text stringlengths 6 976k | token_count float64 677 677 | cluster_id int64 1 1 |
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Web Site Introduction to Algebra Think Algebra is hard? Think again - this site explains the history along with simple equations. Each paragraph scaffolds skills until you get it. Than at th... Curriculum: Mathematics Grades: 3, 4, 5, 6
Web Site Expressions and Equations The activities in each of the four categories are placed in the recommended order that they be taught in class: Simplifying Expressions, Informal or Intuitiv... Curriculum: Mathematics Grades: 8, 9, 10
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Web Site Learning about Rate of Change in Linear Functio... In this two-part example, users can drag a slider on an interactive graph to modify a rate of change (cost per minute for phone use) and learn how modificati... Curriculum: Mathematics Grades: 6, 7, 8
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Web Site Interactive Mathematics National Library of Virtual Manipulatives for Interactive Mathematics related to the NCTM standards for grades 6-8. Some of the tools include graphing and ex... Curriculum: Mathematics Grades: 6, 7, 8
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Web Site Order of Operations and Evaluating Expressions This site provides prep for Mathematics A so students are prepared on Order of Operations. (Keywords: Algebra, McDougall Littel, online tutorials, practice, ... Curriculum: Mathematics Grades: 6
Web Site A+ Math A plus math was designed to help students with their math skills interactively. This website has flashcards, math games, and you can even print worksheets so... Curriculum: Mathematics Grades: K, 1, 2, 3, 4, 5, 6, 7, 8, 9
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Web Site Online Calculus Tutorials From Algebra Review to Multi-Variable Calculus, this website provides step-by-step tutorials for high school and university students. Curriculum: Mathematics Grades: 10, 11, 12, Junior/Community College, University | 677.169 | 1 |
GeoGebra is dynamic mathematics software for all levels of education that brings together geometry, algebra, spreadsheets,...
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GeoGebra is dynamic mathematics software for all levels of education that brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in one easy-to-use package. GeoGebra is a rapidly expanding community of millions of users located in just about every country. GeoGebra has become the leading provider of dynamic mathematics software, supporting science, technology, engineering and mathematics (STEM) education and innovations in teaching and learningGebra to your Bookmark Collection or Course ePortfolio
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The Probability/Statistics Object Library is a virtual library of objects for use by teachers and students of probability and...
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The Probability/Statistics Object Library is a virtual library of objects for use by teachers and students of probability and statistics. The library contains objects of two basic types, applets and components.An applet is a small, self-contained program that runs in a web page. Applets are intended to illustrate concepts and techniques in an interactive, dynamic way. A teacher or student can download an applet, drop it in a web page, and then add other elements of her own choice (such as expository text, data sets, and graphics). The applets in the library contain essentially no mathematical theory and thus can be used by students at various levels. The applets are intended to be small "micro worlds" where students can run virtual versions of random experiments and play virtual versions of statistical games.Components are the building blocks of applets and of other components. The Java objects are of three basic types: virtual versions of physical objects, such as coins, dice, cards, and sampling objects; virtual versions of mathematical objects, such as probability distributions, data structures, and random variables; user-interface objects such as custom graphs and tables. The Java objects can be used by teachers and students with some programming experience to create custom applets or components without having to program every detail from scratch, and thus in a fraction of the usual time. In addition, the components are extensively documented through a formal object model that specifies how the components relate to each other.Each object can be downloaded as a Java "bean" that includes all class and resource files needed for the object. An object in the form of a Java bean can be dropped into a builder tool (such as JBuilder or Visual Cafe) to expose the properties and methods of the object. Each object can also be downloaded in the form of a zip file that includes the source files and resource files for the object/Statistics Object Library to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material Xah's Home Teachers Teaching with Technology - Worldwide to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material Teachers Teaching with Technology - WorldwideKnot a Braid of Links (1996) features a collection of 370+ links to interesting free websites on specific mathematical...
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Knot a Braid of Links (1996) features a collection of 370+ links to interesting free websites on specific mathematical subjects, as well as math journals, math videos, puzzle pages, and reference materials. Each week, Knot a Braid of Links (KaBol) features a "cool math site of the week" as a service to the mathematics community provided by Camel by the Canadian Mathematical Society (CMS Knot a Braid of Links to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material Knot a Braid of Links contains a collection of fully developed high school curriculum modules that use the Internet in significant ways. ...
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This site contains a collection of fully developed high school curriculum modules that use the Internet in significant ways. There are currently 15 modules in Mathematics and 6 modules in Science; also, there are approximately two dozen additional modules that have been created by instructors and/or Education students.The learning modules here are web-based, technology intensive lessons focusing on mathematics and science in an applied context. They have been developed for teachers, by teachers, aligned with the Illinois State Learning Standards and the National Council for Teachers of Mathematics (NCTM) Standards. Some of the lessons are designed to last over several days, some only for a class site was developed to support professional development providers as they design and implement programs for pre-service...
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This site was developed to support professional development providers as they design and implement programs for pre-service and in-service K ? 12 mathematics and science teachers. In this database you will find: (1) A Conceptual Framework - highlights key elements critical to the design and implementation of effective professional development programs, with numerous links to relevant reviews of materials and practitioner essays, and (2) Reviews of Materials - the heart of the database, intended to help K ? 12 mathematics and science professional development providers more readily select materials appropriate for their program goals. Reviews may be searched by Descriptor Search or Keyword Search Education Materials TE-MAT database to your Bookmark Collection or Course ePortfolio
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$34 Algebra - The Complete Course series covers all Algebra core curriculum topics from a basic overview to the more complex algebraic functions. Dr. Monica Neagoy, consultant to the Annenberg Foundation & Public Broadcasting Service, uses concrete examples and practical applications to show how a mastery of fundamental algebraic concepts is the key to success in today's technologically advanced world. Students will learn how to use a model to answer questions and the collection, representation and analysis of data. | 677.169 | 1 |
Intemodino Group s.r.o.
Apps:
Random Number Generator
This simple and easy to use application generates random numbers based on a given length (from 1 to 100 digits) or between the minimum and maximum limits (from 0 to 999999999).
Random Number Generator RNG can generate up to 100 random numbers at a time.
Requirements: Adobe AIR.
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System of equations solver
Quadratic Equation Solver
Solves quadratic equations with real and complex roots using the quadratic formula. Enter the coefficients into the quadratic equation calculator and click Solve to calculate the discriminant and find the roots of your equation.Solve systems of equations
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Requirements: Adobe AIR.
Quadratic Equation Solver ML
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Depending on the discriminant a quadratic equation has one or two distinct real roots, or two distinct complex roots.
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Armed Merchants
Build your NET WORTH! Buy Low, Sell High and defend against pirates.
The year is 1870, you are a merchant trading high value goods between six countries spanning the Atlantic Ocean.
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Equation Master
Equation Master is an APP which contains references to mathematical equations and identities, a solver, and an opportunity to input and store other equations. The equations and identities are organized by area of mathematics for ease of locating. The solver portion is accessed by selecting an equation or a picture.
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11-12 Mathematics syllabus
Mathematics General Stage 6 syllabus - Preliminary and HSC courses - 2012 The purpose of the courses is to provide an appropriate mathematical background for students who wish to enter occupations that require the use of a variety of mathematical and statistical techniques. As well as introducing some new mathematical content, the various Focus Studies within the courses give students the opportunity to apply, and develop further, the knowledge, skills and understanding initially developed in the various Strands: Financial Mathematics, Data and Statistics, Measurement, Probability, and Algebra and Modelling. Through the Focus Studies, students develop the capacity to integrate their knowledge, skills and understanding across the Strands in contemporary contexts chosen for their ongoing relevance to the students' everyday lives and likely vocational pathways. (Syllabus, page 6)
Mathematics and Mathematics Extension 1 Stage 6 syllabus - Preliminary and HSC courses - 1982 The Mathematics course (previously known as 2 Unit Mathematics) is intended to give students an understanding of and competence in some further aspects of mathematics which are applicable to the real world. It is sufficient basis for further studies in mathematics as a minor discipline at tertiary level in support of courses such as the life sciences or commerce.
The Mathematics Extension 1 course (previously known as 3 Unit Mathematics) includes the whole of the Mathematics course and extensions. This course is intended to give students a thorough understanding of and competence in aspects of mathematics including many which are applicable to the real world. Both courses contain fundamental ideas of algebra and calculus.
Mathematics Extension 2 Stage 6 syllabus - Preliminary and HSC courses - 1989 The Mathematics Extension 2 course (previously known as 4 Unit Mathematics) includes the whole of the Mathematics and Mathematics Extension 1 courses and extension. It is designed for students with a special interest in mathematics who have shown a special aptitude for the subject. It represents a distinctly high level of school mathematics involving deep understanding of the fundamental ideas of algebra and calculus. | 677.169 | 1 |
If you want to increase your confidence in mathematics then look no further! Assuming little prior knowledge, this market-leading text is a great companion for those who have not studied mathematics i...
Mathematics for Economics and Business by Rebecca Taylor 9780077107864 (Paperback, 2008) Author:Rebecca Taylor. Mathematics for Economics and Business by Rebecca Taylor 9780077107864 (Paperback, 2008)...
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Author: Bradley, Teresa. Title: Essential Mathematics for Economics and Business (Paperback) Binding: Paperback. By continuing with this checkout and ordering from Hive, you are accepting our current ... | 677.169 | 1 |
This textbook provides a comprehensive introduction to the theory and practice of validated numerics, an emerging new field that combines the strengths of scientific computing and pure mathematics. In numerous fields ranging from pharmaceutics and engineering to weather prediction and robotics, fast and precise computations are essential. Based on the theory of set-valued analysis, a new suite of numerical methods is developed, producing efficient and reliable solvers for numerous problems in nonlinear analysis. Validated numerics yields rigorous computations that can find all possible solutions to a problem while taking into account all possible sources of error--fast, and with guaranteed accuracy.
Validated Numerics offers a self-contained primer on the subject, guiding readers from the basics to more advanced concepts and techniques. This book is an essential resource for those entering this fast-developing field, and it is also the ideal textbook for graduate students and advanced undergraduates needing an accessible introduction to the subject. Validated Numerics features many examples, exercises, and computer labs using MATLAB/C++, as well as detailed appendixes and an extensive bibliography for further reading.
Editorial Reviews
Review
"Beyond obvious practical value, this material offers students an excellent opportunity to revisit and rethink some crucial, fundamental college mathematics."--Choice
"[T]his little book is a very important supplement to existing books on validated numerics. It is a must for researchers working in this field."--G. Alefeld, Mathematical Reviews
"The book contains a lot of exercises, various small programs written in MATLAB code, and four sections with numerous problems provided for experimenting on a computer. It is written at an elementary level corresponding to its aims. But it is also a pleasure for specialists to leaf through the book."--Gunter Mayer, Zentralblatt MATH
"This book is an essential resource for those entering this fast-developing field, and it is also the ideal textbook for graduate students and advanced undergraduates needing an accessible introduction to the subject."--World Book Industry
From the Back Cover
"Validated Numerics contains introductory material on interval arithmetic and rigorous computations that is easily accessible to students with little background in mathematics and computer programming. I am not aware of any other book like it. The exercises and computer labs make it ideal for the classroom, and the references offer a good starting point for readers trying to gain deeper knowledge in this area."--Zbigniew Galias, AGH University of Science and Technology, Kraków
"A significant contribution, particularly since there are not many texts in this area. Validated Numerics will be read by those interested in interval arithmetic, numerical analysis, and ways to make computer simulations more robust and less susceptible to errors. It is well written and well organized."--A. J. Meir, Auburn University | 677.169 | 1 |
Calculus Early Vectors
Browse related Subjects ...
Read More in their curriculum. Stewart begins by introducing vectors in Chapter 1, along with their basic operations, such as addition, scalar multiplication, and dot product. The definition of vector functions and parametric curves is given at the end of Chapter 1 using a two-dimensional trajectory of a projectile as motivation. Limits, derivatives, and integrals of vector functions are interwoven throughout the subsequent chapters. As with the other texts in his Calculus series, in Early Vectors Stewart makes us of heuristic examples to reveal calculus to students. His examples stand out because they are not just models for problem solving or a means of demonstrating techniques - they also encourage students to develop an analytic view of the subject. This heuristic or discovery approach in the examples give students an intuitive feeling for analysis | 677.169 | 1 |
In algebra, you use permutations to count the number of subsets of a larger set. Use permutations when order is necessary. With combinations, you can count the number of subsets when order doesn't matter[more…]
Algebra can help you add a series of numbers (the sum of sequences) more quickly than you would be able to with straight addition. Adding integers, squares, cubes, and terms in an arithmetic or geometric[more…]
A number system in algebra is a set of numbers — and different number systems are used to solve different types of algebra problems. Number systems include real numbers, natural numbers, whole numbers,[more…]
Combinations are another way of counting items. This time, you select some items from a larger group, but you don't care what order they come in. For instance, if you select 6 lottery numbers from a listing[more…]
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Education & Test Prep
Get tips for studying and test-taking on a variety of subjects, including math, science, language arts, and foreign languages, and prepare to ace your standardized tests. | 677.169 | 1 |
Do you know that Differentiation & Integration
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Details about Frames and Bases:
Based on a streamlined presentation of the author s successful work, An Introduction to Frames and Riesz Bases, this book develops frame theory as part of a dialogue between mathematicians and engineers. Newly added sections on applications will help mathematically oriented readers to see where frames are used in practice and engineers to discover the mathematical background for applications in their field. The book presents basic results in an accessible way and includes extensive exercises."
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Rent Frames and Bases 1st edition today, or search our site for other textbooks by Ole Christensen. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Birkhauser Verlag GmbH.
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, easy-to-use guide to every day math. Includes hundreds of sample problems showing you how to use basic concepts of computation, measurement, geometry, algebra, statistics, probability, and more.
Editorial Reviews
About the Author
Brian Burrell is a contributor for Merriam-Webster Inc titles including: 'Merriam-Webster's Guide to Everyday Math, A Home and Business Reference','Merriam-Webster's Pocket Guide to Business and Everyday Math'
I discovered this book several years ago and have since referred it to lots of people. I have purchased at least 10 copies throughout the years, sometimes I tutor teenagers and young adults. All of whom I have given this book and most state they have a better understanding of math now and are not as intimidated. Great reference book.
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I have always had extreme difficulty with math... and it wasn't till recently... my final year of college, that I discovered that math dictionaries even existed. Upon review I must say that this publication covers a wide range of mathematical terminology. It is concise and effective. It would be helpful to anyone else in a similar situation.
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An Introduction to Mathematics for Economics
Contents:
An Introduction to Mathematics for Economics introduces quantitative methods to students of economics and finance in a succinct and accessible style. The introductory nature of this textbook means a background in economics is not essential, as it aims to help students appreciate that learning mathematics is relevant to their overall understanding of the subject. Economic and financial applications are explained in detail before students learn how mathematics can be used, enabling students to learn how to put mathematics into practice. Starting with a revision of basic mathematical principles the second half of the book introduces calculus, emphasising economic applications throughout. Appendices on matrix algebra and difference/differential equations are included for the benefit of more advanced students. Other features, including worked examples and exercises, help to underpin the readers' knowledge and learning. Akihito Asano has drawn upon his own extensive teaching experience to create an unintimidating yet rigorous textbook7604 | 677.169 | 1 |
Materials designed to help teach a "Chance" course or a more standard introductory probability or statistics course. A Chance...
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Materials designed to help teach a "Chance" course or a more standard introductory probability or statistics course. A Chance course is a case study quantitative literacy course designed to make students more informed and critical readers of current news items that use probability and statistics, as reported in daily newspapers. This site contains: Chance News, a monthly newsletter with abstracts of articles from current newspapers and journals, and suggestions for discussion questions for class use, with an archive; video lectures and audio discussions of Chance topics; syllabi of previous Chance courses and articles that have been written about them; a Teacher's Guide and other materials useful for teaching a Chance course; and links to related Internet sources for teaching a probability or statistics course. The Chance team of developers includes: J. Laurie Snell and Peter Doyle of Dartmouth College, Joan Garfield of the University of Minnesota, Tom Moore of Grinnell College, Bill Peterson of Middlebury College, and Ngambal Shah of Spelman Chance to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material Classroom Activities by Texas Instrumentsatorial Math: How to Count Without Counting to your Bookmark Collection or Course ePortfolio
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A comprehensive directory of computational geometry resources both on and off the Internet. General Resources, Literature,...
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A comprehensive directory of computational geometry resources both on and off the Internet. General Resources, Literature, Research and Teaching, Events, Software, other links. Also Geometry Pages to your Bookmark Collection or Course ePortfolio
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Computing Technology for Math Excellence is devoted to resources for teaching and learning mathematics (K-12 and calculus),...
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Computing Technology for Math Excellence is devoted to resources for teaching and learning mathematics (K-12 and calculus), technology integration, and the standards movement in education. CT4ME's major ongoing project is to identify Common Core math resources for teaching and learning.Their resources include links to sites for basic skills mastery, specific upper level subject resources, problem solving and critical thinking, using data, homework assistance, games, simulations, virtual math manipulatives, project-based learning, field trips for math, standardized testing, and more. Resources for teaching mathematics to learners with special needs are provided, including help for struggling readers. Accessibility resources are addressed. An extensive list of software products with potential to raise student achievement Technology for Math Excellence to your Bookmark Collection or Course ePortfolio
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A discussion of the history of conic sections, one of the oldest math subjects studied systematically and thoroughly, with a...
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A discussion of the history of conic sections, one of the oldest math subjects studied systematically and thoroughly, with a description, formulas, properties, a proof, Mathematica notebooks, the ellipse seen as a circle, second degree curves, intersection of circles, orthogonal conics, Pascal's Theorem and Brianchon's Theorem, and related sites. Illustrations, including one showing the intersections of parallel planes and a double cone, forming ellipses, parabolas, and hyperbolas respectively. Hosted by the Math Forumic Sections (Visual Dictionary of Special Plane Curves) to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Conic Sections (Visual Dictionary of Special Plane Curves)
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This website offers its viewers alot of information. Even though it looks like it is mostly for younger kids, it has...
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This website offers its viewers alot of information. Even though it looks like it is mostly for younger kids, it has sections on it for Parents, and Teachers. The "Parents" section offers ways to help out with homework, and ways of making their children want to learn more. The "Teachers" section offers you tips and tricks to make the students intrested, and involved in the discussions in the classroom. For the students, there are games to help them think, but at the same time are very entertaining. The practice problems on this website should interest students as well.It is very simple to navigate through this website. There are a couple tabs across the top of the screen, a broad array of what this site has to offer. Along the left side you can specify which category you want to look through. Down along the body of the website there are various sites that include more games and puzzles.This website also contains links to other fun websites on various subjects Math to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Cool Math
Select this link to open drop down to add material Cool Math Cornell University Library Historical Math Book Collection to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Cornell University Library Historical Math Book Collection
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This site is devoted to the demonstration and support of effective mathematics instruction. Through the use of indexed video...
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This site is devoted to the demonstration and support of effective mathematics instruction. Through the use of indexed video clips and related support materials, viewers can explore specific math concepts at their own pace. Whether you're an educator, a student, a grad student, or a parent this site can serve you. The video clips found on this site have been edited from the interactive COUNTDOWN television show broadcast weekly in Chicago which invites students to call in and help solve math problems. The site is organized according to NCTM standards and also offers downloadable activity sheets to accompany the videoOUNTDOWN to your Bookmark Collection or Course ePortfolio
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Synopses & Reviews
Publisher Comments
Taking only the most elementary knowledge for granted, Lancelot Hogben leads readers of this famous book through the whole course from simple arithmetic to calculus. His illuminating explanation is addressed to the person who wants to understand the place of mathematics in modern civilization but who has been intimidated by its supposed difficulty. Mathematics is the language of size, shape, and order--a language Hogben shows one can both master and enjoy.
Synopsis
"It makes alive the contents of the elements of mathematics."--Albert Einstein
Description
About the Author
Lancelot Hogben was born in Southsea, Hampshire, England, in 1895 and was educated at Trinity College, Cambridge University, where he obtained his M.S. degree. he received his D.Sc. degree from London Unversity and an honorary LL.D. from Birmingham Univesity. His other popular books include Science for the Citizen and (with Frederick Bodmer) The Loom of Language. | 677.169 | 1 |
Concepts inBoundary-valued shape-preserving interpolating splinesLagrange polynomial
In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points and numbers, the Lagrange polynomial is the polynomial of the least degree that at each point assumes the corresponding value (i.e. the functions coincide at each point).
more from Wikipedia
Fortran
Fortran (previously FORTRAN) is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing.
more from Wikipedia
Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them.
more from Wikipedia | 677.169 | 1 |
Teaching Textbooks Algebra 2. Everything is here but the second solution cd (covers lessons 30-51). I have the answer key book, which has all the answers. Book and tests have not been written in. $75 ppd. If interested email me or text me. | 677.169 | 1 |
Here are just a few ways you can use this book:
s Read the book as a stand-alone textbook to learn all the major con-
cepts of calculus. then CliffsQuickReview Calculus is for you!
How to Use This Book
You can use this book in any way that fits your personal style for study and review—you decide what works best with your needs. You can either read the book from cover to cover or just look for the information you want and put it back on the shelf for later. biology. and is explored in Chapters 3 and 4 of this book. as well as some essentials of geometry. or CliffsQuickReview Trigonometry may be valuable starting points for you. Integral calculus handles total changes or areas. then you probably have sufficient background to begin learning calculus.Introduction
is of change. chemistry. this mathematical notion of change is essential to many areas of knowledge. comprehensive reference for calculus?
If so. particularly disciplines like physics. Differential calculus rates of change slopes. and is addressed in Chapters 5 and 6. If some of those are unfamiliar. and economics. then CliffsQuickReview Geometry. Although it is not always immediately obvious. The prerequisites for learning calculus include much of high school algebra and trigonometry.
. If the formulas on the front side of the Pocket Guide (the cardstock page right inside the front cover) and topics covered in Chapter 1 are familiar to you. or just rusty for you.
Why You Need This Book
Can you answer yes to any of these questions?
s Do you need to review the fundamentals of calculus fast? s Do you need a course supplement to calculus? s Do you need a concise. CliffsQuickReview Algebra. Any situation that quantities change over time can be the tools Calculusthatthe mathematicsdeals withunderstood withorinvolvesof calculus.
The site also features timely articles and tips. plus downloadable versions of many CliffsNotes books. Here are a few ways you can search for topics in this book:
s Look for areas of interest in the book's Table of Contents.
s Flip through the book looking for subject areas at the top of each
page. This book defines new terms
and concepts where they first appear in the chapter. and more to enhance your learning.
s Get a glimpse of what you'll gain from a chapter by reading through
the "Chapter Check-In" at the beginning of each chapter. don't hesitate to share your thoughts about this book or any Hungry Minds product. www.
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A great resource.
s Or browse through the book until you find what you're looking
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s Use the Pocket Guide to find often-used formulas. We welcome your feedback!
. or use the
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exam. When you stop by our site.
Functions
A function is defined as a set of ordered pairs (x. x $ 0. This definition establishes the fact that the absolute value of a number must always be nonnegative—that is. x > 0 ] x = [ 0.x. b] = " x ! R: x # b . Each of these definitions also implies that the absolute value of a number must be a nonnegative. the absolute value of a number is the distance. as follows:
Z ] x. written x may be defined in a variety of ways. On a real number line. disregarding direction. there corresponds one and only one second element y. + 3h = " x ! R .3. x = 0 ] ] .3 represents a real number value. The notation f (x) is often used in place of y to indicate the value of the function f for a specific replacement for x and is read "f of x" or "f at x.
Note that an infinite end point ^ !3h is never expressed with a bracket in interval notation because neither + 3 nor . and the range variable is referred to as the dependent variable. A common algebraic definition of absolute value is often stated in three parts.y).4
CliffsQuickReview Calculus
(.3. such that for each first element x. The set of first elements is called the domain of the function."
. x < 0 \
Another definition that is sometimes applied to calculus problems is
x = x2
or the principal square root of x2.
Absolute Value
The concept of absolute value has many applications in the study of calculus. The domain variable is referred to as the independent variable. The absolute value of a number x. that the number is from zero. while the set of second elements is called the range of the function.
^ .
5 If x =. it can be manipulated algebraically to this form. then y = 2 or y =.5 (d) y = f (x) =. then y = 5 or y =. (a) x = y 2 If x = 4. at most.3. The slope of a line indicates whether the line slants up or down to the right or is horizontal or vertical. Many of the key concepts and theorems of calculus are directly related to functions. where a and b are not both zero. then y =+ 3 or y =.4. then y = 4 or y =. The slope is usually denoted by the letter m and is defined in a number of ways:
. (e) y = ! x + 4 If x = 5. Although a linear equation
may not be expressed in this form initially. If a vertical line were to intersect the graph at two or more points.y) represents a function if any vertical line intersects the graph in. Example 1-1: The following are some examples of equations that are functions.5. (f ) x 2 .3
x +4
(f ) y = f (x) = 3 2x + 9 (g) y = f (x) = 6 x (h) y = tan x (i) y = cos 2x Example 1-2: The following are some equations that are not functions.3 (e) y = f (x) = x2.Chapter 1: Review Topics
5
Geometrically. then y = –5 or y = –1 (c) x =. (d) x 2 + y 2 = 25 If x = 0.
Linear Equations
A linear equation is any equation that can be expressed in the form
ax + by = c . then y can be any real number.5. each has an example to illustrate why it is not a function. (a) y = f (x) = 3x + 1 (b) y = f (x) = x 2 (c) y = f (x) = x .2 (b) x = y + 3 If x = 2.5. one point. the graph of a set or ordered pairs (x.y 2 = 9 If x =. which clearly contradicts the definition of a function. the set would have one x value corresponding to two or more y values.
3) .1) . If the slopes of two lines L1 and L2 are m1 and m2. Note that even though these forms appear to be different from one another. hence. Any nonvertical lines are parallel if they have the same slopes. and the horizontal change would be zero. if the product of their slopes is –1. –3).5) =. the lines are perpendicular. then L1 is parallel to L2 if and only if m1 = m2.6
CliffsQuickReview Calculus
rise m = sun vertical change = horizontal change y value change = x value change ∆y = ∆x y1 . then L1 is perpendicular to L2 if and only if m1 ⋅ m2 = –1. Two nonvertical. a negative slope indicating a line slanting down to the right. the x value would remain constant.(4) = (.
. and conversely.7 4
The line.y 2 = x1 .y1 = x 2 . and conversely lines with equal slopes are parallel.x1
Note that for a vertical line. respectively.y1 m = x 2 . If the slopes of two lines L1 and L2 are m1 and m2.
y 2 . has a slope of –7/4. respectively. then. and a slope of zero indicating a horizontal line.x1 (. they can be algebraically manipulated to show they are equivalent.(. Example 1-3: Find the slope of the line passing through (–5.x 2 y 2 .
Some forms of expressing linear equations are given special names that identify how the equations are written. You should note that any two vertical lines are parallel and a vertical line and a horizontal line are always perpendicular. All nonvertical lines have a numerical slope with a positive slope indicating a line slanting up to the right. nonhorizontal lines are perpendicular if the product of their slopes is –1. a vertical line is said to have no slope or its slope is said to be nonexistent or undefined. 4) and (–1.
If a = 0. Example 1-5: Find an equation of the line through the point (3.
y .x 1 ) y .3y = 15 (general form)
.5) 3 3y = 4x .2 x + 6 3 3y =. the equation takes the form y = constant and represents a horizontal line.y 1 = m (x .4 =.b) and slope m. If b = 0.2x + 18 2x + 3y = 18 (general form)
The slope-intercept form of a linear equation is y = mx + b when the line has y-intercept (0.Chapter 1: Review Topics
7
The general or standard form of a linear equation is ax + by = c. Example 1-6: Find an equation of the line that has a slope 4/3 and crosses y-axis at –5.y 1 = m (x .4y = 0 (c) x =.3 (d) y = 6 The point-slope form of a linear equation is y .2 (x . the equation takes the form x = constant and represents a vertical line.15 4x .2 x + 2 3 y =.
y = mx + b y = 4 x + (.4) with slope –2/3. where a and b are not both zero. Example 1-4: The following are some examples of linear equations expressed in general form: (a) 2x + 5y = 10 (b) x .x 1 ) when the line passes through the point (x1.4 =.y1) and has a slope m.3) 3 y .
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CliffsQuickReview Calculus
The intercept form of a linear equation is x/a + y/b = 1 when the line has x-intercept (a. The trigonometric functions sine. tangent. The six basic trigonometric functions may be defined using a circle with equation x 2 + y 2 = r 2 and the angle i in standard position with its vertex at the center of the circle and its initial side along the positive portion of the x-axis (see Figure 1-1).
y x a + b =1 y x -2 + 3 = 1 . and the given number of degrees is multiplied by r/180 to convert to radian measure.3x + 2y = 6 (general form)
Trigonometric Functions
In trigonometry. Example 1-7: Find an equation of the line that crosses the x-axis at –2 and the y-axis at 3. The relationship between these measures may be expressed as follows: 180 % = r radians. the equation 1 radian = 180 % /r is used to change radians to degrees by multiplying the given radian measure by 180/r to obtain the degree measure. To change degrees to radians. secant. angle measure is expressed in one of two units: degrees or radians.b). and cosecant are defined as follows:
. the equivalent relationship 1% = r/180 radi180 ans is used. cosine. cotangent. Similarly.0) and y-intercept (0.
y)
r θ
x
y sin i = r = cos i = x = r
y x2+ y2 x x2+ y2
y tan i = sin i = x cos i cot i = cos i = x sin i y sec i = 1 =r = cos i x x2+ y2 x x2+ y2 y
csc i = 1 = r = sin i y
It is essential that you be familiar with the values of these functions at multiples of 30°. 60°. π/3. 90°. π/2. and 180° (or in radians. π/6. 45°.) You should also be familiar with the graphs of the six trigonometric functions.
y
(x. π/4. Some of the more common trigonometric identities that are used in the study of calculus are as follows:
.Chapter 1: Review Topics
9
Figure 1-1
Defining the trigonometric functions. and π (See Table 1-1.
