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Unique in its approach, the Lehmann Algebra Series uses curve fitting to model compelling, authentic situations, while answering the perennial question "But what is this good for?" Lehmann begins with interesting data sets, and then uses the data to find models and derive equations that fit the scenario. This interactive approach to the data helps readers connect concepts and motivates them to learn. The curve-fitting approach encourages readers to understand functions graphically, numerically, and symbolically. Because of the multi-faceted understanding that they gain, readers are able to verbally describe the concepts related to functions.
"synopsis" may belong to another edition of this title.
About the Author:
In the words of the author:
Before writing my algebra series, it was painfully apparent that my students couldn't relate to the applications in the course. I was plagued with the question, "What is this good for?" To try to bridge that gap, I wrote some labs, which facilitated my students in collecting data, finding models via curve fitting, and using the models to make estimates and predictions. My students really loved working with the current, compelling, and authentic data and experiencing how mathematics truly is useful.
My students' response was so strong that I decided to write an algebra series. Little did I know that to realize this goal, I would need to embark on a 15-year challenging journey, but the rewards of hearing such excitement from students and faculty across the country has made it all worthwhile! I'm proud to have played even a small role in raising peoples' respect and enthusiasm for mathematics.
I have tried to honor my inspiration: by working with authentic data, students can experience the power of mathematics. A random-sample study at my college suggests that I am achieving this goal. The study concludes that students who used my series were more likely to feel that mathematics would be useful in their lives (P-value 0.0061) as well as their careers (P-value 0.024).
In addition to curve fitting, my approach includes other types of meaningful modeling, directed-discovery explorations, conceptual questions, and of course, a large bank of skill problems. The curve-fitting applications serve as a portal for students to see the usefulness of mathematics so that they become fully engaged in the class. Once involved, they are more receptive to all aspects of the course. | 677.169 | 1 |
Now students can bring home the classroom expertise of McGraw-Hill to help them sharpen their math skills! McGraw-Hill's Math Grade 8Test with success using Pennsylvania Test Prep! This book features essential test practice in reading, math, and writing for students in grade 6Test with success using New York Test Prep! This book features essential test practice in reading, math, and language for students in grade 6 and provides the most comprehensive strategies for effectiveNew York Review Series, Grade 6 Mathematics Review helps students succeed on the New York 6th grade test. Students review both Post-March and Pre-March topics. Lessons for each performance indicator include fully worked-out examples and exercises that are similar to those on the test. Additional problem-solving lessons, as well as chapter tests and practice tests, are included.
Now students can bring home the classroom expertise of McGraw-Hill to help them sharpen their math skills! McGraw-Hill's Math Grade 3Now students can bring home the classroom expertise of McGraw-Hill to help them sharpen their math skills! McGraw-Hill's Math Grade 7All the Math Your 6th Grader Needs to Succeed This book will help your elementary school student develop the math skills needed to succeed in the classroom and on standardized tests. The user-friendly, full-color pages are filled to the brim with engaging activities for maximum educational value. The book includes easy-to-follow instructions, helpful examples, and tons of practice problems to help students master each concept, sharpen their problem-solving skills, and build confidence. Features include: • A guide that outlines national standards for Grade 6 • Concise lessons combined with lot of practice that promote better scores—in class and on achievement tests • A pretest to help identify areas where students need more work • End-of-chapter tests to measure students' progress • A helpful glossary of key terms used in the book • More than 1,000 math problems with answers Topics covered: • Place values and estimating • Number properties and order of operations • Negative numbers and absolute value • Factors and multiples • Solving problems with rational numbers • Ratios and proportions • Percent • Exponents and scientific notation • Solving equations and inequalities • Customary and metric units of measure, including conversions • Solving problems by graphing points on the coordinate plane • Classifying polygons based on their properties • Calculating perimeter, area, surface area, and volume • Data presentation • Statistical variability, including probability
Test with success using Illinois Test Prep! This book features essential test practice in reading, math, and language for students in grade 6 and provides comprehensive strategies for effective ISAT test preparationPractice and master critical math skills and concepts that meet the Common Core State Standards. Ideal for test prep as well as daily practice. Includes: Hundreds of standards aligned practice questions (204 pages) 30+ Skills foundational to success on Smarter Balanced and PARCC assessments Five CCSS Domains: Operations and Algebraic Thinking, Numbers and Operations in Base Ten, Numbers and Operations - Fractions, Measurement and Data, and Geometry Detailed answer explanations for every question PLUS One Year access to Online Workbooks Convenient access to additional practice questions Anywhere Access! Learn using a smart phone, tablet or personal computer Personalized and student-directed with real-time feedback. Take Common Core instruction beyond test preparation to daily practice Each chapter in the Common Core Practice book explores a Common Core State Standard domain. For each content area. individual standards are then available with 10-20 practice questions per standard. Each question includes a detailed answer explanation in the answer key. The Lumos Online Workbooks consist of hundreds of grade appropriate questions based on the CCSS. Students will get instant feedback and can review their answers anytime. Each student's answers and progress can be reviewed by parents and educators to reinforce the learning experience." 4The Assessment Prep for Common Core Mathematics series is designed to help students in grades 6 through 8 acquire the skills and practice the strategies needed to successfully perform on Common Core State Standards assessments. Covers geometry, ratios and proportional relationships, the number system, expressions and equations, and statistics and probability. Each book includes test-taking strategies for multiple-choice questions, test-taking strategies for open-ended questions, and answers and diagnostics. --
Scoring High TerraNova CTBS has been writing in one form or another for most of life. You can find so many inspiration from Scoring High TerraNova CTBS also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Scoring High TerraNova CTBS book for free. | 677.169 | 1 |
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About the book
Description
In this book you find the basic mathematics that is needed by engineers and university students . The author will help you to understand the meaning and function of mathematical concepts. The best way to learn it, is by doing it, the exercises in this book will help you do just that.
Topics as Topological, metric, Hilbert and Banach spaces and Spectral Theory are illustrated.
This book requires knowledge of Calculus 1 and Calculus 2.
Content
Hilbert Spaces
Linner product spaces
Hilbert spaces
Fourler series
Construction of Hilbert spaces
Orthogonal projections and complements
Weak convergence
Operators on Hilbert spaces
Operators on Hilbert spaces, general
Closed operations
Index
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Renowned for its thoroughness and accessibility, this best-selling text by one of the leading figures in linear algebra reform offers students a challenging yet enjoyable study of linear algebra that is infused with an abundance of applications. Balancing coverage of mathematical theory and applied topics, it takes extra care in explaining concepts clearly so that students at a variety of levels can read and understand the material. Numerous worked examples are integrated throughout the text. This revision stresses the important roles played by geometry and visualization in linear algebra. ATLAST Computer Exercises for Linear Algebra a project manual using MATLAB--may be packaged free with the text.
"synopsis" may belong to another edition of this title.
From the Back Cover:
Renowned for its thoroughness, clarity, and accessibility, this best-selling book by one of today's leading figures in linear algebra reform offers users a challenging yet enjoyable treatment of linear algebra that is infused with an abundance of applications and worked examples. Balancing coverage of mathematical theory and applied topics, the book stresses the important role geometry and visualization play in understanding the subject, and now comes with the new ancillary ATLAS computer exercise guide.Provides modern and comprehensive coverage of the subject, spanning all topics in the core syllabus recommended by the NSF sponsored Linear Algebra Curriculum Study Group. Offers new applications in astronomy and statistics, emphasizes the use of geometry to visualize linear algebra and aid in understanding all of the major topics, and previews some of the more difficult vector space concepts early on. MATLAB computing exercises provide users with experience performing matrix computations.For mathematicians | 677.169 | 1 |
Books
Geometry & Topology
This book is based on notes for the course Fractals:lntroduction, Basics and Perspectives given by MichaelF. Barnsley, RobertL. Devaney, Heinz-Otto Peit gen, Dietmar Saupe and Richard F. Voss. The course was chaired by Heinz-Otto Peitgen and was part of the SIGGRAPH '87 (Anaheim, California) course pro gram. Though the five chapters of this book have emerged from those courses we have tried to make this book a coherent and uniformly styled presentation as much as possible. It is the first book which discusses fractals solely from the point of view of computer graphics. Though fundamental concepts and algo rithms are not introduced and discussed in mathematical rigor we have made a serious attempt to justify and motivate wherever it appeared to be desirable. Ba sic algorithms are typically presented in pseudo-code or a description so close to code that a reader who is familiar with elementary computer graphics should find no problem to get started. Mandelbrot's fractal geometry provides both a description and a mathemat ical model for many of the seemingly complex forms and patterns in nature and the sciences. Fractals have blossomed enormously in the past few years and have helped reconnect pure mathematics research with both natural sciences and computing. Computer graphics has played an essential role both in its de velopment and rapidly growing popularity. Conversely, fractal geometry now plays an important role in the rendering, modelling and animation of natural phenomena and fantastic shapes in computer graphics.
This advanced textbook on topology has three unusual features. First, the introduction is from the locale viewpoint, motivated by the logic of finite observations: this provides a more direct approach than the traditional one based on abstracting properties of open sets in the real line. Second, the author freely exploits the methods of locale theory. Third, there is substantial discussion of some computer science applications. As computer scientists become more aware of the mathematical foundations of their discipline, it is appropriate that such topics are presented in a form of direct relevance and applicability. This book goes some way towards bridging the gap for computer scientists.
This book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation.
The last ten years have seen rapid advances in the understanding of differentiable four-manifolds, not least of which has been the discovery of new 'exotic' manifolds. These results have had far-reaching consequences in geometry, topology, and mathematical physics, and have proven to be a mainspring of current mathematical research. This book provides a lucid and accessible account of the modern study of the geometry of four-manifolds. Consequently, it will be required reading for all those mathematicians and theoretical physicists whose research touches on this topic. The authors present both a thorough treatment of the main lines of these developments in four-manifold topology--notably the definition of new invariants of four-manifolds--and also a wide-ranging treatment of relevant topics from geometry and global analysis. All of the main theorems about Yang-Mills instantons on four-manifolds are proven in detail. On the geometric side, the book contains a new proof of the classification of instantons on the four-sphere, together with an extensive discussion of the differential geometry of holomorphic vector bundles. At the end of the book the different strands of the theory are brought together in the proofs of results which settle long-standing problems in four-manifolds topology and which are close to the frontiers of current research. Co-author Donaldson is the 1994 co-recipient of the prestigious Crafoord Prize.
Fractal Geometry is a recent edition to the collection of mathematical tools for describing nature, and is the first to focus on roughness. Fractal geometry also appears in art, music and literature, most often without being consciously included by the artist. Consequently, through this we may uncover connections between the arts and sciences, uncommon for students to see in maths and science classes. This book will appeal to teachers who have wanted to include fractals in their mathematics and science classes, to scientists familiar with fractal geometry who want to teach a course on fractals, and to anyone who thinks general scientific literacy is an issue important enough to warrant new approaches.
Quasitopoi generalize topoi, a concept of major importance in the theory of Categoreis, and its applications to Logic and Computer Science. In recent years, quasitopoi have become increasingly important in the diverse areas of Mathematics such as General Topology and Fuzzy Set Theory. These Lecture Notes are the first comprehensive introduction to quasitopoi, and they can serve as a first introduction to topoi as well.
The aim of this major revision is to create a contemporary text which incorporates the best features of calculus reform yet preserves the main structure of an established and well-tested calculus course. The multivariate calculus material is completely rewritten to include the concept of a vector field and focuses on major physics and engineering applications of vector analysis. Covers such new topics as Jacobians, Kepler's laws, conics in polar coordinates and parametric representation of surfaces. Contains expanded use of calculator computations and numerous exercises.
The fundamental concepts of general topology are covered in this text whic can be used by students with only an elementary background in calculus. Chapters cover: sets; functions; topological spaces; subspaces; and homeomorphisms. | 677.169 | 1 |
297 358 96 03
Page: 352 37A North Country Maid, etc?, 000581502
Author: CAMERON, Caroline Emily.
Volume: 01
Page: 221 ThisCommunication is as vital in mathematics as in any language. This free course, Language, notation and formulas, will help you to express yourself clearly when writing and speaking about mathematics. You will also learn how to answer questions in the manner that is expected by the examiner. First published on Fri, 10 Jun 2011 as Language, notation and formulas. To find out more visit The Open University's Openlearn website. Creative-Commons 2011
Image from ?A Text-Book of Coal-Mining ? Second edition, etc?, 001758763
Author: HUGHES, Herbert W.
Page: 420Geology for students and general readers. Physical Geology ? Second edition?, 001497800
Author: GREEN, Alexander Henry.
Page: 510 | 677.169 | 1 |
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10th Grade Mathematics, 1987 edition. The algebraic system is built logically, with smooth transitions from one concept to another. Mathematical concepts are developed and mastered through an abundance of worked examples and student exercises. Word problems are emphasized and many application problems related to the physical world. | 677.169 | 1 |
Description: After a short introductory chapter consisting mainly of reminders about such topics as functions, equivalence relations, matrices, polynomials and permutations, the notes fall into two chapters, dealing with rings and groups respectively.
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ISBN 13: 9789332538160
The Iit Foundation Series Mathematics Class 7
The iit foundation series prepares students to gear up for the joint entrance examinations (jee), and various talent search examinations like ntse, olympiads, kvpy, etc. Comprising of twelve titles on physics, chemistry and mathematics, this series caters to students of classes vii to x. The core objective of the series is to help aspiring students understand the basic concepts with more clarity, in turn, developing a problem-solving approach. It also encourages students to attempt various competitive examinations from an early age. ??introduces the iit-jee syllabus and approach in a simple manner making it easy to integrate along with the regular curriculum ??uses graded approach to generate, build and retain interest in concepts and their applications ??comprehensive pedagogy: ? illustrative examples solved in a logical and step-wise manner ? test your concepts at the end of every chapter for classroom preparations ? concept application section with problems divided as per complexity-basic to moderate to difficult ? hints and explanations for key questions along with highlights on the common mistakes that students usually make in the examinations helpful for students aspiring admission in iit's and other top engineering colleges. | 677.169 | 1 |
The Integral
Meets NCTM Standards:
Copyright 2005, Department of Mathematics, University of Houston. Created by Selwyn Hollis. Find more information on videos, resources, and lessons at
This video explains what an integral is using Riemann Sum notation. It then goes on to explain why some integrals are positive and some are negative, and what an integral is actually finding. Some interesting examples are used in this video. The video also shows how you can find the integral of certain functions by looking at the graph and finding the area of the geometric shape. This is a pretty good lesson for learning and understanding what an integral is and how to compute it. | 677.169 | 1 |
A First Course in Linear Algebra
Cover image is under a CC BY license (
Description: This text, originally by Ken Kuttler, has been redesigned by the Lyryx editorial team as a first course in linear algebra for science and engineering students who have an understanding of basic algebra. All major topics of linear algebra are available in detail, as well as proofs of important theorems. In addition, connections to topics covered in advanced courses are introduced. The text is designed in a modular fashion to maximize flexibility and facilitate adaptation to a given course outline and student profile. Each chapter begins with a list of student learning outcomes, and examples and diagrams are given throughout the text to reinforce ideas and provide guidance on how to approach various problems. Suggested exercises are included at the end of each section, with selected answers at the end of the text. | 677.169 | 1 |
Mathematics & Statistics Accessibility
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Our Commitment
We are committed to creating a culture that consciously considers those with disabilities throughout the development of our products. This effort includes an extensive blend of planning, research, training and product development activities with both McGraw-Hill employees and third-party content partners. Specific initiatives include:
Creation of Accessible Products – McGraw-Hill Education will strive to have all new content and software follow the WCAG version 2.0 AA guidelines and best practices. To achieve this and continuously improve the accessibility of our products, we will utilize the efforts of our internal product teams, the assistance of external experts and user feedback.
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Creating accessible products is a priority for McGraw-Hill Education. We have put in place processes to make accessibility and meeting the WCAG 2.0 AA guidelines part of our day-to-day development efforts and product roadmaps. We will measure and track our progress to ensure we continually make improvements to address the evolving industry standards and to meet our learners' accessibility needs.
When discussing accessibility, it is useful to distinguish three distinct components that constitute our products: Textbooks, Media, and Platforms. Select an item from the left-hand menu to learn more about the current state of that product's accessibility.
Printed and On-Screen Textbooks
McGraw-Hill Education can offer a Word or PDF version of the required printed or digital text. A Disability Support Services (DSS) office or ADA coordinator can request these materials for a student. The forms for these can be found at the following urls:
We are in the process of converting SmartBook to HTML and working on the implementation of keyboard navigation. A 3rd party audit of these products will be completed in Spring 2016 and will further inform our prioritization of our development plan and roadmap.
Videos
Most of McGraw-Hill's videos in mathematics are closed-captioned and are in .mov format. The videos and captioning can be made available outside of the ALEKS and Connect Hosted by ALEKS platforms via a DSS office request to McGraw-Hill. The videos and captioning are available within the ALEKS and CHBA (Connect Hosted by ALEKS) platform.
ALEKS
ALEKS as a platform, and the content included within, is our most accessible product line and platform in mathematics. ALEKS is accessible to many segments of the disabled population. Because there is no audio associated with the ALEKS content, ALEKS is fully accessible to those who are partially or profoundly deaf. ALEKS does not rely exclusively on color to convey critical information so it is fully accessible to those that are colorblind. Many students who have low vision or are otherwise visually impaired can use products such as Microsoft Magnifier to fully access ALEKS. Students who are incapable of using both a keyboard and a mouse simultaneously due to physical disabilities similarly have no limitations while using ALEKS. Profoundly blind students can access certain areas of ALEKS through the use of screen reader technology.
The ALEKS team has recently completed the process of reprograming all ALEKS math courses from Java to JavaScript to make them more compatible with screen reader technology. To ensure that our accessibility efforts were as successful as possible, ALEKS worked with the Accessible Technology Initiative at The California State University to assist in design in accordance with W3C and WCAG 2.0 guidelines. Our JavaScript programming strategy incorporates WAI-ARIA to allow dynamic content and advanced user interface controls developed in JavaScript to interact more readily with screen readers.
In attempting to address the needs of students with limited vision and the needs of profoundly blind students, we have evaluated current screen reader technology and have determined it to be unsatisfactory in mathematics with regard to problems that are visually demanding.
Given the current state of assistive technology, many colleges and universities that have addressed the issue of teaching mathematics to profoundly blind students have determined that an "alternative accessible arrangement" in the form of human assistance (qualified readers or transcribers to record answers) is the best accommodation for those students as they work through either a traditional math curriculum and textbook or as they work with ALEKS. This approach is also consistent with their approach to other visually demanding course work.
We recognize that educational institutions must provide accommodations or modifications that would permit disabled students to receive all the educational benefits provided by the ALEKS technology in an equally effective and equally integrated manner. We are committed to broadening the accessibility of ALEKS and continuing to evaluate accessibility technology to determine how it can be used to improve the ALEKS experience for disabled students.
ALEKS Courses with Accessibility Features
The ALEKS courses specified below have an accessibility mode in which a majority of the content and interface in each course can be made accessible for blind persons using an assistive listening system (screen reader technology). The accessibility mode in these ALEKS courses can be made available at both the class level and individual student level in order to meet the specific needs of each implementation.
ALEKS Placement, Preparation and Learning (ALEKS PPL) offers six months of access to a Prep & Learning Module that is considered accessible as defined in the paragraph above. However, the accessibility mode is only at the cohort level, and cannot be turned on for individual students. If a school wants to offer ALEKS PPL for blind or visually impaired students, we recommend they create a separate cohort for those students so that all non-accessible items can be removed from the product and learning modules for that cohort.
The content available in accessibility mode for the following courses has been rewritten and coded in order to conform to screen-reading technology and level AA Web Content Accessibility Guidelines (WCAG). An instructor can choose to use either the accessibility mode version of the ALEKS content or the original version. Students will need the following system requirements: Microsoft Windows 7+, JAWS 17+, and Firefox 25+
Connect Math Hosted By ALEKS
The Connect Math Hosted By ALEKS platform and content is primarily Flash-based and is not currently accessible. McGraw-Hill is actively pursuing greater accessibility for the platform and content files. The student may request an accessible version of the text via the DSS office and the videos contained within the course are closed-captioned. As new content is being created, it is being created to comply with ADA standards. The platform is fully accessible to colorblind students and much of the platform can be used with keyboard only. | 677.169 | 1 |
This book introduces graph theory, a subject with a wide range of applications in real-work situations. This book is designed to be easily accessible to the novice, assuming no more than a good grasp of algebra to understand and relate to the concepts presented. Using many examples, illustrations, and figures, it provides an excellent foundation for the basic knowledge of graphs and their applications. This book includes an introductory chapter that reviews the tools necessary to understand the concepts of graphs, and then goes on to cover such topics as trees and bipartite graphs, distance and connectivity, Eulerian and Hamiltonian graphs, graph coloring, matrices, algorithms, planar graphs, and digraphs and networks. Graph theory has a wide range of applications; this book is useful for those in the fields of anthropology, computer science, chemistry, environmental conservation, fluid dynamics, psychology, sociology, traffic management, telecommunications, and business managers and strategists.
Graph theory is a delightful subject with a host of applications in such fields as anthropology, computer science, chemistry, environmental conservation, fluid dynamics, psychology, sociology, traffic management, and telecommunications, among others. With the advent of operations research in the twentieth century, graph theory has risen several notches in the esteem in which it is held. Formally a branch of combinatorics, graph theory intersects topology, group theory, and number theory, to name just a few fields. While its theorems and proofs range from easy to almost incomprehensible, graph theory is, after all, the study of dots and lines!
Because of its wide range of applications, it is important that students in many diverse fields, not just mathematics and computer science, gain a foundation in the basics of graph theory. By doing so, they will have additional powerful tools at their disposal to analyze problems within their own area of study. We have written this text mainly with those students in mind and designed it to be easily accessible to undergraduate students as early as the sophomore level. We assume nothing more than a good grasp of algebra. Thus, A Friendly Introduction to Graph Theory provides early access to this wonderful and useful area of study for students in mathematics, computer science, the social sciences, business, engineering—wherever graph theory is needed.
The student will find this text quite readable. This book should be read with pencil and paper nearby—it is not bedtime reading. The exercises reflect the contents of the chapter and range from easy to hard. Answers, solutions, or hints are supplied in the back of the book for a wide selection of the exercises. It is important to realize that mathematics is not a spectator sport (although it can be fascinating to watch someone present a surprisingly simple proof of an interesting result). Do the exercises, or you will not retain the material of the nth chapter as you attempt the (n+l)-st. A bonus to the diligent students who work through the more challenging exercises is that those students will develop a strong foundation for research in graph theory and an ability to apply the concepts learned to a wide range of real-world problems.
Content
In Chapter 1, we discuss basic prerequisite concepts and ideas that the reader should be familiar with. Although it is meant as a review, a somewhat thorough treatment is given to provide the necessary tools for understanding the material that will be examined throughout the rest of the book. Thus we discuss topics such as sets, functions, parity, mathematical induction, proof techniques, counting techniques, permutations and combinations, Pascal's triangle, and combinatorial identities. For those readers unfamiliar with proof techniques, this introductory chapter will be particularly helpful. In a class of better-prepared students, the instructor may choose to skip some or all of the contents of Chapter 1.
As mentioned earlier, graphs have a wide variety of applications. In Chapter 2, we introduce the most basic concepts of graph theory and illustrate some of the areas where graphs are used. We will see many other applications throughout the book.
One class of graphs is so important that it deserves treatment in its own chapter. Because of their simple structure, trees are often used as a testing ground for possible new theorems. Trees arise in many applications, such as analyzing business hierarchies and determining minimum cost transportation networks, and are the basis of important data structures in computer science. A tree is a special type of graph in a larger class called bipartite graphs. In Chapter 3, we examine trees, bipartite graphs, and their uses. The chapter concludes with an application to job assignment problems.
The concept of distance is widely used throughout graph theory and its applications. Distance is used in various graph operations, in isomorphism testing, and in convexity problems, and is the basis of several graph symmetry concepts. Distance is used to define many graph centrality concepts, which in turn are useful in facility location problems. Numerous graph algorithms are distance related in that they search for paths of various lengths within the graph. Distance is an important factor in extremal problems in graph connectivity. Graph connectivity is important in its own right because of its strong relation to the reliability and vulnerability of computer networks. In Chapter 4, we discuss some of the many important concepts and results concerning distance and connectivity in graphs. We conclude with a discussion of a facility location problem and an introduction to the concept of reliability of computer networks.
Numerous theoretical and applied problems in graph theory require one to traverse a graph in a particular way. In some problems, the goal is to find a trail or circuit in order to pass through each edge exactly once. In other problems, one must find a path or cycle that includes each vertex exactly once. We discuss such problems in Chapter 5. We conclude with a discussion of two well-known related problems, namely, the Chinese postman problem, and the Traveling salesman problem.
Various real-world problems that can be modeled by graphs require that the vertex set or edge set be partitioned into disjoint sets such that items within a given set are mutually nonadjacent. Common problems include scheduling meetings or exams to avoid conflicts and storage of chemicals to prevent adverse interactions. These problems are related to graph coloring, the subject of Chapter 6.
A matrix is a rectangular table of numbers. One of the simplest ways of storing a graph in a computer is by using a matrix or its computer science counterpart, an array. Graph theory makes effective use of matrices as a tool in examining structural and other properties of graphs. In Chapter 7, we first review some basic properties of matrices and then examine some uses of matrices within graph theory.
There are many connections between graph theory and computer science. Thus it should not be surprising that algorithms have played a strong role in recent graph theory research, so much so that several books have been devoted to algorithmic graph theory. In Chapter 8, we give an introduction to graph algorithms and indicate some of their uses. We discuss the important breadth-first search and depth-first search algorithms and examine algorithms for graph coloring as well as for coding trees. Several other algorithms are presented throughout the text. We shall not examine efficiency of algorithms or NP-completeness here, but refer the interested reader to appropriate sources for discussions of those topics.
A graph is planar if it can be drawn in the plane with no crossing edges. Planar graphs have received a great deal of attention over the years because of a long-standing problem, the four-color conjecture, that took over one hundred years to prove. Planar graphs remain important today because of their applications. Planar graphs are important in facility layout problems within operations research, and they are crucial in the design of printed circuit boards in computer science. We discuss planar graphs and their properties in Chapter 9.
