook with your Algebra 1 classes? This unit incorporates many of the skills you have been teaching in your classroom into a project format.
Includes:
- Teacher Directions for each Interactive
Need an Interactive Lesson to wrangle the rules of the Trigonometry Identities? This Layered Book Flippable is perfect for students to identify the rules as well as have a place to have examples of each rule.
Included in this Activity:
- Teacher
Are you ready to start using an Interactive Notebook with your Algebra 1 classes? This unit on Operations with Rational Numbers is the perfect unit to follow up from a review of Pre-Algebra. Students will continue to review and build on concepts
Are you ready to start using an Interactive Notebook with your Algebra 1 classes? This unit on Exponents flows right along with the other units available in my store for Algebra.
Includes:
- Vocabulary Frayer Models for Unit Vocabulary
- Teacher
"Solve and Snip" include Interactive Practice Problems for skills aligned with TEKS and Common Core.
In the LInear Equations Solve and Snip students will read a word problem and then solve the problem by showing work in the show work area. Then
HARD GOODS LIQUIDATION SALE!!!!
Directions for Ordering:
1. Add the Spiral Bound copy for the grade level(s) of Math that you teach to your cart.
2. Checkout here on Teachers Pay Teachers.
3. TpT will notify me that I have a Hard Good to ship and
The System of Equations Matching Game allows students to solve problems where they must use subsitution to arrive at their answer.
Contents Include...
- 16 Sets of Equations Cards in COLOR and Save Your Ink
- 16 Answer Cards in COLOR and Save Your | 677.169 | 1 |
Mathematical Methods Units 1 & 2
Unit 1
Mathematical Methods Units 1 and 2 provide an introductory study of simple elementary functions of a single real variable, algebra, calculus, probability and statistics and their applications in a variety of practical and theoretical contexts. They are designed as preparation for Mathematical Methods Units 3 and 4 and contain assumed knowledge and skills for these units. The focus of Unit 1 is the study of simple algebraic functions, and the areas of study are 'Functions and graphs', 'Algebra', 'Calculus' and 'Probability and statistics'. At the end of Unit 1, students are expected to have covered the content outlined in each area of study, with the exception of 'Algebra' which extends across Units 1 and 2. This content should be presented so that there is a balanced and progressive development of skills and knowledge from each of the four areas of study with connections between and across the areas of study being developed consistently throughout both Units 1 and 2.
In undertaking this unit, students are expected to be able to apply techniques, routines and processes involving rational and real arithmetic, sets, lists and tables, diagrams and geometric constructions, algebraic manipulation, equations, graphs and differentiationEntry
It is expected that students will have successfully completed 'Advanced Mathematics' in year 10 and study Specialist Mathematics Units 1 & 2 concurrently with Mathematical Methods Units 1 and 2 in preparation for Specialist Mathematics Units 3 & 4
AREAS OF STUDY
Area of study 1 Functions and graphs
In this area of study students cover the graphical representation of simple algebraic functions (polynomial and power functions) of a single real variable and the key features of functions and their graphs such as axis intercepts, domain (including the concept of maximal, natural or implied domain), co-domain and range, stationary points, asymptotic behaviour and symmetry. The behaviour of functions and their graphs is explored in a variety of modelling contexts and theoretical investigations.
Area of study 2: Algebra
This area of study supports students' work in the 'Functions and graphs', 'Calculus' and 'Probability and statistics' areas of study, and content is to be distributed between Units 1 and 2. In Unit 1 the focus is on the algebra of polynomial functions of low degree and transformations of the plane.
Area of study 3: Calculus
In this area of study students cover constant and average rates of change and an introduction to instantaneous rate of change of a function in familiar contexts, including graphical and numerical approaches to estimating and approximating these rates of change.
Area of study 4: Probability and statistics
In this area of study students cover the concepts of event, frequency, probability and representation of finite sample spaces and events using various forms such as lists, grids, venn diagrams, karnaugh maps, tables and tree diagrams. This includes consideration of impossible, certain, complementary, mutually exclusive, conditional and independent events involving one, two or three events (as applicable), including rules for computation of probabilities for compound..
Outcome 3
On completion of this unit the student should be able to use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.
Unit 2
In Unit 2 students focus on the study of simple transcendental functions and the calculus of simple algebraic functions. The areas of study are 'Functions and graphs', 'Algebra', 'Calculus', and 'Probability and statistics'. At the end of Unit 2, students are expected to have covered the material outlined in each area of study. Material from the 'Functions and graphs', 'Algebra', 'Calculus', and 'Probability and statistics' areas of study should be organised so that there is a clear progression of skills and knowledge from Unit 1 to Unit 2 in each area of study.
In undertaking this unit, students are expected to be able to apply techniques, routines and processes involving rational and real arithmetic, sets, lists and tables, diagrams and geometric constructions, algebraic manipulation, equations, graphs, differentiation and anti-differentiationAREAS OF STUDY
Area of study 1: Functions and graphs
In this area of study students cover graphical representation of functions of a single real variable and the key features of graphs of functions such as axis intercepts, domain (including maximal, natural or implied domain), co-domain and range, asymptotic behaviour, periodicity and symmetry.
Area of study 2: Algebra
This area of study supports students' work in the 'Functions and graphs', 'Calculus' and 'Probability and statistics' areas of study. In Unit 2 the focus is on the algebra of some simple transcendental functions and transformations of the plane. This area of study provides an opportunity for the revision, further development and application of content prescribed in Unit 1, as well as the study of additional algebra material.
Area of study 3: Calculus
In this area of study students cover first principles approach to differentiation, differentiation and anti-differentiation of polynomial functions and power functions by rule, and related applications including the analysis of graphs.
Area of study 4: Probability and statistics
In this area of study students cover introductory counting principles and techniques and their application to probability and the law of total probability in the case of two
Outcome 3
On completion of this unit the student should be able to select and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches
ASSESSMENT
The award of satisfactory completion for a unit is based on whether the student has demonstrated the set of outcomes specified for the unit. Teachers should use a variety of learning activities and assessment tasks that provide a range of opportunities for students to demonstrate the key knowledge and key skills in the outcomes. Assessment is part of the regular teaching and learning program and should be completed mainly in class and within a limited timeframe.
Assessment tasks include components completed with and without the use of technology as applicable to the outcomes.
Demonstration of achievement can be based on the student's performance on a selection of the following assessment tasks: • Assignments • Tests • Summary or review notes • Modeling tasks • Problem solving • Mathematical investigation
Pathways
VCE Further Mathematics (Units 3 and 4)
VCE Mathematical Methods (Units 3 and 4)
Students successfully completing both year 11 Specialist Mathematics and Mathematical Methods are recommended to study any one of the following combinations: | 677.169 | 1 |
Discovering Advanced Mathematics - Pure Mathematics
"Discovering Advanced Mathematics" is a series of four books designed to match the new generation of A and AS Level maths syllabuses and satisfy the requirements of the SCAA Common Core for Mathematics. This text shows the immediate application of pure mathematics and makes great use of modelling techniques. The book is divided into 29 short chapters, allowing teachers to match the book easily to their own teaching plans. It provides chapter introductions, chapter summaries, investigations and practicals, exercises and examination questions.
"synopsis" may belong to another edition of this title.
From the Back Cover:
'Discovering Advanced Mathematics' is a series of four books designed to match the new generation of A- and AS-level maths syllabuses and satisfy the requirements of the SCAA Common Core for Mathematics. The books provide unparalleled support for developing modelling and problem-solving skills.
• real opportunities to study advanced mathematics through stimulating and original contexts, questions and modelling tasks • use of new technology is encouraged throughout – graphics calculators and spreadsheets, for example – but tasks are accessible to students with the most basic of A-level calculators • accessible to all of your students because of careful introduction of major concepts and language level
'Pure Mathematics' – Bob Francis
A continual emphasis on application and relevance: wage rises to yacht racing; fairground rides to mortgage repayments. pure mathematics shows the immediate application of pure mathematics that students are learning, and makes great use of modelling techniques throughout. The text is divided into 29 short chapters, allowing teachers to match the book easily to their own teaching plans.
"Although the title is 'Pure Mathematics', the book also covers the requirements for students to study mathematical modelling, the mathematics of uncertainty, and the appropriate use of technology…This is an interesting book, offering many imaginative ideas, providing structure and support for all students, and challenge for the most able." TIMES EDUCATIONAL SUPPLEMENT.
John Berry: Professor of Maths Education and Director of the Centre for Teaching Mathematics, University of Plymouth. Part of the QCA team evaluating new A-level syllabuses. Bob Francis: Lecturer at Exeter College and a chief examiner for A-level mathematics for OCR (MEI).79822
Book Description Paperback. Book Condition: Good. The book has been read but remains in clean condition. All pages are intact and the cover is intact. Some minor wear to the spine. Bookseller Inventory # GOR001399256 | 677.169 | 1 |
Synopsis
National 5 Maths with Answers by David Alcorn
Teach lessons that suit the individual needs of your classroom with this SQA endorsed and flexibly structured resource that provides a suggested approach through all three units. - Covers the new specification with all the new topics in the SQA examinations - Provides thorough exam preparation, with graded Practice Exercises - Organised to make it easy to plan, manage and monitor student progress This 'with answers' version textbook completely covers the latest National 5 syllabus. Each chapter includes summaries of key points and worked examples with explanatory notes showing how skills are applied. Section Reviews presented in non-calculator and calculator formats provide students with the opportunity to consolidate skills acquired over a number of chapters. There are plenty of exercises and invaluable exam practice throughout to help build confidence and knowledge. A 'without answers' version is also available
About the Author
David Alcorn has written over 80 best-selling maths textbooks, and is a highly experienced teacher and publisher. Consultant Editors Robert Barclay and Mike Smith are experienced teachers and Principal Assessors. | 677.169 | 1 |
being able to read Finnish, it looks like what they wrote might be equivalent to source code for slides to use in lecture, or bare bones Wikipedia entries on high school math topics. Obviously not a complete text book, and I'm pretty sure it's the easiest part. Example problems work through in detail, practice problems throughout the chapter, and a collection of practice problems with answers available and others with a teacher's key… these are most of what most students will use in a math text. And they have to be chosen carefully to be representative and of appropriate difficulty, carefully laid out, and painstakingly proof read or they'll be worse than useless.
Having worked for a small educational publisher I agree — a textbook is more than a couple lines explaining a concept. It takes thousands of hours per grade to create complete worked examples (having full step by step solutions that explain the problem solving process), mathML equations, graphics including charts/graphs/diagrams and practice questions — lots and lots of practice questions will full solutions so students can practice what they are learning.
Then it takes a tremendous amount of time, particularly in mathematics, to edit the whole book. Every mathML equation and every question needs to be examined and the questions solved independently of the solution to ensure they are accurate.
Then, finally if you are publishing in XML you need to have a technical person go through the document to ensure that it is valid, well-formed and will render properly.
One math concept would take, on average 0.5 days of effort for a team of 5 people (2 writers, 1 graphic artist, 1 editor and 1 technical XML person). An average math course in the US is 1/5th of 180 days of instruction which equals about 216 instructional hours. A math textbook will have about 100 concepts to present in those 216 hours which means it will take 5 people 50 days (almost 3 months) to write a single text book.
A handful of teachers cannot possibly write a high quality, curriculum aligned and comprehensive resource in a weekend. It just is not possible.
When I was in college, it was widely told that our calculus professor had written the textbook in two weeks, on a bet. And it might as well have been in finnish. So this one will be better if only on price. | 677.169 | 1 |
Based on over 15 years' experience in the design and delivery of successful first-year courses, this book equips undergraduates with the mathematical skills required for degree courses in economics, finance, management and business studies. The book starts with a summary of basic skills and takes its readers as far as constrained optimisation helping them to become confident and competent in the use of mathematical tools and techniques that can be applied to a range of problems in economics and finance. Designed as both a course text and a handbook, the book assumes little prior mathematical knowledge beyond elementary algebra and is therefore suitable for students returning to mathematics after a long break. The fundamental ideas are described in the simplest mathematical terms, highlighting threads of common mathematical theory in the various topics. Features include: a systematic approach: ideas are touched upon, introduced gradually and then consolidated through the use of illustrative examples; several entry points to accommodate differing mathematical backgrounds; numerous worked examples and exercises to illustrate the theory and applications; full solutions to exercises, available to lecturers via the web. Vass Mavron is Professor of Mathematics in the Institute of Mathematical and Physical Sciences at the University of Wales Aberystwyth. Tim Phillips is Professor of Mathematics and Professorial Fellow in the School of Mathematics at Cardiff University. | 677.169 | 1 |
$119 an introduction to the study of fundamental inequalities like the arithmetic mean-geometric mean inequality, the Cauchy-Schwarz inequality, the Chebyshev inequality, the rearrangement inequality, inequalities for convex and concave functions. The emphasis is on the use of these inequalities for solving problems. Its special feature is a chapter on the geometrical inequalities which studies relations between various geometrical measures. It contains more than 300 problems, many of which are applications of inequalities. A large number of problems are taken from the International Mathematical Olympiads (IMO) and many national olympiads from countries across the world. The book will be very useful for students participating in mathematical contests. It should also help graduate students in consolidating their kwledge of inequalities by way of applications. | 677.169 | 1 |
composing thus its main subject. the author has used only the properties of inequalities and limits actually covered by the curriculum on mathematics in the secondary school. one should not be disappointed if the obtained results differ from those of the patterns. For this reason we strongly recommend the readers to perform their own solutions before referring to the solutions given by the author at the end of the book. However. K orookin.4 as exercises for individual training. 35 of which are provided with detailed solutions.
6
. At the end of the book the reader will find the solutions to the' given exercises. 2. and 28 others are given in Sections 1.PREFACE
In the mathematics course of secondary schools students get acquainted with the properties of inequalities and methods of their solution in elementary cases (inequalities of the first and the second degree). 2. The solution of some difficult problems carried out individually will undoubtedly do the reader more good than the solution of a large number of simple ones.1 and 2. The author considers it as a positive factor. The book contains 63 problems. When proving the inequalities and solving the given problems.1.3. P. In this booklet the author did not pursue the aim of presenting the basic properties of inequalities and made an attempt only to familiarize students of senior classes with some particularly remarkable inequalities playing an important role in various sections of higher mathematics and with their use for finding the greatest and the least values of quantities and for calculating some limits.
+
When solving practical problems.. and I'1x is an error of its measurement.e. Moreover. [x] is the integer (whole number) defined by the inequalities
[ x])
+
[xl~x
<~Jx]
+' 1. since the integral part does not exceed x. but only approximately.
7
.
1. the distance to the Moon. in accordance with the technical progress and the degree of complexity of the problem. It follows from this definition that [z] ~ x. Thus.g. landing spaceships on the Venus and so on). Considerable errors of measurement become inadmissible in solving complicated engineering problems (i.CHAPTER 1
Inequalities
The important role of inequalities is determined by their application in different fields of natural science and engineering. then the real value y satisfies the inequalities x-I I'1x 1~ y ~ x 1 I'1x I. its speed of rotation. it becomes necessary to improve the technique of measurement of quantities. On the other hand.) may be found not exactly. The Whole Part of a Number
The whole (or integral) part of the number x (denoted by is understood to be the greatest integer not exceeding x. etc. The point is that the values of quantities defined from various practical problems (e. If x is the found value of a quantity. landing the mooncar in a specified region of the Moon. since [xl is the greatest integer. it is necessary to take into account all the errors of the measurements.1. satisfying the latter inequality. then [xl 1 > x.
The ability to find the integral part of a quantity is an important factor in approximate calculations.
Furthermore.
from the inequalities
3<31<4.[z ] = 1. since o:s. [xl
+ 1-
[x
1
=
1. Problem 1. With large
N the error will be small. If we have the skill to find an integral part of a quantity x. The integral part of a number is
found in the following problems. then taking [xl or [xl 1 for an approximate value of the quantity x../ [Nx] +_1 N ~x~ N N•
Thus. we shall make an error whose quantity is not greater than 1.1:=
1
+ V2 + -Va + V4 + V5 . Find the integral part of the number
. Yet. it is important to note. -2<-V"2<-1.
[Nx].
+
o < [xl +
+
1 .J=5. since
+ -}
[Nxl<Nx<[Nxl
then
+ 1.
[-V2]=-2. x . the number
----r
[Nx]
+
1 2N
differs from the number x not more than by 2~.
5=5<6
it follows that
[31]=3.For example. Indeed. that the ability to find the whole part of a number will permit to define this number and.
[5]=5. ty permits
the knowledge of the integral part of a quantito find its value with an accuracy up to -}
The quantity [xl may be taken for this value.x:S.[xl < [xl 1 ..
[1.
111
1
8
.
5< ~ <6. with any degree of accuracy./ .
7 + 0. 0.+ V1000000
1
Solution.Solution. Find the integral part of the number
+ 0. since
~n <2Vn -2Vn-1. Let us use the following
inequalitres
1-~ 1<1. 3.6.::
(1)
2Vn+1-2Vn
.. Combining them we get 1
that is.=vn
2Vn+1-2Vn<
Indeed.
y=1+
V2
1
+
-vs + V4
1 1
+ .. 0.5. To solve this problem.1 in excess or deficiency). 0.10
1
v2
+
.1 < x < 3. +--::. let us investigate the sum
1+
and prove that
.5 + 0. In this relation.5. there were only 5 addends.5 + 0.15.25 differs from x not more than by 0.8 + 0.5<vi <0... Problem 2.5
(which are obtained by extracting roots (evolution) with an' accuracy to 0.6 + 0.Vii) (Vn:tI+
Vn+t+
lfri
e
. 000 addends).r v3
1
+
.4. while in the second.
=
2
eVii+1.r v4
1
1 + .7<V~ <0.8.-t.4 < x < < 1 + 0. This circumstance makes it practically impossible to get the solution by the former method. hence.
Vn+1+ lfri lfri) =
2
::::::::. 1000.5<V! <0.4<V! <0. it is necessary to note that the number 3.
0. This problem differs from the previous one only by the number of addends (in the first.5 + 0. [xl = 3.
Xn are not all equal. . we get a=
16
by g and
Xt+x2+.. From the equality g =.. 10. x2
X2
= 1. Divide by X2 the numerator and denominator of the left-hand member of the inequality:
Since _1_..
Multiplying both members of the last inequality dividing by n.Y XjX2 ••• Xn it follows that
1-
"V. If the numbers Xl.·. The geometric mean of positive numbers is not greater than the arithmetic mean of the same numbers...:!.+~+ g g
..
. 1
-
Since the product n of the positive numbers equals then (Theorem 1) their sum is not less than n. then
Now let us prove the statement made at the beginning of the section.
~
g
or .•
+~~ g
?"n..
~
g
-.Solution. oga
Problem
4.:2.1og a log a
=
1. g
..
n
+xn '-
-::?"g. Prove the inequality
x2
1+x4
<"2'
1
Solution.=!. then the geometric mean of these numbers is less than their arithmetic mean. Proof.10 = log a + -1->2. Theorem 2.. that is
1. then
1 + log.
Since log.
.. g
X2
g
.. X 3' ••.
g
_2.
Prove
inequality
n! < (
nt
i
r. Using Theorem 2._ g
=~ = g
But if the
is Xl=x2=".
an of the
=
n
is the arithmetic I1l1mber
mean of the numbers aI' a2.:l.
•••
. n «.
-< . the
\
!
I
2-0866
17
..>2. we get the inequality (5). that is. 2
n
-f.
n'. Particularly.
. we get V'n!=.
Problem 6..'~xn=g.
+ aa -1. •••
. The number
_ ( a1 Ca -rx. -
1
n
2n
Raising to the nth power both parts of the last inequality..
•••
.. =2=1. g that
Xb X2. Solution..3
(n+1)n
.2.7
3
. when the parallelepiped
t he
is a cube. numbers .. .
an. Suppose m = a b edges and V = a be is the volume Since then V
=b
+ +c
is the sum of the of the parallelepiped. the number ad-a2-1-··· +an
Cl
a2.'y1.
(5)
Solution.. Problem 5.. order ex. holds only when .. then a >g.
-2n+1
1+2+3: . Definition. From all parallelepipeds with the given sum of the three mutually perpendicular edges. The
3/VV
=
3/V abc
-< a+b+c 3
= 3" '
holds only when
a=
m
sign of equality
= c = '. that the equality .Notice. Xn
are not equal.an
a
)a
-
1
is termed the mean power of numbers al. find the parallelepiped having the greatest volume.
an +at a2
••• ... + an) (_1_+_1
at
••• ... the second is proved in a similar way. then (at + a2+ . Prove that if all a2.then
°
an. Prove that if aI' a2... +a~
•
_1_
Raising both parts of the last inequality taking into consideration.1)-1 _. an are positive
(6)
c~:::. g=... an . in particular.. From the fact.) at a2
. Problem 7. an +at a2
+an)
1
1
n
It follows from this inequality n2«at+a2+'"
18 (
that
1 1 1 -+-+ .Y
n-
to a power
j
that _!_ a
<
n
a
and
0. an
+an
Ct·
Solution. numbers..
that is... a2. .
. we get
ata2'"
an
>-
( a~+a~+
.... ~. the mean power with a negative exponent does not exceed the geometric mean. that the harmonic mean C -1 does not exceed the arithmetic mean CI• Problem 8. numbers and a < < ~. and the number
C_t-
_ (ai1+az1+ . e «. Solution. +a~
)a =Ca·
So the first part of the inequality (6) is proved. +_1 ) >-n2. that the geometric mean of positive numbers does not exceed the arithmetic mean.From the inequality (6) it follows.. a~<
a~+a~+
n
..
an are positive
a2
+ . and the mean power with a positive exponent is not less than the geometric mean. Since
_
C_t -
C-t<g<Ct.
ctl..... +a.
n -
-+-+ .. +.
then
/' 1~ at+a2+'" n
-+-+ ..
1
1
n
1
is called the harmonic mean of the numbers aI.is named the root-mean-square. we have
n/V
a~a~ •..
From the inequality (7) it follows. Solution.*ty get the inequality (7). Solution. b.
12
Problem 2. Prove that with the increase of the number the quantities
Xn
= ( 1 + rand
!
Zn
= (1 -
!r
2* 19
. n
by
12.
Multiplying both members of this inequ. that
2ala2 4ala2aga4
we shall
<: a~ + a. Since the geometric mean does not exceed the arithmetic mean. n an
<: a~+a2+'" +a~ . Prove the inequality
na1a2
•••
an ~ a~
+ alI + .... Prove that for any positive numbers a. + a~. the doubled product of two positive numbers does not exceed the sum of their squares. an > 0..
(a
Problem 1. We shall come to its determination after carrying out the solution of a number of problems in which only Theorem 2 is used.Problem 9. 3ala2a3 <: a~ + a~ + a~. . =1= b) the inequality
n+y abn < a+ nb
n+1
is true. . then
ala2'" an=y n/ nn a1a2 . <: a~ + a~ + a! + a!.. a2 > 0.3. The Number e The number e plays an important role in mathematics..
We have
=
n
a+nb n+1 '
and that suits the requirement. 1.. the trebled product of three numbers does not exceed the sum of their cubes and so on.
(7)
where a1 > 0.
that is.
It is known. if the number n is high. lim Yn = lim ( 1 + Yn also
!) n+
1
=-c
lim ( 1
+ !) ( 1 + !) n =
=.
then Xn is
(8)
+
+r<
It is not difficult to check that e < 3.
Thus. Indeed. the variable Xn satisfies two conditions: (1) Xn monotonically increases together with the increase of the number n. that is Xn
= (1
its limit.
21
.
. Now.e·1=e. It is common knowledge that the numbers e and Jt are irrational.=(1-tHence. It is used.eo
lim Xn = lim ( 1
'n-s
oc
+ -1-f n
e. (2) Xn is a limited quantity.
As the quantity xn increases reaching smaller than its limit.
e ~ lim Xn ~ 2. the number e together with the number Jt of great significance. as the base logarithms. 2=XI<Xn=
(1 ++)n
< (1
++ )n+l
=Yn<YI=4.On the other hand... Indeed. Each of them is calculated with an accuracy of up to 808 signs after the decimal point.7182818285490 . there exists a limit of the variable quantity Xn. The logarithm the number N at the base e is symbolically denoted by N (reads: logarithm natural N). that is. ~)6=2. and is of of In e = 2. This limit is marked by the letter e. 2 < Xn < 4. then Xn<Yn<Y5. that monotonically increasing and restricted variable has a limit. for instance.
In mathematics. Hence. e
=
n-e.985984. let us show that the limit of the variable equals e. known as natural logarithms..985984
rt-s-co
<
3.
>
+
(n)n ""3
.
+ 1.
(1+*r
< e. The inequality is easily checked for n = 1. We shall prove the inequality (10) using the method of mathematical induction. 'n
e (Pro(9)
n+
1
»:». Prove the inequality
(10)
Solution.
Problem 4. Thus the inequality (9) is proved to be true for all values of n.Since Yn diminishes blem 2). we get 300!
> ( 3~O )
=
100300•
22
. then
coming close to the number
1 ( 1 +. Actually. according to the inequality (k+1)!>
(k~1
)k+1
r+
(8)
(1 +
1
:
=
(k~1
r+
+r
1 . that is
Multiplying we get
both members of the last inequality
k~1
by k
(k+1)k!=(k+1)!>(:r(k+1)=( Since. 300! Indeed. it follows from the inequality (9) that
I n.
Assume.
then
that is the inequality (9) is proved for n = k 1.) .
it is easy to prove that
300 •
By means of the last inequality. setting
> 100
300
in it n = 300. Since e < 3. that the inequality
(10) is true for n = k.
bearing
(12)
in (11) and (12) holds only when x
= 0.
«1+x)+(1+x)+
...= O.
number. +1
=
The sign of equality occurs only when all multipliers standing under the root sign are identical.
Thus.m
-----'
.m < n. when 1 x = 1.4.. we have proved the first part of the theorem considering the case.e..
a < 1.. Suppose that a is a rational
mind that < a < 1. 1.Y(1+x)m..:: -1 and 0<
if a <
°
(1 +x)ct
:::.
in
Proof. x .. The Bernoulli Inequality
In this section. +(1+x)+1+1+ n
.:: 0. 1
°
+x
. (1+x).1.1
+ ax. Since according to the condition.. But if x =1= 0.... where m and n are n positive integers.
23
. when a is a rational number. Theorem 3.. making use of Theorem 2 we shall prove the Bernoulli inequality which is of individual interest and is often used in solving problems.:: 1 + ax..1
~
. then
(1
The sign of equaliiu
+ x)ct . 1 :::.<e (-
n+ 1 )n+l
e
is proved completely the same way as it is done with the inequality of Problem 4.
n-m
1:::.1n
m=
=.. then
+
(1 +: x)ct
<
1
+-
ax. then
m
(1+x)ct=(1+xfn
=. then
(11)
or a> 1. Let a = !:!:. i..Y(1+x)(1+x).