If θ is an angle between π/2 and π. Which of the following equations is not a function? a.Chapter 1: Review Topics
13
Chapter Checkout
Q&A
1. Find an equation in general form of the line with slope –2/5 and
y-intercept (0. 4. e. –4/5
.
3x – 2y = 6 y = sin 3x y = x2 x2 + y2 = 16
y= 3+x
2. d 2.–2) and (–1. Find an equation in general form of the line passing through the points (3. b. what is cos θ? Answers: 1. and sin θ = 3/5. 5. d. x + 2y = –1 5. 3x + 5y = 0 4. c.
3. 2x + 5y = 15 3. Find an equation in general form of the line passing through the
origin and perpendicular to the line 5x – 3y = 6.0).3).
Chapter 2 LIMITS
Chapter Check-In
❑ ❑ ❑
Understanding what limits are Computing limits Determining when a function is continuous
function is essential to calculus.
Intuitive Definition
The limit of a function f (x) describes the behavior of the function close to a particular x value. the x " c function f (x) "approaches" the real number L (see Figure 2-1).
Figure 2-1 The limit of f(x) as x approaches c. It used defining the most concepts in The conceptisof theinlimit of asome offunction.L)
y=f(x)
x
c
. importantthe study of calculus—continuity. It does not necessarily give the value of the function at x. the derivative of a and the definite integral of a function. which means that as x "approaches" c.
y
L
(c. You write lim f (x) = L.
Note that this does not imply that f (c) = L.Chapter 2: Limits
15
In other words. If the function does not approach a real number L as x approaches c. the function value f (x) gets closer to L.f(c)) y=f(x)
x
Figure 2-3
f(c) and lim f (x) are not equal. in fact.
x " c
y
(c.
Figure 2-2 f(c) does not exist. you write lim f (x) DNE (Does Not Exist). the limit does not exist. the function may not even exist at c (Figure 2-2) or may equal some value different than L at c (Figure 2-3). therefore. as the independent variable x gets closer and closer to c.f(c))
.
x " c
y
y=f(x)
x
(c. x " c Many different situations could occur in determining that the limit of a function does not exist as x approaches some value. but lim f (x) does.
x "-3 x Substituting –3 for x yields 0/0. 3 . hence.
x " 2
When x is replaced by 2. 6 . the fraction values will get closer and closer to 1. Some of these techniques are illustrated in the following examples.–6) removed from the graph (see Figure 2-4). simple substitution. which is meaningless.
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Evaluating Limits
Limits of functions are evaluated using many different techniques such as recognizing a pattern. you find that
x "-3 2 (x + 3)( x .
x " 2
-9 Example 2-3: Evaluate lim x + 3 .3) -9 lim x + 3 = lim x x+3 x "-3 = lim (x .3) x "-3
2
=.1) = 5. Factoring first and simplifying.1). 7 2 3 4 Because the value of each fraction gets slightly larger for each term.6
The graph of (x2 – 9)/(x + 3) would be the same as the graph of the linear function y = x – 3 with the single point (–3. . the limit of the sequence is 1. .
Example 2-2: Evaluate lim (3x . . 5 . 2 . hence.
4 5 Example 2-1: Find the limit of the sequence: 1 . and 3x – 1 approaches 5. lim (3x . or using algebraic simplifications. 6 . 3x approaches 6. while the numerator is always one less than the denominator.
Continuity
A function f (x) is said to be continuous at a point (c,f (c)) if each of the following conditions is satisfied: (1) f (c) exists (c is in the domain of f ), (2) lim f (x) exists, and
x " c x " c
(3) lim f (x) = f (c). Geometrically, this means that there is no gap, split, or missing point for f (x) at c and that a pencil could be moved along the graph of f (x) through (c,f (c)) without lifting it off the graph. A function is said to be continuous
Chapter 2: Limits
25
at (c,f (c)) from the right if lim f (x) = f (c) and continuous at (c,f (c))
x " c+
from the left if lim f (x) = f (c). Many of our familiar functions such as x " c linear, quadratic and other polynomial functions, rational functions, and the trigonometric functions are continuous at each point in their domain. A special function that is often used to illustrate one-sided limits is the greatest integer function. The greatest integer function, [x] , is defined to be the largest integer less than or equal to x (see Figure 2-6).
-
hence, f is continuous at x = –3. Many theorems in calculus require that functions be continuous on intervals of real numbers. A function f (x) is said to be continuous on an open interval (a,b) if f is continuous at each point c ∈ (a,b). A function f (x) is said to be continuous on a closed interval [a,b] if f is continuous at each point c ∈ (a,b) and if f is continuous at a from the right and continuous at b from the left. Example 2-26: (a) f (x) = 2x + 3 is continuous on (– ∞,+∞) because f is continuous at every point c ∈ (– ∞,+∞). (b) f (x) = (x – 3)/(x + 4) is continuous on (–∞,–4) and (–4,+∞) because f is continuous at every point c ∈ (–∞,–4) and c ∈ (–4,+∞) (c) f (x) = (x – 3)/(x + 4) is not continuous on (–∞,–4] or [–4,+∞) because f is not continuous on –4 from the left or from the right. (d) f (x) = x is continuous on [0, +∞) because f is continuous at every point c ∈ (0,+∞) and is continuous at 0 from the right. (e) f (x) = cos x is continuous on (–∞,+∞) because f is continuous at every point c ∈ (–∞,+∞). (f ) f (x) = tan x is continuous on (0,π/2) because f is continuous at every point c ∈ (0,π/2). (g) f (x) = tan x is not continuous on [0,π/2] because f is not continuous at π/2 from the left.
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CliffsQuickReview Calculus
(h) f (x) = tan x is continuous on [0,π/2) because f is continuous at every point c ∈ (0,π/2) and is continuous at 0 from the right. (i) f (x) = 2x/(x2 + 5) is continuous on (–∞,+∞) because f is continuous at every point c ∈ (–∞,+∞). (j ) f (x) = x - 2 /( x - 2) is continuous on (–∞,2) and (2,+∞) because f is continuous at every point c ∈ (–∞,2) and c ∈ (2,+∞). (k) f (x) = x - 2 /( x - 2) is not continuous on (–∞,2] or [2,+∞) because f is not continuous at 2 from the left or from the right.
Chapter Checkout Q&A
1. Evaluate the following
ne of the most important applications of limits concept of the derivative of a used Owide variety aoffunction. In calculus, the derivativeisofthefunction isapplyin a problems, and understanding it is essential to ing it to such problems.
Definition
The derivative of a function y = f (x) at a point (x,f (x)) is defined as
Dx " 0
lim
f (x + ∆x) - f (x) ∆x
if this limit exists. The derivative is denoted by f'(x), read "f prime of x" or "f prime at x," and f is said to be differentiable at x if this limit exists (see Figure 3-1).
Figure 3-1 The derivative of a function as the limit of rise over run.
y
f(x + ∆x) (x, f(x)) f(x) y= f(x) x ∆x
(x + ∆x, f(x + ∆x)) f(x + ∆x) − f(x)
x + ∆x
x
5 . (x.x + 5 ∆x 2 = 2x∆x + ∆x ∆x ∆x (2x + ∆x) = ∆x
f (x + ∆x) .–1) is 4. but the derivative at that point may not exist. y = s(t). the function f (x) = x1/3 is continuous over its entire domain or real numbers. One interpretation of the derivative of a function at a point is the slope of the tangent line at this point. which is also continuous over its entire domain of real numbers but is not differentiable at x = –2. As an example. it is defined to be the instantaneous velocity at time t for the function. The relationship between continuity and differentiability can be summarized as follows: Differentiability implies continuity.f (x)) on the graph of y = f (x). where y = s(t). If this limit exists.x 2 + 5 = ∆x ∆x
2 2 2 = x + 2x∆x + ∆x .30
CliffsQuickReview Calculus
If a function is differentiable at x. That is.
f (x + ∆x) . but its derivative does not exist at x = 0. Example 3-1: Find the derivative of f (x) = x2 – 5 at the point (2. The derivative may be thought of as a limit of the average velocities between a fixed time and other times that get closer and closer to the fixed time. it is defined to be the slope of the tangent line at the fixed point. Another interpretation of the derivative is the instantaneous velocity of a function representing the position of a particle along a line at time t.–1). The derivative may be thought of as the limit of the slopes of the secant lines passing through a fixed point on a curve and other points on the curve that get closer and closer to the fixed point. Another example is the function f (x) = x + 2 . then it must be continuous at x. the derivative of f (x) = x2 – 5 at the point (2. but the converse is not necessarily true. but continuity does not imply differentiability.5 . If this limit exists. a function may be continuous at a point.f (x) (x + ∆x) 2 .
.f (x) = 2x + ∆x ∆x f l^ x h = lim (2x + ∆x) = 2x f l^ 2 h = 2 $ 2 = 4
∆x " 0
hence.
2) 4 =.2xy 3 dx y .160
which represents the slope of the tangent line at the point (–1.2xy 3 dy = 2 2 dx 3x y . some equations in x and y do not explicitly define y as a function x and cannot be easily manipulated to solve for y in terms of x.1)[( . at (.10)( . Example 3-19: Find dx if x 2 y 3 . you find that
2xy 3 + x 2 $ 3y 2 $ dy dy .x $ 1 $ =0 dx dx dy dy 3x 2 y 2 -x = y .3x 2 y 2 dy
.–32). you find that
y l= 5 (x 2 .1. even though such a function may exist. Differentiating implicitly with respect to x.3]4 = (.2xy 3 dx dx dy (3x 2 y 2 . it is implied that there exists a function y = f (x) such that the given equation is satisfied.y dy = dx x .
Implicit Differentiation
In mathematics. The technique of implicit differentiation allows you to find the derivative of y with respect to x without having to solve the given equation for y.1y . Because the slope of the tangent line to a curve is the derivative.3) 4 hence.x or 2xy 3 . –32). .xy = 10.x) = y . The chain rule must be used whenever the function y is being differentiated because of our assumption that y may be expressed as a function of x. When this occurs.Chapter 3: The Derivative
37
Example 3-18: Find the slope of the tangent line to a curve y = (x2 – 3)5 at the point (–1.32) y l= 10 (.1) 2 .3) 4 (2x) = 10x (x 2 .
Tangent and Normal Lines
As previously noted.
. imum problems. and acceleration solving related rate problems. the slope of the normal line to the graph of f (x) is –1/f'(x). It may be used and minThe derivative ofsolvingin curve sketching. the derivative of a function at a point. distance. and approximating function values.Chapter 4 APPLICATIONS OF THE DERIVATIVE
Chapter Check-In
❑ ❑ ❑ ❑ ❑
Using the derivative to understand the graph of a function Locating maximum and minimum values of a function Finding velocity and acceleration Relating rates of change Approximating quantities by using derivatives
a function has many applications to problems in calculus. Because the slopes of perpendicular lines (neither of which is vertical) are negative reciprocals of one another. solving maximumproblems. is the slope of the tangent line at this point. velocity. The normal line is defined as the line that is perpendicular to the tangent line at the point of tangency.
4 =.y 1 = m (x .x .
. the equation of the normal line at the point (–1. f (x)) is called a critical point of f (x) if x is in the domain of the function and either f'(x) = 0 or f'(x) does not exist. or does not exist at that point on the curve.44
CliffsQuickReview Calculus
Example 4-1: Find the equation of the tangent line to the graph of f (x) = x 2 + 3 at the point (–1.1 x + 2y = 3
Example 4-2: Find the equation of the normal line to the graph of f (x) = x 2 + 3 at the point (–1.2 = 2x + 2 2x .2). you find that f'(–1) = –1/2 and the slope of the normal line is –1/f'(–1) = 2.x 1 ) y .1 (x + 1) 2 2y .4
Critical Points
Points on the graph of a function where the derivative is zero or the derivative does not exist are important to consider in many application problems of the derivative.x 1 ) y .1/2 $ (2x) 2 x f l(x) = 2 x +3
At the point (–1.
f (x) = (x 2 + 3)1/2 f l(x) = 1 (x 2 + 3) . vertical. hence. From Example 4-1.2 =.y 1 = m (x .2). The geometric interpretation of what is taking place at a critical point is that the tangent line is either horizontal. f'(–1) = –1/2 and the equation of the line is
y . 2).2) is
y . The point (x.y =.2 = 2 (x + 1) y .
Example 4-6: Find the maximum and minimum values of f (x) = x4 – 3x3 – 1 on [–2. Example 4-5: Find the maximum and minimum values of f (x) = sin x + cos x on [0. The largest function value from the previous step is the maximum value. Note the importance of the closed interval in determining which values to consider for critical points. and from Example 4-4. The function is continuous on [0.2π].
f l(x) = 0 & 4x 3 . the maximum function value 39 at x = –2. The theorem is stated as follows. and the minimum function value is –9 at x = 2.2π]. and its derivative is f'(x) = 4x3 – 9x2. The function values at the end points of the interval are f (0)=1 and f (2π)=1.2]. the only critical point occurs at x = 0 which is (0.9x 2 = 0 x 2 (4x .2].46
CliffsQuickReview Calculus
The procedure for applying the Extreme Value Theorem is to first establish that the function is continuous on the closed interval. The function is continuous on [–2. .9) = 0 x = 0. The function values at the endpoints of the interval are f (2) = –9 and f (–2) = 39.–1). hence.2 k. hence. the maximum function value of f (x) is 2 at x=π/4. and the minimum function value of f (x) is . 2 k and a5r/4.2 at x = 5π/4.2]. the critcal points are a r/4. Note that for this example the maximum and minimum both occur at critical points of the function. x = 9 4
Because x = 9/4 is not in the interval [–2.
Mean Value Theorem
The Mean Value Theorem establishes a relationship between the slope of a tangent line to a curve and the secant line through points on a curve at the endpoints of an interval.
. The next step is to determine all critical points in the given interval and evaluate the function at these critical points and at the endpoints of the interval. and the smallest function value is the minimum value of the function on the given interval.
8 .b] and differentiable on an open interval (a.2 3 . the theorem guarantees at least one critical point. The function is continuous on [–2.b) exists such that
f l(c) = f (b) .f (.3 f l(x) =.Chapter 4: Applications of the Derivative
47
If a function f (x) is continuous on a closed interval [a.2) .(.f (a)) and (b. then at least one number c ∈ (a.b).f(a))
x
a c b
Geometrically.3 =.f(c)) y=f(x)
(b. this means that the slope of the tangent line will be equal to the slope of the secant line through (a. Example 4-7: Verify the conclusion of the Mean Value Theorem for f (x) = x2 – 3x – 2 on [–2.2 . Note that for the special case where f (a) = f (b).b).3].3).f(b)) (a.f (a) b-a
Figure 4-1 The Mean Value Theorem.2)
The slope of the tangent line is
f l(x) = 2x .
y
(c.10 = 5 = 5 =.3] and differentiable on (–2. where f'(c) = 0 on the open interval (a.2 2x = 1 x= 1 2
Because 1/2 ∈ [–2.3].f (b)) for at least one point on the curve between the two endpoints. The slope of the secant line through the endpoint values is
f (3) .2 & 2x . the c value referred to in the conclusion of the Mean Value Theorem is c = 1/2
.
then the function is said to be increasing on I. Testing all intervals to the left and right of these values for f'(x) = 4x3 – 16x. If f'(x) > 0. 0) f l(x)< 0 on (0. As noted in Example 4-4. Because the derivative is zero or does not exist only at critical points of the function. f is increasing on (–2. 0. the domain of f (x) is restricted to the closed interval [0. 2) f l(x)> 0 on (2. then. and its critical points occur at x = –2.2. . In determining intervals where a function is increasing or decreasing. you find that
f l(x)< 0 on (. + 3)
hence. you first find domain values where all critical points will occur. Example 4-9: For f (x) = sin x + cos x on [0. Testing all intervals to the left and right of these values for f'(x) = cos x – sin x. it must be positive or negative at all other points where the function exists. As noted in Example 4-3. you find that
. and 2. then f is decreasing on the interval. then the function is said to be decreasing on I. then f is increasing on the interval.–2) and (0. This and other information may be used to show a reasonably accurate sketch of the graph of the function. If f'(x) < 0 at each point in an interval I. If f'(x) > 0 at each point in an interval I.0) and (2. determine all intervals where f is increasing or decreasing.2π].2) f l(x)> 0 on (. Example 4-8: For f (x) = x4 – 8x2 determine all intervals where f is increasing or decreasing.2). and its critical points occur at π/4 and 5π/4.48
CliffsQuickReview Calculus
Increasing/Decreasing Functions
The derivative of a function may be used to determine whether the function is increasing or decreasing on any intervals in its domain. test all intervals in the domain of the function to the left and to the right of these values to determine if the derivative is positive or negative.3.2π]. and if f'(x) < 0.+ ∞) and decreasing on (– ∞. the domain of f (x) is all real numbers.
it is called the First Derivative Test for Local Extrema. Example 4-10: If f (x) = x4 – 8x2. however. Because f'(x) changes from negative to positive around –2 and 2. the function is said to have a local (relative) extremum at that point. . Also.–16). When this technique is used to determine local maximum or minimum function values.5π/4). the function has a local (relative) maximum at the critical point. As noted in Example 4-9.
First Derivative Test for Local Extrema
If the derivative of a function changes sign around a critical point. Note that there is no guarantee that the derivative will change signs. 2 ). f has a local minimum at (5r/4.
.2π] and decreasing on (π/4. the function has a local (relative) minimum at the critical point. 5r m 4 4 f l(x)> 0 on c 5r . If the derivative changes from positive (increasing function) to negative (decreasing function). f (x) has critical points at x = –2. and therefore. 0. and hence. 0.–16) and (2. determine all local extrema for the function. 2. and hence. As noted in Example 4-8. Also f'(x) changes from negative to positive around 5π/4. f'(x) changes from positive to negative around 0.Chapter 4: Applications of the Derivative
49
f l(x)> 0 on. f is increasing on [0. f has a local maximum at (r/4.2 ). 2r E 4
hence. it is essential to test each interval around a critical point. f has a local maximum at (0. Because f'(x) changes from positive to negative around π/4. π/4) and (5π/4. determine all local extrema for the function. Example 4-11: If f (x) = sin x + cos x on [0. f has a local minimum at (–2.0). the derivative changes from negative (decreasing function) to positive (increasing function). f (x) has critical points at x = π/4 and 5π/4. If. r m 4 f l(x)< 0 on c r .2π].
and f"(2) = 32 > 0. and 2. 2 ). you find that f"(–2) = 32 > 0.0).2π] using the Second Derivative Test. Also. revert back to the First Derivative Test to determine any local extrema. the second derivative is difficult or tedious to find.–16). As with the previous situations. f"(0) = –16 < 0. As noted in Example 4-3. If a function has a critical point for which f'(x) = 0 and the second derivative is positive at this point. Another drawback to the Second Derivative Test is that for some functions. the First Derivative Test would have to be used to determine any local extrema.2 and f has a local maximum at (r/4. then f has a local minimum here. This technique is called Second Derivative Test for Local Extrema. As noted in Example 4–4.2π]. then f has local maximum here. and f has a local minimum at (5r/4. Example 4-13: Find any local extrema of f (x) = sin x + cos x on [0. Because f"(x) = 12x2 – 16. . you find that f" (π/4) = . f'(x) = 0 at x = π/4 and 5π/4. the function has a critical point for which f'(x) = 0 and the second derivative is negative at this point. Because f"(x) = – sin x – cos x. Three possible situations could occur that would rule out the use of the Second Derivative Test for Local Extrema:
(1) f l(x) = 0 and f m (x) = 0 doesnot (2) f l(x) = 0 and f m (x) does not exist (3) f l(x) does not exist doesnot
Under any of these conditions. These results agree with the local extrema determined in Example 4-11 using the First Derivative Test on f (x) = – sin x – cos x on [0. Example 4-12: Find any local extrema of f (x) = x4 – 8x2 using the Second Derivative Test. f'(x) = 0 at x = –2.50
CliffsQuickReview Calculus
Second Derivative Test for Local Extrema
The second derivative may be used to determine local extrema of a function under certain conditions. f" (5π/4)= 2. and f has a local minimum (2. 0.2 ). If. and f has a local minimum at (–2.–16). however.
. These results agree with the local extrema determined in Example 4-10 using the First Derivative Test on f (x) = x4 – 8x2. and f has local maximum at (0.
12x .Chapter 4: Applications of the Derivative
51
Concavity and Points of Inflection
The second derivative of a function may also be used to determine the general shape of its graph on selected intervals. it is essential to test each interval around the values for which f"(x) = 0 or does not exist.f (x)) is a point of inflection of the function. If f"(x) changes sign. 2) and f m(x)> 0 on (2.12 f m(x) = 0 & 6x . If a function changes from concave upward to concave downward or vice versa around a point. there is no guarantee that the second derivative will change signs.12 f m(x) = 6x . a function is concave upward on an interval if its graph behaves like a portion of a parabola that opens upward. Geometrically. A function is said to be concave upward on an interval if f"(x) > 0 at each point in the interval and concave downward on an interval if f"(x) < 0 at each point in the interval. f is concave downward on (– ∞. its second derivative will be zero. If the graph of a function is linear on some interval in its domain. and therefore. you find that
f m(x)< 0 on (.2) and concave upward on (2. a function that is concave downward on an interval looks like a portion of a parabola that opens downward. + 3)
hence. Example 4-14: Determine the concavity of f (x) = x3 – 6x2 – 12x + 2 and identify any points of inflection of f (x). its domain is all real numbers. and function has a point of inflection at (2. In determining intervals where a function is concave upward or concave downward. you first find domain values where f"(x) = 0 or f"(x) does not exist.
f l(x) = 3x 2 . Then test all intervals around these values in the second derivative of the function. and it is said to have no concavity on that interval. As with the First Derivative Test for Local Extrema. Likewise.3. Because f (x) is a polynomial function. it is called a point of inflection of the function. then (x.12 = 0 6x = 12 x=2
Testing the intervals to the left and right of x = 2 for f"(x) = 6x – 12.+ ∞).–38)
.
sin x f m(x) =. will suggest which technique is appropriate to use in determining a maximum or minimum value—the Extreme Value Theorem. The restrictions stated or implied for such functions will determine the domain from which you must work. 7r m 4 4 f m(x)< 0 on c 7r . the First Derivative Test. The function that is to be minimized is the surface area (S) while the volume (V ) remains fixed at 108 cubic inches (Figure 4-2). 0. The function.cos x = 0 .52
CliffsQuickReview Calculus
Example 4-15: Determine the concavity of f (x) = sin x + cos x on [0.2π] and identify any points of inflection of f (x).
. Find the dimensions for the box that require the least amount of material.cos x f m(x) = 0 & .0).2π].sin x .sin x . The domain of f (x) is restricted to the closed interval [0. 7r 4 4
Testing all intervals to the left and right of these values for f"(x) = –sin x – cos x.
f l(x) = cos x .2π] and concave upward on (3π/4. 2r m 4
hence.0) and (7π/4.
Maximum/Minimum Problems
Many application problems in calculus involve functions for which you want to find maximum or minimum values.3π/4) and (7π/4. you find that
f m(x)< 0 on . together with its domain. f is concave downward on [0.7π/4) and has points of inflection at (3π/4. Example 4-16: A rectangular box with a square base and no top is to have a volume of 108 cubic inches. 3r E 4 f m(x)> 0 on c 3r . or the Second Derivative Test.sin x = cos x x = 3r .
you find that
h = 6-r 8 6 6h = 48 . Find the maximum volume possible for the inscribed cylinder.8r h=8.r) = 0 r = 0. Letting r = radius of the cylinder and h = height of the cylinder and applying similar triangles.4r 3
Figure 4-3 A cross section of the cone and cylinder for Example 4-17.3
with the domain of f (r) = [0. The cone has 8 cm and radius 6 cm.4 r) 3 2 4 rr 3 f (r) = 8rr .4rr 2 f l(r) = 0 & 16rr . which cannot be greater that the radius of the cone.
8cm h r 6-r 6cm 8cm
Because V = πr2h and h = 8 – (4/3)r.54
CliffsQuickReview Calculus
Example 4-17: A right circular cylinder is inscribed in a right circular cone so that the center lines of the cylinder and the cone coincide. The function that is to be maximized is the volume (V ) of a cylinder inscribed in a cone with height 8 cm and radius 6 cm (Figure 4-3).4rr 2 = 0 4rr (4 . 4
.
f l(r) = 16rr . you find that
V = f (r) = rr 2 (8 .6] because r represents the radius of the cylinder.
then v = s'(t) represents the instantaneous velocity. the derivative of a function representing the position of a particle along a line at time t is the instantaneous velocity at that time. If y = s(t) represents the position function. Velocity. represents the instantaneous acceleration of the particle at time t.6 ft/ sec
.6 s l(2) =. where t is measured in seconds and s is measured in feet. hence.6 (2) .
f (0) = 0 f (4) = 128r 3 f (6) = 0
hence. Find (a) The velocity of the particle at the end of 2 seconds. while a negative velocity indicates that the position is decreasing with respect to time. (b) The acceleration of the particle at the end of 2 seconds. which is the second derivative of the position function.6t . and a negative acceleration implies that the velocity is decreasing with respect to time. A positive velocity indicates that the position is increasing as time increases. The derivative of the velocity. Part (a): The velocity of the particle is
v = s l(t) = 3t 2 .Chapter 4: Applications of the Derivative
55
Because f (r) is continuous on [0.6]. which will occur when the radius of the cylinder is 4 cm and its height is 8/3 cm.6
At t = 2 sec onds s l(2) = 3 (2) 2 . Example 4-18: The position of a particle on a line is given by s(t) = t3 – 3t2 – 6t + 5. If the distance remains constant. If the velocity remains constant on an interval of time.
Distance. Likewise. and a = v'(t) = s"(t) represents the instantaneous acceleration of the particle at time t. then the acceleration will be zero on the interval. a positive acceleration implies that the velocity is increasing with respect to time. use the Extreme Value Theorem and evaluate the function at its critical points and its endpoints. then the velocity will be zero on such an interval of time. the maximum volume is 128π/3 cm3. and Acceleration
As previously mentioned.
v = s'(t) = 0.5 m above the ground.56
CliffsQuickReview Calculus
Part (b): The acceleration of the particle is
a = v l(t) = s m(t) = 6t . Find the rate of change of its volume when the radius is 5 inches. namely time.9. The sign of the rate of change of the solution variable with respect to time will also indicate whether the variable is increasing or decreasing with respect to time.9 (5) 2 + 49 (5) + 15 s (5) = 137.75 in/min. Note that a given rate of change is positive if the dependent variable increases with respect to time and negative if the dependent variable decreases with respect to time.9.
Related Rates of Change
Some problems in calculus require finding the rate of change or two or more variables that are related to a common variable.8t + 49 = 0 . Example 4-20: Air is being pumped into a spherical balloon such that its radius increases at a rate of .6 At t = 2 sec onds v l(2) = s m(2) = 6 (2) . the object will reach its highest point at 137.
.4.9.8t =. To solve these types of problems.5 meters
hence.8t + 49 s l(t) = 0 & . where
v = s l(t) =. How high above the ground will the object reach? The velocity of the object will be zero at its highest point above the ground.49 t = 5 sec onds
The height above the ground at 5 seconds is
s (5) =.9t2 + 49t + 15 gives the height in meters of an object after it is thrown vertically upward from a point 15 meters above the ground at a velocity of 49 m/sec. That is. the appropriate rate of change is determined by implicit differentiation with respect to time.6 v l(2) = s m(2) = 6 ft/ sec 2
Example 4-19: The formula s(t) = –4.
you find that
dV = 4r (5 inches) 2 $ (. At r = 5 inches.75 in/min because the radius is increasing with respect to time. you find that
dV = 4 r $ 3r 2 $ dr 3 dt dt dV = 4rr 2 $ dr dt dt
The rate of change of the radius dr/dt = . Let x = distance traveled by the truck y = distance traveled by the car z = distance between the car and truck The distances are related by the Pythagorean Theorem: x2 + y2 = z2 (Figure 4-4).
. the volume is increasing at a rate of 75π cu in/min when the radius has a length of 5 inches. Example 4-21: A car is traveling north toward an intersection at a rate of 60 mph while a truck is traveling east away from the intersection at a rate of 50 mph.Chapter 4: Applications of the Derivative
57
The volume (V ) of a sphere with radius r is
V = 4 rr 3 3
Differentiating with respect to t.75 in/ min ) dt dV = 75rcu in/ min dt
hence. Find the rate of change of the distance between the car and truck when the car is 3 miles south of the intersection and the truck is 4 miles east of the intersection.
f (x) ∆x
.