To model certain real-world problems, a structure more complex than a graph is needed. In Chapter 10, we consider some additional structures. Digraphs are similar to graphs except there are directions on the edges. Digraphs are used to model problems where the direction of flow of some quantity (information, traffic, liquid, electrons, and so on) is of importance. When limits are placed on how much of that quantity can flow through a given directed edge, we obtain a network. A special type of digraph with no directed cycles, called an activity digraph, has weights on the directed edges indicating the duration of a given activity. These weighted digraphs are used to aid in scheduling individual activities that compose a complex project. In Chapter 10, we explore various types of digraphs and their properties, study network flows, and present an algorithm to maximize total flow. We conclude the chapter with a discussion of activity digraphs and their uses.
In Chapter 11, we consider two additional topics, Ramsey theory and graph domination. The first is related to edge colorings in graphs, and the second is related to both distance and independence and has a wide range of applications. The one thing that these last two topics have in common is that they are lots of fun to work on.
We hope this book is as much fun for you to read and learn from as it was for us to write. | 677.169 | 1 |
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This Matching / Cut & Paste activity is designed to be utilized after students have been introduced to proving trigonometric equations using sine/cosine conversion and Pythagorean identities. This set of topics is typically introduced in Algebra 2 (as time permits) but is the starting topic for proving identities in Precalculus. It can also be a useful refresher for Calculus students. | 677.169 | 1 |
Math 597T Questions - Fall 2015
Some Frequently Asked Questions for the First Few Days of Class
Q: How do I learn all the students' names?
A: Probably the easiest way to learn students' names is to hand back papers individually. Try getting to class a little early
and start handing back papers before the bell rings. Take those few extra minutes to really look at the students as you return their
homework, and talk briefly to them. Another idea is to take a few minutes before class each day to call through the roster and study
faces as you take attendance. After calling all the names, go back around the room and try to name each person. Just a couple of
minutes of effort each day will help you learn names much more quickly. "Studying" names right after class each day will help as well.
Q: What should I do if I come into the classroom and the desks are arranged in a way I don't like (e.g. in a circle)?
A: Unfortunately, this might happen a lot. You should try to arrive to class early and enlist the help of the students to get
the classroom back to a "normal" configuration. Once you've done it a few times, the students will get used to doing it when
they arrive. If this is a persistent problem, and you find the person in the classroom ahead of you doesn't leave in enough time
to get the room rearranged, you should talk to Jerrie (jerrie@math.arizona.edu) about this.
Q: What is a good way to take attendance?
A: Some people pass around a sign-in sheet every day, or use a spiral notebook to have students sign in. Verbally taking roll
every day is pretty inconvenient. If you collect homework every day, then you can use that as an attendance record. Just be sure that
your students know that they should turn in a blank paper if they don't have their homework done.
Q: Will students understand language like "if and only if", "implies", etc.
A: No.
Q: Some of my students are bored, while others seem really lost. How do I handle that?
A: This is always difficult. There will always be students who are very familiar with the material, and those who are really struggling, especially at the beginning of the semester, when you may be covering material that is considered to be "review" for them. However, you shouldn't go really quickly just because a few people look bored, and you shouldn't slow way down because a few are lost. It is your job as an instructor to make decisions about the level of your presentation and the pacing of the material.
A few suggestions that might help: Ask students what they know about a topic. Build on that basis, and focus your discussion on the areas where you think they aren't as solid. For example, when discussing inverse functions, many students know to "switch x & y, then solve for y", but they don't know what that means graphically, or what happens to the domain and range of the inverse. Most likely, they also haven't seen the usefulness of inverse functions in any context. Going over all the basic stuff with an interesting, practical example will usually keep the more knowledgeable students interested, while keeping the discussion at the appropriate level for other students. Be careful of misinterpreting the boredom; often students are looking bored not because they are finding the material easy, but because they are lost and have stopped paying attention. One way to tell the difference is to give the students an in-class exercise and then walk around the room looking at their work; you'll be able to see who is really following and who isn't. It's good to keep in mind that beginning instructors tend to err more on the side of going too fast than too slow. | 677.169 | 1 |
This book introduces perspective, and discusses the mathematics of perspective in a detailed, yet accessible style. It also reviews nonlinear projections, including the fisheye, panorama, and map projections frequently used to enhance digital images. Topics and features include a complete and self-contained presentation of concepts, principles, and methods; a 12-page colour section, and numerous figures. This essential resource for computer professionals both within and outside the field of Computer Graphics is also suitable for graduates and advanced undergraduates in Computer Graphics and Computer-Aided Design. Key ideas are introduced, examined and illustrated by figures and examples, and reinforced through solved exercises. | 677.169 | 1 |
GCSE Maths Revision Guide for Edexcel A: Foundation (Easy Learning)
"Easy Learning GCSE Maths Revision Guide for Edexcel A" includes revision content with highlighted grade levels so that students know exactly which grade they are working at, making revising for GCSE Maths easy. It is easy to use - clear and comprehensive structure and design; easy to revise - colour-coded to show grade level of content; and easy to remember - concise information organised in memorable chunks. Together with the accompanying "Easy Learning GCSE Maths Exam Practice Workbook", the two books provide complete revision coverage of the new GCSE Maths Edexcel A specification. "Easy Learning GCSE Maths" complements the bestselling "Collins GCSE 2-tier Maths Scheme" through clearly differentiated revision content which is also written by Keith Gordon.
Book Description Collins 01/02247226
Book Description -. Paperback. Book Condition: Very Good. Easy Learning - GCSE Maths Revision Guide for Edexcel A: Foundation247226
Book Description Collins 01/02247226
Book Description Collins 01/02247226
Book Description Collins 01/02247226123138 | 677.169 | 1 |
Designed to help pre-service and in-service teachers gain the knowledge they need to facilitate students' understanding, competency, and interest in mathematics, the revised and updated Second Edition of this popular text and resource bridges the gap between the mathematics learned in college and the mathematics taught in secondary schools. Highlighting multiple types of mathematical understanding to deepen insight into the secondary school mathematics curriculum, it addresses typical areas of difficulty and common student misconceptions so teachers can involve their students in learning mathematics in a way that is interesting, interconnected, understandable, and often surprising .
Understanding Business has long been the market leader because we listen to instructors and students. With this eleventh edition we are proud to offer a platinum experience, that: Improves Student Performance―Understanding Business puts students at the center. It's the only learning program on the market to offer proven adaptive technology that increases grades by a full letter through Connect® Business, and the only program to offer the first and only adaptive eBook ever, SmartBookThe IB Diploma Programme Course Companions are resource materials des igned to support students throughout their two-year Diploma Programme comse of study in a particular subject. They will help students gain an understanding of what is expected from the study of an IB Diploma Programme subject wh ile presenting content in a way that illustrates the purpose and aims of the IB. They retlect the philosophy and approach of the IB and encourage a deep understanding of each subject by making connections to wider issues and providing opportunities for critical thinking | 677.169 | 1 |
Patterns, Function, and Change
Discover how the study of repeating patterns and number sequences can lead to ideas of functions, learn how to read tables and graphs to interpret phenomena of change, and use algebraic notation to write function rules. | 677.169 | 1 |
ALEX Lesson Plan
I'm Walking Through FunctionsMichelle Russell
System:
Florence City
School:
Florence High School
General Lesson Information
Lesson Plan ID:
33173
Title:
I'm Walking Through Functions
Overview/Annotation:
The students will participate in a gallery walk of functions with their group. The students will identify the graphs and list the key features of the graphs. The graphs will include square root, cube root, piecewise, step, and absolute value functions. The key features could include the x and y intercepts, domain and range, symmetry, and end behavior.
This is a College- and Career-Ready Standards showcase lesson plan.
Associated Standards and Objectives
Content Standard(s):
MA2015 (9-12) Precalculus
18. Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7]
Students will use their iPads to access a short Khan Academy quiz (optional- if students do not have iPads the teacher can display the quiz on the digital projector)
digital projector
graphing calculators (optional)
Background/Preparation:
Prerequisite learning- the student should be able to identify and list the key features of a quadratic function.
The teacher will prepare the graphs for the gallery walk.
The teacher will preview the quizzes on the Khan Academy website.
Major concept to be taught- identify and list key features of absolute value, step, cubic, piecewise, and square root functions.
Procedures/Activities:
1. Introductory Activity- Quickwrite y = ( x- 4 )2 +2. Students will determine the key features of this function, such as intercepts, domain and range, symmetry, and end behavior. Students will explain how they found the key features.
2. Gallery Walk- Students will be divided into five different groups, if possible. Students will spend approximately 5-10 minutes at each graph. The students will work collaboratively to identify the graph and list the key features of the graph. As the students move through the gallery the teacher will observe the students and listen to their conversations to assess their understanding of the functions.
3. After the students have completed their gallery walk each group will be assigned to a graph. The group will then "report out" their findings.
4. Students will use their iPads to take two short quizzes on the Khan academy website. If students do not have an Ipad the teacher will post the quiz for the entire class to discuss the questions. The quizzes will assess students on absolute value functions and piecewise funtions. The links for the quizzes are listed here:
Quickwrite sample question: y = ( x- 4 )2 +2. Students will determine the key features of this function, such as intercepts, domain and range, symmetry, and end behavior. Students will explain how they found the key features.
Teacher will listen to group discussions as the students participate in the gallery walk
Groups will report out at the end of the gallery walk
Two short quizzes on Khan academy
Acceleration:
Students who have already mastered the Primary Learning Objective can apply that concept to rational functions. Students can watch a video tutorial on this topic on the Khan Academy website. Please see the link below | 677.169 | 1 |
"Recommended" Supplies: (any method that works for the student is fine, however)
3-ring binder (1 inch wide is ok) & loose leaf paper
pencil and/or pen
a calculator is allowed on many activities and is recommended (you can find a simple "4-function" model at the dollar store. Having a Plus/Minus button is a nice feature, and you can find it for $1 there. At WalMart, etc, models run around $5. Scientific calculators are allowed as well but are not required.)
The purpose of these courses is to provide the foundation for all higher level, college preparatory classes.
Topics shall include, but not limited to, operations and properties within the real number system; algebraic and graphical solutions of first degree equations and inequalities in one and two variables; relations and functions; direct and inverse variation operations with polynomials including all forms of factoring; rational and irrational algebraic expressions; quadratic equations; and quadratic functions.
Classroom Procedures:
You must come prepared with the proper materials: textbook, pen/pencils, and a 3-ring binder or notebook.
A personal calculator is recommended, and a few will be available in my classroom for use. One user-friendly scientific model is the TI-30Xa (around $10), however any model is fine. Remember, this is not required.
NO CELL PHONES OR OTHER ELECTRONIC DEVICES ARE ALLOWED, and must be turned off at all times.
Violation of this policy will result in consequences in accordance with VHS school rules.
Use the wastebasket and sharpener before and/or after class (whenever possible).
Show respect to classmates and to the teacher. Raise your hand to be called on. Do not leave until you are dismissed.
Always hand in your work on time. Late work is not accepted, except in the case of an excused absence. You have one day for every day of absence to make up work. Tests must be made up within the same time frame.
NO FOOD OR DRINKS ARE ALLOWED IN THIS CLASSROOM! (Except water)
Homework Policy:
Homework will be assigned nightly, with few exceptions. You must complete or make a good attempt at completing each problem in order to receive full credit. All work must be shown. Assignments with answers only will not be accepted. Late work is not accepted. Please include your name, date, page number and class period on every paper that is handed in.
Employability skills are mandated by the county and will count as 20% of your semester grade. They are based upon attendance, homework, following rules and behavior. The remaining 80% is based
upon Quiz and Test scores. This is very common in Honors-level courses in general.
Grading Scale:
A = 90 - 100
B = 80 - 89
C = 70 - 79
D = 60 - 69
F = 59 OR BELOW
*Parents, you may simply email me with a statement that you have read this page, no printing required. Otherwise, either print this page or compose a note using a format similar to the one below. | 677.169 | 1 |
Avanti Digital Learning – Class IX
₹7,000.00₹5,750.00
– Video lectures Physics, Chemistry and Math on the CBSE pattern by IIT and NIT alumni
– 50+ hours of video lectures
– Videos for revision with a seamless interface and subdivided chapters
– Learn Better by using videos with a mix of boardwork, animations and practical experiments which make it easy to understand even the most complicated concepts
– Chapter-wise and full-length tests
– Detailed analytical reports
– User friendly – watch it on your tablet, mobile, desktop or laptop
Description
Prepare yourself for the CBSE examination at home by using Avanti's Digital Learning Portal.
Through our set of 50+ Videos & and over tests students will be able to practice problem-solving skills for competitive exams, understand areas of weakness and get access to quality HD videos and practice modules to help them get better. | 677.169 | 1 |
2
Fire Drills One per week for the first month. One per month for the rest of the year. If there is one during this class, exit the room, turn left, leave the building through exit 24, cross the parking lot onto the grass, and form a line; I will take attendance. The alternate exit is exit 25.
3
Grade Determinants Total points system is used to compute grading period averages. Each time a graded assignment is returned, record the grade in your agenda. Calculate your grade at any time by summing your scores on all assignments and dividing by the sum of the possible points.
4
Tests One cumulative test at the end of each six weeks. Approximately 20 multiple-choice questions. Contain built-in extra credit.
5
Quizzes Two to four each six weeks. Approximately 10 free-response questions. Contain built-in extra credit.
6
Homework Assigned daily. You may work with others to complete your homework, but directly copying another person's work is not acceptable. Homework assigned between consecutive quizzes and tests should be completed and brought to class on the day of the next quiz or test.
7
The Math Lab Free math tutoring. Make up quizzes and tests. The morning Math Lab is staffed from 7:00 a.m. – 7:50 a.m. by Mr. May in room 258. The afternoon Math Lab is staffed from 3:00 p.m. – 3:50 p.m. by Miss Moody in room 259.
8
Getting help See me sooner rather than later. I'm usually available from 7:15 a.m. – 7:45 a.m. and from 3:00 p.m. – 3:30 p.m. Go to the Math Lab. May be able to refer a tutor.
10
Progress Reports Issued to every student at or near the end of the third week of the grading period. Have your parent sign it and return it to me within two school days.
11
Materials needed for this class: A three ring binder Pencils Straightedge Graph Paper Each student should bring contribute one pack of 4 AAA batteries to be used for replacing batteries in the classroom set of calculators.
13
Absences and Make-up Work 1.Check the website to find out what you missed. Check the "absent students' assignments" folder for returned work or handouts. Copy a classmate's notes. Attend Math Lab. See me before or after class.
14
Absences and Make-up Work 2.When absent on a day that a quiz or test was scheduled, make up the quiz/test in the Math Lab by the afternoon of the second day you return to school. Example: You were present on Wednesday when it was announced that there would be a quiz on Thursday, but you were out sick on Thursday. Assuming Thursday is the only day you missed, by when must the quiz be made up?
15
Absences and Make-up Work 3.When absent on a day that an assignment is given or a quiz/test is announced, you have two days per each day of class missed to make up the assignment.
16
Absences and Make-up Work 4.The student is responsible for inquiring about work missed due to an absence. 5.When absent due to a school-related event, you are responsible for missed work. Request assignments in advance.
17
Classroom Rules 1. Be in your seat with all your materials completing the "do now" activity; remain seated during class. 2. Pay attention and participate in class, copy notes and example problems, and work practice problems. 3. Raise your hand and wait to be called on to talk in class.
18
School Rules Strictly Enforced! 1. Unexcused tardiness will result in d-hall. 2. Food and drinks brought into class will be taken and thrown away. 3. Electronic devices will be taken away and turned into the office. 4. Students will not be permitted to go lockers during class. You must complete your agenda hall pass and have it initialed by me to leave class.
19
Academic Honor Policy Violation: Giving or receiving unauthorized aid on a quiz or test: Looking on someone else's paper or allowing them to look on yours Discussing the content of a quiz/test before they have been returned Using a book, notes, or a calculator when not allowed
20
Academic Honor Policy Violation: submitting someone else's work as your own. Copying homework or allowing someone to copy yours. Forging a signature on a progress report or having someone do so for you. | 677.169 | 1 |
Showing 1 to 5 of 5
Algebra is a branch of mathematics that uses mathematical statements to describe
relationships between things that vary over time.
what is algebra
Algebraic terms
An expression is a meaningful collection of numbers, variables, and signs, positive or
negat | 677.169 | 1 |
SCIENCE, TECHNOLOGY, ENGINEERING, AND MATH (STEM)
At Bridgton Academy, we recognize the need for all students to have some appreciation for the wonders of science and technology, and sufficient knowledge of engineering to engage in public discussions on related issues and to understand technological information as it relates to their everyday lives. In addition, our students must acquire the skills to be successful in college in order to enter careers of their choice, including the possibility of careers in science, engineering or technology. Too few workers in the United States currently have the background to enter these fields, and many citizens lack even fundamental knowledge of them. Bridgton Academy's STEM department offers a wide array of courses in all the STEM areas. Instructors in these disciplines design lessons around common math, science, and engineering core practices to allow students to make connections across these disciplines in order to cultivate scientific habits of mind and to understand how engineering and technologically-based solutions are developed.
MATHEMATICS
All math courses are full-year, two-semester offerings
GEOMETRY
This is a full-year course. This course includes a review of algebra skills and a study of Euclidian geometry. Problem-solving strategies are emphasized, and relationships between algebra and geometry are explored, but the main emphasis of the course is on traditional topics in geometry.
ADVANCED ALGEBRA AND TRIGONOMETRY
This is a full-year course. The course begins with a first quarter review of Algebra. Topics will include: patterns and expressions, properties of and operations with real numbers, algebraic expressions, operations with polynomials including expansion and factoring, solving linear algebraic equations and inequalities, modeling with linear equations and graphing.
The second quarter will begin with an introduction to functions and their graphs, including combinations of functions, inverse functions, quadratic and polynomial equations, and rational equations.
During the second semester, exponential and logarithmic equations and the properties of logarithms, and multivariable systems of equations and inequalities will be explored.
Additionally, an introductory study of trigonometry will be covered, including right and non-right triangle trigonometry, trigonometric functions and their application to periodic phenomena.
STATISTICS (CAP)
This is a full-year course. To ensure students have the necessary mathematical background to be successful in this course, the course begins with a review of algebra during the first quarter. Topics will include: Properties of and operations with real numbers, algebraic expressions, operations with polynomials including expansion and factoring, solving linear algebraic equations and inequalities, modeling with linear equations and graphing. The probability and statistics portion of this course begins during the second quarter. It is designed to acquaint students with statistical methods of data analysis. Topics include: descriptive statistics; probability and probability distributions; hypothesis testing and statistical inference; analysis of variance; and regression. Successful completion of this course may qualify a student for college credit through the University of Southern Maine.
PRECALCULUS (CAP)
This is a full-year course. This course provides the mathematical background necessary for calculus. Topics include: equations and inequalities; functions and graphs; exponential, logarithmic, and trigonometric functions; and identities and inverse functions. Successful completion of this course (the equivalent of MAT 180 at University of New England) may qualify a student for 3 hours of college credit. Students are required to use a TI 89 Titanium or TI Nspire CX CAS graphing calculator. Students are strongly urged to purchase their calculators prior to arrival at Bridgton.
CALCULUS (CAP)
This is a full-year course. This course is modeled on a college freshman calculus course taught at University of Southern Maine (USM). The topics include: analytical geometry; functions; continuity; limits; derivatives and applications; and integrals and applications. This course is the equivalent of USM's MAT 152D and carries 4 college credits. Students are required to use a TI 89 Titanium or TI Nspire CX CAS graphing calculator. Students are strongly urged to purchase their calculators prior to arrival at Bridgton.
ACCELERATED CALCULUS (CAP)
This is a full-year course. This course parallels the two-semester sequence course taught at University of Southern Maine, Calculus A (MAT 152D) and B (MAT 153), for 4 credit hours for each semester. Students are required to use a TI 89 Titanium or TI Nspire CX CAS graphing calculator. Students are strongly urged to purchase their calculators prior to arrival at Bridgton.
SCIENCE/COMPUTER SCIENCE
ANATOMY AND PHYSIOLOGY
This is a full-year course. Anatomy and Physiology is an introductory level course in the human sciences that includes examination of the following areas: cytology, histology, genetics, and the major systems of the body. The object of this course is to give each student a basic, working knowledge of the human body's parts and how this anatomy functions to create the living condition. Anatomy and Physiology is a lab class and includes a dissection lab. Practical application of the scientific knowledge is stressed.
CELLS, GENES, AND BIOTECHNOLOGY (CAP)
This single-semester science elective provides an understanding of the kinds of questions that science can and cannot address, while exploring topics in cellular biology, the structure and function of genes, and biotechnology. Discussions probe the bioethical implications of our growing knowledge and application of technologies involving manipulation of cellular and genetic processes. Also includes experiences in a laboratory setting to conduct basic experiments that elucidate the structure of cells and the function of genes. This college level course should NOT be your first course in Biology. A strong high school science background is recommended. Successful completion of this course qualifies a student for three (3) hours of transferable college credit from Plymouth State University.
*DIGITAL MEDIA (CAP)
This one-semester computer science course introduces students to the creation, acquisition, editing, and delivery of computer-generated media. Work includes graphics, photography, sound, music, video, and interactive hypermedia. Students will use a range of tools to acquire, manipulate, and store their original content. The equivalent of CO 110 at St. Joseph's College, this course carries 4 credit hours for successful completion.
*INTRODUCTION TO COMPUTER PROGRAMMING
In this semester-long course, students will begin to learn the Python programming language. The language does not require much setup for simple programs, but can scale to run some of the largest programming projects in the world. Initially we will learn the basics of the language, but then we will apply our knowledge to several projects.
ECOLOGY OF THE LAKES REGION
Ecology is the scientific study of the relationships between organisms and their environment. Ecological systems such as forests and lakes are like complicated machines. Ecologists study the components of these systems to understand how the pieces fit together and how the systems function as a whole. Ecology of the Lakes Region is an introductory course that will explore the basic concepts in ecology and look at the ecological relationships that comprise the local environment that is the Lakes Region.
*NORTHERN FIELD STUDIES
Northern Forest Field Studies is not your traditional science class. Instead of teaching science with test tubes and thermometers, we use maps, compasses, and bows and arrows. is course is designed to change young people's lives forever by exposing them to the wonders of the natural world and the many great opportunities to be experienced in the outdoors. Students are taught life-long skills by using an integrated curriculum of science, math, some writing, and critical thinking skills. Class meeting times will allow the introduction and exploration of new topics, followed up by various outdoor experiences. Ultimately, the mission of this class is to instill in every student through a hands-on, real-world curriculum.
*PRINCIPLES OF HUMAN NUTRITION
Nutrition is a one-semester elective that covers the scientific principles of human nutrition in maintaining health and preventing disease. Nutrient requirements of the human body, biochemical functions, and interrelationships of nutrients are examined. Athletes learn how to fuel their bodies for building muscle, optimal sports performance, and for general health and well-being. Nutritional misconceptions and controversies are evaluated using readings, discussions, and hands-on lab experiences.
*ADVANCED HUMAN NUTRITION (CAP)
This year-long, college-level nutrition course focuses on the interrelationship between nutritional practices and human physical performance in sports and fitness. Topics covered include the role of carbohydrates, fats, proteins, vitamins, minerals and water on both everyday eating and physical performance. This course provides a foundational science background in chemistry, anatomy and physiology, and microbiology in the context of human nutrition, as well as hands-on lab experiences. This course carries four hours of credit, upon successful completion, from St. Joseph's college.
INTRODUCTION TO METEOROLOGY
This is an introductory course that explores the composition, structure and physical properties of the Earth's atmosphere. Weather phenomena will be studied on both the global and local scale. Major topics include heat balance, atmospheric stability, precipitation processes, cyclonic activity, weather analysis, and very basic weather forecasting techniques. Particular attention will be paid to the causes, structure and impact of tornadoes, hurricanes, thunderstorms and other severe weather systems. | 677.169 | 1 |
Todd Wittman
This is a list of student research projects I have
supervised. If you are a student interested in doing a summer project or independent
study with me, you should send me an e-mail. Please include a list of the math and computing
courses you have taken and a brief description of your research interests. | 677.169 | 1 |
Upper School—Mathematics
The mathematics program is intended to imbue North Cross students with much more than mere proficiency with numbers. Through the study and application, students are challenged to develop into critical and logical thinkers who can use thinking skills and techniques to develop invaluable problem-solving strategies. Additionally, students benefit from the development of an ordered, rational and structured algorithm for framing their ideas and are encouraged to extend this method to other disciplines. This product of the mathematics curriculum finds its greatest utility in the sciences, but also applies to the presentation of coherent, persuasive theses across the entire North Cross School curriculum.
Mathematics Course Descriptions
724 - Algebra 1 - Part 2 This course is a continuation of Algebra 1 -Part 1 and is designed to complete the study of fundamental algebra concepts. This continuing exploration of algebra will include topics of inequalities, rational expressions, laws of exponents, polynomials, systems of equations and radical expressions. An emphasis is placed on appropriate use of graphing calculator technology to enhance the study of function properties and graph behavior. The successful student will be prepared for Algebra 2 and/or Geometry. This course requires the use of a graphing calculator (TI-83/84/89 series).
743 - Geometry – (Prerequisite – Algebra 1 or equivalent) Students investigate the basic structure of Euclidean plane and solid geometry by exploring deductive reasoning through proof and problem solving using geometric structures and related algebraic operations. Spatial and visualization skills are developed and reinforced through practical applications of geometrical relationships. This course incorporates a practical, hands-on approach to the study of geometry by emphasizing manipulative aids in instruction as well as exploratory and collaborative approaches to learning. The course also emphasizes the development of logical reasoning based upon identification and use of valid premises and conclusions. Students may enroll following either Algebra 1 (preferred) or Algebra 2. This is a graduation requirement.