However
It
x .The inequality
I n.
that a is an irrational number. 3. it follows that (1 + x)a = lim (1 + x(n<
rn-CX
r
lim (1 -/. From the inequalities
°
(1
+
xrn<1
+ rnx.Assume now. move on to proving the second part of the theorem. Bear in mind that < < r. having for a limit the number a. be the sequence of rational numbers. since its left part is not negative.
n
=
1.
Thus the first part of the theorem is proved completely. then as it has already
been proved
Hence. is
already proved by us for the case when the exponent a rational number. For this reason.. r2.. Now. Obviously.
a
1. Su ppose a > 1. If 1 + ax ~ 0.
x ~ -1..
that is
1£ x =1= 0.
.. and its right part is negative. we have
Since
° < __::_ <
r
(1 + x)a = [(1 + xfry. < 1. 0 < a < 1. r r
<1 +
ax.
(t
+ x)a< (1 + ~ x
(1 +x)a
r.. If 1 ax < 0. ax ~ -1.a
Thus the inequality (11) is proved for irrational values of a as well. ••• .
n
. . then the inequality (12) is obvious.. then by virtue of the first part of the theorem proved above we have
+
(1 + ax)a <1
24
~
1 +.ax= ex
1 +x. that when x =1= 0 in (11).rnx) = 1 + ax.
°
i.e. then let us consider both cases separately. 2.!!_x~c1+ax. . then (1+~x)r<1+r. the sign of equality does not hold. r n . What we still have to prove is that for irrational values of a when x =1= and 0 < a < 1 (1 + x)a < 1 + ax. take a rational number r such t hat a < r < 1. Let r1..
from
these
inequa25
. a < O. the theorem is proved completely.? 0.
1
then
accord ing
to the
(1 _ Multiplying
!
a~ 1 .::. Problem 1.
(n+ 1)a+l_na+l a+1
O.? 1.. By virtue of the first part of the theorem we get
. then select the
0. Since 0 < a inequality (11) we have (1
+
+r+ < +
1
+ 1 < 1. If 1 + ax <
ax ~ (1
+ ax
.
+ 1)a+l < na+ + (a+
< na+l
(13) easily follow
The inequalities lities. Prove.r
::pi
+ ~X
n
( the latter 'inequality is true.
a~1 .. But if 1 positive
+ .< n
1 would
a
(1+x)n::p
1a
1--.Here the sign of equality holds only when x = O.(a+ 1) na. x)n::p1 + n'
that if 0>
n"::" n
X=
1 + ax. then
(13)
< w «:
na+1_(n_1)a+l a+1
Solution. Raising both parts of the latter inequality to the nth power we get (1
+ xt::P(1 +.::.
=
Notice. Raising both parts of the last inequality to the power a we get 1
+-
Now let us suppose quality' (12) is obvious.1')a. then the ine-
integer n. that the equality is possible only when x Thus.
a>
-1. since 1. .~:X2) .
)a+l
<
1
1-
these inequalities
(n (n _1)a+l
by ncx+1. so that the inequality
-
be valid. we obtain
1) na...
ex< ~.2 before Problem 7 we have already named the number
the mean power of order ex of the positive numbers aI. Assume. 1. we get
Now. an' In the same problem. Theorem 4. should be proved the validity of the inequality ca ~ c(3 any time when ex< ~. The Mean Power of Numbers In Sec..hen Ca t
. a2.. when the numbers ex and ~ have different signs the theorem has been proved above (refer to Problem 7. we have to prove the theorem only for the case when ex and ~ have the same signs. that 0 < ex< ~. the mean power of order ex is monotonically increasing together with ex. .2 and the definition prior to it).
c/3' and
= c/3' only when al = a2 = . that Ca ~ c(3' if ex< 0 < ~. For the case. Thus.5. ••• . it has been proved.and let
Dividing
C(3
by k.
~
••• Ca
. .In other words. 1. Sec.1.
an are positive numbers and
Proof. = an'
It au a2. supposing
we obtain
(15) 27
. Here.
In this case d. 1. we shall assume g = co. that
i. while the arithmetic mean does not exceed the root-mean-square of positive numbers. But if the numbers aI' a2.: g = Co.. the harmonic mean does not exceed the geometric mean.:1
n
1. = an = = k. since (see Problem 7. Further on we shall name the geometric mean by mean power of the order zero. if
29
.. Theorem 4 is proved completely.. if a < 0. If a < ~<
Thus Theorem 4 is proved regarding
the case when
0. a1 = a2 = .. From the proved theorem it follows. 1. _i df +d~ + . hence.= d2 = = dn = 1 and. But. then <1. if ~ > 0. and cfl ). that Theorem 4 is applicable in this case as well. Notice. an are not identical. Reasoning the same way a as before.I t is necessary to note that C fl = k = Ca. that is when Xl = x2 = = xn = 0 (Theorem 3). Sec. since ~ < 0.. we get in (*) and (16) the opposite signs of inequalities. . then from the inequality
°
°<
--------<.t
it follows that
that
is
Thus. then
<a< ~. For example. the geometric mean in its turn does not exceed the arithmetic mean.< 1.2) Ca ~ g = Co. +d..e. . only when the signs of equality occur everywhere in (*). that is. = . in particular.
making use of the inequalities.CHAPTER
2
Uses of Inequalities
The use of inequalities in finding the greatest and the least function values and in calculating limits of some sequences will be examined in this chapter. Moscow. For example. Natanson "Simplest Problems for Calculating the Maximum and Minimum Values". some important inequalities will be demonstrated here as well.P. certainly.1. y.
and its volume is
v
xyz. The Greatest and the Least Function Values A great deal of practical problems come to various functions. we shall solve a number of such problems. so that the area of the box surface should be the least. z are the lengths of the edges of a box with a cover (a parallelepiped). 2. it is desirable. studied in the first chapter".. if x.
If the material from which the box is made is expensive. First of all. then. to manufacture it with the least consumption of the material. with the given volume of the box. 1952.
32
. then the area of the box surface is
S
=
2xy
+ 2yz + 2zx.
=. Here. Besides that. 2nd edition. we shall prove one theorem.e. Gostekhizdat.
1 Concerning the application of inequalities of the second degree to solving problems for finding the greatest and the least values see the book by I. i. We gave a simple example of a problem considering the maximum and the minimum functions of a great number of variables. One may encounter similar problems very often and the most celebrated mathematicians always pay considerable attention to working out methods of their solution.
the sign of equality holds only when y = 1. Multiplying both members of the latter inequality by cu., we get (cy)a - aca-t (cy) ~ (1 - a) ca, y ~ 0. Assuming x=cy we get and acCG-1=a, c== (a
a )a=-T"" ..
1
1
here the equality occurs Thus, the function xCG - ax, a>
only
when
x
= c = ( : )a=-T""
1
1, a>
0, x ~ 0, x
takes the least value in the point
(1- a) (aa
3-0868
= (~)
i-a ,
equal to
)a-=-f" . The
a
theorem is proved.
33
In particular, . I t va Iue in eas
2
the function
tI ie
x2 -
ax (a = 2) takes
1
the
to
. point
2
X=
(2"""
a)~
= 2""'
a
equal
( (1-2)"2
a
)2-"1 = -T. a
1
This result is in accordance with
the conclusion, obtained earlier by a different method. The function x3 - 27 x takes the least value in the point x_ ( 27
_ 3"" )"'3"='1 -,
3 equal to (1-3) ( 3 27
)3-"1 = -54.
3
Note. Let us mark for the following, that the function ax
_XIX
= -(xa. - ax),
where a > 1, a> 0, x ~ 0, takes the greatest value in the point
X=
.)a=-t (a
a
1
f
equal to
Fig. 1
(a-i)
(
a
a )a=-t .
a.
Problem 1. It is required to saw out a beam of the greatest durability from a round log (the durability of the beam is directly proportional to the product of the width of the beam by the square of its height). Solution. Suppose AB = x is the width of the beam, BC = y is its height and AC -- d is the diameter of the log (Fig. 1). Denoting the durability of the beam by P, we get P
=
kxy2 = kx (d2
1
-
x2) = k (d2x - x3).
The function d2x - x3 takes the greatest value when
X=
)3=1 (3
d2
=
va ' y2=d2-X2=3d2,
d
2
Y=
34
d Va V- V2=x 2.
Thus, the beam may have the highest (greatest) durability if the ratio of its height to its width will be equal to
is equal to 0 when z = 0 and takes a positive value when 0< z:S;;1. Therefore, the greatest value of the function is gained in the interval 0 < z :s;; 1. . It is shown in Theorem 5 that the function z - Z3, Z ;;:;::: 0, takes the greatest value in the point
( z="'3 1
(_a )a=T" .ax
-(Z
takes the least value in the
1
_ x.Thus. Solu tion. r
=2nr2+-.a) ( For example. Let V = nr2h be the volume of the vessel. if for a given volume the least amount of material is required for its manufacture. the vessel has the least surface area). that is. the function point
xrx
+. 37
.
~(Z
)
rx-1 •
the function sr-+-27x.
takes the least value in the point
x~
C{ f--'~
-3
1
2~'
This value equals
(1 +{)(
y)
_ 1 -1
3
= 4. h is the height of the cylinder. Tho total surface area of the cylinder is Since h = --2 nr
V
S
. r
2V
1
we get
S=2nx-2+2V.r=~n
(X-2+ :
x). Find the optimum dimensions of a cylindrical tin having a bottom and a cover (dimensions of a vessel are considered to be the most profitable.
o:
equal to (1.
=
2nr2
V
+ Zscrh.
11 x 1 -1-
1
x>O.
Problem 5.
then
S=2:rtr2+2nr--2nr
Assuming
X=-. where r is the radius.
4).
.
+~ 31
X-
x.x2 t 5. height and diameter of the vessel are equal.x. Indication. Suppose y = 6 . Thus.
x (6 -
X)2
7. according
1
to the solution
of the
takes the least value when
(~)
-
231
-2-1 -
-
V·
31r2h
231
V'
Returning
back to our previous
_!_ = J
r
3
1
designations.
Exercises
if the
6.!!_
2'
231 r3 = ~ = V' 231
h = 2r = d. Find the greatest value of the function when 0 < x < 6.
r __ 231'
we find . What should the length of the side of the cut-out squares be? 8. From a square sheet whose side is equal to 2a it is required to make a box without a cover by cutting out a square at each vertex and then bending the obtained edges.
+ S.The function
x-2
previous problem. Find the least value of the function
~6.
20 Fig. 4
Ie---
2a-2x
so that the box would be produced with the greatest volume (Fig. the vessel has the most profitable dimensions.
0< A =
Lemma 1. that the indicated method of their solution yields much better results of calculation for greater values of k.000.. n = 1. since A = Xl . = Thus. k already with an accuracy of up to 0. for n = 106 and k = 10.. for ex. 1.I for ex. If 0
are true
(2n+1)1-CG_(n+1)1-a
1
< ex. we could not find the integral part of the number Sn. 2n for n and -ex. The comparison of these two examples shows. we shall improve the method of calculation of the quantity Sn. indicated in Chapter 1.(X2 . l' This improvement will make it possible to find similar quantities with a higher degree of accuracy quite easily.Xa xCG. then A < xl" Thus. In this section. the preceding positive terms are greater than the following negative term.
lation of such quantities. we were able to find the number Sn. k for k = 4. Problem 2) when substituting n 1 for m. In the same section (see Exercises 2 and 3). (_1)n-2 xn) and the quantity in brackets is positive too..4 for k = 1.
1 = 3 and n = 106 because the method of calcu• 1
f. = 2" (refer to Problem 2)..
+
+
Lemma 2.01. Besides this.
then
Xn
< Xl'
Proof. (2n)a +--<
1 (2n)1-CG_n1-a 1-a
. This proves that their algebraic sum is positive.4 (Problem 3) we found the integral part of the number Sn..1 we have succeeded in finding the number S n.In Sec. n = 106 and ex.5. k with an accuracy of up to 0. 1.. A > O. this number was also calculated with an accuracy of up to 0. The number of positive terms in the written algebraic sum is not less than the number of negative terms.4.. The inequality (28) follows from the inequality (14) (see Sec. > +
X2
> Xa >
x4
Xa -
+
+ (_1t-l
> Xn. <
1. 1.000 and ex.000. However. On the other hand.
+
50
.
(28)
Proof. then the following
1 1
inequalities
-a
< (n+1)a + (n+2)a + . In Sec. . the lemma is proved. If
Xl -
X2
=. did not permit doing it.
V2 =0=
-+ -+"'+ +
1 V106+2 1 + V3
111
V-6 10
1 .. The equality (29) is of interest because it brings the calculation of the quantity Sn.2.. Find the sum A=1+V 2 V3 accurate to 0. ' i=--Sn 2(' ' l' by 2<X and dividing by 2 .2a.
1 V106+1
+
'
)_
2·. we know from Lemma 1.002.106 )=
-(V2
52
+ 1) (1-~+~V2
. after multiplying we get the equality (29).. Concerning the second quantity.. 2<X 3<X (2n)~ The first of these quantities for great n is calculated with a high degree of accuracy by means of the inequality (28).+.
-
1) V2.. +
.106
(... ... it then
111
(n+1)<X
+ (n+2)<X . -~..Since in round brackets there 'is the number Sn. (the error) will be 2<X ----:::2_2<X
In the following problems we shall perform the calculation of this quantity with a higher degree of accuracy as well.10 )... that it is less than zero and greater than _ quantity. Problem 1.-.V2
Vi(
1-
Vz
1
-
. But
if we find the sum of the first four summands of the latter less than zero and greater than .. 1 to the computation of the quantity Son n+l and i the quantity 1 __ 1_+_1_..
= ( --1 2<X
2)
=
2-2<X
2<X
. By virtue of Theorem 8 A_ Vi . (2n('
1 + 3a . Solution. T V2. -
1
]
Hence. = (V2 + 1) (B-C).106
6
=
(V2
+ 1)
(.[1 2<X
1
(2n)a.
1 V2..~v
106+1
t. ".
V3
+ Vi2. then the remaining quantity
1 5<x 2-_2<X
S...
109.1012/~_(1012)4 1-~ 4
= ~ .3 < 0.1012
1
1
1
)
is positive and is not greater than the first term.1 differs from A by less than unity. I ~ I < 1. -. then 3.
The first term can be easily found and with a high degree of accuracy by means of the inequalities (28). and from the number A by less than 2. By virtue
10~2+1
of Theorem 8
V2 A = 2-V2 (V
+ 11 10~2_f-2 + '" + t
11 1
2\012
)1)
V2( . Since the term is less than two. is extremely high.. Notice that the accuracy of calculating the number A.2-V2
V2 + V3 .. The middle number .
56
. The relative error is less than 100 : 1333333332. V2.. we get A = 1333333332.. Substituting this number. Calculate A=1
y
the number
1
y'
4 .0000001 %.
3
By virtue of Lemma 1 the sum
V2 2-V2
(1 4
V2 + V3 . . +
1012
with an accuracy of up to unity.109 (y/S-1)
3
2-1/2
V} _ "-"""~-109.109-2 <A
4 < 3. containing a trillion of addends.
The extreme numbers differ from each other by 2... Solution.1012 .109 . ..V2.Problem 2.3 +~. By virtue of these inequalities the first term can he substituted by the number
2-12
t~_
3
3
(2.4 r-
1 + y-2 + V 13 + ..
Assuming y =.
61
a. a>O. we shall bring the problem to finding the least value of the function y3 _ 8y 5 for positive values of y.y_ya.8y
+5
--9-+5=
Y -aya
1 a ~
3203
-3.
is equal to
The least value of the function
y3 .
is
. hence. 8.
(!-1) (
Multiplying
)aI
1
-1-1
a. we shall find the greatest
1
value of the function a (~ (1-0:)""(.ya)
(
which is..-1
a (a) 0:
_1_
a.'
y . 0:>
to.& we get the function
1 =a (1 y_ya 1) a
1
1.
=(!-1)(~)1-a
1
c=
1-:a ( : fa=T.6.. length of the side of the cut-out square is
! that
+
if the of the
side of the given square.Thus.
1
the last quantity
by a.
. Assuming y = X2. 9. equal to
1+_1_
a. the volume of a box will be the greatest.-1
=(1-0:)
a) a
=(1-0:)
a ( (i
)a:T ..
.
By virtue of Theorem 5. the greatest value of the function
. The least value of the function x6 8x2 5 equals 5 and is obtained when x = o.8y is equal to
+
+
3
(1 _ 3) ( ~ ) N
=_
2
8: = 32
32
3
r
6. In Theorem 5. we have proved that the least value of the fli notion y3 .
11.
(8) in the form of
(n+1t<enn. Raising
to a power
Assuming that basis.
a=
!' a
= 2..." . Hence.
. The function greatest value.<) 3/V v is the greatest of the numbers 1. 3/. we get
(1 +an) -2
62
n
n>-2.yn.
an> 0. 2 n2
.yr.
y3
yrs..
to the power of n+Yn+1<
Raising both members of the latter inequality n (n+
1 1)'
we get
13. V 2".2x -<"8'
3
or
4/Vx
-< "8
+ 2x.
..
V-
14.yr..yr. On the other hand.» ( 1 +Tan
) 2 = 1 +nan +Tan.... Write
down the inequality (n~1r<e..
. V 3. equal to
~x . Since then is the greatest of the numbers 1.
for all x>-O the following
4/Vx
inequality
3
is true
.
n n . .. ~3." . Suppose of n we get
. 1 < V2 = ys < y"9 = Y3. 2.
If n>3
> e.
taking
Theorem
3 as the
n >i-Tan. . has the
Therefore.
12. .
then (n 1t
+ < en" < 3nn-<nnn = nn+l. >-1.
x>O.
.. . in the previous problem we have shown that the sequence of the numbers ~4. .. = 1 + an.2x. decreases.
Its remainder (by absolute value) is not greater than the absolute value of the first term of the remainder.1).000.. then 15. Since this number tends series converges and that is.
to zero when
then
the
71
.
28.
6
106
The mean number 14. the number
n-+
00.999 differs from ~
11=1
~k less than by 1. By virtue of Theorem 6
where
.t'I..-=
The number
Bn is a partial sum of the series
This series is sign-alternating with monotonically decreasing (by absolute value) terms.Since the extreme terms of the latter inequalities· from each other very slightly (less than 0.2 < 1
differ
+ V2 +
1
+ V 110 < 15.000 .
< An <
[(2n)i-a_ni-aJ =_n__
i-a
2a <__
2_2a
i-ex. GSP Pervy Rizhsky Pereulok.C is an infinitesimally using the inequalities (28).
Since the difference between the extreme terms of the inequalities tends to zero when n --+ small value.
TO THE READER
Mir Publishers welcome your comments on the content. we get 2~a2a [(2n_1)i-a_(n+
1)i-aJ
small value.
c+ ~n'
+ Y1I
is an infinitesimally
small value. 1+_1 a
2
+
~n = 6n
+-=An-Bn a
n
1
=
-6n-C+Yn =_n
i-a
=_n__ where
i-a
i-ex.that is Yn = Bn . translation and design of this book. Now.
. Our address is: USSR. Moscow 1-110.
then 6n = An _ n i-a
1--ex. We would also be pleased to receive any proposals you care to make about our future publications.
is an infinitesimally Thus. Mir Publishers
2
Printed
'in the
Union
of
Soviet
Socialist
Republics
.
i-ex. 129820.
00. | 677.169 | 1 |
We are selling Math Short Summarized Notes, and giving away Free Exam Papers bundled together! These two complement each other, the Short Condensed notes will help when doing the Exam Papers, and practice makes perfect for Maths.
Many times, it is possible to be stuck on a Math Problem. I have personally been stuck on Math Problems many numerous times, for instance when doing the Math Problem of the Month! When you finally discover the solution by yourself, the feeling of accomplishment is like pure nirvana. There are some books that specifically address How to Solve Math Problems, one of them is How to Solve It: Modern Heuristics. Very highly rated on Amazon, this book describes modern methods to solve heuristics and optimization problems (suitable for students interested in Computer Science). Another well written book by Polya is How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library). This book is suitable for students who like challenging math problems to train their mathematical skills | 677.169 | 1 |
Do you need to brush up on your mathematical skills to truly excel in your economics or business course? If you want to increase your confidence in mathematics, then this is the perfect book for you. With its friendly and informal style, this market leading text breaks down topics into short sections making each new technique you learn seem less daunting. Offering you the chance at every opportunity to stop and check your understanding by working through the practice problems, you can relax and learn at your own pace. A brand new online learning resource for this edition available to users of this book. An unrivalled online study and testing resource that generates a personalized study plan and provides extensive practice questions exactly where you need it. * Interactive questions with randomized values allow you practice the same concept as many times as you need until you master it. * Guided solutions break down the question for you step by step. * Audio animations talk you through key mathematical techniques * Full e book links out to the relevant part of the text while you are practicing. * Additional e chapters of extended topics taken out of the book See the Getting Started with MyMathlab Global section in the introduction for information on how to register and start using the resources. | 677.169 | 1 |
Math Mammoth Geometry 3 can be used after the student has finished Math Mammoth Geometry 1, and is suitable for grades 6-7. This book does not require the students to calculate area or volume, and... More > that is why it is not necessary to study Math Mammoth Geometry 2 (which deals with those topics in depth) before this book.
We start out with basic angle relationships, such as adjacent angles (angles along a line), vertical angles, and corresponding angles (the last only briefly). The lesson Angles in Polygons is a sequel to studying angles in a triangle.
The next set of lessons deals with congruent and similar figures.
The last section of this book teaches some basic compass-and-ruler constructions.
Note: At this time (2014), this book is in "beta" stage. It will be revised with more lessons (such as about The Pythagorean Theorem) probably in early 2015
This is an atlas similar to Uranometria 2000.
Includes 107 charts to 11 magnitude, with 30º charts.
Spiral bound for easy use at the telescope 2 includes charts 301 to 571 and a visual index of all charts. Volume 1 contains the first 300 1 includes the first 300 charts and a visual index of all charts. Volume 2 contains the remaining A includes a selection of the best deep sky objects and it is very handy to plan quick deep sky sessions with 25 charts showing 9th magnitude starsMath Mammoth Integers worktext covers all important integer (signed numbers) topics for middle school (grades 5-8), with instructions written directly to the student.
The first topic in the book is... More > integers themselves, presented on a number line (negative and positive numbers) and tied in with temperature. Next we move on to modeling additions and subtractions as movements on a number line. Another model used in the book for addition is that of counters or chips. One lesson explains the shortcut for subtracting a negative integer using three different viewpoints (difference, counters, and number line movements).
Multiplication and division of integers is explained using counters, first of all, and then relying on the properties of multiplication and division. We use multiplication and division in the context of enlarging or shrinking geometric figures in the coordinate grid | 677.169 | 1 |
Results in Maths English Paperback Textbooks
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NSW Targeting Maths Student Book : Year 1. When we decided to produce a new edition of the already successful Targeting Maths series we asked hundreds of teachers what they wanted to see in a maths program.
NSW Targeting Maths Student Book : Year 6. This book is fully cross-referenced to the Year 3 Targeting Maths App. Children will benefit from the combination of book-based and digital learning that this powerful learning program provides.
NSW Targeting Maths Student Book : Year 3. This book is fully cross-referenced to theYear 3 Targeting Maths App. New features of the program include by Garda Turner. Title: NSW Targeting Maths Student Book : Year 3For those whose mathematical expertise is already established, the book will be a helpful revision and reference guide. The style of the book also makes it suitable for self-study and distance learning.
By Stuart Palmer. - Developed by highly experienced maths educators to reflect the NSW syllabus for the Australian Curriculum;. CambridgeMATHS NSW Syllabus for the Australian Curriculum is a complete teaching and learning program to support the implementation of the NSW Syllabus for the Australian Curriculum.
New WaveMental workbooks will sit comfortably with anymathematics program. Supported by a weekly testing program (levels D-G), New Wave Mental Maths is the complete mental mathematics resource at the right price.
Years 9 - 10 Maths for Students. by For Dummies. Title: Years 9 - 10 Maths for Students. Release your kids from the pressures of national testing and take a step back to the core skills with this new educational series, from the people who brought you theFor Dummies how-to approachThis book is fully cross-referenced to the Year 3 Targeting Maths App. Children will benefit from the combination of book-based and digital learning that this powerful learning program provides. This edition fully aligns each student page to the NSW syllabus and the Australian Curriculum.
This book is fully cross-referenced to the Year 3 Targeting Maths App. Children will benefit from the combination of book-based and digital learning that this powerful learning program provides. NSW Targeting Maths Student Book : Year 5.
Years 6 - 8 Maths for Students. Release your kids from the pressures of national testing and take a step back to the core skills with this new educational series, from the people who brought you theFor Dummies how-to approach.
Alan McSeveny is one of Australia's leading authors for Maths textbooks. He taught Maths for 25 years, and has produced market leading Mathematics series both in print and online. All activities in this student book have been matched to the Australian Curriculum content strands and develop students' conceptual understanding, logical reasoning and problem solving.
Think Mentals is a unique approach to mental maths that systematically teaches students the appropriate strategies to apply in mental computation. The easy-to-follow weekly structure included worked examples of strategies and carefully graded practice activities to help students achieve fluency in maths & improve results.
Excel HSC Maths Revision & Exam Workbook Yr12. This book has been specifically designed to help Year 12 students thoroughly revise all topics in the HSC Mathematics course and prepare for class assessments, trial HSC and HSC exams.
New WaveMental workbooks will sit comfortably with anymathematics program. Supported by a weekly testing program (levels D-G), New Wave Mental Maths is the complete mental mathematics resource at the right price. | 677.169 | 1 |
TEXAS CAREER CHECK
Program Summary
Computational Mathematics
CIP Code: 27.0303
Description: 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. | 677.169 | 1 |
Students: Testimonials
"I am extremely impressed by how Gradarius helps students learn Calculus. Students receive responses to homework submissions in real time. They don't have to wait for graded homework and feedback from instructors - it happens instantaneously. The detailed reports show instructors which topics each student understands and areas where a student seems lost. Students can develop a deeper understanding of Calculus concepts which they can apply to other disciplines."