Differentials
The derivative of a function can often be used to approximate certain function values with a surprising degree of accuracy. The definition of the derivative of a function y = f (x) as you recall is
f l(x) = lim
∆x " 0
f (x + ∆x) . the distance between the car and the truck is increasing at a rate of 4 mph at the time in question.
x y z
The rate of change of the truck is dx/dt = 50 mph because it is traveling away from the intersection. the concept of the differential of the independent variable and the dependent variable must be introduced.58
CliffsQuickReview Calculus
Figure 4-4 A diagram of the situation for Example 4-21.60 mph) = (5 mi) dz dt 2 dz 20 mi ph = (5 mi) dt dz = 4 mph dt
hence. while the rate of change of the car is dy/dt = –60 mph because it is traveling toward the intersection. To do this. Differentiating with respect to time. you find that
2x dy dy + 2y = 2z dz dt dt dt dx + y dy = z dz x dt dt dt
(4 mi)( 50 mph) + (3 mi)( .
The smaller the change in x. the closer dy will be to ∆y. then the slope of the tangent is approximately the same as the slope of the secant line through (x. ∆x ! 0 hence.f (x) Because ∆y = f (x + ∆x) . provided that the change in x (∆x = dx) is relatively small. written dy.
Figure 4-5 Approximating a function with differentials. 8 f (x + ∆x) . f(x)) y= f(x) ∆x = dx
dy
∆y
x
x + ∆x
x
. That is.
f l(x) . is defined to be
dy = f l(x) $ dx .f (x)B /∆x or equivalently f l(x) $ ∆x . f (x + ∆x) .f (x)
The differential of the independent variable x is written dx and is the same as the change in x. ∆x.f (x) you find that dy = f l(x) dx . f (x + ∆x) .
dx = ∆x. f (x + ∆x) . f(x + ∆x))
(x.
y
(x + ∆x. enabling you to approximate function values close to f (x) (Figure 4-5).f (x)
The differential of the dependent variable y. That is. f l(x) $ dx .f (x)). ∆y
The conclusion to be drawn from the preceding discussion is that the differential of y(dy) is approximately equal to the exact change in y(∆y). If ∆x is very small (∆x ≠ 0).f (x)).Chapter 4: Applications of the Derivative
59
which represents the slope of the tangent line to the curve at some point (x.
.76 cm 2
The area of the square will increase by approximately 2.23 cm.60
CliffsQuickReview Calculus
Example 4-22: Find dy for y = x3 + 5x – 1.
dy = 2 (6 cm)(.23. hence.55 to the nearest thousandth. choose a convenient value of x that is a perfect cube and is relatively close to 26.23. Because the function you are applying is f (x) = 3 x .
Because y = f (x) = x 3 + 5x . you find that ∆x = dx = . 23 cm) dy = 2. namely x = 27. you find that ∆x = dx = –.1 f l(x) = 3x 2 + 5 dy = f l(x) $ dx dy = (3x 2 + 5) $ dx
Example 4-23: Use differentials to approximate the change in the area of a square if the length of its side increases from 6 cm to 6. where y = f (x) = x2.23 cm.55. The area may be expressed as a function of x. hence. The differential dy is
dy = f l(x) dx dy = 1 x .76 cm2 as its side length increases from 6 to 6. Note that the exact increase in area (∆y) is 2. Example 4-24: Use differentials to approximate the value of 3 26. The differential dy is
dy = f l(x) $ dx dy = 2x $ dx
Because x is increasing from 6 to 6.2/3 dx 3 dy = 12/3 dx 3x
Because x is decreasing from 27 to 26.8129 cm2.45. Let x = length of the side of the square.55.
. which rounds to the same answer to the nearest thousandth!
Chapter Checkout Q&A
1.
3
26. Air is being pumped into a spherical balloon such that its volume increases at a rate of 2 in3/sec. 3 .45) 3 (27) 2/3 45 1 = 27 $ .55 .1 60 dy =
which implies that 3 26.9t + 20t + 2 gives the height in meters of an object after it is thrown vertically upward from a point 2 meters above the ground at a velocity of 20 m/sec.. Find the dimensions for the cylinder that require the least amount of material.
x at the x2+ 1
3. 2 4.983 to the nearest thousandth
Note that the calculator value of 3 26.1 60 .100 dy =. How high above the ground will the object reach? 5. A right circular cylinder is to be made with a volume of 100π cubic
inches.983239874. For the function y = x – 5x + 5 on [0. find (a) The maximum and minimum values of the function. (b) All intervals where the function is increasing or decreasing. Find equations for the tangent and normal lines to y =
2 5
point (2.9833
3
26.
. Find the rate of change of its radius when the radius is 6 inches.0167 . 3 .55 will be approximately 1/60 less that 3 27 = 3. 2.Chapter 4: Applications of the Derivative
61
1 $ (.55 .6]. 2. (c) The concavity and any inflection points of the function. ⁄ ). The formula s(t) = –4.55 is 2. hence. 3 2 2.
and H(x) = x3–2 are all antiderivatives of f (x) = 3x2 because F'(x) = G'(x) = H'(x) = f (x) for all x in the domain of f. and C is reffered to as the constant of integration. For example.
Antiderivatives/Indefinite Integrals
A function F(x) is called an antiderivative of a function of f (x) if F'(x) = f (x) for all x in the domain of f. F(x) = x3. where # f ^ x h dx = F ^ x h + C . and H differ only by some constant value and that the derivative of that constant value is always zero. Note that the function F is not unique and that an infinite number of antiderivatives could exist for a given function. The relationship between antiderivatives and definite integrals is discussed later in the chapter with the statement of the Fundamental Theorem of Calculus. if F(x) and G(x) are antiderivatives of f (x) on some interval. G. G(x) = x3 + 5. this means that the graphs of F(x) and G(x) are identical except for their vertical position. The function of f (x) is called the integrand. The expression F(x) + C is called the indefinite integral of
8
.thesecondThese operation may bediscussed in as inverse of one another. The notation used to represent all antiderivatives of a function f (x) is the indefinite integral symbol written ( ). Geometrically. It is clear that these functions F. thought of A long with differentiation. and rules for finding derivatives previous chapters will be useful in establishing corresponding rules for finding antiderivatives. or integration. In other words.Chapter 5 INTEGRATION
Chapter Check-In
❑ ❑ ❑
Understanding and computing basic indefinite integrals Using more advanced techniques of integration Understanding and computing definite integrals
a important operation of calculus is antidifferentiation. then F'(x)= G'(x) and F(x) = G(x) + C for some constant C in the interval.
the inside function of the composition is usually replaced by a single variable (often u).3) dx
Applying formulas (1).66
CliffsQuickReview Calculus
Example 5-5: Evaluate
# (6x
2
+ 5x . you find that
Substitution and change of variables One of the integration techniques that is useful in evaluating indefinite integrals that do not seem to fit the basic formulas is substitution and change of variables. you find that
# (6x
2
x2 x3 + 5x . it must be written in terms of the original variable of integration. substitute with
u= x3+ 1 du = 3x 2 dx 1 du = x 2 dx 3
.3x + C 2
Example 5-6: Evaluate
# x dx 4 . Although this approach may seem like more work initially.
x = 1 arctan 5 + C 5
Using formula (19) with a = 5. Example 5-8: Evaluate
#x
2
(x 3 + 1) 5 dx. you find that Example 5-7: Evaluate
# 25dx x + # 25dx x +
2
2
. + # x dx 4 = ln x + 4 +
+ C.
Because the inside function of the composition is x3 + 1. and (4). (2). (3).3x + C = 2x 3 + 5 x 2 . This technique is often compared to the chain rule for differentiation because they both apply to composite functions.
Using formula (13). it will eventually make the indefinite integral much easier to evaluate.3) dx = 63 + 52 . In this method. Note that for the final answer to make sense. Note that the derivative or a constant multiple of the derivative of the inside function must be a factor of the integrand. The purpose in using the substitution technique is to rewrite the integration problem in terms of the new variable so that one or more of the basic integration formulas can then be applied.
at t = 0. at t = 0. the initial velocity (v0) is
v 0 = v (0) = (. In the discussion of the applications of the derivative.32t + C 1
Now.Chapter 5: Integration
73
Distance. velocity.16 (0) 2 + v 0 (0) + C 2 s0 = C 2
. Using the fact that the velocity is the indefinite integral of the acceleration. the initial distance (s0) is
s 0 = s (0) =. is negative because the velocity is decreasing as the time increases. write v(t) = –32t + v0.32t + v
0
) dt
2 =. In considering the relationship between the derivative and the indefinite integral as inverse operations. In case of a free-falling object.32)( 0) + C 1 v0 = C 1
hence. Velocity. note that the derivative of a distance function represents instantaneous velocity and that the derivative of the velocity function represents instantaneous acceleration at a particular time. the acceleration due to gravity is –32 ft/sec2.32dt
=. you find that
s (t) =
# v (t) dt = # (. note that the indefinite integral of the acceleration function represents the velocity function and that the indefinite integral of the velocity represents the distance function.16t 2 + v 0 t + C 2
Now. Because the distance is the indefinite integral of the velocity.32 $ t2 + v 0 t + C 2
=. because the constant of integration for the velocity in this situation is equal to the initial velocity. you find that
a (t) = s m(t) =. and acceleration.32 v (t) = s l(t) = =
# s m(t) dt
# . and Acceleration
The indefinite integral is commonly applied in problems involving distance. each of which is a function of time. The significance of the negative is that the rate of change of the velocity with respect to time (acceleration).
8. v0 = 0 m/sec because it begins at rest. write s(t) = –16t2 + v0 (t) + s0.32t . and s0 = –35 m because the missile is below ground level.64 ft/ sec s 0 = 512 ft
hence. Example 5-21: A missile is accelerating at a rate of 4t m/sec2 from a position at rest in a silo 35 m below ground level.192 ft/ sec
hence. you find that
v (4) =. the ball will reach the ground 4 seconds after it is thrown. you find that
a (t) =.64t + 512
The distance is zero when the ball reaches the ground or
.74
CliffsQuickReview Calculus
hence.
. Example 5-20: In the previous example. t = 4
hence.16 (t + 8)( t .
v (t) =.16 (t 2 + 4t . what will the velocity of the ball be when it hits the ground? Because v(t) = –32(t) – 64 and it takes 4 seconds for the ball to reach the ground. How long will it take for the ball to reach the ground? From the given conditions. Example 5-19: A ball is thrown downward from a height of 512 feet with a velocity of 64 feet per second.32 (4) . How high above the ground will it be after 6 seconds? From the given conditions.16t 2 .4) = 0 t =. hence.32) = 0 .16t 2 .64 s (t) =. because the constant of integration for the distance in this situation is equal to the initial distance.32 ft/ sec 2 v 0 =.64t + 512 = 0 . the ball will hit the ground with a velocity of –192 ft/sec. you find that a(t) = 4t m/sec2. The significance of the negative velocity is that the rate of change of the distance with respect to time (velocity) is negative because the distance is decreasing as the time increases.64 =.
Definite Integrals
The definite integral of a function is closely related to the antiderivative and indefinite integral of a function. or zero. An arbitrary domain value.35 3 2 (6) 3 .b]. If f (x) < 0 on [a. is chosen in each subinterval. then the Riemann sum will be a negative real number. and you will see that the definite integral will have applications to many problems in calculus.b] is expressed as
S n = f (x 1 ) ∆x + f (x 2 ) ∆x + f (x 3 ) ∆x + $$$ + f (x n ) ∆x or S n = ! f (x i ) ∆x
i=1 n
A Riemann sum may.b].Chapter 5: Integration
75
v (t) =
# 4t dt = 2t # 2t
2
2
dt = 2 t 3 . is determined. if it exists. is a real number value. xi.35m = 109m After 6 seconds. For example. therefore. depending upon the behavior of the function on the closed interval. the missile will be 109 m above the ground after 6 seconds. The product of each function value times the corresponding subinterval length is determined. f (xi).b]. if f (x) > 0 on [a. be thought of as a "sum of n products. Definition of definite integrals The development of the definition of the definite integral begins with a function f (x). The given interval is partitioned into "n" subintervals that. and these "n" products are added to determine their sum.3] using the four subintervals of equal length. then the Riemann sum will be a positive real number.
. where xi is the right endpoint in the ith subinterval (see Figure 5-3). negative. although not necessary. The relationship between these concepts is will be discussed in the section on the Fundamental Theorem of Calculus. you find that s (6) = 3
and
s (t) =
hence. which is continuous on a closed interval [a. can be taken to be of equal lengths (∆x). The Riemann sum of the function f (x) on [a. while the latter two represent an infinite number of functions that differ only by a constant. The primary difference is that the indefinite integral. and its subsequent function value. This sum is referred to as a Riemann sum and may be positive." Example 5-22: Evaluate the Riemann sum for f (x) = x2 on [1.
In other words. The numbers a and b are called the limits of integration with a referred to as the lower limit of integration while b is referred to as the upper limit of integration. is used to define the definite integral of a function on [a.b]. is the same symbol used previously for the indefinite integral of a function. then the definite integral of f (x) on [a.Chapter 5: Integration
77
If the number of subintervals is increased repeatedly. This may be restated as follows: If the number of subintervals increases without bound (n → +∞). Also. As with differentiation. This limit of a Riemann sum. The function f (x) is called the integrand. but the converse is not necessarily true. used with the indefinite integral. Note that the symbol ∫. Unfortunately. the effect would be that the length of each subinterval would get smaller and smaller. the fact that the definite integral of a function exists on a closed interval does not imply that the value of the definite integral is easy to find. The question of the existence of the limit of a Riemann sum is important to consider because it determines whether the definite integral exists for a function on a closed interval. a significant relationship exists between continuity and integration and is summarized as follows: If a function f (x) is continuous on a closed interval [a.
. The reason for this will be made more apparent in the following discussion of the Fundamental Theorem of Calculus. If f (x) is defined on the closed interval [a. then the length of each subinterval approaches zero (∆x →0).b]. if it exists. and the variable x is the variable of integration. keep in mind that the definite integral is a unique real number and does not represent an infinite number of functions that result from the indefinite integral of a function.b] then the definite integral of f (x) from a to b is defined as
# f (x) dx = lim S
b a n "+3
n n
= lim = lim
n "+3
! f (x ) Dx
i i=1 n i
Dx " 0
! f (x ) Dx
i=1
if this limit exits.b] exists and f is said to be integrable on [a. continuity guarantees that the definite integral exists.b].
Those considered in this chapter are areas The definite integral of a function hasrevolution. above the x-axis.b]. volumes of solids of the lengths of arcs of a curve. then the area (A) of the region lying below the graph of f (x).Chapter 6 APPLICATIONS OF THE DEFINITE INTEGRAL
Chapter Check-In
❑ ❑ ❑
Calculating areas with definite integrals Finding volumes with definite integrals Computing arc lengths with definite integrals
applications to many problems in calculus.
Area
The area of a region bounded by a graph of a function. and two vertical boundaries can be determined directly by evaluating a definite integral. and between the lines x = a and x = b is
A=
#
a
b
f ^ x h dx
. andbounded by curves. the x-axis. volumes by slicing. If f (x) > 0 on [a.
the x-axis.b]. below the x-axis.Chapter 6: Applications of the Definite Integral
89
Figure 6-1 Finding the area under a non-negative function. and the lines x = a and x = b would be determined by the following definite integrals:
A= A=
#
a
b
f (x) dx f (x) dx -
#
a
c
# f (x) dx
b c
. then the area (A) of the region lying above the graph of f (x).
y
y= f(x)
a
b
x
If f (x) ≤ 0 on [a. and between the lines x = a and x = b is
A=
# f (x)dx
b a
or A =-
# f (x)dx
b a
Figure 6-2 Finding the area above a negative function. then the area (A) of the region bounded by the graph of f (x).c] and f (x) ≤ 0 on [c.
y
a
b
x
y= f(x)
If f (x) > 0 on [a.b].
it is necessary to determined the position of each graph relative to the graphs of the other functions of the region.
y
y= f(x) a c b
x
Note that in this situation it would be necessary to determine all points where the graph f (x) crosses the x-axis and the sign of f (x) on each corresponding interval.90
CliffsQuickReview Calculus
Figure 6-3 The area bounded by a function whose sign changes. As an example. The points of intersection of the graphs might need to be found in order to identify the limits of integration. if f (x) > g (x) on [a.g (x)B dx
b a
Figure 6-4 The area between two functions.b]. then the area (A) of the region between the graphs of f (x) and g (x) and the lines x = a and x = b is
A=
# 8 f (x) . For some problems that ask for the area of regions bounded by the graphs of two or more functions.
y
y= f(x)
a
b
x
y= g(x)
.
then their areas will be functions of x.
. then their areas will be functions of y.
y
y= x2 8 6 (−2. If the cross sections generated are perpendicular to the x-axis. Because the cross sections are squares perpendicular to the y-axis.b] is
V=
# A (y) dy
b a
Example 6-4: Find the volume of the solid whose base is the region inside the circle x2 + y2 = 9 if cross sections taken perpendicular to the y-axis are squares. the volume (V ) of the solid on [a. provided you know a formula for the region determined by each cross section. denoted by A(y).Chapter 6: Applications of the Definite Integral
93
Figure 6-6 Diagram for Example 6-3.b] is
V=
# A (x) dx
b a
If the cross sections are perpendicular to the y-axis. the area of each cross section should be expressed as a function of y. In this case. The volume (V ) of the solid on the interval [a. The length of the side of the square is determined by two points on the circle x2 + y2 = 9 (Figure 6-7).4) y= 8 − x2
−4
−2
x
2 4
Volumes of Solids with Known Cross Sections
You can use the definite integral to find the volume of a solid with specific cross sections on an interval.4) 4 2 (2. denoted by A(x).
The volume (V ) of a solid generated by revolving the region bounded by y = f (x) and the x-axis on the interval [a. then its radius should be expressed as a function of x.
.b] is revolved about the y-axis. Example 6-6: Find the volume of the solid generated by revolving the region bounded by y = x2 and the x-axis on [2. washers. Disk method If the axis of revolution is the boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution.b] about the x-axis is
V=
#
a
b
r 8 f (x)B dx
2
If the region bounded by x = f (y) and the y-axis on [a.96
CliffsQuickReview Calculus
Volumes of Solids of Revolution
You can also use the definite integral to find the volume of a solid that is obtained by revolving a plane region about a horizontal or vertical line that does not pass through the plane. Because the x-axis is a boundary of the region. then its radius should be expressed as a function of y. then its volume (V ) is
V=
#
a
b
r 8 f (y)B dy
2
Note that f (x) and f (y) represent the radii of the disks or the distance between a point on the curve to the axis of revolution. This type of solid will be made up of one of three types of elements—disks. If a disk is perpendicular to the x-axis. then you use the disk method to find the volume of the solid. or cylindrical shells—each of which requires a different approach in setting up the definite integral to determine its volume. the volume of each disk is its area times its thickness. you can use the disk method (see Figure 6-9). Because the cross section of a disk is a circle with area πr2.3] about the x-axis. If a disk is perpendicular to the y-axis.
y
8 y= x2 6 4 2
f(x)
−4
−2
x
2 4
The volume (V ) of the solid is
V=
#
3
r (x 2 ) 2 dx
3
-2
=r
#
x 4 dx
-2 3 = r . about the x-axis is
. The volume (V ) of a solid generated by revolving the region bounded by y = f (x) and y = g (x) on the interval [a.Chapter 6: Applications of the Definite Integral
97
Figure 6-9 Diagram for Example 6-6. then the area of the washer is πR2 – πr2. and its volume would be its area times its thickness.c -5 m G 5 V = 55r
Washer method If the axis of revolution is not a boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution. Think of the washer as a "disk with a hole in it" or as a "disk with a disk removed from its center.b] where f (x) > g (x). If a washer is perpendicular to the y-axis. you use the washer method to find the volume of the solid. then the radii should be expressed as functions of y. As noted in the discussion of the disk method." If R is the radius of the outer disk and r is the radius of the inner disk. if a washer is perpendicular to the x-axis. 1 x 5E 5
-2
32 = r = 243 . then the inner and outer radii should be expressed as functions of x.
then the radius and height should be expressed in terms of y. about the y-axis is
V=
# 2rx f (x) dx
b a
If the region bounded by x = f (y) and the y-axis on the interval [a.` x 4 + 4x 2 + 4 j D dx
2 4 2 ` . where f (y) > 0.3x + 8x + 12 j dx
-1
=r
#
-1 2 = r . (2πrh). then its volume would be 2πrh times its thickness. Think of the first part of this product. The volume (V ) of a solid generated by revolving the region bounded by y = f (x) and the x-axis on the interval [a. If the cylindrical shell has radius r and height h.Chapter 6: Applications of the Definite Integral
99
Because the x-axis is not a boundary of the region. then the cylindrical shell method will be used to find the volume of the solid. If.b].c . the axis of revolution is horizontal.1 x 5 .b]. you can use the washer method. where f (x) > 0. then the radius and height should be expressed in terms of x.x 3 + 4x 2 + 12x E 5
-1
=r = 128 .` x 2 + 2 j D dx
2
-1 2
r : ` x 2 + 8x + 16 j .x . The f (x) and f (y) factors represent the heights of the cylindrical shells. and the volume (V ) of the solid is
V= =
# #
2
r :^ x + 4 h 2 . then its volume (V ) is
V=
# 2ry f (y) dy
b a
Note that the x and y in the integrands represent the radii of the cylindrical shells or the distance between the cylindrical shell and the axis of revolution. If the axis of revolution is vertical. is revolved about the x-axis. as the area of the rectangle formed by cutting the shell perpendicular to its radius and laying it out flat.34 m G 5 5 V = 162r 5
Cylindrical shell method If the cross sections of the solid are taken parallel to the axis of revolution. however.
. .
e. c. Two nonvertical. d.CQR REVIEW
Use this CQR Review to practice what you've learned in this book. d.
6.
cos(x – y) 1/cot x cos (–x) cos x/sin x csc2x – 1 sin2x + cos2x
.
Chapter 1
1. 4. Any nonvertical lines are parallel if they have _____. b. Which of the following are functions? a. b. you're well on your way to achieving your goal of understanding calculus. e. d. c. Find an equation of the line that has slope 6/5 and crosses the y-axis
at 3. After you work through the review questions. b. nonhorizontal lines are perpendicular if the prod-
uct of their slopes is _____. Complete the trigonometric identity for the following: a. f.
m = rise / run m = (y1 – x1) / (y2 – x2) m = (y1 – y2) / (x1 – x2) m = (y2 – y1) / (x2 – x1) f(x) = 5x + 3 y = cos 5x x=2 f(x) = 6/(x2 + 4) x2 + y2 = 144
2. c.
3. Which of the following can not be used to find the slope of a line? a.
5.
14. Complete the following statements about trigonometric function dif-
ferentiation. In addition to the notation f '(x). f. b. e. The point (x. which the following can be used to
represent the derivative of y = f (x)?
a. d. x " 1 8. then f '(x) = If f (x) = cot x. 12. then f '(x) = If f (x) = sec x. If the derivative of a function is less than zero at each point on an interval I. If the derivative of a function is greater than zero at each point on an
interval I.1 .
11.
Dfx df (x)/dx y' dx/dy
10. Find y' if y = 3 sin x + π.x3 . then f '(x) = If f (x) = tan x. then f '(x) = If f (x) = csc x.CQR Review
105
Chapter 2
7. Evaluate lim
x3. then the function is said to be ______ on I. then the function is said to be _____ on I. f (x)) is called a critical point of f (x) if x is in the ______
of the function.1
1 . c.
If f (x) = sin x.1
Chapter 3
9. then f '(x) = If f (x) = cos x.
a. Find f '(x) if f ^ x h =
Chapter 4
13. then f '(x) =
x 2 + 1. 3 2 x " + 3 3x + x .
.
15. d. c. The _____ is the line that is perpendicular to the tangent line at the
point of tangency. b. and either f '(x) = ______ or ______. Evaluate lim x .
5].
20. b.
. What's wrong with the problem "A rectangular box is to have a vol-
ume of 8 square units.
27.3) and
decreasing on the interval (3. Evaluate
# cos
6
x sin 3 x dx.
18. c. how fast is the depth changing?
22. If
the volume is increasing at a rate of 2 cubic inches per minute. Find the maximum and minimum values of f (x) = x – 3x – 9x + 4
3 2
on the interval [–2. Find the equation of the tangent line to the graph of f (x) = sin(x ) at
2
the point (0. Find the maximum surface area such a box could have.6]. Water is dripping into a cylindrical can with a radius of 3 inches.1/e). If you know that a function is increasing on the interval (0. Find the rate of change of the distance between the two cars when both are 1 mile from the intersection. one heading north at
a rate of 65 mph and the second heading west at a rate of 45 mph.
19.b] is the ______ of any ______ evaluated at the upper limit of integration minus the same antiderivative evaluated at the lower limit of integration.
23.0). x2+ 1
21."
Chapter 5
25.106
CliffsQuickReview Calculus
16. does this imply that the function has a local maximum when x = 3? What sorts of situations are possible?
24. d. Find the equation of the tangent line to the graph of f (x) = x e at
the point (1. What is the acceleration of a free-falling object due to gravity? 26. According to the Fundamental Theorem of Calculus: The value of
the definite integral of a function on [a. Find the maximum value of y =
x on the interval [0. Two cars are traveling toward an intersection. You can not use the Second Derivative Test for Local Extrema in
which of the following situations?
a.
f '(x) = 0 and f"(x) = 0 f '(x) = 0 and f"(x) does not exist f '(x) = –(f"(x)) f '(x) does not exist
2 –x
17.6).
20] about the x-axis.
37. 36. 4 .1 x2
1 -1
=
33. How long will it take for the ball to hit the ground?
32. x dx.3]. Find the volume of the solid generated by revolving the region
bounded by y = 1/x and the x-axis on [1. If a car takes s seconds to accelerate from 0 to 60 (and supposing constant acceleration). Find the area of the region bounded between y = x and y =
2 3
x.1 m
#
1
-1
1 dx =.CQR Review
107
28.. Is this the same as
# f ^ x h dx ?
b a
.
40. Evaluate 29.
38. What's
wrong with the computation 1 . does it matter how far off the roadway the officer sits? (Hint: Think about it as a related rates problem with one of the rates of change being zero.x2
#
2 1
# x ln x dx.1x . 39. The area bounded by the function f (x) and the x-axis between x = a
and x = b is given by (See Chapter 6. Evaluate 30. π] about the y-axis.)
#
a
b
f ^ x h dx .
31.)
34. Find the arc length of the line y = 2x on the interval [0. A rock is thrown upward from a 200 foot cliff with an initial veloc-
ity of 30 feet per second. Find the area of the region bounded by y = x and y = x . If a highway patrol officer is sitting off to the side of a road moni-
toring the speed of approaching traffic.1 m c . over what distance will it travel in the process?
Chapter 6
35. Fast cars are often rated for how quickly they can accelerate from 0
to 60 miles per hour (which is equivalent to 88 feet per second).1 = -2? c. Evaluate
#
0
1
xe x dx. Find the volume of the solid generated by revolving the region
bounded by y = sin x and the x-axis on [0.
polynomials. area.CQR RESOURCE CENTER
CQR Resource Center offers the best resources available in print and online to help you study and review the core concepts of calculus. and other material pre-requisite for calculus. and the basics of coordinate geometry including plotting points. Inc.. Hungry Minds. polynomials.. inequalities.. plus study tips and tools to help test your knowledge. exponential and logarithmic functions. by Jerry Bobrow. sequences and series. by Jerry Bobrow. Kay. vectors. exponents. and inverse functions. decimals. Inc. distances.. graphing. If you need to brush up more of the pre-requisites.. 2001.cliffsnotes. factoring. volume.
Books
This CliffsQuickReview book is just what it's called. CliffsQuickReview Trigonometry. equations. Hungry Minds. CliffsQuickReview Geometry. You can find additional resources. factoring. the Pythagorean theorem. 2001. gives you a review of triangles. CliffsQuickReview Algebra II. 2001. by David A. roots. conic sections. complex numbers. Inc. midpoints. CliffsQuickReview Algebra I. and functions. 30°-60°-90° and 45°-45°-90° triangles. gives you a review of topics including the basics of working with fractions. complex numbers.
. Hungry Minds. or if you want a fuller discussion or other practical advice. Hungry Minds. slopes and equations of lines. gives you a review of topics including solving systems of equations. and an introduction to algebraic expressions and solving equations. gives you a review of topics including sets. a quick review of calculus. gives you a review of topics including perimeter.com. check out these other publications: CliffsQuickReview Basic Math and Pre-Algebra. by Edward Kohn and David Herzog. at www. 2001. powers. trigonometric functions and identities. Inc. Hungry Minds. 2001. by Edward Kohn. Inc. polar coordinates.
by Joel Hass. Cliffs Math Review for Standardized Tests. 2001. can give you all the extra practice problems you want. 1996. The book reviews crucial calculus topics. Hungry Minds. Each topicspecific review section includes a diagnostic test. CliffsAP Calculus AB and BC Preparation Guide. gives a lot of practical tips not just on the subject matter itself. helps you to review. by Elliot Mendelson. 1986. and a section devoted to key strategies. Finney.