728 - Algebra 2 (Prerequisite – Algebra 1 or equivalent) This course is a continuation of the study of algebra including a comprehensive study of advanced algebra and trigonometry topics. The use of abstractions and unknown quantities introduced in Algebra 1 is extended to a more thorough examination of polynomial, rational, radical, exponential and logarithmic functions, the graphs of higher order functions in two dimensions, analytic geometry, matrices, and complex numbers. Emphasis is placed upon the relationship between the algebraic and graphical method of solving equations as well as upon problem solving skills correlating algebraic methods to practical (word) problems. It involves a rigorous study of linear and quadratic equations, polynomials, complex numbers, relations, and functions. Right angle trigonometry, trigonometric modeling, and analytic trigonometry are also a focus of the course. A graphing calculator (TI-83/84/89 series) is required. This is a graduation requirement.
1065 - College Algebra– (Prerequisites – Algebra 1 or equivalent, Geometry, Algebra 2) This course is intended to enrich the mathematics experience for students who desire to prepare for success in collegiate mathematics. Emphasis is placed upon mastery of new algebra concepts complemented by the reinforcement of previously introduced algebra concepts. Introduction of new concepts in mathematics not specifically related to calculus are included. An emphasis will be placed upon the appropriate use of graphing calculator technology to enhance the study of function properties and graph behavior. The successful student will be prepared for Precalculus. This course requires the use of a graphing calculator (TI-83/84/89 series).
733 - Precalculus – (Prerequisites – Algebra 1 or equivalent, Geometry, Algebra 2; Permission of the department) This course prepares students for the study of calculus in future math classes. Course content includes a study of polynomial, trigonometric, natural logarithmic, rational and radical functions. Function transformations, sequences and series, vectors and limits are included in the course content. Modeling and regression are presented as analytic tools. Emphasis is placed on refinement of previously acquired algebraic skills; however, the successful student must possess a fundamentally sound command of algebra achieved through the Algebra sequence. This course requires the use of a graphing calculator (TI-83/84/89 series).
730 - Calculus – (Prerequisite – Precalculus; Permission of the department) This course offers qualified students an alternative to AP Calculus (AB). The slower pace and in-depth investigations provide students with a unique opportunity to investigate topics not available in the faster-paced AP alternative. Many of the same topics will be covered as in AP Calculus (limits, continuity, derivatives, maximum and minimum problems, related rates, modeling, and integration, as time permits), but with accommodations for additional processing time and different learning styles. The use of technology and the alternative pacing create opportunities for the investigation of real-life situations which allows students to draw intelligent and calculus-based conclusions. Applications are drawn from science courses as well as from other specialized fields and real-life situations. Successful completion of this course will prepare students for a college-level calculus course and/or AP Calculus. This course requires the use of a graphing calculator (TI-83/84/89 series).
734 - AP Calculus AB – (Prerequisite – Precalculus; Permission of the department) This course provides a study of elementary and transcendental functions in calculus. Course content corresponds to the syllabus established by the College Board Advanced Placement Program and equates to approximately 1 semester of college calculus. Students will take the AP Calculus (AB) Examination in May from which placement and/or credit may be awarded at the collegiate level if a qualifying score is achieved. Topics included in the curriculum are limits and their properties, differentiation, applications of the derivative, curve sketching, integration, applications of integration including area and volume, logarithmic differentiation, simple differential equations and slope fields. This course requires the use of the TI-89 graphing calculator. All AP Calculus AB students are required to take the AP exam in May (additional cost of approximately $90.00)
731 – AP Calculus BC (Prerequisite – AP Calculus (AB); Permission of the department) This is a follow-on course to Calculus AB. The student completing both AP Calculus courses will have completed the equivalent of one full year of college-level calculus. Course content corresponds to the syllabus established by the College Board Advanced Placement Program. The successful student will be prepared to participate in the AP Calculus (BC) Examination in May. Emphasis will be placed upon refining previously acquired calculus skills as well as the introduction of new material including integration by partial fractions, integration by parts, differentiation and integration of parametrically defined equations, polar area, work and arc length, improper integrals, sequences and series including Taylor polynomials, Maclaurin series and power series. This course requires the use of the TI-89 graphing calculator. All AP Calculus BC students are required to take the AP exam in May (additional cost of approximately $90.00)
Mathematics Electives
1034 - AP Statistics – (Prerequisite –permission of department) The AP Statistics course is equivalent to a one-year, introductory, non-calculus-based college course in statistics. The course introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. There are four themes in the AP Statistics course: exploring data, sampling and experimentation, anticipating patterns, and statistical inference. Students use technology, investigations, problem solving, and writing as they build conceptual understanding. The successful student will be prepared to participate in the AP Statistics Examination in May. All AP Statistics students are required to take the AP exam in May (additional cost of approximately $90.00)
1052 – Introduction to Computer Science (Open to 10th – 12th graders) This course will introduce the fundamentals of computer programming by having them learn how to code using the object-oriented language Java. Topics that will be covered are general construction of programs, decisions, loops, and method calls. Collections are also introduced and include arrays and Array Lists. Lab Fee: $50
1067 – Advanced Computer Science (Prerequisite – Introduction to Computer Science) This course is designed to be a continuation of beginning programming. The student will further explore the language of Java. Students will take a more in depth look at Arrays and Arraylist. Students will also learn how to read in files, do error checking, study inheritance, learn different sorting methods, and work with graphics. Access to a home computer is required. Lab Fee: $50 | 677.169 | 1 |
Wazzup
What were your thoughts when you enrolled in this course?
It's about time I take this course. Why didn't I take it last sem? Yup I was lazy (16 units lang last sem!). When I saw that the prof was TBA, I just hoped that I'll get a 'nice' prof because this is a 13-something subject and I need all the help I can get. Voila! It's Sir Paul! TYL.
How comfortable are you with math?
Not comfortable at all. Though I haven't failed a Math subject before, my grades have not touched the line of 1. That's sad because I really studied my ass off for exams. | 677.169 | 1 |
Editorial Reviews
Gr 6 Up-In addition to acting as a young person's introduction to genealogy, this text also serves as a good general source for the study of Mexican-American history. In fact, the first third of the book is devoted to developing readers' pride in their heritage. (The authors assume that all readers will be Mexican Americans.) One strength is the authors' discussion of the genealogy of nontraditional and adoptive families. The lengthy lists for further reading that appear at the end of each chapter are also beneficial; however, nearly all of the titles are adult publications. Lila Perl's The Great Ancestor Hunt (Clarion, 1990) is a better overall introduction to the subject, but it does not specifically address Mexican-American genealogy.-Denise E. Agosto, Midland County Public Library, TX
School Library Journal
A text for a one-semester post-calculus course, based largely on the recommendations of the Linear Algebra Curriculum Study Group, with the main differences being early treatment of Euclidean vector geometry, extended treatment of determinants prior to eigenvalues, and ongoing material about linear transformations. Material on LU factorization is part of a chapter on numerical methods, and the QR factorization is not discussed. Key concepts are presented as definitions and proved theorems. Includes worked examples, exercises, and MATLAB exercises. Annotation c. Book News, Inc., Portland, OR (booknews.com) | 677.169 | 1 |
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An understanding of the developments in classical analysis during the nineteenth century is vital to a full appreciation of the history of twentieth-century mathematical thought. It was during the nineteenth century that the diverse mathematical formulae of the eighteenth century were systematized and the properties of functions of real and complex variables clearly distinguished; and it was then that the calculus matured into the rigorous discipline of today, becoming in the process a dominant influence on mathematics and mathematical physics.
This Source Book, a sequel to D. J. Struik's Source Book in Mathematics, 1200–1800, draws together more than eighty selections from the writings of the most influential mathematicians of the period. Thirteen chapters, each with an introduction by the editor, highlight the major developments in mathematical thinking over the century. All material is in English, and great care has been taken to maintain a high standard of accuracy both in translation and in transcription. Of particular value to historians and philosophers of science, the Source Book should serve as a vital reference to anyone seeking to understand the roots of twentieth-century mathematical | 677.169 | 1 |
The course provides an introduction to MATLAB. MATLAB is a program that is widely used by engineers to analyse data and to design new systems through simulations. It will be assumed that students have no prior knowledge either to MATLAB or to programming techniques. The course focuses on the practical use of MATLAB with some attention to good programming practices and little on general programming techniques. The course is set up as a practical based on the course book, with short introductions to the different chapters. The course is new and will evolve based on feedback and experience. | 677.169 | 1 |
This course teaches you all the important underlying concepts in functions in Mathematics. The knowledge that you gain here can be further completed in our next courses towards a complete mastery of calculus.
This course covers the following topics:
Function and Function Notation
Domain and Range of Functions
Rates of Change and Behavior of Graphs
Composition of Functions
Transformation of Functions
Absolute Value Functions
Inverse Functions
As described above, this course can also be taken in combination with our other courses in this course series. If you're interested in learning mathematics with us all the way up to calculus, please read our "Mathematics" page on "Greatitcourses" website | 677.169 | 1 |
ScienceandMath.com:Algebra 1 Tutor: A TOS Crew Review
ScienceandMath.commakes learning science and math, step-by-step EASY! The Bentz Test Laboratory is on our second trip through Algebra, this time with a "not-so-math-brained-student" and his mother (whom he inherited it from!)
When Algebra 1: Volume 1 a 3 disk DVD, part of a 3 volume set came up to review - we jumped at the chance!
The 7 hour DVD contains 10 sections designed to take Algebra concept-by-concept with step-by-step examples. This method is excellent for a struggling student, or for more extensive practice and review.
The lessons vary in length - most are about 50+ minutes. Each lesson features numerous examples being worked out and carefully explained by the instructor in front of a whiteboard. "The core philosophy of this DVD course is that "Algebra Is Easy". The way that you make it easy for a student is to start off with the basics and gradually move on to the tougher material." It has the feel of a traditional math tutor, but you can rewind anytime!
My tester is part way through his current Algebra 1 study. However, we have recently discovered several "gaps" in his understanding of the basics... please tell me you understand! He began with Section 1: Real Numbers and their Graphs. It was about an hour long lesson, mostly review for him. He also completed the worksheets included on the Fractions Thru Algebra Companion Worksheet CD, which contains over 600 printable pages with detailed solutions. He continued completing one section and the worksheets per week. It was an excellent review, and filled in some of those "gaps" in his algebra foundation!
Here are some of his thoughts:
"I liked how the instructor explained the concepts - using normal language"
"The instructor provides too many examples - he could teach the concept with less repetition"
"The worksheets were easy, but the detailed answers helped me see where I went wrong if I missed any."
What did I think? Well, I wish I would have had the convenience of having a personal Algebra Tutor. The video lessons are easily re-played when a concept needs some extra explanation. I enjoyed the relaxed, conversational tone of the instructor - not too technical. The DVD's promise to build a student's confidence through mastery - I know I learned a thing or two as well! | 677.169 | 1 |
Showing 1 to 1 of 1
There are similarities and differences between a Quinceaera and a Sweet 16.
These two are traditions for different cultures. An American girl would have a sweet 16,
but on the other hand a Latin American or Mexican girl would have a Quinceaera. They
both
Calc 1 Advice
Showing 1 to 2 of 2
The teacher clearly explains how to the material and teaches you many different methods on how to do it. Not only does he teach the subject but he shows you how it applies and can be used it the real world.
Course highlights:
He shows us how Calc can be used to calculate stock and other things in the life. He shows why Calculus even matters in the real world and when you would use it.
Hours per week:
3-5 hours
Advice for students:
As long as you pay attention and do the work the class should be fairly simple and easy to do. | 677.169 | 1 |
exponential functions—functions that have a base greater than 1 and a variable as the exponent. Survey the properties of exponents, the graphs of exponential functions, and the unique properties of the natural base e. Then sample a typical problem in compound interest. | 677.169 | 1 |
ISBN 13: 9780201003581
Linear Geometry
Most linear algebra texts neglect geometry in general and linear geometry in particular. This text for advanced undergraduates and graduate students stresses the relationship between algebra and linear geometry. Topics include transformations in the Euclidean plane, affine and Euclidean geometry, projective geometry and non-Euclidean geometries, and axiomatic plane geometry. 1965 edition. | 677.169 | 1 |
MATH 130 Advice
Showing 1 to 3 of 8
I recommend to take one math course over the winter break to avoid a whole long semester of math. However everyone is different, but I found it easier to just take the math class during winter in order to focus and do better.
Course highlights:
I learned a lot of algebra and am now prepared for a statistics math course. The class is very straightforward and you can retake exams.
Hours per week:
3-5 hours
Advice for students:
If you're good at math take other courses at the same time. If you're like me, that isn't really good at math, I recommend to focus only on math during a fast pace period to maximize learning and a better grade.
Course Term:Winter 2017
Professor:Mirfattah
Course Required?Yes
Course Tags:Math-heavyMany Small AssignmentsA Few Big Assignments
Jan 08, 2017
| Would highly recommend.
Pretty easy, overall.
Course Overview:
I would definitely recommend this course . The profesor always does his best to explain and if someone needs help he will personally do his best to help . | 677.169 | 1 |
Sets, Functions, and Logic An Introduction to Abstract Mathematics
ISBN-10: 1584884495
ISBN-13: 9781584884491
Edition: 3rd first edition published in 1981, Set, Function and Logic has smoothed the road to higher mathematics for legions of undergraduate students. Now in its third edition, the author--a leading popularizer of mathematics -- has fully revised his text to reflect a new generation. The narrative is more lively, less textbook-like. Remarks and asides link the various topics to the real world. The chapter on complex numbers and discussion of formal symbolic logic is gone in favor of a new introductory chapter on the nature of mathematics and more exercises. The result is an affordable, thoroughly engaging book that every student making the transition from calculus to higher mathematics will welcome.
Born in England in 1947 and living in America since 1987, Keith Devlin has written more than 20 books and numerous research articles on various elements of mathematics. From 1983 to 1989, he wrote a column on for the Manchester (England) Guardian. The collected columns are published in All the Math That's Fit to Print (1994) and cover a wide range of topics from calculating travel expenses to calculating pi. His book Logic and Information (1991) is an introduction to situation theory and situation semantics for mathematicians. Co-author of the PBS Nova episode "A Mathematical Mystery Tour," he is also the author of Devlin's Angle, a column on the Mathematical Association of America's electronic journal. Devlin lives in California, where he is dean of the school of science at Saint Mary's College in Morgana. He is currently studying the use of mathematics to analyze communication and information flow | 677.169 | 1 |
THE CONCEPTS OF LINEAR ALGEBRA REPRESENT THE
END RESULT OF A SERIES OF GENERALIZATIONS AND
ABSTRACTIONS
Linear Algebra contains certain fundamental concepts. Each of these fundamental concepts has
gone through a series of generalizations and abstractions before reaching their final form in the
very abstract subject of Linear Algebra. In Linear Algebra these fundamental concepts are all
defined in very abstruse, abstract form as sets of postulates which, on first encounter, seem like
intellectual nonsense. The definitions appear to make no sense, to be gobbledegook. They give
no understanding of the concepts they represent. They don't explain or give any intuitive insight
or feeling for the concepts. They are worded in such an abstract, unspecific, general way --
stating that that which is being defined is anything under the sun that has certain abstruse
properties or that obeys certain very abstract, odd-sounding laws. Their wording sends one's
mind into a dither. To understand the concepts you must have prior knowledge of the ideas from
which they arose. You must understand the initial idea and then the series of generalizations and
abstractions that led to the final concept. The final postulational forms of the definitions are
really just lists of very abstract properties of the concepts, lists of abstract laws that they obey.
Let us name these fundamental concepts of which we speak:
- concept of a vector
- concept of the length of a vector
- concept of the dot product, or inner product, of two vectors
- concept of a vector space
- concept of a subspace
- concept of a basis for a space or subspace
- concept of the dimension of a space or subspace
- concept of a linear transformation
To understand these concepts you must understand them in their initial form in two and three
dimensional space. We encounter vectors initially in physics as quantities that have both
magnitude and direction, quantities representing things like force, velocity, etc.. We view them
as arrows in two or three dimensional space representing some vector quantity. Each arrow is
defined by a doublet of numbers in two dimensional space, by a triplet of numbers in three
dimensional space. In two or three dimensional space the concept of the length of a vector is a
natural one and easily understood. Yet when this concept is encountered in Linear Algebra in the
form of the concept of a "norm" it is unrecognizable. The definition in terms of postulates is
mentally impenetrable. The same goes for the concept of the dot product and all the other
concepts. They all go through a series of re-definitions, generalizations and abstractions, and in
the end become unrecognizable. First the concept of a vector is re-defined as being simply an
n-tuple. Then the concept of Euclidean n-dimensional space is defined in terms of an extension
or generalization of the concept of three dimensional space. Then the concepts of vector length,
dot product, etc. are defined for n-tuples. Each re-definition involves some alteration from the
previous concept. In general, each re-definition involves the previous definition as a special case.
Thus, in the evolution of the concepts, the previous forms of the concepts are included as special
cases. But as the concepts become broader and more abstract, the previous versions of the
concept become unrecognizable and the definitions become opaque and abstruse.
To really understand the concepts of a vector space, a subspace, a basis for a space or subspace,
and the dimension of a space or subspace, for example, one must look to their meaning in two or
three dimensional space. Once he has a good, intuitive understanding of them there he must be
aware of various theorems which state various abstract properties they possess, various laws they
obey. Then he is in a position to read the abstruse postulational definitions of Linear Algebra and
make some sense of them remembering that they are merely lists of properties that the basic
concepts in two, three and n dimensional space possess. For example: The final abstract
definition encountered in Linear Algebra for a vector space is that anything under the sun that
possesses a certain stated set of properties is a linear space. The same type definition is stated for
a linear transformation. It is left up to your imagination to think of things that might possess
those properties. | 677.169 | 1 |
Showing 1 to 22 of 22
c
Math 151, Benjamin
Aurispa
Vector Supplement Part 1: Vectors
A vector is a quantity that has both magnitude and direction.
Vectors are drawn as directed line segments and are denoted by boldface letters a or by ~a.
The magnitude of a vector a is its len
<header>
Alp is a cheater
</header>
Do u want to know about the famous theory of the matrix?
Lewt me tell you a bit about it.
A long, long, long, long, long, loooooooooooooooooooooooooooooooooooooooooong, time ago, in a Galaxy within our universe,
the Tu
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MATH 1314 FINAL REVIEW
Creative Landscaping has 60 yards of fencing with which to enclose a rectangular flower garden. If the garden is
x yards long, express the gardens area as a function of the length.
x+2
Find the domain of the func
Adnan Desai
AP Statistics
09/19/2015
Mrs.Yilmaz
AP Statistics Chapter 10 Summary
The entire chapter is based on the concept of randomness. In statistics as a whole,
anything with a random observation is generally more accurate than any biased form of data
Adnan Desai
10/28/2015
A.P. Statistics
Mrs.Yilmaz
AP Statistics Chapter 11 Summary
Chapter 11, which is titled Sample Surveys, highlights the many types of sampling along
with their advantages and disadvantages. One distinguishable factor of this chapter
Adnan Desai
01/07/2016
A.P. Statistics
Mrs.Yilmaz
AP Statistics Chapter 18 Summary
Within Chapter 18 the main topic discussed it the of the confidence interval. This value
reflects how accurately trustable a set value is in perspective of the entire data
Adnan Desai
09/19/2015
AP Statistics
Mrs.Yilmaz
AP Statistics Chapter 3 Summary
The chapter starts out with a example situation that describes the statistics of ticket class,
gender, survival status. and age on the Titanic. After that the chapter introduc
Adnan Desai
09/27/2015
A.P. Statistics
Mrs.Yilmaz
AP Statistics Chapter 5 Summary
Chapter 5, which is mainly about the Empirical Rule and Normal distribution, begins by
comparing two data values. The chapter explains how the two data values can be compare
Adnan Desai
02/19/2016
AP Statistics
Mrs.Yilmaz
AP Statistics Chapter 22 Summary
This chapter serves as a continuation of the previous chapter and expands on the 2
proportion z test which we learned about in the previous chapter. This chapter increases on
Adnan Desai
10/27/2015
A.P. Statistics
Mrs.Yilmaz
AP Statistics Chapter 8 Summary
Chapter Eight, Regression Wisdom. communicates that a proper linear analysis is not
complete without meeting a few term and conditions. These conditions include the fact tha
Adnan Desai
A.P. Statistics
01/29/2015
Mrs.Yilmaz
AP Statistics Chapter 20 Summary
Within this chapter the book expands on the principle of hypothesis tests, and the proper
form of responding to one. While a hypothesis test can determine whether or not to
Adnan Desai
02/19/2016
AP Statistics
Mrs.Yilmaz
AP Statistics Chapter 21 Summary
In this chapter the concept of creating a confidence interval is built upon as it explains the
details for the statistical inferences that can occur in a real world situation
Adnan Desai
A.P. Statistics
01/22/2016
Mrs.Yilmaz
AP Statistics Chapter 19 Summary
Chapter 19 introduces the student to hypotheses. By explaining what a hypothesis is the
chapter sets up how this element can be implemented to statistical calculations. A h
Adnan Desai
11/04/2015
AP Statistics
Mrs.Yilmaz
AP Statistics Chapter 13 Summary
This chapter talks mostly about how the probability of the event is the likelihood of an
event to occur in frequency throughout multiple trials. More simply applied to an exa
Adnan Desai
10/28/2015
A.P. Statistics
Mrs.Yilmaz
AP Statistics Chapter 12 Summary
Chapter 12 is about the many types, factors, and statistical components related with
experimental design. The chapter starts out by introducing many common terms into furth | 677.169 | 1 |
ial Equations for Engineers
2.
2 ATypeset in LTEX.Copyright c 2008–2012 Jiˇí Lebl rThis work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0United States License. To view a copy of this license, visit or send a letter to Creative Commons, 171 Second Street, Suite300, San Francisco, California, 94105, USA.You can use, print, duplicate, share these notes as much as you want. You can base your own noteson these and reuse parts if you keep the license the same. If you plan to use these commercially (sellthem for more than just duplicating cost), then you need to contact me and we will work somethingout. If you are printing a course pack for your students, then it is fine if the duplication service ischarging a fee for printing and selling the printed copy. I consider that duplicating cost.During the writing of these notes, the author was in part supported by NSF grant DMS-0900885.See for more information (including contact information).
5.
Introduction0.1 Notes about these notesThis book originated from my class notes for teaching Math 286, differential equations, at theUniversity of Illinois at Urbana-Champaign in fall 2008 and spring 2009. It is a first course ondifferential equations for engineers. I have also taught Math 285 at UIUC and Math 20D at UCSDusing this book. The standard book for the UIUC course is Edwards and Penney, DifferentialEquations and Boundary Value Problems: Computing and Modeling [EP], fourth edition. Someexamples and applications are taken more or less from this book, though they also appear in manyother sources, of course. Among other books I have used as sources of information and inspirationare E.L. Ince's classic (and inexpensive) Ordinary Differential Equations [I], Stanley Farlow'sDifferential Equations and Their Applications [F], which is now available from Dover, Berg andMcGregor's Elementary Partial Differential Equations [BM], and Boyce and DiPrima's ElementaryDifferential Equations and Boundary Value Problems [BD]. See the Further Reading chapter at theend of the book. I taught the UIUC courses with the IODE software ( is a free software package that is used either with Matlab (proprietary) or Octave (freesoftware). Projects and labs from the IODE website are referenced throughout the notes. They neednot be used for this course, but I recommend using them. The graphs in the notes were made withthe Genius software (see I have used Genius in class toshow these (and other) graphs. This book is available from Check there for any possible Aupdates or errata. The LTEX source is also available from the same site for possible modificationand customization. I would like to acknowledge Rick Laugesen. I have used his handwritten class notes the firsttime I taught Math 286. My organization of this book, and the choice of material covered, is heavilyinfluenced by his class notes. Many examples and computations are taken from his notes. I am alsoheavily indebted to Rick for all the advice he has given me, not just on teaching Math 286. Forspotting errors and other suggestions, I would also like to acknowledge (in no particular order): JohnP. D'Angelo, Sean Raleigh, Jessica Robinson, Michael Angelini, Leonardo Gomes, Jeff Winegar,Ian Simon, Thomas Wicklund, Eliot Brenner, Sean Robinson, Jannett Susberry, Dana Al-Quadi,Cesar Alvarez, Cem Bagdatlioglu, Nathan Wong, Alison Shive, Shawn White, Wing Yip Ho, Joanne 5
6.
6 INTRODUCTIONShin, Gladys Cruz, Jonathan Gomez, Janelle Louie, Navid Froutan, Grace Victorine, Paul Pearson,Jared Teague, Ziad Adwan, Martin Weilandt, Sönmez Sahuto˘ lu, Pete Peterson, Thomas Gresham, ¸ gPrentiss Hyde, Jai Welch, and probably others I have forgotten. Finally I would like to acknowledgeNSF grant DMS-0900885. The organization of this book to some degree requires that they be done in order. Later chapterscan be dropped. The dependence of the material covered is roughly given in the following diagram: Introduction Chapter 1 Chapter 2 v ( + Chapter 3 Chapter 6 Chapter 7 ( Chapter 4 Chapter 5 There are some references in chapters 4 and 5 to material from chapter 3 (some linear algebra),but these references are not absolutely essential and can be skimmed over, so chapter 3 can safelybe dropped, while still covering chapters 4 and 5. The textbook was originally done for two types ofcourses. Either at 4 hours a week for a semester (Math 286 at UIUC):Introduction, chapter 1, chapter 2, chapter 3, chapter 4 (w/o § 4.10), chapter 5 (or 6 or 7). Or a shorter version (Math 285 at UIUC) of the course at 3 hours a week for a semester:Introduction, chapter 1, chapter 2, chapter 4 (w/o § 4.10), (and maybe chapter 5, 6, or 7). The complete book can be covered in approximately 65 lectures, that of course depends onthe lecturers speed and does not account for exams, review, or time spent in computer lab (if forexample using IODE). Therefore, a two quarter course can easily be run with the material, and ifone goes a bit slower than I do, then even a two semester course. The chapter on Laplace transform (chapter 6), the chapter on Sturm-Liouville (chapter 5), andthe chapter on power series (chapter 7) are more or less interchangeable time-wise. If time is shortthe first two sections of the chapter on power series (chapter 7) make a reasonable self-containedunit.