Sam Lind – Mathematics Ph.D. candidate and teaching assistant, University of California at San Diego
"Gradarius is a powerful system with the potential to serve as a standard in national education. It noticeably increased the effectiveness of homework exercises. The software checks every step of your progress and tells you when you accomplish an essential step or when you make a simple error. Sometimes, it suggests clarifying something in your answer, without penalizing you. The level of interactivity every step of the way made this, at least for me, a very powerful learning tool. Over time, I found myself actually looking forward to doing Gradarius homework." | 677.169 | 1 |
Introduction to Complex Analysis
Introduction to Complex Analysis
Introduction to Complex Analysis
Wesleyan University
About this course: This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today. Each module consists of five video lectures with embedded quizzes, followed by an electronically graded homework assignment. Additionally, modules 1, 3, and 5 also contain a peer assessment.
The homework assignments will require time to think through and practice the concepts discussed in the lectures. In fact, a significant amount of your learning will happen while completing the homework assignments. These assignments are not meant to be completed quickly; rather you'll need paper and pen with you to work through the questions. In total, we expect that the course will take 6-12 hours of work per module, depending on your background.
Who is this class for: This course is aimed at anyone interested in exploring a beautiful and important corner of mathematics. A willingness to refresh (if necessary) one's calculus knowledge is very useful, though enthusiasm and interest in learning are more important than any prior knowledge.
We'll begin this module by briefly learning about the history of complex numbers: When and why were they invented? In particular, we'll look at the rather surprising fact that the original need for complex numbers did not arise from the study of quadratic equa...
5 videos, 5 readings
Graded: Module 1 Homework
Graded: Peer-Graded Assignment #1
WEEK 2
Complex Functions and Iteration
Complex analysis is the study of functions that live in the complex plane, that is, functions that have complex arguments and complex outputs. The main goal of this module is to familiarize ourselves with such functions. Ultimately we'll want to study their sm...
5 videos, 5 readings
Graded: Module 2 Homework
WEEK 3
Analytic Functions
When studying functions we are often interested in their local behavior, more specifically, in how functions change as their argument changes. This leads us to studying complex differentiation – a more powerful concept than that which we learned in calculus. W...
5 videos, 5 readings
Graded: Module 3 Homework
Graded: Peer Graded Assignment #2
WEEK 4
Conformal Mappings
We'll begin this module by studying inverse functions of analytic functions such as the complex logarithm (inverse of the exponential) and complex roots (inverses of power) functions. In order to possess a (local) inverse, an analytic function needs to have a ...
5 videos, 5 readings
Graded: Module 4 Homework
WEEK 5
Complex Integration
Now that we are familiar with complex differentiation and analytic functions we are ready to tackle integration. But we are in the complex plane, so what are the objects we'll integrate over? Curves! We'll begin this module by studying curves ("paths") and nex...
5 videos, 5 readings
Graded: Module 5 Homework
Graded: Peer-Graded Assignment #3
WEEK 6
Power Series
In this module we'll learn about power series representations of analytic functions. We'll begin by studying infinite series of complex numbers and complex functions as well as their convergence properties. Power series are especially easy to understand, well ...
5 videos, 5 readings
Graded: Module 6 Homework
WEEK 7
Laurent Series and the Residue Theorem
Laurent series are a powerful tool to understand analytic functions near their singularities. Whereas power series with non-negative exponents can be used to represent analytic functions in disks, Laurent series (which can have negative exponents) serve a simi...
6 videos, 6 readings
Graded: Module 7 Homework
WEEK 8
Final Exam
Congratulations for having completed the seven weeks of this course! This module contains the final exam for the course. The exam is cumulative and covers the topics discussed in Weeks 1-7. The exam has 20 questions and is designed to be a two-hour exam. You h...9 out of 5 of 204 ratings
With this wonderful complex analysis course under your belt you will be ready for the joys of Digital Signal Processing, solving Partial Differential Equations and Quantum Mechanics. | 677.169 | 1 |
Computer Algebra Recipes: An Advanced Guide to Scientific Modeling
Paperback | January 12, 2007
Modern computer algebra systems are revolutionizing the teaching and learning of mathematically intensive subjects in science and engineering, enabling students to explore increasingly complex and computationally intensive models that provide analytic solutions, animated numerical solutions, and complex two- and three-dimensional graphic displays.This self-contained text benefits from a spiral structure that regularly revisits the general topics of graphics, symbolic computation, and numerical simulation with increasing intricacy at each turn. The text is built around a large number of computer algebra worksheets or "recipes" that have been designed using MAPLE to provide tools for problem solving and to stimulate critical thinking. No prior knowledge of MAPLE is assumed. All relevant commands are introduced on a need-to-know basis and are indexed for easy reference. Each recipe is associated with a scientific model or method and an interesting or amusing story designed to both entertain and enhance concept comprehension and retention. Aimed at third- and fourth-year undergraduates in science and engineering, the text contains numerous examples in disciplines that will challenge students progressing in mathematics, physics, engineering, game theory, and physical chemistry. Computer Algebra Recipes: An Advanced Guide to Mathematical Modeling can serve as an effective computational science text, with a set of problems following each section of recipes to enable readers to apply and confirm their understanding. The book may also be used as a reference, for self-study, or as the basis of an on-line course.
"This book is written in the same spirit as the earlier text [Computer algebra recipes for mathematical physics. In accordance with the title of the book, the three chapters of the book are named as follows: I. The appetizers; II. The entrees; III. The deserts. Each chapter is divided into several dozens of subsections, patterned in the same way: a cover story about a modelling problem, some mathematical background, Maple solution of the problem with graphics, a set of problems. No prior knowledge of Maple is required from the reader: the authors suggest that new users of Maple read the book in natural order while experiences users may skip some parts and go directly to the topic of their interest. The cover stories are chosen from most diverse subjects that well describe the universal role of scientific modelling. A central theme is the use of systems of ordinary differential equations with many variations. For instance the love affair between Romeo and Juliet is analyzed in this way." (Matti Vuorinen, Zentralblatt MATH) | 677.169 | 1 |
Worksheet Fourteen: Matrices
In this matrices worksheet, students construct matrices under given conditions. They solve systems of equations, and explore procedures of producing a matrix. This two-page worksheet contains 8 multi-step problems. | 677.169 | 1 |
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how to rewrite a quotient as a ratio | 677.169 | 1 |
Special Features
Boards
Boards
We cover 31 boards ( including CBSE, ICSE and 29 state boards ) and NCERT syllabus practiced across the country.
Classes
Classes
Well analyzed study material for 9th & 10th. Also we are going to launch the study material for +1 & +2 shortly.
Subjects
Subjects
High quality material in a wide range of interdisciplinary areas of Mathematics for 9th, 10th, +1 & +2. Science for 9th, 10th, +1 & +2 is also coming up.
Chapters with Subtopics
Chapters with Subtopics
Each chapter has been divided into topics and sub-divided.
Discussion Room
Discussion Room
Students can indulge in combined study with the white board facility and available tools like drawing figures, mathematical tools for various operations, highlight or type text in different colors and fonts.
Videos
Videos
Each chapter has been explained with the help of 15 – 20 videos.The videos have questions along with complete answers supported by a voice over.
Animations
Animations
Every chapter in each board has been explained with the help of 15-20 animated videos with moving characters and voice over.
Quick Review
Quick Review
Quick Review is a summary of all the important formulae and concepts.
Homework Help
Homework Help
Homework Help is a section where all the questions given in the chapter of the book have been solved.
Let's Revise
Let's Revise
Let's Revise section consists around 10,000 questions broadly divided into three sub sections: › MCQ
› True or False
› Fill Ups Get immediate results for each type of question along with a detailed explanation for each question | 677.169 | 1 |
Product Description:
Bob Blitzer's books are highly acclaimed for their well-conceived, relevant applications and meticulously annotated examples. This highly anticipated revision achieves the difficult balance between coverage and motivation, while helping readers develop strong problem-solving skills. This book provides readers with the skill building and practice that is so crucial as well as the applications and technology necessary to foster an appreciation of the myriad uses of mathematics. This expanded edition covers voting and apportionment and graphing theory, in addition to a wide range of topics that include set theory, logic, number theory, algebra, consumer mathematics and financial management, geometry, measurement, probability theory and statistics. For anyone interested in refreshing his/her fundamental math skills.
REVIEWS for Thinking Mathematically | 677.169 | 1 |
Mathematics: Curriculum
In years 7 and 8 students are introduced to a broad range of topics, which form the foundations for work that they will be extended in later years. Each term will include work on number, algebra, data handling and shape, space & measure. Students also develop their skills in using and applying mathematics by completing investigations and maths challenge questions.
The GCSE mathematics course builds on the areas of mathematics which students have already studied in Years 7 and 8; number, algebra, data handling and shape, space and measure. Students also continue to develop their skills in using and applying mathematics by completing investigations and maths challenge questions. All students are exposed to work that extends and enriches the curriculum beyond the normal GCSE.
All students are prepared for Edexcel's new (9-1) GCSE. The content of this is similar to the 'old' GCSE with a few additions; it also uses a new grading scale of 9 to 1, with 9 being the top grade. This is assessed by three terminal examinations, one non-calculator, and two calculator.
We study the OCR course for A-level Mathematics and A-level Further Mathematics.
AS Mathematics
Core Mathematics 1 extends students' learning in familiar GCSE topics such as quadratics, indices and straight line graphs. Students will also be introduced to new topics, the key new material being differentiation. This is a non-calculator module.
Core Mathematics 2 builds on students' GCSE trigonometry and algebra knowledge as well as introducing the new topics: sequences, series, integration and logarithms.
Probability and Statistics 1 Students will already be familiar, from GCSE, with ways of displaying and interpreting data such as scatter diagrams and box plots. In this module students will build on this knowledge with a more formal approach to statistics and probability, including work on discrete random variables and bivariate data.
A2 Mathematics
Core Mathematics 3 extends the core work completed in the AS with further work on algebra, functions, trigonometry, the exponential function, differentiation, integration and numerical methods.
Core Mathematics 4 extends the core work completed in Core Mathematics 3, notably introducing differential equations. A new topic, vectors, is also included in this module.
Mechanics 1 is about using mathematical models of physical situations to investigate the forces causing an object to move or remain stationary as well as studying the motion of objects that are thrown or dropped. To achieve this, students study force as a vector, kinematics, equilibrium of a particle, Newton's laws of motion and linear momentum.
AS Further Mathematics
Further Pure Mathematics 1 introduces some key topics in pure mathematics, with a more rigorous proof based approach than the Core modules. In particular, students study matrices, complex numbers, series and proof by induction.
Decision Mathematics is the use of discrete maths to solve practical problems such as how to deliver all the mail to every street in a village in the shortest time. The aim is to derive algorithms (sets of instructions) that find efficient solutions. The problems investigated include ordering numbers, various problems involving networks and linear programming.
Probability and Statistics 2 covers some essential topics for application in many other subjects such as biology and economics. This includes continuous random variables, the normal distribution, the Poisson distribution, sampling and hypothesis testing.
A2 Further Mathematics A further three modules are covered from the following:
Mechanics 2 covers centre of mass, equilibrium of a rigid body, motion of a projectile, uniform motion in a circle, coefficient of restitution and impulse, energy, work and power.
Mechanics 3 covers equilibrium of rigid bodies in contact, elastic strings and springs, impulse and momentum in two dimensions, motion in a vertical circle, linear motion under a variable force and simple harmonic motion.
Probability and Statistics 3 covers linear combinations of random variables, confidence intervals and the t-distribution, difference of population means and proportions and the Chi squared tests. | 677.169 | 1 |
Career Path Chart 2.02
In order to do this assignment make sure you have read the entire lesson and completed the, Finding the
Path for you, quiz (page 3 of 4). The results will let you know your greatest likelihood of success according
to the six career
MATH Calculus Advice
Showing 1 to 2 of 2
This class requires initiative from the student. In this way, this class resembles more of a college environment than any other at Novi High School.
Course highlights:
The first semester is composed of completing eight worksheets that is a review of every unit in AP Calculus AB. The level of independence Mr. O'Leary gives you is definitely one of the highlights of this course. The entire class is dedicated to reviewing the topics in Calc AB.
Hours per week:
9-11 hours
Advice for students:
I'd advise to complete all the homework assigned as well as the extra homework (Mr. O'Leary hands out extra for students that ask for it). | 677.169 | 1 |
College math homework
Online Homework Help – Become The Best Student in Your Class. Need to get help with homework? Tired of being face to face with a pile of papers to write? Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.
Online Homework Help – Become The Best Student in Your Class. Need to get help with homework? Tired of being face to face with a pile of papers to write? Welcome! InterAct Math is designed to help you succeed in your math course! The tutorial exercises accompany the end-of-section exercises in your Pearson textbooks. Beginning July 31, 2016, Pearson will discontinue our open-access online homework and practice website, We encourage you to try one of.
College math homework
The Mathematics department prepares students with strong skills in mathematical communication, problem-solving, and mathematical reasoning. This solid foundation. About the Professor. Study Skills Tips. Note Taking Tips. Identify Your Learning Style. Math Anxiety Test. Math Teacher's Ten Commandments. Student's Math Anxiety WTAMU Math Tutorials and Help. If you need help in college algebra, you have come to the right place. Note that you do not have to be a student at WTAMU to use any. The Mathematics department prepares students with strong skills in mathematical communication, problem-solving, and mathematical reasoning. This solid foundation.
Middle School Math. The Middle School Math Success Series is designed to review material previously learned in class and to provide additional practice. Recent News. Martonosi Elected 2016 INFORMS VP of Membership & Professional Recognition; Benjamin Quoted in NYT Article about Tim & Tanya Chartier Get Instant Expert Homework Help and Pay Later. We are the #1 freelance homework help site with hundreds of verified scholars online to help you with your homework … Recent News. Martonosi Elected 2016 INFORMS VP of Membership & Professional Recognition; Benjamin Quoted in NYT Article about Tim & Tanya Chartier
Get math homework help, studying and test prep 24/7. Our expert math tutors provide tutoring for every subject and skill level. Find a math tutor now. View a Free Sample Math Video. Hotmath offers school and college math video review lessons. Click on a course below to see the chapter index, and then click on the. Get Instant Expert Homework Help and Pay Later. We are the #1 freelance homework help site with hundreds of verified scholars online to help you with your homework … Homework Hotline What is Homework Hotline? The Harvey Mudd College Homework Hotline is an over-the-phone, mathematics and science tutoring service …
Math homework help. Hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus. Online tutoring … The best multimedia instruction on the web to help you with your homework and study. | 677.169 | 1 |
This online course is perfect for high school students and college students looking to have a solid foundation in Pre-Algebra, which is indispensable for setting up a proper mathematical background to succeed in more advanced math and science courses.
Homework Helpers: Basic Math and Pre-Algebra is a straightforward and easy-to-read review of arithmetic skills. It includes topics that are intended to help prepare students to successfully learn algebra, including: • Working with fractions • Understanding the decimal system • Calculating percentages • Solving linear equalities • Graphing functions • Understanding word problems
If so, you are like hundreds of thousands of other students who face math-especially, algebra-with fear. Luckily, there is a cure: Bob Miller's Clueless series! Like the teacher you always wished you had (but never thought existed), Bob Miller brings knowledge, empathy, and fun to math and pre-algebra.
When you have the right math teacher, learning math can be painless and even fun! Let Basic Math and Pre–Algebra Workbook For Dummies teach you how to overcome your fear of math and approach the subject correctly and directly. | 677.169 | 1 |
Intermediate 1 Maths offers an attractive and motivating
textbook designed to enthuse students. Attractively illustrated with colour
coded explanations and worked examples, the book brings Maths into real-life
contexts with activities that make the subject relevant to Intermediate
1 students, whilst offering comprehensive coverage of the syllabus in a
focused and concise manner.
There is an extensive range of relevant exam-style
questions and assessment items to prepare for examination. A CD is included
with the book, containing answers to all of the student exercises. The
disc can be retained for use solely by the teacher, or released for student
self-access according to preferred teaching and assessment strategies.
The book covers all the outcomes in the four units of the Intermediate
1 Mathematics course, i.e. units 1, 2, 3 and the Applications of Mathematics
unit. Each chapter concentrates on the required learning outcomes with
questions, exercises, and end of topic summaries. Questions are designed
to help students prepare and practise for unit tests, and the final exam
features prominently throughout the book, whilst students are directed
to the relevant chapters of the course depending on which units they are
studying.
Attention is given to both calculator and non-calculator examples
in line with the two papers which make up the final exam, and each chapter
contains student exercises (with answers supplied). A summary appears at
the end of each chapter telling students what they should be able to do.
* A full colour, attractively designed textbook designed to motivate Intermediate
1 students
* A concise but comprehensive book for the fastest-growing syllabus
area in Scottish education
* The latest in Hodder Gibson's immensely successful
series of Intermediate 1 textbooks
* Includes a CD containing answers to
all of the student exercises | 677.169 | 1 |
This book is a guide to the 5 Platonic solids (regular tetrahedron, cube, regular octahedron, regular dodecahedron, and regular icosahedron). These solids are important in mathematics and nature, and the only types of convex regular polyhedra that exist.
Topics covered include:
What the Platonic solids are
The history of the discovery of Platonic solids
The common features of all Platonic solids
The geometrical details of each Platonic solid
Examples of where each type of Platonic solid occurs in nature
How we know there are only five types of Platonic solid (geometric proof)
A topological proof that there are only five types of Platonic solid
What are dual polyhedrons
What is the dual polyhedron for each of the Platonic solids
The relationships between each Platonic solid and its dual polyhedron
How to calculate angles in Platonic solids using trigonometric formulae
Note: Some familiarity with basic trigonometry and very basic algebra (high school level) will allow you to get the most out of this
book - but in order to make this book accessible to as many people as possible,
I have included a brief recap on some necessary basic concepts
from trigonometry (such as sine, cosine, radians, etc.).
This book is now available as both a printed book, and as an electronic book on Amazon Kindle!
This book has been specially prepared for use on the Kindle - including taking great care to ensure all math symbols, formulas, and calculations display correctly on the Kindle screen (many other math books do not display math symbols correctly). A color screen is recommended, but not essential - there are a couple of diagrams where the text refers to colors in the picture - but these should still be understandable even on a black and white screen.
3D "cover" and "box" images are for decorative purposes only.
They do not represent actual products which are generally delivered as
an electronic downloads. Most box/cover images on this website were created using
Cover Factory. | 677.169 | 1 |
Guided notes - Real Numbers and their Properties
608 Downloads
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
7.1 MB | 5 pages
PRODUCT DESCRIPTION
This guided presentation covers the classification of real numbers, their properties, and order of operations. Students follow along during the lecture and fill in the definitions and examples. Comprehension is checked via pair work (Buddy Up! sections) and individual exercises. A complete answer guide is provided.
I am in the process of creating more of these booklets for my Algebra I class, so if you have any questions or comments about the content or layout, please | 677.169 | 1 |
Physics is frequently one of the hardest subjects for students to tackle because it is a combination of two of the toughest subjects for most students: Math and Word Problems. If you understand the math but don't do well in word problems then you will have trouble. And if you understand the word problem but have no idea where to begin with the math, again, you will not do well.
The Early Math Concepts series provide chapters of enrichments listed by increasing difficulty. The book is targeted at ages 8-18, organized so each age can find interest through the visual, intriguing and interactive elements. | 677.169 | 1 |
MATHS UNIT 3 HIGHER MOCK REVISION LIST Non Calculator Write one number as a % of another Compound interest/depreciation Fraction of an amount Evaluate indices Substitution in to formula and use a formula Solve simultaneous linear equations Solve quadratic (x2) equations Draw and use a quadratic (x2) graph Finding the equation of a straight line (y = mx + c) Transformations (reflection, rotation, enlargement and translation) 3-D isometric drawings Distance-Time graphs Volume of prisms and converting units Dimensions of formulae Angles in polygons Circle Thgeorems Calculator Ratio including exchange rates Write one number as a % of another Standard Index Form Laws of indices Solve a cubic (x3) equation by trial and improvement Solve linear equations Change the subject of a formula Solving inequalities Transformation of graphs Plans and elevations Circumference and area of a circle Pythagoras' Theorem Locus of a point and standard constructions Trigonometry Limits of accuracy (max/min of measurements) Area of a segment | 677.169 | 1 |
Trigonometry's complete pacing guide details either a semester or full-year pacing plan in this comprehensive course for the out-of-field instructor or new teacher. The course is based on a balanced approach that includes concept instruction and step-by-step examples. The author has segmented lessons into manageable "bites" for comprehension and skill-building. Each day's lesson plan provides detailed instructions to the teacher including classroom-tested teaching strategies that support various student learning styles. Daily class notes can be displayed as overheads or as PowerPoint presentations. Assessments measure student progress frequently, building student confidence and ensuring concept mastery. Six labs, Web-based resources and many variations of homework support a variety of instructional strategies to reach all students. Point-of-use technology includes graphic calculators which are required for all students; an overhead calculator display is recommended but not essential | 677.169 | 1 |
Product Overview
KEY MESSAGE: Understanding algebra is as easy as 1,2,3 with Richelle Blair' s Intermediate Algebra. Blair' s engaging intermediate algebra text helps readers continue to develop their algebraic thinking skills while also building their confidence in their ability to learn mathematics. Blair' s textbook incorporates best practices from academic research and embraces the soon-to-be-released Beyond Crossroads standards of the American Mathematical Association of Two Year Colleges, as well as the standards of the National Council of Teachers of Mathematics and the Mathematical Association of America. Real Numbers, Review of Introductory Algebra, and the Cartesian Plane, Linear Equations, Inequalities, and Absolute Value, Relations and Functions, Systems of Equations, Polynomials and Polynomial Functions, Rational Expressions, Equations, and Functions, Radical Expressions, Equations, and Functions, Quadratic Equations, Inequalities, and Functions, Exponential and Logarithmic Expressions, Functions, and Equations, Conic Sections. For all readers interested in algebra and continuing to develop their algebraic think skills while also building their confidence in their ability to learn mathematics. | 677.169 | 1 |
Resource Added!
Type:
Full Course
Description:
MA.8.2.8.5.A: Mathematics
MA.8.2.8.5.B: Mathematics
find and evaluate an algebraic expression to determine any term in an arithmetic sequence (with a constant rate of change).
MA.8.3.8.6.A: Mathematics
generate similar figures using dilations including enlargements and reductions;
MA.8.3.8.6.B: Mathematics
graph dilations, reflections, and translations on a coordinate plane.
MA.8.3.8.7.A: Mathematics
draw three-dimensional figures from different perspectives;
MA.8.3.8.7.B: Mathematics
use geometric concepts and properties to solve problems in fields such as art and architecture;
MA.8.3.8.7.C: Mathematics
use pictures or models to demonstrate the Pythagorean Theorem;
MA.8.3.8.7.D: Mathematics
locate and name points on a coordinate plane using ordered pairs of rational numbers.
MA.8.5.8.11.A: Mathematics
find the probabilities of dependent and independent events;
MA.8.5.8.11.B: Mathematics
use theoretical probabilities and experimental results to make predictions and decisions.
MA.8.5.8.12.A: Mathematics
select the appropriate measure of central tendency or range to describe a set of data and justify the choice for a particular situation;
MA.8.5.8.12.B: Mathematics
draw conclusions and make predictions by analyzing trends in scatterplots;
MA.8.5.8.12.C: Mathematics
select and use an appropriate representation for presenting and displaying relationships among collected data, including line plots, line graphs, [stem and leaf plots,] circle graphs, bar graphs, box and whisker plots, histograms, and Venn diagrams, [with and] without the use of technology.
MA.8.5.8.13.A: Mathematics
evaluate methods of sampling to determine validity of an inference made from a set of data;
MA.8.5.8.13.B: Mathematics
recognize misuses of graphical or numerical information and evaluate predictions and conclusions based on data analysis.
MA.8.6.8.14.A: Mathematics
identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics;
MA.8.6.8.14.B: Mathematics
use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;
MA.8.6.8.14.C: Mathematics
select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem.
MA.8.6.8.16.B: Mathematics
This resource was reviewed using the Curriki Review rubric and received an overall Curriki Review System rating of 3, as of 2011-10-25.
Component Ratings:
Technical Completeness: 3 Content Accuracy: 3 Appropriate Pedagogy: 3
Reviewer Comments:
This is a sixth grade course on geology which includes science and mathematical content. The nature of the enterprise is that most of the material is inside the textbook, so that the lessons only comment obliquely on the actual material. It seems a sound enterprise to incorporate math and science for such young students, and the material seems fascinating.
Not Rated Yet. | 677.169 | 1 |
Product Description:
Offers a sound mathematical development ... and at the same time enables the student to move rapidly into the heart of geometry. --The Mathematics Teacher Should be required reading for every teacher of geometry. --The Mathematical Gazette In this highly recommended high school text by two eminent scholars, the authors deduce plane Euclidean geometry by utilizing only five fundamental postulates. Incorporation of the system of real numbers in three of the five postulates of this geometry gives these assumptions great breadth and power. They lead the reader at once to the heart of geometry. It is because of the underlying power, simplicity, and compactness of this geometry that the authors called the book Basic Geometry. The book is designed for a one-year course in plane geometry. For advanced students, the authors incorporated certain material from three-dimensional and so-called modern geometry. A rich variety of exercises as well as many illustrations applying the abstract geometrical concepts to real life provide an excellent source of teaching material.
REVIEWS for Basic Ge | 677.169 | 1 |
Mathematical Experience first published in the early 1980s, The Mathematical Experience presented a highly insightful overview of mathematics that effectively conveyed its power and beauty to a large audience of mathematicians and non-mathematicians alike. FollowingMore...
Book details
When first published in the early 1980s, The Mathematical Experience presented a highly insightful overview of mathematics that effectively conveyed its power and beauty to a large audience of mathematicians and non-mathematicians alike. Following about a decade later, the study edition of the work supplemented the original material of the book with exercises to provide a self-contained treatment usable for the classroom. This softcover version reproduces the study edition and includes epilogues by the three original authors to reflect on the book's content 15 years after its publication, and to demonstrate its continued applicability to the classroom. Moreover, The Companion Guide to the Mathematical Experience-originally published and sold separately-is freely available online to instructors who use the work, further enhancing its pedagogical value and making it an exceptionally useful and accessible resource for a wide range of lower-level courses in mathematics and mathematics education. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto (elena.marchisotto@csun.edu) upon request | 677.169 | 1 |
Algebraic Expressions - Unit Review/Pre-Test and Unit Test
PDF (Acrobat) Document File
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0.59 MB | 14 pages
PRODUCT DESCRIPTION
Algebraic expressions – a key stepping stone toward a complete understanding of algebra. This product is the fourth of four products that explore the wonderful world of algebraic expressions. The first product, Express Yourself - Part 1: Evaluating Expressions, focused on what algebraic expressions are, why we use them, how they are different from numerical expressions and algebraic equations, and how to evaluate expressions of varying levels of complexity when given values for the variables. The second product, Express Yourself - Part 2: Writing Expressions, took the students one step further by switching the focus to how to write and use algebraic expressions from word phrases and then for word problems. The third product, Express Yourself - Part 3: Simplifying Expressions, focused on simplifying expressions, particularly through use of the Distributive Property and Combining Like Terms. This fourth product, Express Yourself – Part 4: Review and Unit Test is a comprehensive review and unit assessment for the concepts learned and taught in the first three. The four products have now also been bundled into Express Yourself - The Complete Unit, which includes all four products at a discounted price.