. The Story of Mathematics. rather than catering primarily to the arcane tastes of math professors. and prepare for standardized math tests. gives a very accessible account of the development of mathematics. Calculus and Analytic Geometry. 3000 Solved Problems in Calculus. Abigail Thompson. Inc. by Jerry Bobrow. 2001. 1997. by David Berlinski. refresh.. 1992.110
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CliffsQuickReview Linear Algebra. is an in-depth look at algebraic equations and inequalities. practice. Hungry Minds. Hungry Minds. from the earliest archeological evidence on. including calculus.. but also on picking teachers and preparing for tests.. and Colin Conrad Adams. Inc. this is the best place to go. gives a complete exploration of what many of the theorems of calculus really mean and a look at how the discipline of calculus is one of the human intellect's most impressive accomplishments. by Kerry King. McGraw-Hill. Inc. W H Freeman & Co. is the most understandable standard calculus textbook available. 1998. Vintage Books.. Leduc. a review test.. and includes sample questions and tests. How to Ace Calculus: The Streetwise Guide. introduces test-taking strategies. A Tour of the Calculus. rules and key concepts. gives you tips and suggestions for getting the most credit you can on the Advanced Placement Calculus AB and BC tests. by Steven A. practice problems. Princeton University Press. glossary. by Richard Mankiewicz and Ian Stewart. by George Brinton Thomas and Ross L. 1985. AddisonWesley Publishing Co. If you want a complete treatment of calculus that's meant more for students to learn from. and analysis for the most common types of standardized questions.
com/~hahn/calc.swarthmore. but it does offer easy-to-use online programs that do the heavy lifting for you—everything from graphing functions and equations to computing limits.com www. integration.utk. derivatives. edu/%7epscrooke/toolkit.math—is an awardwinning site that offers a free question-and-answer service.CQR Resource Center
111
Hungry Minds also has three Web sites that you can visit to read about all the books we publish:
s s s
www. Mathematics—
Internet
Visit the following Web sites for more information about calculus: Ask Dr. Math—forum.html—is a complete calculus help site with entertaining and understandable explanations of most topics.S.O.com www. as well as archives of past questions and answers.math.sosmath.cliffsnotes.shtml—is not the most graphically exciting Web site out there.
. and recommended books.hungryminds.edu/dr. including some animated graphics to demonstrate specific calculus ideas and some sample exams (with solutions).com—is a nice site with a broad range of helpful pages covering algebra through calculus and beyond.edu/visual .dummies.net—is an organized clearinghouse of links to a ton of other pages about math topics. Karl's Calculus Tutor— free help with math problems. S. good links.netsrq.
/archives.vanderbilt.
The MathServ Calculus Toolkit— and sequences and series. calculus@internet—www. limits.calculus/— Visual Calculus —http:/ is an award-winning Web site from the University of Tennessee that offers a wide variety of step-by-step illustrated tutorials on calculus topics including pre-calculus. continuity.
st-and. We created an online Resource Center that you can use today. Help With Calculus For Idiots (Like Me)—ccwf.calculus.html—gives a good yet very brief survey of the origins of many of the major parts of modern calculus.thinkquest.us/www/alvirne.utexas.112
CliffsQuickReview Calculus
The BHS Calculus Project— and AIDS.com—actually computes integrals for you in the blink of an eye. Mathematica Animations—
. and beyond.html—outlines major calculus topics as well as issues in other branches of mathematics.ac.wolfram. nh. Student research and reporting shows how calculus impacts everyday topics such as fractals. AP Calculus Problem of the Week— the second derivative function.uk/ history/HistTopics/The_rise_of_calculus.k12. ice cones. The Integrator—integrals.edu/ ~egumtow/calculus—is another page with explanations of several calculus topics that gives practical advice about what you'll really need to know to get through a calculus class. and Reimann sums.org/20991/calc/index. bicycles. tomorrow.com.seresc. the volume of cones. Next time you're on the Internet. don't forget to drop by short QuickTime movies that illustrate key calculus concepts such as the definition of a derivative. tape decks.htm— serves as an archive of student projects that show calculus's connection to the real world.org/Contributions/ animations.mcs. A History of the Calculus—www-history.bhs-ms.html—offers a different calculus based problem every week.cc. Visitors can also submit their own calculus challenges for future inclusion on the Web site. A fairly active pre-calculus and calculus message board enables visitors to ask and answer thought-provoking questions. cliffsnotes.org/calculus. Math for Morons Like Us: Pre-Calculus & Calculus—http:// library.
The mathematical definition of the derivative is f ^ x + Dx h . done to the inside function.
. concave upward A function is concave upward on an interval if f "(x) is positive for every point on that interval.Glossary
antiderivative A function F (x) is called an antiderivative of a function f (x) if F'(x) = f (x) for all x in the domain of f. concave downward A function is concave downward on an interval if f "(x) is negative for every point on that interval. or in Dx " 0 Dx words the limit of the slopes of the secant lines through the point (x. The derivative is most often d denoted f '(x) or dx . critical point A critical point of a function is a point (x. continuous A function f (x) is continuous at a point x = c when f (c) exists. b denoted # f ^ x h dx. this means that an antiderivative of f is a function which has f for its derivative. To say that a function is continuous on some interval means that it is continuous at each point in that interval. definite integral The definite integral of f (x) between x = a and x = b. lim x " c f ^ x h exists. this means the curve could be drawn without lifting the pencil. and lim x " c f ^ x h = f (c). times the derivative of the inside function. The derivative can be interpreted as the slope of a line tangent to the function. the instantaneous velocity of the function. cylindrical shell method A procedure for finding the volume of a solid of revolution by treating it as a collection of nested thin rings. the d chain rule says dx b f ` g ^ x hj l = f l ` g ^ x hj $ g l ^ x h. with area above the x-axis counting positive and area below the x-axis counting negative. In words. derivative The derivative of a function f (x) is a function that gives the slope of f (x) at each value of x. In words. f (x)) and a second point on the graph of f (x) as that second point approaches the first. the chain rule says the derivative of a composite function is the derivative of the outside function.f ^ x h lim . In words. change of variables A term sometimes used for the technique of integration by substitution. In symbols. f (x)) with x in the domain of the function and either f '(x) = 0 or f '(x) undefined. gives the signed a area between f (x) and the x-axis from x = a to x = b. or the instantaneous rate of change of the function. Critical points are among the candidates to be maximum or minimum values of a function. chain rule The chain rule tells how to find the derivative of composite functions.
. b]. The indefinite integral of f (x) is represented in symbols as # f ^ x h dx . where a and b are not both zero. limit A function f (x) has the value L for its limit as x approaches c if as the value of x gets closer and closer to c. instantaneous velocity One way of interpreting the derivative of a function s(t) is to understand it as the velocity at a given moment t of an object whose position is given by the function s(t). then that point is a local maximum.b).
implicit differentiation A procedure for finding the derivative of a function which has not been given explicitly in the form "f (x) =". Extreme Value Theorem A theorem stating that a function which is continuous on a closed interval [a. and so forth for some function. A function will fail to be differentiable at places where the function is not continuous or where the function has corners. disk method A procedure for finding the volume of a solid of revolution by treating it as a collection of thin slices with circular cross sections. the limit of the average rates of change between a fixed point and other points on the curve that get closer and closer to the fixed point. intercept form The intercept form for the equation of a line is x/a + y/b = 1. higher order derivatives The second derivative. integration by parts One of the most common techniques of integration. third derivative. If a continuous function changes from increasing (first derivative positive) to decreasing (first derivative negative) at a point. general antiderivative If F(x) is an antiderivative of a function f (x). instantaneous rate of change One way of interpreting the derivative of a function is to understand it as the instantaneous rate of change of that function.0) and its y-intercept (the place where the line crosses the y-axis) at the point (0. general form The general form (sometimes also called standard form) for the equation of a line is ax + by = c. b] must have a maximum and a minimum value on [a. then F(x) + C is called the general antiderivative of f (x). If a function changes from decreasing (first derivative negative) to increasing (first derivative positive) at a point. then that point is a local minimum. used to reduce complicated integrals into one of the basic integration forms. where the line has its x-intercept (the place where the line crosses the x-axis) at the point (a.114
CliffsQuickReview Calculus
differentiable A function is said to be differentiable at a point when the function's derivative exists at that point. the value of f (x) gets closer and closer to L. indefinite integral The indefinite integral of f (x) is another term for the general antiderivative of f (x). First Derivative Test for Local Extrema A method used to determine whether a critical point of a function is a local maximum or local minimum.
Glossary
115
Mean Value Theorem If a function f (x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists some c in the interval [a,b] for which f ^ bh - f ^ ah f l ^ ch = . b-a normal line The normal line to a curve at a point is the line perpendicular to the tangent line at that point. point of inflection A point is called a point of inflection of a function if the function changes from concave upward to concave downward, or vice versa, at that point. point-slope form The point-slope form for the equation of a line is y – y1 = m(x – x1), where m stands for the slope of the line and (x1,y1) is a point on the line. Riemann sum A Riemann sum is a sum of several terms, each of the form f (xi)∆x, each representing the area below a function f (x) on some interval if f (x) is positive or the negative of that area if f (x) is negative. The definite integral is mathematically defined to be the limit of such a Riemann sum as the number of terms approaches infinity. Second Derivative Test for Local Extrema A method used to determine whether a critical point of a function is a local maximum or local minimum. If f '(x) = 0 and the second derivative is positive at this point, then the point is a local minimum. If f '(x) = 0 and the second derivative is negative at this point, then the point is a local maximum.
slope of the tangent line One way of interpreting the derivative of a function is to understand it as the slope of a line tangent to the function. slope-intercept form The slopeintercept form for the equation of a line is y = mx + b, where m stands for the slope of the line and the line has its y-intercept (the place where the line crosses the y-axis) at the point (0,b). standard form The standard form (sometimes also called general form) for the equation of a line is ax + by = c, where a and b are not both zero. substitution Integration by substitution is one of the most common techniques of integration, used to reduce complicated integrals into one of the basic integration forms. tangent line The tangent line to a function is a straight line that just touches the function at a particular point and has the same slope as the function at that point. trigonometric substitution A technique of integration where a substitution involving a trigonometric function is used to integrate a function involving a radical. washer method A procedure for finding the volume of a solid of revolution by treating it as a collection of thin slices with cross sections shaped like washers.
Appendix USING GRAPHING CALCULATORS IN CALCULUS
important area that hasn't been the rest of this book is While it's learn and Onethe use of modern technology.ofaddressed inpossible toand pencil, understand calculus without the use tools beyond paper there are many ways that modern technology makes tasks easier or more accurate, and there are also ways that it can give insights that aren't as clear otherwise. Of course, this appendix can't be exhaustive, but it will return to several of the examples from earlier in the book and show how you could apply graphing calculators to them. Because the variety of different calculators available is tremendous, everything here will be done in general terms that should apply to any graphing calculator. For specific details about how to handle your own calculator, you should look at its manual, but this appendix can give you ideas about how that applies to calculus. To keep things general and easy, this appendix usually just gives the calculator's decimal answers to four places, and anything you need to type into your calculator appears in bold, sticking as close as possible to the way things will appear on your calculator keyboard and screen.
Limits
Graphing calculators are ideal tools for evaluating limits. The more sophisticated models have this as a built-in function (consult your manual's index under "limits"), but on any calculator you can at least estimate most limits by looking closely at a graph of the function.
-9 Example 2-3 Revisited: Evaluate lim x + 3 . x "-3 x Graphing the function y=(x^2–9)/(x+3) on a calculator, you can visually estimate that for values of x near –3, the values of y on the graph are
2
Appendix: Using Graphing Calculators in Calculus
117
around –6. Most calculators won't even show the hole in the graph at this point without special effort on your part, since they plot individual points using decimal values that probably don't include exactly –3. If you have trouble judging the y value visually, you can also use the zoom or trace functions on most graphing calculators to get a more accurate estimate. For instance, tracing this graph to an x value near –3, you find that when x = –3.0159, you have y = –6.0159, and from that it's not hard to guess that the limit is around –6.
x+3 Example 2-11 Revisited: Evaluate lim x - 2 . x " 2 Most graphing calculators do a poor job of rendering graphs near vertical asymptotes, but if you know what you're looking for, you can easily get the information you need. In this case, when you graph y=(x+3)/(x–2), the screen should show the curve plunging downward as x approaches 2 from the left and veering upward as x approaches 2 from the right, possibly with a misleading vertical line where the calculator naively tries to connect the two parts. That downward spike as you near –2 from the left is your sign that the limit is –∞.
-
Example 2-14 Revisited: Evaluate lim
x3- 2 . 3 x " + 3 5x - 3x + 2x
4
Graphing the function y=(x^3–2)/(5x^4–3x^3+2x), you look out to the right-hand end of the screen to see what the height of the graph is for the larger values of x. Don't be fooled into thinking there's nothing there, it's just that the y value of the graph is so close to zero that it appears to overlap with the x-axis. If you trace the graph, you can find that when x = 10, you have y = 0.0212, so the limit seems to be 0.
Derivatives
The more sophisticated calculators available today can evaluate derivatives symbolically, giving the same exact values or functions that you can find by hand. Many calculators also have built-in features to numerically compute the value of the derivative of a function at a point. You can consult your calculator's manual for this. You can use any graphing calculator to get at least an approximate value for the derivative of a function at a point, and understanding how this works helps you understand what a derivative really is. Example 3-17 Revisited: Find f '(2) if f ^ x h = 5x 2 + 3x - 1 .
The previous example could be seen that way. Another way to use a graphing calculator is to check answers you get by hand.0.7807) and a point just to the right of x = 2 (like x = 2. after all. if you zoom in towards the point (–1. So.5x+1. you can graph both y1=√(x^2+3) and y2=–. The more work done by hand the more likely most people are to slip. is that it should touch the function and have the same slope. Rearranging this to slope-intercept form. If your graph hadn't turned out like this—if the tangent line hadn't touched the function at the right point.2). you found that x + 2y = 3 is the equation of the tangent line. especially in a longer problem (like in the following example). so near the point of tangency it should be almost impossible to tell the two apart.5. you found the slope of a secant line that's pretty close to the actual tangent line.9048. When this problem was worked in Chapter 4.5 together. y = 4.3.0635. the more the two graphs should appear identical.9048 .1459 1.y 2 m = x1 .1459). the closer you look. In fact. verifying your work can be worthwhile. by just keeping things to two decimal places and still gotten about the right answer. because when you worked it out by hand you got 23/10 and by calculator you got about 2.2). Now use the traditional slope formula to find the slope of the line connecting these two points:
y1 . The whole idea of a tangent line.118
CliffsQuickReview Calculus
Graph the function y=√(5x^2+3x–1) and use the trace feature to find the coordinates of a point just to the left of x = 2 (like x = 1. By picking two x values with the change between them small.x 2 m = 4.2).2.1587 m = 2. y = 5. You could also have been a little bit less careful.3. which is the same value found by hand in Chapter 3.7807 .
. Example 4-1 Revisited: Find the equation of the tangent line to the graph of f ^ x h = x 2 + 3 at the point (–1. and the two graphs should appear to overlap near the point (–1. The reason this works is because the derivative is just the limit that the slopes of the secant lines approach as the change in x goes to zero.3652 m = .3012
So from this. or if they didn't appear to have matching slopes there—you'd know something had gone wrong in your computations and could go back to check them over.0635 . and quicker.0. you can guess that the derivative is about 2.
so the minimum value for your interval appears to happen at x = 2.) It should be easy to see that the function increases for a short interval until the x value reaches π/4. You can now plug x = –2 and x = 2 back into the function to find that the maximum value is 39 and the minimum value is –9. but even without such a feature. If you use the calculator's trace feature. then decreases until 5π/4. quickly doing any of the problems you could work out by hand. However. The lowest point seems to be near x = 2. (Most calculators have a feature that adjusts the viewing window to settings suitable for trig functions—also make sure your calculator is in radian mode rather than degrees.2]). determine all intervals where f is increasing or decreasing. To make it easy on yourself. Many graphing calculators have built-in features for finding maximum or minimum values of functions. as in the following example. you find out that the graph continues to decrease beyond x = 2. and then increases the rest of the way to 2π. have the x-axis tick marks every π/2 units.
. Graph y=sin x + cos x and make sure your viewing window includes x values from 0 to 2π. you can see that the graph is highest at x = –2 (it also gets high on the right. but it's not immediately clear if it happens right at x = 2 itself. graphing calculators make most extreme value problems easy. 2π]. just as you found in Chapter 4—but this time with much less work!
Integrals
Some of the more sophisticated graphing calculators available today can evaluate both definite and indefinite integrals symbolically. If you graph y=x^4–3x^3–1 and make sure the viewing window includes x values from –2 to 2.Appendix: Using Graphing Calculators in Calculus
119
Example 4-6. most graphing calculators don't have this capability and therefore aren't much help with indefinite integrals. so check your manual for details. Revisited: Find the maximum and minimum values of f (x) = x4 – 3x3 – 1 on [–2. Most do have a built-in feature which numerically computes definite integrals. but that's beyond your domain of [–2. Example 4-9 Revisited: For f (x) = sin x + cos x on [0.2]. One final case where graphing calculators and work done by hand can complement each other is finding areas bounded by curves.
Graphing y=x^3+x^2–6x makes it clear that you need to integrate from x = –3 to x = 0.120
CliffsQuickReview Calculus
Example 6-2 Revisited: Find the area of the region bounded by y = x3 + x2 –6x and the x-axis. but determining the limits of integration in the first place can be a nuisance. and then use the negative of the integral from x = 0 to x = 2 where the graph of the function lies below the x-axis.
. The actual integration involved in this problem is straightforward. | 677.169 | 1 |
About this item
Comments: Brand new. We distribute directly for the publisher. Linear algebra is the study of vector spaces and the linear maps between them. It underlies much of modern mathematics and is widely used in applications.A (Terse) Introduction to Linear Algebra is a concise presentation of the core material of the subject--those elements of linear algebra that every mathematician, and everyone who uses mathematics, should know. It goes from the notion of a finite-dimensional vector space to the canonical forms of linear | 677.169 | 1 |
Product Description
The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, students will learn how to derive an algebraic equation using the method of finite differences and how to derive the input/output table of numerical values. Grades 5-9. 30 minutes on DVD.
DVD Playable in Bermuda, Canada, United States and U.S. territories. Please check if your equipment can play DVDs coded for this region. Learn more about DVDs and Videos | 677.169 | 1 |
raphing Calculator by Mathlab
If you're looking for a graphing scientific calculator app that works smoothly and seamlessly, you've found it! Graphing Calculator by Mathlab is a graphing scientific calculator integrated with algebra and is an indispensable mathematical tool for students in elementary school to those in college or graduate school, or just anyone who need…
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2007 Assessment Report 2007 GENERAL COMMENTS
There were 1530 students who sat the Mathematical Methods (CAS) Examination 2 in 2007. Marks ranged from 2 to 78 out of a maximum possible score of 80. Student responses showed that the paper was accessible and that it provided an opportunity for students to demonstrate what they knew. There is some evidence to suggest that, as with the Mathematical Methods cohort, students found the 2007 paper more difficult than the 2006 paper; however, the cohort was much larger, therefore possibly allowing for a larger spread in ability. Of the whole cohort, 13% of students scored 78% or more of the available marks, and 44% scored 54% or more of the available marks. The mean score for the paper was 39.5, with a mean of 14 out of 22 for the multiple-choice section and a mean of 25.8 out of 58 for the extended answer section. The median score for the paper was 41 marks. Only five of the multiple-choice questions were different to those on the Mathematical Methods paper: Questions 3, 5, 10, 17 and 20. Except for Question 3, the other questions were answered correctly by less than 53% of students. The Mathematical Methods (CAS) students performed as least as well as, and generally better than, the Mathematical Methods students on all 17 common multiple-choice questions. In the extended answer section, students sometimes did not give answers in exact form. If a numerical approximation is not asked for in the question, an exact answer must be given. Some students lost marks because they did not give all the answers required in the question. This emphasises the importance of rereading questions before starting the next question. This occurred in Questions 2c., 2g., 3c. and 5fi. Students should take care when drawing graphs – they need to use appropriate scales, make them clearly visible, use the correct domain and use a ruler for linear graphs. It was pleasing to see that students showed their working for questions worth more than one mark. In some cases the formulation of the required computation is the appropriate working; specific examples are given under Section 2 below. An important exception to this is for 'show that' questions, where detailed working must be given. When students present working and develop their solutions, they are expected to use conventional mathematical expressions, symbols, notation and terminology. This was generally well done. However, students must be careful not to write down a CAS output as their final answer if it is not in simplest form.
Marks 0 35 %
1 65
Average 0. Marks 0 1 2 Average 35 38 26 % 0. such as translating the graph of g 3 units to the left or using y = A ( x + 3)( x + 1)( x − 1) . the domain does not have to be given.8
Maths Methods (CAS) GA 3 Exam
Published: 4 April 2008
7
.
Question 3dii. The most common error occurred when students tried to discuss reflections and translations but failed to indicate which graph they were transforming. If only the rule for the inverse function is asked for. to obtain the answer. as they gave instructions to differentiate. Some students were unable to obtain full marks because they rounded up 1 − e −2 to 0. Marks 0 27 %
−2 range = ⎡ ⎣0.2007 Assessment Report
A reflection in the y-axis followed by a translation 1 unit to the right or A translation of 1 unit to the left followed by a reflection in the y-axis Some students did not seem to know the meaning of the word 'transformation'.8647. Students could have used their transformations from Question 3di.1 − e ⎤ ⎦ → R.4
or
y = 2 ( x + 3)( x + 1)( x − 1)
This was a more challenging question and it was not attempted by many students. the domain must be given. where h ( x ) = − log e (1 − x )
When the inverse function is asked for. many of them were unable to explain why b = – 1. where A is a real constant.9 a = 1. (from y = 1) and b = – 1 (goes through origin)
Most students were able to explain why a = 1 by discussing the asymptote.
Question 4b.1 − e ⎤ ⎦
1 12
2 60
Average 1. Others used round brackets instead of square brackets. Marks 0 1 77 10 % y = −2 ( x + 3)( x + 1)( x − 1)
2 13
Average 0.
Question 4cii. Students will be penalised in the future if the domain is left out. however.4
This question was generally well done.7
−2 −1 h −1 : ⎡ ⎣0. Other methods were also acceptable. and substituting in an appropriate point. Others did not consider the domains of h and f. Some students gave the incorrect answer h −1 ( x ) = − log e 1 − x .
Question 4ci.
Question 4a. Marks 0 35 %
1 50
2 15
Average 0.
Question 5d. Marks %
1 11 1 Pr (T ≤ 15 ) ∩ Pr (T ≤ 25 ) 8 1 = = 7 7 Pr (T ≤ 25 ) 8
0 46
2 43
Average 1.3
1 7 ( 30 − t ) dt = ∫ 100 25 8
This question was generally well done. Marks 0 1 2 42 22 37 % ⎛ 7⎞ X ∼ Bi ⎜ 6. Some students did not show the lines along the t-axis. Pr ( X > 4 ) ≈ 0. Marks 0 32 % 1 46 2 22 Average 1. given that it was a conditional probability question.
Question 5c.2007 Assessment Report
Question 5a.0
More students could have accessed method marks for these types of questions if they gave the appropriate distribution with its parameters. 1) were given as the coordinates of the maximum. not (20. as calculator syntax is not acceptable.
Question 5b.
Maths Methods (CAS) GA 3 Exam
Published: 4 April 2008
9
.0
Students need to take care when sketching graphs.. which were more demanding questions. The two oblique lines needed to be drawn to scale along the t-axis and they had to be straight. More students could have accessed method marks for these types of questions if they shaded the appropriate area on a graph. while others only showed the left hand one. Marks 0 35 %
Pr ( f ( t ) < 25 ) = 1 −
30
1 12
2 53
Average 1.1).9709 ⎝ 8⎠
Average 1. Of those who attempted the question. it was pleasing to see that most students used correct mathematical notation. ⎟ . Many students did not attempt Questions 5e.0
This question was generally done. 0. A common error was that (20. and 5f. | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Details about The Queen of Mathematics:
This book takes the unique approach of examining number theory as it emerged in the 17th through 19th centuries. It leads to an understanding of today's research problems on the basis of their historical development. This book is a contribution to cultural history and brings a difficult subject within the reach of the serious reader.
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Rent The Queen of Mathematics 1st edition today, or search our site for other textbooks by Jay Goldman. Every textbook comes with a 21-day "Any Reason" guarantee. Published by A K Peters/CRC Press.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our tutors now. | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Do you remember being hopelessly confused in calculus class? Afterwards, you asked your brainy friend over a cup of coffee. "What was going on in that class?" Your friend then explained it all to you in five minutes flat, making it crystal clear. "Oh", you said, "is that all there is to it?" Later, you wished that friend was around to explain all the lectures to you.
How to Ace Calculus will play the role of that friend. Written by three gifted teachers, it provides brief and highly readable explanations of the key topics of calculus without the technical details and fine print that would be found in a formal text. Capturing the tone of students exchaging ideas among themselves, this unique guide also explains how calculus is taught, how to get the best teachers, what to study, and what is likely to be on exams—all the tricks of the trade that will make learning the material of first-year calculus a piece of cake
Funny, irreverent, and flexible enough to use with any traditional or reform-based calculus text. How to Ace Calculus shows why learning calculus can be not only a mind-expanding experience but also fantastic fun.
Synopsis
Written by three giftedand funnySynopsis
Written by three gifted teachers, this book provides brief and highly readable explanations of the key topics of calculus without the technical details and fine print found in a formal text.
Synopsis
Written by three gifted—and funny——About the Author
Colin Adams is Professor of Mathematics at Williams College. He is the author of The Knot Book and winner of the Mathematical Association of America Distinguished Teaching Award for 1998. Joel Hass is Professor of Mathematics at the University of California at Davis, and Abigail Thompson is also Professor of Mathematics at the University of California at Davis. Both have held fellowships from the Sloan Foundation and the National Science Foundation. | 677.169 | 1 |
Math Help Centre
Are you confident working with fractions? Do you need to review basic algebra? Many Red River College programs require that you have a solid base of math skills. Although this can be intimidating, know there are resources at RRC to help you improve your skills. Look below for links to websites that can help you review your basic skills, or if want to access RRC's online video tutorials.
If you don't see what you need online, remember to visit RRC's Academic Success Centre for help with the specific area of math you need to review. There you can get paired with a peer tutor, or register for one of the many workshops held throughout the year.
Wise Guys
Studying math at midnight? Can't remember how to calculate that formula?
The Academic Success Centre has developed on-line math videos that review common math functions: finding common denominators, direct proportion, scientific notation, working with algebraic expressions, working with calculator in Business Admin, and much more. We continue to add new videos throughout the year, so come back every few months to see what's new. | 677.169 | 1 |
Elementary Algebra is a textbook that covers the traditional topics studied in a modern elementary algebra course. It is...
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Elementary Algebra is a textbook that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques to your Bookmark Collection or Course ePortfolio
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This is 'an introduction to reasoning with propositional and first-order logic, with applications to computer science. It is...
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This is 'an introduction to reasoning with propositional and first-order logic, with applications to computer science. It is Part of the TeachLogic Project ( include:IntroductionPropositional LogicRelations and ModelsFirst-Order LogicConclusion, AcknowledgementsAppendices and Reference Sheets Logic to your Bookmark Collection or Course ePortfolio
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Abstract: The book, Introductory Statistics: Concepts, Models, and Applications, presented in the following pages represents...
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Abstract: The book, Introductory Statistics: Concepts, Models, and Applications, presented in the following pages represents over twenty years of experience in teaching the material contained therein. The high price of textbooks and a desire to customize course material for my own needs caused me to write this material. This Web text and associated exercises is a continuing project. Check back often for updatesctory Statistics: Concepts, Models, and Applications to your Bookmark Collection or Course ePortfolio
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This is a WWW textbook written by Evans M. Harrell II and James V. Herod, both of Georgia Tech. It is suitable for a first...
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This is a WWW textbook written by Evans M. Harrell II and James V. Herod, both of Georgia Tech. It is suitable for a first course on partial differential equations, Fourier series and special functions, and integral equations. Students are expected to have completed two years of calculus and an introduction to ordinary differential equations and vector spaces. For recommended 10-week and 15-week syllabuses, read the preface Methods of Applied Mathematics to your Bookmark Collection or Course ePortfolio
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This is an online textbook in Adobe PDF format. It begins with an introduction to Euclidian Three-Space and vector algebra...
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This is an online textbook in Adobe PDF format. It begins with an introduction to Euclidian Three-Space and vector algebra and covers traditional calculus topics from derivatives up through Stoke's Theorem. Many nice exercisesivariable Calculus to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material Multivariable Calculus Our Ocean Planet: Environmental Science in the 21st Century to your Bookmark Collection or Course ePortfolio
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There can be no more potent demonstration of the trust that Americans place in the rule of law and their confidence in the...