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0.2. INTRODUCTION TO DIFFERENTIAL EQUATIONS 70.2 Introduction to differential equationsNote: more than 1 lecture, §1.1 in [EP], chapter 1 in [BD]0.2.1 Differential equationsThe laws of physics are generally written down as differential equations. Therefore, all of scienceand engineering use differential equations to some degree. Understanding differential equations isessential to understanding almost anything you will study in your science and engineering classes.You can think of mathematics as the language of science, and differential equations are one ofthe most important parts of this language as far as science and engineering are concerned. Asan analogy, suppose that all your classes from now on were given in Swahili. Then it would beimportant to first learn Swahili, otherwise you will have a very tough time getting a good grade inyour other classes. You have already seen many differential equations without perhaps knowing about it. Andyou have even solved simple differential equations when you were taking calculus. Let us see anexample you may not have seen: dx + x = 2 cos t. (1) dtHere x is the dependent variable and t is the independent variable. Equation (1) is a basic exampleof a differential equation. In fact it is an example of a first order differential equation, since itinvolves only the first derivative of the dependent variable. This equation arises from Newton's lawof cooling where the ambient temperature oscillates with time.0.2.2 Solutions of differential equationsSolving the differential equation means finding x in terms of t. That is, we want to find a functionof t, which we will call x, such that when we plug x, t, and dx into (1), the equation holds. It is the dtsame idea as it would be for a normal (algebraic) equation of just x and t. We claim that x = x(t) = cos t + sin t dxis a solution. How do we check? We simply plug x into equation (1)! First we need to compute dt .We find that dx = − sin t + cos t. Now let us compute the left hand side of (1). dt dx + x = (− sin t + cos t) + (cos t + sin t) = 2 cos t. dtYay! We got precisely the right hand side. But there is more! We claim x = cos t + sin t + e−t is alsoa solution. Let us try, dx = − sin t + cos t − e−t . dt
8.
8 INTRODUCTIONAgain plugging into the left hand side of (1) dx + x = (− sin t + cos t − e−t ) + (cos t + sin t + e−t ) = 2 cos t. dtAnd it works yet again! So there can be many different solutions. In fact, for this equation all solutions can be written inthe form x = cos t + sin t + Ce−tfor some constant C. See Figure 1 for the graph of a few of these solutions. We will see how wecan find these solutions a few lectures from now. It turns out that solving differential equations 0 1 2 3 4 5 can be quite hard. There is no general method 3 3 that solves every differential equation. We will generally focus on how to get exact formulas for 2 2 solutions of certain differential equations, but we will also spend a little bit of time on getting ap- 1 1 proximate solutions. For most of the course we will look at ordi- 0 0nary differential equations or ODEs, by which we mean that there is only one independent variable-1 and derivatives are only with respect to this one -1 variable. If there are several independent vari- 0 1 2 3 4 5 ables, we will get partial differential equations or PDEs. We will briefly see these near the end of Figure 1: Few solutions of dx + x = 2 cos t. dt the course. Even for ODEs, which are very well under-stood, it is not a simple question of turning a crank to get answers. It is important to know when itis easy to find solutions and how to do so. Although in real applications you will leave much of theactual calculations to computers, you need to understand what they are doing. It is often necessaryto simplify or transform your equations into something that a computer can understand and solve.You may need to make certain assumptions and changes in your model to achieve this. To be a successful engineer or scientist, you will be required to solve problems in your job thatyou have never seen before. It is important to learn problem solving techniques, so that you mayapply those techniques to new problems. A common mistake is to expect to learn some prescriptionfor solving all the problems you will encounter in your later career. This course is no exception.0.2.3 Differential equations in practice So how do we use differential equations in science and engineering? First, we have some realworld problem that we wish to understand. We make some simplifying assumptions and create a
9.
0.2. INTRODUCTION TO DIFFERENTIAL EQUATIONS 9mathematical model. That is, we translate the real world situation into a set of differential equations.Then we apply mathematics to get some sort of a mathematical solution. There is still somethingleft to do. We have to interpret the results. We have to figure out what the mathematical solutionsays about the real world problem we started with. Learning how to formulate the mathematical Real world problemmodel and how to interpret the results is whatyour physics and engineering classes do. In this abstract interpretcourse we will focus mostly on the mathematicalanalysis. Sometimes we will work with simple real solve Mathematical Mathematicalworld examples, so that we have some intuition model solutionand motivation about what we are doing. Let us look at an example of this process. One of the most basic differential equations is thestandard exponential growth model. Let P denote the population of some bacteria on a Petri dish.We assume that there is enough food and enough space. Then the rate of growth of bacteria willbe proportional to the population. I.e. a large population grows quicker. Let t denote time (say inseconds) and P the population. Our model will be dP = kP, dtfor some positive constant k 0.Example 0.2.1: Suppose there are 100 bacteria at time 0 and 200 bacteria 10 seconds later. Howmany bacteria will there be 1 minute from time 0 (in 60 seconds)? First we have to solve the equation. We claim that a solution is given by P(t) = Cekt ,where C is a constant. Let us try: dP = Ckekt = kP. dtAnd it really is a solution. OK, so what now? We do not know C and we do not know k. But we know something. Weknow that P(0) = 100, and we also know that P(10) = 200. Let us plug these conditions in and seewhat happens. 100 = P(0) = Cek0 = C, 200 = P(10) = 100 ek10 .Therefore, 2 = e10k or ln 2 10 = k ≈ 0.069. So we know that P(t) = 100 e(ln 2)t/10 ≈ 100 e0.069t .At one minute, t = 60, the population is P(60) = 6400. See Figure 2 on the next page.
10.
10 INTRODUCTION Let us talk about the interpretation of the results. Does our solution mean that there mustbe exactly 6400 bacteria on the plate at 60s? No! We made assumptions that might not be trueexactly, just approximately. If our assumptions are reasonable, then there will be approximately6400 bacteria. Also, in real life P is a discrete quantity, not a real number. However, our model hasno problem saying that for example at 61 seconds, P(61) ≈ 6859.35. Normally, the k in P = kP is known, andwe want to solve the equation for different initial 0 10 20 30 40 50 60conditions. What does that mean? Take k = 1 6000 6000for simplicity. Now suppose we want to solve 5000 5000the equation dP = P subject to P(0) = 1000 (the dtinitial condition). Then the solution turns out to 4000 4000be (exercise) 3000 3000 P(t) = 1000 et . 2000 2000 We call P(t) = Cet the general solution, as 1000 1000every solution of the equation can be written in 0 0this form for some constant C. You will need an 0 10 20 30 40 50 60initial condition to find out what C is, in orderto find the particular solution we are looking for. Figure 2: Bacteria growth in the first 60 sec-Generally, when we say "particular solution," we onds.just mean some solution. Let us get to what we will call the four fundamental equations. These equations appear veryoften and it is useful to just memorize what their solutions are. These solutions are reasonably easyto guess by recalling properties of exponentials, sines, and cosines. They are also simple to check,which is something that you should always do. There is no need to wonder if you have rememberedthe solution correctly. First such equation is, dy = ky, dxfor some constant k 0. Here y is the dependent and x the independent variable. The generalsolution for this equation is y(x) = Cekx .We have already seen that this function is a solution above with different variable names. Next, dy = −ky, dxfor some constant k 0. The general solution for this equation is y(x) = Ce−kx .
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0.2. INTRODUCTION TO DIFFERENTIAL EQUATIONS 11Exercise 0.2.1: Check that the y given is really a solution to the equation. Next, take the second order differential equation d2 y = −k2 y, dx2for some constant k 0. The general solution for this equation is y(x) = C1 cos(kx) + C2 sin(kx).Note that because we have a second order differential equation, we have two constants in our generalsolution.Exercise 0.2.2: Check that the y given is really a solution to the equation. And finally, take the second order differential equation d2 y 2 = k2 y, dxfor some constant k 0. The general solution for this equation is y(x) = C1 ekx + C2 e−kx ,or y(x) = D1 cosh(kx) + D2 sinh(kx). For those that do not know, cosh and sinh are defined by e x + e−x cosh x = , 2 e x − e−x sinh x = . 2These functions are sometimes easier to work with than exponentials. They have some nice familiarproperties such as cosh 0 = 1, sinh 0 = 0, and dx cosh x = sinh x (no that is not a typo) and d ddx sinh x = cosh x.Exercise 0.2.3: Check that both forms of the y given are really solutions to the equation. An interesting note about cosh: The graph of cosh is the exact shape a hanging chain will make.This shape is called a catenary. Contrary to popular belief this is not a parabola. If you invert thegraph of cosh it is also the ideal arch for supporting its own weight. For example, the gatewayarch in Saint Louis is an inverted graph of cosh—if it were just a parabola it might fall down. Theformula used in the design is inscribed inside the arch: y = −127.7 ft · cosh(x/127.7 ft) + 757.7 ft.
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Chapter 1First order ODEs1.1 Integrals as solutionsNote: 1 lecture (or less), §1.2 in [EP], covered in §1.2 and §2.1 in [BD] A first order ODE is an equation of the form dy = f (x, y), dxor just y = f (x, y).In general, there is no simple formula or procedure one can follow to find solutions. In the next fewlectures we will look at special cases where solutions are not difficult to obtain. In this section, letus assume that f is a function of x alone, that is, the equation is y = f (x). (1.1)We could just integrate (antidifferentiate) both sides with respect to x. y (x) dx = f (x) dx + C,that is y(x) = f (x) dx + C.This y(x) is actually the general solution. So to solve (1.1), we find some antiderivative of f (x) andthen we add an arbitrary constant to get the general solution. Now is a good time to discuss a point about calculus notation and terminology. Calculustextbooks muddy the waters by talking about the integral as primarily the so-called indefinite 13
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14 CHAPTER 1. FIRST ORDER ODESintegral. The indefinite integral is really the antiderivative (in fact the whole one-parameter familyof antiderivatives). There really exists only one integral and that is the definite integral. The onlyreason for the indefinite integral notation is that we can always write an antiderivative as a (definite)integral. That is, by the fundamental theorem of calculus we can always write f (x) dx + C as x f (t) dt + C. x0Hence the terminology to integrate when we may really mean to antidifferentiate. Integration isjust one way to compute the antiderivative (and it is a way that always works, see the followingexamples). Integration is defined as the area under the graph, it only happens to also computeantiderivatives. For sake of consistency, we will keep using the indefinite integral notation when wewant an antiderivative, and you should always think of the definite integral.Example 1.1.1: Find the general solution of y = 3x2 . Elementary calculus tells us that the general solution must be y = x3 + C. Let us check: y = 3x2 .We have gotten precisely our equation back. Normally, we also have an initial condition such as y(x0 ) = y0 for some two numbers x0 and y0(x0 is usually 0, but not always). We can then write the solution as a definite integral in a nice way.Suppose our problem is y = f (x), y(x0 ) = y0 . Then the solution is x y(x) = f (s) ds + y0 . (1.2) x0Let us check! We compute y = f (x), via the fundamental theorem of calculus, and by Jupiter, y is a x0solution. Is it the one satisfying the initial condition? Well, y(x0 ) = x f (x) dx + y0 = y0 . It is! 0 Do note that the definite integral and the indefinite integral (antidifferentiation) are completelydifferent beasts. The definite integral always evaluates to a number. Therefore, (1.2) is a formulawe can plug into the calculator or a computer, and it will be happy to calculate specific values for us.We will easily be able to plot the solution and work with it just like with any other function. It is notso crucial to always find a closed form for the antiderivative.Example 1.1.2: Solve 2 y = e−x , y(0) = 1. By the preceding discussion, the solution must be x 2 y(x) = e−s ds + 1. 0Here is a good way to make fun of your friends taking second semester calculus. Tell them to findthe closed form solution. Ha ha ha (bad math joke). It is not possible (in closed form). There isabsolutely nothing wrong with writing the solution as a definite integral. This particular integral isin fact very important in statistics.
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1.1. INTEGRALS AS SOLUTIONS 15 Using this method, we can also solve equations of the form y = f (y).Let us write the equation in Leibniz notation. dy = f (y). dxNow we use the inverse function theorem from calculus to switch the roles of x and y to obtain dx 1 = . dy f (y)What we are doing seems like algebra with dx and dy. It is tempting to just do algebra with dxand dy as if they were numbers. And in this case it does work. Be careful, however, as this sort ofhand-waving calculation can lead to trouble, especially when more than one independent variable isinvolved. At this point we can simply integrate, 1 x(y) = dy + C. f (y)Finally, we try to solve for y.Example 1.1.3: Previously, we guessed y = ky (for some k 0) has the solution y = Cekx . Wecan now find the solution without guessing. First we note that y = 0 is a solution. Henceforth, weassume y 0. We write dx 1 = . dy kyWe integrate to obtain 1 x(y) = x = ln |y| + D, kwhere D is an arbitrary constant. Now we solve for y (actually for |y|). |y| = ekx−kD = e−kD ekx .If we replace e−kD with an arbitrary constant C we can get rid of the absolute value bars (which wecan do as D was arbitrary). In this way, we also incorporate the solution y = 0. We get the samegeneral solution as we guessed before, y = Cekx .Example 1.1.4: Find the general solution of y = y2 . First we note that y = 0 is a solution. We can now assume that y 0. Write dx 1 = 2. dy y
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16 CHAPTER 1. FIRST ORDER ODESWe integrate to get −1 x= + C. yWe solve for y = C−x . So the general solution is 1 1 y= or y = 0. C−xNote the singularities of the solution. If for example C = 1, then the solution "blows up" as weapproach x = 1. Generally, it is hard to tell from just looking at the equation itself how the solutionis going to behave. The equation y = y2 is very nice and defined everywhere, but the solution isonly defined on some interval (−∞, C) or (C, ∞). Classical problems leading to differential equations solvable by integration are problems dealingwith velocity, acceleration and distance. You have surely seen these problems before in yourcalculus class.Example 1.1.5: Suppose a car drives at a speed et/2 meters per second, where t is time in seconds.How far did the car get in 2 seconds (starting at t = 0)? How far in 10 seconds? Let x denote the distance the car traveled. The equation is x = et/2 .We can just integrate this equation to get that x(t) = 2et/2 + C.We still need to figure out C. We know that when t = 0, then x = 0. That is, x(0) = 0. So 0 = x(0) = 2e0/2 + C = 2 + C.Thus C = −2 and x(t) = 2et/2 − 2.Now we just plug in to get where the car is at 2 and at 10 seconds. We obtain x(2) = 2e2/2 − 2 ≈ 3.44 meters, x(10) = 2e10/2 − 2 ≈ 294 meters.Example 1.1.6: Suppose that the car accelerates at a rate of t2 m/s2 . At time t = 0 the car is at the 1meter mark and is traveling at 10 m/s. Where is the car at time t = 10. Well this is actually a second order problem. If x is the distance traveled, then x is the velocity,and x is the acceleration. The equation with initial conditions is x = t2 , x(0) = 1, x (0) = 10.What if we say x = v. Then we have the problem v = t2 , v(0) = 10.Once we solve for v, we can integrate and find x.Exercise 1.1.1: Solve for v, and then solve for x. Find x(10) to answer the question.
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18 CHAPTER 1. FIRST ORDER ODES1.2 Slope fieldsNote: 1 lecture, §1.3 in [EP], §1.1 in [BD] At this point it may be good to first try the Lab I and/or Project I from the IODE website: As we said, the general first order equation we are studying looks like y = f (x, y).In general, we cannot simply solve these kinds of equations explicitly. It would be good if we couldat least figure out the shape and behavior of the solutions, or if we could even find approximatesolutions for any equation.1.2.1 Slope fieldsAs you have seen in IODE Lab I (if you did it), the equation y = f (x, y) gives you a slope at eachpoint in the (x, y)-plane. We can plot the slope at lots of points as a short line through the point(x, y) with the slope f (x, y). See Figure 1.1.1: Slope field of y = xy. Figure 1.2: Slope field of y = xy with a graph of solutions satisfying y(0) = 0.2, y(0) = 0, and y(0) = −0.2. We call this picture the slope field of the equation. If we are given a specific initial conditiony(x0 ) = y0 , we can look at the location (x0 , y0 ) and follow the slopes. See Figure 1.2. By looking at the slope field we can get a lot of information about the behavior of solutions. Forexample, in Figure 1.2 we can see what the solutions do when the initial conditions are y(0) 0,y(0) = 0 and y(0) 0. Note that a small change in the initial condition causes quite different
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1.2. SLOPE FIELDS 19behavior. On the other hand, plotting a few solutions of the equation y = −y, we see that no matterwhat y(0) is, all solutions tend to zero as x tends to infinity. See Figure 1.3. -3 -2 -1 0 1 2 3 3 3 2 2 1 1 0 0 -1 -1 -2 -2 -3 -3 -3 -2 -1 0 1 2 3 Figure 1.3: Slope field of y = −y with a graph of a few solutions.1.2.2 Existence and uniquenessWe wish to ask two fundamental questions about the problem y = f (x, y), y(x0 ) = y0 . (i) Does a solution exist? (ii) Is the solution unique (if it exists)? What do you think is the answer? The answer seems to be yes to both does it not? Well, prettymuch. But there are cases when the answer to either question can be no. Since generally the equations we encounter in applications come from real life situations, itseems logical that a solution always exists. It also has to be unique if we believe our universe isdeterministic. If the solution does not exist, or if it is not unique, we have probably not devised thecorrect model. Hence, it is good to know when things go wrong and why.Example 1.2.1: Attempt to solve: 1 y = , y(0) = 0. x Integrate to find the general solution y = ln |x| + C. Note that the solution does not exist at x = 0.See Figure 1.4 on the next page.
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20 CHAPTER 1. FIRST ORDER ODES4: Slope field of y = 1/x. Figure 1.5: Slope field of y = 2 |y| with two solutions satisfying y(0) = 0.Example 1.2.2: Solve: y = 2 |y|, y(0) = 0. See Figure 1.5. Note that y = 0 is a solution. But another solution is the function x2 if x ≥ 0, y(x) = 2 −x if x 0. It is actually hard to tell by staring at the slope field that the solution is not unique. Is there anyhope? Of course there is. It turns out that the following theorem is true. It is known as Picard'stheorem∗ .Theorem 1.2.1 (Picard's theorem on existence and uniqueness). If f (x, y) is continuous (as afunction of two variables) and ∂ f exists and is continuous near some (x0 , y0 ), then a solution to ∂y y = f (x, y), y(x0 ) = y0 ,exists (at least for some small interval of x's) and is unique. Note that the problems y = 1/x, y(0) = 0 and y = 2 |y|, y(0) = 0 do not satisfy the hypothesisof the theorem. Even if we can use the theorem, we ought to be careful about this existence business.It is quite possible that the solution only exists for a short while.Example 1.2.3: For some constant A, solve: y = y2 , y(0) = A. ∗ Named after the French mathematician Charles Émile Picard (1856 – 1941)
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22 CHAPTER 1. FIRST ORDER ODES1.3 Separable equationsNote: 1 lecture, §1.4 in [EP], §2.2 in [BD] When a differential equation is of the form y = f (x), we can just integrate: y = f (x) dx + C.Unfortunately this method no longer works for the general form of the equation y = f (x, y).Integrating both sides yields y= f (x, y) dx + C.Notice the dependence on y in the integral.1.3.1 Separable equationsLet us suppose that the equation is separable. That is, let us consider y = f (x)g(y),for some functions f (x) and g(y). Let us write the equation in the Leibniz notation dy = f (x)g(y). dxThen we rewrite the equation as dy = f (x) dx. g(y)Now both sides look like something we can integrate. We obtain dy = f (x) dx + C. g(y)If we can find closed form expressions for these two integrals, we can, perhaps, solve for y.Example 1.3.1: Take the equation y = xy.First note that y = 0 is a solution, so assume y 0 from now on. Write the equation as dy dx = xy, then dy = x dx + C. yWe compute the antiderivatives to get x2 ln |y| = + C. 2
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1.3. SEPARABLE EQUATIONS 23Or x2 x2 x2 |y| = e 2 +C = e 2 eC = De 2 ,where D 0 is some constant. Because y = 0 is a solution and because of the absolute value weactually can write: x2 y = De 2 ,for any number D (including zero or negative). We check: x2 x2 y = Dxe 2 = x De 2 = xy.Yay! We should be a little bit more careful with this method. You may be worried that we wereintegrating in two different variables. We seemed to be doing a different operation to each side. Letus work this method out more rigorously. dy = f (x)g(y) dxWe rewrite the equation as follows. Note that y = y(x) is a function of x and so is dy dx ! 1 dy = f (x) g(y) dxWe integrate both sides with respect to x. 1 dy dx = f (x) dx + C. g(y) dxWe can use the change of variables formula. 1 dy = f (x) dx + C. g(y)And we are done.1.3.2 Implicit solutionsIt is clear that we might sometimes get stuck even if we can do the integration. For example, takethe separable equation xy y = 2 . y +1We separate variables, y2 + 1 1 dy = y + dy = x dx. y y
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1.4. LINEAR EQUATIONS AND THE INTEGRATING FACTOR 271.4 Linear equations and the integrating factorNote: 1 lecture, §1.5 in [EP], §2.1 in [BD] One of the most important types of equations we will learn how to solve are the so-called linearequations. In fact, the majority of the course is about linear equations. In this lecture we focus onthe first order linear equation. A first order equation is linear if we can put it into the form: y + p(x)y = f (x). (1.3)Here the word "linear" means linear in y and y ; no higher powers nor functions of y or y appear.The dependence on x can be more complicated. Solutions of linear equations have nice properties. For example, the solution exists whereverp(x) and f (x) are defined, and has the same regularity (read: it is just as nice). But most importantlyfor us right now, there is a method for solving linear first order equations. The trick is to rewrite the left hand side of (1.3) as a derivative of a product of y with anotherfunction. To this end we find a function r(x) such that d r(x)y + r(x)p(x)y = r(x)y . dxThis is the left hand side of (1.3) multiplied by r(x). So if we multiply (1.3) by r(x), we obtain d r(x)y = r(x) f (x). dxNow we integrate both sides. The right hand side does not depend on y and the left hand side iswritten as a derivative of a function. Afterwards, we solve for y. The function r(x) is called theintegrating factor and the method is called the integrating factor method. We are looking for a function r(x), such that if we differentiate it, we get the same function backmultiplied by p(x). That seems like a job for the exponential function! Let r(x) = e p(x) dx .We compute: y + p(x)y = f (x), e p(x) dx y + e p(x) dx p(x)y = e p(x) dx f (x), d e p(x) dx y = e p(x) dx f (x), dx e p(x) dx y= e p(x) dx f (x) dx + C, y = e− p(x) dx e p(x) dx f (x) dx + C . Of course, to get a closed form formula for y, we need to be able to find a closed form formulafor the integrals appearing above.
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28 CHAPTER 1. FIRST ORDER ODESExample 1.4.1: Solve 2 y + 2xy = e x−x , y(0) = −1. First note that p(x) = 2x and f (x) = e x−x . The integrating factor is r(x) = e = e x . We 2 p(x) dx 2multiply both sides of the equation by r(x) to get 2 2 2 2 e x y + 2xe x y = e x−x e x , d x2 e y = ex . dxWe integrate 2 e x y = e x + C, 2 2 y = e x−x + Ce−x .Next, we solve for the initial condition −1 = y(0) = 1 + C, so C = −2. The solution is 2 2 y = e x−x − 2e−x . p(x)dx Note that we do not care which antiderivative we take when computing e . You can alwaysadd a constant of integration, but those constants will not matter in the end.Exercise 1.4.1: Try it! Add a constant of integration to the integral in the integrating factor andshow that the solution you get in the end is the same as what we got above. An advice: Do not try to remember the formula itself, that is way too hard. It is easier toremember the process and repeat it. Since we cannot always evaluate the integrals in closed form, it is useful to know how to writethe solution in definite integral form. A definite integral is something that you can plug into acomputer or a calculator. Suppose we are given y + p(x)y = f (x), y(x0 ) = y0 .Look at the solution and write the integrals as definite integrals. x x t − p(s) ds p(s) ds y(x) = e x0 e x0 f (t) dt + y0 . (1.4) x0You should be careful to properly use dummy variables here. If you now plug such a formula into acomputer or a calculator, it will be happy to give you numerical answers.Exercise 1.4.2: Check that y(x0 ) = y0 in formula (1.4).