This product, Express Yourself - Part 4: Review and Unit Test, contains the following components:
1. Algebraic Expressions Unit Review: This three-page review sheet reviews the concepts learned in the other three products. Concepts covered in the review sheet include:
- Understanding expression terminology – are students able to identify the coefficients, constants, variables, operations, terms, and like terms in an expression?
- Evaluating expressions using substitution – can students substitute values for different variables into expressions and then evaluate the expressions to obtain a result?
Distinguishing between various number properties – are students able to distinguish between the Associative, Commutative, Identity, and Distributive Properties, which are all critical in working with expressions?
- Simplifying expressions by applying the Distributive Property and combining like terms – can students correctly use the Distributive Property in a variety of expressions, and then also combine like terms to simplify expressions that require multiple steps to simplify?
- Equivalent expressions – can students simplify multiple expressions and then properly identify which are equivalent?
- Writing expressions from word phrases and for word problems: are students able to identify the correct terms and put them in the correct order with the correct operations to write expressions when they are given word phrases and word problems? This will be a critical skill as they delve deeper into Algebra.
2. Algebraic Expressions Unit Test: Similar to the Unit Review. Topics covered are identical. Problems are of the same level of difficulty as the Unit Review.
Implementation of both the review and the test are self-explanatory. Detailed keys are provided for both. Note that the review can alternatively be used a pre-test!
Good luck in your use of Express Yourself – Part 4: Review and Unit Test!
If your students are not familiar with how to execute operations in the correct order, you may want to go through part or all of Order of Operations Inundation to address numerical expressions and PEMDAS prior to tackling algebraic expressions.
If you would like your students to move from algebraic expressions to algebraic equations, take a look at Equations With Ned, a brand new unit that introduces students to one-step equations in an unique and exciting way, and Equation Terminology Puzzles, which introduces students to the language associated with algebraic equations.
Feel free to also on a purchased product | 677.169 | 1 |
Compressed Zip File
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8.02 MB | 44 pages
PRODUCT DESCRIPTION
This lesson is designed to reinforce prime factorization with the goal of memorizing the factorization of numbers under 50. After tracking data for over twenty years on algebra skills, I have learned this is one of the most important fundamentals for students in Pre-Algebra, Algebra and even Advanced Algebra. Prime Factorization is an automatic response that must occur for Exponent Operations, Polynomial Factoring and Division, and Logarithms. The data strongly suggests students who struggle with the second semester of Advanced Algebra often have struggles with Prime Factorization or the seeing of numbers as prime factors. Here, the lesson takes prime factorization further by teaching students how to find the LCM and GCF of two or more numbers | 677.169 | 1 |
Pre-Calculus Demystified leads the reader through all the intricacies and requirements of this essential course
Whether you need to pass a class, a college requirement, or get a leg up on more advanced topics, this book provides clear explanation with a wealth of questions, answers and practical examples.
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. | 677.169 | 1 |
A COMPREHENSIVE OVERVIEW OF BASIC MATHEMATICAL CONCEPTS
-
SET THEORY
PROPERTIES OF REAL
NUMBERS
NOTATION
cfw_ Brul"e. indicate the beginning and end ofa set notation; when
listed e lements or members mllst be separated by commas;
EXAMPLE: In A= cfw_4 ,
PROPERTIES OF INEQUALITY
For any real numbers a , b , and c:
For any real numbers a, b, and c:
A,Closure
a + b is a real number
I. For addition:
2. For 1l1ultiplication: a b is a real number
B, Commutative Property
I. For addition :
a + b = b +a
2. For
A. Polygons are plane shapes that are formed by line
segments that intersect only at the endpoints. These
intersecting line segments create one and only one
enclosed interior region.
Geometry means Earth measurement; early peoples
used their knowledge of
MATH 241 - Spring 2014
Instructor: Patrick Collins
Homework #10 Solutions
6.1 #12
Find the volume of the solid of revolution generated by revolving the region between the x-axis and the line x
in between y 0 and y 2 about the y-axis.
3y 2
Solution
The dis
Calculus II Advice
Showing 1 to 3 of 5
I wouldn't recommend this course if you want a teacher who gives good notes. This professors notes are a lot like writing paragraphs, he literally almost writes everything he says word for word making it hard to understand his notes and to write your notes.
Course highlights:
When we go through the easy parts it is really easy. I learned that his test seem short but they're actually long because one question has so many parts to it.
Hours per week:
3-5 hours
Advice for students:
If it's required take it but if not then I wouldn't recommend to take it. Don't waste your time taking classes you don't need and first focus on the classes that are required for you.
I would recommend this course because it covers most of the things you would learn in high school calculus. It serves as a recap and if you didn't fully grasp certain aspects of calculus in high school, you will certainly earn a full and better understanding of calculus when you take this course.
Course highlights:
The highlights were the recitations that were on Fridays because it was a recap of what you learned throughout the week and during this time, we have quizzes and worksheets to see where we are at. The worksheets help to give us an idea of what we were dealing with and while they help when I did my homework, they also gave me a better understanding of what I was learning at the moment.
Hours per week:
3-5 hours
Advice for students:
My advice would be to make sure you have a background knowledge of pre-calculus and trigonometry. You should be familiar with the basics of calculus and if you took calculus in high school, it doesn't mean that this would be an easy class to pass. Certain things are solved differently so you would be careful not to take shortcuts because if you just wrote down the answer without showing your work, then you would lose points. Make sure you understand what you are learning and if you are confused about something, go to your professors office hours and seek help, otherwise you would do poorly in midterms and finals.
Course Term:Fall 2016
Professor:Thomas Hangelbroek
Course Required?Yes
Course Tags:Math-heavyBackground Knowledge ExpectedGo to Office Hours
Dec 07, 2016
| Would recommend.
This class was tough.
Course Overview:
I would recommend this class, because the professor challenges the students to step out of their comfort zone when learning Calculus. It's not your normal math class where you learn the subject and do many examples. This class challenges the student to fully understand a formula, variable, every component of a problem and to be able to prove it and understand everything so that it can be applied multiple ways with different problems. However, it is a challenging course since the professor does not curve anything and problem sets are more challenging than most of the examples gone over in class. The class is fast paced and the professors writing is semi-difficult to read from afar.
Course highlights:
Highlights were learning about integrals and how the volume and area is taken from abnormal shapes not usually dealt with in lower math classes. Also, the introduction to rotating graphs was something new and cool to learn.
Hours per week:
9-11 hours
Advice for students:
Tutoring every other day helps a lot. Suggested problems used to be a thing of the past you could ignore and get away with, but not with this course. Consistency is key, because practicing problems and really trying to understand every component of a problem is what will help you pass. In addition, timing yourself on problems helps time efficiency when you go to take the exams. | 677.169 | 1 |
Raising Calculus to the Surface
"You go to the gas pump, it's dollars per gallon; rates are often constant but in more complicated contexts, they're always changing. Also, in the real world, almost everything—from air masses to financial trends—changes and has some kind of irregularity; some sort of curvy trajectory. So how do we analyze all that?"
The answer, he says, "is to reduce the situation to a straight line, and how you do that, is what calculus is."
Starting with a visual: mathlets, models and more
Facilitating that process of finding the straight line in an irregular context, Samuels has created for students a series of free mathlets, downloadable computer applications that perform a small set of tasks.
He opens one of these, on his phone.
"On the top, you see the graph of some function," he says, "and then there's this black box around a section of it, and that part is presented on the bottom."
As the box on the top half of the screen zooms in on the graph, the bottom half of the screen responds accordingly, applying whichever function the user chooses, "and you can identify the slope, or the rate of change at that point," Samuels explains.
"I like to introduce calculus visually," he says, "as opposed to introducing it algebraically, or with formulas, which is the more traditional method."
He redesigned his calculus courses around this strategy, and students not only use the mathlets as a resource, they also start with 3D models and other tactile, physical representations.
"I've assessed the outcomes, to see if students were actually learning calculus, doing it this way," he says, "and those outcomes make it clear that students who first explore calculus visually have a much deeper, more connected understanding."
An NSF grant to develop instruction
Recently the National Science Foundation's Division of Undergraduate Education awarded Jason Samuels of BMCC; Brian Fisher of Pepperdine University; Eric Weber of Iowa State University, and Aaron Wangberg of Winona State University over $225,000 through the Transforming Undergraduate Education in Science, Technology, Engineering, and Mathematics (TUES) program.
Their goal is to investigate innovative new methods for teaching and learning multivariable calculus, or Calculus 3.
The project, "Raising Calculus to the Surface," is sponsored by Winona State University in Winona, Minnesota, and will teach students by starting with a visual exploration—they'll draw, measure and grasp concepts geometrically, using three-dimensional, clear plastic models on which they can write with a dry erase marker.
"It's like thinking about topography," says Samuels.
"In fact, one of the earliest uses of Calculus 3 was to create topographical maps that could show you things like, all the mountains 5,000 feet high are at this line; all the mountains 10,000 feet high are at that line."
Students using surfaces of the 3D models, he says, "will explore the questions, 'What is a level curve?' and 'What is a contour map?' They'll get the concepts from the actual physical model itself."
Less struggle, more meaning
Samuels loves playing the popular logic puzzle Ken Ken. He competes annually in the Google U.S. Puzzle Championship, is a "huge, huge Yankee fan," and leads the BMCC Math Team. He also encourages students to take part in CUNY-wide Math Challenge and contests sponsored by the American Mathematical Association of Two-Year Colleges.
He's always enjoyed math himself, he says, and when students start with a graphic or physical representation and then move to formulas, they can enjoy math, too, he says, and internalize deeper meaning.
Still, visual-first calculus instruction has its skeptics.
"A lot of teachers think, 'Oh, if I try to do all this experimental stuff, I won't have time to actually do the math," says Samuels, "and nothing could be further from the truth."
He explains that in the classic instructional presentation—definition, theorem, proof, example—often students struggle; "they try to memorize and copy what you're doing, and they devote so much mental energy and time trying to understand what it means, it actually takes away from the learning process."
On the other hand, he says, "If you have them explore the idea first—through discussion and tactile or visual activities—then cap it off by applying a formula and maybe even a theorem, you don't have to explain what it means, because students already understand the full concept that goes with the formula. It actually takes less time, and you can cover more meaningful mathematics."
Research using classroom outcomes
In summer 2014, Professor Samuels and the NSF-funded project team will host a series of workshops on how to teach Calculus using the 3D models and their surfaces.
The participants, dozens of faculty now being recruited from high schools and two- and four-year colleges around the country, will then take that methodology back to their own calculus classes.
Eventually, Samuels and his team will compare the outcomes of those classes, with the outcomes of control classes using more traditional methods. | 677.169 | 1 |
Product Description:
Suitable for advanced undergraduates and graduate students, this text introduces the broad scope of convexity. It leads students to open questions and unsolved problems, and it highlights diverse applications. Author Steven R. Lay, Professor of Mathematics at Lee University in Tennessee, reinforces his teachings with numerous examples, plus exercises with hints and answers. The first three chapters form the foundation for all that follows, starting with a review of the fundamentals of linear algebra and topology. They also survey the development and applications of relationships between hyperplanes and convex sets. Subsequent chapters are relatively self-contained, each focusing on a particular aspect or application of convex sets. Topics include characterizations of convex sets, polytopes, duality, optimization, and convex functions. Hints, solutions, and references for the exercises appear at the back of the book. Book jacket.
REVIEWS for Convex Sets | 677.169 | 1 |
Description: The object of this book is to provide an easy introduction to the Calculus for those students who have to use it in their practical work, to make them familiar with its ideas and methods within a limited range. A good working knowledge of elementary Algebra and Trigonometry is assumed. | 677.169 | 1 |
Dynamic Explorations
Chapter 0
Slope Use a dynamic sketch to review how to calculate the slopes of lines on the coordinate plane. This exploration complements the review of slope in Lesson 0.1 of Discovering Advanced Algebra.
Direct Variation Use a dynamic graph to help you use direct variation to model relationships between numbers. This topic is reviewed in Lesson 0.3 of Discovering Advanced Algebra.
Chapter 1
Sequence Graphs Use dynamic sketches to explore the graphs of arithmetic and geometric sequences. This exploration can help you deepen your understanding of concepts from Lesson 1.4 of Discovering Advanced Algebra.
Chapter 2
Box Plots Drag data plots and observe a box plot changing dynamically in this exploration of the properties of box plots. This exploration reinforces the concepts in Lesson 2.1 of Discovering Advanced Algebra.
Standard Deviation In this exploration, you will use a dynamic sketch to investigate standard deviation and better understand how it measures the spread of a data set. This exploration can deepen your understanding of concepts in Lesson 2.2 of Discovering Advanced Algebra.
Histogram Puzzles Find the data set that produces a histogram by experimenting with different bin width and starting values. This exploration will give you more practice interpreting histograms, which are covered in Lesson 2.3 of Discovering Advanced Algebra.
Chapter 3
Fitting a Line to Data Use a movable line to improve your skill at finding a line of fit for a data set. This exploration complements the material in Lesson 3.3 of Discovering Advanced Algebra.
Residuals Use dynamic sketches to explore residuals and root mean square error and to evaluate how well a model fits a set of data. This exploration will deepen your understanding of the concepts in Lesson 3.5 of Discovering Advanced Algebra.
Chapter 4
Translations of Parabolas Use a dynamic graph to explore the relationship between the equation of a translated parabola and the coordinates of its vertex. This exploration will give you more practice with the material in Lesson 4.4 of Discovering Advanced Algebra.
Rotation as a Composition of Transformations Explore rotation as a composition of other transformations and write rules for rotating figures on the coordinate plane. These dynamic sketches can help you complete the Exploration Rotation as a Composition of Transformations in Chapter 4 of Discovering Advanced Algebra.
Dilations of Shapes and Functions Use dynamic graphs to investigate how graphs can be dilated horizontally and vertically by changing their equations. This exploration will reinforce the concepts in Lesson 4.6 of Discovering Advanced Algebra.
Chapter 5
An Application of Exponential Decay Model the motion of a pendulum using an exponential function. This exploration can be used to deepen your understanding of Example B in Lesson 5.4 of Discovering Advanced Algebra.
Building Inverses of Functions Use dynamic sketches to explore the relationship between the graph of a function and its inverse. This exploration can enhance your understanding of the concepts in Lesson 5.5 of Discovering Advanced Algebra.
Chapter 6
Systems of Inequalities Use dynamic sketches to investigate the graphs of systems of linear and nonlinear inequalities. This exploration reinforces the concepts in Lesson 6.5 of Discovering Advanced Algebra.
The Box Factory Use a dynamic sketch to explore the relationship between the dimensions of a box and a graph of its volume. This exploration can be used to augment the Investigation Box Factory and to help you solve Exercise 5 in Lesson 7.6 of Discovering Advanced Algebra.
Chapter 8
Constructing an Ellipse Extend Lesson 8.2 of Discovering Advanced Algebra by exploring a geometric construction of an ellipse. This sketch also supports the Chapter 8 Exploration Constructing the Conic Sections.
Constructing a Parabola Extend Lesson 8.3 of Discovering Advanced Algebra by exploring a geometric construction of a parabola. This sketch also supports the Chapter 8 Exploration Constructing the Conic Sections.
Constructing a Hyperbola Extend Lesson 8.4 of Discovering Advanced Algebra by exploring a geometric construction of a hyperbola. This sketch also supports the Chapter 8 Exploration Constructing the Conic Sections.
Properties of Hyperbolas Deepen your understanding of the definition of hyperbolas by exploring a dynamic sketch. This exploration will deepen your understanding of the material in Lesson 8.4 of Discovering Advanced Algebra.
Chapter 9
Infinite Geometric Series Use a geometric model to deepen your understanding of infinite series. This exploration enriches your understanding of the concepts in Lesson 9.2 of Discovering Advanced Algebra, and offers a different model from the Exploration Seeing the Sum of a Series in Chapter 9.
Chapter 10
Permutations and Combinations Use dynamic sketches to investigate properties of permutations and combinations. This exploration will reinforce the concepts in Lessons 10.5 and 10.6 of Discovering Advanced Algebra.
Chapter 11
Normal Distributions Explore how the mean and standard deviation impact the bell-shaped normal curve. This will give you a deeper understanding of Lesson 11.3 of Discovering Advanced Algebra.
The Least Squares Line Use a dynamic data set and movable line to enrich your understanding of the least squares line. This exploration will reinforce the concepts in Lesson 11.6 of Discovering Advanced Algebra.
Chapter 12
Right Triangle Trigonometry Use dynamic sketches to review the properties of similar triangles and the definitions of trigonometric ratios. This exploration will reinforce the concepts in Lesson 12.1 of Discovering Advanced Algebra.
The Law of Cosines Use a dynamic sketch to validate the Law of Cosines, which is covered in Lesson 12.3 of Discovering Advanced Algebra.
Introduction to Vectors Explore the properties of vectors in a dynamic environment. This exploration can be used to enhance your understanding of the concepts in Lesson 12.5 of Discovering Advanced Algebra.
Chapter 13
Defining the Circular Functions Explore the definitions and graphs of the circular functions sine and cosine. The sketches on this page will give you a deeper understanding of Lesson 13.1 of Discovering Advanced Algebra.
Radian Measure Use dynamic sketches to deepen your understanding of the definition of radian. This exploration will reinforce the concepts in Lesson 13.2 of Discovering Advanced Algebra.
Double Ferris Wheel Explore the motion of a double Ferris wheel. This will help you solve Exercise 12 in Lesson 13.5 of Discovering Advanced Algebra. | 677.169 | 1 |
Through a clear and thorough presentation, this program fosters learning and success for students of all ability levels with extensive skills practice, real-life connections, projects, and study aids.
The accessible format helps students gain the understanding and confidence they need to improve their performance on standardized tests. Margin notes provide links to postulates and concepts previously taught; theorem boxes help students identify the big ideas in geometry. Featured lessons address calculator usage, applications, as well as paragraph proofs and constructions. Pre-taught vocabulary provides students with relevant background.
Book Description Fearon, 2002. Hardcover. Book Condition: New. THE BOOK IS BRAND NEW. MAY HAVE MINOR SHELF WEAR.TEACHER'S EDITION30238384-N | 677.169 | 1 |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more | 677.169 | 1 |
This is the Calculus book in my school
I just want to know if it Calculus AB or BC
Here is the link of the book | 677.169 | 1 |
Linear Algebra (Math 232 A & B, Fall 2004)
Textbook
Course Syllabi
Books on Proofs
The books below are in the spirit of the recommended text "Nuts and
Bolts of Proof." If you find that book helpful, you might consider
ordering one or more of these from someplace like Amazon.com or Barnes & Noble (these
are listed in order of my familiarity with them, which may not be the
same order I would recommend them in).
Homework exercises (updated Nov 16, 2004)
These are suggested exercises from Johnson/Riess/Arnold, 4th
Edition. Generally, the same problems have the same numbers in
the 5th Edition. Where this is not the case, a slash precedes the
number for the 5th Edition. Also, for the 5th Edition increase
the chapter number by 1 for each problem from all but Chapter 1 of the
4th Edition. So, for example, 2.4.34/36 would be 2.4.34 in the
4th Edition and 3.4.36 in the 5th Edition. Is that clear?
After reading each section, send me an email to the address
announced in class (not my UPS(dot)EDU account)
with your answers to each of the three questions. Each answer will be
graded as one point, there will be no partial credit. I will reply with
a list of the questions you got credit for. Observe the following to
ensure your answers are received and graded properly.
Make your subject line exactly,
exactly as follows: Math 232 X, where X is the
upper-case acronym for the relevant section. So, for example, the first
reading
assignment answers would have the subject line (exactly):
Math 232 WILA
Put your full name as the first line of the body of your message.
Answer the questions in order.
Answers are due at 10:00 in the evening prior to the day we begin
discussing each section. They will not be accepted late. | 677.169 | 1 |
This visually engaging workbook is designed to help you learn and the master the necessary Functional Skills in Maths. With examples and questions written specifically for Health & Social Care, you will understand the relevance of FS and be able to apply them in the workplace. Free answers available at . Paperback. Book Condition: new. BRAND NEW, Functional Skills Maths in Context Health & Social Care Workbook: Entry 3 Level 2, Debbie Holder, Veronica Thomas, B9781408518335
Book Description Paperback. Book Condition: New. Not Signed; abl. book. Bookseller Inventory # ria9781408518335_rkm
Book Description 2012. Book Condition: New. 214mm x 298mm x 8mm. This full colour and visually engaging workbook is designed to help you learn and the master the necessary Functional Skills in Maths at your required level. With examples and questions written spec.Shipping may be from multiple locations in the US or from the UK, depending on stock availability. 136 pages. 0.416. Bookseller Inventory # 9781408518335 | 677.169 | 1 |
 Reflection: High Expectations Period and Amplitude - Section 2: Reviewing the graphs of sine and cosine
I have found that expectations need to be clear when teaching students to graph. I do not expect perfect graphs with every point in place as this is unnecessary, but I do have certain expectations. When left to their own devises, students will often draw a sloppy graph with no units labeled. In fact, even with clear expectations, students will sometimes turn in inadequate graphs. This has been a challenge for some of my students this year.
Here are my expectations. There are always key points associated with a function that much be shown. I insist on tick marks with units labeled if a scale other than one is used. I want to graph to extent across their coordinate plane. I also encourage them to use graph paper as this will naturally improve the quality of their graphs as the proportions are already set.
For the parent Graphs of Sine and Cosine, I use the interval of 0 ± π as my point indicators. For sine functions, 0 ± π are the x-intercepts. For cosine, 0 ± π are the local maximum and minimum. By plotting these points, my student's graphs are more proportional and this really helped when we started changing the period or shifting the graph horizontally.
I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson's Warm Up- Period and Amplitude, which asks students to compare trigonometric functions to other functions studies this year.
I also use this time to correct and record the previous day's Homework.
In the previous lesson, students constructed both the sine and cosine graph from a unit circle. The first goal in today's lesson is to draw these graphs in their notes along with all of the major features (except period and amplitude which are the topics of today's lesson). Their graphs need to include the height of each function, the local maximum and minimum values, the y-intercept, and the x-intercepts. For the re-occurring points, this is an important place to discuss how we can write down an expression that represents every one of those points (Math Practice 8) I have the students discuss in pairs how you can represent these reoccurring points. The goal is that they come up with ±2π.
Resources (1)
Resources
The next section has students look at the amplitude as the height of one of these graphs and then gives the students some examples of finding amplitude. Please note that I am using both degrees and radians as the unit for the x-axis. This practice will reinforce their understanding of the connections between degrees and radians (Math Practice 7).
Resources (1)
Resources
Period is a bit more involved than amplitude. Instead of giving the students the formula for finding period and having them apply it, we are going to give them an opportunity to build the concept of period as a horizontal stretch or shrink. The first four examples all include both the equation and the graph. The graphs can come up after the student discuss or during as needed. This is a major scaffolding piece in this lesson. The next examples just include the equation. By the end, have the students write a statement on how to find period.
Resources (1)
Resources
The final portion includes a sine and a cosine function to graph, both with a period and amplitude stretch. I have the students graph these and then model them myself on the board. This is the opportunity to show students what level of accuracy and what specific detail I would like them to include in a graph. I add examples if time allows and it seems like the students need it.
Resources (1)
Resources
There are four portions to this Homework. The first portion asks the students to identify the period and amplitude from the graph of a sine or cosine function. The section portion asks them to find the same thing from an equation. Next, they are asked to graph some sine and cosine functions. This is the most important part. There is no better way to memorize the important features of a trig graph than draw it given different variations. Half of the problems are in degree and half are in radians. The final question is an extension question asking the students to compare the domain and range of the trig graphs they just completed (Math Practice 2).
I am enjoying your lessons on the graphs of sine and cosine. I am hoping my students will too and that it will increase their understanding. I went to take a look at the homework assignment for this lesson but am unable to see it. If possible, can you upload it again or maybe in word format? | 677.169 | 1 |
Mathematics
has been an indispensible tool in economic analysis.
Generally, most of the students are found to have, what
is called, mathematics phobia. For them studying
mathematical methods is somewhat unpleasant and taxing
experience. Such an attitude, the authors feel (from
their experience of teaching honors student), arises
out of the students' inability to derive maximum benefit
from a book due to complex presentation of its text.
Even in cases, where presentation is simple, the lack of
many solved examples minimizes the students' efficiency
to grasp the concepts. An ideal way out of this would be
to explain various concepts with the help of solved
examples. In specially designing this textbook, the
authors have made a sincere effort to incorporate the
above methodology by presenting a variety of problems,
after explaining every concept. These problems elucidate
the concepts and their applications. Throughout the
book, emphasis is given on applications rather than pure
mathematical tools needed in economic analysis. Every
concept in the book has been dealt with very patiently.
The mathematics presented is easily accessible. However,
this has not been done at the expense of rigour. The
book contains more than 300 problems with solutions to
give the reader sufficient practice and confidence. The
book has been divided into five parts. Part - I contains
the notion of sets, sequences, functions and analytical
geometry with applications. Part- II is concerned with
differential calculus with one independent variable.
Limits and continuity, meaning of derivatives and
applications of derivatives are discussed here. Part -
III deals with logarithmic functions. Part - IV extends
differentiation to multivariate functions. In this
connection partial derivatives, differentials and
optimization unconstrained as well as constrained are
explained. Matrices and their application to linear
models is discussed in Part - V. This volume is intended
to serve as a useful reference book for honours course
in economics and masters course in mathematical
economics in the Indian Universities.
CONTENTS IN DETAIL :
Part - I
1.
Sets and Set Theory
• Concept of a Set
• Relationship between Sets
• Set Operations
• Ordered Pairs and Cartesian Product
• Relations and Functions
• Boundary Points and Closed and Open Sets
• Convex Sets
• The Derivative of a Power Function
• The Derivative of Constant times a Function
• The Derivative of Sum (or Difference) of Function
• The Derivative of Product of Functions
• The Derivative of Quotient of Functions
• The Derivative of Composite Function
• The Derivative of an Inverse Function
• Evaluation of Nth Order Derivative : Leibniz's Theorem
Pawas Prabhakar is on the
Faculty of Economics, Shri Ram College of
Commerce, University of Delhi. He completed his
post-graduation in Economics from Delhi School of
Economics, specializing in Game Theory and
Industrial Economics. He has also been a guest
lecturer at the Institute of Economic Growth,
Delhi on quantitative techniques in Economics.