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There can be no more potent demonstration of the trust that Americans place in the rule of law and their confidence in the U.S. legal system.The pages that follow survey that system. Much of the discussion explains how U.S. courts are organized and how they Outline of the US Legal System to your Bookmark Collection or Course ePortfolio
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This is an Internet-based E-Book for advanced-placement (AP) statistics educational curriculum. The E-Book is initially...
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ThisThere are 4 novel features of this specific Statistics EBook – it is community-built, completely open-access (in terms of use and contributions), blends concepts with technology and is multi-lingual (via a translation by page automation Probability and Statistics eBook to your Bookmark Collection or Course ePortfolio
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This course is composed of a range of different free, online materials. These materials include audio and video lectures, as...
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This course is composed of a range of different free, online materials. These materials include audio and video lectures, as well as more traditional textbook-type materials. A sizeable portion of the video and audio lectures is found in the iTunes University academic library: A Student Friendly Approach to your Bookmark Collection or Course ePortfolio
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'Geography touches every aspect of our lives. At its simplest, geography is concerned with where something is at, why it's...
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'Geography touches every aspect of our lives. At its simplest, geography is concerned with where something is at, why it's there, and how it relates to things around it. Geography influences where we live, affects our economic prosperity, has dictated the outcome of significant historical events, and shapes our local, regional, and global relationships with each other. This textbook focuses on physical geography and can answer many questions that you might have about the natural world and how it relates to your life. Ever wondered why present-day volcanoes occur on the west coast of the United States but not on the east coast? Why tornadoes seem to be unique to the United States? Why palm trees are found in Florida and pine trees in Maine? Where to go for a break from cold winter weather, or to escape a hot summer day? Physical geography can answer these questions, and more.''Like many educators, I have a hard time finding a textbook that is to my liking. Publishers have been rather slow to embrace online textbooks. Most textbooks these days include a CD-ROM and companion web site. Having three distinct modes of information delivery makes it difficult to effectively integrate the material. A student must jump from their textbook to a CD-ROM and then to the web. The Physical Environment seamlessly integrates all learning resources together. Those digital textbooks that are available tend to be print versions simply converted to digital. Most lack the interactivity and utilization of the rich resources available online Physical Environment: Intro to Physical Geography to your Bookmark Collection or Course ePortfolio
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Algebra 1 Subject Kit (3rd ed.)
Help develop your teenager's thinking skills with the Algebra 1 educational materials from BJU Press. Algebra 1 provides the thinking skills and experience required for further education and future careers. The educational materials will encourage your homeschooler to think biblically about the applications of algebra. It presents algebra as an important tool that your child can use in exercising dominion over the earth as God commanded. The educational materials also aims to teach for understanding since it is important that your teen understands the concepts being presented.
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Rated 5 out of 5Â by Kiki21 Excellent math program
This book is easy to use and teach. Excellent examples and good structure. Love the review questions at the end of each lesson. I was not sure I would be able to efficiently teach Algebra I but the teachers edition has made this easy. So glad we have continued using this math program.
September 13, 2014
Rated 3 out of 5Â by dlgo Teacher help for tests
This is a straight forward answer guide. The problem I have is that only the answers are given. The problems are not worked out. As a non-math person, it would be helpful to both the teacher and student to see how the problems are worked out to help see exactly where the student made an error.
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'This module consisst of two units, namely Introduction to Ordinary differential equations and higher order differential...
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'This module consisst of two units, namely Introduction to Ordinary differential equations and higher order differential equations respectively. In unit one both homogeneous and non-homogeneous ordinary differential equations are discussed and their solutions obtained with a variety of techniques.Some of these techniques include the variation of parameters, the method of undetermined coefficients and the inverse operators. In unit two series solutions of differential equations are discussed .Also discussed are partial differential equations and their solution by separation of variables. Other topics discussed are Laplace transforms, Fourier series, Fourier transforms and their applications.Outline: SyllabusUnit 1: Introduction to Ordinary Differential EquationsLevel 2. Priority A. Calculus 3 is prerequisite.First order differential equations and applications. Second order differential equations. Homogeneous equations with constant coefficients. Equations with variable coefficients. Non-homogeneous equations. Undetermined coefficients. Variations of parameters. Inverse differential operators.Unit 2: Higher Order Differential Equations and ApplicationsLevel 2. Priority B. Differential Equations 1 is prerequisite.Series solution of second order linear ordinary differential equations. Special functions. Methods of separation of variables applied to second order partial differential equations. Spherical harmonics. Laplace transform and applications. Fourier series, Fourier transform and applications Differential Equations to your Bookmark Collection or Course ePortfolio
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Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and...
see more preparation in linear algebra. In writing this book I have been guided by the these principles: • An elementary text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student's• An elementary text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student's place, and have chosen to err on the side of too much detail rather than not enough. definitions, preferring to deal with concepts in a more conversational way, copiously illustrated by 299 completely worked out examples. Where appropriate, concepts and results are depicted in 188 figures Differential Equations with Boundary Value Problems to your Bookmark Collection or Course ePortfolio
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The applet displays the direction field for the differential equation ...
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The applet displays the direction field for the differential equation dy/dx = axm + bxn .You choose the parameters a, b, m, n, by using the sliders or by typing directly in the right-hand control panels. The applet draws the direction field. Source codeuler to your Bookmark Collection or Course ePortfolio
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This is a free, online textbook offered by Bookboon.com. "A free book with a collection of examples of how to solve linear...
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Quoted from the site: "IDEA is an interdisciplinary effort to provide students and teachers around the world with computer based activities for differential equations in a wide variety of disciplines." Currently, IDEA contains nearly twenty activities. These are applications of differential equations to areas as diverse as bungee jumping and salmon migration. Some of these applications are presented as text with illustrations, but others include interactive IDEA: Internet Differential Equations Activities to your Bookmark Collection or Course ePortfolio
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'This is a simple application that will factor quadratic equations, if possible. It's quick, handy, and easy. Example:...
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'This is a simple application that will factor quadratic equations, if possible. It's quick, handy, and easy. Example: 6x² + 13x + 6 = (2x+3)(3x+2).If you need to *solve* quadratics as well as factor them, you should check out iFactor and Solve Quadratics in the app store. It can handle any quadratic, including ones with complex rootsFactor Quadratics App for iOS to your Bookmark Collection or Course ePortfolio
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Synopses & Reviews
Publisher Comments
If you think algebra has to be boring, confusing and unrelated to anything in the real world, think again Written in a humorous, conversational style, this book gently nudges students toward success in pre-algebra and Algebra I. With its engaging question/answer format and helpful practice problems, glossary and index, it is ideal for homeschoolers, tutors and students striving for classroom excellence. It features funky icons and lively cartoons by award-winning Santa Fe artist Sally Blakemore, an Emergency Fact Sheet tear-out poster, and even an Algebra Wilderness board game guaranteed to help students steer clear of Negatvieland--and have fun.The Algebra Survival Guide is the winner of a Paretns' Choice award, and it meets the Standards 2000 of the National Council of Teachers of Mathematics. Its 12 content chapters tackle all the trickiest topics: Properties, Sets of Numbers, Order of Operations, Absolute Value, Exponents, Radicals, Factoring, Cancelling, Solving Equations, the Coordinate Plane and yes even those dreaded word problems. The Guide is loaded with practice problems and answers, and its 288 pages give students the boost they need in a style they'll enjoy to master the skills of algebra.
Synopsis
Praised by leading educational journals, the Algebra Survival Guide uses humor, cartoons, and crystal- clear explanations to help students tackle the trickiest topics of pre-algebra and Algebra I. The text features a lively question-and-answer format, step-by-step instructions, and practice problems with answer. Also includes a tear-out "Emergency Fact Sheet" poster and a board game. 288 pages + posters + board game. | 677.169 | 1 |
Sherman Chottiner
This textbook is one third pre-calculus (algebra, analytical geometry) and the rest introductory calculus presented in just over four hundred pages of content. The presentation is often overtly jocular. This style could be friendly and even entertaining to some: "Are you having so much trouble with derivatives that you are thinking about jumping from the nearest tall building? Well, stop, because you'll only come under the influence of other derivatives…" Ignoring the jokes, the book, complete with exercises and examples, is a suitable textbook for a first semester of calculus, especially at the high school level. The text integrates with Freemathhelp.com and Calc101.com through suggested content-relevant tasks.
The calculus portion is only a few hundred pages that also offer plenty of large images and many exercises. Thus, there is not much overall ground covered as compared to most other calculus texts at this level. However, much of the content is lucidly explanatory. I especially appreciate the time taken to derive basic derivative rules, such as the chain rule. I also applaud the author for presenting the development of integration rules for a wide range of cases as well as a rich section on applied optimization approaches and elasticity. There are also fairly detailed applications from sports, state budgets, and tax equalization rates that take the reader to the verge of ODEs. There is an enlightening graphical emphasis throughout these applications.
The author chooses mnemonic variable usage, such as "s" for slope over the common "m", but this approach may confuse some students of the elementary topics, such as the "i" used for intercept values presented to a student who has probably learned a little about complex numbers. The author's jolly nature and willingness to use unconventional phrasing come together best in the introduction of utility assessment as "happiness optimization".
Tom Schulte prefers a dry humor delivering lectures to mathematics students at Oakland Community College in Auburn Hills, Michigan. | 677.169 | 1 |
Mathematics for Economics and Business / Edition 6
Overview students are encouraged to stop and check their understanding along the way by working through practice problems.
FEATURES
Many worked examples and business related problems.
Core exercises now have additional questions, with more challenging problems in starred exercises which allow for more effective exam preparation.
Answers to every question are given in the back of the book, encouraging students to assess their own progress and understanding.
Wide-ranging topic coverage suitable for all students studying for an Economics or Business degree.
Mathematics for Economics and Business is the ideal text for any student taking a course in economics, business or management.
IAN JACQUES was formerly a senior lecturer at Coventry University. He has considerable experience teaching mathematical methods to students studying economics, business and accounting. | 677.169 | 1 |
no-nonsense practical guide to statistics, providing concise summaries, clear model examples, and plenty of practice, making this workbook the ideal complement to class study or self-study, preparation for exams or a brush-up on rusty skills.
About the Book Established as a successful practical workbook series with over 20 titles in the language learning category, Practice Makes Perfect now provides the same clear, concise approach and extensive exercises to key fields within mathematics. The key to the Practice Makes Perfect series is the extensive exercises that provide learners with all the practice they need for mastery.
Not focused on any particular test or exam, but complementary to most statistics curricula
Deliberately all-encompassing approach: international perspective and balance between traditional and newer approaches.
Large trim allows clear presentation of worked problems, exercises, and explained answers.
Features
No-nonsense approach: provides clear presentation of content.
Over 500 exercises and answers covering all aspects of statistics
Successful series: "Practice Makes Perfect" has sales of 1,000,000 copies in the language category – now applied to mathematics
Workbook is not exam specific, yet it provides thorough coverage of the statistics skills required in most math tests. | 677.169 | 1 |
How good one must be in math for Games Development?
Just curious, i'm looking to write a simple 2D game and i know math just not so good at it, i heard that Math isn't really required when you're writing 2D games they're most used when writing 3D games, is that true? also can someone please recommend me some math concepts i should learn for Games development? i would really appreciate it, thanks!
Math is needed for both 2D and 3D games (to make objects move and stuff), i don't write plenty of games but when it comes to games development i think you need to know Algebra, Geometry, Trigonometry and calculus well, you don't have to be fluent at them or anything but you need to understand how they work, as mentioned in the previous topic you can learn all of them right here for calculus i think you're best off buying a book, i recommend this book (specifically written for games developers) it's well over 600 pages and i really recommend you read it.
It really depends on what games you are writing. If you're writing a platform game or a horizontal scrolling shooter, then no, you don't need to be particularly good at maths. If you're writing a 2d rts game or board game, then the maths can get fairly hard.
also can someone please recommend me some math concepts i should learn for Games development?
To start with, just make sure you are definitely very comfortable with algebra.
"i heard that Math isn't really required when you're writing 2D games they're most used when writing 3D games, is that true?"
No. A 2-D game can just as mathematically intense as a 3-D game. But, like what Mats said, it depends on what type of game your making. If a game does not utilise a vast amount of mathematics, it doesn't mean the game will be any less fun. For example, look at the early Pokemon games; they were 2-D, no physics were involved and mathematics in general was limited, but it was still a popular gaming series.
If your main objective is to build 3-D games, start with 2-D games. If your mathematical skills are limited, build 2-D games will little mathematics. Then, learn what needs to be learnt. Just take little steps. It unnecessary work learning all of those mathematical fields what Uk Marine stated; just learn what your game calls for. No more, no less. Don't be afraid of expanding your mathematical comfort zone with each game you make, or else you'll never improve.
Think of it like this: Game programming is a good excuse to learn math. It's not like the whole process of programming a game is math. You come across situations while you are doing it that require math. So when you need to know some math to implement a feature, you just look it up or ask someone for some help. It's pretty much just like any other part of the programming process, whether it be knowing how to work with something in the STL, or how to build a good code structure. In that sense, I don't think that the math part is any more difficult than the programing component, but if you don't know it, there are more steps involved and it will take you more time. But the math components in games are for the most part, a small subset of the math you learn in school. After so much experience in game programming, you will have learned, or built up the resources, to solve most of your problems.
Knowing math just helps you figure out a lot in your head. It honestly doesn't hurt to learn more math though.
Lumpkin wrote:
If you can program and can do math I'm not sure why you're not already there?
That is way too simplistic. It takes more than understanding programming and math to make games even less in some cases. Depending on what you are wanting to make it would also require understanding graphics and their coordinate system to display graphics properly. Networking, requires understanding of something else. The more you want to do in a game requires more skill sets than just programming and math. Honestly, if you are doing console applications, you don't even need math, just programming. | 677.169 | 1 |
Find a Harbor Acres, NY StatisticsAlgebra 1 is a textbook title or the name of a course, but it is not a subject. It is often the course where students become acquainted with symbolic manipulations of quantities. While it can be confusing at first (eg "how can a letter be a number?"), it can also broaden your intellectual scope | 677.169 | 1 |
Algebra Made SimpleOverviewAlgebra Made Simple was written utilizing the principles and standards for school mathematics published by the National Council of Teachers of Mathematics (NCTM). These standards are the cornerstone of basic math principles that ensure the highest quality of learning for students.
Specially formatted for the Nook e-book reader, this book is easy to read on the Nook and other ebook readers.
Related Subjects
Meet the Author
Kara Monroe is a graduate of Ball State University with a degree in Mathematics Education. Kara taught high school mathematics in Lakeland, Florida. Kara went on to join Ivy Tech Community College Richmond as the manager of instructional technology. While at Ivy Tech, Kara has earned her MBA from Jones International University and a Ph.D. in Higher Education Leadership from Capella University. Kara serves as the Executive Director of Finance, Facilities, and Information Technology at Ivy Tech Community College Richmond. However, her passion remains with students in the classroom and she seeks every opportunity to teach and develop courses in mathematics, computer information systems, business, and education leadership.
Most Helpful Customer Reviews
This handbook did not cover many of the topics we are encountering in a standard algebra textbook. I wish i coudl ge tmy four bucks back!
314LCK159
More than 1 year ago
THIS 'BOOK' IS ONLY 15 PAGES LONG. CERTAINLY NOT WORTH $4.
Anonymous
More than 1 year ago
Anonymous
More than 1 year ago
Igor_Math
More than 1 year ago
There is brief, but very interesting overview of Algebra I topics. Solving linear equations with one unknown is described on several ways, such it looks so simple.
But, also linear equations of two unknowns are well explained. That is extremely important to understand there are infinitely many solutions. Using carefully chosen examples, the slope-intersect form of a line is explained on the most intuitive way. The reader is going to figure out the connection between algebra and geometry and that is the best way to gain intuition for understanding equations with infinitely many solutions.
The author is using interesting (tricky) way to describe some methods for factoring polynomial expression. There are lots of exercises and the reader could have practice to do factoring by himself. All the solutions of exercises are explained step-by-step. | 677.169 | 1 |
However, I omitted the solutions for educational purposes (you can ask for it. Like my mentor, I also hate spoon feeding the student. Originally, this compilation was a work of 5 person, I and 4 of my classmates. We failed this subject. I don't care why they failed. (OMG this is so simple. If you want to know why I also failed, just contact me in my e-mail.) Well, as part of the requirement to complete the unit, we were asked by our professor to research and compile these problems with their solution. After we submitted the drafts for her to check, I thought our task was complete. After 3 months of checking the solution (can you imagine that? She burned 3 months of our college life waiting for her approval.), she returned the papers noting that we have to change some of us have the same problems solved. (Take note, when she returned the paper it was only 10 days before closing the curtain.) So there we were digging again the net hoping to find another set of problems that will save our grades. It's not an easy task, but we managed to collect the needed problems. After so much effort in editing, I managed to print this and pass it to her. She refused to accept this because we're late. Indeed, we're 6 days late but you can't blame us for it really is a tiresome task—searching for problems that we don't have idea if it really exist, unless we make our own problems (that is our last resort.) After so much persuading, she accepted the pile and said that she'll think about giving our grade. (F*CK! Ka-pasamba mo!) After a couple of weeks, she asked again for our drafts for her to check (WHAT THE F*CK)... well to cut this crap short, three of my classmates passed her subject in the end while I and my best bud didn't. The problems here, like I said, are compiled, came from different sources. You can find the reference books used at the end of every section. I also provided some definition of every section. I didn't put extra lecture because I assume that you have acquired that knowledge from your professor. If you have any questions, suggestions, comments or anything and everything, feel free to contact me @ ijotika@hotmail.com. MABUHAY!!!
P.S. I uploaded this document because I want to help you guys that like me suffer/suffers/suffered from the wrath of our prof. this is for you guys. P.P.S. By the way I came from the BICOL UNIVERSITY COLLEGE OF ENGINEERING here in Legaspi City, Albay, Philippines. So if ever you, BU CEngians, came across this page, you already knew which professor I'm talking about. P.P.P.S. Please excuse my grammar. I am not goooooooooood in English. aaenlay tsaltoX aiyeuXX fukremr cmirieh (#$%!@)
ijotika
Chemical Calculations 1
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MATERIAL BALANCE
A material balance is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique. The exact conservation law used in the analysis of the system depends on the context of the problem but all revolve around mass conservation, i.e. that matter cannot disappear or be created spontaneously. Therefore, mass balances are used widely in engineering and environmental analyses. For example mass balance theory is used to design chemical reactors, analyze alternative processes to produce chemicals as well as in pollution dispersion models and other models of physical systems. Closely related and complementary analysis techniques include the population balance, energy balance and the somewhat more complex entropy balance. These techniques are required for thorough design and analysis of systems such as the refrigeration cycle. The general form quoted for a mass balance is The mass that enters a system must, by conservation of mass, either leave the system or accumulate within the system . Mathematically the mass balance for a system without a chemical reaction is as follows:
Strictly speaking the above equation holds also for systems with chemical reactions if the terms in the balance equation are taken to refer to total mass i.e. the sum of all the chemical species of the system. In the absence of a chemical reaction the amount of any chemical species flowing in and out will be the same. This gives rise to an equation for each species in the system. However if this is not the case then the mass balance equation must be amended to allow for the generation or depletion (consumption) of each chemical species. Some use one term in this equation to account for chemical reactions, which will be negative for depletion and positive for generation. However, the conventional form of this equation is written to account for both a positive generation term (i.e. product of reaction) and a negative consumption term (the reactants used to produce the products). Although overall one term will account for the total balance on the system, if this balance equation is to be applied to an individual
Chemical Calculations 1
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species and then the entire process, both terms are necessary. This modified equation can be used not only for reactive systems, but for population balances such as occur in particle mechanics problems. The amended equation is given below. Note that it simplifies to the earlier equation in the case that the generation term is zero.
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MATERIAL BALANCE PROBLEMS WITHOUT REACTION
1. A lacquer plant must deliver 1000 lb of an 8% nitrocellulose solution. They have in stock a 5.5% solution. How much dry nitrocellulose must be dissolved in the solution to fill the order?1 M (lb) N= 100% F (lb) N= 0.055 S= 0.945 1.00
P (1000lb) N= 0.08 S= 0.92 1.00
2. A liquid adhesive, which is used to make laminated boards, consists of a polymer dissolved in a solvent. The amount of polymer in the solution has to be controlled for this application. When the supplier of the adhesive receives an order for 3000kg of an adhesive solution containing 13% by weight polymer, all it has on hand is (1) 500kg of a 10% by weight solution, (2) a very large quantity of a 20 wt% solution, and (3) pure solvent. Calculate the weight of each of the three stocks that must be blended together to fill the order. Use all of the 10% by weight solution. 2
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3. Hydrogen pre-carbon in the form of core is burned. a. With complete combustion using theoretical air. b. With complete combustion using 50% excess air. c. Using 50% excess air but with10% of the carbon burning to CO only IN EACH CASE CALCULATE THE GAS ANALYSIS THAT WILL BE FOUND BY TESTING THE FLUE GASES WITH AN ORSAT APPARATUS. 3 4. An evaporator is concentrating solutions coming from two different sources. The solutions from the first source containing 10% NaCl and 10% NaOH flows at the rate of 50 kg/min. the other solution containing 8% NaCl and 12% NaOH flows at the rate of 70 kg/min. The two streams are fed directly to the evaporator. If 50% of the total water is to be evaporated, calculate the composition and the flow rate of the product. 4 50% H2O evaporated
MATERIAL BALANCE WITH CHEMICAL REACTION
1. A natural gas consisting entirely of Methane (CH4) is burned with an oxygen enriched air of composition 40% O2 and 60% N2. The Orsat analysis of the product gas is reported by the laboratory is CO2 : 20.2%, O2 : 4.1%, and N2 : 75.7%. Can the reported analysis be correct? Show all calculations.1
W
H4 100% P CO2= 20.2 O2= 4.1 N2= 75.7
A enriched air O2 = 40 N2= 60 2. Plants in Europe sometimes use the mineral pyrites (the desired compound in the pyrites is FeS2) as a source of SO2 for the production of sulfite pulping liquor. Pyrite rock containing 48% sulfur is burned completely by flash combustion. All of the iron turns Fe3O4 in the cinder (the solid product), and a negligible amount of SO 3 occurs in either the cinder or the product gas. The gas from such as furnace is passed through milk of lime (CaO in water) absorbers to produce bisulfate pulping liquor. The exit gas from the absorber analyzes: 0.07% SO2, 2.9% O2, 96.4% N2. (Refer to the figure) Calculate the kg of air supplied to the burner per kg of the pyrite burned. 2 3FeS2 + 8 O2 Fe3O4 + 6 SO2
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3. A gas containing 80% CH4 and 20% He is sent through a quartz diffusion tube to recover the Helium. 20% by weight of the original gas is recovered, and its composition is 50% He. Calculate the composition of the waste gas if 1ooKmol of gas are processed per minute. The initial gas pressure is 120kPa, and the final gas pressure is 115kPa. The barometer reads 740mmHg. The temperature of the process is 22°C.
4. Pyrites ore is used for the production of the sulfuric acid. The pyrites is 80% FeS 2 and 20% inert materials. The FeS2 is burned with 30% excess air based on the reaction, 4 FeS2 + 11 O2 2 Fe2O3 + 8 SO2 Only 90% of the FeS2 follows this reaction. The rest goes as follows: 4 FeS2 + 15O2 2 Fe2O3 + 8 SO3 What is the composition of the gases resulting in the combustion? SO2 SO3 O2 N2 Inerts
B 80% FeS2 U 20% inert materials R 30% excess air N E R
5. A fed containing A, B and inerts enters a reactor. The reaction taking place is: 2A + B C The product stream leaving the reactor is having the following composition by mole: A=23.08%, B=11.54%, C=46.15, inerts= 19.23%. Find the analysis of feed on mole basis. Basis: 100 kmol of product stream
RECYCLE
In a reactive process, there is generally some unreacted feed material found in the product. In order to reduce cost and increase efficiency, the unreacted material is often separated and reused in a recycle loop.
Recycle may also be used to recover an expensive catalyst, maintain feed concentration below certain levels, or in a totally closed loop to pump a cooling or heating fluid. Keep in mind that while flow rates of the process effluent, process product, and recycle vary, all three have identical composition.
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RECYCLE PROBLEMS WITHOUT CHEMICAL REACTION
1. Examine the given figure. What is the quantity of the recycle stream in kg/hr? 1
C
Crystals carry off 4% H2O (4 kg H2O per kg total crystals +H2O) 2. Based on the process drawn in the diagram, what is the kg recycle per kg feed if the amount of W waste is 100 kg? The known compositions are inserted on the process diagram.2
X
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3. A planting plant has a waste stream containing Zn and Ni in quantities in excess of that allowed to be discharged into the sewer. The proposed processes to be used as a first step in reducing the concentration of several of the streams are listed in the table, what is the flow ( ℎ ) of the recycle stream R32 if the feed is 1( ℎ ).3 TABLE CONCENTRATION : g/L Stream Zn F 100 Po 190.1 P2 3.50 R32 4.35 W 0 D 0.10 Ni 10.0 17.02 2.19 2.36 0 1.00
4. A fresh feed (contains 5,000 kg of wet material) with 60% moisture is needed to be dried. To facilitate the operation, a part of dried product contains 5% water is recycled and mixed with the feed. The mixing stream of the recycle and the fresh feed contains 30% water. Calculate the water removed and the recycle to feed ratio. 4
(P) Desalinized water 500 ppm salt Given the data in the figure, determine; (a) the rate of waste brine removal, (b) the rate of desalinized (potable water) production; (c) the fraction of the brine leaving the reverse osmosis cell (which acts in essence as a separator) that is recycled. 5
1. Boron trichloride (BCl3) gas can be fed into a gas stream and for doping silicon. The simplest reaction (not the only one) is 4 BCl3 + Si 3 SiCl4 + 4 B
If all the BCl3 not reacted is recycled, what is the mole ratio of recycle to SiCl 4 exiting the separator? The conversion on one pass through the reactor is 87% and 1 mole per hour of BCl3 is fed to the reactor.1 Unreacted BCl3 BCl3 1mol/hr
87% REACTOR
SiCl4 BCl3 SiCl4 SEPARATOR
2. Many chemicals generate emissions of volatile compounds that needed to be controlled. In the process shown in the accompanying figure, the CO in the exhaust is substantially reduced by separating it from the reactor effluent and recycling the unreacted CO together with the reactant. Although the product is proprietary, the information is provided that the fresh feed stream contains 40% reactant, 50% inert and 10% CO, and that on reaction 2 moles of reactant yield 2.5moles of product. Conversion of the reactant to product is 73% on one pass through the reactor and 90% for the overall process. The recycle stream contains 80% CO and 20% reactant. Calculate the ratio of moles of the recycle stream to moles of the product stream. 2
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X
REACTOR
SEPARATOR
3. Nitroglycerine, a widely used high explosive, when mixed with wood flour is called "dynamite'. It is made by mixing high-purity glycerine (99.9+% pure) with nitration acid, which contains 50% H2SO4, 43% HNO3 and 7% H2O by weight. The reaction is: C3H8O3 + 3HNO3 + (N2SO4) C3H5O3(NO2)3 + 3H2O + (H2SO4)
The H2SO4 does not take part in the reaction, but it is present to "catch" the H2O formed. Conversion of Glycerine is complete in the nitrator, and there are no side reactions, so all of the Glycerine fed into the nitrator forms into Nitroglycerine. The mixed acid entering the nitrator (stream G) contains 20% excess HNO 3 to assure that all the Glycerine reacts. After nitration, the mixture of Nitroglycerine and spent acid (HNO3, H2SO4, and H2O) goes to a separator (a settling tank). The Nitroglycerine is insoluble in the spent acid and its density is less, so it rises to the top. It is carefully drawn off as product stream P and sent to wash tanks for purifications. The spent acid is withdrawn from the bottom of the separator and sent to an acid recovery tank, where the HNO 3 and H2SO4 are separated. The H2SO4- H2O mixture is stream W, and is concentrated and sold for industrial purposes, the recycle stream to the nitrator is a 70% by weight solution of HNO3 in H2O. In the diagram, product stream P is 96.5% Nitroglycerine and 3.5% H 2O by weight. To summarize: Stream F = 50% by weight H2SO4, 43% HNO3, 7% H2O Stream G = 20% excess HNO3 Stream P = 96.5% by weight Nitroglycerine, 3.5% by weight H2O Stream R = 70% by weight HNO3, 30% by weight H2O a.) If 1x103 kg of Glycerine/hr is fed to the Nitrator, How many kg/hr of Stream F will be the result? b.) How many kg/hr are in the recycle stream?
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c.) How many kg of fresh feed, Stream F are fed per hour? d.) Stream W is how many kg/hr? What is its analysis in weight %?