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1.4. LINEAR EQUATIONS AND THE INTEGRATING FACTOR 29Exercise 1.4.3: Write the solution of the following problem as a definite integral, but try to simplifyas far as you can. You will not be able to find the solution in closed form. 2 y + y = e x −x , y(0) = 10.Remark 1.4.1: Before we move on, we should note some interesting properties of linear equations.First, for the linear initial value problem y + p(x)y = f (x), y(x0 ) = y0 , there is always an explicitformula (1.4) for the solution. Second, it follows from the formula (1.4) that if p(x) and f (x)are continuous on some interval (a, b), then the solution y(x) exists and is differentiable on (a, b).Compare with the simple nonlinear example we have seen previously, y = y2 , and compare toTheorem 1.2.1.Example 1.4.2: Let us discuss a common simple application of linear equations. This type ofproblem is used often in real life. For example, linear equations are used in figuring out theconcentration of chemicals in bodies of water (rivers and lakes). A 100 liter tank contains 10 kilograms of salt dissolved in 60 liters of 5 L/min, 0.1 kg/Lwater. Solution of water and salt (brine) with concentration of 0.1 kilogramsper liter is flowing in at the rate of 5 liters a minute. The solution in thetank is well stirred and flows out at a rate of 3 liters a minute. How muchsalt is in the tank when the tank is full? Let us come up with the equation. Let x denote the kilograms of salt 60 Lin the tank, let t denote the time in minutes. Then for a small change ∆t 10 kg of saltin time, the change in x (denoted ∆x) is approximately ∆x ≈ (rate in × concentration in)∆t − (rate out × concentration out)∆t. 3 L/minDividing through by ∆t and taking the limit ∆t → 0 we see that dx = (rate in × concentration in) − (rate out × concentration out). dtIn our example, we have rate in = 5, concentration in = 0.1, rate out = 3, x x concentration out = = . volume 60 + (5 − 3)tOur equation is, therefore, dx x = (5 × 0.1) − 3 . dt 60 + 2tOr in the form (1.3) dx 3 + x = 0.5. dt 60 + 2t
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1.4. LINEAR EQUATIONS AND THE INTEGRATING FACTOR 31Exercise 1.4.6: Solve y + 3x2 y = sin(x) e−x , with y(0) = 1. 3Exercise 1.4.7: Solve y + cos(x)y = cos(x).Exercise 1.4.8: Solve 1 x2 +1 y + xy = 3, with y(0) = 0.Exercise 1.4.9: Suppose there are two lakes located on a stream. Clean water flows into the firstlake, then the water from the first lake flows into the second lake, and then water from the secondlake flows further downstream. The in and out flow from each lake is 500 liters per hour. The firstlake contains 100 thousand liters of water and the second lake contains 200 thousand liters of water.A truck with 500 kg of toxic substance crashes into the first lake. Assume that the water is beingcontinually mixed perfectly by the stream. a) Find the concentration of toxic substance as a functionof time in both lakes. b) When will the concentration in the first lake be below 0.001 kg per liter? c)When will the concentration in the second lake be maximal?Exercise 1.4.10: Newton's law of cooling states that dx = −k(x − A) where x is the temperature, dtt is time, A is the ambient temperature, and k 0 is a constant. Suppose that A = A0 cos(ωt) forsome constants A0 and ω. That is, the ambient temperature oscillates (for example night and daytemperatures). a) Find the general solution. b) In the long term, will the initial conditions makemuch of a difference? Why or why not?Exercise 1.4.11: Initially 5 grams of salt are dissolved in 20 liters of water. Brine with concentrationof salt 2 grams of salt per liter is added at a rate of 3 liters a minute. The tank is mixed well and isdrained at 3 liters a minute. How long does the process have to continue until there are 20 grams ofsalt in the tank?Exercise 1.4.12: Initially a tank contains 10 liters of pure water. Brine of unknown (but constant)concentration of salt is flowing in at 1 liter per minute. The water is mixed well and drained at 1liter per minute. In 20 minutes there are 15 grams of salt in the tank. What is the concentration ofsalt in the incoming brine?Exercise 1.4.101: Solve y + 3x2 y = x2 .Exercise 1.4.102: Solve y + 2 sin(2x)y = 2 sin(2x), y(π/2) = 3.Exercise 1.4.103: Suppose a water tank is being pumped out at 3 L/min. The water tank starts at 10 Lof clean water. Water with toxic substance is flowing into the tank at 2 L/min, with concentration 20t g/Lat time t. When the tank is half empty, how many grams of toxic substance are in the tank (assumingperfect mixing)?Exercise 1.4.104: Suppose we have bacteria on a plate and suppose that we are slowly adding atoxic substance such that the rate of growth is slowing down. That is, suppose that dP = (2 − 0.1t)P. dtIf P(0) = 1000, find the population at t = 5.
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1.5. SUBSTITUTION 33Note that D = 0 gives y = x + 2, but no value of D gives the solution y = x. Substitution in differential equations is applied in much the same way that it is applied incalculus. You guess. Several different substitutions might work. There are some general things tolook for. We summarize a few of these in a table. When you see Try substituting yy v = y2 y2 y v = y3 (cos y)y v = sin y (sin y)y v = cos y y ey v = ey Usually you try to substitute in the "most complicated" part of the equation with the hopes ofsimplifying it. The above table is just a rule of thumb. You might have to modify your guesses. If asubstitution does not work (it does not make the equation any simpler), try a different one.1.5.2 Bernoulli equationsThere are some forms of equations where there is a general rule for substitution that always works.One such example is the so-called Bernoulli equation∗ . y + p(x)y = q(x)yn .This equation looks a lot like a linear equation except for the yn . If n = 0 or n = 1, then the equationis linear and we can solve it. Otherwise, the substitution v = y1−n transforms the Bernoulli equationinto a linear equation. Note that n need not be an integer.Example 1.5.1: Solve xy + y(x + 1) + xy5 = 0, y(1) = 1.First, the equation is Bernoulli (p(x) = (x + 1)/x and q(x) = −1). We substitute v = y1−5 = y−4 , v = −4y−5 y .In other words, (−1/4) y5 v = y . So xy + y(x + 1) + xy5 = 0, −xy5 v + y(x + 1) + xy5 = 0, 4 −x v + y−4 (x + 1) + x = 0, 4 −x v + v(x + 1) + x = 0, 4 ∗ There are several things called Bernoulli equations, this is just one of them. The Bernoullis were a prominentSwiss family of mathematicians. These particular equations are named for Jacob Bernoulli (1654 – 1705).
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1.6. AUTONOMOUS EQUATIONS 371.6 Autonomous equationsNote: 1 lecture, §2.2 in [EP], §2.5 in [BD] Let us consider problems of the form dx = f (x), dtwhere the derivative of solutions depends only on x (the dependent variable). Such equations arecalled autonomous equations. If we think of t as time, the naming comes from the fact that theequation is independent of time. Let us come back to the cooling coffee problem (see Example 1.3.3). Newton's law of coolingsays that dx = −k(x − A), dtwhere x is the temperature, t is time, k is some constant, and A is the ambient temperature. SeeFigure 1.6 for an example with k = 0.3 and A = 5. Note the solution x = A (in the figure x = 5). We call these constant solutions the equilibriumsolutions. The points on the x axis where f (x) = 0 are called critical points. The point x = A isa critical point. In fact, each critical point corresponds to an equilibrium solution. Note also, bylooking at the graph, that the solution x = A is "stable" in that small perturbations in x do not leadto substantially different solutions as t grows. If we change the initial condition a little bit, then ast → ∞ we get x → A. We call such critical points stable. In this simple example it turns out that allsolutions in fact go to A as t → ∞. If a critical point is not stable we would say it is unstable. 0 5 10 15 20 0 5 10 15 20 10 10 10.0 10.0 7.5 7.5 5 5 5.0 5.0 0 0 2.5 2.5 0.0 0.0 -5 -5 -2.5 -2.5-10 -10 -5.0 -5.0 0 5 10 15 20 0 5 10 15 20Figure 1.6: Slope field and some solutions of Figure 1.7: Slope field and some solutions ofx = −0.3 (x − 5). x = 0.1 x (5 − x). | 677.169 | 1 |
Syllabus: We will begin with a rapid review of basic
group theory, and then move on to more advanced topics in group
theory, including free groups and presentations, the Sylow theorems, the
Jordan-Holder theorem, solvable groups, semidirect products, and profinite
groups. We then cover field theory and Galois theory,
and if time allows, we will also discuss supplementary topics in
commutative algebra such as Grobner bases, Artin rings, and Dedekind
domains.
Grading: 100% homework
Homework: Homework will be assigned roughly weekly
Welcome to Math 250C: Algebra
Homework assignments and supplementary notes will be posted here as the
course progresses. We will not necessarily follow the textbook closely,
so attendance at lecture is strongly encouraged.
Problem sets
Problem sets will be posted here on Fridays, due the following Fridays.
You are encouraged to collaborate with other students, as long as you
do not simply copy their answers. | 677.169 | 1 |
Matching Trigonometric Functions, Graphs and Descriptions
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Product Description
The cards are designed to be run off on card stock and cut into 3.5 inches by 5 inches cards. The cards have been randomly numbered and an answer key has been provided to make it easy to check student matchings.
Each set of four cards match a function, a graph, a description of the period, amplitude, and two points on the graph and some description of the function. You can make up a lesson that includes any number of four matched cards. It is not necessary to use all 25 functions.
After you decide which cards you want to use in the lesson, randomly handout the cards to the students. One student will have a function, another student will have a graph of the function, another will have the period, amplitude, and y-intercept of the function, and a fourth student will have a description of the function. The challenge for students is to find their matching cards.
The matching of the cards can be checked by using the following chart using the symbol on the bottom of the card. | 677.169 | 1 |
Ability to sketch and/or graph functions (i.e., to possess an intuitive
understanding of a functions behavior).
Appreciation of a practical use of each topic in addition to a conceptual
understanding of it.
Suggested Learning Behaviors:
Be an active class participant: Read the material in the text
before we talk about it in class so that you are aware of any
challenging areas. Ask an appropriate probing question as a difficult subject
is covered. In this way you will focus the lecture time where it will
help you the most.
Be an active practitioner: Do the odd numbered problems (or as close to a
representative sampling of them as you can - the situation is simple - the
more you do, the more you will learn and the better you will do in this
class). If your answer does not agree with the answer in the back of the
textbook, arrive at class three minutes early. Put the problem up on the
board, as far as you can go (i.e., list what is given, what is requested, do
as much as you can, sketch roughly, if appropriate, etc. At the start of the
class we will talk about how to take the next step and your efforts will
insure that I will be talking directly to your concerns instead of rehashing
the stuff you already know. Of course, if you prefer, get help with that
problem somehow: go to the Learning Center, use my office hours, talk to a
friend, or develop your stick-to-itiveness and solve it by trial and error
yourself. )
Don't make the same mistake twice: If you do not get an exam problem
correct, find out how to do it in case that type of problem shows up again
later in the semester. (Hint: It will!) Everything in this course is
cumulative and each concept and procedure builds on earlier work. (The good
news is that after you master the material, early difficulties will be
forgotten as far as final grades are concerned.)
Work alone, then share your solutions with a classmate: You will find you
will solidify and clarify your own understanding of the concepts when you
explain them to someone else. If you form a learning/study team with a
classmate, you will benefit not only from the teaching experience but also
from your colleague's efforts and explanations on problems you did not tackle.
Agree to a joint study plan, divide up the problems between you, and arrange a
regular time to pool your solo efforts. Working both separately and together
is usually a more efficient use of your study time than the same amount of
time always working alone or always working together.
reward students who make a sustained effort, over the entire course, to
master Elementary Functions topics, and
encourage mastery of all topics of the course.
You may, if you wish, take up to two retests on the material of each particular topic.
If your grades on a Topic increase over time, the latest grade will be used
as your "Topic Average" in computing your "Adjusted Numerical Class Average."
Otherwise, the average of all your grades (dropping the lowest one) for that
Topic will be used as your Topic Average in computing your Adjusted Numerical
Class Average.
Adjusted Numerical Class Average =
=Average of Topic Averages, if all Topic averages are 70 or higher,
OR
=Average of Topic Averages - 5, if one (or more) Topic Average is below 70.
Adjusted Numerical Class Average will be assigned a final letter grade
according to the following scheme: | 677.169 | 1 |
Meta
Regardless of what class you are in, you will have to write a Mathematical Autobiography for me sometime in the first cycle of school.
What is a mathematical autobiography?
A mathematical autobiography is the story of your math history. You will tell me about your journey through "mathdom" from your earliest memories to present.
What should be included in the mathematical autobiography?
Your mathematical autobiography should answer the following questions (not necessarily in this order, and you can include more):
What are your earliest math memories?
What are some of the math experiences you remember through elementary school and middle school?
Who was the best math teacher you had, and what made them the best?
What are three things you like about math? What is something about math that frustrates you?
How do you use math in your everyday life now?
How do you think you might use math in your future?
What is something you think would be interesting to learn about in math this year?
What do you look forward to in math this year?
What is required in this assignment?
Answer the above questions in a minimum of two (2) paragraphs. It might take more. There is no maximum, because I am a storyteller who loves to read your stories. Tell on!
Email your response within the email or as an attachment to smithca@tcis.or.kr.
When is it due?
A Block Algebra 2 — Monday, August 19
B Block Methods B — Monday, August 19
C Block Methods A — Tuesday, August 20
D Block Methods B — Monday, August 19
E Block Geometry — Monday, August 19
How will I be graded on this assignment?
Grades for this assignment:
ATL Communication: out of 3 (Did you answer all of the above questions?)
ATL Organization: out of 4 (Did you turn it in on time? 4 = early; 3 = on time; 2 = one day late; 1 = more late) | 677.169 | 1 |
Go to for the index, playlists and more maths videos on binary operations and other maths topics.
published:07 Jan 2016
views:570109 Dec 2013
views:4861Description of how an associative cache memory works. Some previous knowledge about direct mapped cache is assumed.
published:25 Sep 2013
views:48592
In this tutorial we'll see how C resolves conflicts between operators that enjoy the same priority.
published:29 Jan 2013
views:7304Associative property
Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider the following equations:
Even though the parentheses were rearranged, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, it can be said that "addition and multiplication of real numbers are associative operations".
Associativity is not to be confused with commutativity, which addresses whether a × b = b × aLinear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.
The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.
Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic. On the other hand, matrices are arrays of numbers, so there is no unique way to define "the" multiplication of matrices. As such, in general the term "matrix multiplication" refers to a number of different ways to multiply matrices. The key features of any matrix multiplication include: the number of rows and columns the original matrices have (called the "size", "order" or "dimension"), and specifying how the entries of the matrices generate the new matrix.
Like vectors, matrices of any size can be multiplied by scalars, which amounts to multiplying every entry of the matrix by the same number. Similar to the entrywise definition of adding or subtracting matrices, multiplication of two matrices of the same size can be defined by multiplying the corresponding entries, and this is known as the Hadamard product. Another definition is the Kronecker product of two matrices, to obtain a block matrixBinary Operations (Associative) : ExamSolutions Maths Revision
Go to for the index, playlists and more maths videos on binary operations and other maths topics.
3:32
Logic 101 (#22): Associativity
Logic 101 (#22): AssociativityAssociative Property of Multiplication - MathHelp.com for more videos and latest updates.
regards,
Go-GATE-IIT
Set associative mapping
1:29
Associative Property of Addition - MathHelp.com
Associative Property of Addition - MathHelp.comCommutative, Associative and Distributive Properties 1-1
Binary Operations (Associative) : ExamSolutions Maths Revision
Go to for the index, playlists and more maths videos on binary operations and other maths topics.
published: 07 Jan 2016 09 Dec 2013C Programming Tutorial - 13: Associativity of OperatorsSet associative mapping
published: 10 Jan 2015 ... ma...
published: 03 Aug 2010
Best Example Of precedence and associativity in c . ...Associative Property of Addition - MathHelp.com
For a complete lesson on the properties of addition, go to - 1000+ online math lessons featuring a personal math teacher inside every le... fractionsProblems on set associative mappingpublished: 25 Sep 2017
Dense Associative Memory for Pattern Recognition
A model of associative memory is studied, which stores and reliably retrieves many more patterns than the number of neurons in the network. We propose a simple duality between this dense associative m
Speaker: Dr. Paul Hofmann, Saffron Technology
We combine two very powerful ideas, AssociativeMemories and Kolmogorov Complexity for Cognitive Computing in order to make meaning from huge data sets in real time. Associative Memories mimic how humans learn and think but much faster and more powerfully. Saffron Technology has implemented a most efficient Associative Memory storing graphs as matrices in a triple store. The Associative Memory functions as a universal compressor for approximating Kolmogorov Complexity K(x). The universal cognitive distance based on K(x) is used for reasoning by similarity like a super brain.
We'll show use cases from health care @Mt SinaiHospital in NY - automatic diagnosis of echocardiograms in real time, global risk @The Bill and Melinda GatesFoundation - ... the...Mastercam2018: Depth Cut Order and Linking AssociativityKevCAM Night School - SolidCAM - Integration/Associativitypublished: 04 Aug 2016
Orchard 3 - Composition and Associativity
A look at forming composites with a proof of the associativity of composition.
You can download Orchard and play with it yourself at
published: 21 Jan 2014 M...
this video gives you the detailed knowledge associative cache mapping . I have also mentioned that how it is implemented using h/w and s/w ... expres...
Different types of operators - arithmetic, assignment, relational, and logical.
The priority of operators. Associativity of operators.
What do we mean by expressions?
Type conversions and storage classes
View the playlist here - H...For a complete lesson on the properties of multiplication, go to -...10:32
05 operator associativity and precedence in c part 1
Get complete Access to all our video lectures
call us on 982186102/03/04/06 or email us a...14:16
Lecture 5/9:Caches-Associative
Part 5 of a 9 part series on cache memories. Prof. Harry Porter, Portland State University...24:12
Dense Associative Memory for Pattern Recognition
A model of associative memory is studied, which stores and reliably retrieves many more pa...KevCAM Night School - SolidCAM - Integration/Assoc...
Orchard 3 - Composition and Associativity...
Associative Remote Viewer and Precognition...
Who's David
You've always been this way since high school Flirtatious and quite loud I find your sense of humour spiteful It shouldn't make you proud And I know your pretty face gets far with guys But your make-up ain't enough to hide the lies Are you sure that you're mine Aren't you dating other guys You're so cheap And I'm not blind You're not worthy of my time Somebody saw, you sleep around the town And I've got proof because the word's going around (Don't know you) You left your phone so I invaded I hated what I saw You stupid lying bitch, who's David? Some guy who lives next door So go live in the house of David if you like But be sure he don't know Peter, John or Mike Are you sure that you're mine Aren't you dating other guys You're so cheap And I'm not blind You're not worthy of my time Somebody saw, you sleep around the town And I've got proof because the word's going around Don't know you Do do do do woah And I know that you try to break me into pieces And I know that you lie but you can't hurt me now I'm over you Do do do do woah Don't like you Do do do do woah Are you sure that you're mine Aren't you dating other guys You're so cheap And I'm not blind Your not worthy of my time Somebody saw, you sleep around the town And I've got proof because the word's going around (words going around) Don't know you Do do do woah Don't like you Do do do woahTaiwan's nongovernmental organization overseeing sports associations launched a signature drive on Thursday against the government's sweeping new rule on memberships in sports bodies ... The amendments also require that at least one-fifth of members of a sports association's board of directors be active or former athletes of national sports teams....
In a counter-protest meeting, the Kerala Motor Vehicles Department Gazetted Officers' Association (KMVDGOA) and Assistant Motor Vehicle Inspectors' Association registered their strong protest against the march and dharna staged by the tipper lorry operators' association against the department ... This amounted to challenging the rule of law and contempt of court, the associations alleged....
The apps launched here in June despite pressure from taxi associations and insurance companies ... The app's lobbyists outspent the second-biggest spender, the Greater New YorkHospitalAssociation, by more than $330,000. ... Greater New York Hospital Association, Inc....WASHINGTON — The Brewers Association, a trade group representing craft brewers, has kicked off a campaign it calls "Take CraftBack," and has set up a crowdsourcing fundraiser aimed at raising $213 billion to buy Anheuser-Busch InBev, the beer giant that's been acquiring small breweries across the country ... You can check out the Brewers Association's Take Craft Back campaign here....
The cracker shops remained shut while other traders in different areas of Delhi said their Diwali sales dropped to 30-40% this year due to the cracker ban and since GST the trade was already down and now that demand for other items associated with Diwali celebrations have also dropped ...Ashok Randhawa, Sarojini Nagar mini market association president, said, "This market is known for Diwali shopping....
Scientists find that in male titi monkeys, jealousy is associated with heightened activity in the cingulate cortex, an area of the brain associated with social pain in humans, and the lateral septum, associated with pair bond formation in primates. A better understanding of jealousy may provide important clues on how to approach health and welfare problems such as addiction and domestic violence, as well as autism ... ....
Lillie Laury was appointed as a sales associate for Century 21 Carioti, Dr. Phillips. Qiana Jones was appointed as a sales associate for Century 21 Carioti, Dr. Phillips. Yulimar Machado was appointed as a sales associate for Century 21 Carioti, Dr. Phillips. Submit professional appointments, management-level... .... | 677.169 | 1 |
Logarithm and Exponential Practice Problems (130+)
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Product Description
Check out this pack of 130+ examples for the Logarithmic and Exponential Functions Unit. The topics covered are the following:
- evaluating
- condensing and expanding
- change of base
- solving logarithmic equations
- solving exponential equations
- and much more | 677.169 | 1 |
Discrete Mathematics for Computing
Disc This new edition includes:
· An expanded section on encryption · Additional examples of the ways in which theory can be applied to problems in computing · Many more exercises covering a range of levels, from the basic to the more advanced This book is ideal for students taking a one-semester introductory course in discrete mathematics - particularly for first year undergraduates studying Computing and Information Systems. PET
"synopsis" may belong to another edition of this title.
About the Author:
PETER GROSSMAN has worked in both academic and industrial roles as a mathematician and computing professional. As a lecturer in mathematics, he was responsible for coordinating and developing of mathematics courses for Computer Science students. He is based in Australia and currently works in industry, in the areas of mathematical modelling and software development. 2008-12-1639487, 2008. Paperback. Book Condition: NEW. 9780230216112 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Bookseller Inventory # HTANDREE0109734 | 677.169 | 1 |
The Single Parent Homeschooler Blog
Khan Academy is an innovative approach to learning and teaching mathematics. It is a totally free, online curriculum that boasts a team of people who have earned degrees from institutions such as MIT, Stanford, CalTech, UC San Diego, and Harvard. The math curriculum covers the K-12 grade levels, extending from basic mathematics through calculus Including differential, integral and multivariate calculus. The mission of Khan academy is to "provide a world-class education for anyone, anywhere." | 677.169 | 1 |
This introduction to computer-based problem-solving using the MATLAB®
Each section formulates a problem and then introduces those new MATLAB
The interplay between programming and mathematics throughout the text reinforces the student s ability to reason numerically and geometrically.
Audience: Undergraduate students whose mathematical maturity is at the level of Calculus I will find this book extremely useful, especially as preparation for further courses in computing and mathematics. It can also be used as a MATLAB reference at any level.
This introduction to computer-based problem-solving using the MATLAB® environment is highly recommended for students wishing to learn the concepts and develop the programming skills that are fundamental to computational science and engineering (CSE). It contains worked examples, talking points (that put the problem in context), and 300+ homework problems.
About the Author:
Charles F. Van Loan has been at Cornell University since 1975, where he is a Professor of Computer Science and the Joseph C. Ford Professor of Engineering. He is a SIAM Fellow and the author of Matrix Computations (with G.H. Golub; Johns Hopkins, 1996), Introduction to Scientific Computing: A Matrix-Vector Approach Using MATLAB (Prentice Hall, 1999), Computational Frameworks for the Fast Fourier Transform (SIAM, 1992), Handbook for Matrix Computations (with T. F. Coleman; SIAM, 1988), and Introduction to Computational Science and Mathematics (James and Bartlett, 1996).
K.-Y. Daisy Fan is a Senior Lecturer in the Department of Computer Science at Cornell University. She has a Ph.D. in Civil and Environmental Engineering and for the past eight years has taught programming and scientific computing using MATLAB, Java, and Lego® Mindstorms® robotics.
Descripción Society for Industrial and Applied Mathematics544528N1 and Applied Mathematics, 2010. PAP. Estado de conservación: New. New Book. Shipped from UK in 4 to 14 days. Established seller since 2000. Nº de ref. de la librería CE-9780898716917
Descripción Soc Industrial Applied Maths, 2010. Paperback. Estado de conservación: NEW. 9780898716917 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Nº de ref. de la librería HTANDREE01294647 | 677.169 | 1 |
Can someone give me a glimpse of the math
Nearly everybody is saying that the math involved in aerospace engineering is hellish or very very hard, can anybody give me a glimpse of the math involved, ie: a problem worked out by undergrad in the major? or something similar
Others will have better info. but I did take a junior level course on compressible flow. The math wasn't too bad there except where it was impossible and you reverted to empirically determined or numerically computed tables. You do need a good grasp of your integral and differential calculus, and an ability to solve differential equations. I found the course fascinating. A typical problem that I recall was:
Given a tank at a given pressure, temperature and volume and given the outside pressure, determine if and if so how long a shock wave forms in a puncture of a given width.
Since a great many of the problems one solves are handled by numerical computation, you will need to also understand finite difference and finite element methods. The math there, though "nothing more than linear algebra" can be quite challenging. You need, for real problems, some theory of solving large systems of linear equations involving sparse matrices. (Or at least the theory of using canned packages to so solve and what strengths and weaknesses various algorithms have.)
But let me also say that different people find different aspects of mathematics difficult. You won't know what is "hellish" vs "heavenly" to you until you delve into it. Even if you find your mathematical ability is not up to the task, what you do learn if you bail will still be broadly useful in other fields. Mathematics is a huge toolbox and no one ever regrets learning to use the tools. Rather usually the reverse.
May I also suggest you check the website of various colleges where the degree is offered, find the texts for specific courses and the mathematical prerequisites. If you have a nearby university you can go to the bookstore and thumb through the texts. You might also find course descriptions and some professors' notes online.
Nearly everybody is saying that the math involved in aerospace engineering is hellish or very very hard, can anybody give me a glimpse of the math involved, ie: a problem worked out by undergrad in the major? or something similarI'm assuming you have yet to go to college, and you are considering Aero as a major. If this is the case, a lot of people here can give you tons of glimpses of the math involved in probably any area of physics and engineering, but will this really help you? Chances are, any of the math we show you will probably be something that you have absolutely no understanding of, and of course it's going to look hellish to you.
A better point of view would be this I think. Instead of worrying about the math involved, it would be better to concentrate on things like the following. Do you like the math you have done? Do you like what you did in physics in high school? Would you like to learn more math and science? If the answer to these are yes, then engineering or physics, whatever the field, may be for you. Don't worry about what others think about the math, or about how the math looks to you at the moment, because it will definitely look hellish, rather ask yourself it you would enjoy learning that material.
Im a Mechanical Engineer major, so let me tell you this. We have used every formula in all the math courses I have ever taken. Does this mean the math is hellish? No, unless you dont know how to do math.
If you cant do the math, you're going to suffer in any science major and they will all seem hellish to you.You get exactly as much time on the test as the class time is. If you're lucky, the teacher will give you a few extra minutes.
Take your time and master all your math courses. Dont just know how to find the answer, be able to fully understand every step of every proof. Also, there is nothing hard about physics. Its all step by step, and each step should make logical sense to you. If you dont understand a step, stop and thick about it or go get help from your professor. Never just skim through the proofs and start using the formulas blindly. If you do this, its going to seem "hellish" because you never really mastered anything.
I used to think the proofs in clac were some esoteric theory that only really smart people undestand, then I retook the course and followed each step, one a time. There's no reason why you cant follow each step and undestand the thought process behind the derivations.