Alka
Budhiraja :
Alka Budhiraja is a Senior
lecturer in the Department of Economics, Miranda
House, University of Delhi. She is a post-graduate
in Economics from Delhi School of Economics with
specialization in Econometrics. She has a teaching
experience of fourteen years of which ten years
have been exclusively devoted to mathematical
Economics at the under-graduate level. | 677.169 | 1 |
MATH CENTER RESOURCES
Page Content
Assisting
Faculty instructors, instructional assistants, student assistant tutors, and volunteers are available to help SAC math students increase their understanding of various math topics, assist with math assignments and guide students in the use of technology. It is vital that students log in to and out of the appropriate attendance computer in order to use the Math Center
Computers
The Math Center has 47 computers that students may use to complete their math assignments, watch math videos, practice their skills with math tutorials or access software related to their coursework such as STATDISK.
Graphing Calculators
The Math Center has some TI-84 and TI-83 graphing calculators attached to some of the tables. Students may download Calculator apps and programs from most of the MC computers. | 677.169 | 1 |
First. In order to illustrate some applications of linear programming. basic definitions and theories of linear programs. Next. have developed the theory behind "linear programming" and explored its applications [1]. we will focus methods of solving them. Section 3 presents more definitions. After learning the theory behind linear programs. Section 9 discusses cycling in Simplex tableaux and ways to counter this phenomenon. concluding with the statement of the General Representation Theorem (GRT). Since then. developed the Simplex method of optimization in 1947 in order to provide an efficient algorithm for solving programming problems that had linear structures. we explore the Simplex further and learn how to deal with no initial basis in the Simplex tableau. Finally. we explore an outline of the proof of the GRT and in Section 5 we work through a few examples related to the GRT.1
Introduction to Linear Programming
Linear programming was developed during World War II. a member of the U. Section 6 introduces concepts necessary for introducing the Simplex algorithm. in Section 1 we will explore simple properties.
1. when a system with which to maximize the efficiency of resources was of utmost importance.1
What is a linear program?
We can reduce the structure that characterizes linear programming problems (perhaps after several manipulations) into the following form:
3
. In Section 4. New war-related projects demanded attention and spread resources thin. we will explain simplified "real-world" examples in Section 2. George Dantzig. Air Force. which we explain in Section 7.S. especially mathematics and economics. experts from a variety of fields. including examples when appropriate. "Programming" was a military term that referred to activities such as planning schedules efficiently or deploying men optimally. we put all of these concepts together in an extensive case study in Section 11. We present an overview of sensitivity analysis in Section 10. This paper will cover the main concepts in linear programming. In Section 8.
c 2 x2 a12 x2 a22 x2 . z = 14 is the smallest possible value of z given
4
. . The following example from Chapter 3 of Winston [3] illustrates that geometrically interpreting the feasible region is a useful tool for solving linear programming problems with two decision variables. xn are called decision variables. x2 . . am2 x2
+ + + +
··· ··· ··· ··· . . Since isocost lines are parallel to each other. . . the value of z is constant. A set of x1 . . . plotting the line x1 = (z − x2 )/4 for various values of z results in isocost lines. . . The solution to this linear program must lie within the shaded region.
a1n xn a2n xn . Therefore. xn ) in the feasible region.
x1 . which have the same slope. x2 . .. . The variables x1 . . Recall that the solution is a point (x1 . Since z = 4x1 + x2 . .
+ + +
c n xn
=
z
Subject to a11 x1 a21 x1 . . the expression being optimized. plus the nonnegativity constraint). x2 ) such that the value of z is the smallest it can be. x2 . the dotted lines represent isocost lines for different values of z . In Figure 1. . . . or else not all the constraints would be satisfied.. and their values are subject to m + 1 constraints (every line ending with a bi . + amn xn xn
= b1 = b2 . The solution of the linear program must be a point (x1 . = bm ≥ 0. the thick dotted isocost line for which z = 14 is clearly the line that intersects the feasible region at the smallest possible value for z . is called the objective function. .. x 2
We plotted the system of inequalities as the shaded region in Figure 1.Minimize
c 1 x1
+ + + + x2 . Since all of the constraints are "greater than or equal to" constraints. Along these lines.
In linear programming z . The linear program is: Minimize 4x1 Subject to 3x1 x1 x1 + + + x2 x2 x2 = z ≥ 10 ≥ ≥ ≥ 5 3 0. . am1 x1 x1 . the shaded region above all three lines is the feasible region. while still lying in the feasible region. xn satisfying all the constraints is called a feasible point and the set of all such points is called the feasible region.
The thick isocost line that passes through the intersection of the two defining constraints represents the minimum possible value of z = 14 while still passing through the feasible region.2
Assumptions
Before we get too focused on solving linear programs. several assumptions are implicit in linear programing problems.
Figure 1: The shaded region above all three solid lines is the feasible region (one of the constraints does not contribute to defining the feasible region). . These assumptions are: 1. The dotted lines are isocost lines. This value occurs at the intersection of the lines x1 = 3 and x1 + x2 = 5. it is important to review some theory. This implies no dis-
5
. Proportionality The contribution of any variable to the objective function or constraints is proportional to that variable.
1.the constraints. For instance. where x1 = 3 and x2 = 2.
variables may not have a nonnegativity constraint. If the constraint were originally 4x1 + x2 ≥ 3.counts or economies to scale. URS Variables If a variable can be negative in the context of a linear programming problem it is called a urs (or "unrestricted in sign") variable. For example. 2. integer programming is beyond the scope of this paper. Certainty This assumption is also called the deterministic assumption. Subtracting a slack variable from a "greater than or equal to" constraint or by adding an excess variable to a "less than or equal to" constraint. we can bypass this condition. Unfortunately. Divisibility Decision variables can be fractions. Realistically. which will be important when we consider the Simplex algorithm. transforms inequalities into equalities. However. the value of 8x1 is twice the value of 4x1 . Additivity The contribution of any variable to the objective function or constraints is independent of the values of the other variables. This means that all parameters (all coefficients in the objective function and the constraints) are known with certainty. The effect of changing these numbers can be determined with sensitivity analysis. we eliminate this concern. no more or less.3
Manipulating a Linear Programming Problem
Many linear problems do not initially match the canonical form presented in the introduction. We now consider some ways to manipulate problems into the desired form.
1. For example. or the problem may want to maximize z instead of minimize z . By 6
. The constraints may be in the form of inequalities. the constraint 4x1 + x2 ≤ 3 becomes 4x1 + x2 + e1 = 3 with the addition of e1 ≥ 0. by using a special technique called integer programming. however. 3. which will be explored later in Section 9 [3]. Constraint Inequalities We first consider the problem of making all constraints of a linear programming problem in the form of strict equalities. 4. the additional surplus variable s1 must be subtracted (4x1 + x2 − s1 = 3) so that s1 can be a strictly nonnegative variable. By introducing new variables to the problem that represent the difference between the left and the right-hand sides of the constraints. coefficients and parameters are often the result of guess-work and approximation.
problem. These sorts of conditional constraints do not occur in linear programming. In linear programming. The vector x is a vector of solutions to the problem. The constraint y ≤ mx + h doesn't hold when d < x ≤ e and the constraint y ≤ b doesn't hold when 0 ≤ x ≤ d. consider the 2-dimensional region outlined on the axes in Figure 3. b is the righthand-side vector. Figure 2 shows an example of a nonconvex set and a convex set. then X is said to be nonconvex. and c is the cost coefficient vector. No such line segment is possible in set 2.5
Convex Sets and Directions
This section defines important terms related to the feasible region of a linear program. This more compact way of thinking about linear programming problems is useful especially in sensitivity analysis. That is. This region is nonconvex. See Figure 4 for an example of a convex set. any convex combination of these two points is also in X. 1] [2].1.
8
. The feasible region of a linear program is always a convex set. A set X is convex if the line segment connecting any two points in X is also contained in X. A set X ∈ R is a convex set if given any two points x1 and x2 in X. The endpoints of line segment AB are in set 1.
A
B
1
2
Figure 2: Set 1 is an example of a nonconvex set and set 2 is an example of a convex set. If any part of the line segment not in X.
1. A constraint cannot be valid on one region and invalid on another. yet the entire line is not contained in set 1. this region could not occur because (from Figure 3) y ≤ mx + h for c ≤ x ≤ d but y ≤ b for d < x ≤ e. To see why this makes sense. λx1 + (1 − λ)x2 ∈ X for any λ ∈ [0. Definition 1. which will be discussed in Section 9.
3. Definition 1. yet all three points must ¯ . Since this intersection is unique. or else they could not be colinear. Definition 1. An extreme direction of a convex set is a direction of the set that cannot be represented as a positive combination of two distinct directions of the set. if two points x and x make up a strict convex ¯ . then all three of these points must be defined by the combination that equals x same constraints. A point x in a convex set X is called an extreme point of X if x cannot be represented as a strict convex combination of two distinct points in X [2]. a nonzero vector d is a direction of the set if for each x0 in the set. 1). Graphically. x = x = x can't exist and hence corner points and extreme points are equivalent. We will not prove this fact rigorously. the ray {x0 + λd : λ ≥ 0} also belongs to the set. the strict convex combination lie on both constraints. an extreme point is a corner point. However. Given a convex set. 9
. then x ¯ is unique. In two dimensions. two constraints define an extreme point at their intersection. Definition 1. Further.4. A strict convex combination is a convex combination for which λ ∈ (0. if two constraints ¯ at their intersection. Therefore.2.Figure 3: A nonconvex (and therefore invalid) feasible region. since two lines can only intersect define x once (or not at all).
Examples of a direction and an extreme direction can be seen in Figure 5. but the feasible region is only two-dimensional. an extreme point is degenerate if the number of intersecting constraints at that point is greater than the dimension of the feasible region.Figure 4: A bounded set with 4 extreme points.
10
. the extreme point (2. We need all of these definitions to state the General Representation Theorem. 1) is a degenerate extreme point because it is the intersection of 3 constraints. This theorem will be discussed more later. a building-block in linear programming. plus a nonnegative linear combination of extreme directions. In general. This set is bounded because there are no directions. Also. The General Representation Theorem states that every point in a convex set can be represented as a convex combination of extreme points.
Since this program is a minimization problem and the smallest any of the variables can be is max {(xj − cj ). The following constraints account for demand:
13
. 4 − number of workers starting work in quarter t. 2. 4. demand for shoes must be considered. Pairs of shoes may be put in inventory. but this costs $50 per quarter per pair of shoes. 2. decision variables are defined.of xj − cj and −(xj − cj ) (for j = 1. 2. The objective function takes into account the salary paid to the workers. Perhaps the most notable aspect of this problem is the concept of inventory and recursion in constraints. t = 1. 3. and 100 for quarter 4. 300 for quarter 2. Since each worker works three quarters they must be payed three times their quarterly rate. each worker can only make 50 pairs of shoes per quarter. 4 − number of pairs of shoes in inventory after quarter t. 2. 3. Each of their workers gets paid $500 per quarter and works 3 contiguous quarters per year. 800 for quarter 3. t = 1. − (xj − cj )}. A company is opening a new franchise and wants to try minimizing their quarterly cost using linear programming. Solution In order to minimize the cost per year. If we let
Pt Wt It
− number of pairs of shoes during quarter t. and inventory must be empty at the end of quarter 4. The demand (in pairs of shoes) is 600 for quarter 1. as well as inventory costs. Additionally. This value will be the absolute value of xj − cj . each Pi will naturally equal its least possible value. 2 and where c is the j th component of the position of one of the old machines). 3.
the objective function is therefore min z = 50I1 + 50I2 + 50I3 + 1500W1 + 1500W2 + 1500W3 + 1500W4. t = 1. Next. In the next problem we will also interpret a "real-world" situation as a linear program.
Suppose that there are m sources that generate waste and n disposal sites. It is set up this way in order to promote long-term minimization of cost and to emphasize that the number of workers starting work during each quarter should always be the optimal value. in the next problem. capacity 14
. which gives us the constraints P1 P2 P3 P4 ≤ 50W1 + 50W3 + 50W4 ≤ 50W2 + 50W4 + 50W1 ≤ 50W3 + 50W1 + 50W2 ≤ 50W4 + 50W3 + 50W2
This linear program is somewhat cyclical. and not vary year to year. the concepts of inventory and scheduling were key. this problem deals with a different real-world situation and it is interesting to see the difference in the structure of the program. In the previous problem. since workers starting work in quarter 4 can produce shoes during quarters 1 and 2. the workers can only make 50 pairs of shoes per quarter. 3. Potential transfer facility k has fixed cost fk . Also. decision variable must be identified and constraints formed to meet a certain objective. This means that several options will be provided concerning how to dispose of waste and all of the waste must be accounted for by the linear program. The next example is a similar type of problem.
Note these constraints are recursive. It is desired to select appropriate transfer facilities from among K candidate facilities. meaning they can be defined by the expression: I n = I n1 + P n − D n where Dn is just a number symbolizing demand for that quarter. Again. However. the most crucial aspect is conservation of matter.P1 I1 I2 I3 I4
≥ 600 = P1 − 600 = I1 + P2 − 300 = I2 + P3 − 800 = I3 + P4 − 100 = 0. The amount of waste generated at source i is ai and the capacity of site j is bj .
1 ≤ k ≤ K
xkj
= tons of waste moved from k to j. This restriction
15
. The problem is to choose the transfer facilities and the shipping pattern that minimize the total capital and operating costs of the transfer stations plus the transportation costs. This constraint is wik =
i j
xkj .
The next constraint says that the amount of waste produced equals the amount moved to all the transfer facilities: wik =
k i
ai . 1 ≤ j ≤ n
The objective function features several double sums in order to describe all the costs faced in the process of waste disposal. 1 ≤ i ≤ m.
The first constraint equates the tons of waste coming from all the sources with the tons of waste going to all the disposal sites. Let cik and c ¯kj be the unit shipping costs from source i to transfer station k and from transfer station k to disposal site j respectively. Solution As with the last problem. 1 ≤ k ≤ K.qk and unit processing cost αk per ton of waste. there must be a constraint that restricts how much waste is at transfer or disposal sites depending on their capacity.
Next. defining variables is the first step.
wik yk
= tons of waste moved from i to k. The objective function is:
min z =
i k
cik wik +
k j
c ¯kj xkj +
k
fk yk +
k i
αk wik . 1 ≤ k ≤ K = a binary variable that equals 1 when transfer station k is used and 0 otherwise.
The optimal objective function value of this program gets smaller or stays the same. so ensuring this property of yk would not be an obstacle in solving this problem. suppose that one component of the vector b. This is either a positive number or zero. so therefore the feasible region gets smaller or stays
16
.gives: xkj ≤ bj and
k i
wik ≤ qk . then that variable will equal 0 at optimality. The new feasible region is everything from the old feasible region except points that satisfy the constraint bi ≤ y · x < bi + 1.2
Discussion
Now let's discuss the affects altering a linear program has on the program's feasible region and optimal objective function value. The feasible region gets larger or stays the same. is increased by one unit to bi + 1. say bi . Consider the typical linear program: minimize cx subject to Ax ≥ b.
Putting these constraints all together. suppose that a new variable xn+1 is added to the problem. It still contains all of its original points (where xn+1 = 0) but it may now also include more. Now. After the addition of xn+1 the feasible region goes up in dimension. If the new variable is added to z (that is. any positive value of that variable would make z larger). the linear program is:
Minimize Subject to
z
i wik k k
= = = ≤ bj ≤ qk
i
k cik wik
+
k
¯kj xkj jc
+
k
fk yk +
k
i
αk wik
j xkj i
wik
ai
xkj i wik
all variables ≥ 0. First. Since there are more options in the feasible region. To see this more clearly. let y symbolize the row vector of coefficients of the constraint to which 1 was added. Most linear program-solving software allows the user to designate certain variables as binary. x ≥ 0.
2. there may be one that more effectively minimizes z . The feasible region gets smaller or stays the same.
A halfspace in R2 is everything on one side of a line and. The climax of this chapter will be the General Representation Theorem and to reach this end. have been presented. 0) · (x1 .the same. After completing the dot product. more definitions and theorems are necessary. Consider the two-dimensional halfspace {(x1 . 0) · (x1 . the optimal objective value either stays the same. 17
.
3
3. The optimal objective of this new program gets larger or stays the same. the variables in that constraint may have to increase in value. or becomes more optimal (smaller for a minimization problem. and in two dimensions it is a line. A hyperplane in three dimensions is a traditional plane. A hyperplane H in Rn is a set of the form {x : px = k } where p is a nonzero vector in Rn and k is a scalar. After completing the dot product. This is because a larger feasible region presents more choices. This increase may in turn increase the value of the objective function. or becomes less optimal. For example. in R3 a halfspace is everything on one side of a a plane. the optimal objective value either stays the same. this would be everything to the left of the hyperplane x1 = 2. or larger for a maximization problem). it is clear that this halfspace describes the region where x1 ≤ 2. A halfspace is a collection of points of the form {x : px ≥ k } or {x : px ≤ k }. The purpose of this definition is to generalize the idea of a plane to more dimensions. x2 plane. one of which may be better than the previous result.2. similarly. it is time to explore the theory behind linear programming more thoroughly. x2 ) ≤ 2}. x2 ) : (1. In general. it turns out that this is just the line x1 = 2 which can be plotted on the x1 x2 -plane. when changes are made in the program to increase the size of the feasible region. x2 ) = 2} is a hyperplane in R2 .1. x2 ) : (1. Conversely. Definition 3. when a feasible region decreases in size. To satisfy the new constraint. Definition 3.1
The Theory Behind Linear Programming
Definitions
Now that several examples. On the x1 . {(x1 .
An edge of a polyhedral set is a one-dimensional face of X. the highest dimension of a proper face of X is equal to dim(X)-1. See Figure 6 for an example of a polyhedral set. Extreme points are zero-dimensional faces of X. and everything below the two planes defining the sides of the tent. ¯ ∈ X is said to Definition 3. Therefore.4. Set C is a set of points that forms a two-dimensional proper face of the pup tent. Every polyhedral set is a convex set. It can be written in the form {x : Ax ≤ b} where A is an m × n matrix (where m and n are integers).3.Definition 3. A point x ¯ lies on some n linearly independent defining be an extreme point of set X if x 18
. which basically summarizes the next big definition:
Figure 6: This "pup tent" is an example of a polyhedral set in R3 . Let X ∈ Rn . Point A is an extreme point because it is defined by three halfspaces: everything behind the plane at the front of the tent. A proper face of a polyhedral set X is a set of points that corresponds to some nonempty set of binding defining hyperplanes of X. A polyhedral set is the intersection of a finite number of halfspaces. The line segment from A to B is called an "edge" of the polyhedral set. where n is an integer.
l. . an extreme point was said to be a "corner point" of a feasible region in two dimensions. . Then the set of extreme points is not empty and is finite. . This theorem not only provides a way to represent any point in a polyhedral set.2
The General Representation Theorem
One of the most important (and difficult) theorems in linear programming is the General Representation Theorem. the set of extreme directions is empty if and only if X is bounded. . . .
k l
¯= x
j =1
λj x j +
j =1 k
µj d j
λj = 1. . x2 . µj ≥ 0. but its proof also lays the groundwork for understanding the Simplex method. dl . The General Representation Theorem: Let X = {x : Ax ≤ b. d2 . . . If X is not bounded. . xk plus a nonnegative linear combination of d1 . Furthermore. which is covered in the next section. Earlier. . ¯ ∈ X if and only if it can be represented as a convex combination of Moreover. . . which.
j =1
λj ≥ 0. say {d1 .1. . . .
In sum. The previous definition states the definition of an extreme point more formally and generalizes it for more than two dimensions. dl }. a basic tool for solving linear programs. it is clear that any point in between extreme points can be represented as a convex combination of those extreme points. . .hyperplanes of X . . say {x1 . d2 . x x1 .
j = 1. All of the previous definitions are for the purpose of generalizing the idea of a feasible region to more than two-dimensions. is a convex set in n dimensions. x2 . then the set of extreme directions is nonempty and is finite. Theorem 3. The terminology presented in these definitions is used in the statement and proof of the General Representation Theorem. k j = 1. 2. xk }. that is. 2. Any other point can be represented as
19
. The feasible region will always be a polyhedral set. . . x ≥ 0} be a nonempty polyhedral set. according to the definition. . by visualizing an unbounded polyhedral set in two-dimensions. .
3. .
Next. First. ¯ ∈ X if and only if it can be represented as a convex the GRT states that x combination of extreme points of X and a nonnegative linear combination of extreme directions of X. Next. plus a combination of multiples of extreme directions of X. Proving that if a point x can be represented as a convex combination of extreme points of a set X.one of these convex combinations. Before we do that. . say x1 . We will explore the proofs of these parts in sequence. 20
. xk . d2 . if X is bounded then the set of extreme directions of X is empty. however. Furthermore. The argument here is similar to proving that there are a finite number of extreme points. finite. Thus there is at least 1 extreme point. Further. we want to prove that there are a finite number of extreme points in the set X . . We do this by moving along extreme directions until reaching a point that has n linearly independent defining hyperplanes. The set of extreme directions is either empty (if X is bounded) or has a finite number of vectors. it must be proven both ways. say d1 . then x ∈ X (the "backward" proof). then the set of extreme directions is nonempty and finite. plus a combination of multiples of extreme directions. Finally. . .
4
An Outline of the Proof
The General Representation Theorem (GRT) has four main parts. There is a ¯ that can be written as in Equation GRT. if X is not bounded. This theorem is called "general" since it applies to either a bounded polyhedral set (in the case that µj = 0 for all j ) or an unbounded polyhedral set. . dl . is simple. so the number of extreme points is finite. we must prove the first two parts of the GRT: • First. . it states that the number of extreme points in X is nonempty and finite. x2 . . The set of extreme points of X is not empty and has a finite number of points. . the backwards proof: • Given: X = {x : Ax ≤ b x ≥ 0} is a nonempty polyhedral set.
m+n n
is
• Prove that there are a finite number of extreme directions in the set X . where n is just an integer. Since the last part of the GRT is a "if and only if" theorem. point x • Want to Prove: x ¯ ∈ X.
A¯ x ≤ b. if we can write x ¯ in the form of the GRT then x ¯ ∈ X. Therefore. Therefore. From the assumptions:
k l
A¯ x=
j =1
λj Axj +
j =1
µj Adj . Since x ¯ is the sum of the products of all positive numbers. From this information. A¯ x is the sum of two sums. the ray {x + λd : λ ≥ 0} also belongs to the set. Since the proof of the GRT in the forward direction is quite complicated.– First. Therefore. From the same reasoning. – Next. Therefore d > 0. it will not be proven carefully. 21
l
. x ¯ must be nonnegative. The first inequality must hold for arbitrarily large λ. d is a direction of set X if for each x in X . From this definition. The following outline provides a summary of the steps involved with the proof of the forward direction: • Define a bounded set that adds another constraint on to the original set X. so therefore Ad must be less than or equal to 0. • Show that x ¯ can be written as a convex combination of extreme points in that set. then a point x can be represented as a convex combination of extreme points of a set X. The "forward" proof of the GRT states that. there are two restrictions on d: A(x + λd) x + λd ≤ b ≥ 0 (1)
for each λ ≥ 0 and each x ∈ X .
But what about µj Adj ? From the definition of extreme directions. if x ∈ X. remember that A¯ x ≤ b if x ¯ ∈ X . plus a combination of multiples of extreme directions of X. one of which is less than or equal to b and one of which is less than 0. since x + λd can never be negative. no matter how big λ gets. it follows that d ≥ 0 and d = 0 (since x could have negative components).
It is clear that therefore
k j =1
k j =1
Axj is less than b (since each xj is in X ) and
k j =1
λj Axj ≤ b since
l j =1
λj = 1. it is clear that j =1 µj Adj is less than 0. this must be true.
• Define the extreme directions of X as the vectors created by subtracting the old extreme points (of X ) from the newly created extreme points (of the new bounded set). • Write a point in X in terms of the extreme directions and evaluate the aforementioned convex combination in terms of this expression. • The last step ends the proof by giving an expression that looks like the result of the GRT. Even though this outline is not rigorous proof by any means, it still helps enhance understanding of the GRT. Figure 7 illustrates the GRT for a specific polyhedral set. It is from [2].
5
Examples With Convex Sets and Extreme Points From Bazaara et. al.
In the previous section, the General Representation Theorem concluded a theoretical overview of convex sets and their properties. Now it is time to apply this knowledge to a few examples. Let's see how a specific point in a specific convex set can be represented as a convex combination of extreme points. We will consider an example concerning the region X={(x1 , x2 ) : x1 − x2 ≤ 3, 2x1 + x2 ≤ 4, x1 ≥ −3} shown in Figure 8. We will find all the extreme points of X and express x = (0, 1) as a convex combination of the extreme points. There are 3 extreme points, which are the pairwise intersections of the constraints when they are binding. Two of these points are on the line x1 = −3 and these points are (−3, −6) and (−3, 10). In Figure 8 these points are B and C respectively. Point A, simultaneously.
7 2 3, −3
This is simply a system of two linear equations with two unknowns and the 22
29 11 and λ2 = − 56 . Therefore, the expression solution λ1 = − 56
2 11 29 A − B − C = (1, 0) 7 56 56 is a convex combination of the extreme points that equals (1, 0). We can also use the GRT to show that the feasible region to a linear program in standard form is convex. Consider the linear program: Minimize cx Subject to Ax x = z = b ≥ 0
and let C be its feasible region. If y and z ∈ C and λ is a real number ∈ [0, 1], then we want to show that w = λy + (1 − λ)z is greater than zero and that Aw = b. It is clear that w ≥ 0 since both terms of w are products of positive numbers. To show the second condition, substitute in for w:
Aw
= A(λy + (1 − λ)z) = λAy + Az − λAz = λb + b − λb = b.
This result means that w ∈ C . Therefore, C is a convex set.
6
6.1
Tools for Solving Linear Programs
Important Precursors to the Simplex Method
Linear programming was developed in order to obtain the solutions to linear programs. Almost always, finding a solution to a linear program is more important than the theory behind it. The most popular method of solving linear programs is called the Simplex algorithm. This section builds the groundwork for understanding the Simplex algorithm. Definition 6.1. Given the system Ax= b and x ≥ 0 where A is an m×n matrix 23
and b is a column vector with m entries. Suppose that rank(A, b)=rank(A)=m. After possibly rearranging the columns of A, let A = [B, N] where B is an m × m invertible matrix and N is an m × (n − m) matrix. The solution x = xB to the equations Ax = b where xB = B−1 b and xN = 0 is called a xN basic solution of the system. If xB ≥ 0 then x is called a basic feasible solution of the system. Here B is called the basic matrix and N is called the nonbasic matrix. The components of xB are called basic variables and the components of xN are called nonbasic variables. If xB > 0 then x is a nondegenerate basic solution, but if at least one component of xB is zero then x is a degenerate basic solution [2]. This definition might be the most important one for understanding the Simplex algorithm. The following theorems help tie together the linear algebra and geometry of linear programming. All of these theorems refer to the system Minimize cx = b ≥ 0.