The process is to be designed for 85% overall conversion of propane. The reaction products are separated into two streams: the first, which contains H 2, C3H6 and 0.656% of the propane that leaves the reactor, is taken off as product, the second stream, which contains the balance of unreacted propane and 5% of the propylene in the product stream, is recycled to the rector. Calculate the composition of the product, the ration (moles recycled/ moles fresh feed) and the single-pass conversion. Illustration: Basis Fresh Feed 100 mol C3 H8
1 100 + Xr Yr
100 mole of fresh feed A Reactor
2 Xp Yp Zp
Separation unit
3 X Y 85% conversion 0.656%
Xr Yr
Z 5%
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5. Reactors that involve biological materials (bioreactors) use living organisms to produce variety of products. Bioreactors are used for producing ethanol, antibiotics and proteins for dietary supplements and medical diagnosis. The overall conversion of the proprietary component in the fresh feed to product is 100% in recycle bioreactor. The conversion of the proprietary component to product per pass in the reactor is 40%. Determine the amount of recycle and the mass percent of component in the recycle stream if product contains 90% product, and the feed to the reactor contains 3% wt of component. (PS) Product Stream
10% water, 90% product
Consider a juice concentration process in which the dehydration process runs most efficiently by removing more water than is desired. A portion of the feed may be directed around the dehydrator in a bypass loop, to be mixed with unprocessed feed. The figure below illustrates a bypass loop.
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BYPASS PROBLEMS WITHOUT CHEMICAL REACTION
1. Air at 310K saturated with water vapor is dehumidified by cooling to 285K and by consequent condensation of water vapor. Air leaving the dehumidifier, saturated at 285K is mixed with a part of the original air which is bypassed. The resulting air stream is reheated to 320K. It is desired that the final air contains water vapor not more than 0.03kg per kg of dry air. Calculate the mass of dry air (in kg) bypassed per kg of dry air sent through the dehumidifier.1
2. The figure shows a 3-stage separation process. The ratio of P 3/D3 is 3, the ratio of P2/D2 is 1, and the ratio of A to B in stream P2 is 4:1. Calculate the composition and percent of each component in stream E.2
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3. A process needs an air supply which should contain 0.12 kmol / kmol da exactly. 1500 m3/ min of air at 25 °C and 101.3 k Pa is to be treated. Part of this air goes to a spray chamber. Where the air picks up water goes out with the humidified air to produce a 1850-kg mixture containing 0.12 kmol H2O/ kmol da. What is the water consumption (kg/min)? What is the ratio of the flowmeter readings of the bypass stream and the stream to the water spray?3 m3 bypass m1 1500 m3 /min 25°C, 101.kPa m2 spray chamber m4 m5 1,850 kg air mixer 0.12 kmol H2O/kmol da 0.3 kmol H2O/ kmol d.a
4. In a textile industry, it is desired to make a 24% solution (by mass) of caustic soda for a mercerization process. Due to the very high heat of dissolution of caustic soda in water, the above solution is prepared by two step process. First, in a dissolution tank, caustic soda is dissolved in the correct quantity of water to produce 50% solution. After complete dissolution and cooling, the solution is taken to dilution tank where some more water is added to produce 24% solution. Assuming no evaporation loss in dissolution tank, calculate the mass ratio W 1/W2.
BYPASS PROBLEMS WITH CHEMICAL REACTION
1. Perchloric acid (HClO4) can be prepared from Ba(ClO4)2 and HClO4 , as shown in the diagram below. Sulfuric acid is supplied 20% in excess in reactor 1 that converts 80% of the Ba(ClO4)2. 6125lb of the feed is bypassed to the second reactor where 10% in excess H2SO4 is supplied and converts 60% of Ba(ClO4)2. Between the adjacent reactors is a stream that contains 63% HClO4 by mole over the time period. Calculate;2 a. lbmole of waste H2SO4 per lbmole of BaSO4 that exits the separator b. lb HClO4 in G per lb feed c. over-all conversion of Ba(ClO4)2.
.
. The over-all reaction produces 7400 lb HClO4
2. TiCl4 can be formed by reacting Titanium Dioxide with Hydrochloric Acid. 1200Kg feed with 64% TiO2, 3% inerts, 20% HCl, and 13%H2O is the initial feed. 800Kg of it is fed in to unit 1 that removes all the inerts from the original feed but unfortunately, also removes 20% of TiO 2 from the feed. That is why a bypass stream is needed with an amount of 400Kg. The stream leaving unit 1 together with the bypass is sent to unit 2 which converts all TiO 2 into TiCl4. A separator unit isolates the desired product from the rest. 3 Required: A. amount of TiCl4 produced B. amount of W and its composition
Chemical Calculations 1
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Chemical Calculations 1
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PURGE
When a process uses a recycle loop, there can often be a buildup of some undesired material within the system. By using a purge, a fraction of the recycle loop material is removed. This purge fraction is generally only a few percent of the recycle flow rate.
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PURGE PROBLEMS WITHOUT CHEMICAL REACTION
1. Caustic soda, among the largest producing product in Chemical Industry, is produced by electrolysis of common salt, NaCl, using membrane cell electrolyzers. The NaCl solution feed contains 30% by vol. of the solute. The NaOH produced is collected on storage tanks and the unreacted material containing 25% NaCl is being recycled. A purge stream is used to separate/remove the Cl 2 from the recycle stream. Calculate the following: a. Recycle stream flow rate/ recycle stream and product rate. b. Purge rate if it contains 40% Cl2 to produce 32.5 mol of NaOH, equivalent to 65% by vol. in the final product.
2. To save energy, stack gas from a furnace is used to dry ice. The flow sheet and known data are shown in the figure below. What is the amount of recycle gas ( in lb mole) per 100 lb of P if the concentration of water in the gas stream entering the dryer is 5.20%?
B
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26
3. One half of the high-pressure steam after being utilized for power generation is sent to the plant as process steam. The other half is returned to the boiler as condensate carrying 50 ppm solids.
To keep the solid level in the boiler below 1600 ppm solids, a part of the boiler water is blowdown (purge) continuously. the fresh boiler feed is found to contain 500ppm solids. if steam produced is free of solids, calculate the weight ratio of feed water to the blowdown (purge) water. 4. An evaporator is concentrating solutions from the two different sources. The solution from the first source containing 20% NaCl and 15% NaOH, flows at the rate of 40 kg/min. The other solution containing 8% NaCl and 12% NaOH flows at the rate of 70 kg/min. The two streams are fed directly to the evaporator, in this process, 60% of the water evaporated. In order to obtain the desired concentration,80% NaOH and 65% of NaCl were recycled. However, 20% of H2O was mixed on the recycle stream; to avoid this, purging of H2O must be done. Calculate the composition of NaOH and NaCl in the recycle stream and the product in the evaporator.
Chemical Calculations 1
27
Illustration: 60% H2O 40 kg/min 20% NaCl 15% NaOH
EVAPORATOR
Solution: 70 kg/min 8% NaCl 12% NaOH
80% NaOH 65% NaCl
20% H2O
5. In the production of NH3 from hydrogen and nitrogen, the nitrogen, the NH 3 produced is condensed for the reactor (convertor) product stream and the unreacted material is recycled. If the feed contains 0.2% Argon (from the nitrogen separation process), calculate the purge rate required to hold the argon in the recycle stream below 5.0%. R Purge
PURGE PROBLEMS WITH CHEMICAL REACTION
1. Ethylene dioxide is produced by oxidation of ethylene as per the following reaction: C2H4 + ½ O2 C2H4O Fresh feed containing ethylene and air is mixed with recycle feed and mixed feed enters into a reactor. The proportion of C2H4 : O2 : N2 in mixed feed is 1:0.5:5.65 (on mole basis) 50% per pass conversion is achieved in the reactor are feed to the absorber where only all C 2H4O formed removed. The gases from the absorber containing C 2H4, O2 and N2 are recycled each. To avoid built up of N2 in the system, small portions of recycle stream is continuously purged based on 100 mol ethylene is mixed feed, calculate the fresh feed to the process, the purge stream, recycle ratio, combined feed ratio and overall conversion of ethylene.
Chemical Calculations 1
30
2. It is proposed to produce ethylene oxide ((CH 2)2O) by the oxidation of ethane (C2H6) in the gas phase; +
(CH2)2O + H2O
The ratio of the air to the C2H6 in the gross feed into the reactor is 10 to 1, and conversion of C2H6 on one pass through the reactor is 18%. The unreacted ethane is separated from the reactor products and recycled as shown in the figure below. What is the ratio of the recycle stream to the feed stream and calculate the composition of purge stream and the stream exiting the reactor.
The fresh feed to the process contains CO 2 and H2 in stoichiometric proportions, 0.5mol% Inerts(I). The reactor effluent passes to a condenser which removes essentially all of theCH3OH and water formed, none of the reactants or inerts. The latter substances are recycled back to the reactor to avoid build-up of the inerts in the system; a purge stream is withdrawn fron the recycle. The feed to the reactor contains 2% inerts and the single pass conversion is 60%. Calculate the molar flow rates of the fresh feed, the total feed to the reactor and the purge stream for CH 3OH production rate of 1000mol/hr.
Chemical Calculations 1
32
4. The fresh feed to an ammonia production process contains N2, H2 and 0.125 mole %. The feed is combined with the recycle stream and the combined stream contains: 24.00 mol% nitrogen, 74 mol% hydrogen and 2% inerts, then it is feed to the reactor, in which 25% single-pass conversion of nitrogen is achieved. The product pass through a condenser in which essentially all the ammonia is removed and the remaining gases are recycled. However, to prevent buildup of the inert in the system, a purge stream must be taken off. Calculate the overall conversion of nitrogen, the ratio (moles purge/mole of gas leaving the condenser) and the ratio (moles fresh feed/moles feed to the reactor).
The conversion of CO entering the reactor is 20%. A feed stream consisting 33% CO, 66.5% H2, and 0.5%CH4 is mixed with a recycle stream and sent to a reactor. The methane leaving the reactor is separated and the unconverted gases are recycled. To prevent the accumulation of CH 4 and keep its concentration, in the recycle stream at 3%, a portion of the recycle stream is blown off. For 100 moles of fresh feed, determine the following: a. recycle stream (moles), b. purge stream (moles) c. composition of purge (moles) and d. moles of methanol. | 677.169 | 1 |
Grading policy: Your grade in this course will be based
on your performance in the
computational/programming assignments which you will be
doing in this course.
You will do eight assignments, from which
the best seven will be counted. There may be extra credit
opportunities.
Letter grades for the course will be decided using the following:
Score above
90%
80%
65%
50%
otherwise
Letter grade
A
B
C
D
F
Please note that the syllabus is subject to change by announcement.
Learning Goals
Upon completion of this class you should
be able to use Matlab's extensive linear algebra
capabilities,
be able to program in Matlab efficiently,
be aware of the flexible file I/O capabilities
provided in Matlab,
know how to utilize the extensive 2D and 3D
graphics capabilities in Matlab,
know how to use Matlab specific programming features
such as logical subscripting and vectorization,
be aware of the great number of built-in numerical
methods in Matlab,
be able to produce presentable Matlab output.
List of the Topics Covered
The first half of each class will be used to present new material.
The second half of class will be for working on the assignments,
with the opportunity to ask the instructor for help.
Lecture
Day
Date
Main Topic(s)
Chapter(s)
1
Fri
01/03/14
Snow Cancellation
2
Mon
01/06/14
A Tutorial Introduction to Matlab
1, 2, 3
3
Wed
01/08/14
Matrix Algebra in Matlab
4, 5
4
Fri
01/10/14
An Introduction to Matlab Programming
6, 7
5
Mon
01/13/14
Matlab Programming: Input and Output
13
6
Wed
01/15/14
Intermediate Matlab Programming
10
7
Fri
01/17/14
3D Graphics in Matlab
Mon
01/20/14
Martin Luther King Day (no class)
8
Wed
01/22/14
Effective Programming and Data Types in Matlab
9
Fri
01/24/14
Numerical Methods in Matlab (Makeup Class For Snow Cancellation)
11, 12
UMBC Academic Integrity Policy
By enrolling in this course, each student assumes the responsibilities of
an active participant in UMBC's scholarly community in which everyone's
academic work and behavior are held to the highest standards of honesty.
Cheating, fabrication, plagiarism, and helping others to commit these acts
are all forms of academic dishonesty, and they are wrong.
Academic misconduct could result in disciplinary action that may include,
but is not limited to, suspension or dismissal.
To read the full Student Academic Conduct Policy, consult the
UMBC Student Handbook, the Faculty Handbook, the UMBC Integrity
webpage
or the Graduate School website | 677.169 | 1 |
Description
A Mathematical Dictionary for Schools contains over 500 definitions of technical terms found within GCSE syllabuses. Key words and phrases are explained in clear, simple language with illustrations to aid understanding of more difficult terms. It has been written for key stage 4/GCSE students but is also suitable for key stage 3 and is the ideal companion for coursework and revision. | 677.169 | 1 |
And the rest of your post is a bunch of flim-flam that is avoiding the issue like everyone else does, that many students fail to understand the simplicity and certainty of algebraic reasoning. And they would fail to understand everything you have listed here as well. I think Richard's latest Dan-post showed it the best. There is no way any one of us here would have questioned that teacher's conjecture except for the simple fact that the algebra looked wrong. No computer. No popcorn. If you really wanted kids to do all these other things, then that is where you start. With algebra.
> On Mon, Jun 3, 2013 at 12:13 PM, Robert Hansen <bob@rsccore.com> wrote: > > On Jun 3, 2013, at 2:01 PM, kirby urner <kirby.urner@gmail.com> wrote: > >> There should be no teaching of algebra as it was taught in say 1964. They had no decent way to program personal computers back then, no Google Earth, no GPS. > > > Algebra has nothing, zero, zip, to do with computers. Algebra is entirely about thinking. This is what I mean about the purpose lost. > > That's entirely incorrect Bob. Plus you're quite inconsistent as you've written here a number of times that 'Mathematics for the Digital Age and Programming in Python' is a respectable math-learning text as text books go. It had better be, as families frequently pay the big bucks to have their students go to such schools that use it. > > Computer programs (as much as on-paper forms as "human computers" used to fill out, such as when computing a geodesic domes angles and edges) are in part about standardizing algorithms -- meaning you need to know right away what's a constant and what's a variable. Then you learn about what variations in input produce what variations in output. Again, that's functional analysis and ratio-ing deltas, the basis of diffy-Q. > > Whipping out programs is a great way to study the properties of mathematical objects, including Pascal's Triangle, which has everything to do with the Binomial Theorem (still part of pre-calc algebra the last I checked). > > Computers are a tool, like calculators. Nowadays it's still about using calculators. That's part of what's criminal. Teachers are at least a decade behind on average and getting further behind by the day. SQL relates to boolean algebra or has "boolean algebra" nothing to do with "algebra" despite the truth tables the alegbra books commonly feature. > > The algebra you had as a kid is not something I'm keen to perpetuate. Conway's and Guy's 'The Book of Numbers' is more in the ballpark, but then so is 'Godel Esher Bach' in providing a more "lexical" idea of precise rule following (the kind of appreciation helps make chess a part of maths, as much as Conway's Game of Life, which it isn't only for ethnic reasons, no "pure reason" at work -- English has a way of dumpstering a lot of good thinking thanks to mind-numbing antibodies). > > But lets just say for the sake of argument that there's nothing in the newer STEM curriculum that's recognizably "algebra" the way you remember it, with just such a list of topics, just such a table of contents. > > That would be, is going to be, a very good thing, a blessing. My thesis is we can't afford to hold kids back to the degree you would like to, for reasons of nostalgia perhaps. > > When you have a language that permits operator overloading so you-the-student get to define a meaning for multiplication, addition etc., and think about properties of multiplicative and additive inverse in the abstract, you're inherently in a better position to really learn algebra than some late 1900s 8th grader with no access to such tools. > > Kirby > > > Bob Hansen > | 677.169 | 1 |
Kirkus Discoveries
Mathematician and consultant Shestopaloff thoroughly explores the world of financial mathematics in a volume that will be valuable to anyone in the field. Beginning with interest and considering annuities, mortgages, and investment and risk measurement methods, Shestopaloff uncovers the complexities of investment mathematics with clear, understandable text accompanied by numerous derivations, examples, graphs and tables. Topics studied include the internal rate of return-which the author considers in a lengthy discussion that includes its relationship with similar
calculations-and nominal and effective interest rates. He also considers compounding using various computational methods and linking-a more accurate alternative to geometric linking, which is applied to financial trading. Shestopaloff discusses measurement of risk with details of the various risks and quantifying methods that are involved in investing, such as risks in interest rate, volatility, operational risk, downside risk and more. He briefly explains the probabilistic calculations involved. The introductory text includes definitions of all terms and rapidly advances through equations to allow mathematicians of different skill levels to follow the explanations. An associated software package is available, and the author briefly reviews computation methods, as well as the accuracy obtained by different methods. Shestopaloff ends with a caution that - although software may make many of these calculations invisibly and easily - it is still imperative to understand the mathematics behind the software. His explanations are thorough without excessive wordiness and the text smoothly accompanies equations and derivations. The author helpfully analyzes business consequences alongside the mathematics. The detailed index and table of contents, with paged references to subtopics, make this a very convenient reference book. Although additional editing could have corrected minor linguistic issues, readers will find the text easy to comprehend. Shestopaloff has presented many of these topics in previous peer-reviewed journal papers, but academics, students and professionals - from programmers to financial mathematicians - will find this a convenient one-volume guide, well-written and seamless. A valuable addition to the financial mathematician's library.
Midwest Book Review
For dedicated mathematicians, there is as much art and beauty as there is science in their calculations, formulas, precepts, concepts, and expositions. There is also utility, practicality, insight, and value in the application of mathematical principals to financial systems and the economy which are complex compilations of factors that mathematicians develop models to explain otherwise inexplicable and seemingly random phenomena. That's why Yuri Shestopaloff's "Science of Inexact Mathematics: Investment Performance Measurement, Mortgages and Annuities, Computing Algorithms, Attribution, Risk Valuation" is such a seminal work in the field of applied mathematics to financial issues and economic performances with respect to investment strategies and interpretations. Offering detailed computing algorithms (including software implementation), the informed and informative text is enhanced with numerical examples, graphical and tabular illustrations throughout. A work of impressive scholarship. ... is especially recommended for academic, governmental, and professional library collections. | 677.169 | 1 |
Math Center
The Math Center is a non-credit, Community Education class which provides assistance
in mathematics as a completely free service. Current Allan Hancock College students
as well as other individuals who are 18 years or older may fill out a simple registration
form and attend as frequently as they want. Registration forms may be found in the
Math Center or at Community Education in Building S.
The goal of the Math Center (sometimes called the Math Lab) is to assist students
in the successful completion of any Allan Hancock College mathematics class by providing
additional instructional resources. The Math Center offers many resources, including
one-on-one, drop-in tutoring by our staff of instructors and student tutors. Please
see the full list of resources below:
Handouts on math topics, including content from various math courses as well as information
on overcoming math anxiety and preparing for and taking math tests
Two private study rooms
Make-up testing
Joining the math center group
Current students may access more detailed information by entering their myHancock
portal and joining the Math Center Group. Details may include information such as
the current schedule of instructors and student tutors who work in the Math Center, helpful
handouts on math topics, upcoming workshops, etc.
To join the Math Center Group:
Enter myHancock
Look at the center of the Home page in the box titled "My Groups." Click on "View
All Groups" at the bottom of the box.
Click on "Groups Index"
Click on "Departments"
Join the Math Center Group
LOCATION
Building M, Room 101
TELEPHONE
Telephone: 805-922-6966 x3463
HOURS
Fall and Spring Semesters:
Monday-Thursday
9 a.m. to 8 p.m.
Friday:
9 a.m. to 2 p.m.
Saturday:
11 a.m. to 3 p.m.
Summer Session:
Monday:
10 a.m. to 4 p.m.
Tuesday:
10 a.m. to 5:30 p.m
Wednesday:
10 a.m. to 5:30 p.m.
Thursday:
10 a.m. to 4 p.m.
Friday:
10 a.m. to 2 p.m.
STAFF
Achieve success at the math center | 677.169 | 1 |
Note that this
section contains some general tips on making the most out of your
homework. The next section contains tips
on actually working homework problems.
Understand the Purpose of Homework. Instructors do not give you homework
assignments to make your life miserable (well some might, but most don't!). Homework assignments are given to help
you to learn the material in the class and to develop good reasoning and
problems solving skills.
Mathematics is just not a subject that most people will instantly
understand every single topic after hearing the instructors
lecture. Most people need to work
on some problems in order to really start to understand the topic. That is the point of the homework. It gives you a set of problems that will
help you to understand the topics.
Remember that it almost always seems easier to watch an instructor doing
problems on the board than it actually is.
You won't know if you truly understand the material and can do the
problem if you don't attempt the homework.
Have The Actual Assignment. This may seem like a silly tip, but
make sure that you accurately write down the assignment and due date. This is one of the more common mistakes
that students make with homework.
Do the Assignment Promptly. You should always do the assignment as
soon after the lecture as possible while the lecture is still fresh in
your mind. Do not wait until the
last minute to do the whole assignment when comments made by the
instructor are no longer fresh in your mind.
Often these little asides that an instructor makes won't seem
important at the time, but when it comes time to doing the homework the
reason for making the comment will become clear.
Be Organized. When you start working on homework make
sure that you've got all the materials that you'll need to do the homework
such as notes and textbook.
Review. Go back over the lecture for each
section and review any examples that the instructor worked to make sure
that you understand the ideas from that section. Make note of any common errors that your
instructor may have mentioned. Do
the same with the text book. Read
the section and note examples worked and common errors mentioned in the
text book.
One of the biggest roadblocks in doing homework that I've seen in many
students is that they completely ignore the notes and/or text. They look at a problem and if they can't
see how to do it they give up and go to the next problem. Often there will be a similar problem in
the notes and/or text that can help you to get started!
Read/Follow the Directions. Make sure that you read and follow all
the directions for both the homework set and the individual problems.
Be Neat. Make sure that you write neatly. This will help the instructor as he/she
is grading the assignment and you when you are going over the assignment
in preparation for an exam.
Show All Work. Make sure that you show all of your
work. Do not just give the answer. Many instructors will not accept
homework that consists only of answers and no work. Also, do not skip large chunks of the
work. Instructors aren't mind
readers and so they won't know how you got from Step 1 to Step 3 unless
you also show Step 2. This is also
important if you made a mistake in Step 2.
If the instructor can't determine how you got from Step 1 to Step 3
you're liable to lose far more points than you would have lost for the
error in Step 2 had you shown it.
Showing your work will also help you when you are reviewing for the exam.
Check Your Work. Always go back over your work and make
sure that you've not made any simple arithmetic/sign errors. | 677.169 | 1 |
Essential Matlab for Engineers and Scientists essential guide to MATLAB as a problem solving tool - This text presents MATLAB both as a mathematical tool and a programming language, giving a concise and easy to master introduction to its potential and power. The fundamentals of MATLAB are illustrated throughout with many examples from a wide range of familiar scientific and engineering areas, as well as from everyday life. The new edition has been updated to include coverage of Symbolic Math and SIMULINK. It also adds new examples and applications, and uses the most recent release of Matlab. * New chapters on Symbolic Math and SIMULINK provide complete coverage of all the functions available in the student edition of Matlab. * New: more exercises and examples, including new examples of beam bending, flow over an airfoil, and other physics-based problems * New: A bibliography provides sources for the engineering problems and examples discussed in the text * A chapter on algorithm development and program design · Common errors and pitfalls highlighted * Extensive teacher support on solutions manual, extra problems, multiple choice questions, PowerPoint slides * Companion website for students providing M-files used within the book. | 677.169 | 1 |
We are currently experiencing an issue on the site. You may not find the product that you are looking for. Please check back tomorrow. Please note that we are working to correct this issue and apologize for the inconvenience.
Web Codes
These authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan's Precalculus: Enhanced with Graphing Utilities gives students a model for success in mathematics | 677.169 | 1 |
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Using Real-World Data in Your Class
Carlo Barbieri
Learn how both Mathematica and Wolfram|Alpha can investigate extensive data about the world. This Wolfram Technologies for STEM Education: Virtual Conference for Education talk demonstrates how using real-world data in the class can be used effectively with Wolfram technologies.
An electrical and computer engineering graduate student researcher shares his insights from academics and industry about how he uses Mathematica and the Wolfram Language, and how it compares to Matlab. | 677.169 | 1 |
Mathematics
Preparation for Success
High school students interested in a major in Mathematics should take four years of high school mathematics including a year of mathematics their senior year. Those planning to take a math course in college should also take a full year of mathematics as a senior. Taking four years of math is highly recommended for all high school students. | 677.169 | 1 |
Geocadabra is dynamic geometry software that supports students learning 2D and 3D geometry, functions and curves (with analysis), and probability.
The software was developed in Holland, and is avaiThis is a quiz on two-step algebraic equations with a focus on algebraic expressions. There are 15 multiple-choice questions with built in answer checking, and some helpful hints from the aliens, miss...A function machine with two operations, and options to hide or reveal numbers and set all operations and numbers. A notepad is available for recording inputs and outputs, and a loop function takes the... More: lessons, discussions, ratings, reviews,...
Solve a simple addition equation and click the circle containing the answer. The trick: the circles move. Four different difficulty levels refer to the speed of the circles rather than the complexity ... More: lessons, discussions, ratings, reviews,...
Using this virtual manipulative you may: graph a function; trace a point along the graph; dynamically vary function parameters; change the range of values displayed in the graph; graph multiple functi... More: lessons, discussions, ratings, reviews,...
Investigate the famous Locker problem. Invent your own version! This applet has a feature that allows students to investigate some of their own versions of the locker problem. For example, instead of ... | 677.169 | 1 |
Business mathematics is mathematics used by commercial enterprises to record and manage business operations. Commercial organizations use mathematics in accounting, inventory management, marketing, sales forecasting, and financial analysis. Mathematics typically used in commerce includes elementary arithmetic, elementary algebra, statistics and probability. Business management can be made more effective in some cases by use of more advanced mathematics such as calculus, matrix algebra and linear programming. In academia, "Business Mathematics" includes mathematics courses taken at an undergraduate level by business students. These courses are slightly less difficult and do not always go into the same depth as other mathematics courses for people majoring in mathematics or science fields. The two most common math courses taken in this form are Business Calculus and Business Statistics. Examples used for problems in these courses are usually real-life problems from the business world.
Importance of business mathematics in the field of business:
There are many factors involved in starting and operating a business. This is true regardless of the size of the business and regardless of whether it operates form a local location or worldwide on the Internet. Any business owner should have some training in various areas such as business math or make sure that that the right people are hired that have the required training.
The importance of business math can't be overstated. It is not only necessary to keep good records of sales and expenditures so the owner knows where the business stands, but governments at all level require that proper tax records be kept. Records must be kept of sales taxes collected and owed, unemployment and social security taxes paid out and of course the income of the business. It is not necessary to have someone with a degree in accounting to handle these matters although that may be desirable for larger businesses.
Good training in business math can be obtained in a variety of ways. Most university business degree programs have business math courses as part of the requirements as do business schools that offer associate degrees. There are also good courses available from different sources on the Internet. Many of these courses are general business math courses, but there are also specific courses available for individual
Whatever the source, all businesses require someone that is thoroughly familiar with business math. This is necessary for the protection of the owner, the business, and even the employees. However, getting the training needed for someone does not have to be expensive or overly time consuming.
Some major importance of business mathematics in business:
Mathematics have been one of the primary elements of business. Since have to take advantage of every opportunity for profit, making frequent statistical market analyses a necessity.
Money Transactions:
The four basic math operations are essential to understand transactions and calculate profits and losses. On every transaction, from paying your grocery bill to making an investment, a certain amount of money is removed from one budget to another. Hence, a transaction always includes a subtraction from the buyer's budget and an addition to seller's budget. In cases of mass payments, such as during a monthly payday or when people buy tickets for a concert, you can multiply the value of individual tickets or paychecks by the number...
YOU MAY ALSO FIND THESE DOCUMENTS HELPFULAssignment questions:
Specification of Assessment
LO1 (Task 01): Understand the organisational purposes of business
1.1
Identify the purposes of different types of organisation
1.2
Describe the extent to which Siemens meets the objectives of different stakeholders
D1
Critically evaluate the responsibilities of Siemens to meet stakeholders objectives.
1.3
Explain the responsibilities of Siemens and strategies employed to meet stakeholders.
M1
Identify and...
...Business mathematics is mathematics used by commercial enterprises to record and manage business operations. Commercial organizations use mathematics inaccounting, inventory management, marketing, sales forecasting, and financial analysis. Mathematics typically used in commerce includes elementary arithmetic,elementary algebra, statistics and probability. Business management can be made more effective in some cases by use of more advanced mathematics...cash on hand or any tangible or intangible item that can be converted quickly and easily into cash, typically within 20 days, without losing much of their value. These assets are among the most basic types of financial resources used by consumers, business and investors. Cash and checking accounts are the two most obvious forms of liquid assets.
Currency:
Legal tender for purchases and to settle outstanding debts, currency remains the most common type of liquid asset used...
...
Assignment
On
Business Environment
Contents
Introduction
Business environment refers to the combination of external and internal factors influence the organization of the operating conditions. There are several factors of the customer and the business environment, such as suppliers, technology, law and government activity and the market, social and economic trends competitions and owner, is improved....
...
Lecture one:
What is international business?
What is globalisation?
Is domestic business immune to the forces of globalisation?
What are the causes/factors facilitating the growth of international business?
What are the indicators of the growth of IB?
How internationally integrated have economies become?
Who gains, who loses from globalisation
Outline
1. Globalisation – nature and factors
What is Globalisation?
Globalisation refers to...