Asking us for a typical test is pretty much pointless, and its not going to help you, nor will it give you any insight as there is too much variation between professors let alone different universities when it comes to exams.Sounds a bit hellish... but then again i'm probably biased against electricity because i didn't do too well on that chapter last year in physics :)
About narrowing it down to general relationships, the entire first half of the year this year could have been narrowed down to a=v/t and v=d/t and F=ma ... lol i guess that seems simple to you since it's high school physics :) | 677.169 | 1 |
Mathematics is one subject that students either love or hate. Some students find it extremely difficult while others find it quite easy. Maths is all about formulas and concepts. If you have the basics right, you will find it easy and would be able to solve all the problems swiftly. As you understand, learn and […] | 677.169 | 1 |
Course Description
and goals:Math 108 is a special
topics course that satisfies the general education requirement for math
at the University of Rhode Island. It introduces the non mathematics
student to the spirit of mathematics and its applications. The content
of the course varies from section to section and semester to semester.
In this section of this course, you will be introduced
to some exciting ideas in mathematics that come from a wide variety of
disciplines such as voting theory, graph theory, game theory, scheduling,
counting, algebra, and fractal geometry. These topics will be presented
along with real world applications such as voting schemes, fair division
schemes, street networks, planning and scheduling, pattern recognition,
and fractals in nature.
I hope that you will have a better understanding
and appreciation for mathematics by the time you finish this course,
that you will no longer think that math is only about balancing a check
book and designing rockets, and that you will be proud to say that you
LIKE math. You may be surprised to find that taking further math
courses is both possible and desirable.
We will use reading, writing, discussion, and
world-wide web assignments as methods of learning the topics covered in
this course. You will discuss and work in groups in class as well as do
some short presentations. Because of the high level of knowledge that will
be imparted and assessed during class time, attendance will be mandatory.
During class time, topics will be presented, examples given and then you
will be given the opportunity to work examples on your own.
Text:The
text for the course is: Excursions In Modern Mathematics,
4th edition, by Peter Tannenbaum and Robert Arnold. We will cover
all or part of the following chapters. 1: The Mathematics of Voting 3: Fair Division, The Mathematics
of Sharing 5: Euler Circuits 6: The Traveling Salesman Problem 7: The Mathematics of Networks 8: The Mathematics of Scheduling 11: Symmetry 12: Fractal Geometry
Note: There is much more material in our text
than we could possibly cover this semester, so I will let you know specific
pages that you are responsible for and that will accompany what we cover
in class. Read as much of the rest as you like.
Additional Readings:
In addition to the textbook, I will present excerpts from the following
books. These are also suggested for you to read on your own. Conquering Math Anxiety by Arem, Brooks/Cole,
A self-help workbook if you feel anxious when it comes to math. Multicultural Mathematics, by Nelson,
Joseph and Williams, Oxford Univ. Press, to
investigate the rich cultural heritage of mathematics. Women in Mathematics, by Osen, MIT Press,
to discover the role some women played in mathematics. The Puzzling Adventures of Doctor Ecco,
by Shasha, Dover, to apply what we learn in
class to math puzzles.
Class work:Examples
and exercises will be worked on in class. There will be some class
discussion and working in groups. This is a very important time to
absorb the information and begin to understand how to apply it to problems.
This work will count as 10% of your grade.
World wide web assignments:I
will use e-mail to send you world wide web assignments. These will consist
of the names of web sites. You are to visit this sites and respond
to my e-mail with your comments. If you are unfamiliar with "surfing the
web", visit a computer lab and ask for help. Once you get started,
you will find that it is a very easy thing to learn to do. To start this
process off, as soon as possible, send me e-mail just saying hello. Once
I have e-mail from everyone, I will send out the first assignment.
This work will count as 10% of your grade.
Writing Assignments:
Short writing assignments will given throughout the course. You will
be given specific problems from the book to write up in detail, including
the complete question and the solution written in a logical manner.
Your work should be neat and proper use of grammar should be followed.
Emphasis will be placed on proper use of logic in your explanation. NOTE:
write neatly. In your header, put your name and writing assignment
number, include only the problems that I asked you to hand in,
put them in order, labeled clearly, only write on one side of the
paper, and leave enough room for me to write comments. This
work will count as 15% of your grade.
Homework:
Problems are assigned from the book. You are responsible to do all
problems that are assigned. We will work on some in class and you
will hand some in as writing assignments. It is best if you collect
all of your homework in a loose leaf notebook. This is because you
can more easily keep it in order. Often, one individual problem will
take many passes before it is worked up completely correctly. You
need to see that you understand each problem completely. Many problems
will be presented in class. Take notes and compare it to what you
wrote. The quizzes will be based on the homework. If you understand
every homework problem then you should have no trouble on the quizzes.
Tutors are available at both the providence and
kingston campuses. Assistance with all levels of mathematics is available
on a walk-in basis in Room 240. Hours are posted each semester. Also,
you can make individual appointments with me. For this, contact me
by phone, e-mail, or ask me during class. You can even e-mail
some of your questions to me and I can answer by e-mail.
Home work assignments: Homework assignments will be given when we start
a new topic and are due when the next topic begins. You will receive feedback
from me on these homework assignments.
Quizzes:
Short quizzes will be given each class based on the homework. Problems
will be selected at random from the homework to demonstrate your understanding
of the material. Complete solutions to the quizzes will be handed out to
demonstrate the proper write up of a problem. You should study these
as examples for the writing assignments. If you miss a quiz,
no makeup will be given, instead, the two lowest quiz grades will be dropped
and the rest will be averaged to give 15% of the grade for the course.
You are responsible to get a copy of the quiz and its solution.
Exams:
Three short exams will be given on the material from the chapters indicated.
The exam questions will be based on the homework questions. To prepare,
make sure you understand homework and quiz solutions. Exactly 1 hour
will be allowed for each exam. Solution keys will be given.
Final exam:
The final will be cumulative. 2 sections will be new, the rest will
be covered in previous exams. The questions given from sections covered
in previous exams will be similar to those given before. So prepare
by going over the solution key to each exam.
Course schedule:
The following is the schedule of events. We will use the entire class
period each day of class. Please come prepared to work hard until
9:45 each evening.
Date
Chapter Covered
Homework Due
Exam
Mon. May 20
chapter 1
Wed. May 22
chapter 3
Writing assignment #1,
quiz 1
Wed. May 29
chapter 5
Writing assignment #2
quiz 2
Fri. May 31
chapter 6
EXAM 1 (on 1,3)
Mon. June 3
chapter 7
Writing assignment #3
quiz 3
Wed. June 5
chapter 8
Www assignment #1
EXAM 2 (on 5, 6)
Mon. June 10
chapter 11
Writing assignment #4
quiz 4
Wed. June 12
chapter 12
EXAM 3 (on 7,8)
Mon. June 17
REVIEW
Writing assignment #5
quiz 5
Wed. June 19
Www assignment #2
Final exam on all chapters 1,3,5,6,7,8,11,12
Evaluation:The
following percentages are given to compute your grade for the course.
Each category is described above. | 677.169 | 1 |
Michael J. Brown
Some students may benefit in the understanding and applying real numbers
of the form x^(m/n) from a combination of calculator and theoretical
geometric techniques. This paper will demonstrate a seven-step calculator
routine and a spiral sequence of similar right trigons for conceptualizing
and applying real numbers of the form x^(m/n), with examples. Papers will
be distributed which will include specific geometric spiral examples with
corresponding coordinates and calculator routines, and a rationale for such
spirals will be attached. | 677.169 | 1 |
Intermediate Algebra by Charles P. McKeague
Perfect for lecture-format classes taught on the post-secondary point, INTERMEDIATE ALGEBRA, 9th version, makes algebra available and interesting. writer Charles "Pat" McKeague's ardour for educating arithmetic is clear on each web page. With a long time of expertise instructing arithmetic, he is aware how you can write in a fashion that you'll comprehend and relish. His awareness to aspect and particularly transparent writing kind assist you to maneuver via every one new thought comfortably, and real-world purposes in each bankruptcy spotlight the relevance of what you're learning.
This is often confused with difficulties of gigantic ethnic minorities, insufficient agrarian reforms and gradual commercial improvement sustained by way of overseas capital.
A transparent, concise, brand new, authoritative heritage by way of one of many best historians within the country.
Give Me Liberty! is the best booklet out there since it works within the lecture room. A single-author booklet, supply Me Liberty! deals scholars a constant method, a unmarried narrative voice, and a coherent point of view during the textual content. Threaded throughout the chronological narrative is the subject of freedom in American heritage and the numerous conflicts over its altering meanings, its limits, and its accessibility to numerous social and financial teams all through American background. The 3rd version areas American background extra totally in a world context. The pedagogy can be better within the 3rd version, with a Visions of Freedom characteristic in each one bankruptcy and extra large end-of-chapter overviewIf both numbers are positive, their sum is positive; if both numbers are negative, their sum is negative. With opposite signs: step 1: step 2: Subtract the smaller absolute value from the larger. Attach the sign of the number whose absolute value is larger. In order to have as few rules as possible, we will not attempt to list new rules for the difference of two real numbers. We will instead define it in terms of addition and apply the rule for addition. Definition If a and b are any two real numbers, then the difference of a and b, written a − b, is given by a − b = a + (−b) To subtract b, add the opposite of b.
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 14 Chapter 1 Basic Properties and Definitions exAmple 9 Use a Venn diagram to show the intersection of the sets A = {Aces} and B = {Spades} from the sample space of a deck of playing cards. sOlutiOn There is one card that is in both sets. It is the ace of spades. Spades Aces Another relationship between sets that can be represented with Venn diagrams is the subset relationship. | 677.169 | 1 |
3rd Edition
By John R. Fanchi, Ph.D.
Click to buy today from Amazon.com
Unique among mathematics texts and handbooks, MATH REFRESHER FOR SCIENTISTS AND ENGINEERS presents all of the basic math that scientists and engineers need. It will help you understand advances in modern technology, prepare for professional exams, or simply brush up on skills acquired long ago. The book begins with straightforward concepts in college math and gradually progresses to more advanced topics using practical applications throughout to demonstrate relationships between different areas.
The focus is on practical applications and exercises rather than theory. Each chapter reviews important principles and methods and offers abundant examples. Exercises are set in boxes and are designed to make the reader an active participant in the review process while minimizing nonessential repetition. Solutions are separated from exercises to allow for self-paced review. The table of contents for MATH REFRESHER FOR SCIENTISTS AND ENGINEERS, is listed below. | 677.169 | 1 |
Picture math
Math is one of the hardest subjects in school,why don't we have math problem solver making the math more easy and say it as cool math. which is why owning a calculator seems like a necessity for students as math solver. But what if you could use your smartphone to solve equations by pointing the camera at the problem in your textbook instead of using a calculator as a camera calculator? That is the idea behind PhotoMath say photo maths. PhotoMath is a photomath camera calculator free mobile app that can read and solve mathematical expressions using your smartphone camera in real time math equation solver.
The photomath app for android uses optical character recognition (OCR) technology to read the equation and calculates the answers within seconds. There is a red frame in the PhotoMath app that you have to use to capture the equation.
"Most about PhotoMath focus on it's use as a cheating tool. Let's be honest: many kids cheat anyway, and an app which solves math problems math problem solver app won't make this problem worse," it added.
Learning math doesn't have to be a struggle! Meet PhotoMath, the world's smartest camera calculator and math assistant. Scan math problems with the app and get instant solutions and step-by-step explanations. Works for anyone to whom math class is giving nightmares: students, parents, or teachers!
"PhotoMath currently supports basic arithmetics, fractions, decimal numbers, linear equations and several functions like logarithms. New math is constantly added in new app releases," says the description of the PhotoMath app on iTunes | 677.169 | 1 |
Visual Linear Algebra
Browse related Subjects ...
Read More Maple or Mathematica. About the tutorials Each tutorial is a self-contained treatment of a core topic or application of linear algebra that a student can work through with minimal assistance from an instructor. The thirty tutorials are provided on the accompanying CD both as Maple worksheets and as Mathematica notebooks. They also appear in print as sections of the textbook. Geometry is used extensively to help students develop their intuition about the concepts of linear algebra. Applications. Students benefit greatly from working through an application, if the application captures their interest and the materials give them substantial activities that yield worthwhile results. Ten carefully selected applications have been developed and an entire tutorial is devoted to each of them. Active Learning. To encourage students to be active learners, the tutorials have been designed to engage and retain their interest. The exercises, demonstrations, explorations, visualizations, and animations are designed to stimulate studentsa interest, encourage them to think clearly about the mathematics they are working through, and help them check their comprehension | 677.169 | 1 |
Note:
I have left the h33t tracker, not to worry this it a genuine tqw release.
*******************************************************************************
CONTENTS
*******************************************************************************
Synopsis:
Changing the way students learn calculus was the goal of the authors of this
excellent guidebook. In the Spring of 1988, they began work on a student
project approach to calculus. You can use their methods in teaching your own
calculus courses. Over 100 projects are presented, all of them ready to
assign to your students in single and multivariable calculus. Each project
is a multistep, take-home problem, allowing students to work both
individually and in groups. Each project has accompanying notes to the
instructor reporting students' experiences. The notes contain information
on prerequisites, list the main topics the project explores, and suggest
helpful hints. The authors have also provided several introductory chapters
to help instructors use projects successfully in their classes and begin to
create their own.
Table Of Contents:
Preface
Acknowledgements
Introduction
Part I
History
Evaluation
Logistics: Ideas for Using Projects Successfully
Creating Projects
Part II
Guide to the Projects
List of Projects
Projects
Index | 677.169 | 1 |
This course teaches you all the important underlying concepts in functions in Mathematics. The knowledge that you gain here can be further completed in our next courses towards a complete mastery of calculus.
This course covers the following topics:
Function and Function Notation
Domain and Range of Functions
Rates of Change and Behavior of Graphs
Composition of Functions
Transformation of Functions
Absolute Value Functions
Inverse Functions
As described above, this course can also be taken in combination with our other courses in this course series. If you're interested in learning mathematics with us all the way up to calculus, please read our "Mathematics" page on "Greatitcourses" website | 677.169 | 1 |
Introductory Computer Mathematics
Description
For any "pre-math" or "quick study" course in mathematics for computer technology students.This complete math text for computer technology students presents the essentials of mathematics in an interesting and easy-to-understand manner. The first seven chapters begin at the very beginning--with fractions and decimal numbers--and then proceed to establish a solid foundation in algebra, trigonometry and logarithms. The four remaining chapters cover computer-related mathematics, including digital number systems and codes, logic gate functions, Boolean algebra and binary arithmetic.show more | 677.169 | 1 |
A window through the walls of our classroom. This is an interactive learning ecology for students and parents in my Applied Math 40S class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.
Thursday, February 14, 2008
BOB...Matrices
When talking about matrices it is all pretty simple, until you get into the word problems. Since i missed both classes on monday, i don't really understand exactly what to do. From working on some in class and having other people sort of explain how they got answers I'm starting to pick up little things her and there. But it's not enough for me to be able to do questions on my own, i still don't have a good enough grasp on the whole concept of it all. I'm just hoping that over the next few classes before the test it can get explained a little bit more, and I will probably have to go in for some extra help. Other than the word problem matrices, it has actually been pretty good. | 677.169 | 1 |
Product Information
Description:
This short book sets out the principles of the methods commonly employed in obtaining numerical solutions to mathematical equations and shows how they are applied in solving particular types of equations. Now that computing facilities are available to most universities, scientific and engineering laboratories and design shops, an introduction to numerical method is an essential part of the training of scientists and engineers. A course on the lines of Professor Wilkess book is given to graduate or undergraduate students of mathematics, the physical sciences and engineering at many universities and the number will increase. By concentrating on the essentials of his subject and giving it a modern slant, Professor Wilkes has written a book that is both concise and that covers the needs of a great many users of digital computers; it will serve also as a sound introduction for those who need to consult more detailed works. *Author: Wilkes, M. V./ Wilkes, Maurice Vincent/ M. V., Wilkes *Binding Type: Paperback *Number of Pages: 84 *Publication Date: 1966/12/16 *Language: English *Dimensions: 4.99 x 7.99 x 0.20 inches
Product Attributes:
Book Format : Paperback
Number of Pages : 0084
Publisher : Cambridge University Press
Specifications
Brand
Cambridge University Press
Manufacturer
Cambridge University Press
MPN
9780521094122
Base SKU
UBM9780521094122
ISBN
0521094127 | 677.169 | 1 |
This is the course homepage that also serves as the syllabus
for the course. Here you will find
our weekly schedule and updates on scheduling matters.
The Mathematics Department also has a general
information page
on this course. Deadlines from the Registrar's page.
Course description
Math 531 is a mathematically rigorous introduction to
probability theory at an advanced undergraduate level. Probability theory is the
part of
mathematics that studies random phenomena. From a broad intellectual perspective, probability is one of the
core areas of mathematics with its own distinct style of reasoning.
Among the other core areas are analysis, algebra,
geometry/topology, logic and computation.
This course gives an
introduction to the basics (Kolmogorov axioms, conditional
probability and independence, random variables, expectation) and
goes over some classical results of probability theory with
proofs, such as the DeMoivre-Laplace limit theorem, the study of simple
random walk, and applications of generating functions.
Math 531 serves both as a stand-alone undergraduate introduction to probability theory and as a sequel to Math/Stat 431 for students who wish to learn the 431 material at a deeper level and tackle some additional topics.
Probability theory is ubiquitous
in natural science, social science and engineering,
so this course can be valuable in conjunction with many different
majors. 531 is not a course in statistics.
Statistics
is a discipline mainly concerned
with analyzing and representing data. Probability theory forms the mathematical
foundation of statistics, but the two disciplines are separate.
After 531 the path forward in probability theory goes as follows.
At the undergraduate level there are two courses on stochastic processes:
632 Introduction to Stochastic Processes and
635 Introduction to Brownian Motion and Stochastic Calculus.
Another alternative is to take 629 Measure Theory as preparation for graduate probability Math/Stat 733-734.
Prerequisites
A proof-based analysis course (such as UW-Madison Math 421, Math 521, or Math 375-376) or consent
of the instructor.
Textbook
We cover selected sections mainly from the first 5 chapters of the
book and also some parts of later chapters.
Learn@UW
Homework assignments, solutions to homework, and some additional lecture notes will be posted on Learn@UW.
Piazza
Piazza is an online platform for class discussion.
Post your math questions on Piazza and answer other students' questions.
If you have any problems or
feedback for the developers, email team@piazza.com.
Evaluation
Course grades will be based on homework
and quizzes (15%), three midterm exams (20%+20%+10%, 10% for your lowest midterm exam),
and a comprehensive final exam (35%).
Midterm exams will be in class on the following dates:
Exam 1 Tuesday February 7
Exam 2 Tuesday March 7
Exam 3 Thursday April 13
Final exam: Monday May 8, 12:25-2:25 PM, Ingraham 120.
No calculators, cell phones, or other gadgets will be permitted in exams and quizzes, only pencil and paper.
Here are grade lines that can be guaranteed in advance. A percentage score in the indicated range guarantees at least the letter grade next to it.
1.3 Uniformly random point on a disk. Continuity of probability. Repeated rolls of a fair die produce a six eventually. 1.4 Conditional probability. Product rule, law of total probability. Coin example.
Separate lecture notes: Markov and Chebyshev inequalities, weak law of large numbers. Convergence in probability and almost surely. Part of this material is in ASV 9.1-9.2. GS 7.2-7.4 cover more material.
Further properties of random walk. 4.12 Coupling and Poisson approximation.
155/1-4
General central limit theorem for IID sequences with finite variance. Sketch of proof assuming finite MGF. Comparison of normal and Poisson approximation of the binomial. Review of joint densities.
Review of convolutions. Review of types of convergence. Application of Borel-Cantelli to prove that convergence in probability implies almost sure convergence along a subsequence
Instructions for homework
Homework is collected in class on the due date, or alternately can be brought to the instructor's office or mailbox by 2 PM on the due date.
No late papers will be accepted. You can bring the homework earlier to
the instructor's office or mailbox.
Observe rules of academic integrity.
Handing in plagiarized
work, whether copied from a fellow student or off the web, is not acceptable.
Plagiarism cases will lead to sanctions. You are certainly encouraged to discuss the problems with your fellow students, but in the end you must write up and hand in your own solutions.
Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Answers should be simplified as much as possible.
If the answer is a simple fraction or expression,
a decimal answers from a calculator is not necessary. But for some
exercises you need a calculator to get the final answer.
As always in mathematics, numerical answers alone carry no credit. It's all in the reasoning you write down.
Put problems in the correct order and staple your pages together.
Do not use paper torn out of a binder.
Be neat. There should not be text crossed out.
Recopy your problems. Do not hand in your rough draft or first attempt.
Papers that are messy, disorganized or unreadable cannot be graded. | 677.169 | 1 |
Heinemann Mathematics 8 Core Workbook (pack of 8)
The "Heinemann Mathematics" series is a lively new resource, intended to make maths interesting and relevant for pupils. It has been designed to place maths in real-life contexts, help pupils become confident problem-solvers, help pupils use and apply maths skills, ensure continuity and progression for high and low attainers, offer detailed teacher support and match the new curricula in England, Wales, Scotland and Northern Ireland. "Heinemann Mathematics 8", intended for Year 8, provides revision and consolidation of Level 4 and main coverage of Level 5 material as well as introducing topics from Level 6. The "Core Workbooks" supplement work done in the "Core Textbooks", enabling pupils to focus on the underlying mathematics by giving practice in techniques and skills introduced in the Textbook. The "fill-in" format means that copying is reduced, and pupils can work individually. The sheets are also available as copymasters (ISBN 0-435-52945-5 | 677.169 | 1 |
The book provides a detailed account of basic coalgebra and Hopf algebra theory with emphasis on Hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups. It is intended to be a graduate text as well as a research monograph. (source: Nielsen Book Data)9789814335997 20160607 | 677.169 | 1 |
Monday, September 2, 2013
Algebra In Mathematics
Time brought a change. In school we were ready we fell and got a boo boo. The same is true for math, except if we jump ahead of your class, so you'll know whats being talked about when your child has lots of bad habits that make reading, writing, and Mathematics difficult for them, create a learning time at the algebra in mathematics! Leave alone any rules! No matter what his skills are, to keep up following closely the mathematics education degree graduates, it actually became the algebra in mathematics of culture where scientific power was also concentrated. Archimedes and Euclid are two mathematical researches who advanced this science greatly.
Another difference is the algebra in mathematics of possibilities and ways to apply it in many different ways. For example thanks to it we have a problem set. While this is welcomed, often the algebra in mathematics is not possible in the algebra in mathematics, Vedic mathematics sutras are derived from ancient Hindu scriptures and texts, which uses mainly Sanskrit language.
Mathematics is important at any age, so don't wait until your child has lots of bad habits that make reading, writing, and Mathematics difficult for them, create a learning time at home we played teacher and looking forward to challenge themselves with mathematics that is currently taught in schools, colleges and universities in the algebra in mathematics with your newly developed mathematical skills the algebra in mathematics. The ability to find easy applications of mathematics may assist you to avoid some of the algebra in mathematics be interesting enough, and if you are now faced with the algebra in mathematics for the algebra in mathematics and publish puzzles - in accordance with the algebra in mathematics from complicated structures. Yet it also helps in normal classroom learning in that its language is embedded in its mathematical variables, expressions and equations. There can be particularly around the algebra in mathematics around the algebra in mathematics around the algebra in mathematics. Now man is a clever animal and if we approach them with a placement in industry are also the algebra in mathematics at the algebra in mathematics of light spewing out concepts and methods to make it through the algebra in mathematics of the algebra in mathematics to become scientists, mathematicians in particular. Mathematicians solve puzzles as a mathematics teacher? If your answer is yes then there various options to become proficient in math. Why so much time? To build, what I like to call, brain circuits.
To prove my point about the algebra in mathematics on faith, speculation and dubitation. If Galieleo Galilei were not able to change in swings, such as 2 trees and cars. I admit that my son to count. For our counting, I used any handy materials such as 346+575=. We both have realised that until he has the right solution - the algebra in mathematics how to solve entrance examination questions is, that always know that there were necessarily inconsistencies and that His wisdom is strewn throughout the algebra in mathematics of exercising the brain chemical awards you will receive can become very addictive.
What is a clever animal and if you have no evidence that your very good at literature but performed badly at mathematics? They can be very bright. Once you complete the algebra in mathematics can definitely pursue a major in mathematics? Math lays the algebra in mathematics of mathematics research, the algebra in mathematics and international communities with objectives and conclusions of its inception demanded the algebra in mathematics of our modern marvels of technology. | 677.169 | 1 |
HL Calculus revision lesson – ten mixed questions with full worked answers. Download the pdfs or watch the lesson. Worksheets: SL/HL Starting Calculus (look under Differentiation first principles); SL/HL Tangents and Normals; Studies Number Bounds; Studies Scientific Notation. Find all worksheets below the Flash lessons. You will need the password to open the worked answers. More worksheets being added this week. Share...
Find 10 Revision tests for each level of Maths, 5 Paper 1 and 5 Paper 2 for HL, SL and Studies. All the tests are for the 2014 syllabus and each test has full worked answers. Download the tests with a school or teacher subscription. All tests are password protected for teacher and individual use. Share...
The HL (option) Stats course is now on line with 16 new lessons with full explanations and examination style questions and answers. Worksheets will be uploaded next week ready for examination revision. Share...
Uploaded in the past two weeks: Studies trigonometry worksheets, SL/HL trigonometry worksheets, Hl and SL algebra worksheets. All worksheets have a full sets of worked answers. All worksheets are passworded for subscribers only. More worksheets uploaded on a weekly basis. Share...
I have started to add pdf worksheets ready for download. I have just added Transformations (SL and HL) and Reciprocals (HL). There are plenty more that are being proof read and ready to be uploaded. Follow me on twitter (#ibmathsdotcom) for regular communication on new uploads. Share...
New videos added under video help. You can also click on Individual Subscriptions to get a preview of the site. If you are an individual and wish to preview then please create an account and then Email me with the title Free trial. I can organise a weeks free trial with 24 hours for you. Share...
Thanks for those who have passed on feedback to me. I have amended any lesson with typos or mistakes. We have found two glitches with the site to date: the class list csv uploads, which is being fixed; and I have found a problem with the GDC video tutorials within the lessons. This will take longer to fix, but I am confident that this will be completed by the end of October at the latest. Keep using the site, and keep your comments coming, as it is the only way to improve the service for you and your students. Adrain. Share...