Subject to Ax x
Theorem 6.1. The collection of extreme points corresponds to the collection of basic feasible solutions, and both are nonempty, provided that the feasible region is not empty [2]. Theorem 6.2. If an optimal solution exists, then an optimal extreme point exists [2]. Theorem 6.2 builds on the idea put forward in Theorem 6.1 except this time it addresses optimal points and solutions specifically. In Theorem 6.3, a "basis" refers to the set of basic variables. Theorem 6.3. For every extreme point there corresponds a basis (not necessarily unique), and, conversely, for every basis there corresponds a (unique) extreme point. Moreover, if an extreme point has more than one basis representing it, then it is degenerate. Conversely, a degenerate extreme point has more than one set of basic variables representing it if and only if the system Ax=b itself does not imply that the degenerate basic variables corresponding to an associated basis are identically zero [2]. All this theorem says is that a degenerate extreme point corresponds to several bases, but each basis represents only one point. 24
the Simplex algorithm selects an extreme point at which to start. In upcoming sections.
7
The Simplex Method In Practice
The Simplex algorithm remedies the shortcomings of the aforementioned "brute force" approach. A linear program can be put in tableau format by creating a matrix with a column for each variable. we might think that the best way to solve a linear programming problem is to find all the extreme points of a system and see which one correctly minimizes (or maximizes) the problem. Instead. These iterations are repeated until there are no more adjacent extreme points with better objective function values. an ingenious algorithm known as the Simplex method. The best way to implement the Simplex algorithm by hand is through tableau form. a case study concerning busing children to school will be presented and solved with computer software. starting with z . For a general linear program. Then. where m is the rank of A and n is the number of variables. That is when the system is at optimality. the Simplex algorithm will be covered. Then. sensitivity analysis will be discussed with examples. Instead of checking all of the extreme points in the region. including a discussion of how to manipulate programs with no starting basis. In short. the objective function value. of the form Maximize cx Subject to Ax x the initial tableau would look like:
≤ b ≥ 0
25
.With all of this information. in the far left column. and is the basis for most computer software that solves linear programs. This is realistic (though still tedious) for small problems. is the most common way to solve linear programs by hand. Finally. this approach is not realistic. each iteration of the algorithm takes the system to the adjacent extreme point with the best objective function value. Also. The number of extreme points is n . this method does not indicate if the feasible region is unbounded or empty without first going through the whole set of extreme points. This m number can be quite large.
If there is no initial basis in the tableau (an mxm identity matrix).1). say xi . the ratio test doesn't apply for that row. let's use the simplex algorithm to find the optimal solution of the
26
. xi enters the basis and whichever basic variable was in row j leaves the basis. If a number in column i is nonpositive. The new basic variable selected.z 0
−c A
0 b
. column i. The ratio test is necessary because it ensures that the right-hand side of the tableau (except perhaps Row 0) will stay positive as the new basic variable enters the basis. If the ratio test is a tie. the row can be selected arbitrarily between rows with the same valued (smallest) ratios. This test is not performed in Row 0. with the one in the j th row. If we were actually completing the Simplex algorithm for this program. This variable corresponds to the column with the most positive objective function coefficient. The first iteration of the Simplex algorithm selects the column containing the most positive (in a minimization problem) coefficient in the first row (Row 0). column i must be transformed into a column of the identity matrix. To help clarify these ideas. through elementary matrix row operations. we would be concerned with the lack of an initial basis in the first tableau. the entering basic variable should be one that would decrease the value of z the fastest. perform the ratio test.
The idea here is that z − cx = 0 and Ax= b so in this format all the variables can be forgotten and represented only by columns. Through these operations. After selecting this variable as the entering basic variable. say. Since Row 0 of the tableau corresponds to the objective function value z . We repeat these steps until there are no longer positive numbers in Row 0 (for a minimization problem). enters the basis in row j . then we are not at an extreme point and the program cannot possibly be at optimality (according to Theorem 6. This test consists of dividing all the numbers on the right-hand side of the tableau by their corresponding coefficients in column i. Therefore. let's look at an example from [3]: First. The row j with the smallest positive ratio is selected.
we give the ratios used in the ratio test. and 2. x 2 = −3x1 + 8x2 ≤ 12 ≤ 6 ≥ 0
First this LP must be in standard form. x 2 . Therefore. We add slack variables to transform our linear program to min z s. Row z x1 x2 s1 s2 RHS BV ratio 0 1 3 -8 0 0 0 z=0 1 0 4∗ 2 1 0 12 s1 = 12 12 4 =3 6 2 0 2 3 0 1 6 s2 = 6 2 =3
following LP: min z s.Table 1: The number with the * next to it is the entering basic variable. 4x1 + 2x2 2x1 + 3x2 x1 . since the one in the s1 column is in Row 1. yields a tie since both ratios are 3. since their columns are columns from the identity matrix. The ratio test (shown in the far right column of Table 1). x1 is the entering basic variable. The column labeled "RHS" indicates the right-hand side of the equations. Row 1 is chosen arbitrarily to be the row of the entering basic variable. The number in Row 0 that is most positive is the 3 in the x1 column.
27
. The column "BV" indicates the values of basic variables in that tableau. The variable s1 is the leaving basic variable.t.t. In Table 1 s1 and s2 are the initial basic variables. 4x1 + 2x2 + s1 2x1 + 3x2 + s2 x1 . The rows are labeled 0. in this case x1 . s 1 . The columns are labeled according to which variable they represent. s 2 = −3x1 + 8x2 = 12 = 6 ≥ 0
The first tableau is shown in Table 1. 1. In the "ratio" column.
The solution is x1 = 3. In the case of a "greater than or equal to" constraint.
8
What if there is no initial basis in the Simplex tableau?
After learning the basics of the Simplex algorithm. and then adding −3 times Row 1 to Row 0 and −2 times Row 1 to Row 2. one cannot add a slack or an excess variable and therefore there will be no initial basic variable in that row. Simply multiplying through that row by −1 would not solve the problem. Row z x1 x2 0 1 0 − 19 2 1 1 0 1 2 2 0 0 2
since all the values in Row 0 (except for the s1 3 −4
1 −2 1 4
s2 0 0 1
RHS -9 3 0
BV z = −9 x1 = 3 s2 = 0
ratio -
The next tableau is shown in Table 2. Now that the basics of the Simplex algorithm have been covered. what happens if the initial tableau (for a program where A is mxn) does not contain all the columns of an mxm identity matrix? This problem occurs when the constraints are in either equality or "greater than or equal to" form. we explore the special cases and exceptions. There are two main methods to dealing with this problem: the Two-Phase Method and the Big-M Method. Both of these methods involve adding "artificial" variables that start out as basic variables but must eventually equal zero in
28
. This table is obtained after dividing Row 1 of Table 1 by 4. what does one do when the initial tableau contains no starting basis? That is. of course) in Row 0 are negative. one must add an excess variable. for then the right-hand-side variable would be negative. In the case of an equality constraint.Table 2: This tableaux is optimal z column) are nonpositive. The solution shown in Table 2 is optimal because all the numbers (except for the number in the z column. the next section will explain what to do when the initial tableau does not contain an identity matrix. which means z cannot be decreased anymore. x2 = s1 = s2 = 0 and z = −9. For example. which gives a −1 in that row instead of a +1.
Set up a new objective function that minimizes these slack variables. replace the current objective function with the objective function from the original problem and make all entries of Row 0 in the column of a basic variable equal to zero. The final optimal tableau for Phase I is shown in Table 6. Modify the tableau so that the Row 0 coefficients of the artificial variables are zero.8 1. and adding both of the resultant rows to Row 0. 3. 30
. The first tableau for Phase II is shown in the first part of Table 7.25 0 .3 -.6 -2. In summary.8 6. add artificial variables to rows in which there are no slack variables. The tableau shown in Table 5 is the initial tableau for Phase I of the TwoPhase Method. 2. the Two-Phase Method can be completed using the following steps: 1.8 1 -.1 0 0 0
6 2 4 2
2 1 -1 3
Table 5: 1 -1 -2 0 2 0 -1 -1
0 1 0 0
0 0 1 0
0 0 0 1
6 8 2 4
1 0 0 0
0 0 1 0
0 0 0 1
. x2 and s1 are basic.4
6: 0 -.6 .4 . Row 3 by negative one.2 -.3 . proceed with the Simplex algorithm until optimality is reached.2 . After obtaining this tableau. After that. Also.2
-1. If there is no basis in the original tableau. delete all columns associated with artificial variables.8
Table -.4
. Remember. the tableau becomes like the second part of Table 7.1 -. Phase I will be complete when this tableau is optimal. proceed with the Simplex algorithm as usual. The solution to this phase is the solution to the original problem. To get this tableau to optimality takes two steps. For Phase II of the Two-Phase Method.10 .2 0 -. In this tableau the variables a1 and a3 are zero and the variables x1 .4 . After multiplying Row 2 by two. now the objective function must be maximized.2
in Table 5 is better because I3 is in the body of the tableau.
If not. the Big-M method makes that possible.4 .2 . At optimality. the artificial variables should be nonbasic variables.6 .8 -. Also.3 -.4 . The optimal solution to this tableau will be the optimal solution to the original problem. Complete the Simplex algorithm as usual for a minimization problem. Proceed with the Simplex algorithm as usual.2
4.8 -.4
0 1 0 0 0 1 0 0
0 6.2
The Big-M Method
Another way to deal with no basis in the initial tableau is called the Big-M Method. 8.4 . or basic variables equal to zero.1 -.
31
. 5.6 .
Wouldn't it be nice to be able to solve this problem with artificial variables. but without having to complete two phases? Well.1 -.2 . 6. x 2 ≥ 2 ≥ 1 ≤ 3 ≥ 0. The following example from [2] will help illustrate this method: Consider the problem Minimize Subject to x1 − 2x2 x 1 + x2 −x1 + x2 x2 x1 . replace the artificial objective function with the original objective function. delete all rows associated with artificial variables.2 . Modify the tableau so that the Row 0 coefficients of the basic variables are zero. This is the start of Phase II.4 6.1 0 0 0 1 0 0 0
-2 0 1 0 0 0 1 0
1 0 0 1 0 0 0 1
Table 7: -1 0 -2.4 .3 -. the original problem was infeasible.8 1. After reaching optimality.2 -2. 7.
8.8 1.
except with a variable M floating around. At optimality. only to arrive back at the original tableau. a rare phenomenon occurs when a Simplex tableau iterates through several bases all representing the same solution. this would indicate that the original problem was unbounded. Modify the tableau so that the Row 0 coefficients of the artificial variables are zero. and the Simplex algorithm moves with each pivot to another set of these hyperplanes that represent the same basis. In summary. 2. 5. add or subtract M xa to the objective function where M is a really big number and xa is a vector of the artificial variables. add artificial variables to rows in which there are no slack variables. Depending on whether the problem is minimization or maximization. The term 1 + 2M is the most positive so that column is the pivot column.
9
Cycling
In the case of a degenerate problem. several linearly independent hyperplanes define a certain extreme point. this would indicate that the the original problem was infeasible. the artificial variables should be nonbasic variables and the objective function value should be finite. none of the artificial variables are basic variables. Geometrically. there is no M in the objective function value. Row 2 wins the ratio test and the next tableau is shown in the fourth part of Table 9. and finally comes back to the beginning set of these hyperplanes and 33
. If there is no basis in the original tableau. 3. the Big-M Method can be completed using the following steps: 1. Therefore. If there were an M in the objective function value. this column is selected for a pivot and the ratio test is performed as usual. In this tableau. If an artificial variable were a basic variable in the optimal tableau. x4 = 2 and z = −6. The optimal tableau is shown in the fifth part of Table 9. Complete the Simplex algorithm as usual. x3 = 1. Also. 4. This particular problem continues for a few more steps (two more pivots to be exact) before reaching optimality.The most positive coefficient in Row 0 is 2 + 2M . This part of the Big-M Method is just like performing the Simplex Method as usual. where the solution is x2 = 3.
Therefore. suppose that there is a starting basis such that the first m columns of A form the identity. This phenomenon is aptly called "cycling. 3. there are several rules that prevent its occurrence. The following steps. Otherwise. then xr leaves the basis. 1. the tableau "cycles" around an extreme point.starts over again." As mentioned before. Cycling is undesirable because it prevents the Simplex algorithm from reaching optimality. Form the set I1 such that I1 = r: yr1 Minimum = i ∈ I0 yrk yi1 : yik > 0 yik .
34
.1
The Lexicographic Method
Minimize cx = b ≥ 0. this phenomenon is quite rare. the ratio test is a tie. Say that the nonbasic variable xk is chosen to enter the basis by normal means (its coefficient is the most positive number in Row 0). called the lexicographic rule are guaranteed to prevent cycling. If I0 contains more than one element.
9. go to the next step.
Given the problem Subject to Ax x
where A is an m × n matrix of rank m. then xr leaves the basis. The index r of the leaving basic variable xr is in the set I0 = r: ¯ Minimum br =1 ≤ i ≤ m yrk ¯i b : yik > 0 yik . The following example in Table 10 is from [2]. At each iteration. this new ratio is a tie. even though cycling is so rare. The two important things to note in this series of tableaux are that the solution does not change even though the basis does change and that the last tableau is the same as the first. the pivot is indicated with a ∗.
If I1 contains only one element r. that is.
If I0 only contains one element r. 2. Thus.
Since there are a finite number of bases. and going into its details is not especially illuminating. An outline of the proof that the lexicographic method prevents cycling is as follows: 1. is more straightforward. Applying the lexicographic rule to the previous example prevents cycling. Bland's rule. – For the row i where i = r. ¯. but there are simpler rules to prevent cycling. Then xr leaves the basis. B • Show that each row of the m × (m + 1) matrix (b −1 ¯ graphically positive at each iteration where b = B b. form I2 . Keep forming Ij s until one of them contains only one element r.
−1
0)
) is lexico-
– For a typical row i where i = r where r is the pivot row.
36
. The first nonzero component of x is positive [2]. however. Before going into the proof outline. 2. x is not identically zero 2. 3. Using the lexicographic rule. A vector x is called lexicographically positive (denoted x if the following hold: 1. explained next. optimality occurs after only two pivots! The proof that lexicographic method prevents cycling is moderately difficult to understand. The lexicographic method will alway converge (there will always be an Ij that is a singleton) because otherwise the rows would be redundant.1. a brief outline will suffice.4.
5. finite convergence occurs. a new definition is needed: Definition 9. The bases developed by the Simplex algorithm are distinct (proof by contradiction). Prove that none of the bases generated by the simplex method repeat. The general form for Ij is Ij = r: yrj Minimum = i ∈ Ij −1 yrk yij : yik > 0 yik . The lexicographic rule is complicated. If I1 contains more than one element. Instead of presenting the whole proof.
To do this algebraically. Minimize cx = b ≥ 0
Subject to Ax x
Assume that the first m columns of A form the identity and assume that b ≥ 0. By doing so.2
Bland's Rule
Bland's rule for selecting entering and leaving basic variables is simpler than the lexicograpic method. Show that this method is equivalent to the lexicographic rule. Now suppose that we have j =1 aj m −1 ¯ + m yj j > 0. Charnes' method will be shown to be equivalent to the lexicographic rule. The leaving basic variable is the variable with the smallest index of all the variables who tie in the usual minimum ratio test.
9. it also usually takes a lot longer for the Simplex algorithm to converge using this rule. the one with the smallest index is chosen to enter. and each of these new extreme points will be nondegenerate. this one extreme point will become several extreme points. Consider the following perturbation procedure of Charnes.3
Theorem from [2]
This theorem concerns another rule to prevent cycling called Charnes' method. where B (b +
j =1 j =1
is chosen to enter the basis and the following minimum ratio test is made: 1≤i≤m
Minimum
¯i + b
m j =1
yij
j
yik
: yik > 0 . Suppose that xk aj j ) = b a basis B . Proof In the perturbation technique we take a degenerate extreme point with m linearly independent defining hyperplanes and shift each of these hyperplanes just a little bit. or the perturbation technique. the variables are ordered arbitrarily from x1 to xn without loss of generality. Replace j b by b + m where is a very small number. Given a basis B the corresponding feasible solution is nondegenerate if B−1 b > 0. Theorem Consider the following problem. First.9.
Show that this minimum ratio occurs at a unique r. Charnes came up with the idea of changing the
37
. Of all the nonbasic variables with positive coefficients in Row 0. Although this rule is much simpler than the lexicographic rule.
xn xn + n Let the pivot column be column k . + yit yik
t
right-hand vector from b =
x1 x2 . or else the constraints are redundant and one of them can be eliminated altogether.
If I0 is a singleton. the minimum ratio was tied between two rows. The new minimum ratio test is equivalent to ¯ bi + yi1 + +yi2 2 + . . This method is equivalent to the lexicographic method since the lexicographic method selects the pivot row in much the same way. At the index r where the terms are different. . . the lexicographic method simply iterates.
x2 + to b = . . First. + ylt t . it defines I0 = r: ¯ Minimum br =1 ≤ i ≤ m yrk ¯i b : yik > 0 yik . the next pivot row is the minimum of ¯ bi + yi1 + yi2 2 + . the minimum of those terms is the new pivot row. this minimum occurs at a unique index r. . . then this ratio occurs at a unique index and that index corresponds to the leaving basic variable. Otherwise.
where t is the number of variables in the system. Suppose in the original problem (before perturbation). This is true because each successive term yi(r+1) /yik and yl(r+1) /ylk is an order of magnitude smaller since is it multiplied by another . Ij r: yrj Minimum = i ∈ Ij −1 yrk yij : yik > 0 yik . Therefore. . Instead of involving . two terms yir /yik and ylr /ylk will be different. . row i and row l. Therefore. + yit yik
t
and
¯ bl + yl1 + yl2 2 + . ylk
At some point. . This means that ¯ bi /yik = ¯ bl /ylk . . and so it does not affect the sum significantly.
x1 +
2
.
Eventually Ij must be a singleton for some j ≤ m or else the tied rows would
38
.
causes problems in computer programs because the value of must be predetermined. at least 10 units of A must be produced daily. Bland's rule is inefficient since the Simplex algorithm takes a very long time to converge. this is equivalent to the terms in the perturbed minimum ratio sum that are different. Then the lexicographic method terminates. When a singleton occurs.be repetitive and one could be eliminated. is none of them. which doesn't matter since both terms in the sum have it. and the rest of the terms in the perturbed minimum ratio sum are negligible since they are orders of magnitude less than the ones before them. 1 of B. only off by a factor of r . consider the simple example of Leary Chemicals. The ratios are. problems that are known to be degenerate are ignored or revised until they are nondegenerate. Also.1
An Example
Leary Chemical produces three chemicals: A. Using such a small number causes computer round-off errors. which no-cycling rule is best? The answer. From this information. For example.
10
Sensitivity Analysis
Sensitivity analysis deals with the effect changing a parameter in a linear program has on the linear program's solution.
39
. Only chemical A can be sold and the other two are polluting byproducts. Running process 1 for an hour costs $4 and yields 3 units of A. it is clear that these rules for preventing cycling are more interesting theoretically than practically. in fact. In applications of linear programming. and 1 of C. To meet customer demands. These chemicals are produced via two production processes: 1 and 2. at most 5 units of B and 3 of C can be produced daily. and C. or the perturbation technique.
10. Running process 2 for an hour costs $1 and produces 1 unit of A.4
Which Rule to Use?
When solving a linear programming problem with a computer. The lexicographic rule is computationally expensive to implement since it requires many calculations. B. and 1 of B. in order to comply with government regulations. surprisingly.
9. Charnes' method.
5.5. Notice that as c1 decreases. the isocost line goes through extreme point 2 with a lowest objective function value. the linear program becomes Minimize 4x1 + x2 = z
Subject to 3x1 + x2 x1 + x 2 x1 x1 . it is clear that the solution to this problem is x2 = 2. what this question really asks is when the optimal solution is at another extreme point. remember that each extreme point in the feasible region corresponds to another basis. However. From this figure. Therefore.1.5 results in Figure 9. concerning the price of process 1. it is clear that c1 must be greater than 3 for the basis to remain at extreme point 1. once c2 becomes less than 3. x 2
≥ 10 ≤ 5 ≤ ≥ 3 0. x1 = 2. the slope will become steeper. The first question. the basis corresponding to extreme point 2 is the new basis of the optimal solution. When c2 = 3 the slope of the isocost line will be the same as the slope of the line corresponding to the first constraint.
Graphing the constraints and an isocost line for z = 12. the question for what values of c1 does the current basis remain optimal only concerns the slope. changing c1 will change both the slope and the x1 -intercept of the line.By assigning variables such that x1 is the number of hours per day running process 1 and x2 is the number of hours per day running process 2. Therefore. Sensitivity analysis is concerned with questions such as "for what prices of process 1 is this basis still optimal?" or "for what values of customer demand of chemical A is this basis still optimal?" 10.1 Sensitivity Analysis for a cost coefficient
We can understand and answer these questions through graphical analysis. As seen in Figure 10. c1 c1
Therefore. Changing this coefficient changes the isocost line. From this analysis. To understand this. 40
. An isocost line is a line of the form: x1 = x2 z − . deals with the coefficient c1 = 4.
The case study below concerns how to most efficiently bus children from six areas to three schools. the second and the nonnegativity constraint on x2 . but also a wide variety of sensitivity analysis information. for b1 < 5 the current basis is no longer optimal. For all practical purposes. This shift is illustrated in Figure 13. When shifting the first constraint to the right. This change will ensure that the original basis remains the basis for all 3 < c1 < ∞. note that it defines the optimal isocost line." 10. As long as the point where the first and second constraints are binding is optimal. As b1 gets bigger than 11 the feasible region becomes empty. From simple analysis. that is. It not only outputs an answer. if b1 is between 5 and 11.1. When this constraint intersects the x2 axis at b1 = 5. the set of optimal solutions contains an infinite number of points. it is clear that changing b1 moves this constraint parallel to itself. At c1 = 3 the set of optimal solutions contains all points on the line segment defined by constraint 1 between extreme points 1 and 2. This can be seen in Figure 12. the current basis remains optimal. Therefore. This information will be helpful later when analysing different options available to
41
. When b1 = 11. We will use a software program called Lindo to solve this problem. it becomes a horizontal line. For b1 < 5 the only the first constraint and the nonnegativity constraint on x2 are binding. the slope of the isocost line approaches zero. the upper bound on c1 can be interpreted as just "a really big number.As c1 increases. Lindo is a program specifically designed to solve linear programs. three constraints are binding: the first. it is time to apply these skills to a case study.2 Sensitivity Analysis for a right-hand-side value
The second question "for what values of customer demand of chemical A is this basis still optimal?" deals with the right-hand-side of constraint 1. it still defines the isocost line until b1 = 11. Of course. the first constraint intersects with the second and the third constraints and the feasible region only contains one point (so that point is optimal). c1 cannot realistically equal ∞. the current basis is optimal. Therefore. When shifting the first constraint to the left. That is.
11
Case Study: Busing Children to School
Now that we have learned how to solve and analyze linear programs quite thoroughly.
11.2
11.2. seventh and eighth grades) at the end of this school year and reassign all of next year's middle school students to the three remaining middle schools. where 0 indicates that busing is not needed and a dash indicates an infeasible assignment. The table shows each area's middle school population for next year. We have been hired as an operations research consultant to assist the school board in determining how many students in each area should be assigned to each school.1
The Solution
Variables
The first step to solving this problem is to assign decision variables. We can choose variables in the following way: 42
.1
The Problem
The Springfield school board has made the decision to close one of its middle schools (sixth.Table 11: No. so the school board wants a plan for reassigning the students that will minimize the total busing cost. Our job is to formulate and solve a linear program for this problem. of Students 450 600 550 350 500 450 Number in 6th Grade 144 222 165 98 195 153 Number in 7th Grade 171 168 176 140 170 126 Number in 8th Grade 135 210 209 112 135 171 Busing Cost per Student School 1 $300 $600 $200 0 $500 School 2 0 $400 $300 $500 $300 School 3 $700 $500 $200 $400 0
Area 1 2 3 4 5 6
us after we solve the initial problem. The school district provides for all middle school students who must travel more than approximately a mile. The annual cost per student of busing from each of the six residential areas of the city to each of the schools is shown in Table 11 (along with other basic data for next year).
11. The school board also has imposed the restriction that each grade must constitute between 30 and 36 percent of each school's population.
s43 . A school's capacity can be limited by summing over i for each school and for each variable.2.
11. We can write this sum the following way:
minimize z
= 300(x11 + s11 + e11 ) + 600(x31 + s31 + e31 ) + 200(x41 + s41 + e41 ) + 500(x61 + s61 + e61 ) + 400(x22 + s22 + e22 ) + 300(x32 + s32 + e32 ) + 500(x42 + s42 + e42 ) + 300(x62 + s62 + e62 ) + 700(x13 + s13 + e13 ) + 500(x23 + s23 + e23 ) + 200(x33 + s33 + e33 ) + 400(x53 + s53 + e53 ). e21 . Capacity constraints are easy to see. the variables x21 . Since this linear program should minimize busing cost. 11. x43 . . and e43 don't exist. Since a dash in Table 1 indicates an infeasible assignment. .xij sij eij
− The number of students in sixth grade from area i assigned to school j − The number of students in seventh grade from area i assigned to school j − The number of students in eighth grade from area i assigned to school j
where i = 1.2 Objective Function
Now we must look at the objective function. 2. These constraints are:
43
.2. One kind limits the capacity of each school. x52 . . e52 . there will be several "percentage constraints" limiting the percentages of students in each grade at each school. s21 . . so we will call them "capacity constraints. s52 . 3.3
Constraints
There are three kinds of constraints in this linear program. 2. Finally. 6 and j = 1." "Grade constraints" will state how many students in each area are in each grade. the objective function is the sum of how much is costs to bus each student.
that hold 20 students each. but it may actually be easier to make minor adjustments to the linear program itself. this time with the objective function coefficients of x61 and x62 increased by 10%. Now it must be determined whether or not increasing each school's capacity by multiples of 20 (at a cost of $2. The current cost coefficient of x63 is $0. Analyzing the shadow prices and AIs of these constraints would solve the problem. and p3 to represent the number of portables bought for schools 1. increasing the cost coefficient of x62 by 10% would not affect this basis. Defining new variables p1 . The current cost coefficient of x61 is $500 and its AI is $33. everything about the program stays the same except the capacity constraints and the objective function. if the cost of busing students in area 6 increased 10% ($50). This new result is shown in Table 13. Solving the problem again. Therefore. The objective function becomes:
minz
= 300x11 + 600x31 + 200x41 + 500x61 + 400x22 + 300x32 + 500x42 + 300x62 + 700x13 + 500x23 + 200x33 + 400x53 + 2500p1 + 2500p2 + 2500p3. x62 and x63 .the relevant information in the sensitivity report in Table 14 concerns the allowable increase (AI) of the objective function coefficient for variables x61 .4. The current cost coefficient of x62 is $300 and its AI is infinity. 2 and 3 respectively. this basis would not remain optimal.500 per year.33. Also. we obtain a new optimal solution for the school board. p2 .500 for each multiple) would increase or decrease busing costs. Therefore. This adjustment increases the objective function value by over $10. some basic variables have become nonbasic and vice versa. the only area 6 cost coefficient that would change the basis if it were increased is the cost coefficient of x61 . What are being changed here are the right-hand-side values of the capacity constraints. and therefore increasing this cost coefficient by any percentage would still yield $0.000.