...Chapter 5
Interest Rates
5-1. Your bank is offering you an account that will pay 20% interest in total for a two-year deposit. Determine the equivalent discount rate for a period length of
a. Six months.
b. One year.
c. One month.
a. Since 6 months is [pic] of 2 years, using our rule [pic]
So the equivalent 6 month rate is 4.66%.
b. Since one year is half of 2 years [pic]
So the equivalent 1 year rate is 9.54%....
3371 Words |
16 Pages
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offers most accurate, extensive and varied set of assessment questions of any course management program in addition to all questions including some form of question assistance including answer specific feedback to facilitate success. The text also offers multimedia presentations (videos and animations) of much of the material that provide an alternative pathway through the material for those who struggle with reading scientific exposition. Furthermore, the book includes math review content in both a self-study module for more in-depth review and also in just-in-time math videos for a quick refresher on a specific topic. | 677.169 | 1 |
comprehensive self-paced student tutorial on the topic of dimensional analysis. It is organized into seven sections, each containing a quiz (answers included). The tutorial initiates with an introduction to physical quantities and SI units, then progresses into dimensional analysis and its use to determine if an equation is dimensionally correct. A full post-test is included to gauge student understanding of the topic.
This is part of series of tutorials on physics and mathematics used in physics classes.
Disclaimer: ComPADRE offers citation styles as a guide only. We cannot offer interpretations about citations as this is an automated procedure. Please refer to the style manuals in the Citation Source Information area for clarifications. | 677.169 | 1 |
Book Description Brooks/Cole Pub Co, 2003. Book Condition: New. Brand New, Unread Copy in Perfect Condition. A+ Customer Service! Summary: Provides completely worked-out solutions to all odd-numbered exercises within the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer. Bookseller Inventory # ABE_book_new_0534393330 | 677.169 | 1 |
Find a Dania Pre-Algebra is a basic skills course which includes factoring and set theory. | 677.169 | 1 |
Find a BaytownThe Socratic Method is a "guided question" methodology that allows students to see how a problem is solved rather then just mindlessly applying a formula without a conceptual understanding of the problem. A lot of students have difficulties with math and the math found in other subjects such as physics. Math is an interesting subject with a myriad of techniques for finding an answer. | 677.169 | 1 |
Practice with graphing lines in the coordinate plane is included. A Pre - Calculus course begins with a rigorous review of Algebra II. These topics include the properties of the Real Number System, the more difficult factoring problems, and solutions of equations and inequalities | 677.169 | 1 |
Seeing Structure in Expressions HSA-SSE.B.3
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Your students should be gaining fluency in mathematics, and be able to write and rewrite expressions. The spells of nausea, hyperventilation, and paranoia surrounding the beautiful language of mathematics should have subsided by now. If they haven't, you may want to consult the school nurse.
Rather than rewriting expressions for the fun of it (and it is fun, isn't it?), students should understand what these different expressions can tell us about the quantities they represent. These mathematical expressions can tell us the zeros (or roots or x-intercepts) and the maximum and minimum values of a function, and have plenty of other applications in the real world. Most of them involving money. | 677.169 | 1 |
One of the 18th century's greatest mathematicians, Lagrange made significant contributions to analysis and number theory. He delivered these lectures on arithmetic, algebra, and geometry at the École Normale, a training school for teachers. An exemplar among elementary expositions, they feature both originality of thought and elegance of expression series of lectures given by Lagrange at the École Normale in 1795 complete with a biographical sketch by Thomas McCormack. The lectures cover arithmetic, algebra, equation solving and geometry among other areas. At the time they were considered among the finest introductory materials and I can see why. Even translated Lagrange covers each apparently elementary topic with clarity and there is a depth you may not have expected; almost immediately offering alternatives to the usual methods in which to multiply and divide, for instance.
What is interesting to note are the differences between this and a modern textbook. Modern textbooks are great but here you get mote depth than the usual elementary treatment and a few interesting methods that you may not have previously encountered or employed. As an accompaniment to existing textbooks I would recommend this to an enthusiast or serious gcse and A level student aspiring to further study. It is 5 stars in its time, the only reason for 'losing' a star is in terms of layout and examples - but of course these were lectures not the lessons & practice associated with a modern text so perhaps it is a little unfair to make those comparisons.
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07645449CliffsQuickReview Math Word Problems gives you a clear, concise, easy-to-use review of the basics of solving math word problems. Introducing each topic, defining key terms, and carefully walking you through each sample problem gives you insight and understanding to solving math word problems. You begin by building a strong foundation in translating expressions, inserting parentheses, and simplifying expressions. On top of that base, you can build your skills for solving word problems:
Discover the six basic steps for solving word problems
Translate English-language statements into equations and then solve them
Solve geometry problems involving single and multiple shapes
Work on proportion and percent problems
Solve summation problems by using the Board Method
Use tried-and-true methods to solve problems about money, investments, mixtures, and distance
CliffsQuickReview Math Word Problems acts as a supplement to your textbook and to classroom lectures. Use this reference in any way that fits your personal style for study and review you decide what works best with your needs. Here are just a few ways you can search for information:
View the chapter on common errors and how to avoid them
Get a glimpse of what you ll gain from a chapter by reading through the Chapter Check-In at the beginning of each chapter
Use the Chapter Checkout at the end of each chapter to gauge your grasp of the important information you need to know
Test your knowledge more completely in the CQR Review and look for additional sources of information in the CQR Resource Center
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With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades | 677.169 | 1 |
Find a Kennesaw ACTThis algebra deals mostly with linear functions. Algebra 2 is a more advanced, more complex version of algebra 1. Here we get more involved with non-linear functions as well as imaginary and complex numbers | 677.169 | 1 |
Details about Geometry:
This Geometry workbook makes the fundamental concepts of geometry accessible and interesting for college students and incorporates a variety of basic algebra skills in order to show the connection between Geometry and Algebra. Topics include: A Brief History of Geometry 1. Basic Geometry Concepts 2. More about Angles 3. Triangles 4. More about Triangles: Similarity and Congruence 5. Quadrilaterals 6. Polygons 7. Area and Perimeter 8. Circles 9. Volume and Surface Area 10. Basic Trigonometry
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Rent Geometry 1st edition today, or search our site for other textbooks by Alan Bass. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our Geometry tutors now. | 677.169 | 1 |
Details about Mathematics for the Health Sciences:
Students will learn basic math skills, the use of measurement systems, and strategies of problem solving needed in health science courses. This text is designed for active learning--students are asked to answer questions that follow the introduction of each new topic. Students can compare their responses with the answers provided in the margins to know if they are ready to go on to the next subsection. Exercise sets and self-tests, with their answers, are also provided. Proportions are used extensively; dimensional analysis is emphasized.
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Rent Mathematics for the Health Sciences 1st edition today, or search our site for other textbooks by Keith Roberts. Every textbook comes with a 21-day "Any Reason" guarantee. Published by CENGAGE Learning. | 677.169 | 1 |
Computational science is an exciting new field at the intersection of the sciences, computer science, and mathematics because much scientific investigation now involves computing as well as theory and experiment. This textbook provides students with a versatile and accessible introduction to the subject. It assumes only a background in high school algebra, enables instructors to follow tailored pathways through the material, and is the only textbook of its kind designed specifically for an introductory course in the computational science and engineering curriculum. While the text itself is generic, an accompanying website offers tutorials and files in a variety of software packages.
This fully updated and expanded edition features two new chapters on agent-based simulations and modeling with matrices, ten new project modules, and an additional module on diffusion. Besides increased treatment of high-performance computing and its applications, the book also includes additional quick review questions with answers, exercises, and individual and team projects.
The only introductory textbook of its kind--now fully updated and expanded
Features two new chapters on agent-based simulations and modeling with matrices first edition of this book had received very positive feedback and was warmly welcomed by the mathematical community. It is very good news for all us is that the second revised edition is even better!"--Svitlana P. Rogovchenko, Zentralblatt MATH
Praise for the previous edition: "The heart of Introduction to Computational Science is a collection of modules. Each module is either a discussion of a general computational issue or an investigation of an application. . . . [This book] has been carefully and thoughtfully written with students clearly in mind."--William J. Satzer, MAA Reviews
Praise for the previous edition: "Introduction to Computational Science is useful for students and others who want to obtain some of the basic skills of the field. Its impressive collection of projects allows readers to quickly enjoy the power of modern computing as an essential tool in building scientific understanding."--Wouter van Joolingen, Physics Today
Praise for the previous edition: "A masterpiece. I know of nothing comparable. I give it five stars."--James M. Cargal, UMAP Journal
Praise for the previous edition: "This is an important book with a wonderful collection of examples, models, and references."--Robert M. Panoff, Shodor Education Foundation
Praise for the previous edition: "This is a very good introduction to the field of computational science."--Peter Turner, Clarkson University
"Despite its substantial weight, the book is extremely user friendly. . . . There are many different courses that one could build with this book as foundation, and it is an indispensible resource for anyone seeking to bring modeling projects into their classes."--David M. Bressoud, UMAP Journal
About the Author
Angela B. Shiflet is the Larry Hearn McCalla Professor of Mathematics and Computer Science and director of computational science at Wofford College. George W. Shiflet is the Larry Hearn McCalla Professor of Biology at Wofford College.
I found the book very useful in its conceptualization of simulation as a new form of synthesis for acquiring knowledge and helping human being make decisions. Simulation is a form of communication in that empirically-based models could be used to view the on-going processes that are cognitively beyond the capacity of human mind to untangle.
Furthermore, I found it admirable that the authors had instructed the readers in the art of model building using widely available simulation tools or even tools such as MS Excel that are not built specifically for the purpose of simulations.
I especially liked the tutorials with their wide selection of interesting material.
Regrettably, the subject matter of the book - simulation & (dynamic model building) - does not fit well within the traditional physics curriculum: Mechanics, Electromagnetism, and Quantum Mechanics. Numerical methods, including simulations, are not emphasized in such courses and normally one spends much of one's time studying well-known and solvable (in closed, analytical form) problems.
That does not mean that there is no room in physics for modeling and simulation: fractals, galaxy formation, dynamics of globular clusters, etc. are all areas that we are dependent on our numerical models and their fitness to observed phenomena to understand the processes of Nature. However, these are usually advanced topics not covered in undergraduate curricula.
And then the physicists tend to want to build their own tools rather that use COTS packages.
I think it is difficult to find a home in traditional university departments for a course on simulation based on this book. The fundamental reason, in my opinion, is that the personal computer as a scientific instrument is not valued or appreciated. And from that follows the lack of interest in simulations as venues for gaining scientific knowledge in Physics, in Chemistry & Biology. Conceivably a course in simulations might be of interest to engineers but then we would be leaving all those "soft"-science majors such as Ecology or Public Health behind. And those soft-science students are among some the people who could benefit the most from this book.
For example, the gene<->protein<->enzyme interactions, with their feedback loops and multiple pathways, are so complex that no human mind could expect to grasp all that goes on inside a cell. So Module 6.3 covering enzyme dynamics is absolutely on the right track from a scientific perspective; taking simulation out of the "hard" science world and into biology. Only through simulations and modeling are we going to develop a synthetic understanding of the cell in all its complexity.
I do not know if you have seen the book, "Historical Dynamics" by Peter Turchin in which he presents and discusses mathematical models of the evolution of agrarian states on the Eurasian land mass. His models are informed by empirical data collected from historical sources and do capture many aspects of historical reality. There is clearly a very important paradigm here at work but which is not as enunciated as I believe it should be; namely that simulations extend our scientific evidentiary-based knowledge into hitherto for dark realms of empirical experience.
Yet, Turchin did not use a simulation tool; he rolled his own and wrote much of his code in APL (A Programming Language) which is quite obscure. So readers must redo the models themselves. And his book is not about simulations, it is about what simulations tell us about history. I think that there are many fields of study in which the students could benefit from parts of this book; history, sociology, ecology, and natural resources comes to mind. There the exposition must be based heavily on using Commercial Off-the-Shelf (COTS) packages to teach the students how to build useful models.
So, for students in "soft" departments, the material in modules 5.2, 5.3, 5.4 could be skipped. In fact, even for students of physics and engineering, the understanding of the basis of the numerical simulations is not as important as learning how to build a system and observing how it behaves. On the other hand, the module 2, in my opinion, is important for all students to understand and to master so that they may interpret the results of their simulations correctly.
In my own case, I would love to be able to use a COTS package in which I could put galaxies - using a visual palette tool - on a 3-dimensional grid and observe their evolution in time as I changed the metric of space-time and/or the equation of state of the matter field.
Or consider the equations of stellar evolution, one would love to be able to run them again and again by changing parameters of these models knowing that the fundamental equations and their integration were worked out 70 years ago. As it is today, there are no such user-friendly generic approaches available to students or researchers, all such things must be painfully hand-crafted almost from scratch, barring some software libraries.
On a few occasions that I imagined writing such a book myself, I realized how difficult it was to do justice to the breadth and depth of the field of simulations from its hard-core physical scieces and engineering to ecology and wild-life management in a single book. While I might quibble with the inclusion of this or that topic or technique, I really cannot come up with a better design. There is an enormous amount of material here that may or may not be of interest to all audiences but there is a lot of material that is of interest to special audiences with focus on this or that scientific field.
The authors, if I understand them correctly, are positing that simulations (Computational Sciences) as a new way of knowledge discovery. In this they are right, in my opinion and are in the company of such luminaries as Dr. Steven Wolfram of Mathematic fame. But the problem is that "Introduction to Simulations" used to be taught in Industrial Engineering departments and then moved to places like the RAND Corporation and then the Pentagon. It is not viewed as a subject worthy of study in its own right (although such simulations as Halo or the World of War Craft sell millions of copies and make a few people quite wealthy).
Which brings me to another topic related to simulations and to this book; namely computer games. Computer games are also simulations but with the caveat that they do not follow the Laws of Physics, Chemistry, and Biology and therefore are on somewhat of a tangent to "scientific" simulations such as those covered in this book. Gamers perform simulations, cosmologists do simulations, climatologists do simulations, agronomists do simulations, and many others but there is not set curriculum, no defined or unified approach and indeed no place to go to get an introduction to the art and science of simulation in a typical undergraduate college. This book is an attempt at just that.
I think non-science majors in Liberal Arts colleges could benefit also from a simulation course based on this book. Some colleges have a unified natural science department without the traditional divisions among Physics, Chemistry, Mathematics, and Biology. In such an environment, a course based on this book may offer the non-science majors - a la Dr. Turchin's approach - the prospect of gaining confidence in building quantitative models of Reality and making inferences about the world on basis of such models.
This is a good book and a path-breaking book and I hope it finds the traction that it so righty deserves.
This book has the potential to be great but not as is. It is kind of a compilation of other authors work right now. Also, the authors leave too much to the reader. Basically the book needs to be completed. The tutorials need to have FULL solutions. Also the works need to be original not just take a bunch of stuff from my differential equations book and others and slap it in there and site it...But, I would definitely recommend the authors devote everything to this and not to stop because it could become the future textbook for computational science
If you have a solid background in math (calculus, differential equations), physics, or computer science, you will probably find this book to be very elementary, and you will likely hate the approach it uses (visual methods for understanding differential equations, annoying pseudo code, etc). It only touches on the subjects it broaches, with terribly simple and poorly articulated exercises, and "projects" that in general should be exercises (thus there aren't really "projects"). This book was used for a graduate level course I took, and in my opinion is barely suitable for an undergraduate course. | 677.169 | 1 |
Burbank, CA Algebra 2Introduction to Probability and Statistics We will master the following topics:
1. Operating with real numbers
2. Solving word problems
3 | 677.169 | 1 |
To succeed in Algebra II, start practicing now Algebra II builds on your Algebra I skills to prepare you for trigonometry, calculus, and a of myriad STEM topics. Working through practice problems helps students better ingest and retain lesson content, creating a solid foundation to build on for future success. Algebra II Workbook For Dummies,...The CliffsStudySolver workbooks combine 20 percent review material with 80 percent practice problems (and the answers!) to help make your lessons stick. CliffsStudySolver Algebra I is for students who want to reinforce their knowledge with a learn-by-doing approach. Inside, you'll get the practice you need to tackle numbers and operations with... more...
From angles to functions to identities - solve trig equations with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear - this hands-on-guide focuses on helping you solve the many types of trigonometry equations you encounter in a focused, step-by-step... more...
Covers percentages, probability, proportions, and more Get a grip on all types of word problems by applying them to real life Are you mystified by math word problems? This easy-to-understand guide shows you how to conquer these tricky questions with a step-by-step plan for finding the right solution each and every time, no matter the kind or... more...
With its use of multiple variables, functions, and formulas algebra can be confusing and overwhelming to learn and easy to forget. Perfect for students who need to review or reference critical concepts, Algebra I Essentials For Dummies provides content focused on key topics only, with discrete explanations of critical concepts taught in a typical... more...
Passing grades in two years of algebra courses are required for high school graduation. Algebra II Essentials For Dummies covers key ideas from typical second-year Algebra coursework to help students get up to speed. Free of ramp-up material, Algebra II Essentials For Dummies sticks to the point, with content focused on key topics only. It provides... more...
From signed numbers to story problems — calculate equations with ease Practice... more... | 677.169 | 1 |
I came to a career in software development with a degree in English, rather than Computer Science or another science/engineering background. I have gone a long way on my self-taught basis, but after 10+ years of doing this, I want to go back and fill in the gaps, particularly with the math.
The obvious place to give myself a Comp-Sci education is to go through The Art of Computer Programming. However, as I didn't take all that much math and my last math class in college was in 1995, I need some brushing up and augmenting to even be able to read the math notation in TAOCP.
My thought was to go to Khan Academy and work through the necessary topics as a remedial prereq to reading TAOCP. However, in a Catch 22, I'm trying to figure out which topics do I actually need to go through as prep.
So, what I'm wondering is, if someone basically only had high school math (I've got a bit more than that, but I think it's a valid question for someone to approach this with just high school as a background), what math "classes" does one need from somewhere like Khan Academy in order to start TAOCP prepared to read and understand the included math?
Echoing the others, a discrete mathematics class is what to aim for. One of the strengths of Knuth's books is the extensive algorithm analysis in the text and in the exercises. A undergraduate sequence in calculus will be needed to understand some of the analysis. And "Seminumerical Algorithms" would be best appreciated I think with a undergraduate number theory course. Plus number theory is fun in its own right! | 677.169 | 1 |
Thinking Mathematically / Edition 3
Overview experience of mathematical thinking processes.
Most Helpful Customer Reviews
Thinking Mathematically is a book that has no answers...there is no way to look at the answers. It teaches how to solve problems, how to apply this to life, and change the problem into a learning experience. It is packed with many puzzles and variations - enjoy! | 677.169 | 1 |
Factor fearlessly, conquer the quadratic formula, and solve linear equations. There's no doubt that algebra can be easy to some while extremely challenging to others. If you're vexed by variables, Algebra I For Dummies, 2nd Edition provides the plain-English, easy-to-follow guidance you need to get the right solution every time! Now with 25% new and revised content, this easy-to-understand reference not only explains algebra in terms you can understand, but it also gives you the necessary tools to solve complex problems with confidence. You'll understand how to factor fearlessly, conquer the quadratic formula, and solve linear equations.
Calculus For Dummies, 2nd Edition makes calculus manageable—even if you're one of the many students who sweat at the thought of it. By breaking down differentiation and integration into digestible concepts, this guide helps you build a stronger foundation with a solid understanding of the big ideas at work. This user-friendly math book leads you step-by-step through each concept, operation, and solution, explaining the "how" and "why" in plain English instead of math-speak. Through relevant instruction and practical examples, you'll soon learn that real-life calculus isn't nearly the monster it's made out to be.
Do you need to get up and running on bookkeeping basics and the latest tools and technology used in the field? You've come to the right place! Bookkeeping All-In-One For Dummies is your go-to guide for all things bookkeeping, covering everything from learning to keep track of transactions, unraveling up-to-date tax information recognizing your assets, and wrapping up your quarter or your year. Bringing you accessible information on the new technologies and programs that develop with the art of bookkeeping, it cuts through confusing jargon and gives you friendly instruction you can put to use right away.
A beautiful landscape reflects well on your house, making it a welcome part of a neighborhood or native terrain. And it dramatically increases your home's value. Landscaping Basics For Dummies gets you started on turning the little patch of earth you call your own into a personal paradise.
Simplify your small business accounting with confidence! Managing the books for a small business can be a challenging task just ask any of the countless business owners and managers who have spent hour after hour hunched over multiple spreadsheets. QuickBooks 2016 All–In–One For Dummies takes the pain out of managing your small business′ finances through one essential reference., 2nd Edition helps you learn Algebra II by doing Algebra II. | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Strategies
and tools for problem solving, including computer use, will be applied
to specific problems from number theory, geometry, analytic geometry, algebra,
discrete mathematics, logic, and calculus.
OBJECTIVES:
The
course is intended to be a first graduate course in mathematics for students
in all of these programs.As such,
it will provide a common mathematical foundation for students in all of
the programs, drawing upon the full range of undergraduate courses in mathematics.Mathematical
connection will be emphasized in the course, allowing students to relate
topics studied separately to one another.Also
stressed will be mathematical reasoning and communication skills, as applied
to mathematics.This course will
permit students to build upon and share knowledge already acquired while
pointing out areas in which additional study may be needed.In
addition, it will develop the communication skills and understanding of
the process of doing mathematics necessary for graduate-level study. | 677.169 | 1 |
College Algebra
9780132402866
ISBN:
0132402866
Edition: 8 Pub Date: 2007 Publisher: Prentice Hall
Summary: The Eighth Edition of this highly dependable book retains its best features-accuracy, precision, depth, and abundant exercise sets-while substantially updating its content and pedagogy. Striving to teach mathematics as a way of life, Sullivan provides understandable, realistic applications that are consistent with the abilities of most readers. Chapter topics include Graphs; Polynomial and Rational Functions; Conics;... Systems of Equations and Inequalities; Exponential and Logarithmic Functions; Counting and Probability; and more. For individuals with an interest in learning algebra as it applies to their everyday lives.[read more]
Ships From:Multiple LocationsShipping:Standard, ExpeditedComments:ALTERNATE EDITION: hardcover teacher edition same as student edition but contains all answers and the isbn is differ... [more]ALTERNATE EDITION: hardcover teacher edition same as student edition but contains all answers and the isbn is different it may have highlights book only no other supplements will ship immediately [less] | 677.169 | 1 |
This ebook is available for the following devices:
iPad
Windows
Mac
Sony Reader
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iRiver Story
This lucid and accessible text provides an introductory guide to projective geometry, an area of mathematics concerned with the properties and invariants of geometric figures under projection. Including numerous worked examples and exercises throughout, the book covers axiomatic geometry, field planes and PG(r, F), coordinating a projective plane, non-Desarguesian planes, conics and quadrics in PG(3, F). Assuming familiarity with linear algebra, elementary group theory, partial differentiation and finite fields, as well as some elementary coordinate geometry, this text is ideal for 3rd and 4th year mathematics undergraduates. | 677.169 | 1 |
Details about College Algebra:
Accessible to students and flexible for Accessible to students and flexible for instructors, COLLEGE ALGEBRA, SEVENTH EDITION, uses the dynamic link between concepts and applications to bring mathematics to life. By incorporating interactive learning techniques, the Aufmann team helps students to better understand concepts, work independently, and obtain greater mathematical fluency. The text also includes technology features to accommodate courses that allow the option of using graphing calculators. Additional program components that support student success include tutorial practice, online homework, Live Online Tutoring, and Instructional DVDs. The authors' proven Aufmann Interactive Method allows students to try a skill as it is presented in example form. This interaction between the examples and Try Exercises serves as a checkpoint to students as they read the textbook, do their homework, or study a section. In the Seventh Edition, Review Notes are featured more prominently throughout the text to help students recognize the key prerequisite skills needed to understand new concepts. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
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Rent College Algebra 7th edition today, or search our site for other textbooks by Richard N. Aufmann. Every textbook comes with a 21-day "Any Reason" guarantee. Published by CENGAGE Learning. | 677.169 | 1 |
Algebra and Trigonometry with Modeling and VisualizationGary Rockswold teaches algebra in context, answering the question, "Why am I learning this?" By experiencing math through applications, students see how it fits into their lives, and they become motivated to succeed. Rockswoldrs"s focus on conceptual understanding helps students make connections between the concepts and as a result, students see the bigger picture of math and are prepared for future courses. Introduction to Functions and Graphs; Linear Functions and Equations; Quadratic Functions and Equations; More Nonlinear Functions and Equations; Exponential and Logarithmic Functions; Trigonometric Functions; Trigonometric Identities and Equations; Further Topics in Trigonometry; Systems of Equations and Inequalities; Conic Sections; Further Topics in Algebra For all readers interested in college algebra and trigonometry. | 677.169 | 1 |
This first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a Euclidean space of three dimensions. Written by an outstanding teacher and mathematician, it explains the material in the most effective way, using vector notation and technique. It also provides an introduction to the study of Riemannian geometry. Suitable for advanced undergraduates and graduate students, the text presupposes a knowledge of calculus. The first nine chapters focus on the theory, treating the basic properties of curves and surfaces, the mapping of surfaces, and the absolute geometry of a surface. The final chapter considers the applications of the theory to certain important classes of surfaces: surfaces of revolution, ruled surfaces, translation surfaces, and minimal surfaces. Nearly 200 problems appear throughout the text, offering ample reinforcement of every subject.
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Regrettably, I have to report that this book "Differential Geometry" by William Caspar Graustein is of little interest to the modern reader. I had hoped that it would throw some light on the state of differential geometry in the 1930s, but the modernity of this book is somewhere between Gauß and Darboux. Nevertheless, I'm grateful to Dover for keeping it in print. It's good to keep the old books alive.
Graustein only very peripherally acknowledges the work of Riemann, Christoffel, Levi-Civita and Weyl in two pages, almost at the end of the book. He gives a very brief mention of Weyl's idea of an affine connection, which generalizes the Riemannian metric.
You will find almost no modern differential geometry in this book. Even the idea of a "tangent vector" is introduced only as a unit vector tangent to a curve. There are no tensors, no affine connection or Riemannian metric (apart from a very brief allusion), no topology, no differential forms, etc. etc. It's just the geometry of surfaces embedded in Euclidean 3-space, in the 19th century idiom of Gauss and Darboux. However, I do not regret buying it. It's good to have a book which gives me the Gaussian differential geometry in a well-presented compact format. | 677.169 | 1 |
The program provides detailed, step-by-step solution in a tutorial-like format to the following problem: Given a 2x2 matrix, or a 3x3 matrix, or a 4x4 matrix, or a 5x5 matrix. Find its inverse matrix by using the Gauss-Jordan elimination method. The check of the solution is given. The program is designed for university students and professors. | 677.169 | 1 |
Daily Warm-Ups: Algebra, Common Core State StandardsOverviewMaterials include
Reproducible teacher book
More than 100 varied problems directly addressing CCSS
Includes CD-ROM with detailed correlations, student problems ideal for projecting within the classroom, and an answer key | 677.169 | 1 |
Editorial Reviews
This award-winning series helps students to build confidence in their ability to "do math" and develop the knowledge and skills required to become mathematically literate. Since Algebra I is the next step for a student who has successfully completed Basic Math, this last lesson acts as an introduction to that subject. Algebra is discussed as a generalization of arithmetic. The arithmetic of polynomials is related to the arithmetic of whole numbers. The vital nature of Algebra I is discussed and hope is expressed that the student will see Algebra I with a new awareness after completing Basic Math. Finally, a review is conducted of the important concepts of arithmetic learned in this series. Students will learn to relate algebra to arithmetic and to explain the importance of algebra. | 677.169 | 1 |
Product Description
This two-book teacher's guide accompanies ACSI Math Student Worktext Grade 4, and contains everything you need to teach either a homeschool or public/private school student. Two large binders contain three-ring-punched inserts. One binder contains blackline masters for all lessons, the other contains an entire replica of the student book in full color, and the teacher's guide. The Teacher's guide contains objectives, lists of materials needed, an introduction, directed step-by-step instruction, sidebar notes on the lesson objective, and suggestions for reinforcement. Instructions include illustrations, and student questions are reproduced with the correct answers overlaid. 375 page teacher's guide. A Packet is included to update this teacher's guide for the 2007 updated ACSI Math 4 course | 677.169 | 1 |
Simplex method
Simplex method, Standard technique in linear programming for solving an optimization problem, typically one involving a function and several constraints expressed as inequalities. The inequalities define a polygonal region (seepolygon), and the solution is typically at one of the vertices. The simplex method is a systematic procedure for testing the vertices as possible solutions.
collection of mathematical principles and methods used for solving quantitative problems in many disciplines, including physics, biology, engineering, economics, and business. The subject grew from a realization that quantitative problems in manifestly different disciplines have important...
in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The | 677.169 | 1 |
Product Description
The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, students will learn the effect of the numerical coefficient a and coefficient b along with the use of parallel and perpendicular lines in graphical re presentations. Grades 5-9. 30 minutes on DVD.
DVD Playable in Bermuda, Canada, United States and U.S. territories. Please check if your equipment can play DVDs coded for this region. Learn more about DVDs and Videos | 677.169 | 1 |
Algebra is a branch of mathematics that uses letters or other symbols to represent unknown quantities, called variables. These variables and number values are combined to form equations.
The rules of these equations follow the exact same rules as arithmetic, such as the commutative and associative laws for addition and multiplication.