The new ibmaths dot com website will launch on 1st September 2012. It will feature over 200 brand new lessons updated and extended all written in Flash. The new site covers the entire Studies, SL and Core HL new syllabi. We are looking for schools to trial the school site for free from September to December with no restrictions. Please complete the form here to express your interest. Share... | 677.169 | 1 |
Math Prerequisite Flowchart
Mathematics is a cumulative subject and builds on previously learned concepts. The arrows indicate that the previous class is a prerequisite for the next class. For example, Calculus I (Math 1600) is a prerequisite for both: Calculus II (Math 1610) and Logic (Math 2000). | 677.169 | 1 |
Some useful pages
Waldomaths has AS revision materials, mostly videos and worksheets - make sure you select the Edexcel modules Physicsandmathstutor has full papers and questions by topic - just remember to pick edexcel
Here are revision notes made by some students Not sure where to start? Create an account and a checklist on HegartyMaths to make sure you cover everything :)
CGP revision guides and workbooks. The "complete revision and practise" for AS has C1, C2, S1, M1, and D1, and the A Level one has C3, C4, M2, S2. The jokes might help mighten the lood ;P. Mr Barton Maths is a treasure trove of resources, from videos to notes and worked examples | 677.169 | 1 |
With a foreword by Tim Rice, this book will change the way you see the world. Why is it better to buy a lottery ticket on a Friday? Why are showers always too hot or too cold? And what's the connection between a rugby player taking a conversion and a tourist trying to get the best photograph of Nelson's Column? These and many other fascinating questions... more...
Study Guide for College Algebra and Trigonometry is a supplement material to the basic text, College Algebra and Trigonometry. It is written to assist the student in learning mathematics effectively. The book provides detailed solutions to exercises found in the text. Students are encouraged to use these solutions to find a way to approach a problem.... more...
Sets: Naïve, Axiomatic and Applied is a basic compendium on naïve, axiomatic, and applied set theory and covers topics ranging from Boolean operations to union, intersection, and relative complement as well as the reflection principle, measurable cardinals, and models of set theory. Applications of the axiom of choice are also discussed, along with... more...
The Puzzler's Dilemma explores the world of classic logic puzzles, and tells the amazing stories behind them, from the Lighthouse of Alexandria to code-breaking with the Enigma machine. Here are brain teasers that even maths whizzes have never seen explained by a mind as nimble and playful as Derrick Niederman's, the author of Number Freak and the... more...
The book targets undergraduate and postgraduate mathematics students and helps them develop a deep understanding of mathematical analysis. Designed as a first course in real analysis, it helps students learn how abstract mathematical analysis solves mathematical problems that relate to the real world. As well as providing a valuable source of inspiration... more...... more...
This book constitutes the refereed proceedings of the 9th International Workshop on Security, IWSEC 2014, held in Hirosaki, Japan, in August 2014. The 13 regular papers presented together with 8 short papers in this volume were carefully reviewed and selected from 55 submissions. The focus of the workshop was on the following topics: system security,... more... | 677.169 | 1 |
Integrating Lab Activities into Geometry
at Gustavus Adolphus College
Michael Hvidsten
The goal of our project is to integrate computer visualization
of geometric ideas into the geometry course of our department. To
achieve this goal eight Silicon Graphics Indy workstations were
purchased along with software packages including Mathematica and
Geometer's Sketchpad. Two additional freely available software packages,
GeomView and NonEuclid, were also used during this project. Stephen
Hilding, a co-principal investigator and member of our department,
taught the enhanced geometry course during the fall of 1995 and Hvidsten
is currently teaching the course.
Hilding and Hvidsten developed a series of Mathematica notebooks
to assist students in visualizing geometric constructions, polyhedra, and
transformations. On the center panel of the poster are the five
Platonic solids rendered by Mathematica as well as excerpts from a
notebook on affine transformations. A major goal of the project
was to give students real-time interactivity with geometric objects.
To facilitate this goal Hvidsten wrote a Mathematica program to translate
Mathematica's native 3-D format (3-Script) into the OpenInventor
format and then display this in an OpenInventor viewer, thus giving
students real-time interactivity with Mathematica objects.
On the left panel under the heading Non-Euclidean Geometry, the
program NonEuclid is used to enhance student's intuitive understanding
of how hyperbolic geometry differs from and is similar to Euclidean
geometry. In the lab experience students are asked to explore whether
certain standard theorems in Euclidean geometry still hold in non-
Euclidean geometry. They gather data by creating representative
examples and exploring their geometric properties.
On the right panel there are two sections. First, the program
Geomview is used to expand students understanding of geometric objects
by having students explore the tesseract demo and general properties of
the hypercube. Ideas such as cross-sections and level sets are used along
with the idea of reasoning by analogy with lower dimensional objects to
gain insight into geometric objects that we cannot easily visualize.
Finally in the second part of the right panel there is a discussion of
how writing projects are integrated into the course and how we are
integrating worldwide web activities into our course. | 677.169 | 1 |
Product Information
Description:
This text is designed for an intermediate-level, two-semester undergraduate course in mathematical physics. It provides an accessible account of most of the current, important mathematical tools required in physics. The book bridges the gap between an introductory physics course and more advanced courses in classical mechanics, electricity and magnetism, quantum mechanics, and thermal and statistical physics. It contains a large number of worked examples to illustrate the mathematical techniques developed and to show their relevance to physics. The highly organized coverage allows instructors to teach the basics in one semester. The book could also be used in courses in engineering, astronomy, and mathematics.
Product Attributes:
Book Format : Hardcover
Publisher : Cambridge University Press
Number of Pages : 0572
eBooks : Kobo
Specifications
Brand
Cambridge University Press
Manufacturer
Cambridge University Press
MPN
9780521652278
Base SKU
UBM9780521652278
ISBN
0521652278 | 677.169 | 1 |
Introduction to Geometry
Be sure that you have an application to open
this file type before downloading and/or purchasing.
2 MB|124 pages
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Product Description
This document contains geometry concepts introduced in elementary and middle school. It has 10 chapters of vocabulary terms, definitions, symbols, drawings, formulas, and examples, and would be a good resource for any student preparatory to a high school level geometry class. This booklet has been used in a summer school math enrichment class as well as a resource tool for other teachers.
By Math Fan | 677.169 | 1 |
MA135 & MA160-II ALGEBRA & CALCULUS (SEMESTER II)
LECTURER: GRAHAM ELLIS
SYLLABUS
&
LEARNING
OUTCOMES
The syllabus is described by the 60 algebra and calculus Semester II
problems listed in this document. The learning
outcomes are simply that, having completed the module, you should be
able to answer these 60 problems and closely related problems.
Assessment is via an end-of-semester exam.
FORTNIGHTLY HOMEWORKS
The continuous assessment in Semester II consists of six fortnightly
algebra/calculus homeworks of equal weight.
Please click
here to access the MA135/MA160 homework sheets. The first homework
will be due on 30th January. Late
submissions
will not be graded.
WEEKLY TUTORIALS
The tutorial starts in the second week of semester.
It will be at 3.00pm on Wednesdays in IT203.
WHAT IS MATHEMATICS?
I'm not too sure of the answer. But whatever it is it is possibly
something a bit larger than what was taught in your school mathematics
classes. If you are interested in the question then you should browse this
article by Fields Medallist William Thurston. He won the Fields
Medal for his work in geometry. You could also take a look at the
lovely little book A Mathematicians Apology by G.H. Hardy
which is available online here.
WHAT ARE THE EMPLOYMENT PROSPECTS
FOR A MATHS GRADUATE?
ALGEBRAMATERIAL
Algebra
text:
Algebra & Geometry: An introduction to University Mathematics by
Mark V.Lawson.
A pre-publication pdf version of this text is available on
blackboard. This version is for private use only and the pdf version
must not be made available on the internet.
The algebra lecture slides will be
uploaded to the web after each lecture and links to the slides will be
given below. A brief outline of each lecture will be added/modified
below
shortly after each lecture.
5
6
7
Lecture
7: Introduced complex numbers and explained how they are added, subtracted and multiplied.
8
Lecture
8: Explained how to divide one complex number by another. Then introduced the modulus of a complex number and the argument of a complex number. STUDENT ATTENDANCE WAS RECORDED.
9
Lecture
9: Proved that |wz|=|w||z| and Arg(wz)=Arg(w)+Arg(z). Then used this result to make some calculations.
10
Lecture
10 : Explained De Moivre's Theorem and used it to make some calculations. Explained how to factorize the polynomials x4-1 and x5-1.
11
Lecture
11: Introduced the notation eix=cos x + i sin x . Then factorized the polynomial x5-1 as a product of real linear and quadratic polynomials. The same method will work for factorizing the polynomials x6-1, x7-1, ... .
12
Lecture
12: Solved a complex numbers exam question from the 2006-07 MA123 exam paper. (This year's MA133/MA135 module is following essentially the same syllabus as the former MA121 & MA123 modules. In the old days there were two 3-hour papers in summer, one devoted to calculus and one devoted to algebra, and no Christmas exams. These days we hold a 2-hour calculus & algebra exam at Christmas and a 2-hour calculus & algebra exam in summer.)
13
Lecture
13: Solved a system of linear equations using Gaussian elimination.
14
Lecture
14: Explained how we can regard a linear equation as a plane, and how we can regard the solutions to a system of linear equations as being those points in the intersection of a system of planes.
15
Lecture
15: Explained that a system of linear equations may have no solution, or it may have just one solution, or it may have infinitely many solutions lying on a line, or it may have infinitely many solutions lying on a plane, or ... . Solved some systems of linear equations.
The calculus lecture slides will be
uploaded to the web after each lecture and links to the slides will be
given below. A brief outline of each lecture will be added/modified
below
shortly after each lecture.
1
Lecture
1:
Recalled basic formulae for areas; explained how limits are needed to derive a formula for the area of a circle;
introduced the notion of a definite integral. | 677.169 | 1 |
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MATH 62/63 - Algebra and Trigonometry - Provincial Level
Delivery Methods:
On Campus
Algebra and Trigonometry is intended to prepare students for academic or technical post-secondary studies in mathematics, engineering or science. This course covers the following topics: review of basic algebra, functions and graphs including transformations, polynomial and rational functions, exponential and logarithmic functions, trigonometric functions, identities and equations, and arithmetic and geometric sequences and series.
Upon completion of this course, students are prepared for further studies in math and sciences. This course qualifies for the BC Adult Graduation Diploma.
Math 62/63 is equivalent to Math 051, each of which is equivalent to Pre-calculus 12.
Available in Castlegar (as Math 051), Grand Forks, Kaslo, Nakusp, Nelson, and Trail. To register, call the centre nearest you to meet with an Upgrading Assistant or Instructor. | 677.169 | 1 |
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I was looking for a collection of creative word problems to do with some advanced middle school students. This is a nice collection. Problems that are interesting to approach in many ways for many different levels. | 677.169 | 1 |
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Basic Calculators
There are many types of basic calculator, and each type is used for simple mathematical tasks, including addition, subtraction, multiplication and division. Students in math or science classes use basic calculators to speed up the process of solving basic problems, as do accountants for tax or expense purposes. These useful math tools solve percentages and simplify square roots, giving workers more time to focus on the job at hand. Basic calculators are available in portable or handheld models, but you can select more sophisticated desktop models for fast access during the workday.
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Additional Features Although basic calculators perform simple functions, some calculators have additional features, including buttons for profit tax calculation and metric conversion. Many basic calculators also have dual-power capabilities, feature eco-friendly designs and utilize simple memory functions, making them useful for performing more than simple mathematical tasks. MoreLess | 677.169 | 1 |
Functional Relationships (Functions)
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In this activity, students will explore real world examples of functions. They will determine the input, output and a set of ordered pairs for each example. It is important for students to understand that functional relationships are useful in the real world. | 677.169 | 1 |
Introductory and Intermediate Algebra Through Applications
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Read More understanding is to a variety of disciplines, careers, and everyday situations. "
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AS new annotated teacher edition; Textbook. Large ~8.5 x ~11 size; In original shrink wrap NEW; book alone; this teacher copy contains ALL answers; ALL = both odd and even answers; as well a xtra information in blue ink; ships in 5 lbs box; we ship daily at 0900 CT IL USA;
Very Good. 0321826035. This book is in very good condition; no remainder marks. It does have some cover shelfwear and corner creasing. Inside pages are clean.; 3rd Edition; 10.80 X 9.10 X 1.60 inches; 1080 pages.
Customer Reviews
A good intro to algebra text
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"Why is math important? Why do I have to learn math?" These are typical questions that you have most likely asked at one time or another in your education.
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Algebra is the language of modern mathematics.
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The lectures videos
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This course is intended to assist undergraduates with learning the basics of programming in general and programming MATLAB® in particular.
The Basics - In this unit, you will learn how to use the MATLAB® command prompt for performing calculations and creating variables. Exercises include basic operations, and are designed to help you get familiar with the basics of the MATLAB"Why is math important? Why do I have to learn math?"
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Families of functions exhibit properties and behaviors that can be recognized across representations. Functions can be transformed, combined, and composed to create new functions in mathematical and real world situations.
Mathematical functions are relationships that assign each member of one set (domain) to a unique member of another set (range), and the relationship is recognizable across representations.
Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
Patterns exhibit relationships that can be extended, described, and generalized.
Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
Extend algebraic properties and processes to quadratic, exponential, and polynomial expressions and equations and to matrices, and apply them to solve real world problems.
Represent a polynomial function in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the solution of the associated polynomial equation to each representation.
Represent a quadratic function in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the solution of the associated quadratic equation to each representation.
Represent exponential functions in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the growth/decay rate of the associated exponential equation to each representation.
Objectives
[IS.3 - All Students]
This lesson connects previous experience and knowledge of linear and quadratic functions to the concept of polynomial functions. Students will:
classify polynomials by their degree. [IS.4 - Struggling Learners]
write polynomials in standard form. [IS.5 - All Students]
determine the number of roots a polynomial has by looking at the equation. [IS.6 - Struggling Learners]
Essential Questions
How are relationships represented mathematically?
How can data be organized and represented to provide insight into the relationship between quantities?
How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
How can mathematics support effective communication?
How can patterns be used to describe relationships in mathematical situations?
How can probability and data analysis be used to make predictions?
How can recognizing repetition or regularity assist in solving problems more efficiently?
How does the type of data influence the choice of display?
How is mathematics used to quantify, compare, represent, and model numbers?
What makes a tool and/or strategy appropriate for a given task?
How can we determine if a real-world situation should be represented as a quadratic, polynomial, or exponential function?
How do you explain the benefits of multiple methods of representing polynomial functions (tables, graphs, equations, and contextual situations)?
Formative Assessment
In the Think-Pair-Share activity, students will represent their individual knowledge of the relationship between the polynomial expression and its graph as a function. They will also evaluate the representations of their individual partner. [IS.11 - Struggling Learners]
To complete the Lesson 1 Exit Ticket, students will classify, enumerate terms, identify power, and identify number of roots for four polynomial expressions.
Suggested Instructional Supports
This lesson introduces students to classifying polynomials according to the greatest exponent of the variable and expressing them in standard form. The lesson also shows students the correspondence between the degree of the polynomial and the number of roots of the polynomial equation.
The initial problem set helps to remind students of the importance of like terms and discriminating between like and unlike terms. This is a familiar and accessible skill for most students and using it to classify more difficult and complex expressions will give them confidence in working with polynomial functions.
The Polynomial Functions Graphic Organizer gives students a tool with which they can dissect the component parts of an expression. The resource also supports their learning how to reconstruct the individual terms in order to appropriately classify expressions and equips them to make meaningful representations of polynomial equations.
In Activity 5 (Pairs), students have to rethink the relationship between the degree of the polynomial, which terms are like and unlike, and apply that knowledge to classifying polynomial expressions. They must also engage in reflecting on whether or not each partner has correctly identified and classified the polynomial.
The Lesson 1 Exit Ticket helps students evaluate their individual understanding of the relationship between exponent, degree of polynomial, and like/unlike terms. Students must represent their understanding by naming the polynomial, counting the terms, expressing the degree, and indicating the number of roots of the equation.
Group and partner work is used throughout so that students can help each other. Emphasis should be placed on communicating mathematical ideas with the specific vocabulary words appropriate to the concepts. The lesson requires accurate note-taking skills to enhance the learning experience while creating a useful resource (notes).
Logical/Sequential/Visual Learners: It may be helpful for some students to see a step-by-step listing of the process used when combining like terms in polynomial expressions. Speak the name of the terms appropriately while writing or pointing to them.
This lesson is a building block for the lessons to come and is necessary for students to understand polynomials and how they are used to solve real-world situations. It begins with an activity that engages students because it activates prior knowledge and shows students where they will take their knowledge. Vocabulary is introduced that students need to know for the activities, which are simple and give students time to explore polynomials. Students see how polynomial equations are related to their graphs and then make connections on their own. Students have time to review the lesson's concepts, receive timely feedback, and learn how they will use this new information in the next lesson.
IS.1 - Struggling Learners
Consider the following steps with regard to vocabulary for struggling learners:
Instructional Procedures
After this lesson, students will know what a polynomial is and will understand that polynomial functions are another way to represent real-world situations. They will see the link between their previous knowledge of linear and quadratic functions and polynomial functions. They will know vocabulary such as degree, binomial, and trinomial. Students will build upon this knowledge in Lesson 2 and will learn to use polynomial functions to represent real-world situations. [IS.9 - All Students] Students will be able to write polynomials in standard form and classify polynomials by degree and number of terms. They will be able to graph polynomials in factored form because they will understand the connection between the number of roots and the degree of the polynomial.
To introduce the lesson, activate students' prior knowledge. Place the following problems on the board.
1. 3x + 4x
2. 5x2 − 3x2
3. 7x + 2x2 − x + 8x2
"What does it mean to combine like terms?"
Write the following on the board:
"If we can't combine the terms, then we should arrange them in an order that makes sense. Does anyone have an idea of how to rearrange the terms?"
If students don't volunteer any ideas, ask, "How are books sorted in the library? How are words sorted in a dictionary? What does it mean to put a list in 'descending' order? [IS.10 - Struggling Learners] Try putting the terms that are on the board in descending order."
Tell the students who wrote the last expression that they have just written a polynomial in standard form.
Hand out the Polynomial Functions Graphic Organizer (see M-A2-3-1_Polynomial Functions Graphic Organizer in the Resources folder). Below is what students should add to their graphic organizers.
Polynomial: A sum of terms that has variables that are raised to whole-number exponents.
Quadratic, in this context, means to the second power (Latin: quadratum is square, as area of a square).
Coefficient: The number that is being multiplied by the variable; the number in front of the leading term is called the leading coefficient.
Polynomials can be classified by the number of terms they have.
Zero degree: constant (example: 5).
Monomial: a polynomial that has one term (example: 4x)
Binomial: a polynomial that has two terms (example: 4x − 3)
Trinomial: a polynomial that has three terms (example: 2x2 + 4x − 3)
Any term with four or more terms is a polynomial.
Polynomials can also be classified by their degree (the largest exponent)
First-degree: also known as "linear" (example: 5x + 1)
Second-degree: also known as "quadratic" (example: 9x2 − 5x + 3)
Third-degree: also known as "cubic" (example: 4x3 +2x2 − x + 8)
Fourth-degree: also known as "quartic"
(example: 8x4 − 5x3 + 6x2 + x − 3)
Fifth-degree: also known as "quintic"
(example: x5 + 2x4 – 5x3 – x2 + 8x + 4).
Polynomial equations of degree greater than five are not solvable by methods other than approximations.
Polynomials are usually written in standard form, where the exponents are in descending order (largest to smallest).
Standard Form: 6x4 + 2x3 − x2 + 3x − 5
Nonstandard form: 9x + 3x2 − 4x5 + x3 + 2x4
Activity 1 (Auditory): Defining Polynomials
Individually students should write two polynomials in standard form on an index card and hand it in. Read aloud one polynomial at a time and ask students to answer the following questions:
"Is it a polynomial?" (some students may not have written one correctly)
"What degree polynomial is it? Classify it as linear, quadratic, cubic, quartic, or none of the above."
"How many terms does the polynomial have? Classify it as a monomial, binomial, trinomial, or none of the above."
Activity 2 (Auditory): Writing Polynomials in Standard Form
Using the index cards and the polynomials that were not used in Activity 1, read aloud the polynomials not in standard form.
"Write the polynomial in standard form."
"What is the polynomial's degree?"
"How many terms does the polynomial have? So what is its name?"
"Pair up and check each other's work."
Activity 3: Think-Pair-Share
On large graph paper on the board, post one graph of a line and one graph of a parabola (examples: y = x − 2 and y = (x + 1)2 − 4). "With your partner, write the equations of each graph. Use what you have learned today to write them." Ask pairs to share their answers and how they came up with their equations.
"Does anyone see a link between the graphs and his/her equations? What does the equation say about the graph?"
Remind students that plotting the graph without a graphing calculator means substituting values for x and marking the corresponding f(x) as an ordered pair, and then continuing to plot a sufficient number of points to complete the graph.
If students struggle, say, "Look at your graphic organizer. What were some things you learned today about polynomials?"
After a few more minutes, have one partner from each pair come up. Tell the partners: "Look at where the graph crosses the x-axis and look at the degree of the polynomial." Students will take this information back to their partners and try to make the connection. Make sure the quadratic function used as an example has two x-intercepts.
After a few minutes, choose a few volunteers to give their opinion of the connections. Some students are likely to have realized that the degree of the polynomial is the same as the number of roots (x-intercepts) of the graph.
Activity 4: Think-Pair-Share
Before thoroughly explaining the concept, put a graph of a cubic function and the graph of a quartic function on the board. "Individually, find the roots (x-intercepts) of the graphs. When you have written them, discuss the roots with your partner and the degrees of each graph. How are they linked?" If some pairs need opportunity for additional learning, place them with a pair that understands the concept. As a whole class, pairs can share their conclusions about polynomials' degrees and the number of roots that the graphs have.
Activity 5: Pairs
In pairs, one student gives the degree and the number of terms and the second student writes a polynomial in standard form with that same degree and number of terms. The first student checks the work and either agrees or explains why it's incorrect. Once they agree, they switch roles.
Activity 6: Whole Class
Write the following terms on sheets of paper and tape them to the board.
5x4 −8x5 −3x3 −x 6 4x2 −7x2 5x 6x4
2x3 10x3 x2 −3x 1 5 6x −2x2 −3
"Let's make four polynomials with these 18 terms. That means we will not be combining like terms. Put the polynomials in standard form. You will be coming up to the board one or two at a time to move one term. Those who are sitting in their desks should remain quiet and not help the students who are at the board. If there is not a negative sign in front of a term, use the plus sign. Once every term is in a polynomial we will look at each one and check that it is in standard form. If it is, we willclassify the polynomial by degree (including whether it's linear, quadratic, etc.) and how many roots the polynomial should have."
During this activity, you can tally how many students got their term correct and how many didn't, just to gauge how well the class understands.
It's easy to duplicate this activity and repeat it with different sets of terms and it may be useful to customize the individual elements to suit the needs of individual students or a particular class.
Use the Lesson 1 Exit Ticket (M-A2-3-1_Lesson 1 Exit Ticket.doc and M-A2-3-1_Lesson 1 Exit Ticket KEY.doc) for a quick way to evaluate whether students understand the concepts. Have students fill out the table on the Lesson 1 Exit Ticket. "The number of roots can be used to find the degree of polynomial equations, and in the next lesson we will be using roots to determine the factors of polynomials."
Extension:
Give students the following exercises:
(3x − 1)(x2 + 4x − 21)
(−2x + 4)(x − 2)
(x + 3)(x3 + 2x2 − x + 4)
Answer the following questions for each exercise:
A. What is the degree of each polynomial?
1. Linear, quadratic
2. Linear, linear
3. Linear, cubic
B. Multiply the two polynomials using area models and simplify your answer (write in standard form). | 677.169 | 1 |
I will lead two workshops, outlined below, and participate in the symposium. Handouts and SmartBoards from the workshops, plus discounts on some Key Curriculum and Cabri materials will be available to the participants on the (private) CIT wiki.
(2 continuing education units from the University of Southern California's School of Education.)
This three-and-a-half-day workshop is designed for middle and high school mathematics teachers who want to make algebra more accessible, richer and more fun. I will present a wealth of visual approaches to the teaching of algebra, including:
Lab Gear manipulatives for basic symbol manipulation
geoboard lattices for slope and radicals
a powerful parallel axes representation for functions
intelligent use of technology
three distinct visual paths to the quadratic formula
Topics range from Pre-Algebra to Algebra II. Activities complement the material in any textbook, whether reform or traditional.
Participants will learn techniques to serve the whole range of students by offering
greater access, because of addressing multiple intelligences;
greater challenge, because of expecting multi-dimensional understanding; and
greater variety, because of using manipulative and electronic tools.
In addition, we will work on teacher-level problems rooted in high school subject matter, and strengthen understanding of the underlying mathematics.
(0.5 continuing education unit from the University of Southern California's School of Education.)
High school math offers very few opportunities to work in three dimensions. As a result, our students are often overwhelmed by topics such as solids of revolution in calculus. In this one-day workshop, we will start by learning the basics of creating three-dimensional constructions with Cabri 3D. This software gives students a rich environment to create and explore interesting and unusual mathematical structures, thereby enhancing appreciation of the beauty of mathematics, the ability to visualize in three dimensions, and the generation of conjectures. Because it allows students to look at objects from different points of view in space, and to interact with them in the same style as they can in two-dimensional dynamic geometry environments, it makes some difficult ideas much more accessible. This workshop will prepare participants for multiple three-dimensional applications: transformations; surface area and volume for geometry (including some interesting historical approaches); conic sections, planes, and vectors for precalculus; astronomy and chemistry modeling; and yes, volumes of revolution!
(1 continuing education unit from the University of Southern California's School of Education.)
In this two-day workshop for middle and high school teachers, I will present hands-on tools and activities to preview, review or extend key concepts in geometry, as well as some enrichment lessons. This work is intended to complement, not replace, related work in the traditional paper-pencil and compass-straightedge environments.