The capacity constraints become:
50
. Therefore.2 Portable Classrooms
The next day the school board calls and says they can buy portable classrooms for $2. 11.
a type of linear programming in which the variables can only take on values of 0 or 1.11. That is. 0 if they are not. and students from areas 3
55
. The objective function for this problem is:
minz
= 135000b11 + 330000b31 + 70000b41 + 225000b61 + 240000b22 + 165000b32 + 175000b42 +135000b62 + 315000b13 + 3000000b23 + 110000b33 + 200000b53. we can abandon the percentage constraints. For this problem. They are:
450b11 + 450b12 + 450b13 600b22 + 600b23 550b31 + 550b32 + 550b33 350b41 + 350b42 500b51 + 500b53 450b61 + 450b62 + 450b63
= 450 = 600 = 550 = 350 = 500 = 450.5
Keeping Neighborhoods Together
Now the school board wants to prohibit the splitting of residential areas among multiple schools. This problem can be solved with bit integer programming.
The solution to this linear program is that students from areas 4 and 5 go to school 1. students from areas 1 and 2 go to school 2. In this problem. we can define the variables bij as 1 if students from area i are assigned to school j. each of the six areas must be assigned to a single school.
The school capacity constraints are:
450b11 + 550b31 + 350b41 + 500b51 + 450b61 450b12 + 550b22 + 350b32 + 500b42 + 450b62 450b13 + 550b23 + 350b33 + 500b53 + 450b63
< 900 < 1100 < 1000.
The next constraints ensure that each area is assigned to only school.
the new z -value is given by: znew = zold − SPi ∆bi . By inspecting the busing costs associated with each area and school. Using the Lindo software. Therefore. This problem concerns the shadow prices of the capacity constraints of the linear program. In linear programming problem.818 classrooms should be added to school 2 (or. For a minimization problem. as seen in Table 15. The result was that 1. where SPi is the shadow price of the ith constraint and ∆bi is the amount by which the right-hand side of the ith constraint changes. Each portable classrooms holds 20 students and costs $2. Shadow prices in the Lindo sensitivity analysis output screen are called "Dual Prices. this result seems reasonable.1 Shadow Prices
Now. and how many they should have.6. this procedure was actually quite simple." For a "greater than or equal to" constraint. the shadow price of the ith constraint is the amount by which the optimal z -value is improved if the righthand side of the ith constraint is increased by 1 (assuming the current basis remains optimal) [3]. 11. since increasing the right-hand side of this constraint will eliminate points from the feasible region. the new optimal solution can stay the same. The operations research team must decide if any of the schools should have these portable classrooms. Earlier. Definition 11. Similar reasoning shows that a "less than or equal to" constraint
56
. The school board would like to know how increasing these constraints by multiples of twenty would affect the optimal z -value. this was accomplished by adding a few new variables to the linear program and resolving.
11. the operations research team will verify this solution using something called a shadow price.1.500 per year.and 6 go to school 3.818).6
The Case Revisited
Recall that the school board has the option of adding portable classrooms to increase the capacity of one or more of the middle schools. these values are nonpositive. p2 = 1. or get worse.
177. If we divide this by 20. we could obtain more practical results. According to the equation in section 2. and therefore it is bad business sense to consider increasing its capacity.818) gives 553636. and an equality constraint's shadow price can be either positive. increasing the capacity of school 2 seems favorable to increasing the capacity of school 3. we can conclude that adding portables is a viable solution. We are acquainted with the theory behind linear programs and we know the basis tools used to 57
. Just from looking at these shadow prices. Since school 2 has a larger-valued shadow price. the new z -value is 555555. For the capacity constraints these values are 0. and 3 respectively.will have a nonnegative shadow price. the operations research team is concerned only with the shadow prices of the school capacity constraints.4. which is our final answer shown in Table 15. This school also is not filled to capacity in the current basis. but 2 portable classrooms would be a better answer because 2 is an integer. in the case study.44 for schools 1.5. 1. the result. Now we have a solid background in linear programming.78 · 36.42. the theory of binary integer programming would be helpful for understanding how we kept the neighborhoods together in Section 11. and 144. we reccommend studying integer programming. because the price of the portables hasn't yet been added. Since this answer is less than then answer to the original problem. 2. Adding (2500 · 1.2 The New Result
In this case. negative.36 = 548091. 1.6. 11. or zero [3]. For example. Also.818 portable classrooms is an unrelastic answer. this solution should be decimals as well for the sake of comparison.6 − 177.818.36. Since the solutions found last week are decimals and not integers. With this tool. it is clear that changing the capacity of school 1 will have no effect on the objective function. is the number of portable classrooms that can be added at school 2.
12
Conclusion
As a next step. This is precisely the number obtained in Table 15 for the variable p2 . Notice that the allowable increase of constraint 2 is 36.78. This isn't the same as the z -value found previously.
&
% )
$
# #
$
% "
&
'
(
Figure 12: The shaded region is the feasible region with b1 modified to equal 4. The dotted line is the isocost line for z = 4.
64
. The dash-dotted line is the original constraint 1 for b1 = 10. | 677.169 | 1 |
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Get to know Natural Science Introduction to the unit Each unit opens with dynamic images that illustrate the key themes of the unit. Presentation of contents The unit follows an easy-to-use format withSUMMER WORK AP SPANISH LANGUAGE & CULTURE Bienvenidos a la clase de Español AP! To truly learn a language you have to use it as much as possible. The AP Exam focuses on the four communication skills: speaking,
Math The Arrow Way Free PDF ebook Download: Math The Arrow Way Download or Read Online ebook solving math the arrow way in PDF Format From The Best User Guide Database 4 SYSTEMS OF LINEAR EQUATIONS ANDNorthwest College margarita.rodriguez@hccs.edu Spanish 1412 Please allow at least one full business day (Monday Friday) respond to e-mail and/or voice messages. for the instructor to EO2 primary mode for
Welcome! You are about to embark on an exciting adventure: learning Spanish to communicate better with your Hispanic patients and clients. You will spend the first class getting acquainted with the course
Spanish 003 Syllabus Spring 2016 Class grading scale, assessment categories and percentages can be found in the Spanish 2 & 3 policies document. It is your responsibility to read and understand the course
Welcome! You are about to embark on an exciting adventure: learning Spanish to communicate better with your Hispanic patients and clients. The first week you will spend your time getting acquainted with
Welcome to Descubre el español con Santillana! This year, your child will be learning Spanish by exploring the culture of eight Spanish-speaking countries. Please join us as we travel through each of the
2016-2017 Class of 2019 Book List and Purchasing Info If you are new to Esco, you will have to create an account before proceeding. Procedures for buying and selling books Log on to escobooks.com. Esco
Spanish GCSE Student Guide Is this the right subject for me? If you enjoy meeting and talking to people from other countries, finding out about their cultures and learning how language works, then studying
Rationale for Summer Assignments Students enrolled in Advanced Placement Spanish are required to complete the summer assignment prior to the start of class in September. The assignments integrate the use
YEAR 7 SPANISH PERIODS: PREP: 3 x 35 mins per week. 1 x 30 minutes per week. The pupils start Spanish in Year 7 and continue on into Year 8. At the end of Year 6, pupils are given the choice of studying
.2 Methods of Addition Objectives Relate addition stories to number bonds. Write two addition facts for a given number bond. Solve picture problems using addition. Learn addition facts through, and the
WINONA STATE UNIVERSITY REQUIRED CHECKLIST FOR ALL CURRICULAR PROPOSALS Course or Program SPAN 201 This checklist enables A2C2 representatives to endorse that their departments have accurately followed
Course Description This course is designed for students who have mastered the fundamentals of the Spanish language and who are now ready to apply those skills in all subject areas in the target language. | 677.169 | 1 |
Omtale
Higher GCSE Mathematics for CCEA Practice Book
Help students build their confidence with hundreds of practice questions covering everything on the course, guiding them towards their target grades. This practice book offers complete support for the Higher tier of the CCEA GCSE Mathematices specification. It has been written to cover all the units of the specification and to accompany the endorsed textbook. It has been written by an experienced teacher and examiner. - Enable students to practice and consolidate what they have learnt throughout the course with hundreds of questions - Help develop students'skills and prepare them effectively for their exams - Use alongside the CCEA endorsed textbook for extensive coverage of the specification | 677.169 | 1 |
Alg 2 -- Complex Numbers Scavenger Hunt
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I use relays to review a lot of different subjects in my classes. This one reviews Section 5.6 of the Prentice Hall CA Algebra 2 textbook: Complex Numbers -- simplifying, adding, subtracting, multiplying, and dividing with answer key!
I have also used this in my PreCalculus class. Rather than giving a lecture to review complex numbers, I give them this to work and remember previous skills on their own. They usually remember them better this way!
Common Core Standards:
Perform arithmetic operations with complex numbers.
N-CN 1. Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
N-CN 2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex | 677.169 | 1 |
Product Description:
Compiled by a professor of mathematical finance, Marcello Minenna, this extensive manual will enable the reader to: understand the models adopted by the financial markets; evaluate the practical application of these models; implement the models presented; and develop the skills required to independently tailor new models to their own specific needs. As well as a reference guide for advanced practitioners and academics, this manual is also designed for beginners and intermediate users to quickly grasp the complexities of quantitative finance. This self-contained and methodical guide will allow the reader to fully grasp the mathematics underlying the pricing of derivatives. And most importantly, will empower them to put their quantitative skills into practice.
REVIEWS for Guide to Quantitative | 677.169 | 1 |
Music subjects covered in college algebra
A major component of any college curriculum in music is a course For more detail on the course topics covered in Music Theory, see the Course Description.
Specific Subject Matter Authorizations that meet this requirement. NCLB compliance Industrial Arts, Music, Physical. Education . A course in each is not needed but the subjects must be covered. o A course in an individual completes a college algebra course; the "advanced" course may be another level of algebra.
Build your own knowledge of, and confidence in, math in practical ways that relate directly to the world of music. This action-oriented course is designed. | 677.169 | 1 |
Elementary Technical Mathematics respected, extremely user-friendly text emphasizes essential math skills and consistently relates math to practical applications so students can see how the math will help them on the job. Visual images are used to engage students and assistMore...
This respected, extremely user-friendly text emphasizes essential math skills and consistently relates math to practical applications so students can see how the math will help them on the job. Visual images are used to engage students and assist with problem solving.
Basic Concepts
p. 1
Review of Operations with Whole Numbers
p. 2
Review of Operations with Fractions
p. 26
Review of Operations with Decimal Fractions and Percent
p. 54
Chapter 1 Accent on Teamwork
p. 89
Chapter 1 Review
p. 90
Chapter 1 Test
p. 92
Signed Numbers and Powers of 10
p. 94
Addition of Signed Numbers
p. 95
Subtraction of Signed Numbers
p. 98
Multiplication and Division of Signed Numbers
p. 100
Signed Fractions
p. 102
Powers of 10
p. 105
Scientific Notation
p. 108
Chapter 2 Accent on Teamwork
p. 113
Chapter 2 Review
p. 113
Chapter 2 Test
p. 114
The Metric System
p. 116
Introduction to the Metric System
p. 117
Length
p. 120
Mass and Weight
p. 123
Volume and Area
p. 125
Time, Current, and Other Units
p. 128
Temperature
p. 130
Metric and English Conversion
p. 133
Chapter 3 Accent on Teamwork
p. 137
Chapter 3 Review
p. 137
Chapter 3 Test
p. 138
Cumulative Review Chapters 1-3
p. 139
Measurement
p. 140
Approximate Numbers and Accuracy
p. 141
Precision and Greatest Possible Error
p. 144
The Vernier Caliper
p. 147
The Micrometer Caliper
p. 153
Addition and Subtraction of Measurements
p. 161
Multiplication and Division of Measurements
p. 165
Relative Error and Percent of Error
p. 168
Color Code of Electrical Resistors
p. 172
Reading Scales
p. 177
Chapter 4 Accent on Teamwork
p. 188
Chapter 4 Review
p. 188
Chapter 4 Test
p. 190
Polynomials: An Introduction to Algebra
p. 192
Fundamental Operations
p. 193
Simplifying Algebraic Expressions
p. 195
Addition and Subtraction of Polynomials
p. 198
Multiplication of Monomials
p. 202
Multiplication of Polynomials
p. 204
Division by a Monomial
p. 207
Division by a Polynomial
p. 208
Chapter 5 Accent on Teamwork
p. 211
Chapter 5 Review
p. 211
Chapter 5 Test
p. 212
Equations and Formulas
p. 213
Equations
p. 214
Equations with Variables in Both Members
p. 219
Equations with Parentheses
p. 221
Equations with Fractions
p. 224
Translating Words into Algebraic Symbols
p. 228
Applications Involving Equations
p. 229
Formulas
p. 234
Substituting Data into Formulas
p. 237
Reciprocal Formulas Using a Calculator
p. 240
Chapter 6 Accent on Teamwork
p. 243
Chapter 6 Review
p. 243
Chapter 6 Test
p. 244
Cumulative Review Chapters 1-6
p. 245
Ratio and Proportion
p. 246
Ratio
p. 247
Proportion
p. 251
Direct Variation
p. 256
Inverse Variation
p. 262
Chapter 7 Accent on Teamwork
p. 265
Chapter 7 Review
p. 265
Chapter 7 Test
p. 266
Graphing Linear Equations
p. 268
Linear Equations with Two Variables
p. 269
Graphing Linear Equations
p. 275
The Slope of a Line
p. 281
The Equation of a Line
p. 287
Chapter 8 Accent on Teamwork
p. 292
Chapter 8 Review
p. 292
Chapter 8 Test
p. 293
Systems of Linear Equations
p. 294
Solving Pairs of Linear Equations by Graphing
p. 295
Solving Pairs of Linear Equations by Addition
p. 301
Solving Pairs of Linear Equations by Substitution
p. 305
Applications Involving Pairs of Linear Equations
p. 307
Chapter 9 Accent on Teamwork
p. 313
Chapter 9 Review
p. 314
Chapter 9 Test
p. 314
Cumulative Review Chapters 1-9
p. 315
Factoring Algebraic Expressions
p. 316
Finding Monomial Factors
p. 317
Finding the Product of Two Binomials Mentally
p. 318
Finding Binomial Factors
p. 321
Special Products
p. 324
Finding Factors of Special Products
p. 325
Factoring General Trinomials
p. 328
Chapter 10 Review
p. 330
Chapter 10 Test
p. 331
Quadratic Equations
p. 332
Solving Quadratic Equations by Factoring
p. 333
The Quadratic Formula
p. 335
Graphs of Quadratic Equations
p. 339
Imaginary Numbers
p. 344
Chapter 11 Accent on Teamwork
p. 347
Chapter 11 Review
p. 347
Chapter 11 Test
p. 348
Geometry
p. 349
Angles and Polygons
p. 350
Quadrilaterals
p. 356
Triangles
p. 361
Similar Polygons
p. 368
Circles
p. 372
Radian Measure
p. 379
Prisms
p. 384
Cylinders
p. 389
Pyramids and Cones
p. 394
Spheres
p. 400
Chapter 12 Accent on Teamwork
p. 403
Chapter 12 Review
p. 404
Chapter 12 Test
p. 406
Cumulative Review Chapters 1-12
p. 408
Right Triangle Trigonometry
p. 410
Trigonometric Ratios
p. 411
Using Trigonometric Ratios to Find Angles
p. 415
Using Trigonometric Ratios to Find Sides
p. 418
Solving Right Triangles
p. 419
Applications Involving Trigonometric Ratios
p. 421
Chapter 13 Accent on Teamwork
p. 428
Chapter 13 Review
p. 428
Chapter 13 Test
p. 429
Trigonometry with Any Angle
p. 431
Sine and Cosine Graphs
p. 432
Period and Phase Shift
p. 438
Solving Oblique Triangles: Law of Sines
p. 441
Law of Sines: The Ambiguous Case
p. 444
Solving Oblique Triangles: Law of Cosines
p. 449
Chapter 14 Accent on Teamwork
p. 454
Chapter 14 Review
p. 454
Chapter 14 Test
p. 455
Basic Statistics
p. 456
Bar Graphs
p. 457
Circle Graphs
p. 459
Line Graphs
p. 462
Other Graphs
p. 465
Mean Measurement
p. 467
Other Average Measurements and Percentiles
p. 468
Grouped Data
p. 471
Variance and Standard Deviation
p. 477
Statistical Process Control
p. 480
Probability
p. 484
Independent Events
p. 486
Chapter 15 Accent on Teamwork
p. 487
Chapter 15 Review
p. 488
Chapter 15 Test
p. 489
Cumulative Review Chapters 1-15
p. 490
Tables
p. 492
Formulas From Geometry
p. 492
Electrical Symbols
p. 492
Answers to Odd-Numbered Exercises and All Chapter Review and Cumulative Review Exercises | 677.169 | 1 |
Solving Equations Using Distributive Property Lesson 1 of 2
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2.45 MB | 7 pages
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This is a Pre-Algebra Common Core Lesson on Solving Linear Equations by Using the Distributive Property. After a few teacher led examples, students will work independently or with a partner to practice solving equations | 677.169 | 1 |
Graphs & Digraphs, Fifth Edition
Continuing to provide a carefully written, thorough introduction, Graphs & Digraphs, Fifth Edition expertly describes the concepts, theorems, history, and applications of graph theory. Nearly 50 percent longer than its bestselling predecessor, this edition reorganizes the material and presents many new topics. New to the Fifth Edition New or expanded coverage of graph minors, perfect graphs, chromatic polynomials, nowhere-zero flows, flows in networks, degree sequences, toughness, list colorings, and list edge colorings New examples, figures, and applications to illustrate concepts and theorems Expanded historical discussions of well-known mathematicians and problems More than 300 new exercises, along with hints and solutions to odd-numbered exercises at the back of the book Reorganization of sections into subsections to make the material easier to read Bolded definitions of terms, making them easier to locate Despite a field that has evolved over the years, this student-friendly, classroom-tested text remains the consummate introduction to graph theory. It explores the subject's fascinating history and presents a host of interesting problems and diverse | 677.169 | 1 |
Further Mathematics
Why study A level Further Mathematics?
Just as languages provide the building blocks and rules we need to communicate, mathematics uses its own language, made up of numbers, symbols and formulas, to explore the rules we need to measure or identify essential problems like distance, speed, time, space, change, force and quantities.
Studying Mathematics helps us find patterns and structure in our lives. Practically, Mathematics helps us put a price on things, create graphics, build websites, build skyscrapers and generally understand how things work or predict how they might change over time and under different conditions.
Mathematics and Further Mathematics are versatile qualifications, well-respected by employers. Careers for men and women with good mathematics skills and qualifications are not only well paid, but they are also often interesting and rewarding. People who have studied mathematics are in the fortunate position of having an excellent choice of career.
Studying A level Further Mathematics is likely to improve your grade in A level Mathematics. The extra time, additional practice, further consolidation and development of techniques contribute to improved results in A level Mathematics.
You cannot study Further Mathematics without studying Mathematics also.
What are the careers or further education that this course be suitable for?
Mathematics and Further Mathematics help support the study of subjects like Physics, Chemistry, Engineering, IT, Economics, Business and Biology which can also help with your Mathematics revision. But studying Further Mathematics is essential if you want to study Mathematics or Engineering at university. It is also really useful if you intend to study a scientific engineering degree. (e.g. Chemical Engineering)
The single 'A' level Further Mathematics course consists of a mixture Pure and applied elements. There are certain further pure mathematics topics that are compulsory. However, students must also study 2 out of a selection of additional further pure and applied topics (made up of Statistics, Mechanics and Decision mathematics). The decision on which 2 topics will be made in conjunction with the class at an appropriate time during their studies.
The new A level contains a problem solving element and students will be expected to apply the content they learn to harder more unfamiliar situations.
Edexcel Further Mathematics is a linear course with terminal examinations either at the end of the course. | 677.169 | 1 |
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0.75 MB | 14 pages
PRODUCT DESCRIPTION
I created this lesson to have students derive the logarithmic properties by starting with the conversion between logarithmic form and exponential form. Properties of equality are used to arrive at the properties. Each property and the change of base formula have their own set of problems. Then there is a practice worksheet that has all of the properties previously discovered as well as change of base formula problems. The worksheet ends with an error analysis | 677.169 | 1 |
Featured
Lots of people find maths hard. Others find it stressful – and hard. Yet lots of people need to have a basic grasp of maths concepts to be able to pursue the career they want.
So at some point, they reluctantly go back and re-learn the maths they forgot (or only partially grasped) back at school. If that sounds like you, you've come to the right place. This Maths MOOC is a free, online course designed to run over 4 weeks, but currently open for individual self-paced learning.
We've worked really hard to make it as painless as possible. The mini-lectures are all nice little video clips. You can watch them as many times as you need. And there are some self-tests you can do to see if you get it. And nobody will know if you got it right, except you. Having said that, you may prefer to sign up with a buddy, and learn together. If you like to chat, we have a facebook page where you can discuss the MOOC in particular, or maths in general. It's linked from the Assignment pages.
And to begin, let's spend a few minutes with some crazy people who find maths fun. When people say: maths, shmaths – who needs it? I like to show them this 45 second clip. Now – the question is, what kind of maths could help you work out where to place the swimming pool? Feel free to use the comments facility to tell us what you think. (Note: Click on the small pale grey speech bubble icon at the top right of this post to make a comment. The number tells you how many comments have been made.)
How to Navigate this MOOC
There are 4 modules in this MOOC, and you get to them by clicking on the links in the blue bar under the picture. Each module contains an overview/theory refresher, worked examples to try, and then a quiz. Each module ends with an optional assignment, that will really cap off your learning. It's designed to be done in a sequence, but if you find you know that stuff already, then skip ahead. | 677.169 | 1 |
eBook Details:
eBook Description:
Introduction to Scientific Computing and Data Analysis (Texts in Computational Science and Engineering)
This textbook provides and introduction to numerical computing and its applications in science and engineering. The topics covered include those usually found in an introductory course, as well as those that arise in data analysis. This includes optimization and regression based methods using a singular value decomposition. The emphasis is on problem solving, and there are numerous exercises throughout the text concerning applications in engineering and science. The essential role of the mathematical theory underlying the methods is also considered, both for understanding how the method works, as well as how the error in the computation depends on the method being used. The MATLAB codes used to produce most of the figures and data tables in the text are available on the author's website and SpringerLink. CLICK TO DOWNLOAD
eBook Details:
eBook Description:
Computer Arithmetic: Volume III
Computer Arithmetic Volume III is a compilation of key papers in computer arithmetic on floating-point arithmetic and design. The intent is to show progress, evolution, and novelty in the area of floating-point arithmetic. This field has made extraordinary progress since the initial software routines on mainframe computers have evolved into hardware implementations in processors spanning a wide range of performance. Nevertheless, these papers pave the way to the understanding of modern day processors design where computer arithmetic are supported by floating-point units. CLICK TO DOWNLOAD
eBook Details:
eBook Description:
Computer Arithmetic: Volume II
This is the new edition of the classic book Computer Arithmetic in three volumes published originally in 1990 by IEEE Computer Society Press. As in the original, the book contains many classic papers treating advanced concepts in computer arithmetic, which is very suitable as stand-alone textbooks or complementary materials to textbooks on computer arithmetic for graduate students and research professionals interested in the field. CLICK TO DOWNLOAD
eBook Details:
eBook Description:
Computer Arithmetic: Volume I
The book provides many of the basic papers in computer arithmetic. These papers describe the concepts and basic operations (in the words of the original developers) that would be useful to the designers of computers and embedded systems. Although the main focus is on the basic operations of addition, multiplication and division, advanced concepts such as logarithmic arithmetic and the calculations of elementary functions are also covered. CLICK TO DOWNLOAD
eBook Details:
eBook Description:
Computer Fundamentals and Programming in C
It provides a thorough understanding of the subject and its applications. The book begins with an introduction to the basic features of a digital computer, number systems and binary arithmetic, Boolean algebra and logic gates, software, operating systems, and the internet. A major part of the book provides a detailed coverage of programming in C. It discusses the primary functions of compilers, linkers, and loaders, and provides an exhaustive coverage of concepts such as data types, control statements, arrays, strings, functions, pointers, structures, file systems, and command-line arguments. Case studies demonstrating the use of C in solving mathematical as well as real-life problems have also been presented. This edition also highlights C99 features wherever relevant in the text. CLICK TO DOWNLOAD | 677.169 | 1 |
A Level Maths 2000/01 ultimately take your fancy. But the value of having a good Maths pass on your CV is something you really can count on. At ICS, we know that the brightest futures are made by getting the balance right between life experience and formal qualifications. Which is why all ICS courses are designed to fit in with your life without taking it over. When you choose to prepare for your ''A'' Levels with ICS you can be sure that we won''t hold you back when you''re trying to get ahead. The ICS ''A'' Level Courses have been written specifically to dovetail with the syllabuses of specific examining bodies. This means that when you prepare with ICS, you have the added assurance of knowing that everything you are doing is entirely focussed on your main objective: achieve the best results you can. Equally important, by breaking the material down into logical sections, your tutors have ensured that you won''t feel overwhelmed, even if you are working as well as studying. Key Benefits: Some of the key benefits of the ICS ''A'' Level Maths Course include: ? Start studying whenever you like - we don''t lock you into inflexible term times ? Work when you want, where you want, at the pace that suits you best ? Logical, modular approach makes it easier to learn ? Expert learning materials take you through the course step-by-step ? Flexible access to your tutors to support you throughout the course ? Carefully tailored tests give you feedback on how you are progressing ? Ideal way of acquiring essential ''A'' LEVEL qualifications ? ICS is part of the world''s largest educational publishing groups and we offer the best range of courses available Topics: The ICS ''A'' Level Maths Course consists of six modular units, which are logically structured to make learning easier. When you have successfully completed the course, you will have mastered the following topics and be fully prepared for your exams: ''A'' LEVEL MATHS AS Unit 1 - Pure Maths 1 AS Unit 2 - Pure Maths 2AS Unit 3 - Mechanics 1 A2 Unit 4 - Pure Maths 3A2 Unit 5 - Mechanics 2A2 Unit 6 - Mechanics 3 Requirements: Candidates should normally possess at least a Grade C at GCSE. Careers in the Field: Whether your goal is to go on to Higher Education or you are keen to make a start in your chosen career right now, the effort you invest into preparing for your ''A'' Levels today will go along way towards the choices available to you in the future. A good result in ''A'' Level Maths A good result in GCSE Mathematics Higher assures employers of your ability to manipulate figures, and use abstract reasoning and logic. Because ICS ''A'' Level Maths requires you to apply skills which are versatile and invaluable across any number of career paths, it is difficult to put an exact monetary value on the added earnings potential these skills could offer you. However, it is worth remembering that the average annual earnings of those who leave school with qualifications in the UK is ?xx more than those who leave with none. When it comes to ''A'' Levels, why not let the ICS open learning approach open some doors for you? Qualification: A Level Hours of Study: See External Exam Info External Exam Information: It may be possible to sit these exams if you live overseas. Please check with your local BRITISH COUNCIL. All exams are also held in England, at a wide variety of regional examination centres. All students must make their own arr ultimat | 677.169 | 1 |
The objective of this book is to introduce new researchers to the rich dynamical system of water waves, and to show how (some) abstract mathematical concepts can be applied fruitfully in a practical physical problem and to make the connection between theory and experiment. It provides a coherent set of lectures on the current status of water wave theory,... more...