Functions are a special type of equation, where one variable can be uniquely defined in terms of the other.
Another part of this topic is graphing of equations and functions using the Cartesian coordinate graph or polar coordinates.
Also, covered in this topic is set theory or what constitutes a grouping of numbers.
Sites21
Contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. Intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
"It is the best text of its type that I have come across to date...an excellent resource for anyone involved in mathematical practice up to and including degree standard...it is a pleasant experience to flick through it at leisure, dropping in here and there on some of the thousands of results the book holds. It is beautifully illustrated and set out...we have here a comprehensive volume whose worth will not readily fade with time...If you feel the need to own a mathematical reference to see you through school and university mathematics to graduation, you couldn't do much better than to buy this one." --Mathematics Today A complete desk-top reference for working scientists, engineers, and students, this handbook serves as a veritable math toolbox for rapid access to a wealth of mathematics information for everyday use in problem solving, examinations, homework, etc. Compiled by professional scientists, engineers, and lecturers and internationally renowned for its clarity and completeness, The Handbook includes hundreds of tables of frequently used functions, formulae, transformations, and series, plus many applications. The layout, structured table of contents, and index make finding the relevant information quick and painless.
Synopsis
Mathematicians, students and researchers constantly refer to various sources for different formulas, equations, etc. "The Handbook of Mathematics and Computational Science" puts up-to-date equations, formulas, tables, illustrations, and explanations in one invaluable reference volume. Fully up-to-date, this handbook will quickly become the standard reference for mathematicians and students. 545 illus.
Synopsis
A reference handbook for modern mathematics. The Handbook of MathematA reference handbook for modern mathematics. The Handbook ofMathemat | 677.169 | 1 |
A new ANGLE to learning GEOMETRY Trying to understand geometry but feel like you're stuck in another dimension? Here's your solution. Geometry Demystified , Second Edition helps you grasp the essential concepts with ease. Written in a step-by-step format, this practical guide begins with two dimensions, reviewing points, lines, angles, and distances,... more...
The word barycentric is derived from the Greek word barys (heavy), and refers to center of gravity. Barycentric calculus is a method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. Hence, in particular, barycentric calculus provides excellent insight into triangle centers.... more...
The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies.... more... | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
The third edition of Cynthia Young's College Algebra brings together all the elements that have allowed instructors and learners to successfully "bridge the gap" between classroom instruction and independent homework by overcoming common learning barriers and building confidence in students' ability to do mathematics. Written in a clear, single voice that speaks to students and mirrors how instructors communicate in lecture, Young's hallmark pedagogy enables students to become independent, successful learners. Varied exercise types and modeling projects keep the learning fresh and motivating. Young continues her tradition of fostering a love for succeeding in mathematics by introducing inquiry-based learning projects in this edition, providing learners an opportunity to master the material with more freedom while reinforcing mathematical skills and intuition.
The seamless integration of Cynthia Young's College Algebra 3rd edition with WileyPLUS, a research-based, online environment for effective teaching and learning, continues Young's vision of building student confidence in mathematics because it takes the guesswork out of studying by providing them with a clear roadmap: what to do, how to do it, and whether they did it right.
WileyPLUS sold separately from text.
Synopsis
Cynthia Young's 3rd Edition of College Algebra focuses on revisions and additions including hundreds of new exercises, more opportunities to use technology, and themed modeling projects that help connect content to real-world issues. The text builds upon the previous two editions with more in-depth and enhanced coverage on ways to help overcome common learning barriers in algebra and build confidence for readers. The text features truly unique, strong pedagogy and as with the previous two issues, is written in a clear, single voice. | 677.169 | 1 |
Lesson Starters for Algebra for Common Core State Standards : A PowerPoint Teaching Tool
Description
Having trouble engaging your algebra students? Lesson Starters for Algebra for Common Core State Standards is a set of PowerPoint slides that highlights student-centered situations to teach algebra. Comes on a CD-ROM containing PowerPoint slides Includes complete teacher notes for each slide Complements the Glencoe Algebra I textbook or nhances any Algebra I program Rigorous enough to require true problem solving and accessible enough to allow all students to progress toward a solution Correlated to Common Core State Standards | 677.169 | 1 |
Algebra Helper 1
Algebra Helper is a great tool for any students from upper elementary to college. It provides concise flip cards to summarize important algebraic formulas so you don't need to worry about the formula you always forget. Algebra Helper includes a powerful tool, Quadratic equation solve, which has amazing capabilities to solve any quadratic equations, even with complex roots. Scratch Pad is also included so you would no longer need to scramble for paper and pencil while doing your math. In additional to solve math problems, Algebra Helper also provides a sophisticated organizing tool. Of course, you are going to have some fun with voice recognition toy so that you and your friends can have some fun while doing your homeworkLion Lite
MathLion Lite allows you to ditch the pencil and paper and solve algebra problems on your Android device. Avoid all the simple mistakes and focus on arriving at the solution by rearranging the equations by touch. Here are some of the things you can do with MathLion Lite.
MathLion Lite Features
- Combine terms (add, subtract, multiply, and divide)…Algebra Helper 2+
Algebra Helper 2+ is a comprehensive Algebra reference book and a collection of necessary tools for algebra learning and teaching. Algebra Helper 2+ can be used by any students, parents and teachers.
Algebra Goals and Examples section is a wonderful and necessary resource for any teacher, parents and students at school and at home. Goals and Exam…
Algebra Tests & Solutions
• Algebra Tests and Solutions is an incredible algebra learning, testing and teaching resources for any parents, teachers and students.
• Algebra Tests and Solutions is a comprehensive Algebra reference book and a collection of necessary tools for algebra learning and teaching. Algebra Tests and Solutions can be used by any students, parents and…
Algebra Helper 1+
Algebra Helper 1+ is a comprehensive Algebra reference book and a collection of necessary tools for algebra learning and teaching. Algebra Helper 1 can be used by any students from elementary school to college.
Algebra Goals and Examples section is a wonderful and necessary resource for any teacher, parents and students at school and at home. GoalMATEK - Math Camera Solver
MATEK is an innovative STEM application that teaches and helps you solve simple or complex mathematical equations by snapping a picture of a written or printed equation or just by handwriting it on your touchscreen.
Providing a step-by-step human logic explanation of an equation, MATEK promotes a can-do attitude towards solving mathematics by offe… | 677.169 | 1 |
After a few initial chapters on the basics of Mathematica, the logic of this book is controlled by group theory.
The book has three major parts. Part I begins with the most elementary symmetry concepts, showing how to express them in terms of matrices and permutations, before moving on to the construction of mathematical groups. In Part II, mathematical group theory is presented with motivating questions and experiments first, and theorems that answer those questions second. In Part III, the projection operators that flow from the Great Orthogonality are automated and applied to chemical and spectroscopic problems, which are now seen to fall within a unified intellectual framework.
Intended for students of chemistry and molecular physics, the book may be read either independently or on a computer screen with Mathematica running behind it. The included CD-ROM presents the entire content of the book plus interactive examples using Mathematica notebooks for problem-solving and learning. | 677.169 | 1 |
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INTRODUCTION OF ADDITIONAL MATHEMATICS PROJECT WORK1/2011 The aims of carrying out this project work are to enable students to : a)Apply mathematics to everyday situations and appreciate theimportance and the beauty of mathematics in everyday lives b)Improve problem-solving skills, thinking skills , reasoning andmathematical communication c)Develop positive attitude and personalities and instrinsicmathematical values such as accuracy , confidence andsystematic reasoning d)Stimulate learning environment that enhances effectivelearning inquiry-base and teamwork e)Develop mathematical knowledge in a way which increase students¶ interest and confidence.
Introduction of integration
In mathematics,integration is a technique of finding a function g(x) the derivative of which, Dg(x), is equal to a given function f(x). This is indicated by the integral sign ³œ,´ as in œf(x), usually called the indefinite integral of the function. (The symbol dx is usually added, which merely identifies x as the variable.) The definite integral, written with a and b called the limits of integration, is equal to g(b) í g(a), whereDg(x) = f(x).Some antiderivatives can be calculated by merely recalling which function has a given derivative, but the techniques of integration mostly involve classifying the functions according to which types of manipulations will change the function into a form the antiderivative of which can be more easily recognized. For example, if one is familiar with derivatives, the function 1/(x + 1) can be easily recognized as the derivative of loge(x + 1). The antiderivative of (x2 + x + 1)/(x + 1) cannot be so easily recognized, but if written as x(x + 1)/(x + 1) + 1/(x + 1) = x + 1/(x + 1), it then can be recognized as the derivative of x2/2 + loge(x + 1). One useful aid for integration is the theorem known as integration by parts. In symbols, the rule is œfDg = fg í œgDf. That is, if a function is the product of two other functions, f and one that can be recognized as the derivative of some function g, then the original problem can be solved if one can integrate the product gDf. For example, if f = x, and Dg = cos x, then œx·cos x = x·sin x í œsin x = x·sin x í cos x + C. Integrals are used to evaluate such quantities as area, volume, work, and, in general, any quantity that can be interpreted as the area under a curve.
Definition The process of finding a function, given its derivative, is called anti-differentiation (or integration). If F'(x) = f(x), we say F(x) is an anti-derivative of f(x). Examples F(x) =cos x is an anti-derivative of sin x, and ex is an anti-derivative of ex. Note that if F(x) is an anti-derivative of f(x) then F(x) + c, where c is a constant (called the constant of integration) is also an anti-derivative of F(x), as the derivative of a constant function is 0. In fact they are the only anti-derivatives of F(x).
We write f(x) dx = F(x) + c. if F'(x) = f(x) . We call this the indefinite integral of f(x) . Thus in order to find the indefinite integral of a function, you need to be familiar with the techniques of differentiation.
HISTORY
Over 2000 years ago, Archimedes (287-212 BC) found formulas for the surface areas and volumes of solids such as the sphere, the cone, and the paraboloid. His method of integration was remarkably modern considering that he did not have algebra, the function concept, or even the decimal representation of numbers. Leibniz (1646-1716) and Newton (1642-1727) independently discovered calculus. Their key idea was that differentiation and integration undo each other. Using this symbolic connection, they were able to solve an enormous number of important problems in mathematics, physics, and astronomy. Fourier (1768-1830) studied heat conduction with a series of trigonometric terms to represent functions. Fourier series and integral transforms have applications today in fields as far apart as medicine, linguistics, and music. Gauss (1777-1855) made the first table of integrals, and with many others continued to apply integrals in the mathematical and physical sciences. Cauchy (1789-1857) took integrals to the complex domain. Riemann (1826-1866) and Lebesgue (1875-1941) put definite integration on a firm logical foundation. Liouville (1809-1882) created a framework for constructive integration by finding out when indefinite integrals of elementary functions are again elementary functions. Hermite (1822-1901) found an algorithm for integrating rational functions. In the 1940s Ostrowski extended this algorithm to rational expressions involving the logarithm. In the 20th century before computers, mathematicians developed the theory of integration and applied it to write tables of integrals and integral transforms. Among these mathematicians were Watson, Titchmarsh, Barnes, Mellin, Meijer, Grobner, Hofreiter, Erdelyi, Lewin, Luke, Magnus, Apelblat, Oberhettinger, Gradshteyn, Ryzhik, Exton, Srivastava, Prudnikov, Brychkov, and Marichev. In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published his work on the general theory and practice of integrating elementary functions. His algorithm does not automatically apply to all classes of elementary functions because at the heart of it there is a hard differential equation that needs to be solved. Efforts since then have been directed at handling this equation algorithmically for various sets of elementary functions. These efforts have led to an increasingly complete algorithmization of the Risch scheme. In the 1980s some progress was also made in extending his method to certain classes of special functions. The capability for definite integration gained substantial power in Mathematica, first released in 1988. Comprehensiveness and accuracy have been given strong consideration in the development of Mathematica and have been successfully accomplished in its integration code. Besides being able to replicate most of the results from well-known collections of integrals (and to find scores of mistakes and typographical errors in them), Mathematica makes it possible to calculate countless new integrals not included in any published handbook.
Part 5
Oil Reserves - Top 20 Nations (% of Global) Saudi Arabia has 261,700,000,000 barrels (bbl) of oil, fully 25% of the world's oil. The United States has 22,450,000,000 bbl.
The United States government recently declared Alberta's oil sands to be 'proven oil reserves.' Consequently, the U.S. upgraded its global oil estimates for Canada from five billions to 175 billion barrels. Only Saudi Arabia has more oil. The U.S. ambassador to Canada has said the United States needs this energy supply and has called for a more streamlined regulatory process to encourage investment and facilitate development. - CBC Television - the nature of things - when is enough enough
Oil Production & Consumption, Top 20 Nations by Production (% of Global) Here are the top 20 nations sorted by production, and their production and consumption figures. Saudi Arabia produces the most at 8,711,000.00 bbl per day, and the United States consumes the most at 19,650,000.00 bbl per day, a full 25% of the world's oil consumption.
Exports & Imports Here's export and imports for all the nations listed in the CIA World Factbook, sorted alphabetically as having exports and imports. Conspicuously missing is the United States, but I can tell you that we consume 19,650,000.00 bbl per day, and produce 8,054,000.00, leaving a discrepancy of 11,596,000.00 bbl per day. This compares to the European Union, which produces 3,244,000.00 bbl per day and consumes 14,480,000.00 bbl per day for a discrepancy of 11,236,000.00 per day. Basically, about the same.
World Oil Market and Oil Price Chronologies: 1970 - 2003
Further Exploration
Petroleum engineers work in the technical profession that involves extracting oil in increasinglydifficult situations as the world's oil fields are found and depleted. Petroleum engineers searchthe world for reservoirs containing oil or natural gas. Once these resources are discovered, petroleum engineers work with geologists and other specialists to understand the geologicformation and properties of the rock containing the reservoir, determine the drilling methods to be used, and monitor drilling and production operations. Low-end Salary: $58,600/yr Median Salary: $108,910/yr High-end Salary: $150,310/yr EDUCATION:
Engineers typically enter the occupation with a bachelor s degree in mathematics or anengineering specialty, but some basic research positions may require a graduate degree. Mostengineering programs involve a concentration of study in an engineering specialty, along withcourses in both mathematics and the physical and life sciences. Engineers offering their servicesdirectly to the public must be licensed. Continuing education to keep current with rapidlychanging technology is important for engineers. MATH REQUIRED: College AlgebraGeometryTrigonometryCalculus I and IILinear AlgebraDifferential EquationsStatistics WHEN MATH IS USED: Improvements in mathematical computer modeling, materials and the application of statistics, probability analysis, and new technologies like horizontal drilling and enhanced oil recovery,have drastically improved the toolbox of the petroleum engineer in recent decades. POTENTIAL EMPLOYERS: About 37 percent of engineering jobs are found in manufacturing industries and another 28 percent in professional, scientific, and technical services, primarily in architectural, engineering,and related services. Many engineers also work in the construction, telecommunications, andwholesale trade industries. Some engineers also work for Federal, State, and local governmentsin highway and public works departments. Ultimately, the type of engineer determines the typeof potential employer. FACTS: Engineering diplomas accounted for 12 of the 15 top-paying majors, with petroleum engineeringearning the highest average starting salary of $83,121.
Conclusion I have done many researches throughout the internet anddiscussing with a friend who have helped me a lot in completing this project. Through the completion of this project, I have learned many skills and techniques. This project really helps me to understand more about the uses of progressions in our daily life. This project also helped expose the techniques of application of additional mathematics in real life situations. While conducting this project, a lot of information that I found.Apart from that, this project encourages the student to work together and share their knowledge. It is also encourage student to gather information from the internet, improve thinking skills and promote effective mathematical communication.Last but not least, I proposed this project should be continue because it brings a lot of moral values to the student and also test the students understanding in Additional Mathematics.
Reflection
After spending countless hours,day and night to finish this Additional Mathematics Project,here is what I got to say:
Doing this project makes me realize how important Additional Mathematics is.Also, completing this project makes me realize how fun it is and likable is Additional Mathematics | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Graph theory is used today in the physical sciences, social sciences, computer science, and other areas. Introductory Graph Theory presents a nontechnical introduction to this exciting field in a clear, lively, and informative style.
Author Gary Chartrand covers the important elementary topics of graph theory and its applications. In addition, he presents a large variety of proofs designed to strengthen mathematical techniques and offers challenging opportunities to have fun with mathematics.
A useful Appendix covers Sets, Relations, Functions, and Proofs, and a section devoted to exercises — with answers, hints, and solutions — is especially valuable to anyone encountering graph theory for the first time.
Undergraduate mathematics students at every level, puzzlists, and mathematical hobbyists will find well-organized coverage of the fundamentals of graph theory in this highly readable and thoroughly enjoyable book.
Synopsis
Clear, lively style covers all basics of theory and application, including mathematical models, elementary concepts of graph theory, transportation problems, connection problems, party problems, diagraphs and mathematical models, games and puzzles, graphs and social psychology, planar graphs and coloring problems, and graphs and other mathematics.
Description
Includes bibliographies and index.
About the Author
Six Degrees of Paul Erdos
Contrary to popular belief, mathematicians do quite often have fun. Take, for example, the phenomenon of the Erdos number. Paul Erdos (1913-1996), a prominent and productive Hungarian mathematician who traveled the world collaborating with other mathematicians on his research papers. Ultimately, Erdos published about 1,400 papers, by far the most published by any individual mathematician.
About 1970, a group of Erdos's friends and collaborators created the concept of the "Erdos number" to define the "collaborative distance" between Erdos and other mathematicians. Erdos himself was assigned an Erdos number of 0. A mathematician who collaborated directly with Erdos himself on a paper (there are 511 such individuals) has an Erdos number of 1. A mathematician who collaborated with one of those 511 mathematicians would have an Erdos number of 2, and so on — there are several thousand mathematicians with a 2.
From this humble beginning, the mathematical elaboration of the Erdos number quickly became more and more elaborate, involving mean Erdos numbers, finite Erdos numbers, and others. In all, it is believed that about 200,000 mathematicians have an assigned Erdos number now, and 90 percent of the world's active mathematicians have an Erdos number lower than 8. It's somewhat similar to the well-known Hollywood trivia game, Six Degrees of Kevin Bacon. In fact there are some crossovers: Actress-mathematician Danica McKellar, who appeared in TV's The Wonder Years, has an Erdos number of 4 and a Bacon number of 2.
This is all leading up to the fact that Gary Chartrand, author of Dover's Introductory Graph Theory, has an Erdos number of 1 — and is one of many Dover authors who share this honor.
Table of Contents
Chapter 1 Mathematical Models
1.1 Nonmathematical Models
1.2 Mathematical Models
1.3 Graphs
1.4 Graphs as Mathematical Models
1.5 Directed Graphs as Mathematical Models
1.6 Networks as Mathematical Models
Chapter 2 Elementary Concepts of Graph Theory
2.1 The Degree of a Vertex
2.2 Isomorphic Graphs
2.3 Connected Graphs
2.4 Cut-Vertices and Bridges
Chapter 3 Transportation Problems
3.1 The Königsberg Bridge Problem: An Introduction to Eulerian Graphs
3.2 The Salesman's Problem: An Introduction to Hamiltonian Graphs
Chapter 4 Connection Problems
4.1 The Minimal Connector Problem: An Introduction to Trees
*4.2 Trees and Probability
4.3 PERT and the Critical Path Method
Chapter 5 Party Problems
5.1 The Problem of Eccentric Hosts: An Introduction to Ramsey Numbers
5.2 The Dancing Problem: An Introduction to Matching
Chapter 6 Games and Puzzles
6.1 "The Problem of the Four Multicolored Cubes: A Solution to "Instant Insanity"
6.2 The Knight's Tour
6.3 The Tower of Hanoi
6.4 The Three Cannibals and Three Missionaries Problem
Chapter 7 Digraphs and Mathematical Models
7.1 A Traffic System Problem: An Introduction to Orientable Graphs
7.2 Tournaments
7.3 Paired Comparisons and How to Fix Elections
Chapter 8 Graphs and Social Psychology
8.1 The Problem of Balance
8.2 The Problem of Clustering
8.3 Graphs and Transactional Analysis
Chapter 9 Planar Graphs and Coloring Problems
9.1 The Three Houses and Three Utilities Problem: An Introduction to Planar Graphs | 677.169 | 1 |
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Numerical Analysis: Mathematics of Scientific Computing
Publisher:
American Mathematical Society
Number of Pages:
788
Price:
89.00
ISBN:
9780821847886
This book introduces students to the theoretical underpinnings of numerical analysis. Applications are referenced along the way, but not explored in detail. Open to a wide spectrum of readers, the book is largely self-contained. Each chapter has an introductory section that covers relevant preliminaries at the undergraduate level. The result is a thorough compendium on scientific computing suitable as a reference source or course textbook.
The chapters feature theorems clearly explained proofs. Roughly once per section basic algorithms are given in pseudo-code. This makes it is easy to implement them in any programming language. Section examples are basic and illustrative, as befits the introductory nature of the text. The sections also have approachable Computer Problems for algorithm implementation. Reading the text and doing just the applied Computer Problems should give a wide, practical introduction to numerical analysis, but may not justify the cover price.
The chapter problem sets are at higher level and will prove much more difficult to independent readers. Combine this with the absence of any solutions to the problem sets and an unprepared solo reading of this text may prove unfruitful. However, as a text for a course in which lectures bridge the gap between section material and problems, I am sure this works very well. Self-starters would be better off with a recent edition of the authors' Numerical Mathematics and Computing.
Including a twelve-page appendix on mathematical software, this text covers the theory of numerical analysis in eleven fairly rigorous chapters broken down into concise, well-organized sections. Topics covered include number representation and error propagation in computers, solving systems of equations, approximation theory, numeric differentiation and integration, numeric ODEs and PDEs, and optimization.
Tom Schulte teaches mathematics at Oakland Community College and is a Senior System Engineer for Plex Systems. | 677.169 | 1 |
Essentials of Discrete138.53
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About the Book
The Second Edition of David Hunter's Essentials of Discrete Mathematics is the ideal text for a one-term discrete mathematics course to serve computer science majors, as well as students from a wide range of other disciplines. The material is organized around five types of mathematical thinking: logical, relational, recursive, quantitative, and analytical. This presentation results in a coherent outline that steadily builds upon mathematical sophistication. Graphs are introduced early and are referred to throughout the text, providing a richer context for examples and applications. Students will encounter algorithms near the end of the text, after they have acquired enough skills and experience to analyze them properly. The final chapter contains in-depth case studies from a variety of fields, including biology, sociology, linquistics, economics, and music. | 677.169 | 1 |
Price: £39.99 (Excluding VAT at 20%)
This material is the second of three volumes which have been developed to help teachers deliver topics from the "Mathematics for Technicians" Unit for BTEC Nationals in Engineering.
It forms part of a planned library at BTEC First, National and Higher levels in Engineering and Construction.
It comprises a series of well-structured PowerPoint presentations. Each focusing on one particular topic whose selection has been based on the ability of PowerPoint to particularly enhance its presentation relative to other methods of delivery. The presentations will aid learners to meet various pass and merit criteria.
The 70 slides explain concepts, methods and problem-solving techniques in a unique, clear and consistent manner. The presentations are colour coded to distinguish between the presentation of concepts, setting of problems, problem-solving, though processes and emphasis.
A step-by-step, no frills, method of delivery is employed with several slides exploiting the ease with which PowerPoint can introduce diagrams and graphs. Very many PowerPoint animations are used in a straightforward manner.
The topics are the laws of logarithms, the solution of simultaneous linear equations by the method of elimination, the solution of quadratic equations by factorisation and by the formula, the radian, the plotting of trig curves and the areas and volumes of regular figures such as the cylinder, sphere and cone.
The presentations allow the teacher to introduce the topic flexibly at a pace consistent with the ability of any particular group of learners.
The series aims to provide an efficient platform for delivery from which inexperienced and experienced teachers alike will benefit.
Once purchased, the CD can be freely copied and networked throughout the school.
SPECIAL OFFER: THERE ARE THREE VOLUMES IN THIS SERIES - BUY ALL THREE FOR £89.99 PLUS VAT) | 677.169 | 1 |
For junior- to senior-level courses in Graph Theory taken by majors in Mathematics, Computer Science, or Engineering or for beginning-level graduate courses.
Once considered an "unimportant" branch of topology, graph theory has come into its own through many important contributions to a wide range of fields ― and is now one of the fastest-growing areas in discrete mathematics and computer science. This new text introduces basic concepts, definitions, theorems, and examples from graph theory. The authors present a collection of interesting results from mathematics that involve key concepts and proof techniques; cover design and analysis of computer algorithms for solving problems in graph theory; and discuss applications of graph theory to the sciences. It is mathematically rigorous, but also practical, intuitive, and algorithmic very good introductory book on Graph Theory. If you don't want to be overwhelmed by Doug West's, etc., and yet receive a decent introduction to the topic, this book is your best bet. It covers all the topics required for an advanced undergrad course or a graduate level graph theory course for Math, engineering, operations research or computer science students in good depth and details. There are good examples and interesting exercises; some computer codes (JAVA) are also available in the book implementing some of the algorithms.
I would say O.R. and CS people will benefit a lot from it both as a reference or a textbook if adapted for a one semester graduate course. The only drawback is the price! | 677.169 | 1 |
Introduction to Graph Theory158.62
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$132.72141 fills a need for a thorough introduction to graph theory that features both the understanding and writing of proofs about graphs. Verification that algorithms work is emphasized more than their complexity. An effective use of examples, and huge number of interesting exercises, demonstrate the topics of trees and distance, matchings and factors, connectivity and paths, graph coloring, edges and cycles, and planar graphs. For those who need to learn to make coherent arguments in the fields of mathematics and computer science. | 677.169 | 1 |
Although there are a few things (e.e summary of main arguments,a good collection of problems) that would be useful to the general understanding of the underlying mathematics but lack of pedagogical insights leave the book out of range for an ordinary undergraduate or even graduate student. It demands a good understanding of real analysis to understand certain concepts related to ODE etc, which in itself is alright if one happens to have a somewhat pure mathematical background otherwise there are far better books in the market to start and gain working knowledge of mathematics of physics.
Arfken and Weber is one such book that stays closer to physics than the above which most of times gives the feel of a badly written mathematical textbook. Mary Boas is the place to start and for the PostGraduate mathematical physics Sadri Hassani gives a much wider perspective.
Overall, IMO skip this book if you are also interested in physics rather than doing mathematics alone from a book with cluttered printing and awkward notations.
Accessories are good but at times the clippers and front attachments give a cheap plastic feel. overall, the ceramic blades do not need oiling and a trimmer bundled is also alright (not impressive). Mind you that the trimmer does not come with a cell. you need to buy AA battery to use it.
The book is intended to give you a grasp of basic working knowledge of calculus in a story telling mode. If you enjoy stories, then you will learn mathematics (calculus) on the way towards the end. Its a relaxed and friendly approach to give the reader the applied underpinnings of calculus. However, remember that its not a substitute for your textbook on calculus. This book should be useful for those who suffer from math anxiety. Otherwise, you should go for Stewart's or Thomas' calculus Textbooks which are the foundation of university curriculum of bachelor in Physics or Mathematics. Overall, the product is good but not a textbook substitute.
...lots of solved examples to make you feel confidant to handle the complex numbers, functions and its calculus. the matter is explained with suitable examples and relevant exercises to imbibe what you have learned. This is a quick way to learn and to be used along the standard more intractable texts and a support.
I just started with the first chapter and am pleasantly surprised with the lucidity and the presentation style of this book :)
This is a must have for any serious student (advanced undergrad or a graduate) of physics who want to use it as self study guide and should be in the offing for ready reference or recap of ideas already covered in the classroom.
you won't suffer reading and working through the book and then move on confidently to "Lie Groups for the pedestrians" by H J Lipkin for an expanded reading and classic references mentioned therein.
having bought the new Xbox 360 Slim, i can assuredly say that its the best gaming console in the market with competitive price and loads of features. those of you who have 360 elite need not buy this one as there is not much difference in the hardware apart from the cosmetics. the console runs silently and the feature to install 4 USB channel and 250 gig HD is very well thought of....the new xbox beats PS3 hands down...i have both thats why ;) | 677.169 | 1 |
All Content Items (745)
Given information about composite fiugres, the student will determine the area of composite 2-dimensional figures comprised of a combination of triangles and parallelograms using appropriate units of measure.
Given a graph and/or verbal description of a situation (both continuous and discrete), the student will identify mathematical domains and ranges and determine reasonable domain and range values for the given situations.
Given a real-world situation that can be modeled by a linear function or a graph of a linear function, the student will determine and represent the reasonable domain and range of the linear function using inequalities.
Given algebraic, tabular, and graphical representations of linear functions, the student will determine the intercepts of the function and interpret the meaning of intercepts within the context of the situation.
Given algebraic, tabular, graphical, or verbal representations of linear functions in problem situations, the student will determine the meaning of slope and intercepts as they relate to the situations. | 677.169 | 1 |
This manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text. This gives students the information as to how these problems are solved.
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Rent Student Solutions Manual for Gustafson/Frisk's Beginning and Intermediate Algebra: An Integrated Approach, 5th 5th edition today, or search our site for other textbooks by Peter D Frisk. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Cengage.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our tutors now. | 677.169 | 1 |
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