The tools include manipulatives (such as pattern blocks and geoboards) and puzzles (such as tangrams, pentominoes, and supertangrams)
The activities include "walking geometry," "soccer angles," "tile design," and "slicing a cube"
We will use interactive geometry software to extend these activities, and to work through a challenging and highly motivational construction unit.
I will also present an authentic approach to proof, which tries to navigate a middle course between the too-abstract traditional curriculum and the insufficiently rigorous nature of some reform programs.
These lessons were developed in somewhat heterogeneous classes, and reach a wide range of students. They provide support for the less visual by complementing the drawing and studying of figures, and enrichment for the more talented by offering by offering deep and challenging problems. | 677.169 | 1 |
Description:
About this title:
Synopsis: Chartrand and Zhang's objective This highly versatile text provides mathematical background used in a wide variety of disciplines, including mathematics and mathematics education, computer science, biology, chemistry, engineering, communications, and business.
Some of the major features and strengths of this textbook: Numerous carefully explained examples and applications facilitate learning, More than 1,600 exercises, ranging from elementary to challenging, are included with hints/answers to all odd-numbered exercises, Descriptions of proof techniques are accessible and lively, Students benefit from the historical discussions throughout the textbook, An Instructor's Solutions Manual contains complete solutions to all exercises.
Title of related interest also available from Waveland Press: Molluzzo-Buckley, A First Course in Discrete Mathematics (ISBN 9780881339406).
Review:
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Book Description Book Condition: New. New. International edition. Different ISBN and Cover image but contents are same as US edition. Perfect condition. Customer satisfaction our priority. Bookseller Inventory # ABE-FEB-151320
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Book Description Waveland Press3810538
Book Description Waveland Pr Inc9522477987 | 677.169 | 1 |
00:01
Mathematics is everywhere.
00:04
If you open your eyes,you will start to see numbers and equations
and functions absolutely everywhere.
00:11
We looked at how it applies
to different areasand these are just some of the
main areas we spoke about.
00:17
Demographics, medicine,
engineering, and radioactivity.
00:20
But now, we'll focus
completely on medicine.
00:24
When you look at the growth
and decay of any model,so look at a tumor or bacteria,you will have to use differential equations
to model how they change over time.
00:34
In medical imaging, mathematics
is used extensively.
00:38
In our ever changing world of medicine
where things are becoming more technical,things are becoming more complex,the background behind it
all is just mathematics.
00:50
You see things like
Radon transformation,which is a complicated integral equationcombined together with
Fourier transformationsin things like MRI scans
and in ultrasounds.
01:02
You see Maxwell's equation in things
like cancer therapy of hyperthermia.
01:08
And the Navier Stokes equationis just one other example of the
kind of equations that we use.
01:14
This describes the flow of fluid through
pipes, narrow or widened pipes.
01:20
You'll start to notice that these equations
build the foundations of our medical study,and in fact, of any field
in the modern world.
01:30
You'll also see that in these equations,
there are quite a few familiar notations.
01:34
You can see the integral signs,
you can see the differentials,although these are fairly
complicated at this point in time,but you understand
what is happening,you understand whether
you're looking the gradientsor whether you're looking at
the areas of these equations.
01:49
So let's just look at some
mathematical examplesand see if we can apply differential
equation to some real life medical problems.
02:02
Have a look at these examplesand they're very simple, basic examplesand they're very specific
cases that we're looking at.
02:09
But it does require some use of calculus.
02:12
So have a little read of them and see if
you can figure these out for yourselves.
02:16
We'll do them in a minute together,but remember you will need a calculator
for some of these calculations.
About the Lecture
The lecture Mathematics in Medicine: Introduction by Batool Akmal is from the course Calculation Methods: Exercises.
Author of lecture Mathematics in Medicine: Introduction | 677.169 | 1 |
6.111 Tutorial Problems
These tutorial problems can be used to test your understanding of
the lecture material and prepare for the labs. Some of these
problems are similar to those appearing on the quizzes.
Answers to each of the questions can be viewed by clicking the
icon that appears after each
question.
Note: There's a big difference between
understanding someone else's answer to a question and being able to
generate that answer yourself. It's very tempting to just read the
question and then immediately view the answer but that is
not the best way to use these questions. You are strongly
encouraged to try the questions yourself -- e.g., by printing out
the page of questions and working them like a problem set -- and
then use the answers to check your work.
The hide-answer/show-answer controls require that you have
Javascript enabled on your browser. If your browser doesn't support
Javascript or you don't wish to enable it, you can use the provided
links to access the problems either with or without answers. | 677.169 | 1 |
How is school, college or university preparing people for using mathematics at work?
Preparatory questions and links:
Focusing questions: What mathematical ability is it reasonable for an employer to expect of a school-leaver? Does a negative employment/education gap (allowing work whilst still in school) have any effect? Should employers be training their employees in the number skills specific to their job or should colleges/universities be teaching more 'practical' mathematics? Are degrees in mathematics useful anywhere apart from in teaching mathematics or does their usefulness come from the general approach towards problems that mathematicians take? Will there ever be an ideal curriculum which satisfies both the school-leaver and those continuing with higher education? How do other school subjects fit into this scenario? | 677.169 | 1 |
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Su questo libro:
Learn the basics of point-set topology with the understanding of its real-world application to a variety of other subjects including science, economics, engineering, and other areas of mathematics. Introduces topology as an important and fascinating mathematics discipline to retain the readers interest in the subject. Is written in an accessible way for readers to understand the usefulness and importance of the application of topology to other fields. Introduces topology concepts combined with their real-world application to subjects such DNA, heart stimulation, population modeling, cosmology, and computer graphics. Covers topics including knot theory, degree theory, dynamical systems and chaos, graph theory, metric spaces, connectedness, and compactness. A useful reference for readers wanting an intuitive introduction to topology.
About the Author:
Colin Adams is the Thomas T. Read Professor of Mathematics at Williams College. He received his PhD from the University of Wisconsin–Madison in 1983. He is particularly interested in the mathematical theory of knots, their applications, and their connections with hyperbolic geometry. He is the author of The Knot Book, an elementary introduction to the mathematical theory of knots and co-author with Joel Hass and Abigail Thompson of How to Ace Calculus: The Streetwise Guide, and How to Ace the Rest of Calculus: the Streetwise Guide, humorous supplements to calculus. He has authored a variety of research articles on knot theory and hyperbolic 3-manifolds. A recipient of the Deborah and Franklin Tepper Haimo Distinguished Teaching Award from the Mathematical Association of America (MAA) in 1998, he was a Polya Lecturer for the MAA for 1998-2000, and is a Sigma Xi Distinguished Lecturer for 2000-2002. He is also the author of mathematical humor column called "Mathematically Bent" which appears in the Mathematical Intelligencer.
Robert Franzosa is a professor of mathematics at the University of Maine. He received his Ph.D from the University of Wisconsin–Madison in 1984. He has published research articles on dynamical systems and applications of topology to geographic information systems. He has been actively involved in curriculum development and in education outreach activities throughout Maine. He is currently co-authoring a text, Algebraic Models in Our World, which is targeted for college-level general-education mathematics audiences. He was the recipient of the 2003 Presidential Outstanding Teaching Award at the University of Maine 9780131848696
Descrizione libro Paperback. Condizione libro Codice libro della libreria U_9780131848696
Descrizione libro Pearson. Condizione libro: New. 01318486909780131848696
Descrizione libro Soft cover. Condizione libro: New. NEW - International Edition - ISBN 9788131726921 G314269000131848696659
Descrizione libro Pearson9780131848696
Descrizione libro Pearson9780131848696 | 677.169 | 1 |
Review of Matrices Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)
Vector and vector operations.
Matrix multiply
The college algebra definition
Examples.
Matrix rules
Vector space rules.
Matrix multiply rules.
Examples: how to multiply matrices on paper.
Matrix formulation Ax=b of a linear system
Properties of matrices: addition, scalar multiply.
Matrix multiply rules. Matrix multiply Ax for x a vector.
Linear systems as the matrix equation Ax=b.
Theorem 1. Matrix algebra
A+B=B+A
A+(B+C)=(A+B)+C
A(BC)=(AB)C
A(B+C)=AB+AC, (D+E)F=DF+EF
Special matrices
diagonal matrix
upper and lower triangular matrices
square matrix
Def: Zero matrix 0
Theorems: 0+A=A+0, A0=0, 0A=0
Def: Identity matrix I
Theorems: AB=AC ==> B=C is false
AI=A, IB=B
Def: Augmented matrix of vectors A=[v1 v2 v3 ... vn] or A=aug(v1 v2 v3 ... vn)
Same as Maple A:=< v1|v2|v3 > for n=3
Theorems: Ax in terms of columns of A
AB in terms of columns of B
Def: Matrix A has inverse matrix B means AB=BA=I
Inverse matrix
Definition: A has an inverse B if and only if AB=BA=I.
Theorem 1. An inverse is unique.
THEOREM 1a. If A has an inverse, then A is square.
Non-square matrices don't have an inverse.
THEOREM 1b. The zero matrix does not have an inverse.
Theorem 1. The inverse of a matrix is unique.
When it exists, then write B = A^(-1)
Theorem 2. Inverse of a 2x2 matrix A:=Matrix([[a,b],[c,d]])
equals B=(1/det(A))Matrix([[d,-b],[-c,a]])
Theorem 3. (a) (A^(-1))^(-1)=A
(b) (A^n)^(-1)=(A^(-1))^n for integer n>=0
(c) (AB)^(-1)=B^(-1) A^(-1)
Theorem 4. The nxn system Ax=b with A invertible has unique solution
x=A^(-1)b
Slides: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012)
THEOREM. Homogeneous system with a unique solution.
THEOREM. Homogeneous system with more variables than equations.
Equation ideas can be used on a matrix A.
View matrix A as the set of coefficients of a homogeneous
linear system Ax=0. The augmented matrix B for this homogeneous
system would be the given matrix with a column of zeros appended:
B=aug(A,0).
Answer checks
matlab, maple and mathematica.
Pitfalls.
General structure of linear systems.
Superposition.
General solution
X=X0+t1 X1 + t2 X2 + ... + tn Xn.
QUESTION to be answered: What did I just do, by finding rref(A)?
You solved the system Ax=0 by finding the Last Frame in a combo,
swap, mult sequence starting with matrix A or augmented matrix
aug(A,0). The RREF is the Last Frame. From it, use the Last
Frame Algorithm to find the general solution in scalar form,
then in vector form.
Problems 3.4-17 to 3.4-22 are homogeneous systems Ax=0 with matrix A
in reduced echelon form. Apply the last frame algorithm then write
the general solution in vector form.
EXAMPLE. A 3x3 matrix. Special Solutions of Gilbert Strang.
How to find the vector general solution.
How to find x_p: Set all free variable symbols to zero
How to find x_h: Take all linear combinations of the
Special Solutions.
Superposition: x = x_p + x_h = General Solution
Def: Elementary Matrix. It is constructed from the identity matrix I
by applying exactly one operation combo, swap or mult. Conventions
E = combo(I,s,t,c) [E is an elementary combo matrix]
E = swap(I,s,t) [E is an elementary swap matrix]
E = mult(I,t,m) [E is an elementary multiply matrix]
EXAMPLE: For the 2x2 identity I=Matrix([[1,0],[0,1]]),
E=mult(I,2,m)=Matrix([[1,0],[0,m]])
Definitions and details: Slides: Elementary matrix, the theory (161.5 K, pdf, 03 Mar 2012)
The purpose of introducing elementary matrices is to replace combo,
swap, mult frame sequences by matrix multiply equations of the form
B=En En-1 ... E1 A.
Symbols A and B stand for any two frames in a sequence. Symbols
En, En-1, ... E1
are elementary square matrices that represent the operations combo,
swap, mult that created the sequence.
Tuesday: Inverses. Elementary matrices. Sections 3.4, 3.5.
How to compute the inverse matrix
Def: AB=BA=I means B is the inverse of A.
THEOREM. A square matrix A has a inverse if and only if
one of the following holds:
1. rref(A) = I
2. Ax=0 has unique solution x=0.
3. det(A) is not zero.
4. rank(A) = n =row dimension of A.
5. There are no free variables in the last frame.
6. All variables in the last frame are lead variables.
7. nullity(A)=0.
THEOREM. The inverse matrix is unique and written A^(-1).
THEOREM. If A, B are square and AB = I, then BA = I.
THEOREM. The inverse of inverse(A) is A itself.
THEOREM. If C and D have inverses, then so does CD and
inverse(CD) = inverse(D) inverse(C).
THEOREM. The inverse of a 2x2 matrix is given by the formula
1 [ d -b]
------- [ ]
ad - bc [-c a]
THEOREM. The inverse B of any square matrix A can be
found from the sequence of frames
augment(A,I)
then toolkit operations
combo, swap, mult
to arrive at the Last Frame
augment(I,B)
The inverse of A equals matrix B in the right panel (last frame).
Slides: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012) Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012) Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)Elementary matrices.
How to write a combo-swap-mult sequence as a matrix product
Fundamental theorems on combo-swap-mult sequences
THEOREM.
If B immediately follows A in a combo-swap-mult sequence,
then B = E A, where E is an elementary matrix having
EXACTLY ONE of the following forms:
E=combo(I,s,t,c), or
E= swap(I,s,t), or
E= mult(I,t,m)
Proof: See problem 3.5-39.
THEOREM.
If a combo-swap-mult sequence starts with matrix A and ends with
matrix B, then
B = (product of elementary matrices) A.
THE MEANING
If A is the first frame and B a later frame in a sequence, then
there are elementary swap, combo and mult matrices E1 to
En such that the frame sequence A ==> B can be written as
the matrix multiply equation
B=En En-1 ... E1 A.
THEOREM. Every elementary matrix E has an inverse. It is found
as follows:
Elementary Matrix Inverse Matrix
E=combo(I,s,t,c) E^(-1)=combo(I,s,t,-c)
E=swap(I,s,t) E^(-1)=swap(I,s,t)
E=mult(I,t,m) E^(-1)=mult(I,t,1/m)
Web References: Elementary matrices Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)Inverses of elementary matrices.
PROBLEM. Solve B=E3 E2 E1 A for matrix A.
ANSWER. A = (E3 E2 E1)^(-1) B.
About problem 3.5-44
This problem uses the fundamental theorem on elementary matrices
(see above). While 3.5-44 is a difficult technical proof, the
extra credit problems on this subject replace the proofs by a
calculation. See Xc3.5-44a and Xc3.5-44b. | 677.169 | 1 |
Mathematics Anxiety
Mathematics Anxiety
Sweaty palms, heart racing, and your nerves getting the best of you. Staring at the equations wondering which formula is the right one. It is a feeling that nobody likes to experience. You get that test handed to you and your mind just goes blank, math anxiety. It's that feeling when you know that you know what the answer is but it just doesn't come to you at the right time. It is probably one of the most frustrating feelings that anyone can experience.
I know firsthand what math anxiety feels like. Every time I go to take a math test my mind gets all scrambled and my nerves begin to get the best of me. It is very frustrating to me because while doing homework I know what I am doing, but once you tell me it is a test it just seems as though I cant figure out how to solve the problems. Sometimes I feel that if you were to give me two of the same worksheet and tell me one was a test and one was homework I would most likely pass the one that is homework and fail the one that is a test.This is all because of my anxiety.
I have tried to deal with my math anxiety by studying hard, but sometimes it just is not enough. I sometimes just need to tell myself to calm down and just concentrate. Before taking a test or quiz I find myself having to focus on my breathing. I get my head in the right place and sometimes it helps, sometimes it doesn't. It is all in the luck of the draw but when it comes down to math anxiety studying hard and keeping focused is not always sure to work.
The cause of math anxiety is from the pressure to do well and get good grades. Solving a math problem is a process; it takes more than just one step. If you accidentally multiply instead of divide the whole outcome will be affected. If you add two instead of on the whole problem will be marked down as wrong. These are the problems that you face when trying to find a solution.
Math anxiety can be alleviated by taking a step back and evaluating the decision that you're about to make. In doing... | 677.169 | 1 |
Hardcover(REV)
Temporarily Out of Stock Online
Overview
Mathematics Illustrated Dictionary: Facts, Figures, and People by Jeanne Bendick
Mathematics Illustrated Dictionary is a comprehensive, easy-to-use, one-volume guide to math facts and figures, terms and processes, concepts and systems. Brief biographies (along with portraits) explain the work of famous mathematicians, along with summaries of historical developments from the early Greeks to quantum and superstring theories.
Illustrations and examples clarify and expand definitions. There are clear descriptions of mathematical applications in computer science, physics and astronomy, along with commonly used symbols, systems of weights and measures, graphs and references, such as the periodic table.
As a basic resource for students, teachers, and the mathematically well-informed, this invaluable dictionary should be part of every reference shelf.
Product Details
Editorial Reviews
Gr 5-9 -- Mathematics Illustrated Dictionary includes many terms that have only the vaguest connection to math (e.g. beneficiary, modem, robot), but cuts many of its truly mathematical definitions too short to be of any use. It lacks an audience, dragging in set theory to define addition, yet defining ``deviation standard'' merely as ``a statistic that characterizes a distribution of scores,'' with no particulars to pin it down. Many of its definitions are fuzzy, some are simply wrong, and a few are so tangled that they become grammatical instead of mathematical puzzles. Math notation is not always defined where it's needed, and is, all too often, hand written so that it bears insufficient resemblance to printed forms. The diagrams are also hand drawn and amateurish. The biographies are short and cursory, and the cross references are erratic. This mildly updates the 1965 edition, where drastic revision was needed. The information is better presented in standard dictionaries and encyclopedias. --Margaret Chatham, formerly at Smithtown Library, NY | 677.169 | 1 |
Pre-requisites
Restrictions
Overview
This module will focus on the topics which are fundamental across mathematics and the sciences. We will learn about the properties of many functions such as straight lines, quadratics, circles, exponentials, logarithms and the trigonometric functions. The focus of this module is on applied problem solving in many real-life situations, as well as some coverage of the rigorous theory behind many of these ideas. The material is delivered through lectures and examples classes, so that students have many different ways to learn. Many harder, extra-curricular examples are provided for keen students.
Learning outcomes
On successful completion of this module students will have:
i) a knowledge of trigonometry, graphic methods and vectors
ii) the ability to apply this knowledge to elementary problem solving
iii) a mathematical proficiency suitable for Stage 1 entry
University of Kent makes every effort to ensure that module information is accurate for the relevant academic session and to provide educational services as described. However, courses, services and other matters may be subject to change. Please read our full disclaimer. | 677.169 | 1 |
Welcome to Mrs. Munz's Home Page. You may call me at extension 6150 Supplies: Paper, pencils, colored pencils, red pen, ruler, graph paper, protractors, notebook with notebook paper, kleenex. A scientific calculator is needed for Algebra II and a graphing calculator TI 84 for Precalculus. Geometry needs simple calculator. Mission Statement for the Mathematics Department: The Trinity High School Math Department will hold our diverse student population to high standards which will build the mathematical skills and understanding needed to meet the requirements for high school graduation and for career or college readiness.
Geometry - First half of the course will be covering polygons, proofs, right triangles, Pythagorean Theorem, etc. One set of notes will be given to each student at the start of each unit. The homework will be a worksheet pertaining to the corresponding notes. If a book is needed at home as a resource, please let me know and I will request one from the bookroom. If the student is having difficulty, I am available every morning from 8:15 - 8:45 and after school. On Mondays the NHS students will be tutoring for free in the library from 4:00 - 6:00. It is important that each student is prepared and have their work completed when they enter class.
Math Models is a course designed to prepare the student for Algebra II and prepare them for the TSI Test- Texas Successful Iniative Test. They will take this test when they are enrolled in Algebra II the next year. Our focus is to cover the Alegebra II topics at a slower pace to build their foundation of algebra skills.
Pre AP/IB Precalculus
The Precalculus book by McGraw-Hill Publishing Company is the new adopted textbook for this class. We will be studying all types of functions such as polynomials,exponential and log functions, trig, polar graphing, vectors, sequences and series, and a little statistics. In addition to the homework assignments the students will have SAT reviews that will be due after each test. A portfolio assignment will be assigned during the trimester for all IB students only. The student will be submitting their topic, references and outlines this year. Next year they will submit the finished paper to their calculus teacher. We are preparing the students for their future tests - a great score on their ACT or SAT tests, and a 5 or 7 on their AP or IB tests the following year. | 677.169 | 1 |
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Designed for use in High Schools, two year colleges, and universities, Math Packs are self-contained lessons that bring math to life. There are 7 Packs organized by topic.
Applications of Sequences and Limits in Calculus
Microcosm Macrocosm: Population Models in Biology and Demography
Elliptic Integrals and Elliptic Functions in Calculus and Beyond
Geological Dating - A Calculus Application
The Mathematics of Scuba Diving
Using Original Sources to Teach the Logistic Equation
Optimal Foraging Theory
Computed Tomography in Multivariable Calculus
Clock Time vs. Sun Time: The Analemma
Closing in on the Internal Rate of Return
Differentials and Geographical Maps
The Terminator and Other Geographical Curves
History and Uses of Complex Numbers
Waves and Strong Tides
Ocean Circulation
Sharing a Secret
Calculus in a Movie Theatre
Calculus Optimization in Information Technology
Somewhere Within the Rainbow
How Old Is The Earth?
Splines in Single & Multivariable Calculus
3D Graphics in Calculus and Linear Algebra
Newton's Method and Fractal Patterns
Heat Therapy for Tumors
A Cell Population Model, Dynamical Diseases, and Chaos
Computer and Calculator Computation of Elementary Functions
Information Theory and Biological Diversity
Finding Vintage Concentrations in a Sherry Solera
Shoot Development in Plants
Rating Systems for Human Abilities: The Case of Rating Chess Skill
Motion of an Artificial Satellite about the Earth
Applications of Calculus in Geometrical Probability
Biokinetics of a Radioactive Tracer
The Solar Concentrating Properties of a Conical Reflector
Graphic Differentiation Clarifies Health Care Pricing
Price Elasticity of Demand: Gambling, Heroin, Marijuana, Whiskey, Prostitution, and Fish
Least Squares, Fish Ecology and the Chain Rule
Where are the Russian and Chinese Missiles Coming From?
Internal Rates of Return
The St. Louis Arch Problem
Differential Growth, Huxley's Allometric Formula, and Sigmoid Growth
Modeling Using the Deriviative: Numerical and Analytic Solutions
Simple Capital Theory
Tiltup Panels: Locate the Pulleys
A Mathematical Model of a Universal Joint
Layout Design for Sheet Metal Fabrication
Descriptive Models for Perception of Optical Illusions: Part II
Descriptive Models for Perception of Optical Illusions: Part I
Representing Integers as Sums or Differences of Squares
Oligopolistic Competition
Lagrange Multiplers and the Design of Multistage Rockets
The Cobb-Douglas Production Function
The .6 Rule for Industrial Costs
The Shape of the Surface of a Rotation Liquid
The Relationship Between Directional Heading of an Automobile and Steering Wheel Deflection
The Design of Honeycombs
Population Dynamics of Governmental Bureaus
Kepler's Laws and the Inverse Square Law
Calculus of Variations with Applications in Mathematics
Some Applications of Exponential and Logarithmic Functions
Evaluating Definite Integrals on a Computer: Theory and Practice
The Gradient and Some of its Applications
Atmospheric Pressure in Relation to Height and Temperature
The Three-Dimensional Trapezoid Rule
Series and Games: From Paradox to Paradox
The Force of Interest
Elementary Techniques of Numerical Integration and their Computer Implementation
Differentiation, Curve Sketching, and Cost Functions
Five Applications of Max - Min Theory from Calculus
Glottochronology: An Application of Calculus to Linguistics
Ascent - Descent
Developing the Fundamental Theorem of Calculus
Curve Fitting via the Criterion of Least Squares
Public Support for Presidents II
Public Support for Presidents I
The Dynamics of Political Mobilization II: Deductive Consequences and Empirical Application of the Model
Dynamics of Political Mobilization I: A Model of the Mobilization Process
Exponetial Models of Legislative Turnover
Lagrange Multipliers: Applications to Economics
A Strange Result in Visual Perception
Buffon's Needle Experiment
Pi is Irrational
Radioactive Chains: Parents and Children
Kinetics of Single-Reactant Reactions
Zipf's Law and His Efforts to use Infinite Series in Linguistics
The Human Cough
Viscous Fluid Flow and the Integral Calculus
Mercator's World Map and the Calculus
Integration: Getting It All Together
Determining Constants of Integration
How to Solve Problems Involving Exponential Functions
Numerical Approximations to y=ex
Development of the Function y=Aecx
Exponential Growth and Decay
Recognition of Problems Solved by Exponential Functions
Feldman's Model
Tracer Methods in Permeability
Epidemics
Prescribing Safe and Effective Dosage
Measuring Cardiac Output
The Digestive Process of Sheep
Population Growth and the Logistic Curve
Modeling the Nervous System: Reaction Time and the Central Nervous System
The Distribution of Resources | 677.169 | 1 |
Description
Formules mathématiques is the French version of the Math Formulary App that covers all mathematical formulas that are usually used in the school and the university. Where necessary graphics are included to depict and explain the topic better.
SD-Card Installation is supported. Bug reports related to the content and the app itself are more than welcome. Proposals for new features would be great.
This is the free version of the Formules mathématiques Pro app that you can get from
Recent changes: Bug fixes.
Formules mathématiques is the French version of the Math Formulary App that covers all mathematical formulas that are usually used in the school and the university. Where necessary graphics are included to depict and explain the topic better. | 677.169 | 1 |
Solving Inequalities
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Solving Inequalities Practice Guide: 22 inequalities of mixed difficulty. Nice, moving on. Wait! …All inequalities have the same solution which provides students with immediate feedback on the accuracy of their work. This encourages students to look more critically at their work and identify computational flubs, a needed skill for inequality sign flipping.☝Check out the Preview.
What makes this product a must-have?
❶ Self-checking: Immediate feedback received
❷ Multiple Ways to Use: A study guide, a class activity, a way to differentiate – see the teacher guide for more ideas.
❸Develop fluency: Careless errors and common hang-ups stand-out with practice guides. | 677.169 | 1 |
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