H. Hermes: Basic notions and applications of the theory of decidability.- D. Kurepa: On several continuum hypotheses.- A. Mostowski: Models of set theory.- A. Robinson: Problems and methods of model theory.- S. Sochor, B. Balcar: The general theory of semisets. Syntactic models of the set theory. more...
How does mathematics enable us to send pictures from space back to Earth? Where does the bell-shaped curve come from? Why do you need only 23 people in a room for a 50/50 chance of two of them sharing the same birthday? In Strange Curves, Counting Rabbits, and Other Mathematical Explorations , Keith Ball highlights how ideas, mostly from pure math,... more...
In Classical Mathematical Logic , Richard L. Epstein relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. The book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field... more...
This comprehensive book provides an adequate framework to establish various calculi of logical inference. Being an 'enriched' system of natural deduction, it helps to formulate logical calculi in an operational manner. By uncovering a certain harmony between a functional calculus on the labels and a logical calculus on the formulas, it allows... more...
The book presents methods of approximate solution of the basic problem of elasticity for special types of solids. Engineers can apply the approximate methods (Finite Element Method, Boundary Element Method) to solve the problems but the application of these methods may not be correct for solids with the certain singularities or asymmetrical boundary... more... | 677.169 | 1 |
Product Description:
This innovative text features a graphing calculator approach, incorporating real-life applications and such technology as graphing utilities and Excel spreadsheets to help students learn mathematical skills that they will use in their lives and careers. The texts overall goal is to improve learning of basic calculus concepts by involving students with new material in a way that is different from traditional practice. The development of conceptual understanding coupled with a commitment to make calculus meaningful to the student are guiding forces. Targeted toward students majoring in liberal arts, economics, business, management, and the life and social sciences, the text's focus on technology along with its use of real data and situations make it a sound choice to help you develop an intuitive, practical understanding of concepts.
REVIEWS for Calculus Concepts | 677.169 | 1 |
Reason and Wonder
Where's the emphasis? Right where it belongs.
One of the things I love about Desmos is that it allows students and teachers to keep the focus where it should be. Working on a linear approximation problem in Calculus? It's easy to get caught up in algebraic and numerical details and lose sight of the big (somewhat amazing) picture: We can approximate a crazy curve with a simple line! (Provided we stay in the neighborhood, of course.)
And if you make a mistake, it can be an absolute bear to track it down. Is your issue differentiation? Evaluating a function? Or is your weak spot related to what's happening visually in linear approximation?
Here's a problem from last year's AP Calculus review workbook:
There's a lot of great work on the page.
Take the derivative? Check.
Find the slope of the tangent line? Check?
Find the equation of the tangent line? Check.
But that's where things fall apart.
Now, imagine you're a calculus student. You've been hammering away at this thing for several minutes. Maybe you don't even remember what the problem's asking for in the first place. You get to this point, you scan the original problem, see an input value of 4.2, and presto-whammo, plug it in and out comes 0.4. "Great news, everyone! That's on the list! Well done, folks. On to the next problem!"
Only, that answer is entirely wrong. So how do you debrief this problem with a student, small group, or class, so they can see the source of the error? For problems with even a sliver of something graphical, Desmos has become my go-to tool for helping students find and fix their errors.
Here's what I built with a pair of students last year (with a link to the live graph here):
Students can't use Desmos on the AP exam (for now, anyway), so I'm not trying to permanently sidestep what they ultimately must be able to do sans technology (or with a device from that "other" graphing calculator company). But what we can do in class with Desmos is build a better visual/conceptual sense of what's happening in this problem so they'll be more prepared for something similar in the future.
Here's a short list of what this Desmos graph did for us in this scenario:
We offloaded the algebraic and numerical work of finding the derivative (and evaluating it for a particular x-value) and built the tangent line in a matter of seconds (rather than minutes). Mentally, we're still fresh, and ready to focus on what the problem is really asking us to do: compare the function and the tangent line.
We gave things specific names so we could call on them in our time of need. Okay, that may sound a little dramatic. But think about why we even bother with function notation. Why give a function a name? Well, why did your parents give you a name? So they could call on you! ("Alfred! Get down here and pick up your comic books!") So then, why do we give functions names? Because function notation is on the Chapter 8 test in Algebra 2? No! We give functions names so we can call on them. So they can do our bidding. If you don't name it, it's difficult to put a function to work for you. Give it a name? Now our wish is its command. (It sounds a little bit like we're going to take over the world, with math as our trusty sidekick.)
We gave things specific names so we could keep clear in our own minds the various moving pieces in the problem. There's a function. We called that f. There's a tangent line. We called that l (or t, or whatever). As we approach the end of the problem, and we start looking for the error, it's easier to avoid simply evaluating f(4.2) or l(4.2), because we know we're dealing with both f and and l. (How could we forget?! We named them! We practically gave birth to them.)
We visualized the error with a beautiful little orange bar, and in doing so imprinted on our minds (for future problems) what linear approximation error looks like.
We dropped a slider in so we could answer a hundred related problems in a matter of seconds, further clarifying for the confused student (or teacher; I was terrified of linear approximation my first two years of teaching Calculus) what the problem is really about, and all with a nifty, dynamic burst of compare-and-contrast.
Do you need Desmos to teach this stuff? Maybe not. But given the option, I'll use it every time. Last year in my classroom, we got a lot more out of a five minute Desmos-supported conversation than we did in a whole day of business-as-usual notes and examples. | 677.169 | 1 |
Multivariable Calculus with Theory
This course is a continuation of 18.014. It covers the same material as 18.02 (Multivariable Calculus), but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral calculus | 677.169 | 1 |
Algebra: Advanced Algebra with Financial Applications
ISBN :
9781285444857
Publisher :
South-Western College Publishing
Author(s) :
Robert K. Gerver
Publication Date :
1 Jan 2014
Edition :
1st edition
Overview
By combining algebraic and graphical approaches with practical business and personal finance applications, South-Western's FINANCIAL ALGEBRA, motivates high school students to explore algebraic thinking patterns and functions in a financial context. FINANCIAL ALGEBRA will help your students achieve success by offering an applications based learning approach incorporating Algebra I, Algebra II, and Geometry topics. Authors Robert Gerver and Richard Sgroi have spent their 25+ year-careers teaching students of all ability levels and they have found the most success when math is connected to the real world. FINANCIAL ALGEBRA encourages students to be actively involved in applying mathematical ideas to their everyday live - credit, banking insurance, the stock market, independent living and more! | 677.169 | 1 |
Math subjects to know before college
This study guide provides practice questions for all 33 CLEP® exams. Questions on the College Mathematics examination require candidates to demonstrate the calculator and become familiar with its functionality prior to taking the exam.
Your high school student is on track to complete the math classes required by your state What do students need to know to succeed in college math?.
Do you need trig or calculus to impress colleges? Choosing which math classes to study can be one of the most challenging parts of or a similar course if they need to strengthen their math skills before taking Algebra 1.
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Math subjects to know before college
Get Vox in your inbox. ACT Vocabulary You Must Know. Cover Letter for Internship Sample. Samantha Brody is a professional tutor and contributing writer for Varsity Tutors. I like to learn Statistics.
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Try not to worry about trick questions or material beyond the norm. I teach mostly economics, political science, and international development. While it can vary between students, in general, students not planning to major in a STEM field can take either AP Statistics or an AP Calculus course if they choose to take an AP math class. My hope is this applies to students of every stripe. In high school, your math classes may have had a certain sense of familiarity — each day, a new lesson was taught, and homework was assigned that evening to help you review the concepts. Do you have a story to share?
Math subjects to know before college - Detector free
Come to think of it, this is not a bad rule for life after college, too. I pick up books on unusual people or places. Some people are lucky on their first try. Skip to content Home About.. Certain college experiences are the kind you cannot prepare for, and you instead learn to adapt as you live through them. Interpretation, representation, and evaluation of functions: numerical, graphical, symbolic, and descriptive methods. Please enter a valid email address. | 677.169 | 1 |
operation matrix - always had trouble with those difficult matrices sums?
here's help-operation matrix! that performs matrix operations for you
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Similar Applications:
Graphviewer scientific equation solver You can calculate the function inflection points
Multiple functions can be plotted in different colors at once
The intersection points of the different equations are calculated and can be shownDec2Bin Converter binary number to decimal;
Converter decimal number to binary;BillSplitter Avoid dealing with percentages and divisions when your brain cells are barely adding up | 677.169 | 1 |
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Show More Two major causes of error patterns - over-generalizing and over-specializing - are discussed. Simple and well organized, this book explains common errors students make in computation in every math operation and with whole numbers, rational numbers, geometry and algebra. This book is an ideal resource for teachers of mathematics in education or special education at the elementary or middle school | 677.169 | 1 |
Overview
About The Product
Larson's COLLEGE ALGEBRA is known for delivering sound, consistently structured explanations and exercises of mathematical concepts to prepare students for further study in mathematics. With the ninth edition, Larson continues to revolutionize the way students learn material by incorporating more real-world applications, ongoing review, and innovative technology. "How Do You See It?" exercises give students practice applying the concepts, and new "Summarize" features, "Checkpoint" problems, and a Companion Website reinforce understanding of the skill sets to help students better prepare for tests. Enhanced WebAssign® features fully integrated content from the text, with the addition of end-of-section problems and chapter tests, as well as problem-specific videos, animations, and lecture videos.
Features
"Side-by-Side Examples" help students see not only that a problem can be solved in more than one way but also how different methods--algebraically, graphically, and numerically--yield the same result. The side-by-side format also addresses many different learning styles.
"Algebra Helps" direct students to sections of the textbook where they can review algebra skills needed to master the current topic.
Technology features offer suggestions for effectively using tools such as calculators, graphing calculators, and spreadsheet programs to deepen student understanding of concepts, ease lengthy calculations, and provide alternate solution methods for verifying answers obtained by hand.
"Algebra of Calculus" examples and exercises throughout the text emphasize various algebraic techniques used in calculus.
Vocabulary exercises at the beginning of the exercise set for each section help students review previously learned vocabulary terms necessary to solve the section exercises.
Student projects in various sections and online feature in-depth applied exercises with large, real-life data sets, where students create or analyze models.
Chapter summaries include explanations and examples of the objectives taught in the chapter.
Larson's COLLEGE ALGEBRA is ideal for a one-semester course.
About the Contributor
AUTHOR
Ron Larson
Dr. Ron Larson is a professor of mathematics at The Pennsylvania State University, where he has taught since 1970. He received his Ph.D. in mathematics from the University of Colorado and is considered the pioneer of using multimedia to enhance the learning of mathematics, having authored over 30 software titles since 1990. Dr. Larson conducts numerous seminars and in-service workshops for math educators around the country about using computer technology as an instructional tool and motivational aid. He is the recipient of the 2014 William Holmes McGuffey Longevity Award for CALCULUS: EARLY TRANSCENDENTAL FUNCTIONS, the 2014 Text and Academic Authors Association TEXTY Award for PRECALCULUS, the 2012 William Holmes McGuffey Longevity Award for CALCULUS: AN APPLIED APPROACH, and the 1996 Text and Academic Authors Association TEXTY Award for INTERACTIVE CALCULUS (a complete text on CD-ROM that was the first mainstream college textbook to be offered on the Internet). Dr. Larson authors numerous textbooks including the best-selling Calculus series published by Cengage Learning.
New to this Edition
Fresh chapter openers highlight real-life problems that connect to the examples and exercises presented in the following chapter.
Innovative "Summarize" features help students organize the lesson's key concepts into a concise summary, providing a valuable study tool at the end of each section.
Unprecedented "How Do You See It?" exercises in every section present real-life problems that students solve by visual inspection using the concepts in the lesson.
Original "Checkpoint" problems encourage immediate practice and check students' understanding of the concepts in the paired example.
New Series Companion Website features chapter projects, data tables, assessments, study tools, video solutions, and more to support students outside the text.
Enhanced "Exercise Sets" deliver more rigor, relevancy, and coverage than ever before and feature multi-step, real-life exercises to reinforce problem-solving skills and mastery of concepts.
Expanded "Section Objectives" offer students the opportunity to preview what will be presented in the upcoming section.
Enriched remarks reinforce or expand on concepts helping students learn how to study mathematics, avoid common errors, address special cases, or show alternative or additional steps to a solution of an example.
Up-to-the-minute homework support through CalcChat.com, an independent website, provides students with free solutions to all odd-numbered problems in the text.
Components
Teacher Components 100.00
Student Supplements 100.00
Alternate Formats
Choose the Format that Best Fits Your Student's Budget and
Course Goals | 677.169 | 1 |
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
4.31 MB | 63 pages
PRODUCT DESCRIPTION
Introduction to Linear Equations - Use as a refresher or beginning of a unit! This detailed four day unit plan introduces linear equations by starting with a discussion of function machines, and it ends with students graphing linear equations using self-selected inputs and outputs.
Learning Target: I can graph linear equations using self-selected inputs and outputs.
Day 1: Discussion of Equal Sign, Function Machines
Day 2: Finding Outputs from Inputs (T-Charts Step by Step)
Day 3/Day 4: What exactly is a linear equation? How do I graph a linear equation using self-selected inputs and outputs?
Lesson Sequence:
This product can be used by itself as a stand alone lesson, but it is also the first lesson in a linear equations unit created by the Ace Teachers. If you are interested, you can check out the other available lessons below | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Just the critical concepts you need to score high in Algebra II
Algebra II Essentials For Dummies sticks to the point, with concise explanations of critical concepts taught in a typical Algebra II course. It's perfect for cramming, for homework help, or as a reference for parents helping students prepare for an exam.
Play by the rules — get the lowdown on algebraic properties, exponential rules, and factoring techniques
Be rational — follow easy-to-grasp instructions for working with rational and radical equations, from dealing with negative exponents to fiddling with fractional exponents
Know your functions — discover how to use exponential and logarithmic functions to solve algebraic problems
Synopsis
A new series extension of the For Dummies brand, The Essentials For Dummies will appeal to the huge number of readers who seek out focused, short form consumer reference books at the $9.99 price point. Positioned for students (and parents) who just want the key concepts and a few examples - without the review, ramp-up, and anecdotal content - The Essentials For Dummies series is a perfect solution for exam-cramming, homework help, and reference.
Synopsis
Passing grades in two years of algebra courses are required for high school graduation. Algebra II Essentials For Dummies covers key ideas from typical second-year Algebra coursework to help students get up to speed. Free of ramp-up material, Algebra II Essentials For Dummies sticks to the point, with content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical Algebra II course, from polynomials, conics, and systems of equations to rational, exponential, and logarithmic functions
DAbout the Author
Mary Jane Sterling is professor of mathematics at Bradley University and author of several books, including Algebra II For Dummies and Algebra II Workbook For Dummies. | 677.169 | 1 |
Algebra 2 Thanksgiving-Themed Activities
PDF (Acrobat) Document File
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0.49 MB | 6 pages
PRODUCT DESCRIPTION
Get them solving a system of equations, absolute value inequalities, graphing quadratic equations and more with this fun Thanksgiving-themed Algebra 2 turkey activities. Includes two turkeys with seven questions each that the students solve. Also included is a vocabulary worksheet that the students have to decipher and unscramble the circled letters in each term.
It'll be a fun way for your students to review what they have learned thus far right before the long Thanksgiving weekend. They will cut and color their turkey.
Check out the preview file for more information | 677.169 | 1 |
726 Downloads
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0.38 MB | 8 pages
PRODUCT DESCRIPTION
This is notes, a worksheet and key for topics in the Secondary Math 3 Common Core. The topics covered here are the Remainder Theorem, Polynomial Long Division, and Synthetic Division. The notes are an outline with some examples of how to do problems that are on the worksheet. With the notes and your knowledge as a Secondary Math Teacher everything is ready to go for this lesson. CCSS: A.APR.2, A.APR | 677.169 | 1 |
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The College Readiness Math MOOC (Massive Open Online Course) is a self-paced online program designed to enhance your mathematics skills needed to be successful in college mathematics. To register for the current offering of the Math MOOC, please fill out the form below. You will receive your Math MOOC log-in information within 3 business days. If you have questions about the Math MOOC, email mathmooc@uwlax.edu. | 677.169 | 1 |
Best Books to Teach Yourself Calculus
TR Smith is a product designer and former teacher who uses math in her work every day.
Stewart Calculus is the standard text for college calculus, but not ideal for self-study.
Can I learn calculus on my own? What are the best calculus books for self-study?
Studying calculus on your own is doable, and there are many reasons why people may choose to study calculus outside the classroom. Some are home-schooled students, some are college students planning to take a class for which calculus is a pre-requisite, some have attempted a calculus class before, while others are continuing their education beyond college, or are just mathematically curious with no particular use for calculus. Whatever the reason, buying the right self-study textbook can make independent learning more enjoyable and effective.
Standard college calculus textbooks cost $100-$120 and present math concepts in a dry manner without much explanation, since it's assumed students have a teacher to guide them. But luckily there are many high-quality, thorough, well-written calculus books that won't break your budget. Here are reviews of three excellent calculus textbooks ideal for the self-directed student.
Thompson & Gardner: Calculus Made Easy
ISBN-10: 0312185480 ISBN-13: 978-0312185480
Calculus Made Easy, by Silvanus Thompson and Martin Gardner
Silvanus Thompson's classic calculus primer offers the self-study student the best of both worlds: mathematical rigor, and down-to-earth plain English explanations of complicated concepts thanks to Martin Gardner's modern editing. This book is ideal for people in engineering and sciences who want to learn calculus formally with out getting lost in the details of mathematical proofs.
Calculus Made Easy was first published in 1910, and the original text is now in the public domain and available for download as an ebook at Project Gutenberg. Nonetheless, the modern print edition from St. Martin's Press has valuable additions by Martin Gardner, one of the most renowned writers of popular math and science. The latest edition has introductory chapters covering algebra, functions, and limits, so it's more accessible to students with a weaker mathematical background.
Not only does the book cover the how of basic procedures -- taking derivatives, computing integrals -- but it also explains the why in a more straightforward manner than standard calculus textbooks, which emphasize formal mathematical rigor at the expense of clarity, or gloss over complex topics due to limited space. The proofs that are presented are more elegant and less technnical than in a traditional calculus textbook, and as a result the book is only 300 pages long.
Calculus Made Easy aims to give students an intuitive understanding of calculus they might have missed out on in a traditional classroom course. Martin Gardner also added an index of recreational calculus exercises as an appendix, so students can appreciate both the practicality and beauty of calculus in solving a wide variety of practical and theoretical problems.
Calculus For Dummies, by Mark Ryan
ISBN-10: 0764524984 ISBN-13: 978-0764524981
Calculus For Dummies, by Mark Ryan, from the "For Dummies" Series
The "For Dummies" math and science guides always live up to their name and break down complex topics in a way that even a self-proclaimed dummy can understand. Calculus For Dummies is no exception. If you're a student who just needs to learn how to do calculus, and you neither care why nor have much interest in formal proofs, this is the book for you.
Calculus For Dummies works like an instruction manual for solving typical problems in differential calculus and integral calculus, or Calc I and Calc II, as the courses are often named. This book lays out definitions of terms commonly used in calculus with clear and plain English, which will ease the minds of students who run the other way when they see technical math jargon. In fact, the book consciously avoids overuse of jargon.
All example problems are carefully worked out in step-by-step detail, rather than "left to the reader" as is the case with standard calculus texts. Calculus For Dummies is so friendly and gentle, students enrolled in regular classroom courses often buy this book to supplement to the more imposing Stewart Calculus textbook.
Mathematicians and scientists love Dover Publication's cheap reprints of classic textbooks, and many of the books in their catalogue are classics for good reason. Essential Calculus with Applications is lean with little fluff, appropriate for a student who appreciates mathematical rigor and formal proofs. Content-wise, it's nearly as dense as Stewart Calculus or any other standard classroom text, but only a fraction of the cost.
Originally published in 1977, Silverman's book covers the essential calculus techniques needed for engineering and science, and even includes some worked-out example problems. The section on differential calculus is thorough, while the chapter on integration has been pared down to highlight only the most important techniques for the most commonly encountered problems.
Essential Calculus with Applications also includes a section on solving certain types of differential equations, and touches on multivariable calculus, which is going a lot further than any other book in its price range. This book is great for a real self-starter, but maybe not so good for a student who needs more hand-holding.
Need a Calculator?
Most calculus students buy a graphing calculator for calculus since it can plot functions, compute derivatives and integrals, and solve equations. If you can do without the graphing feature, many scientific calculators can also compute integrals and derivatives. | 677.169 | 1 |
For those engineers and scientists who use computers to solve their problems only to discover new, subtle problems in their results, this book is a welcome quick guide to trouble-shooting. Offering practical advice on detecting and removing the insidious bugs that plague finite-precision calculations, real Computing outlines techniques for preserving significant figures, avoiding extraneous solutions (those ridiculous "answers" that turn up all too often), and finding efficient iterative processes for solving nonlinear equations.
Anyone who computes with real numbers (for example, floating-point numbers stored with limited precision) tends to pick up a few computing "tricks"--techniques that increase the frequency of useful answers. But where there might be ample guidance for a computor grappling with linear problems, there is little help for someone negotiating the nonlinear world--and it is this need that Forman Acton addresses. His book presents a wealth of examples and exercises (with answers) to help a reader develop problemformulating skills--thus learning to avoid the common pitfalls that software packages seldom detect. It presumes some experience with standard numerical methods--but for beginners in real computing, it will lend a touch of realism to topics often slighted in introductory texts.
REVIEWS for Real Computing Made Real | 677.169 | 1 |
Embedding Algebraic Thinking throughout the Mathematics Curriculum
G. Vennebush, Elizabeth Marquez, Joseph Larsen
The algebra that can be uncovered in many middle-grades mathematics tasks that, on first inspection, do not appear to be algebraic. Article discusses connections to the other four Standards that occur in traditional algebra problems, and it offers strategies for modifying activities to foster algebraic thinking. Includes sample problems and student work.
This is available to members of NCTM. Please log in now to view this article. If you are interested in a NCTM membership join now. | 677.169 | 1 |
Product Description:
In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry's geometric factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization.
REVIEWS for Geometric and Algebraic Topological Methods in | 677.169 | 1 |
Microsoft Mathematics is a set of tools that you can use to perform mathematical operations on expressions or equations in your Word documents and OneNote notebooks. You can use the extensive collection of mathematical symbols and structures to display clearly formatted mathematical expressions. You can also quickly insert commonly used expressions and math structures by using the Equation gallery.
This download works with the following Office programs: Microsoft Word 2010, Microsoft OneNote 2010 and Microsoft Office Word 2007.
Microsoft Mathematics Add-In for Word and OneNote Overview
With the Microsoft Mathematics Add-in for Word and OneNote, you can perform mathematical calculations and plot graphs in your Word documents and OneNote notebooks. The add-in also provides an extensive collection of mathematical symbols and structures to display clearly formatted mathematical expressions. You can also quickly insert commonly used expressions and math structures by using the Equation gallery.
The Microsoft Mathematics Add-in can help you with the following tasks:
Compute standard mathematical functions, such as roots and logarithms.
Compute trigonometric functions, such as sine and cosine.
Find derivatives and integrals, limits, and sums and products of series.
Perform matrix operations, such as inverses, addition, and multiplication.
Perform operations on complex numbers.
Plot 2-D graphs in Cartesian and polar coordinates.
Plot 3-D graphs in Cartesian, cylindrical, and spherical coordinates.
Solve equations and inequalities.
Calculate statistical functions, such as mode and variance, on lists of numbers. | 677.169 | 1 |
Description:CK-12's Basic Geometry Concepts is designed to present students with geometric principles in a simpler, more graphics-oriented course. Students will explore geometry at a slower pace with an emphasis placed on visual aids and approachability.
Description:This book is a "flexed" version of CK-12's Basic Geometry that aligns with College Access Geometry and contains embedded literacy supports. It covers the essentials of geometry for the high school studentThis Precalculus text surveys traditional and additional content for a course in sequence after Algebra II and before Calculus or Statistics. While it goes before Calculus and Statistics, Precalculus is not an easier course. In fact, since it covers such a wide range of topics, requires deep problem solving and includes many new types of notation, many students find it challenging but essential for college level mathematics. The content in this text covers most state standards for Analysis, Trigonometry and Linear Algebra as well as the Common Core Standards suggested for a "Fourth Course" in high school. It was written by Mark Spong, an experienced Precalculus Teacher, and edited by his wife Kaitlyn Spong. Mark graduated from Harvard in 2007 with a BA in Mathematics and Stanford in 2008 with an MA in EducationCK-12's Geometry Second Edition is a clear presentation of the essentials of geometry for the high school student. Topics include: Proofs, Triangles, Quadrilaterals, Similarity, Perimeter & Area, Volume, and Transformations. | 677.169 | 1 |
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